###### Abstract

We employ non-perturbative renormalisation group methods to compute the full
momentum dependence of propagators in quantum gravity in general dimensions.
We disentangle all different graviton and Faddeev-Popov ghost modes and find
qualitative differences in the momentum dependence of their propagators. This
allows us to reconstruct the form factors quadratic in curvature from first
principles, which enter physical observables like scattering cross sections.
The results are qualitatively stable under variations of the gauge fixing
choice.

###### keywords:

Quantum Gravity; Functional Renormalisation; Momentum Dependence; Propagator

xx 1 5 Non-perturbative propagators in quantum gravity Benjamin Knorr 1,∗ and
Marc Schiffer 2 Benjamin Knorr and Marc Schiffer Correspondence:
<EMAIL_ADDRESS>(B.K.)

###### Contents

  1. 1 Introduction
  2. 2 Functional Renormalisation Group
  3. 3 Momentum dependence in quantum gravity
    1. 3.1 Fluctuation approach
    2. 3.2 General structure of the RG flows
    3. 3.3 On the relation between form factors and anomalous dimensions
    4. 3.4 Momentum-dependent anomalous dimension versus wave function renormalisation
  4. 4 Momentum-dependent fluctuation RG flow: analytical structure
    1. 4.1 Behaviour at small momentum
    2. 4.2 Behaviour at large momentum
      1. 4.2.1 Spin two sector
      2. 4.2.2 Spin zero sector
      3. 4.2.3 Ghost sector
    3. 4.3 Limit of large dimension
    4. 4.4 The flow for positive $\mu$
  5. 5 Momentum-dependent fluctuation RG flow: numerical results
    1. 5.1 Numerical strategy
    2. 5.2 Dimensional dependence
    3. 5.3 Gap dependence
    4. 5.4 Gauge dependence
    5. 5.5 Form factors and the derivative expansion
  6. 6 Momentum-dependent background RG flow
  7. 7 Comparing background and fluctuation results
  8. 8 Summary and Outlook
  9. A Asymptotic expansion of form factors
  10. B Calculation of the background form factors induced by GR

##  1 Introduction

The unification of gravity with quantum mechanics is a notoriously hard
problem in theoretical physics. Even a century after the development of
quantum theory and general relativity, breakthroughs in the understanding of
quantum properties of spacetime are few and far between. This is not least so
because of the typical scale that we expect quantum gravity effects to be
important at - the Planck scale, which is about $10^{-35}$ m. To illustrate
this fantastically small number, measuring the typical size of a human to an
accuracy of a Planck length is roughly comparable to measuring the extension
of the Milky Way with an accuracy of an atomic nucleus. This emphasises why
experimental data on quantum gravity effects is hard to come by, and
accordingly quantum gravity theories presently mostly rely on theoretical
considerations, and can only be confronted with consistency tests.

A conservative strategy, and potentially a path to success in formulating a
theory of quantum gravity, is to build on well-established theories valid at
larger length scales, and only add as few extra assumptions as possible to
extend the theory to include new phenomena. In the context of quantum gravity,
such an approach is the Asymptotic Safety program Weinberg (1979). It embraces
the importance of symmetries in the understanding of the Standard Model and
adds an interacting ultraviolet completion of gravity in the form of a quantum
realisation of scale symmetry. An enormous advantage of this approach is its
closeness to standard quantum field theory notions, and the straightforward
connection to low energy physics. Especially in the matter sector, this allows
to confront the scenario with many observational consistency tests. A
disadvantage is that the existence of such a scale invariant regime is hard to
prove. In practice, this is however less of a problem. Many interacting fixed
points have been found with satisfactory precision in other contexts, e.g. in
statistical physics and condensed matter systems Gies and Scherer (2010); Gies
et al. (2010); Braun et al. (2011); Gies et al. (2013). By now, there is ample
evidence that a suitable fixed point indeed exists not only in various
approximations on the dynamics of pure quantum gravity Reuter (1998); Souma
(1999); Reuter and Saueressig (2002); Lauscher and Reuter (2002); Litim
(2004); Machado and Saueressig (2008); Codello et al. (2009); Benedetti et al.
(2009); Machado and Percacci (2009); Manrique and Reuter (2010); Manrique et
al. (2011); Groh and Saueressig (2010); Eichhorn and Gies (2010); Benedetti et
al. (2011); Manrique et al. (2011); Reuter and Saueressig (2012); Benedetti
and Caravelli (2012); Rechenberger and Saueressig (2013); Christiansen et al.
(2014); Dietz and Morris (2013); Ohta and Percacci (2014); Falls et al. (2013,
2016); Christiansen et al. (2016); Becker and Reuter (2014); Christiansen et
al. (2015); Morris and Slade (2015); Ohta et al. (2015, 2016); Gies et al.
(2015); Demmel et al. (2015); Biemans et al. (2017); Gies et al. (2016); Denz
et al. (2018); Platania and Saueressig (2018); Falls et al. (2018); Knorr and
Lippoldt (2017); Christiansen et al. (2018); De Brito et al. (2018); Falls et
al. (2019); Kluth and Litim (2020); Falls et al. (2020); Knorr (2021), but
also in the presence of matter Narain and Percacci (2010); Shaposhnikov and
Wetterich (2010); Donà et al. (2014); Meibohm et al. (2016); Donà et al.
(2016); Oda and Yamada (2016); Eichhorn and Lippoldt (2017); Wetterich and
Yamada (2017); Biemans et al. (2017); Christiansen et al. (2018); Hamada and
Yamada (2017); Eichhorn et al. (2018); Eichhorn and Held (2018); Eichhorn et
al. (2018); Alkofer and Saueressig (2018); Eichhorn et al. (2018, 2019a,
2019b); Pawlowski et al. (2019); Knorr et al. (2019); Bürger et al. (2019);
Eichhorn and Schiffer (2019); Reichert and Smirnov (2020); Kurov and
Saueressig (2020); Daas et al. (2020); Eichhorn and Pauly (2020, 2021); Ali et
al. (2020). Phenomenological implications of the interacting fixed point have
been discussed in the context of particle physics Shaposhnikov and Wetterich
(2010); Harst and Reuter (2011); Eichhorn and Gies (2011); Christiansen and
Eichhorn (2017); Eichhorn and Held (2018); Eichhorn and Versteegen (2018);
Gies and Martini (2018); Eichhorn and Held (2018); Eichhorn and Schiffer
(2019); Reichert and Smirnov (2020); Alkofer et al. (2020); de Brito et al.
(2021); Gies and Salek (2021), cosmology Bonanno and Platania (2015); Alkofer
et al. (2016); Bonanno et al. (2017, 2018a); Platania and Saueressig (2018);
Bonanno et al. (2018b); Gubitosi et al. (2018); Platania (2019, 2020), and
black holes Bonanno and Reuter (1999); Falls and Litim (2014); Koch and
Saueressig (2014a, b); Bonanno et al. (2018); Adeifeoba et al. (2018);
Platania (2019); Bosma et al. (2019); Held et al. (2019). For recent reviews
and introductions, see Percacci (2017); Eichhorn (2019); Reuter and Saueressig
(2019); Reichert (2020); Pawlowski and Reichert (2020), and for a critical
discussion of the status of the field, see Donoghue (2020); Bonanno et al.
(2020). Evidence for asymptotic safety in gravity has also been found in
lattice formulations of quantum gravity, namely Euclidean and Causal Dynamical
Triangulations Ambjorn et al. (2005); Laiho and Coumbe (2011); Ambjorn et al.
(2012); Coumbe and Laiho (2015); Laiho et al. (2017); Ambjørn et al. (2018);
Loll (2020); Ambjorn et al. (2021); Bassler et al. (2021). On the lattice, a
scale invariant regime can be realised if a second-order phase transition
exists that gives rise to consistent low-energy physics.

The physical renormalisation group running of couplings is one of the key
objects of study in quantum field theories. This translates into the momentum
dependence of correlation functions, which are the basic building blocks of
observables like scattering cross sections. The easiest non-trivial
correlation function is the propagator, which is the inverse of the two-point
correlation function. It stores important information about the unitarity and
causality of the theory as it is related to the spectral function (if it
exists), see e.g. Bonanno et al. (2021). An accurate description of the
propagator is thus crucial to decide whether Asymptotic Safety provides a
theory of quantum gravity which is compatible with these notions.

In this work we compute the full momentum dependence of the propagators of the
graviton and its accompanying Faddeev-Popov ghost with non-perturbative
renormalisation group techniques. A key advance is that we distinguish between
all the different modes of the graviton and the ghost. As is well-known from
representation theory, the graviton splits into a gauge-invariant spin two
mode (the transverse-traceless mode), a gauge-invariant spin zero mode, and a
pure gauge vector mode (which can be split into a spin one transverse, and a
spin zero longitudinal component). The ghost is a vector field itself, and
also splits into a transverse and a longitudinal mode. The physical
information is stored in the spin two and gauge-invariant spin zero mode. The
momentum dependence of their propagators can be mapped to the form factors
which are quadratic in curvature tensors.

The central results can be summarised as follows:

  * •

The spin two and spin zero modes of the graviton feature qualitative and
quantitative differences in their momentum dependence.

  * •

The overall gauge and gap dependence is small, see Figure 4 and Figure 5.

  * •

Within our approximation, only the spin two mode shows the property of
momentum locality Christiansen et al. (2015), and furthermore only in four
dimensions and with equal three- and four-graviton coupling, see (65).

  * •

A derivative expansion of the form factors shows alternating signs, indicating
that computations relying on such an expansion are inherently unstable and
invariably introduce fictitious poles Platania and Wetterich (2020), see (85).

  * •

The propagators of the two ghost modes are related for asymptotic momenta, but
differ for finite momenta, see (60) and (70).

  * •

Quantum corrections to the free propagator decrease exponentially with
increasing dimension, see (78) and Figure 3.

This work is structured as follows. Our renormalisation group tool of choice
is the so-called functional renormalisation group, which we briefly review in
section 2. In section 3 we discuss our setup and some general aspects of
momentum-dependent correlation functions. Section 4 and appendix A comprise a
discussion of analytical properties of the momentum dependence of the
propagators, whereas in section 5 we discuss numerical results. Sections 6 and
7 as well as appendix B contain a comparison of results obtained in different
approximation schemes. Finally, we conclude with a summary and an outlook in
section 8.

##  2 Functional Renormalisation Group

To extract the scale dependence of the system, we employ the functional
renormalisation group (FRG). It is based on the Wetterich equation Wetterich
(1993); Ellwanger (1994); Morris (1994), describing the flow of the scale-
dependent action $\Gamma_{k}$,

$\partial_{t}\Gamma_{k}[h,\bar{g}]=\frac{1}{2}\mathrm{STr}\left[\left(\Gamma^{(2)}_{k}[h,\bar{g}]+\mathfrak{R}_{k}[\bar{g}]\right)^{-1}\partial_{t}\mathfrak{R}_{k}[\bar{g}]\right]\,,$
(1)

which was adapted to gravity in Reuter (1998). Here, we denote $t=\log k$ as
the “RG-time”, $\Gamma_{k}^{(2)}$ is the second functional derivative of
$\Gamma_{k}$ with respect to fluctuation fields, and the super-trace
$\mathrm{STr}$ contains a sum over all fields and indices, as well as an
integration over the continuous coordinates. The regulator $\mathfrak{R}_{k}$,
which enters the generating functional as a momentum-dependent mass term for
the fluctuation, provides an infrared (IR) regularisation of modes with
$p^{2}<k^{2}$, and thereby ensures IR finiteness. The factor
$\partial_{t}\mathfrak{R}_{k}$ ensures ultraviolet (UV) finiteness by cutting
off modes with $p^{2}>k^{2}$. Overall, the regulator term together with its
derivative implement the Wilsonian idea of momentum-shell wise integration of
quantum fluctuations. As a result, the scale-dependent effective action
$\Gamma_{k}$ interpolates between the classical action $S$ when no quantum
fluctuations are integrated out, i.e. in the limit $k\to\infty$, and the full
quantum effective action $\Gamma$, when $k\to 0$. A central role is played by
the fixed points of the flow, which realise quantum scale invariance, where
all couplings measured in units of the scale $k$ are scale-independent. For
reviews of the FRG, see Berges et al. (2002); Pawlowski (2007); Gies (2012);
Berges and Mesterhazy (2012); Dupuis et al. (2020); Reichert (2020). Due to
the momentum-shell wise integration of quantum fluctuations, the Wetterich
equation (1) is formulated on Euclidean backgrounds, since only then the
squared momentum gives definite information on whether a certain mode is a UV
or an IR mode. In the following, we will therefore assume a Euclidean
background and hence discuss Euclidean quantum gravity. The generalisation of
the functional RG to Lorentzian spacetimes is one of the important challenges
of the approach Bonanno et al. (2020), and first steps towards investigations
of Lorentzian spacetimes have been presented in Manrique et al. (2011);
Rechenberger and Saueressig (2013); Demmel and Nink (2015); Biemans et al.
(2017); Houthoff et al. (2017); Knorr (2019); Baldazzi et al. (2019); Eichhorn
et al. (2020); Nagy et al. (2019).

Due to the formulation as a local coarse graining and the necessity of a gauge
fixing, the formal introduction of a background metric $\bar{g}$ is hard to
avoid. In this spirit, the full metric $g$ is split into background metric
$\bar{g}$ and a (not necessarily small) metric perturbation $h$ according to

$g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu}\,.$ (2)

Other parameterisations have been investigated for example in Kawai et al.
(1993); Aida et al. (1995); Nink (2015); Demmel and Nink (2015); Percacci and
Vacca (2015); Gies et al. (2015); Labus et al. (2016); Ohta and Percacci
(2016); Ohta et al. (2015); Falls (2015); Ohta et al. (2016); Donà et al.
(2016); Ohta et al. (2016); Falls and Ohta (2016); Percacci and Vacca (2017);
Knorr (2018); Alkofer and Saueressig (2018). The practical advantage of the
background field method is that one can retain background diffeomorphism
invariance at each step. Within the FRG, the background metric $\bar{g}$ only
serves as a technical tool to allow for a momentum-shell wise integration of
quantum fluctuations, and in principle never has to be specified. Without
approximations of the dynamics of the theory, the physical results obtained at
$k\to 0$ therefore are entirely independent of the specific choice of
$\bar{g}$. However, in the following sections, we will use a concrete choice
for $\bar{g}$, since this significantly reduces the computational complexity.

Since the scale-dependent effective action $\Gamma_{k}$ contains all operators
that are consistent with the symmetries of the system, practical computations
of RG flows of couplings require the truncation of $\Gamma_{k}$ to a
manageable set of operators. Especially in gauge theories, the momentum
dependence of scale-dependent correlation functions provides crucial
information on the system, such as on unitarity. Thus, in approximations, the
resolution of momentum dependence is unavoidable to obtain reliable results. A
scale identification which allows to translate the RG running of operators
into their momentum dependence is, in general, insufficient. For example,
higher order $n$-point correlation functions generically depend on $n$
different momenta independently, such that the momentum dependence cannot be
captured by the dependence on just one scale $k$. It is however possible that
for special momentum configurations, the RG running qualitatively agrees with
the physical momentum dependence Bonanno et al. (2021). Furthermore, the RG
scale $k$ is an artificial scale introduced via the regulator
$\mathfrak{R}_{k}$ to regularise the path integral. Physical quantities, such
as scattering amplitudes, can only be extracted in the limit $k\to 0$, where
all contributions from the regulator $\mathfrak{R}_{k}$ vanish. In this limit,
the UV behaviour of a given theory is described by the momentum dependence of
operators. The momentum-dependent evaluation of $\beta$-functions is thus
necessary to extract the UV physics of a system.

In the context of asymptotically safe quantum gravity, different expansion
schemes are used, which allow to extract momentum-dependent flow equations. On
the one hand, a vertex expansion in terms of metric fluctuations $h$ is
employed Christiansen et al. (2014, 2016). In this so-called fluctuation
approach, see Pawlowski and Reichert (2020) for a review, the starting point
is a seed action $S$, which is then expanded in terms of fluctuation fields
$h$, typically around a flat background, see Christiansen et al. (2018);
Bürger et al. (2019) for expansions around non-flat backgrounds. Consequently,
the different vertices are labelled with scale-dependent couplings, whose flow
is evaluated for different values of the external momenta, allowing to extract
momentum-dependent flows of $n$-point correlation functions. In this approach,
the momentum-dependent RG running of vertex correlation functions of the
graviton two- Christiansen et al. (2016), three- Christiansen et al. (2015)
and four-point functions Denz et al. (2018), as well as three-point functions
in gravity-matter systems Meibohm et al. (2016); Christiansen et al. (2018);
Eichhorn et al. (2018, 2019a, 2019b) have been investigated. As an important
approximation, all computations in the fluctuation approach identify the scale
dependence of different tensor structures of the graviton with the scale
dependence of the purely transverse-traceless part of the corresponding
$n$-point correlator. On the other hand, the form factor expansion Bosma et
al. (2019); Knorr et al. (2019) is based on diffeomorphism invariant
operators, where the corresponding form factors are general functions of
covariant derivatives. In this expansion, the scale dependence of the entire
form factor can be extracted within the background field approximation. As a
caveat, the background field approximation ignores that the regulator and the
gauge fixing break the full diffeomorphism symmetry, which causes the scale-
dependent effective action $\Gamma_{k}$ to depend on $\bar{g}$ and $h$
individually Reuter and Weyer (2009); Manrique and Reuter (2010); Manrique et
al. (2011, 2011); Donkin and Pawlowski (2012); Christiansen et al. (2014);
Codello et al. (2014); Christiansen et al. (2016); Becker and Reuter (2014,
2014); Demmel et al. (2015); Dietz and Morris (2015); Christiansen et al.
(2015); Meibohm et al. (2016); Safari (2016); Wetterich (2018, 2017); Morris
and Preston (2016); Safari and Vacca (2017, 2016); Labus et al. (2016); Morris
(2016); Denz et al. (2018); Percacci and Vacca (2017); Meibohm and Pawlowski
(2016); Knorr and Lippoldt (2017); Christiansen et al. (2018); Ohta (2017);
Nieto et al. (2017); Eichhorn et al. (2018, 2019a, 2019b); Bürger et al.
(2019); Knorr et al. (2019); Pawlowski and Reichert (2020); Bonanno et al.
(2021). Comparing fluctuation and background results at the same level of
approximation thus gives information about how much the respective modified
Ward identities are broken.

In the present work, we provide a comparison of both expansion schemes on the
level of the two-point function. For this purpose, we will set the stage and
provide more details on momentum dependence in section 3. In sections 4 and 5,
we derive and analyse the momentum-dependent flow equation for the graviton
two-point correlator for general dimension $d$. Importantly, we distinguish
all different tensor structures at the level of the two-point function. After
gauge fixing, we compute the scale dependence of four different tensor
structures for the graviton and ghost two-point functions. Additionally, in
section 6, we compute the corresponding form factors on the background field
level which contribute to the gravitational two-point function. We show that
on this level in the expansion and under certain assumptions, there is a one-
to-one correspondence between vertex correlation functions and form factors.
Furthermore, in section 7 we compare both expansions based on the analysis of
asymptotic behaviours of form factors and correlation functions.

##  3 Momentum dependence in quantum gravity

In this section we discuss our setup to resolve momentum-dependent correlation
functions. We start with a discussion of the correlators themselves. Next, we
present the general structure of the RG flow. Finally, we relate the form
factor expansion to the fluctuation correlation functions.

###  3.1 Fluctuation approach

For the computation of vertex correlation functions, we choose a flat
Euclidean background

$\bar{g}_{\mu\nu}=\delta_{\mu\nu}\,,$ (3)

which is a technical choice simplifying the computations significantly. In
principle, each diffeomorphism invariant operator in the seed action is
labelled by a single coupling. However, under the presence of the regulator
and gauge fixing, the scale dependence of different $n$-point functions,
originating from the same operator in the seed action, generally differ Denz
et al. (2018); Christiansen et al. (2018); Eichhorn et al. (2018, 2019b,
2019a). Therefore, we will introduce separate couplings for each operator in
the vertex expansion. Furthermore, on the level of the two-point function, we
will distinguish between the transverse-traceless (TT) mode, and the gauge-
invariant scalar mode. Specifically, the seed action we will use in the
following reads

$S=S_{\mathrm{grav}}+S_{\mathrm{gh}}+S_{\mathrm{gf}}\,,$ (4)

where we approximate $S_{\mathrm{grav}}$ by the Einstein-Hilbert action
$S_{\mathrm{EH}}$ describing classical gravity:

$S_{\mathrm{EH}}=-\frac{1}{16\pi
G_{\mathrm{N}}}\int\mathrm{d}^{d}x\sqrt{g}\left(R-2\Lambda\right)\,.$ (5)

Here, $G_{\mathrm{N}}$ and $\Lambda$ are the Newton coupling and the
cosmological constant, respectively. Computing the effect of quantum
fluctuations of gravity requires the inclusion of a gauge fixing condition
$F^{\mu}=0$, where we choose

$F^{\mu}=\left(\bar{g}^{\mu\kappa}\bar{D}^{\lambda}-\frac{1+\beta_{h}}{d}\bar{g}^{\kappa\lambda}\bar{D}^{\mu}+\frac{\gamma_{h}}{d}\bar{D}^{\mu}\bar{D}^{\kappa}\frac{1}{\bar{D}^{2}}\bar{D}^{\lambda}\right)h_{\kappa\lambda}\,.$
(6)

Here, $\bar{D}^{\mu}$ refers to the background covariant derivative, and
$\beta_{h}$ and $\gamma_{h}$ are gauge parameters. The presence of
$\gamma_{h}$ allows to compute the scale dependence of all tensor structures
on the level of the graviton two-point function. The gauge fixing condition is
implemented into the action via the inclusion of the gauge fixing action

$S_{\mathrm{gf}}=\frac{1}{32\pi\alpha_{h}G_{\mathrm{N}}}\int\mathrm{d}^{d}x\sqrt{\bar{g}}\,F^{\mu}\bar{g}_{\mu\nu}F^{\nu}\,.$
(7)

The gauge parameter $\alpha_{h}$ controls how strongly we implement the gauge
fixing condition in the path integral. A strict implementation corresponds to
the Landau limit $\alpha_{h}\to 0$. The resulting Faddeev-Popov determinant is
taken care of by introducing ghost fields $c^{\mu}$ and $\bar{c}^{\nu}$, with
ghost action $S_{\mathrm{gh}}$:

$S_{\mathrm{gh}}=\frac{1}{16\pi
G_{\mathrm{N}}}\int\mathrm{d}^{d}x\sqrt{\bar{g}}\,\bar{c}_{\mu}\frac{\delta
F^{\mu}}{\delta h_{\sigma\kappa}}\mathcal{L}_{c}g_{\sigma\kappa}\,.$ (8)

The Lie derivative $\mathcal{L}_{c}g_{\sigma\kappa}$ of the full metric
$g_{\mu\nu}$ along the direction of the ghost field is given by

$\mathcal{L}_{c}g_{\sigma\kappa}=2\bar{g}_{\rho(\sigma}\bar{D}_{\kappa)}c^{\rho}+c^{\rho}\bar{D}_{\rho}h_{\sigma\kappa}+2h_{\rho(\sigma}\bar{D}_{\kappa)}c^{\rho}\,.$
(9)

Round brackets indicate a normalised symmetrisation.

With the seed action specified and using the parameterisation (2), we expand
the scale-dependent effective action in powers of the fluctuation fields, via
the vertex expansion Christiansen et al. (2014, 2016)

$\Gamma_{k}[\Phi,\bar{g}]=\sum_{n=0}^{\infty}\frac{1}{n!}\Gamma^{(n)}_{k\,\,A_{1}\dots
A_{n}}[0,\bar{g}]\Phi^{A_{n}}\dots\Phi^{A_{1}}\,,$ (10)

where we have introduced the superfield $\Phi$ as a collection of all
dynamical fields in our system,

$\Phi^{A}=\left(h_{\mu\nu}(x),c^{\mu}(x),\bar{c}_{\mu}(x)\right)\,.$ (11)

The Einstein summation convention over the superindex A implies the summation
over discrete indices as well as an integration over the coordinates.
Furthermore, $\Gamma^{(n)}_{k}$ refers to the $n$-th functional derivative
with respect to the superfield $\Phi$. For convenience, we rescale the
vertices according to

$\Gamma^{(n)}_{k\,\,A_{1}\dots
A_{n}}[\Phi,\bar{g},G_{\mathrm{N}},\Lambda]\to\left(\,k^{-2}g_{n}\right)^{n/2}\Gamma^{(n)}_{k\,\,A_{1}\dots
A_{n}}[\Phi,\bar{g},k^{-2}g_{n},k^{2}\lambda_{n}]\,.$ (12)

As mentioned at the beginning of the section, we have introduced individual
dimensionless couplings $g_{n}$ and $\lambda_{n}$ that label the different
$n$-point functions. The vertices introduced in (12) also satisfy flow
equations similar to (1), see e.g. Denz et al. (2018). Due to the breaking of
diffeomorphism invariance by the gauge fixing term and the regulator, the
scale dependence of these different couplings does not necessarily agree. In
addition to the distinction of different dimensionless $n$-point couplings
$g_{n}$ and $\lambda_{n}$, also the couplings of pure gravity and ghost-
gravity, or gravity-matter $n$-point vertices will differ, and should be
distinguished. Therefore, for vertices originating from the ghost action (8),
one generally has to replace the coupling $g_{n}$ in (12) by $g_{c}$. For the
course of this work, we will however work under the assumption that the
differences between those couplings, caused by the breaking of diffeomorphism
invariance, is negligible, and therefore assume that $g_{n}=g_{c}=g$, and
$\lambda_{n}=\lambda$, for $n\geq 3$, and only distinguish the two-point
functions from higher-order vertices. In this way, we assume the exact
realisation of _effective universality_ Denz et al. (2018); Eichhorn et al.
(2018, 2019a, 2019b), which refers to the semi-quantitative agreement of the
scale dependence of different $n$-point vertices.

In the present work, we will focus on the two-point function of the pure
gravity system. For the graviton two-point function, there are five
independent tensor structures, three of them associated with the gauge fixing
parameters $\alpha_{h}$, $\beta_{h}$ and $\gamma_{h}$. The two remaining
tensor structures, which will be unaffected by the choice of gauge, are
related to the transverse-traceless mode $h^{\rm TT}_{\mu\nu}$ which satisfies

$\bar{D}^{\mu}h^{\rm TT}_{\mu\nu}=0\,,\qquad\bar{g}^{\mu\nu}h^{\rm
TT}_{\mu\nu}=0\,,$ (13)

and the scalar mode $h_{0}$ defined as

$h^{0}_{\mu\nu}=\Pi_{\phantom{0}\mu\nu}^{0\phantom{\mu\nu}\alpha\beta}\,h_{\alpha\beta}\,,$
(14)

where the scalar projector $\Pi^{0}$ is orthogonal to the gauge fixing action
and the transverse-traceless projector, i.e.,

$\Pi_{\phantom{0}}^{0\phantom{}}\cdot
S^{(2)}_{\text{gf}}=S^{(2)}_{\text{gf}}\cdot\Pi_{\phantom{0}}^{0\phantom{}}=0\,,\qquad\Pi_{\phantom{0}}^{0\phantom{}}\cdot\Pi_{\phantom{\text{TT}}}^{\text{TT}\phantom{}}=0\,.$
(15)

Explicitly, the transverse-traceless projector in momentum space is given by

$\displaystyle\Pi_{\phantom{\text{TT}}\mu\nu}^{\text{TT}\phantom{\mu\nu}\rho\sigma}$
$\displaystyle=\delta_{(\mu}^{\phantom{(\mu}\rho}\delta_{\nu)}^{\phantom{\nu)}\sigma}-\frac{1}{d-1}\bar{g}_{\mu\nu}\bar{g}^{\rho\sigma}-\frac{2}{p^{2}}\delta_{(\mu}^{\phantom{(\mu}(\rho}p_{\nu)}^{\phantom{)}}p_{\phantom{)}}^{\sigma)}$
(16)
$\displaystyle\qquad+\frac{1}{d-1}\frac{1}{p^{2}}\left(\bar{g}_{\mu\nu}p^{\rho}p^{\sigma}+p_{\mu}p_{\nu}\bar{g}^{\rho\sigma}\right)+\frac{d-2}{d-1}\frac{1}{p^{4}}p_{\mu}p_{\nu}p^{\rho}p^{\sigma}\,.$

The projector on the scalar mode $h_{0}$, referred to in (15) reads

$\Pi_{\phantom{0}\mu\nu}^{0\phantom{\mu\nu}\rho\sigma}=\frac{B^{2}}{C}\left(\bar{g}_{\mu\nu}+\frac{A}{B}\frac{p_{\mu}p_{\nu}}{p^{2}}\right)\left(\bar{g}^{\rho\sigma}+\frac{A}{B}\frac{p^{\rho}p^{\sigma}}{p^{2}}\right)\,,$
(17)

with

$\displaystyle A$ $\displaystyle=\left(d\beta_{h}-\gamma_{h}\right)\,,$ (18)
$\displaystyle B$ $\displaystyle=(d-\beta_{h}-1+\gamma_{h})\,,$ (19)
$\displaystyle C$
$\displaystyle=(d-1)\left(\gamma_{h}(-2\beta_{h}+\gamma_{h}-2)+d^{2}+d\left(\beta_{h}^{2}+2\gamma_{h}-1\right)\right)\,.$
(20)

The projectors $\Pi_{\phantom{\text{TT}}}^{\text{TT}\phantom{}}$ and
$\Pi_{\phantom{0}}^{0\phantom{}}$ can also be formulated in curved spaces, see
e.g. Knorr (2021). We will now discuss how we resolve the two-point function.
We start with the momentum-independent parts of the correlator. We can
introduce two gaps $\mu_{\text{TL}}$ and $\mu_{0}$ which correspond to the
traceless and the trace sector, respectively. These gaps $\mu_{x}$ are related
to the different $n$-point couplings introduced in (12) via
$\mu_{x}=-2\lambda_{2,x}$, where the subscript $x$ labels the respective mode.
Explicitly, we introduce them by adjusting the two-point function via Knorr
and Lippoldt (2017)

$\displaystyle\Gamma^{(2)\,\mu\nu\rho\sigma}_{h}$
$\displaystyle=S_{\mathrm{EH}}^{(2)\,\mu\nu\rho\sigma}\big{|}_{\Lambda=0}+S^{(2)\,\mu\nu\rho\sigma}_{\mathrm{gf}}$
(21)
$\displaystyle\qquad+\frac{k^{2}}{32\pi}\left(\Pi^{\mathrm{TL}\,\mu\nu\rho\sigma}\,\mu_{\text{TL}}-\frac{\Pi^{\mathrm{Tr}\,\mu\nu\rho\sigma}}{(d-1)(d+\gamma_{h})^{2}}\left(A^{2}\mu_{\text{TL}}+d(d-2)B^{2}\mu_{0})\right)\right)\,,$

where $\Pi^{\mathrm{Tr}}$ and $\Pi^{\mathrm{TL}}$ are the trace and traceless
projectors, respectively:

$\Pi_{\phantom{\text{Tr}}\mu\nu}^{\text{Tr}\phantom{\mu\nu}\rho\sigma}=\frac{1}{d}\bar{g}_{\mu\nu}\bar{g}^{\rho\sigma}\,,\qquad\Pi_{\phantom{\text{TL}}\mu\nu}^{\text{TL}\phantom{\mu\nu}\rho\sigma}=\delta_{(\mu}^{\phantom{(\mu}\rho}\delta_{\nu)}^{\phantom{\nu)}\sigma}-\Pi_{\phantom{\text{Tr}}\mu\nu}^{\text{Tr}\phantom{\mu\nu}\rho\sigma}\,.$
(22)

The introduction of the dimensionless quantities $\mu_{\text{TL}}$ and
$\mu_{0}$ according to (21) ensures that, in the Landau limit, the two
propagating modes feature a standard propagator with mass parameters
$\mu_{\text{TL}}$ and $\mu_{0}$, respectively. We emphasise that equation (21)
is only a rewriting of the two-point function which conveniently allows to
introduce the individual gaps $\mu_{\text{TL}}$ and $\mu_{0}$.

To capture the full momentum dependence of the propagators, we introduce
independent momentum-dependent wave function renormalisations for the
transverse-traceless and the gauge-invariant scalar mode. Specifically, we
rescale $h$ according to

$h_{\mu\nu}\to\mathcal{Z}_{h\,\mu\nu}^{\phantom{h\,\mu\nu}\rho\sigma}\,h_{\rho\sigma}\,,$
(23)

where the wave function renormalisation tensor $\mathcal{Z}_{h}$, given by

$\mathcal{Z}_{h\mu\nu}^{\phantom{h\mu\nu}\rho\sigma}=\delta_{(\mu}^{\phantom{(\mu}\rho}\delta_{\nu)}^{\phantom{\nu)}\sigma}+\left(\sqrt{Z_{h^{\mathrm{TT}}}(p^{2})}-1\right)\Pi_{\phantom{\text{TT}}\mu\nu}^{\text{TT}\phantom{\mu\nu}\rho\sigma}+\left(\sqrt{Z_{h^{0}}(p^{2})}-1\right)\Pi_{\phantom{0}\mu\nu}^{0\phantom{\mu\nu}\rho\sigma}\,.$
(24)

The rescaling (23) entails that the two-point function (21) needs to be
multiplied with $\mathcal{Z}_{h}$ from the left and the right:

$\Gamma^{(2)}_{h}\to\mathcal{Z}_{h}\cdot\Gamma^{(2)}_{h}\cdot\mathcal{Z}_{h}\,.$
(25)

The scale dependence of the graviton wave function renormalisation is encoded
in the anomalous dimensions

$\eta_{\text{TT}}(p^{2})=-\partial_{t}\ln
Z_{h^{\mathrm{TT}}}(p^{2})\,,\qquad\eta_{0}(p^{2})=-\partial_{t}\ln
Z_{h^{0}}(p^{2})\,.$ (26)

We choose a spectrally adjusted regulator, i.e., a regulator which is
proportional to the momentum-dependent part of the two-point functions,

$\mathfrak{R}^{h\mu\nu\rho\sigma}_{k}=\Gamma^{(2)\mu\nu\rho\sigma}_{h}\big{|}_{\mu_{\text{TL}}=\mu_{0}=0}\mathcal{R}_{k}(p^{2})\,,$
(27)

which ensures that mass-like terms are not regularised Gies (2002); Pawlowski
(2007); Benedetti et al. (2011); Gies et al. (2015). The regulator function
$\mathcal{R}_{k}$ implements the momentum-shell wise integration. Furthermore,
we will choose the Landau gauge $\alpha_{h}\to 0$, which is a fixed point for
all gauge parameters Litim and Pawlowski (2002); Knorr and Lippoldt (2017).
With these choices, the graviton propagator reads

$G_{h}=\frac{32\pi}{Z_{h^{\text{TT}}}(p^{2})}\frac{1}{p^{2}+\mathcal{R}_{k}(p^{2})+\mu_{\text{TL}}k^{2}}\Pi_{\phantom{\text{TT}}}^{\text{TT}\phantom{}}-\frac{32\pi}{Z_{h^{0}}(p^{2})}\frac{1}{\left(d-2\right)\left(d-1\right)}\frac{C}{B^{2}}\frac{1}{p^{2}+\mathcal{R}_{k}(p^{2})+\mu_{0}k^{2}}\Pi_{\phantom{0}}^{0\phantom{}}\,,$
(28)

where the coefficients $B$ and $C$ are defined in (19) and (20), respectively.

Similarly, for the ghost, we will distinguish between the longitudinal and the
transverse mode, and rescale the corresponding modes with momentum-dependent
wave function renormalisations $Z_{c_{\mathrm{L}}}$ and $Z_{c_{\mathrm{T}}}$,
respectively. Specifically, the rescaling of the ghost will be

$c_{\mu}\to\mathcal{Z}_{c\,\mu}^{\phantom{c\,\mu}\alpha}\,c_{\alpha},\qquad\bar{c}_{\mu}\to\mathcal{Z}_{c\,\mu}^{\phantom{c\,\mu}\alpha}\,\bar{c}_{\alpha}\,,$
(29)

with

$\mathcal{Z}_{c\mu}^{\phantom{c\mu}\nu}=\sqrt{Z_{\text{c}^{\text{T}}}}\,\Pi_{\phantom{\text{T}}\mu}^{\text{T}\phantom{\mu}\nu}+\sqrt{Z_{\text{c}^{\text{L}}}}\,\Pi_{\phantom{\text{L}}\mu}^{\text{L}\phantom{\mu}\nu}\,.$
(30)

The longitudinal and transverse projectors are defined in the usual way,

$\Pi_{\phantom{\text{L}}\mu}^{\text{L}\phantom{\mu}\nu}=\frac{p_{\mu}p^{\nu}}{p^{2}}\,,\qquad\Pi_{\phantom{\text{T}}\mu}^{\text{T}\phantom{\mu}\nu}=\delta_{\mu}^{\phantom{\mu}\nu}-\Pi_{\phantom{\text{L}}\mu}^{\text{L}\phantom{\mu}\nu}\,.$
(31)

The scale dependence of the ghost wave function renormalisations is encoded in
the anomalous dimensions

$\eta_{\text{c}^{\text{L}}}(p^{2})=-\partial_{t}\ln
Z_{\text{c}^{\text{T}}}(p^{2})\,,\qquad\eta_{\text{c}^{\text{T}}}(p^{2})=-\partial_{t}\ln
Z_{\text{c}^{\text{L}}}(p^{2})\,.$ (32)

Therefore, after distinguishing all different modes on the level of the two-
point function and choosing Landau gauge, there are four momentum-dependent
anomalous dimensions and two gaps, completely parameterising the flow of the
two-point correlation function.

###  3.2 General structure of the RG flows

Figure 1: Diagrams that contribute to the scale dependence of the graviton
two-point function. Double lines indicate gravitons, dashed lines stand for
the Faddeev-Popov ghosts, and the circled cross denotes the regulator
insertion $\partial_{t}\mathcal{R}_{k}$.

Since we aim at investigating the flow of physical $n$-point functions, we
employ the Landau limit, where the scale dependence of all gauge parameters
vanishes. Focusing on the graviton sector in this limit, only the two physical
projectors $\Pi_{\phantom{\text{TT}}}^{\text{TT}\phantom{}}$ and
$\Pi_{\phantom{0}}^{0\phantom{}}$ contribute to the propagator, see (28). When
projecting the external legs of $n$-point functions on the physical modes, the
full gauge dependence in the gravity sector is contained in the projector on
the scalar mode $\Pi_{\phantom{0}}^{0\phantom{}}$. For this projector, as
defined in (17), we observe that

$\Pi_{\phantom{0}}^{0\phantom{}}[\beta_{h},\gamma_{h}]=\Pi_{\phantom{0}}^{0\phantom{}}\left[\frac{\dim\beta_{h}-\gamma_{h}}{\dim+\gamma_{h}},0\right]\,.$
(33)

This shows that, for the scale dependence of couplings induced by the graviton
sector the gauge parameter $\gamma_{h}$ is redundant, since it can be removed
by a rescaling of $\beta_{h}$.

While the cancellations in the ghost sector are more involved, and require
non-trivial cancellations between contributions from propagators and vertices,
the same rescaling for $\beta_{h}$ as given in (33) eliminates the
$\gamma_{h}$-dependence of the ghost-induced flows. Therefore, in the
following we will employ $\gamma_{h}=0$ and keep $\beta_{h}$ general. Despite
this redundancy of gauge parameters on the level of the scale dependence of
couplings in the Landau limit, the tensor structure corresponding to
$\gamma_{h}$, (6), is nevertheless an independent tensor structure.

With the structure of the graviton propagator as given in (28), together with
the regulator defined in (27), we can compute the product of the regulator
insertion and two propagators, which enters every diagram of the flow as one
of its building blocks, see Figure 1 and Figure 2. For the gravitational
sector and in the Landau limit, it reads

$\displaystyle G_{h}\cdot\left(\partial_{t}\mathcal{R}_{k}^{h}\right)\cdot
G_{h}$
$\displaystyle=\frac{32\pi}{Z_{h^{\text{TT}}}(p^{2})}\frac{\partial_{t}\mathcal{R}_{k}(p^{2})-\eta_{\text{TT}}(p^{2})\,\mathcal{R}_{k}(p^{2})}{\left(p^{2}+\mathcal{R}_{k}(p^{2})+\mu_{\text{TL}}k^{2}\right)^{2}}\Pi_{\phantom{\text{TT}}}^{\text{TT}\phantom{}}$
(34)
$\displaystyle\qquad-\frac{32\pi}{Z_{h^{0}}(p^{2})}\frac{1}{(d-2)(d-1)}\frac{C}{B^{2}}\frac{\partial_{t}\mathcal{R}_{k}(p^{2})-\eta_{0}(p^{2})\,\mathcal{R}_{k}(p^{2})}{\left(p^{2}+\mathcal{R}_{k}(p^{2})+\mu_{0}k^{2}\right)^{2}}\Pi_{\phantom{0}}^{0\phantom{}}\,.$

A similar expression can be computed for the ghost sector, where the product
can be spanned by a transverse and a longitudinal part. Importantly, in the
Landau limit, both the graviton propagator and the above product decay into a
sum of the two projectors $\Pi_{\phantom{\text{TT}}}^{\text{TT}\phantom{}}$
and $\Pi_{\phantom{0}}^{0\phantom{}}$. This feature is due to a cancellation
of gauge modes contained in the regulator with contributions from the
propagators. It entails that the flow of $n$-point functions projected on
gauge-invariant modes is only driven by the gauge-invariant modes themselves,
and no gauge modes drive their scale dependence. It also significantly
decreases the number of different tensor structures that need to be computed,
when generalising the present computation to higher $n$-point correlators. Due
to the decomposition into the orthogonal projectors
$\Pi_{\phantom{\text{TT}}}^{\text{TT}\phantom{}}$ and
$\Pi_{\phantom{0}}^{0\phantom{}}$, vertices with one or more gauge mode will
not contribute to the scale dependence of any physical $n$-point correlator
Pawlowski and Reichert (2020).

In the graviton sector, we project onto the different tensor structures of the
two-point function by contracting the indices of (21) with
$\Pi_{\phantom{\text{TT}}}^{\text{TT}\phantom{}}$ and
$\Pi_{\phantom{0}}^{0\phantom{}}$, respectively. Structurally, after
projection, the flow of the TT-part of the graviton two-point function reads
Christiansen et al. (2016); Meibohm et al. (2016):

$-\left(y+\mu_{\text{TL}}\right)\eta_{\text{TT}}\left(y\right)+\partial_{t}\mu_{\text{TL}}+2\mu_{\text{TL}}=\Pi_{\phantom{\text{TT}}\mu\nu}^{\text{TT}\phantom{\mu\nu}\rho\sigma}\,\dot{\Gamma}^{(2)\phantom{\rho\sigma}\mu\nu}_{\phantom{(2)}\rho\sigma}\equiv\mathrm{flow}_{\mathrm{TT}}\left(y\right)\,,$
(35)

where the last term on the left-hand side comes from the scale derivative
acting on the $k^{2}$ coming with the dimensionless gap $\mu_{\text{TL}}$, see
(21). Furthermore, the right-hand side is obtained by evaluating and
projecting the diagrams in Figure 1. We also introduced the shorthand
$y=\frac{p^{2}}{k^{2}}$.

Figure 2: Diagrams that contribute to the scale dependence of the ghost two-
point function. Double lines indicate gravitons, dashed lines stand for the
Faddeev-Popov ghosts, and the circled cross denotes the regulator insertion
$\partial_{t}\mathcal{R}_{k}$.

In complete analogy, the flow equation for the scalar mode of the graviton
two-point function reads

$-\left(y+\mu_{0}\right)\eta_{0}\left(y\right)+\partial_{t}\mu_{0}+2\mu_{0}=\Pi_{\phantom{0}\mu\nu}^{0\phantom{\mu\nu}\rho\sigma}\,\dot{\Gamma}^{(2)\phantom{\rho\sigma}\mu\nu}_{\phantom{(2)}\rho\sigma}\equiv\mathrm{flow}_{0}\left(y\right)\,.$
(36)

Each of the equations (35) and (36) is disentangled by evaluating the right-
hand-side at $y=-\mu_{\text{TL}}$ and $y=-\mu_{0}$, respectively Christiansen
et al. (2016):

$\partial_{t}\mu_{\text{TL}}=-2\mu_{\text{TL}}+\mathrm{flow}_{\mathrm{TT}}\left(-\mu_{\text{TL}}\right)\,,\quad\partial_{t}\mu_{0}=-2\mu_{0}+\mathrm{flow}_{0}\left(-\mu_{0}\right)\,,$
(37)

and

$\eta_{\text{TT}}\left(y\right)=-\frac{\mathrm{flow_{\mathrm{TT}}\left(y\right)}-\mathrm{flow}_{\mathrm{TT}}\left(-\mu_{\text{TL}}\right)}{y+\mu_{\text{TL}}}\,,\quad\eta_{0}\left(y\right)=-\frac{\mathrm{flow}_{0}\left(y\right)-\mathrm{flow}_{0}\left(-\mu_{0}\right)}{y+\mu_{0}}\,.$
(38)

In the ghost sector, the equations determining the anomalous dimensions are

$y\,\eta_{\text{c}^{\text{T}}}\left(y\right)=\Pi_{\phantom{\text{T}}\mu}^{\text{T}\phantom{\mu}\nu}\,\dot{\Gamma}^{(2)\phantom{\nu}\mu}_{\phantom{(2)}\nu}\equiv\mathrm{flow}_{c^{\mathrm{T}}}\left(y\right)\,,\quad
y\,\eta_{\text{c}^{\text{L}}}\left(y\right)=\Pi_{\phantom{\text{L}}\mu}^{\text{L}\phantom{\mu}\nu}\,\dot{\Gamma}^{(2)\phantom{\nu}\mu}_{\phantom{(2)}\nu}\equiv\mathrm{flow}_{c^{\mathrm{L}}}\left(y\right)\,,$
(39)

where the diagrammatic representation of $\mathrm{flow}_{c}\\!\left(y\right)$
is shown in Figure 2. The transverse and longitudinal parts of the flow are
extracted by contraction with the corresponding projectors (31).

To evaluate the diagrams shown in Figure 1 and Figure 2, we used the
Mathematica packages _xAct_ Brizuela et al. (2009); Martín-García et al.
(2007, 2008); Martín-García (2008); Nutma (2014). The complete derivation is
included in the attached notebook. We validated the correctness of the results
with an independent code based on _xAct_ as well as _FormTracer_ Cyrol et al.
(2017).

###  3.3 On the relation between form factors and anomalous dimensions

We will now briefly discuss the relation between an action containing form
factors and the momentum-dependent wave function renormalisations. On a flat
background, the complete information on the graviton propagator is included in
the action

$\Gamma\simeq\frac{1}{16\pi
G_{N}}\int\text{d}^{d}x\,\sqrt{g}\,\left[2\Lambda-R-\frac{1}{4}\frac{d-2}{d-1}R\,f_{R}(\Delta)\,R+\frac{1}{4}\frac{d-2}{d-3}C^{\mu\nu\rho\sigma}\,f_{C}(\Delta)\,C_{\mu\nu\rho\sigma}\right]\,,$
(40)

where $f_{R}$ and $f_{C}$ are form factors. The normalisation of the form
factors ensures a unit prefactor in the propagator, see (41) below. This
action, which is based on a curvature expansion of the full effective action,
is an expansion in terms of diffeomorphism invariant operators. The form
factors $f_{R}$ and $f_{C}$ capture the full momentum dependence of the
graviton propagator. In comparison to a derivative expansion, they do not
necessarily introduce new poles into the propagator. In a derivative
expansion, one would in general expect the emergence of additional,
potentially spurious poles of the propagators, see Platania and Wetterich
(2020), and subsection 5.5. However, computations based on curvature
expansions usually make use of the background-field approximation, where the
difference between the background and the fluctuation propagators are
neglected. This approximation potentially suffers from severe background
dependence, which is lifted in the fluctuation expansion of the effective
action (10).

Since gauge fixing and regulator both break diffeomorphism invariance, the
expansion in terms of form factors as in (40), and in terms of fluctuation
vertex correlation functions do not agree. Therefore, the question arises
whether it is possible to extract the diffeomorphism invariant part – the form
factors – from the scale dependences of the fluctuation two-point functions,
without explicitly computing and solving modified Ward identities.

In the following, we will present a mapping between the form factors $f_{R}$
and $f_{C}$, and the fluctuation two-point functions. For this, we will
compare the propagators in both expansions, and neglect that these propagators
are different due to the breaking of diffeomorphism invariance. A motivation
for this assumption is the feature of effective universality Eichhorn et al.
(2018), which suggests that, at least in the vicinity of the fixed point
solution, the breaking of diffeomorphism invariance is mild. Therefore, under
this assumption, we investigate the possibility to extract the form factors
$f_{R}$ and $f_{C}$ from the fluctuation propagators.

For this, we will briefly indicate the form of the (unregularised) flat
background propagator that arises from the action (40). We will also set the
cosmological constant to zero for the moment. In this case, the scalar parts
of the spin two and zero background propagators read

$\bar{G}^{\text{TT}}(p^{2})\propto\frac{1}{p^{2}\left(1+p^{2}f_{C}(p^{2})\right)}\,,\qquad\bar{G}^{0}(p^{2})\propto\frac{1}{p^{2}\left(1+p^{2}f_{R}(p^{2})\right)}\,.$
(41)

This can be compared with the unregularised fluctuation propagator (28) at
vanishing gaps, and suggests the following relation between form factors and
wave function renormalisation:

$\frac{Z_{h^{\text{TT}}}(p^{2})}{Z_{h^{\text{TT}}}(0)}=1+p^{2}f_{C}(p^{2})\,,\qquad\frac{Z_{h^{0}}(p^{2})}{Z_{h^{0}}(0)}=1+p^{2}f_{R}(p^{2})\,.$
(42)

For finite values of the cosmological constant and the gaps, the situation is
more complicated. The cosmological constant enters the two parts of the
background propagators in a specific (gauge-dependent) ratio. By contrast, in
general the fluctuation gaps need not to have any particular ratio. It is thus
in general not possible to map the propagators in a one-to-one way without
non-local terms in at least one of the form factors. If we nevertheless want
to insist on an equivalence and allow for inverse powers of the momentum in at
least one of the form factors, we can still map the propagators one-to-one.
For example, we could identify the gap in the spin two sector with the actual
cosmological constant, and introduce a term $\sim 1/p^{4}$ in the form factor
$f_{R}$. Interestingly, such a term has been observed in the context of
reconstructing an effective action from numerical lattice data obtained within
causal dynamical triangulations Knorr and Saueressig (2018). Such a term also
has interesting cosmological applications Belgacem et al. (2018, 2020). We
will however not further discuss this issue here, since in the end, a proper
evaluation of the problem necessarily involves the solution of the modified
Ward identities.

###  3.4 Momentum-dependent anomalous dimension versus wave function
renormalisation

As a final point in this section, we will discuss the relation of the fixed
point condition for the dimensionless wave function renormalisation and the
momentum-dependent anomalous dimensions. In the literature on momentum
dependence in asymptotic safety, one commonly uses momentum-dependent
anomalous dimensions rather than the wave function renormalisation. This is
because only the anomalous dimensions are relevant in the flow equations - by
construction all factors of the wave function renormalisation drop out, see
e.g. (35). Since the anomalous dimension is related to the scale derivative of
the dimensionful wave function renormalisation, one does not impose the fixed
point condition directly. We can however easily translate between this
language and a formulation where we require the dimensionless wave function
renormalisation to be constant at the fixed point with respect to the RG scale
$k$, see also Eichhorn et al. (2019b).

The definition of the anomalous dimension $\eta$ in terms of the dimensionful
wave function renormalisation $Z$ reads

$\eta\left(y\right)=-\partial_{t}\ln Z\left(y\right)\,,$ (43)

see (26) and (32). We want to express the right-hand side in terms of the
dimensionless wave function renormalisation $z$ for which we require a fixed
point, and which is given by 111We refer to $z(y)$ as the dimensionless wave
function renormalisation, since its canonical mass dimension and the anomalous
dimension cancel at the fixed point.

$z(y)=k^{\eta(0)}Z(y)\,.$ (44)

In that process, we have to account both for the fact that we can normalise
the wave function renormalisation, giving rise to an anomalous running induced
by the anomalous dimension evaluated at zero momentum, as well as the scaling
of the argument. In that way, we find

$\eta\left(y\right)=\eta(0)-\frac{\dot{z}\left(y\right)}{z\left(y\right)}+2y\frac{z^{\prime}\left(y\right)}{z\left(y\right)}\,.$
(45)

Here, the overdot indicates the logarithmic scale derivative with respect to
the intrinsic $k$-dependence, excluding the trivial scaling of the argument.
At a fixed point, where $\dot{z}_{\ast}=0$, we have

$\eta_{\ast}(y)=\eta_{\ast}(0)+2y\,\frac{z_{\ast}^{\prime}(y)}{z_{\ast}(y)}\,.$
(46)

We can solve this differential equation for the wave function renormalisation:

$z_{\ast}(y)=z_{\ast}(0)e^{\int_{0}^{y}\text{d}s\,\frac{\eta_{\ast}(s)-\eta_{\ast}(0)}{2s}}=z_{\ast}(0)e^{\int_{0}^{1}\text{d}\omega\,\frac{\eta_{\ast}(\omega\,y)-\eta_{\ast}(0)}{2\omega}}\,.$
(47)

Assuming that the momentum-dependent anomalous dimension is bounded, this has
the consequence that for large $y$, the wave function renormalisation scales
as

$z_{\ast}(y)\propto
y^{\frac{\eta_{\ast}(\infty)-\eta_{\ast}(0)}{2}}\,,\qquad\text{as
}y\to\infty\,.$ (48)

Notably, only if the anomalous dimension vanishes asymptotically, that is, if
it fulfils _momentum locality_ 222A correlation function is called momentum-
local in this context if the ratio of its flow to the correlator itself
vanishes for large momenta. Christiansen et al. (2015), the standard fall-off
behaviour of the propagator in terms of the anomalous dimension at zero
follows, see (28), namely

$G(y)\propto\frac{1}{y^{1-\frac{\eta(0)}{2}}}\,,\qquad\text{as
}y\to\infty\,.\qquad\text{(momentum locality)}$ (49)

If momentum locality is not fulfilled, we need non-local information on the
momentum dependence, and the formula reads

$G(y)\propto\frac{1}{y^{1+\frac{\eta(\infty)-\eta(0)}{2}}}\,,\qquad\text{as
}y\to\infty\,.\qquad\text{(no momentum locality)}$ (50)

Having direct access to the fixed point wave function renormalisation via
(47), we can calculate the fixed point form factor from the corresponding
momentum-dependent anomalous dimension by inverting (42),

$f_{\ast}(y)=\frac{e^{\int_{0}^{y}\text{d}s\,\frac{\eta_{\ast}(s)-\eta_{\ast}(0)}{2s}}-1}{y}\,.$
(51)

Consequently, the behaviour at large momentum is

$f_{\ast}(y)\sim\begin{cases}cy^{\frac{\eta_{\ast}(\infty)-\eta_{\ast}(0)}{2}-1}\,,\qquad&\eta_{\ast}(\infty)-\eta_{\ast}(0)>0\,,\\\
-\frac{1}{y}+cy^{\frac{\eta_{\ast}(\infty)-\eta_{\ast}(0)}{2}-1}\,,\qquad&\eta_{\ast}(\infty)-\eta_{\ast}(0)\leq
0\,,\end{cases}\qquad\text{as }y\to\infty\,,$ (52)

where $c$ is given by

$c=e^{\int_{0}^{\infty}\text{d}s\,\left[\frac{\eta_{\ast}(s)-\eta_{\ast}(0)}{2s}-\frac{\eta_{\ast}(\infty)-\eta_{\ast}(0)}{2(1+s)}\right]}\,.$
(53)

For a derivation of this, see appendix A.

Let us finally make the connection to the derivative expansion, see also
Christiansen et al. (2016). To quadratic order in $y$, we find

$\frac{z_{\ast}(y)}{z_{\ast}(0)}=1+\frac{1}{2}\eta_{\ast}^{\prime}(0)\,y+\frac{1}{8}\left(\eta_{\ast}^{\prime}(0)^{2}+\eta_{\ast}^{\prime\prime}(0)\right)y^{2}+\mathcal{O}(y^{3})\,.$
(54)

This translates into the form factor

$f_{\ast}(y)=\frac{1}{2}\eta_{\ast}^{\prime}(0)+\frac{1}{8}\left(\eta_{\ast}^{\prime}(0)^{2}+\eta_{\ast}^{\prime\prime}(0)\right)y+\mathcal{O}(y^{2})\,.$
(55)

Let us also emphasise at this point that in theories with more than one mode,
and correspondingly more than one anomalous dimension, the derivative
expansion is potentially unstable, the more so the more modes one has Knorr
(2021). This comes from the fact that for a well-defined flow, one needs that
for all modes $z_{\ast}(y)>0$. In particular, this implies that all the
highest retained coefficient of all form factors need to have positive sign to
avoid turning one of the modes into a ghost. For general theories, there is
however no reason why all these coefficients need to be positive. One thus
likely faces the dilemma that for a given order in the derivative expansion,
the highest order coefficient of some mode might be negative. With a large
number of modes, it is thus more likely that at no finite order, a fixed
point, which is physically viable and does not necessarily feature ghost
modes, can be found in a derivative expansion. This makes it clear that in
complicated systems, the full resolution of the momentum dependence is not
optional. In fact, as we will see in section 5, we find indications that
quantum gravity with its two off-shell modes already shows such an alternating
pattern for low orders of the derivative expansion.

##  4 Momentum-dependent fluctuation RG flow: analytical structure

Now, we will discuss several analytical properties of the momentum-dependent
fluctuation two-point correlation functions which were introduced in section
3. For a clearer notation, we will work with dimensionless momenta, i.e. we
make the identification $\frac{p^{2}}{k^{2}}\to p^{2}$, such that no explicit
factors of $k$ will appear in the following discussion.

As has been found earlier, the Landau limit $\alpha_{h}\to 0$ induces a fixed
point for all gauge parameters Knorr and Lippoldt (2017). Due to this fact, we
only consider this limit for the fluctuation flows in this work. Additionally,
there is the constraint

$\beta_{h}<d-1\,,$ (56)

which ensures that the ghost kinetic operator has the correct sign and is
invertible. The most popular gauge choice in the literature is

$\beta_{h}=\frac{d}{2}-1\,,$ (57)

which we will focus on in our analysis. Other choices that have been argued
for to be preferred are the transverse gauge $\beta_{h}=-1$ Wetterich (2017,
2018) and the singular choice $\beta_{h}\to-\infty$ which has been considered
in e.g. Percacci and Vacca (2015); Labus et al. (2016); Gies et al. (2015);
Knorr and Lippoldt (2017). The choice $\beta_{h}=0$ in the Landau limit allows
for a particularly simple regularisation in a curved spacetime Knorr (2021).

The analytical properties which we will discuss in the following will provide
important cross-checks to test the numerical evaluation discussed in section
5.

###  4.1 Behaviour at small momentum

Let us first discuss the behaviour of the flow of the two-point function at
small momentum. A key observation is that the flow contains terms of the form

$\frac{1}{(p^{2}+2pqx+q^{2})}\,,\qquad\frac{1}{(p^{2}+2pqx+q^{2})^{2}}\,,$
(58)

where $p$ is the external momentum, $q$ is the loop momentum, and $x$ is the
cosine of the angle between the two momenta. These terms look like
unregulated, gapless propagators. Their origin lies in the projectors
occurring in the propagators, see (16) and (17), and are thus simply a feature
of gravity. The same phenomenon happens for spin one flows, where the
transverse and longitudinal projectors also introduce such terms. These terms
cause a number of technical difficulties, see also the discussion below in
section 4.4. For small external momenta $p$, they are the reason that the
derivative expansion does not commute with performing the loop integral. This
is because expanding these terms in small $p$ is actually an expansion in
powers of $p^{2}/q^{2}$, so that higher orders of the external momentum
introduce higher negative powers of the loop momentum. At a critical order,
the negative powers cancel the measure term $q^{d-1}$ and potential extra
powers from the vertex factors. For higher orders the integrals thus do not
converge as they suffer from IR divergences. This problem has been encountered
before Christiansen (2016), where momentum-independent contributions of
vertices have been neglected to define a finite, but ultimately inconsistent,
flow. Even at a level where the integrals converge, strong instabilities have
been found Christiansen et al. (2016) which might prevent one from obtaining
conclusive and robust results.

Note that once the integral over the loop momentum has been performed, the
flow appears to be smooth, see the numerical results presented below in
section 5. This emphasises that reliable approximations must involve momentum
dependence beyond a local expansion.

Besides this general discussion, one can investigate whether there are
relations between the different flows at vanishing momentum. In the graviton
sector, we find the relation

$\beta_{h}\to-\infty:\qquad\left.\frac{\text{flow}_{\text{TT}}}{\text{flow}_{0}}\right|_{p^{2}=0}=\frac{2}{d}-1<0\,.$
(59)

This limit is formally singular, in part due to how we have to introduce the
gap in the spin zero sector, see (21). In this limit, our ansatz formally
diverges, indicating that we actually write down a gap for the spin zero gauge
mode. This is consistent as this gauge choice projects out the trace mode.

In the ghost sector, we find that

$\left.\frac{\text{flow}_{\text{c}^{\text{T}}}}{\text{flow}_{\text{c}^{\text{L}}}}\right|_{p^{2}=0}=\frac{d-1-\beta_{h}}{d-1}>0\,.$
(60)

The divergence for $\beta_{h}\to-\infty$ indicates that the longitudinal flow
vanishes at $p=0$ for this gauge choice. Due to the constraint (56), the
flows, and thus the ghost anomalous dimensions, have the same sign at
vanishing momentum.

###  4.2 Behaviour at large momentum

We will now discuss the large momentum behaviour of the flow of the two-point
function. We emphasise that all of the following aspects a priori rely on our
truncation of the three- and four-point function - a dynamical implementation
of their flow can alter these results. In the following, we also assume that
the regulator falls off exponentially. Similar conclusions hold for regulators
with compact support, in which case no exponentially suppressed corrections
appear. Some of the aspects that we present in the following have been
previously discussed in Reichert (2018); Pawlowski and Reichert (2020).

As a general feature, we note that in the large momentum limit, the self-
energy diagrams simplify. This is because of our assumption on the regulator,
so that we can neglect it in the propagator which carries both the loop and
the external momentum, up to exponentially small corrections. As a
consequence, we can perform the angular integration exactly. The relevant
integrals are

$\int_{-1}^{1}\text{d}x\left(1-x^{2}\right)^{\frac{d-3}{2}}\left(p^{2}+2pqx+q^{2}\right)^{k}\left(p^{2}+2pqx+q^{2}+\mu\right)^{-1}\,,\qquad
k\in\\{-2,\dots,6\\}\,,$ (61)

which give rise to hypergeometric functions. We will now discuss the different
parts of the flow in this limit in turn.

####  4.2.1 Spin two sector

It has been noted before that in $d=4$ and with identified three- and four-
graviton couplings, the spin two two-point function shows momentum locality
Christiansen et al. (2016, 2015); Denz et al. (2018). This means that the flow
of the two-point function goes to a constant at large momentum, in contrast to
the naive expectation of a quadratic behaviour. This is due to a cancellation
of the self-energy and the tadpole diagrams.

We found that this cancellation only happens in $d=4$ \- in higher dimensions,
the two-point function rises quadratically at large momentum. We can also
confirm that this qualitative behaviour is independent of the choice of gauge
parameters and regulators. When we discuss numerical results below, this
manifests itself by $\eta_{\text{TT}}$ going to zero for large arguments in
$d=4$, but staying finite in this limit in other dimensions, see Figure 3.

There are three contributions to the flow of the spin two two-point function:
the graviton tadpole, the graviton self-energy, and the ghost self-energy
diagram, see Figure 1. The tadpole diagram has the exact form

$\text{flow}_{\text{TT}}^{\text{tadpole}}(p^{2},\mu_{\text{TL}},\mu_{0})=g_{4}\left(A_{\text{TT,0}}^{\text{tadpole}}[\eta_{\text{TT}},\eta_{0}](\mu_{\text{TL}},\mu_{0})+A_{\text{TT,2}}^{\text{tadpole}}[\eta_{\text{TT}},\eta_{0}](\mu_{\text{TL}},\mu_{0})p^{2}\right)\,.$
(62)

The coefficients $A_{\text{TT,i}}^{\text{tadpole}}$ depend functionally on the
graviton anomalous dimensions and the regulators, as well as on the gaps,
gauge parameters and the dimension. The structure of the self-energy diagrams
is more involved due to the propagator depending on the sum of external and
loop momentum. In the large external momentum limit, neglecting the regulator
depending on the sum of the two momenta, we find that

$\displaystyle\text{flow}_{\text{TT}}^{\text{hSE}}(p^{2},\mu_{\text{TL}},\mu_{0})\sim
g_{3}$
$\displaystyle\int\frac{\text{d}^{d}q}{(2\pi)^{d}}\sum_{i=-2}^{6}\Bigg{[}A_{\text{TT,i}}^{\text{hSE}}[\eta_{0}](p^{2},q^{2})\frac{\left(p^{2}+2pqx+q^{2}\right)^{i}}{p^{2}+2pqx+q^{2}+\mu_{0}}$
(63) $\displaystyle\hskip
71.13188pt+B_{\text{TT,i}}^{\text{hSE}}[\eta_{\text{TT}}](p^{2},q^{2})\frac{\left(p^{2}+2pqx+q^{2}\right)^{i}}{p^{2}+2pqx+q^{2}+\mu_{\text{TL}}}\Bigg{]}\,,\qquad\text{as
}p\to\infty\,,$

for the graviton self-energy diagram, and

$\text{flow}_{\text{TT}}^{\text{cSE}}(p^{2})\sim
g_{c}\int\frac{\text{d}^{d}q}{(2\pi)^{d}}\sum_{i=-2}^{4}A_{\text{TT,i}}^{\text{cSE}}[\eta_{\text{c}^{\text{T}}},\eta_{\text{c}^{\text{L}}}](p^{2},q^{2})\left(p^{2}+2pqx+q^{2}\right)^{i}\,,\qquad\text{as
}p\to\infty\,.$ (64)

for the ghost self-energy diagram. Performing the integrals, we find the
behaviour

$\text{flow}_{\text{TT}}(p^{2},\mu_{\text{TL}},\mu_{0})\sim
p^{2}\Bigg{[}(d-4)g_{3}\mathcal{I}_{\text{TT}}^{1}[\eta_{\text{TT}},\eta_{0}](\mu_{\text{TL}},\mu_{0})+(g_{4}-g_{3})\mathcal{I}_{\text{TT}}^{2}[\eta_{\text{TT}},\eta_{0}](\mu_{\text{TL}},\mu_{0})\Bigg{]}\,,\,\text{as
}p\to\infty\,.$ (65)

The $\mathcal{I}_{\text{TT}}^{i}$ are gauge- and dimension-dependent
functions. This parameterisation exemplifies that only for $g_{3}=g_{4}$, the
flow is momentum-local in $d=4$. The ghost diagram does not contribute to the
leading order behaviour.

More generally, one can ask if there are other (integer) dimensions where
there is a choice $g_{4}=cg_{3}$ for which the flow exhibits momentum
locality. This is indeed the case for

$g_{4}=\frac{7}{4}g_{3}\,,\qquad d=6\,.$ (66)

This is the only other combination of coupling identifications and integer
dimension which gives rise to momentum locality.

####  4.2.2 Spin zero sector

The general structure of the spin zero sector is the same as that of the spin
two sector. In particular, asymptotically we have an expansion as in (62),
(63) and (64) for the different diagrams. The main difference is that we do
not find momentum locality in $d=4$. The only combination of coupling
identification and integer dimension which gives rise to a momentum-local spin
zero sector is

$g_{4}=-2g_{3}\,,\qquad d=6\,.$ (67)

From this we conclude that, at least in our setup, there is no situation where
both spin two and spin zero sector are momentum-local. However, if also higher
$n$-point functions are distinguished according to their tensor structures,
this situation might change.

One can further ask the question, for $d\neq 4$, whether there is a specific
ratio between the behaviour of the flows of the two sectors at large momentum.
For this analysis, we assume again that $g_{4}=g_{3}$. In general, there is no
such relation: the contributions including the spin zero propagator to each of
the flows is generally different from the contributions including the spin two
propagator. There are three exceptions to this. Two of them are independent of
the choice of the gauge parameter $\beta_{h}$:

$\displaystyle d$
$\displaystyle=3:\qquad\frac{\text{flow}_{\text{TT}}}{\text{flow}_{0}}(p^{2},\mu_{\text{TL}},\mu_{0})\sim
1\,,\qquad\text{as }p\to\infty\,,$ (68) $\displaystyle d$
$\displaystyle=6:\qquad\frac{\text{flow}_{\text{TT}}}{\text{flow}_{0}}(p^{2},\mu_{\text{TL}},\mu_{0})\sim-\frac{1}{4}\,,\qquad\text{as
}p\to\infty\,.$

The third exception is the gauge choice $\beta_{h}\to-\infty$, so that

$\beta_{h}\to-\infty:\qquad\frac{\text{flow}_{\text{TT}}}{\text{flow}_{0}}(p^{2},\mu_{\text{TL}},\mu_{0})\sim-\frac{(d-4)(d^{3}-d^{2}+8d-12)}{(d-2)(d+2)(3d^{2}-11d+12)}\,,\qquad\text{as
}p\to\infty\,.$ (69)

####  4.2.3 Ghost sector

The flow in the ghost sector differs structurally from the flow in the
graviton sector in that the tadpole diagram vanishes for a linear
parameterisation of the metric fluctuations Eichhorn and Gies (2010), see also
(8), since in the present approximation the ghost action is linear in $h$.
Neither of the two ghost modes shows momentum locality in our setup for any
choice of dimension and gauge. However, independent of dimension and gauge
choice, the two flows agree asymptotically,

$\frac{\text{flow}_{\text{c}^{\text{T}}}}{\text{flow}_{\text{c}^{\text{L}}}}(p^{2},\mu_{\text{TL}},\mu_{0})\sim
1\,,\qquad\text{as }p\to\infty\,.$ (70)

In this limit, only one of the two self-energy diagrams in Figure 2
contributes, namely the first one, where the regulator is inserted on the
graviton line. Together with the relation of the two flows at vanishing
momentum, this gives an estimate of their typical disagreement.

###  4.3 Limit of large dimension

Let us discuss aspects of the limit $d\to\infty$. In this limit, we can
perform the remaining angular integral exactly, and also comment on the radial
integral. To see the first statement, note that this integral structurally
looks like

$\int_{-1}^{1}\text{d}x\,\left(1-x^{2}\right)^{\frac{d-3}{2}}f(x)\,,$ (71)

where $f(x)$ is one of the integrands in the flow. We will now make the
assumption that the dependence on $x$ is smooth, so that $f$ admits a
convergent expansion in Chebyshev polynomials,

$f(x)=\sum_{n\geq 0}f_{n}T_{n}(x)\,.$ (72)

Inserting this expansion into (71), and assuming that we can swap the order of
summation and integration, we arrive at

$\displaystyle\int_{-1}^{1}\text{d}x\,\left(1-x^{2}\right)^{\frac{d-3}{2}}f(x)$
$\displaystyle=\sum_{n\geq
0}f_{n}\int_{-1}^{1}\text{d}x\,\left(1-x^{2}\right)^{\frac{d-3}{2}}T_{n}(x)$
(73) $\displaystyle=\sum_{k\geq
0}f_{2k}\left(-\frac{1}{2}\right)^{k}\frac{\sqrt{\pi}\,\Gamma\left(\frac{d-1}{2}\right)}{\Gamma\left(\frac{d}{2}+k\right)}\frac{(d-2)!!}{(d-2k-2)!!}$
$\displaystyle\sim\sqrt{\frac{2\pi}{d}}\sum_{k\geq
0}(-1)^{k}f_{2k}=\sqrt{\frac{2\pi}{d}}f(0)\,,\qquad\text{as }d\to\infty\,.$

In the last line we expanded the expression to leading order in the limit of
large $d$. This result is intuitively clear: for large $d$ the measure
suppresses all values of $x$ except $x=0$.

To deal with the radial integral, we will assume that we have a regulator
which falls off like an exponential, so that also all integrands in the flow
fall off exponentially. More precisely, with $z$ denoting the square of the
loop momentum, we assume that any integrand is of the form

$f(z)=g(z)(1+z)^{k}e^{-az}\,,\qquad a>0\,,\,k\geq 0\,.$ (74)

We included an additional power law behaviour so that we can assume that $g$
is bounded on the whole interval. As a consequence, we can expand it in a
series of rational Chebyshev functions,

$g(z)=\sum_{n\geq 0}g_{n}T_{n}\left(\frac{z-1}{z+1}\right)\,.$ (75)

Combining this with the measure, we have integrals of the form

$\displaystyle\sum_{n\geq
0}g_{n}\int_{0}^{\infty}\text{d}z^{\frac{d}{2}+k}\,T_{n}\left(\frac{z-1}{z+1}\right)\,e^{-az}$
$\displaystyle\sim\sum_{n\geq
0}g_{n}a^{-\frac{d}{2}-1-k}2\sqrt{\pi}\left(\frac{d}{2}\right)^{\frac{d}{2}+1+k}e^{-\frac{d}{2}}$
(76)
$\displaystyle=2\sqrt{\pi}g(\infty)e^{-\frac{d}{2}}\left(\frac{d}{2a}\right)^{\frac{d}{2}+1+k}\,,\qquad\text{as
}d\to\infty\,.$

If we combine these two results with the factor from the angular integration,

$\int\text{d}\Omega=\frac{1}{2^{d}\pi^{\frac{d+1}{2}}\Gamma\left(\frac{d-1}{2}\right)}\,,$
(77)

we get the overall scaling of the flow as

$\text{flow}\propto\left(\frac{1}{4\pi a}\right)^{\frac{d}{2}}d^{k+1}\,.$ (78)

This result shows that there are two outcomes for the behaviour at large $d$,
depending on the fall-off of the regulator. For

$a>\frac{1}{4\pi}\,,$ (79)

the dimensional factor goes to zero, whereas for

$a\leq\frac{1}{4\pi}\,,$ (80)

the flow diverges in this limit.

###  4.4 The flow for positive $\mu$

As a last point in this section, we will discuss a technical challenge related
to the projection procedure for the anomalous dimensions, see (35) and (36).
For positive gaps $\mu>0$, we have to evaluate the flow at negative squared
momentum. This is in general challenging, for two reasons. First, we have to
define a regulator for, in general, complex momenta. Second, the factors (58)
induce poles in the integration domain for $p^{2}<0$.

We choose a regulator in the following way. First, to avoid having to define
it in the entire complex plane, we take the real part of the argument. Second,
we have to define the desired behaviour of the regularised propagator for
$p^{2}=-\mu$, that is we have to define

$\left.\frac{1}{p^{2}+\mathcal{R}_{k}(p^{2})+\mu}\right|_{p^{2}=-\mu}=\frac{1}{\mathcal{R}_{k}(-\mu)}\,.$
(81)

In the limit of large masses, we require that this expression behaves like a
standard regularised propagator,

$\mathcal{R}_{k}(-\mu)\sim 1+\mu\,,\qquad\text{as }\mu\to\infty\,.$ (82)

Third, we clearly also need that the regularised propagator does not introduce
any poles in the integration region. A choice which fulfills all these
requirements is

$\mathcal{R}_{k}(y)=\frac{e^{-\tilde{y}}}{1+e^{-2\tilde{y}}}+\frac{1-\tilde{y}}{1+e^{2\tilde{y}}}\,,\qquad\tilde{y}=\text{Re
}y\,.$ (83)

For (large) positive arguments, this regulator is just of exponential type
since the second term vanishes in this limit.

Let us now discuss how we deal with the unregularised pole structures (58)
coming from the projectors. The poles are situated at $x=0$ and
$q^{2}=-p^{2}=\mu$. We define the integration over these poles by splitting
the integration domain into a disk centred around the pole, and the rest.
Inside of the disk, we choose radial coordinates. One can show analytically
that then the Jacobian of the coordinate transformation and the angular
integration remove all potential poles, so that the integral is well-defined.
Let us finally mention that only simple poles arise in approximations which do
not resolve the difference between the two graviton anomalous dimensions and
gaps.

##  5 Momentum-dependent fluctuation RG flow: numerical results

In this section we discuss numerical results on the momentum dependence of the
propagators, and the influence of the dimension, the choice of gauge and the
size and sign of gaps. We will also discuss the derivative expansion of the
form factors. All results are obtained with the regulator (83). As a generic
choice, we set the gravitational coupling to one, $g=1$, and use the Landau
limit $\alpha_{h}\to 0$.

Figure 3: Dimensional dependence of anomalous dimensions for the choices
$g=1,\mu_{\text{TL}}=0,\mu_{0}=0,\beta_{h}=\frac{d}{2}-1$ and the regulator
(83). For clarity, we have reinstated $k$ in the momentum argument.

###  5.1 Numerical strategy

Before we present the actual numerical results, we give a short discussion of
how we obtained them. In the previous section, we found that the anomalous
dimensions are bounded both at zero and infinite argument. We will further
assume that they are bounded on the entire positive real line. In that case,
they can be expanded in a series of rational Chebyshev functions Boyd (1987),

$\eta(y)=\sum_{n\geq 0}\eta_{n}T_{n}\left(\frac{y-1}{y+1}\right)\,.$ (84)

Such an expansion is equivalent to compactifying the domain and employing a
standard expansion in Chebyshev polynomials. This expansion shows desirable
convergence properties if the function that is represented by the series is
smooth, see e.g. Boyd (2000) for an in-detail discussion. In the context of
functional renormalisation group flows, they have been systematically
discussed in Borchardt and Knorr (2015, 2016); Grossi and Wink (2019). For
applications in quantum gravity, see also Knorr (2018); Bosma et al. (2019);
Knorr et al. (2019).

In practice, we will truncate (84) at a finite order, insert the expansion
into the integral equations, and evaluate the equations at a set of
collocation points. The set of integral equations for the anomalous dimensions
then reduces to an _algebraic_ set of equations for the expansion
coefficients, which can be solved by standard linear algebra methods. The
integrals have been performed with Mathematica’s numerical integration
routine.

To judge whether the numerical precision of the solution is high enough, we
verify that the analytic relations between the different anomalous dimensions
found in the last section are satisfied.

Figure 4: Gap dependence of the dynamical anomalous dimensions for the choices
$d=4,g=1,\beta_{h}=1$ and the regulator (83). For clarity, we have reinstated
$k$ in the momentum argument.

###  5.2 Dimensional dependence

First we will study the dimensional dependence of the anomalous dimensions.
For the gauge choice $\beta_{h}=\frac{d}{2}-1$ and vanishing gaps
$\mu_{\text{TL}}=\mu_{0}=0$, this is shown in Figure 3. There are a few
general features. First, the overall magnitude of the anomalous dimensions
decreases with increasing dimension. This is in agreement with our analytical
estimate in the previous section, see (78). Second, we observe that while
$\eta_{0}$ stays positive at these coordinates in all dimensions,
$\eta_{\text{TT}}$ shifts to negative values quickly. Third, while there is
some quantitative difference between the two ghost anomalous dimensions in
$d=4$, they agree more and more the larger the dimension is. Fourth, as we
have seen analytically in subsection 4.2, $\eta_{\text{TT}}$ is momentum-local
only in $d=4$. Finally, we observe a general flattening for large $d$: the
momentum dependence of all anomalous dimensions is essentially trivial so that
they can be very well approximated by a constant. However, this conclusion
only holds in the current approximation, where higher-order curvature
operators were neglected in the seed action. Since some of these operators are
expected to be relevant in higher dimensions, their presence might alter the
momentum dependence of the anomalous dimensions.

###  5.3 Gap dependence

Let us now discuss the dependence of the anomalous dimensions on the gaps
$\mu_{\text{TL}}$ and $\mu_{0}$. For this we fix $d=4$ and the gauge
parameters $\beta_{h}=1$. The anomalous dimensions are shown in Figure 4 for
the gaps $\mu_{x}=\pm 1/4$.

Generally, we see that the gaps mostly influence the overall magnitude and the
behaviour at small momenta. Rather small variations in the gaps can shift the
value of $\eta_{0}(0)$ rather drastically. By contrast, the overall shape of
the other anomalous dimensions is rather unaffected. We take this as evidence
that the qualitative momentum dependence in large parts of theory space looks
approximately similar to the one at the specific points that we present in
this work.

Figure 5: Gauge dependence of the dynamical anomalous dimensions for the
choices $d=4,g=1,\mu_{\text{TL}}=0,\mu_{0}=0$ and the regulator (83). For
clarity, we have reinstated $k$ in the momentum argument.

###  5.4 Gauge dependence

Now we will discuss the gauge dependence of the anomalous dimensions. Once
again we choose vanishing gaps and $d=4$. The anomalous dimensions for the
choices $\beta_{h}\in\\{-1,0,1\\}$ are shown in Figure 5. We find that overall
the dependence on $\beta_{h}$ is mild, and mostly at the quantitative level.
This is a promising indication that even though off-shell quantities like
$\beta$-functions and propagators are inherently gauge-dependent, this
dependence is well-controlled. As analytically expected, we find that the two
ghost anomalous dimensions agree at $p=0$ for $\beta_{h}=0$. Also, their value
at infinity is numerically close to their value at zero. As a consequence, for
this gauge choice a single, constant ghost anomalous dimension is an accurate
approximation. Notably, also $\eta_{\text{TT}}$ and $\eta_{0}$ are numerically
close at $p=0$ for this choice of gauge.

###  5.5 Form factors and the derivative expansion

Let us finally discuss the form factors and their derivative expansion
obtained via (51). As parameters, we choose $d=4$ and
$\beta_{h}=1,\mu_{\text{TL}}=\mu_{0}=0$. While this is in general not a fixed
point, we want to illustrate the general form of the fluctuation form factors.
Since the general shape of the anomalous dimensions seems to be stable under
variation of the gaps, see subsection 5.3, we expect that the qualitative
picture presented here is correct for at least some part of theory space.

Figure 6: Fluctuation form factors reconstructed from the dynamical anomalous
dimensions according to (51) for the choices
$g=1,\mu_{\text{TL}}=0,\mu_{0}=0,\beta_{h}=1$ and the regulator (83). The
dashing indicates that the function is negative. For clarity, we have
reinstated $k$ in the momentum argument.

The fluctuation form factors, which were reconstructed according to (51), are
shown in Figure 6. We find that the $R^{2}$ form factor
$f_{R}^{\mathrm{fluc}}$ is positive while the $C^{2}$ form factor
$f_{C}^{\mathrm{fluc}}$ is negative. Due to the positivity of
$f_{R}^{\mathrm{fluc}}$, there is no additional pole in the propagator of the
gauge-invariant scalar mode. Similarly, for the transverse-traceless
propagator, there is no additional pole. Both go to a constant value for small
momenta and fall off with a power law at large momenta. Interestingly, the
$C^{2}$ form factor $f_{C}^{\mathrm{fluc}}$ agrees qualitatively with that
obtained in a background approximation of conformally reduced gravity Bosma et
al. (2019).

For small momenta, we find

$\displaystyle f^{\mathrm{fluc}}_{R}(y)$ $\displaystyle\approx
0.464+0.426y-6.49y^{2}+\mathcal{O}(y^{3})\,,$ (85) $\displaystyle
f^{\mathrm{fluc}}_{C}(y)$
$\displaystyle\approx-0.0941-0.213y+3.16y^{2}+\mathcal{O}(y^{3})\,.$

This entails that the worst-case scenario for the derivative expansion
explained in subsection 3.4 and in Knorr (2021) is actually realised for this
point of theory space: the Taylor coefficients of the two form factors have
alternating signs, so a local expansion would not be able to resolve this
point accurately. This means that in any derivative expansion in gravity which
is currently technically feasible, viable fixed points might be missed due to
the expansion.

##  6 Momentum-dependent background RG flow

In this section we provide results from a computation within the background
field approximation. The minimal way to obtain the background form factors is
to calculate their forms induced by the Einstein-Hilbert action. For
simplicity, we will restrict ourselves to four dimensions and a harmonic gauge
fixing, $\alpha_{h}=\beta_{h}=1$. This choice simplifies the calculation so
tremendously to justify violating our preference of using the Landau limit.
The detailed calculation is shown in appendix B. In particular, the resulting
flow equations are given in eqs. (122) - (125).

For the form factors, the fixed point equations are linear first order
differential equations, so that we can immediately calculate the solution. The
free integration constant can be identified with the value of the functions at
infinite momentum. With our choice of regulator, at $\Lambda=0$ we can avoid
additional poles in the propagator for

$f_{R}(\infty)\gtrsim 2.95\,,\qquad f_{C}(\infty)\gtrsim 0.611\,.$ (86)

To illustrate the solutions, we plot both form factors for two different
choices of integration constants in Figure 7 and Figure 8. The first satisfies
the above bounds and avoids new poles, whereas the second introduces new
poles. We set the cosmological constant to zero and the dimensionless Newton’s
constant to one. This is in general not a fixed point, but allows a more
direct comparison to the results of the fluctuation calculation.

Figure 7: Background form factors $f_{R}$ and $f_{C}$ for the choices
$g=1,\Lambda=0,f_{R}(\infty)=5,f_{C}(\infty)=2$ and the regulator (83). The
dashing indicates that the function is negative in that regime. This choice of
integration constants avoids additional poles in the propagator. For clarity,
we have reinstated $k$ in the momentum argument.

##  7 Comparing background and fluctuation results

We will now compare the results obtained from the background and the
fluctuation computation at the level of the form factors in $d=4$. As our
parameters we choose $g=1,\Lambda=\mu_{\text{TL}}=\mu_{0}=0$. For the
background computation we choose the harmonic gauge $\alpha_{h}=\beta_{h}=1$,
whereas for the fluctuation calculation we use $\alpha_{h}=0,\beta_{h}=1$. The
form factors are shown in Figure 6, Figure 7 and Figure 8.

Since the background computation is essentially one-loop, the form factors
show a logarithmic behaviour at small momenta. By contrast, the fluctuation
form factors go to finite values at vanishing momentum. The universal one-loop
logarithms can be expected to come out correctly once we study the flow in the
limit $k\to 0$, at least in some part of theory space where effective field
theory around flat spacetime is valid, see also Bonanno et al. (2021).

For large momenta, the background form factors are either finite or zero,
depending on the choice of integration constant. This is again due to
restricting to one-loop. The correct behaviour will be fixed dynamically once
the backreaction of these form factors onto the flow is taken into account.
The fluctuation form factors show a power law fall-off according to (52). This
entails that

$f_{R}^{\text{fluc}}(y)\propto y^{-0.85}\,,\qquad
f_{C}^{\text{fluc}}(y)+\frac{1}{y}\propto y^{-1.17}\,,\qquad\text{as
}y\to\infty\,.$ (87)

Regarding additional poles in the propagator, we have to choose the
integration constants for the background form factors according to (86) to
avoid them. For the fluctuation propagators, no new poles appear at the point
that we investigated. The absence of additional poles in the propagators is
crucial for the theory to be unitary. At the investigated point, the
fluctuation results suggest that indeed no new poles arise, indicating that
asymptotically safe quantum gravity might be unitary Becker et al. (2017);
Draper et al. (2020a, b); Platania and Wetterich (2020). Within the background
field approximation however, the presence of additional poles is not yet
conclusive, since in the present setup, they can only be avoided by choosing
the integration constant appropriately. Avoiding this choice requires a more
consistent computation, which includes the backreaction of the form factors.

Finally, the general shape and the signs differ between the two ways of
computation. For the background computation, $f_{R}$ is negative for small
momenta, whereas $f_{C}$ is positive, in both cases due to the logarithm. For
larger momenta, the signs can, but need not change, depending on the
integration constant. The fluctuation form factors have a definite sign:
positive for $f^{\mathrm{fluc}}_{R}$ and negative for $f^{\mathrm{fluc}}_{C}$.
They do not show any non-trivial feature besides what is dictated by the
asymptotics: they show an approximately constant regime for small momenta, a
cross-over at around $p\approx k$, leading to the power law fall-off at large
momenta.

It is difficult to predict which of these features are generic in most parts
of theory space, and which are special to the point of investigation. We leave
this investigation to future work, together with a detailed analysis of the
momentum-dependent flow of interaction vertices.

Figure 8: Background form factors $f_{R}$ and $f_{C}$ for the choices
$g=1,\Lambda=0,f_{R}(\infty)=0,f_{C}(\infty)=0$ and the regulator (83). The
dashing indicates that the function is negative in that regime. This choice of
integration constants introduces additional poles in the propagator. For
clarity, we have reinstated $k$ in the momentum argument.

##  8 Summary and Outlook

In this work we discussed the full non-perturbative momentum dependence of the
graviton and ghost propagator in quantum gravity. This includes the two
different graviton modes with spins zero and two, and the two ghost modes,
with spins zero and one. We obtained the results with the help of functional
renormalisation group equations, and resolved both gauge and dimensional
dependence.

A key result is that the propagators of the different graviton modes behave
qualitatively differently. In four dimensions and within our approximation,
the spin two mode is momentum-local, whereas the spin zero mode does not show
this property. This qualitative result is independent of the gauge choice, and
thus potentially has physical significance Christiansen et al. (2015);
Pawlowski and Reichert (2020). By contrast, the two ghost modes agree
qualitatively and partially even quantitatively, and their anomalous
dimensions are approximately momentum-independent.

The dependence on the choice of gauge as well as on the gaps is largely
quantitative rather than qualitative. We take this as a sign that the selected
examples presented in this work are indeed representative of larger parts of
theory space. Furthermore, the weak dependence of the anomalous dimensions on
the choice of gauge is a promising indication that, even in truncations, the
FRG gives rise to reliable and stable results.

In dimensions larger than four, we find that all anomalous dimensions quickly
go to zero. Assuming a regulator which falls off quickly enough, this
suppression for large dimensions is exponential in the dimension $d$.

All these results have been obtained from calculations of dynamical
correlation functions. To also compare to a computation in a background field
approximation, we have calculated the background form factors induced by an
Einstein-Hilbert action to quadratic order in the curvature. To facilitate
this comparison, we created a dictionary relating the form factors to
anomalous dimensions, both evaluated at a fixed point. We find that the one-
loop structure of our approximation in the background sector makes such a
comparison difficult, since many features are dictated by the one-loop form
rather than dynamically, e.g., the asymptotic behaviour for both large and
small momenta. A dynamical computation of the background form factors with
backreaction is desirable but postponed to the future.

The present work represents a major step in a complete and systematic
computation of correlation functions in quantum gravity. The natural next step
is to resolve the three-graviton vertex. Since the complete vertex is rather
complicated, two possible paths are the resolution of the full momentum
dependence of selected tensor structures, or the complete resolution of all
tensor structures in a derivative expansion. While the former strategy allows
to extract the full momentum dependence in one specific sector, it is not
guaranteed that such an approximation has desirable features the full theory
should have Knorr (2021). In the latter strategy, all tensor structures are
taken into account, however, as we have seen in subsection 5.5, a derivative
expansion might not be able to resolve the fixed points of the theory
accurately.

Finally, we expect that disentangling the different graviton modes will play
an important role in the discussion of spectral functions Bonanno et al.
(2021). In particular, the two sectors will have individual spectral
functions, and their different high-momentum behaviour will have a non-trivial
effect.

All authors contributed equally to this article. All authors have read and
agreed to the published version of the manuscript.

The research of B.K. has been supported by Perimeter Institute for Theoretical
Physics. Research at Perimeter Institute is supported in part by the
Government of Canada through the Department of Innovation, Science and
Economic Development and by the Province of Ontario through the Ministry of
Colleges and Universities. M.S. has been supported by the German Academic
Scholarship Foundation and gratefully acknowledges hospitality at Syracuse
University and at CP3-Origins, University of Southern Denmark, during
different stages of this project.

###### Acknowledgements.

We would like to thank Stefan Lippoldt, Alessia Platania and Manuel Reichert
for insightful discussions and Alessia Platania and Manuel Reichert for
constructive feedback on the manuscript. The authors declare no conflict of
interest. The funders had no role in the design of the study; in the
collection, analyses, or interpretation of data; in the writing of the
manuscript, or in the decision to publish the results. yes

##  Appendix A Asymptotic expansion of form factors

In this appendix we derive the asymptotic expansion (52). For this, we
consider the integral in the exponent of (51),

$\mathfrak{I}(y)=\int_{0}^{y}\text{d}s\,\frac{\eta_{\ast}(s)-\eta_{\ast}(0)}{2s}\,.$
(88)

We will assume that $\eta_{\ast}$ is a bounded function on the whole positive
real line, which is indeed true in our case. With that, we can first show that
the leading order behaviour of (88) at large $y$ is logarithmic. For that, let
us take the derivative of (88):

$\mathfrak{I}^{\prime}(y)=\frac{\eta_{\ast}(y)-\eta_{\ast}(0)}{2y}\,.$ (89)

Since $\eta_{\ast}$ is bounded, for large $y$, we find that

$\mathfrak{I}^{\prime}(y)\sim\frac{\eta_{\ast}(\infty)-\eta_{\ast}(0)}{2y}\,,\qquad\text{as
}y\to\infty\,.$ (90)

As a consequence, the integral itself is asymptotic to a logarithm,

$\mathfrak{I}(y)\sim\frac{\eta_{\ast}(\infty)-\eta_{\ast}(0)}{2}\ln
y\,,\qquad\text{as }y\to\infty\,.$ (91)

We still have to determine the sub-leading behaviour. For that, let us take
the difference of the integral and the logarithm above, and take the limit of
$y$ going to infinity. We will find that this limit exists, so that the sub-
leading part is a constant. To achieve that, let us rewrite the logarithm via

$\ln y=\left[\int_{0}^{y}\text{d}s\,\frac{1}{1+s}\right]-\ln\frac{1+y}{y}\,.$
(92)

The virtue of this rewriting is that the logarithm on the right-hand side
vanishes in the limit of large $y$. We can then combine the integrals so that

$\mathfrak{I}(y)-\frac{\eta_{\ast}(\infty)-\eta_{\ast}(0)}{2}\ln
y=\frac{\eta_{\ast}(\infty)-\eta_{\ast}(0)}{2}\ln\frac{1+y}{y}+\int_{0}^{y}\text{d}s\,\Bigg{[}\frac{\eta_{\ast}(s)-\eta_{\ast}(0)}{2s}-\frac{\eta_{\ast}(\infty)-\eta_{\ast}(0)}{2(1+s)}\Bigg{]}\,.$
(93)

We can now take the limit $y\to\infty$. The integral converges in this limit,
since the terms falling off like $1/s$ at large $s$ cancel, so that finally we
get

$\mathfrak{I}(y)\sim\frac{\eta_{\ast}(\infty)-\eta_{\ast}(0)}{2}\ln
y+\int_{0}^{\infty}\text{d}s\,\Bigg{[}\frac{\eta_{\ast}(s)-\eta_{\ast}(0)}{2s}-\frac{\eta_{\ast}(\infty)-\eta_{\ast}(0)}{2(1+s)}\Bigg{]}\,,\qquad\text{as
}y\to\infty\,.$ (94)

This completes the derivation of the asymptotic formula (52).

##  Appendix B Calculation of the background form factors induced by GR

In this appendix we derive the flow equations for $G_{N}$ and $\Lambda$ as
well as the two-curvature form factors induced by the Einstein-Hilbert
truncation in a background field approximation and in four dimensions. To make
our life easier, we will choose the harmonic gauge condition

$\alpha_{h}=\beta_{h}=1\,,\qquad\gamma_{h}=0\,.$ (95)

Even though we argued in the main text that only the Landau limit allows for a
clean flow, in a background field calculation this gauge choice simplifies the
calculation tremendously. The reason for this is that in this gauge, the two-
point function is diagonal and only depends on Laplacians and curvatures,
while it does not involve uncontracted derivatives. Splitting the graviton
into traceless and trace parts,

$h_{\mu\nu}=h^{\text{TL}}_{\mu\nu}+\frac{1}{4}\bar{g}_{\mu\nu}h\,,\qquad\bar{g}^{\mu\nu}h^{\text{TL}}_{\mu\nu}=0\,,$
(96)

the individual two-point functions at vanishing fluctuation fields read

$\displaystyle\frac{\delta^{2}\Gamma_{k}}{\delta h^{\text{TL}\mu\nu}\delta
h^{\text{TL}}_{\rho\sigma}}$
$\displaystyle\propto\frac{1}{G_{N}}\left[{\bar{\Delta}}_{2\mu\nu}^{\phantom{2\mu\nu}\rho\sigma}-2\Lambda\Pi_{\phantom{\text{TL}}\mu\nu}^{\text{TL}\phantom{\mu\nu}\rho\sigma}\right]\,,$
(97) $\displaystyle\frac{\delta^{2}\Gamma_{k}}{\delta h^{2}}$
$\displaystyle\propto\frac{1}{G_{N}}\left[\bar{\Delta}-2\Lambda\right]\,,$
$\displaystyle\frac{\delta^{2}\Gamma_{k}}{\delta\bar{c}_{\mu}\delta c_{\nu}}$
$\displaystyle\propto{\bar{\Delta}_{c}}^{\mu\nu}\,.$

Here we introduced the operators

$\displaystyle{\bar{\Delta}}_{2\mu\nu}^{\phantom{2\mu\nu}\rho\sigma}$
$\displaystyle=\left(\bar{\Delta}-\frac{2}{3}\bar{R}\right)\Pi_{\phantom{\text{TL}}\mu\nu}^{\text{TL}\phantom{\mu\nu}\rho\sigma}-2\bar{C}_{(\mu}{}^{\rho}{}_{\nu)}{}^{\sigma}\,,$
(98) $\displaystyle{{\bar{\Delta}}_{c}}^{\mu\nu}$
$\displaystyle=\bar{\Delta}\bar{g}^{\mu\nu}-\bar{R}^{\mu\nu}\,.$

These are also the operators that we regularise. This yields the very simple
flow

$\displaystyle\left.\dot{\Gamma}_{k}\right|_{h=0}$
$\displaystyle=\frac{1}{2}\text{Tr}\Bigg{[}\Pi^{\text{TL}}\frac{\left(2-\frac{\dot{G}_{N}}{G_{N}}\right)\mathcal{R}_{k}(\bar{\Delta}_{2})-2\bar{\Delta}_{2}\mathcal{R}_{k}^{\prime}(\bar{\Delta}_{2})}{\bar{\Delta}_{2}+\mathcal{R}_{k}(\bar{\Delta}_{2})-2\Lambda}\Bigg{]}+\frac{1}{2}\text{Tr}\Bigg{[}\Pi^{\text{Tr}}\frac{\left(2-\frac{\dot{G}_{N}}{G_{N}}\right)\mathcal{R}_{k}(\bar{\Delta})-2\bar{\Delta}\mathcal{R}_{k}^{\prime}(\bar{\Delta})}{\bar{\Delta}+\mathcal{R}_{k}(\bar{\Delta})-2\Lambda}\Bigg{]}$
(99)
$\displaystyle\quad-\text{Tr}\Bigg{[}\frac{2\mathcal{R}_{k}(\bar{\Delta}_{c})-2\bar{\Delta}_{c}\mathcal{R}_{k}^{\prime}(\bar{\Delta}_{c})}{\bar{\Delta}_{c}+\mathcal{R}_{k}(\bar{\Delta}_{c})}\Bigg{]}\,.$

Here, we used the same shape function in all modes. For later convenience, we
introduce the shorthands

$\displaystyle
f_{h}(x)=\frac{\left(2-\frac{\dot{G}_{N}}{G_{N}}\right)\mathcal{R}_{k}(x)-2x\mathcal{R}_{k}^{\prime}(x)}{x+\mathcal{R}_{k}(x)-2\Lambda}\,,\qquad
f_{c}(x)=2\frac{\mathcal{R}_{k}(x)-x\mathcal{R}_{k}^{\prime}(x)}{x+\mathcal{R}_{k}(x)}\,.$
(100)

In this notation, the flow reads

$\left.\dot{\Gamma}_{k}\right|_{h=0}=\frac{1}{2}\text{Tr}\left[\Pi_{\text{TL}}f_{h}(\bar{\Delta}_{2})\right]+\frac{1}{2}\text{Tr}\left[\Pi_{\text{Tr}}f_{h}(\bar{\Delta})\right]-\text{Tr}f_{c}(\bar{\Delta}_{c})\equiv\mathcal{T}_{\text{TL}}+\mathcal{T}_{\text{Tr}}+\mathcal{T}_{c}\,.$
(101)

To improve the readability, in the following we will refrain from using an
overbar for background quantities.

##### Early time heat kernel expansion

To evaluate the traces, we will employ the early time expansion of the heat
kernel. For an operator of Laplace type $\Delta+\mathbbm{E}$, where
$\mathbbm{E}$ is an endomorphism with the appropriate bundle structure, the
expansion reads Codello and Zanusso (2013)

$\displaystyle\text{Tr}\,e^{-s(\Delta+\mathbbm{E})}$
$\displaystyle\simeq\frac{1}{(4\pi
s)^{d/2}}\int\text{d}^{d}x\,\sqrt{g}\,\text{tr}\Bigg{\\{}\mathbbm{1}-s\mathbbm{E}+\frac{s}{6}R\,\mathbbm{1}+s^{2}\Big{[}\mathbbm{1}\,R_{\mu\nu}\,f_{Ric}(s\Delta)\,R^{\mu\nu}+\mathbbm{1}\,R\,f_{R}(s\Delta)\,R$
(102)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+R\,f_{RE}(s\Delta)\,\mathbbm{E}+\mathbbm{E}\,f_{E}(s\Delta)\,\mathbbm{E}+\mathcal{F}_{\mu\nu}\,f_{F}(s\Delta)\,\mathcal{F}^{\mu\nu}\Big{]}\Bigg{\\}}\,.$

In this expression, Tr is a functional trace, tr is the trace over the bundle
indices, $\mathcal{F}$ is the bundle curvature

$\mathcal{F}_{\mu\nu}=[D_{\mu},D_{\nu}]\,,$ (103)

which depends on the index structure of the field traced over, and

$\displaystyle f_{Ric}(x)$ $\displaystyle=\frac{f(x)-1+\frac{x}{6}}{x^{2}}\,,$
(104) $\displaystyle f_{R}(x)$
$\displaystyle=\frac{f(x)}{32}+\frac{f(x)-1}{8x}-\frac{f(x)-1+\frac{x}{6}}{8x^{2}}\,,$
(105) $\displaystyle f_{RE}(x)$
$\displaystyle=-\frac{f(x)}{4}-\frac{f(x)-1}{2x}\,,$ (106) $\displaystyle
f_{E}(x)$ $\displaystyle=\frac{f(x)}{2}\,,$ (107) $\displaystyle f_{F}(x)$
$\displaystyle=-\frac{f(x)-1}{2x}\,,$ (108)

are heat kernel form factors. The universal heat kernel function $f$ is
defined by

$f(x)=\int_{0}^{1}\text{d}\xi\,e^{-x\xi(1-\xi)}=2\int_{0}^{\frac{1}{2}}\text{d}\xi\,e^{-x\xi(1-\xi)}=\sqrt{\frac{\pi}{x}}e^{-\frac{x}{4}}\operatorname{erfi}\left(\frac{\sqrt{x}}{2}\right)=1-\frac{x}{6}+\mathcal{O}(x^{2})\,.$
(109)

The Taylor expansion shows that all form factors are regular at zero.

To bring the traces (99) into the form of the standard heat kernel trace
(102), we use the inverse Laplace transform. For a function $g$ of an operator
$\mathcal{O}$, this entails that we can write

$g(\mathcal{O})=\int_{0}^{\infty}\text{d}s\,\tilde{g}(s)\,e^{-s\mathcal{O}}\,.$
(110)

To be in line with the basis of the main text, we can use the relation

$\int\text{d}^{d}x\,\left[R_{\mu\nu}\,\Delta^{n}\,R^{\mu\nu}\right]=\int\text{d}^{d}x\,\left[\frac{1}{3}R\,\Delta^{n}\,R+\frac{1}{2}C_{\mu\nu\rho\sigma}\,\Delta^{n}\,C^{\mu\nu\rho\sigma}\right]+\mathcal{O}(\mathcal{R}^{3})\,,\qquad
n>0\,,$ (111)

to replace any occurrence of Ricci tensors by Ricci scalars and Weyl tensors.
In the following we will also neglect the Euler characteristic, which
corresponds to the case $n=0$ in the above equation.

##### Some integral transformations in d=4

To bring the flow equations into a convenient form, we will use the formula

$\int_{0}^{\infty}\text{d}s\,\tilde{g}(s)s^{-n}=\frac{1}{\Gamma(n)}\int_{0}^{\infty}\text{d}z\,z^{n-1}g(z)\,,$
(112)

which holds for $n>0$. Also, the following identities hold true:

$\displaystyle\int_{0}^{\infty}\text{d}s\,\tilde{g}(s)\,f(s\Delta)$
$\displaystyle=2\int_{0}^{\frac{1}{4}}\text{d}u\,\frac{1}{\sqrt{1-4u}}\,g(u\,\Delta)\,,$
(113)
$\displaystyle\int_{0}^{\infty}\text{d}s\,\tilde{g}(s)\,\frac{f(s\Delta)-1}{s\Delta}$
$\displaystyle=-\int_{0}^{\frac{1}{4}}\text{d}u\,\sqrt{1-4u}\,g(u\,\Delta)\,,$
(114)
$\displaystyle\int_{0}^{\infty}\text{d}s\,\tilde{g}(s)\,\frac{f(s\Delta)-1+\frac{s\Delta}{6}}{s^{2}\Delta^{2}}$
$\displaystyle=\frac{1}{6}\int_{0}^{\frac{1}{4}}\text{d}u\,(1-4u)^{3/2}g(u\,\Delta)\,.$
(115)

In all these equations, $\tilde{g}$ is the inverse Laplace transform of an
arbitrary function $g$, see (110), and $f$ is the universal heat kernel
function (109). The equations can be shown by inserting the integral form of
$f$ and performing repeated variable transformations while exchanging the
order of integrals.

##### Trace contribution of the spin zero part

We start with the contribution of the trace part of the graviton to the flow.
The trace projector makes it a scalar trace, with

$\mathcal{F}_{\mu\nu}=0\,,\qquad\mathbbm{E}=0\,.$ (116)

Using the inverse Laplace transform (110), then applying the general trace
formula (102), and using the integral identities (112)-(115) for this case and
in $d=4$ gives

$\displaystyle\mathcal{T}_{\text{Tr}}$
$\displaystyle\simeq\frac{1}{2}\frac{1}{16\pi^{2}}\int\text{d}^{4}x\,\sqrt{g}\,\Bigg{\\{}\int_{0}^{\infty}\text{d}z\left[zf_{h}(z)+\frac{1}{6}Rf_{h}(z)\right]+\int_{0}^{\frac{1}{4}}\text{d}u\,\Bigg{[}\frac{1}{12}C_{\mu\nu\rho\sigma}(1-4u)^{3/2}f_{h}(u\,\Delta)C^{\mu\nu\rho\sigma}$
(117) $\displaystyle\hskip
113.81102pt+R\left\\{\frac{1}{16}\frac{1}{\sqrt{1-4u}}-\frac{1}{8}\sqrt{1-4u}+\frac{5}{144}(1-4u)^{3/2}\right\\}f_{h}(u\,\Delta)R\Bigg{]}\Bigg{\\}}\,.$

##### Trace contribution of the ghost

Next, we present the contribution of the ghost. In this case,

$\mathbbm{E}_{\mu\nu}=-R_{\mu\nu}\,,\qquad\text{tr}\mathcal{F}_{\mu\nu}\,g(\Delta)\,\mathcal{F}^{\mu\nu}=-R_{\mu\nu\rho\sigma}\,g(\Delta)\,R^{\mu\nu\rho\sigma}\,.$
(118)

With this, the trace is

$\displaystyle\mathcal{T}_{c}$
$\displaystyle\simeq-\frac{1}{16\pi^{2}}\int\text{d}^{4}x\,\sqrt{g}\Bigg{\\{}\int_{0}^{\infty}\text{d}z\left[4zf_{c}(z)+\frac{5}{3}Rf_{c}(z)\right]$
(119) $\displaystyle\hskip
78.24507pt+\int_{0}^{\frac{1}{4}}\text{d}u\,\Bigg{[}C_{\mu\nu\rho\sigma}\left\\{\frac{1}{2}\frac{1}{\sqrt{1-4u}}-\sqrt{1-4u}+\frac{1}{3}(1-4u)^{3/2}\right\\}f_{c}(u\,\Delta)C^{\mu\nu\rho\sigma}$
$\displaystyle\hskip
78.24507pt+R\left\\{\frac{13}{12}\frac{1}{\sqrt{1-4u}}-\frac{7}{6}\sqrt{1-4u}+\frac{5}{36}(1-4u)^{3/2}\right\\}f_{c}(u\,\Delta)R\Bigg{]}\Bigg{\\}}\,,$

##### Trace contribution of the spin two part

Finally, we present the contribution of the traceless spin two component of
the graviton. With

$\displaystyle\mathbbm{E}_{\mu\nu\rho\sigma}$
$\displaystyle=\frac{2}{3}R\,\Pi_{\text{TL}\mu\nu\rho\sigma}-C_{\mu\rho\nu\sigma}-C_{\nu\rho\mu\sigma}\,,$
(120)
$\displaystyle\text{tr}\Pi_{\text{TL}}\mathcal{F}_{\mu\nu}\,g(\Delta)\,\mathcal{F}^{\mu\nu}$
$\displaystyle\simeq-6C_{\mu\nu\rho\sigma}\,g(\Delta)\,C^{\mu\nu\rho\sigma}+2R\,g(\Delta)\,R-12R_{\mu\nu}\,g(\Delta)\,R^{\mu\nu}\,,$

we find

$\displaystyle\mathcal{T}_{\text{TL}}$
$\displaystyle\simeq\frac{1}{2}\frac{1}{16\pi^{2}}\int\text{d}^{4}x\,\sqrt{g}\Bigg{\\{}\int_{0}^{\infty}\text{d}z\left[9zf_{h}(z)-\frac{9}{2}Rf_{h}(z)\right]$
(121) $\displaystyle\hskip
78.24507pt+\int_{0}^{\frac{1}{4}}\text{d}u\,\Bigg{[}C_{\mu\nu\rho\sigma}\left\\{3\frac{1}{\sqrt{1-4u}}-6\sqrt{1-4u}+\frac{3}{4}(1-4u)^{3/2}\right\\}f_{h}(u\,\Delta)C^{\mu\nu\rho\sigma}$
$\displaystyle\hskip
78.24507pt+R\left\\{\frac{25}{16}\frac{1}{\sqrt{1-4u}}+\frac{7}{8}\sqrt{1-4u}+\frac{5}{16}(1-4u)^{3/2}\right\\}f_{h}(u\,\Delta)R\Bigg{]}\Bigg{\\}}\,.$

##### Background flow equations

Now that we have computed all traces, we can read off the beta functions of
$G_{N}$, $\Lambda$ and the two-curvature form factors by comparing the
coefficients of the scale derivative acting on the action (40), with the
results for (101), given in (117), (119) and (121). We will present them for
the dimensionless quantities $g,\lambda,f_{R}$ and $f_{c}$. They read

$\displaystyle\dot{g}$
$\displaystyle=g\frac{2-\frac{g}{3\pi}\int_{0}^{\infty}\text{d}z\,\left[13\frac{2\mathcal{R}_{k}(z)-z\mathcal{R}_{k}^{\prime}(z)}{z+\mathcal{R}_{k}(z)-2\lambda}+10\frac{\mathcal{R}_{k}(z)-z\mathcal{R}_{k}^{\prime}(z)}{z+\mathcal{R}_{k}(z)}\right]}{1-\frac{13g}{6\pi}\int_{0}^{\infty}\text{d}z\frac{\mathcal{R}_{k}(z)}{z+\mathcal{R}_{k}(z)-2\lambda}}\,,$
(122) $\displaystyle\dot{\lambda}$
$\displaystyle=\left(-4+\frac{\dot{g}}{g}\right)\lambda-\frac{5\dot{g}}{2\pi}\int_{0}^{\infty}\text{d}z\,z\,\frac{\mathcal{R}_{k}(z)}{z+\mathcal{R}_{k}(z)-2\lambda}$
$\displaystyle\qquad+\frac{g}{\pi}\int_{0}^{\infty}\text{d}z\,z\,\left[5\frac{2\mathcal{R}_{k}(z)-z\mathcal{R}_{k}^{\prime}(z)}{z+\mathcal{R}_{k}(z)-2\lambda}-4\frac{\mathcal{R}_{k}(z)-z\mathcal{R}_{k}^{\prime}(z)}{z+\mathcal{R}_{k}(z)}\right]\,,$
(123) $\displaystyle\dot{f}_{R}(z)$
$\displaystyle=\frac{\dot{g}}{g}f_{R}(z)+2z\,f_{R}^{\prime}(z)+\frac{3\dot{g}}{\pi}\int_{0}^{\frac{1}{4}}\text{d}u\,\mu_{h}^{R}(u)\frac{\mathcal{R}_{k}(u\,z)}{u\,z+\mathcal{R}_{k}(u\,z)-2\lambda}$
$\displaystyle\qquad-\frac{6g}{\pi}\int_{0}^{\frac{1}{4}}\text{d}u\,\Bigg{[}\mu_{h}^{R}(u)\frac{2\mathcal{R}_{k}(u\,z)-u\,z\mathcal{R}_{k}^{\prime}(u\,z)}{u\,z+\mathcal{R}_{k}(u\,z)-2\lambda}+\mu_{c}^{R}(u)\frac{\mathcal{R}_{k}(u\,z)-u\,z\mathcal{R}_{k}^{\prime}(u\,z)}{u\,z+\mathcal{R}_{k}(u\,z)}\Bigg{]}\,,$
(124) $\displaystyle\dot{f}_{C}(z)$
$\displaystyle=\frac{\dot{g}}{g}f_{C}(z)+2z\,f_{C}^{\prime}(z)-\frac{\dot{g}}{\pi}\int_{0}^{\frac{1}{4}}\text{d}u\,\mu_{h}^{C}(u)\frac{\mathcal{R}_{k}(u\,z)}{u\,z+\mathcal{R}_{k}(u\,z)-2\lambda}$
$\displaystyle\qquad+\frac{2g}{\pi}\int_{0}^{\frac{1}{4}}\text{d}u\,\Bigg{[}\mu_{h}^{C}(u)\frac{2\mathcal{R}_{k}(u\,z)-u\,z\mathcal{R}_{k}^{\prime}(u\,z)}{u\,z+\mathcal{R}_{k}(u\,z)-2\lambda}+\mu_{c}^{C}(u)\frac{\mathcal{R}_{k}(u\,z)-u\,z\mathcal{R}_{k}^{\prime}(u\,z)}{u\,z+\mathcal{R}_{k}(u\,z)}\Bigg{]}\,.$
(125)

Here, we introduced the combined measures

$\displaystyle\mu_{h}^{R}(u)$
$\displaystyle=\frac{13}{8}\frac{1}{\sqrt{1-4u}}+\frac{3}{4}\sqrt{1-4u}+\frac{25}{72}(1-4u)^{3/2}\,,$
(126) $\displaystyle\mu_{c}^{R}(u)$
$\displaystyle=-\frac{13}{6}\frac{1}{\sqrt{1-4u}}+\frac{7}{3}\sqrt{1-4u}-\frac{5}{18}(1-4u)^{3/2}\,,$
$\displaystyle\mu_{h}^{C}(u)$
$\displaystyle=3\frac{1}{\sqrt{1-4u}}-6\sqrt{1-4u}+\frac{5}{6}(1-4u)^{3/2}\,,$
$\displaystyle\mu_{c}^{C}(u)$
$\displaystyle=-\frac{1}{\sqrt{1-4u}}+2\sqrt{1-4u}-\frac{2}{3}(1-4u)^{3/2}\,.$

##### Fixed point structure

Let us now analyse the fixed point structure of these background flow
equations. Employing the fixed point condition

$\dot{g}=\dot{\lambda}=\dot{f}_{R}=\dot{f}_{C}=0\,,$ (127)

we find that the form factor equations are first order linear ordinary
differential equations of the form

$2z\,f^{\prime}(z)=\int_{0}^{\frac{1}{4}}\text{d}u\,\mu(u)\mathcal{K}(u\,z)\,,$
(128)

for some kernel $\mathcal{K}$. The explicit solution to this equation reads

$f(\infty)-f(\Delta)=\int_{0}^{\frac{\Delta}{4}}\text{d}x\int_{\Delta}^{\infty}\text{d}z\,\frac{1}{2z^{2}}\,\mu\left(\frac{x}{z}\right)\,\mathcal{K}(x)+\int_{\frac{\Delta}{4}}^{\infty}\text{d}x\int_{4x}^{\infty}\text{d}z\,\frac{1}{2z^{2}}\,\mu\left(\frac{x}{z}\right)\,\mathcal{K}(x)\,.$
(129)

We chose to impose the boundary condition at $\Delta=\infty$ since for this
choice the integrals on the right-hand side converge for all $\Delta\geq 0$.
Inserting the explicit expressions for the kernel, we can also perform the
$z$-integration. With the shorthand notation

$\nu(a,b,c|\omega)=a\left(1-(1-\omega)^{1/2}\right)+\frac{b}{3}\left(1-(1-\omega)^{3/2}\right)+\frac{c}{5}\left(1-(1-\omega)^{5/2}\right)\,,$
(130)

the explicit induced background fixed point form factors are

$\displaystyle f_{R}(\Delta)$
$\displaystyle=f_{R}(\infty)-\frac{3g}{2\pi}\Bigg{[}\int_{0}^{1}\frac{\text{d}\omega}{\omega}\,\nu\left(\frac{13}{8},\frac{3}{4},\frac{25}{72}\,\bigg{|}\,\omega\right)\,\frac{2\mathcal{R}_{k}(\frac{\omega\,\Delta}{4})-\frac{\omega\,\Delta}{4}\mathcal{R}_{k}^{\prime}(\frac{\omega\,\Delta}{4})}{\frac{\omega\,\Delta}{4}+\mathcal{R}_{k}(\frac{\omega\,\Delta}{4})-2\lambda}$
$\displaystyle\qquad\qquad\qquad\qquad+\int_{1}^{\infty}\frac{\text{d}\omega}{\omega}\,\nu\left(\frac{13}{8},\frac{3}{4},\frac{25}{72}\,\bigg{|}\,1\right)\,\frac{2\mathcal{R}_{k}(\frac{\omega\,\Delta}{4})-\frac{\omega\,\Delta}{4}\mathcal{R}_{k}^{\prime}(\frac{\omega\,\Delta}{4})}{\frac{\omega\,\Delta}{4}+\mathcal{R}_{k}(\frac{\omega\,\Delta}{4})-2\lambda}$
$\displaystyle\qquad\qquad\qquad\qquad+\int_{0}^{1}\frac{\text{d}\omega}{\omega}\,\nu\left(-\frac{13}{6},\frac{7}{3},-\frac{5}{18}\,\bigg{|}\,\omega\right)\,\frac{\mathcal{R}_{k}(\frac{\omega\,\Delta}{4})-\frac{\omega\,\Delta}{4}\mathcal{R}_{k}^{\prime}(\frac{\omega\,\Delta}{4})}{\frac{\omega\,\Delta}{4}+\mathcal{R}_{k}(\frac{\omega\,\Delta}{4})}$
$\displaystyle\qquad\qquad\qquad\qquad+\int_{1}^{\infty}\frac{\text{d}\omega}{\omega}\,\nu\left(-\frac{13}{6},\frac{7}{3},-\frac{5}{18}\,\bigg{|}\,1\right)\,\frac{\mathcal{R}_{k}(\frac{\omega\,\Delta}{4})-\frac{\omega\,\Delta}{4}\mathcal{R}_{k}^{\prime}(\frac{\omega\,\Delta}{4})}{\frac{\omega\,\Delta}{4}+\mathcal{R}_{k}(\frac{\omega\,\Delta}{4})}\Bigg{]}\,,$
(131) $\displaystyle f_{C}(\Delta)$
$\displaystyle=f_{C}(\infty)+\frac{g}{2\pi}\Bigg{[}\int_{0}^{1}\frac{\text{d}\omega}{\omega}\,\nu\left(3,-6,\frac{5}{6}\,\bigg{|}\,\omega\right)\,\frac{2\mathcal{R}_{k}(\frac{\omega\,\Delta}{4})-\frac{\omega\,\Delta}{4}\mathcal{R}_{k}^{\prime}(\frac{\omega\,\Delta}{4})}{\frac{\omega\,\Delta}{4}+\mathcal{R}_{k}(\frac{\omega\,\Delta}{4})-2\lambda}$
$\displaystyle\qquad\qquad\qquad\qquad+\int_{1}^{\infty}\frac{\text{d}\omega}{\omega}\,\nu\left(3,-6,\frac{5}{6}\,\bigg{|}\,1\right)\,\frac{2\mathcal{R}_{k}(\frac{\omega\,\Delta}{4})-\frac{\omega\,\Delta}{4}\mathcal{R}_{k}^{\prime}(\frac{\omega\,\Delta}{4})}{\frac{\omega\,\Delta}{4}+\mathcal{R}_{k}(\frac{\omega\,\Delta}{4})-2\lambda}$
$\displaystyle\qquad\qquad\qquad\qquad+\int_{0}^{1}\frac{\text{d}\omega}{\omega}\,\nu\left(-1,2,-\frac{2}{3}\,\bigg{|}\,\omega\right)\,\frac{\mathcal{R}_{k}(\frac{\omega\,\Delta}{4})-\frac{\omega\,\Delta}{4}\mathcal{R}_{k}^{\prime}(\frac{\omega\,\Delta}{4})}{\frac{\omega\,\Delta}{4}+\mathcal{R}_{k}(\frac{\omega\,\Delta}{4})}$
$\displaystyle\qquad\qquad\qquad\qquad+\int_{1}^{\infty}\frac{\text{d}\omega}{\omega}\,\nu\left(-1,2,-\frac{2}{3}\,\bigg{|}\,1\right)\,\frac{\mathcal{R}_{k}(\frac{\omega\,\Delta}{4})-\frac{\omega\,\Delta}{4}\mathcal{R}_{k}^{\prime}(\frac{\omega\,\Delta}{4})}{\frac{\omega\,\Delta}{4}+\mathcal{R}_{k}(\frac{\omega\,\Delta}{4})}\Bigg{]}\,.$
(132)

Some of the integrals can be performed analytically:

$\int_{1}^{\infty}\frac{\text{d}\omega}{\omega}\,\nu\left(a,b,c\,\bigg{|}\,1\right)\,\frac{\mathcal{R}_{k}(\frac{\omega\,\Delta}{4})-\frac{\omega\,\Delta}{4}\mathcal{R}_{k}^{\prime}(\frac{\omega\,\Delta}{4})}{\frac{\omega\,\Delta}{4}+\mathcal{R}_{k}(\frac{\omega\,\Delta}{4})}=\left(a+\frac{b}{3}+\frac{c}{5}\right)\ln\left(1+\frac{\mathcal{R}_{k}\left(\frac{\Delta}{4}\right)}{\frac{\Delta}{4}}\right)\,,$
(133)

while the others have to be evaluated numerically.

References yes

## References

  * Weinberg (1979) Weinberg, S.  Ultraviolet divergences in quantum theories of gravitation.  General Relativity: An Einstein centenary survey, Eds. Hawking, S.W., Israel, W; Cambridge University Press 1979, pp. 790–831.
  * Gies and Scherer (2010) Gies, H.; Scherer, M.M.  Asymptotic safety of simple Yukawa systems.  Eur. Phys. J. C 2010, 66, 387–402, [arXiv:hep-th/0901.2459].  doi:black10.1140/epjc/s10052-010-1256-z.
  * Gies et al. (2010) Gies, H.; Rechenberger, S.; Scherer, M.M.  Towards an Asymptotic-Safety Scenario for Chiral Yukawa Systems.  Eur. Phys. J. C 2010, 66, 403–418, [arXiv:hep-th/0907.0327].  doi:black10.1140/epjc/s10052-010-1257-y.
  * Braun et al. (2011) Braun, J.; Gies, H.; Scherer, D.D.  Asymptotic safety: a simple example.  Phys.Rev. 2011, D83, 085012, [arXiv:hep-th/1011.1456].  doi:black10.1103/PhysRevD.83.085012.
  * Gies et al. (2013) Gies, H.; Rechenberger, S.; Scherer, M.M.; Zambelli, L.  An asymptotic safety scenario for gauged chiral Higgs-Yukawa models.  Eur. Phys. J. C 2013, 73, 2652, [arXiv:hep-th/1306.6508].  doi:black10.1140/epjc/s10052-013-2652-y.
  * Reuter (1998) Reuter, M.  Nonperturbative evolution equation for quantum gravity.  Phys.Rev. 1998, D57, 971–985, [arXiv:hep-th/hep-th/9605030].  doi:black10.1103/PhysRevD.57.971.
  * Souma (1999) Souma, W.  Nontrivial ultraviolet fixed point in quantum gravity.  Prog. Theor. Phys. 1999, 102, 181–195, [arXiv:hep-th/hep-th/9907027].  doi:black10.1143/PTP.102.181.
  * Reuter and Saueressig (2002) Reuter, M.; Saueressig, F.  Renormalization group flow of quantum gravity in the Einstein-Hilbert truncation.  Phys. Rev. 2002, D65, 065016, [arXiv:hep-th/hep-th/0110054].  doi:black10.1103/PhysRevD.65.065016.
  * Lauscher and Reuter (2002) Lauscher, O.; Reuter, M.  Ultraviolet fixed point and generalized flow equation of quantum gravity.  Phys.Rev. 2002, D65, 025013, [arXiv:hep-th/hep-th/0108040].  doi:black10.1103/PhysRevD.65.025013.
  * Litim (2004) Litim, D.F.  Fixed points of quantum gravity.  Phys. Rev. Lett. 2004, 92, 201301, [arXiv:hep-th/hep-th/0312114].  doi:black10.1103/PhysRevLett.92.201301.
  * Machado and Saueressig (2008) Machado, P.F.; Saueressig, F.  On the renormalization group flow of f(R)-gravity.  Phys.Rev. 2008, D77, 124045, [arXiv:hep-th/0712.0445].  doi:black10.1103/PhysRevD.77.124045.
  * Codello et al. (2009) Codello, A.; Percacci, R.; Rahmede, C.  Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group Equation.  Annals Phys. 2009, 324, 414–469, [arXiv:hep-th/0805.2909].  doi:black10.1016/j.aop.2008.08.008.
  * Benedetti et al. (2009) Benedetti, D.; Machado, P.F.; Saueressig, F.  Asymptotic safety in higher-derivative gravity.  Mod. Phys. Lett. 2009, A24, 2233–2241, [arXiv:hep-th/0901.2984].  doi:black10.1142/S0217732309031521.
  * Machado and Percacci (2009) Machado, P.F.; Percacci, R.  Conformally reduced quantum gravity revisited.  Phys. Rev. 2009, D80, 024020, [arXiv:hep-th/0904.2510].  doi:black10.1103/PhysRevD.80.024020.
  * Manrique and Reuter (2010) Manrique, E.; Reuter, M.  Bimetric Truncations for Quantum Einstein Gravity and Asymptotic Safety.  Annals Phys. 2010, 325, 785–815, [arXiv:gr-qc/0907.2617].  doi:black10.1016/j.aop.2009.11.009.
  * Manrique et al. (2011) Manrique, E.; Reuter, M.; Saueressig, F.  Bimetric Renormalization Group Flows in Quantum Einstein Gravity.  Annals Phys. 2011, 326, 463–485, [arXiv:hep-th/1006.0099].  doi:black10.1016/j.aop.2010.11.006.
  * Groh and Saueressig (2010) Groh, K.; Saueressig, F.  Ghost wave-function renormalization in Asymptotically Safe Quantum Gravity.  J. Phys. 2010, A43, 365403, [arXiv:hep-th/1001.5032].  doi:black10.1088/1751-8113/43/36/365403.
  * Eichhorn and Gies (2010) Eichhorn, A.; Gies, H.  Ghost anomalous dimension in asymptotically safe quantum gravity.  Phys. Rev. 2010, D81, 104010, [arXiv:hep-th/1001.5033].  doi:black10.1103/PhysRevD.81.104010.
  * Benedetti et al. (2011) Benedetti, D.; Groh, K.; Machado, P.F.; Saueressig, F.  The Universal RG Machine.  JHEP 2011, 1106, 079, [arXiv:hep-th/1012.3081].  doi:black10.1007/JHEP06(2011)079.
  * Manrique et al. (2011) Manrique, E.; Rechenberger, S.; Saueressig, F.  Asymptotically Safe Lorentzian Gravity.  Phys. Rev. Lett. 2011, 106, 251302, [arXiv:hep-th/1102.5012].  doi:black10.1103/PhysRevLett.106.251302.
  * Reuter and Saueressig (2012) Reuter, M.; Saueressig, F.  Quantum Einstein Gravity.  New J.Phys. 2012, 14, 055022, [arXiv:hep-th/1202.2274].  doi:black10.1088/1367-2630/14/5/055022.
  * Benedetti and Caravelli (2012) Benedetti, D.; Caravelli, F.  The Local potential approximation in quantum gravity.  JHEP 2012, 1206, 017, [arXiv:hep-th/1204.3541].  doi:black10.1007/JHEP06(2012)017, 10.1007/JHEP10(2012)157.
  * Rechenberger and Saueressig (2013) Rechenberger, S.; Saueressig, F.  A functional renormalization group equation for foliated spacetimes.  JHEP 2013, 03, 010, [arXiv:hep-th/1212.5114].  doi:black10.1007/JHEP03(2013)010.
  * Christiansen et al. (2014) Christiansen, N.; Litim, D.F.; Pawlowski, J.M.; Rodigast, A.  Fixed points and infrared completion of quantum gravity.  Phys.Lett. 2014, B728, 114–117, [arXiv:hep-th/1209.4038].  doi:black10.1016/j.physletb.2013.11.025.
  * Dietz and Morris (2013) Dietz, J.A.; Morris, T.R.  Asymptotic safety in the f(R) approximation.  JHEP 2013, 1301, 108, [arXiv:hep-th/1211.0955].  doi:black10.1007/JHEP01(2013)108.
  * Ohta and Percacci (2014) Ohta, N.; Percacci, R.  Higher Derivative Gravity and Asymptotic Safety in Diverse Dimensions.  Class. Quant. Grav. 2014, 31, 015024, [arXiv:hep-th/1308.3398].  doi:black10.1088/0264-9381/31/1/015024.
  * Falls et al. (2013) Falls, K.; Litim, D.; Nikolakopoulos, K.; Rahmede, C.  A bootstrap towards asymptotic safety, [arXiv:hep-th/1301.4191].
  * Falls et al. (2016) Falls, K.; Litim, D.F.; Nikolakopoulos, K.; Rahmede, C.  Further evidence for asymptotic safety of quantum gravity.  Phys. Rev. 2016, D93, 104022, [arXiv:hep-th/1410.4815].  doi:black10.1103/PhysRevD.93.104022.
  * Christiansen et al. (2016) Christiansen, N.; Knorr, B.; Pawlowski, J.M.; Rodigast, A.  Global Flows in Quantum Gravity.  Phys. Rev. 2016, D93, 044036, [arXiv:hep-th/1403.1232].  doi:black10.1103/PhysRevD.93.044036.
  * Becker and Reuter (2014) Becker, D.; Reuter, M.  En route to Background Independence: Broken split-symmetry, and how to restore it with bi-metric average actions.  Annals Phys. 2014, 350, 225–301, [arXiv:hep-th/1404.4537].  doi:black10.1016/j.aop.2014.07.023.
  * Christiansen et al. (2015) Christiansen, N.; Knorr, B.; Meibohm, J.; Pawlowski, J.M.; Reichert, M.  Local Quantum Gravity.  Phys. Rev. 2015, D92, 121501, [arXiv:hep-th/1506.07016].  doi:black10.1103/PhysRevD.92.121501.
  * Morris and Slade (2015) Morris, T.R.; Slade, Z.H.  Solutions to the reconstruction problem in asymptotic safety.  JHEP 2015, 11, 094, [arXiv:hep-th/1507.08657].  doi:black10.1007/JHEP11(2015)094.
  * Ohta et al. (2015) Ohta, N.; Percacci, R.; Vacca, G.P.  Flow equation for $f(R)$ gravity and some of its exact solutions.  Phys. Rev. 2015, D92, 061501, [arXiv:hep-th/1507.00968].  doi:black10.1103/PhysRevD.92.061501.
  * Ohta et al. (2016) Ohta, N.; Percacci, R.; Vacca, G.P.  Renormalization Group Equation and scaling solutions for f(R) gravity in exponential parametrization.  Eur. Phys. J. 2016, C76, 46, [arXiv:hep-th/1511.09393].  doi:black10.1140/epjc/s10052-016-3895-1.
  * Gies et al. (2015) Gies, H.; Knorr, B.; Lippoldt, S.  Generalized Parametrization Dependence in Quantum Gravity.  Phys. Rev. 2015, D92, 084020, [arXiv:hep-th/1507.08859].  doi:black10.1103/PhysRevD.92.084020.
  * Demmel et al. (2015) Demmel, M.; Saueressig, F.; Zanusso, O.  A proper fixed functional for four-dimensional Quantum Einstein Gravity.  JHEP 2015, 08, 113, [arXiv:hep-th/1504.07656].  doi:black10.1007/JHEP08(2015)113.
  * Biemans et al. (2017) Biemans, J.; Platania, A.; Saueressig, F.  Quantum gravity on foliated spacetimes: Asymptotically safe and sound.  Phys. Rev. 2017, D95, 086013, [arXiv:hep-th/1609.04813].  doi:black10.1103/PhysRevD.95.086013.
  * Gies et al. (2016) Gies, H.; Knorr, B.; Lippoldt, S.; Saueressig, F.  Gravitational Two-Loop Counterterm Is Asymptotically Safe.  Phys. Rev. Lett. 2016, 116, 211302, [arXiv:hep-th/1601.01800].  doi:black10.1103/PhysRevLett.116.211302.
  * Denz et al. (2018) Denz, T.; Pawlowski, J.M.; Reichert, M.  Towards apparent convergence in asymptotically safe quantum gravity.  Eur. Phys. J. 2018, C78, 336, [arXiv:hep-th/1612.07315].  doi:black10.1140/epjc/s10052-018-5806-0.
  * Platania and Saueressig (2018) Platania, A.; Saueressig, F.  Functional Renormalization Group Flows on Friedman–Lemaître–Robertson–Walker backgrounds.  Found. Phys. 2018, 48, 1291–1304, [arXiv:hep-th/1710.01972].  doi:black10.1007/s10701-018-0181-0.
  * Falls et al. (2018) Falls, K.; King, C.R.; Litim, D.F.; Nikolakopoulos, K.; Rahmede, C.  Asymptotic safety of quantum gravity beyond Ricci scalars.  Phys. Rev. 2018, D97, 086006, [arXiv:hep-th/1801.00162].  doi:black10.1103/PhysRevD.97.086006.
  * Knorr and Lippoldt (2017) Knorr, B.; Lippoldt, S.  Correlation functions on a curved background.  Phys. Rev. 2017, D96, 065020, [arXiv:hep-th/1707.01397].  doi:black10.1103/PhysRevD.96.065020.
  * Christiansen et al. (2018) Christiansen, N.; Falls, K.; Pawlowski, J.M.; Reichert, M.  Curvature dependence of quantum gravity.  Phys. Rev. 2018, D97, 046007, [arXiv:hep-th/1711.09259].  doi:black10.1103/PhysRevD.97.046007.
  * De Brito et al. (2018) De Brito, G.P.; Ohta, N.; Pereira, A.D.; Tomaz, A.A.; Yamada, M.  Asymptotic safety and field parametrization dependence in the $f(R)$ truncation.  Phys. Rev. 2018, D98, 026027, [arXiv:hep-th/1805.09656].  doi:black10.1103/PhysRevD.98.026027.
  * Falls et al. (2019) Falls, K.G.; Litim, D.F.; Schröder, J.  Aspects of asymptotic safety for quantum gravity.  Phys. Rev. D 2019, 99, 126015, [arXiv:gr-qc/1810.08550].  doi:black10.1103/PhysRevD.99.126015.
  * Kluth and Litim (2020) Kluth, Y.; Litim, D.F.  Fixed Points of Quantum Gravity and the Dimensionality of the UV Critical Surface, [arXiv:hep-th/2008.09181].
  * Falls et al. (2020) Falls, K.; Ohta, N.; Percacci, R.  Towards the determination of the dimension of the critical surface in asymptotically safe gravity.  Phys. Lett. B 2020, 810, 135773, [arXiv:hep-th/2004.04126].  doi:black10.1016/j.physletb.2020.135773.
  * Knorr (2021) Knorr, B.  The derivative expansion in asymptotically safe quantum gravity: general setup and quartic order, [arXiv:hep-th/2104.11336].
  * Narain and Percacci (2010) Narain, G.; Percacci, R.  Renormalization Group Flow in Scalar-Tensor Theories. I.  Class.Quant.Grav. 2010, 27, 075001, [arXiv:hep-th/0911.0386].  doi:black10.1088/0264-9381/27/7/075001.
  * Shaposhnikov and Wetterich (2010) Shaposhnikov, M.; Wetterich, C.  Asymptotic safety of gravity and the Higgs boson mass.  Phys. Lett. 2010, B683, 196–200, [arXiv:hep-th/0912.0208].  doi:black10.1016/j.physletb.2009.12.022.
  * Donà et al. (2014) Donà, P.; Eichhorn, A.; Percacci, R.  Matter matters in asymptotically safe quantum gravity.  Phys.Rev. 2014, D89, 084035, [arXiv:hep-th/1311.2898].  doi:black10.1103/PhysRevD.89.084035.
  * Meibohm et al. (2016) Meibohm, J.; Pawlowski, J.M.; Reichert, M.  Asymptotic safety of gravity-matter systems.  Phys. Rev. 2016, D93, 084035, [arXiv:hep-th/1510.07018].  doi:black10.1103/PhysRevD.93.084035.
  * Donà et al. (2016) Donà, P.; Eichhorn, A.; Labus, P.; Percacci, R.  Asymptotic safety in an interacting system of gravity and scalar matter.  Phys. Rev. 2016, D93, 044049, [arXiv:gr-qc/1512.01589].  [Erratum: Phys. Rev.D93,no.12,129904(2016)], doi:black10.1103/PhysRevD.93.129904, 10.1103/PhysRevD.93.044049.
  * Oda and Yamada (2016) Oda, K.y.; Yamada, M.  Non-minimal coupling in Higgs–Yukawa model with asymptotically safe gravity.  Class. Quant. Grav. 2016, 33, 125011, [arXiv:hep-th/1510.03734].  doi:black10.1088/0264-9381/33/12/125011.
  * Eichhorn and Lippoldt (2017) Eichhorn, A.; Lippoldt, S.  Quantum gravity and Standard-Model-like fermions.  Phys. Lett. 2017, B767, 142–146, [arXiv:gr-qc/1611.05878].  doi:black10.1016/j.physletb.2017.01.064.
  * Wetterich and Yamada (2017) Wetterich, C.; Yamada, M.  Gauge hierarchy problem in asymptotically safe gravity–the resurgence mechanism.  Phys. Lett. B 2017, 770, 268–271, [arXiv:hep-th/1612.03069].  doi:black10.1016/j.physletb.2017.04.049.
  * Biemans et al. (2017) Biemans, J.; Platania, A.; Saueressig, F.  Renormalization group fixed points of foliated gravity-matter systems.  JHEP 2017, 05, 093, [arXiv:hep-th/1702.06539].  doi:black10.1007/JHEP05(2017)093.
  * Christiansen et al. (2018) Christiansen, N.; Litim, D.F.; Pawlowski, J.M.; Reichert, M.  Asymptotic safety of gravity with matter.  Phys. Rev. 2018, D97, 106012, [arXiv:hep-th/1710.04669].  doi:black10.1103/PhysRevD.97.106012.
  * Hamada and Yamada (2017) Hamada, Y.; Yamada, M.  Asymptotic safety of higher derivative quantum gravity non-minimally coupled with a matter system.  JHEP 2017, 08, 070, [arXiv:hep-th/1703.09033].  doi:black10.1007/JHEP08(2017)070.
  * Eichhorn et al. (2018) Eichhorn, A.; Hamada, Y.; Lumma, J.; Yamada, M.  Quantum gravity fluctuations flatten the Planck-scale Higgs potential.  Phys. Rev. 2018, D97, 086004, [arXiv:hep-th/1712.00319].  doi:black10.1103/PhysRevD.97.086004.
  * Eichhorn and Held (2018) Eichhorn, A.; Held, A.  Top mass from asymptotic safety.  Phys. Lett. 2018, B777, 217–221, [arXiv:hep-th/1707.01107].  doi:black10.1016/j.physletb.2017.12.040.
  * Eichhorn et al. (2018) Eichhorn, A.; Lippoldt, S.; Skrinjar, V.  Nonminimal hints for asymptotic safety.  Phys. Rev. D 2018, 97, 026002, [arXiv:hep-th/1710.03005].  doi:black10.1103/PhysRevD.97.026002.
  * Alkofer and Saueressig (2018) Alkofer, N.; Saueressig, F.  Asymptotically safe $f(R)$-gravity coupled to matter I: the polynomial case.  Annals Phys. 2018, 396, 173–201, [arXiv:hep-th/1802.00498].  doi:black10.1016/j.aop.2018.07.017.
  * Eichhorn et al. (2018) Eichhorn, A.; Labus, P.; Pawlowski, J.M.; Reichert, M.  Effective universality in quantum gravity.  SciPost Phys. 2018, 5, 031, [arXiv:hep-th/1804.00012].  doi:black10.21468/SciPostPhys.5.4.031.
  * Eichhorn et al. (2019a) Eichhorn, A.; Lippoldt, S.; Pawlowski, J.M.; Reichert, M.; Schiffer, M.  How perturbative is quantum gravity?  Phys. Lett. 2019, B792, 310–314, [arXiv:hep-th/1810.02828].  doi:black10.1016/j.physletb.2019.01.071.
  * Eichhorn et al. (2019b) Eichhorn, A.; Lippoldt, S.; Schiffer, M.  Zooming in on fermions and quantum gravity.  Phys. Rev. 2019, D99, 086002, [arXiv:hep-th/1812.08782].  doi:black10.1103/PhysRevD.99.086002.
  * Pawlowski et al. (2019) Pawlowski, J.M.; Reichert, M.; Wetterich, C.; Yamada, M.  Higgs scalar potential in asymptotically safe quantum gravity.  Phys. Rev. 2019, D99, 086010, [arXiv:hep-th/1811.11706].  doi:black10.1103/PhysRevD.99.086010.
  * Knorr et al. (2019) Knorr, B.; Ripken, C.; Saueressig, F.  Form Factors in Asymptotic Safety: conceptual ideas and computational toolbox.  Class. Quant. Grav. 2019, 36, 234001, [arXiv:hep-th/1907.02903].  doi:black10.1088/1361-6382/ab4a53.
  * Bürger et al. (2019) Bürger, B.; Pawlowski, J.M.; Reichert, M.; Schaefer, B.J.  Curvature dependence of quantum gravity with scalars, [arXiv:hep-th/1912.01624].
  * Eichhorn and Schiffer (2019) Eichhorn, A.; Schiffer, M.  $d=4$ as the critical dimensionality of asymptotically safe interactions.  Phys. Lett. 2019, B793, 383–389, [arXiv:hep-th/1902.06479].  doi:black10.1016/j.physletb.2019.05.005.
  * Reichert and Smirnov (2020) Reichert, M.; Smirnov, J.  Dark Matter meets Quantum Gravity.  Phys. Rev. D 2020, 101, 063015, [arXiv:hep-ph/1911.00012].  doi:black10.1103/PhysRevD.101.063015.
  * Kurov and Saueressig (2020) Kurov, A.; Saueressig, F.  On characterizing the Quantum Geometry underlying Asymptotic Safety.  Front. in Phys. 2020, 8, 187, [arXiv:hep-th/2003.07454].  doi:black10.3389/fphy.2020.00187.
  * Daas et al. (2020) Daas, J.; Oosters, W.; Saueressig, F.; Wang, J.  Asymptotically safe gravity with fermions.  Phys. Lett. B 2020, 809, 135775, [arXiv:hep-th/2005.12356].  doi:black10.1016/j.physletb.2020.135775.
  * Eichhorn and Pauly (2020) Eichhorn, A.; Pauly, M.  Safety in darkness: Higgs portal to simple Yukawa systems, [arXiv:hep-ph/2005.03661].
  * Eichhorn and Pauly (2021) Eichhorn, A.; Pauly, M.  Constraining power of asymptotic safety for scalar fields.  Phys. Rev. D 2021, 103, 026006, [arXiv:hep-th/2009.13543].  doi:black10.1103/PhysRevD.103.026006.
  * Ali et al. (2020) Ali, P.; Eichhorn, A.; Pauly, M.; Scherer, M.M.  Constraints on discrete global symmetries in quantum gravity, [arXiv:gr-qc/2012.07868].
  * Harst and Reuter (2011) Harst, U.; Reuter, M.  QED coupled to QEG.  JHEP 2011, 1105, 119, [arXiv:hep-th/1101.6007].  doi:black10.1007/JHEP05(2011)119.
  * Eichhorn and Gies (2011) Eichhorn, A.; Gies, H.  Light fermions in quantum gravity.  New J.Phys. 2011, 13, 125012, [arXiv:hep-th/1104.5366].  doi:black10.1088/1367-2630/13/12/125012.
  * Christiansen and Eichhorn (2017) Christiansen, N.; Eichhorn, A.  An asymptotically safe solution to the U(1) triviality problem.  Phys. Lett. 2017, B770, 154–160, [arXiv:hep-th/1702.07724].  doi:black10.1016/j.physletb.2017.04.047.
  * Eichhorn and Versteegen (2018) Eichhorn, A.; Versteegen, F.  Upper bound on the Abelian gauge coupling from asymptotic safety.  JHEP 2018, 01, 030, [arXiv:hep-th/1709.07252].  doi:black10.1007/JHEP01(2018)030.
  * Gies and Martini (2018) Gies, H.; Martini, R.  Curvature bound from gravitational catalysis.  Phys. Rev. D 2018, 97, 085017, [arXiv:hep-th/1802.02865].  doi:black10.1103/PhysRevD.97.085017.
  * Eichhorn and Held (2018) Eichhorn, A.; Held, A.  Mass difference for charged quarks from asymptotically safe quantum gravity.  Phys. Rev. Lett. 2018, 121, 151302, [arXiv:hep-th/1803.04027].  doi:black10.1103/PhysRevLett.121.151302.
  * Alkofer et al. (2020) Alkofer, R.; Eichhorn, A.; Held, A.; Nieto, C.M.; Percacci, R.; Schröfl, M.  Quark masses and mixings in minimally parameterized UV completions of the Standard Model.  Annals Phys. 2020, 421, 168282, [arXiv:hep-ph/2003.08401].  doi:black10.1016/j.aop.2020.168282.
  * de Brito et al. (2021) de Brito, G.P.; Eichhorn, A.; Schiffer, M.  Light charged fermions in quantum gravity.  Phys. Lett. B 2021, 815, 136128, [arXiv:hep-th/2010.00605].  doi:black10.1016/j.physletb.2021.136128.
  * Gies and Salek (2021) Gies, H.; Salek, A.S.  A curvature bound from gravitational catalysis in thermal backgrounds, [arXiv:hep-th/2103.05542].
  * Bonanno and Platania (2015) Bonanno, A.; Platania, A.  Asymptotically safe inflation from quadratic gravity.  Phys. Lett. B 2015, 750, 638–642, [arXiv:gr-qc/1507.03375].  doi:black10.1016/j.physletb.2015.10.005.
  * Alkofer et al. (2016) Alkofer, N.; D’Odorico, G.; Saueressig, F.; Versteegen, F.  Quantum Gravity signatures in the Unruh effect.  Phys. Rev. 2016, D94, 104055, [arXiv:gr-qc/1605.08015].  doi:black10.1103/PhysRevD.94.104055.