# Infrared behavior of Weyl Gravity

Lesław Rachwał 1∗ and Stefano Giaccari 2 1\. Departamento de Física – ICE,
Universidade Federal de Juiz de Fora, MG, Brazil  2\. Department of Sciences,
Holon Institute of Technology (HIT), 52 Golomb St., Holon 5810201, Israel
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper, we introduce and motivate the studies of Quantum Weyl Gravity
(also known as Conformal Gravity). We discuss some appealing features of this
theory both on classical and quantum level. The construction of the quantum
theory is described in detail to the one-loop level. To facilitate
computations we use only physical degrees of freedom, which are singled out
through the York decomposition. At the one-loop level we compute the partition
function around a general Einstein space. Next, the functional renormalization
group of couplings in Quantum Weyl Gravity is investigated. We reproduce
completely previous results obtained on maximally symmetric and Ricci-flat
backgrounds. Finally, we comment on further directions and on the issue of
conformal anomaly.

* - speaker at IARD 2020

##  1 Introduction

Conformal gravity [1] is a field-theoretical system with a very peculiar form
of relativistic dynamics. The evolution of Green functions in this model is
governed by two principles of the fundamental character. First is the
principle of general relativity since this model describes gravitational
interactions. Therefore we have the freedom to choose an arbitrary coordinate
system to describe the physics and nothing observable depends on this choice.
The second principle is the invariance under local conformal transformations
since this is a conformal gravity, where the conformal symmetry is realized in
the local version [2]. In turn, again we have a freedom to choose an arbitrary
system of units and scales to measure lengths, times, energies, etc. In
conformal gravity nothing observable depends on this choice of mass scale.
When these two principles are realized, the resulting dynamical system is of a
very special nature.

Relativistic dynamics is a very broad topic. In general, it describes theories
where the fundamental dynamical principles governing the dynamics of the
system are Lorentz invariant. When the Lorentz symmetry is made local (gauged)
then we end up with theories describing the dynamical structure and evolution
of spacetime itself. The requirement of local Lorentz invariance puts very
strong dynamical constraints on the possible form of the relativistic theory
of gravitation. Similarly, by gauging conformal symmetry we produce a very
particular gravitational theory, which is unique in four-dimensional
spacetime. The symmetry in the global version describes matter systems and
models which are in particular scale-invariant, so they do not possess any
characteristic invariant mass scale and their dynamics looks the same at any
scale of observation. In conformal gravity we describe the dynamics of the
relativistic gravitational field which does not contain any mass scale (like
Planck scale for example). This situation is very special regarding
gravitational theories and one can realize that new additional constraints are
put on the dynamics of the gravitational field. At the end, the system is very
constrained, but not contradictory and instead it leads to unique very
restrictive predictions for gravitational observables. This is why it is
interesting to investigate this relativistic system closer.

Although conformal symmetry was first used by Weyl in what would be called now
a gravitational context, it was rediscovered in models of particle physics in
the second half of the last century [3]. It was easier to find conformal high
energy models without gravity than with it. Conformal symmetry as well as
scale invariance was first better understood in massless and scaleless models
of particle physics. In the realm of quantum field theories (QFT) some very
special models show conformality also on the quantum level [4]. Since then
this symmetry has been realized and investigated both on the classical and
quantum level of matter models [5, 6, 7, 8, 9, 10, 11, 12]. However, somehow
in parallel, conformal methods found their applications in classical research
on gravitational theories [13, 14, 15, 16, 17, 18]. This put back conformal
symmetry to its natural gravitational setup, where, however, it was most times
analyzed in sufficient details only in the classical domain. As it is well
known, the question concerning the definition of a consistent quantum
gravitational theory is still open although there are very interesting
proposals on the market nowadays. The quantum role of conformal symmetry in
the gravitational setup is currently being actively investigated [19, 20, 21,
22, 23, 24, 25, 26]. The constraints mentioned above select a unique theory in
$d=4$ dimensions which bears the name of Weyl gravity or conformal gravity
(although most of the preliminary works in this theory were done by Bach in
1920’s [1]).

Quantum Weyl Gravity (QWG) [27] is a proposal for a fundamental theory
realizing symmetries of relativity and conformality in the local version both
on the classical and quantum level. We will discuss the status of this
theoretical model, its overall consistency and its current problems below a
bit. Most of them are related to the ultraviolet (UV) regime of energies. One
can assume hypothetically that they are solved by an existence of some UV
fixed point (FP) realizing also tentative UV-completion of the theory. The
situation on the other side of the spectrum, namely in the infrared (IR) part
of energies, is also interesting. In this contribution we review the results
obtained previously for the IR renormalization group (RG) flows in QWG. We
also put them in a perspective relating them with a very special relativistic
dynamics happening in QWG.

##  2 Motivations for QWG

First, we discuss here the motivations for Weyl gravity (WG) on the classical
level. Later we add also some special features visible only on the quantum
level. Both particle models (without gravity) and gravitational models with
conformal symmetry are discussed briefly.

The first motivation for conformal symmetry in our Universe comes from the
rough observation of cosmic microwave background radiation (CMB). Its spectrum
(giving us some clues about the dynamics of the very early universe) is very
close to be scale-invariant [28]. The spectral index of CMB is

$n_{s}=1.00-\varepsilon\quad{\rm with}\quad\varepsilon\ll 1.$ (1)

Hence as a first approximation we can assume that early Universe is described
by a scale-invariant gravitational theory coupled also to scale-invariant
matter models (like electromagnetic radiation). As it is well known Maxwell
theory of electromagnetic interactions is not only globally scale-invariant,
but is invariant under full local conformal transformations (depending on the
spacetime location). The same extension can be performed on the level of
gravitational dynamics to end up with $4$-dimensional Weyl gravity. Moreover,
it is also a common wisdom that in a fully quantized theory involving
relativistic and dynamical gravity, all global symmetries must be made local.
And this procedure when applied to scale-invariance results in full invariance
under local conformal transformations. So the early Universe must be described
by a theory enjoying local (gauged) conformal dynamical symmetry. This is one
of the first phenomenological or observational motivation for WG.

Conformal symmetry has its deep roots in the differential geometry of two-
dimensional curved surfaces. There it was first observed that the following
transformations

$g_{\mu\nu}\to g_{\mu\nu}^{\prime}=\Omega^{2}(x)g_{\mu\nu}$ (2)

make any curved $2$-dimensional manifold flat. (This is why all $2$-manifolds
are conformally flat). The transformations in (2) are not induced by any
coordinate transformation $x\to x^{\prime}=x^{\prime}(x)$, hence the
differential structure of the manifolds is not preserved. Under the conformal
transformations the metric is “charged” whereas the coordinates remain
untouched, so that the invariant infinitesimal line element in differential
geometry is not preserved and scales according to the law

$ds^{2}\to ds^{\prime 2}=\Omega^{2}(x)ds^{2}.$ (3)

From this we derive that distances (lengths, times, energies, etc.) are not
invariant. They are relative. And they depend on the choice of the gauge; in
this case we speak about a conformal gauge. We conclude that by performing
conformal transformations we abandon the absolute meaning of physical
distances and scales [29]. With physical relativity we know that velocities
are relative, here we add the conformal gauge-dependence of scales. But one
element of the geometrical description remains invariant under the conformal
transformations, even when their parameter $\Omega=\Omega(x)$ depends on the
location on the manifold. These are angles between vectors. The angle between
two vectors $\vec{u}$ and $\vec{v}$ could be defined as

$\measuredangle(\vec{u},\vec{v})=\arccos\left(\frac{\vec{u}\cdot\vec{v}}{\sqrt{|\vec{u}|^{2}|\vec{v}|^{2}}}\right).$
(4)

This is an example of conformal invariant since it does not depend on the
scales.

In Minkowskian framework (when we include temporal relations to our
differential geometry description of curved spacetime), the structure which is
left invariant under conformal transformations is a causal structure on the
spacetime manifold. The light-like trajectories of massless particles remain
such since their defining condition $ds^{2}=0$ is untouched by conformal
rescalings (2). This was one of the reasons for the success of methods of
conformal transformation during the golden years of general relativity (GR).
In the gravitational framework, conformal methods revealed secrets of black
holes and their horizons as well as they were instrumental in flattening of
cosmologies. (All Friedmann-Lemaître-Robertson-Walker gravitational spacetimes
are conformally flat.) Only the causal or conformal structure of the spacetime
manifold is preserved after the process of conformal change of the metric.
Moreover, in the gravitational context global scale symmetry (equivalent to
absence of mass dimension parameter in the theory, so all the couplings are
dimensionless) can be gauged like other global gauge symmetries, which can be
made local and dynamical, for example in some models of particle physics. For
the gravitational dynamics, the local conformal symmetry (invariance with
respect to transformations (2) with the spacetime-dependent parameter
$\Omega(x)$) can be an additional gravitational symmetry constraining the
dynamics even more strongly, as we will show below (both on the quantum and
classical level). The conformal symmetry can be treated like a gauge symmetry
of gravity and can be placed on the same level as local Lorentz symmetry (so
invariance under local boosts together with local translations) in the full
gravitational theory.

Conformal symmetry should be the last dynamical symmetry (making impacts on
the dynamics of the system) to be discovered in particle physics. For its
verification one needs to go to arbitrarily high energy scales. However, the
first time this symmetry was realized on the quantum level was in theoretical
models without gravity and on flat Minkowski spacetime backgrounds [4].

In high energy models, one intuitively knows that almost every time one deals
with massless particles or excitations, then the conformal symmetry emerges in
the description of the system. Even if the system is not intended to possess
some conformal properties, the limit where all particles become effectively
without mass, leads to a miraculous enhancement of the symmetries of the
considered model. And there are some properties that could be only explained
by invoking conformal symmetry. Hence conformal symmetry is a quite frequently
an emergent approximate symmetry of models of QFT, when the regime of high
energy is analyzed. The particles do not have to be strictly with vanishing
mass, their mass can be effectively very small compared to other energy scales
in the very high energy approximation. For the classical considerations done
on flat Minkowski spacetime background, conformal symmetry can be seen just as
global scale-invariance. However, full conformal group contains more
generators than just dilatation generator from scale-invariance. And also its
predictions are more firm and it constrains the dynamical system enjoying
invariance under the full conformal group more.

The full constraining power of conformal symmetry is seen on the Green
functions of the model enjoying it. It is more than just scale-invariance,
which forbids only the presence of dimensionful parameters in Green functions.
For global scale-invariance dimensionful momenta $p$, energies $E$, and fields
$\phi$ can still be present on the legs of Green functions and can enter into
the corresponding expression for these Green functions in arbitrary
combinations not constrained by any further symmetry principle. Full conformal
symmetry changes this last fact. The dynamics of a Quantum Field Theory (QFT)
with conformal symmetry at quantum level is so strongly constrained that it is
customary to refer to this special class of theories under the name of
conformal field theories (CFT). Quantum Green functions are very strongly
constrained in such models. For example, the form of all $2$-point and
$3$-point functions is completely fixed and $4$-point functions at arbitrary
loop order depends only on one arbitrary function of very specially
constructed conformal ratios [30]. Quantum conformality enjoyed by the
theories in this class means that all quantum fluctuations look the same at
all energy scales. So we do not have standard suppression of quantum
fluctuations in IR or in UV. In quantum CFT models they show the same strength
in every energy regime. But quantum conformality is more than quantum scale-
invariance since these two group of symmetries are not isomorphic to each
other. Moreover, as a next piece of motivation, one can notice that the
current standard model (SM) of particle physics is a classically almost scale-
invariant theory and the breaking of scale-invariance is related to the
spontaneous breaking of electroweak symmetry (so it is proportional to the
vacuum expectation value of the Higgs field, or in turn to its mass).

Relativistic models with scale-invariance enjoy a symmetry under an
$11$-parameter group, which includes the generator of dilatations, apart from
those from the Poincaré group. Instead, full conformal group in $d=4$ contains
$15$ generators because of $4$ additional generators of conformal boosts. To
require invariance under these additional transformations of all Green
functions (also on the quantum level) is a big constraint, which signifies
that there is an additional symmetry on the quantum level. This is of course
also a symmetry of the full quantum effective action. Conformal symmetry as
additional symmetry on the quantum level constrains further (besides
relativistic constraints imposed by symmetry under transformations from the
Lorentz group) quantum dynamics in such a way that UV-divergences are
completely avoided in such models. Since there cannot be any scale or mass
generated in quantum dynamics of CFT model, then there is no need to
renormalize the quantum theory by introducing an arbitrary renormalization
scale $\mu$. And the fact of need for doing infinite UV-renormalization is
tightly related to the presence of infinities in the UV regime of the theory.
This additional symmetry is essential for avoiding quantum singularities in
CFT models. However, the similar arguments can be also applied to classical
gravitational theory enjoying conformal symmetry to solve the ubiquitous
problems with classical spacetime singularities. This is a next motivation for
classical Weyl gravity, which should keep its conformal properties also on the
quantum level for this argument to work.

The CFT matter models considered above are indeed very special QFT’s since in
them there are no UV-divergences. This implies that there are no
$\beta$-functions both on the perturbative (to all orders) level as well as
non-perturbatively due to conformal algebra in which we must have the energy-
momentum tensor operator of CFT. If we do not need to use the renormalization
scale $\mu$ to define properly theory on the quantum level, then also the form
of the quantum effective action is very special. One can view the quantization
procedure ${\cal Q}$ as a several mapping in theory space ${\cal T}$:

${\cal Q}:{\cal T}_{{\rm cl}}\to{\cal T}_{q},$ (5)

where both the classical theory ${\cal T}_{{\rm cl}}$ and the quantum theory
${\cal T}_{q}$ (regularized and renormalized with some physical
renormalization conditions) are elements of the theory space ${\cal T}$. The
quantum theory ${\cal T}_{q}$ is specified completely by quantum effective
action $\Gamma$ (regularized and renormalized), similarly like ${\cal T}_{{\rm
cl}}$ is defined by classical action functional $S_{{\rm cl}}$. The conformal
symmetry constrains the effective action of the quantum theory ${\cal T}_{q}$
so much that ${\cal T}_{q}={\cal T}_{{\rm cl}}$ (up to a finite
renormalization of couplings). In other words, the quantization procedure
${\cal Q}$ finds its mathematical fixed point for very special initial
elements ${\cal T}_{{\rm cl}}$, which leads to conformal symmetry both on the
classical level. And what is more important it leads also to conformality on
the quantum level which is preserved by quantum corrections. In a mathematical
sense this gives a fully idempotent definition of the quantum effective action
$\Gamma$ in CFT. Quantization process ${\cal Q}$ produces quantum corrections
to the classical dynamics, but they are scale-invariant, so their dynamics is
completely captured by the original classical action, if it is properly
conformally invariant. We conclude that quantum effective action $\Gamma$ is
the same (up to finite dimensionless coupling rescalings) as the classical
conformally invariant action $S_{{\rm cl}}$.

The conformal symmetry of CFT is explicitly realized on the form of all
correlation functions of the theory. But not only there. Due to conformal
symmetry the special set of Ward identities is satisfied in CFT relating
different $n$-point function of the theory. They are quantum operatorial
manifestation of the conformal symmetry present in the quantum effective
action $\Gamma$. They encode on the quantum level, for example, the fact that
the trace of the effective energy-momentum tensor $T=g^{\mu\nu}T_{\mu\nu}$ of
the full theory is off-shell zero even after inclusion of quantum corrections.
And this statement has the operatorial meaning, not only in the sense of
expectation values. Generally, the set of all conformal Ward identities
satisfied by any CFT (including these models when gravity is dynamical) is an
expression of the fact that the quantum system is highly constrained and that
its relativistic dynamics is not an ordinary one. Moreover, quantum CFT models
are finite (no UV divergences), so these are very well behaved quantum models
which do not need regularization for their definitions on the quantum level.
The conformal symmetry in CFT’s is very constraining such that the form of
classical action describing the starting point of the theory $S_{{\rm cl}}$ is
very restrictive, provided that the number of spacetime dimension $d$ is
fixed. Another specification of the model is the listing of all fields in it.
However, not any composition is allowed. There are also restrictions on this.
For example, if we desire too construct a CFT with gauge interactions, this
must take a form of a very special theory of ${\cal N}=4$ supersymmetric Yang-
Mills theory. If we want to add also gravitational interactions, this must be
a model of ${\cal N}=4$ conformal supergravity first discovered in 1985 by
Fradkin and Tseytlin. Other actions, when conformal symmetry is preserved on
the quantum level (so it is not broken) are not possible in $d=4$.

We see that in CFT there are no $\beta$-functions of (necessarily)
dimensionless coupling parameters. So they are really coupling constants, in
the old sense, they are not RG running constants. There is no RG flow in
models of CFT. The only difference from the classical field theory is that the
dimension of operators may receive some finite corrections (though not related
to any RG running), so there could be some anomalous dimensions of various
operators in CFT’s. The fact that there is no RG flow is equivalent to the
statement that conformal field theories sit at the FP of the RG flows. They
describe completely the situation (scaling dimensions, also known as critical
exponents) of all operators present in a given CFT. For a different point of
view, see [31]. The analysis of FP’s of RG flows is a very important topic for
its own interest. However, knowing the general structure of CFT’s in a form of
CFT data (conformal algebra, set of primary operators, their scaling
dimensions) helps a lot in describing the FP’s of RG flows which are very
special points in theory space ${\cal T}$. As it is well known FP’s of RG are
starting (or ending) points of RG flows since the situation at the FP’s could
be seen as very extraordinary. But the existence of non-trivial RG flows
breaks explicitly conformal symmetry of the CFT model. To do this in a more
gentle manner, one can imagine turning smoothly various deformation operators
and then the breaking of conformal symmetry could be achieved in a spontaneous
way. In this way CFT’s are starting points in the theory space ${\cal T}$ for
various non-conformal theory modifications, in which there is RG flow, but
still there is some remnant and memory of the constraints on the dynamics that
were imposed at the FP. Therefore this kind of relativistic dynamics enjoys RG
flows and only some remnants of full conformal symmetry. The full symmetry is
recovered in the UV limit of the theory. In the next sections of this
contribution, we will present one quantum computation which is done in the
theory, which originates from full CFT with gravitational interactions, but in
which conformal symmetry is spontaneously broken and the RG flows towards IR
is possible. As we will see even in such partially broken model, the amount of
constraints which were left after fully realized conformal symmetry is
significant and the ensuing relativistic dynamics is constrained and highly
interesting.

##  3 Weyl gravity

Now we proceed with the construction of Weyl gravity in $d=4$ spacetime
dimensions. The number of dimensions here has to be fixed and even integer. In
quantum theory we will not allow playing with the “effective” dimensionality
of spacetime similarly to what happens for example in dimensional
regularization (DIMREG) scheme). This requirement has to do with the
construction of the classical action of gravity which is supposed to be
conformally invariant. For such a construction we must use Weyl tensors as
building blocks, and if we require locality of the action, then they can
appear only in positive integer powers. Hence the dimension $d$ of spacetime
has to be even integer, and we choose $d=4$ to be consistent with our every
day experience of relativistic physics.

The main building block for the action is the Weyl tensor (tensor of conformal
curvature), which is defined in $d=4$ as

$C_{\mu\nu\rho\sigma}=R_{\mu\nu\rho\sigma}-2R_{[\mu[\rho}g_{\sigma]\nu]}+\frac{1}{3}g_{[\mu\rho}g_{\nu]\sigma}R,$
(6)

where $R_{\mu\nu\rho\sigma}$, $R_{\mu\nu}$ and $R$ denote the Riemann tensor,
Ricci tensor and Ricci scalar respectively. Its main useful property is the
conformal transformation law

$C_{\mu\nu\rho\sigma}\to\Omega^{2}C_{\mu\nu\rho\sigma}$ (7)

or for versions with different positions of indices (such position matters!)

$C_{\mu\nu\rho}{}^{\sigma}\to C_{\mu\nu\rho}{}^{\sigma},\quad{\rm and}\quad
C^{\mu\nu\rho\sigma}\to\Omega^{-6}C^{\mu\nu\rho\sigma}$ (8)

Of course, Weyl tensor plays the role of the gravitational curvature and for
example the vanishing of all its components for some given spacetime implies
it is conformally flat. Then such a spacetime can be mapped to a completely
flat one by a conformal transformation (2) of the metric tensor. Generally, in
conformal gravity different spacetimes differ only by their conformal
structures (so how the Weyl tensor $C_{\mu\nu\rho\sigma}$ looks like), and the
details about the differential metric structure are irrelevant. Moreover,
metrics related by conformal transformations are in the same conformal
equivalence class and hence they describe the same gravitational spacetime
from the point of view of conformal symmetry. Not only we have to be
independent of the particular choice of the coordinate system, but also of the
choice of the conformal gauge, so of the rescaling factor $\Omega^{2}$ in the
definition of conformal transformation for metric (2). This conformal factor
$\Omega^{2}$ (known also as a scale factor in cosmology) is strictly non-
negative due to the usage of the square of a completely arbitrary real scalar
function $\Omega(x)$ of the point $x$ of the spacetime. To describe the
gravitational field in conformal gravity, one must exploit conformal
invariants, which are both diffeomorphism invariants (scalars) as well as they
do not transform under general conformal transformations. The set of them is
quite restricted. The gravitational action must be one of them.

This action [1] in $d=4$ is constructed exclusively with the help of Weyl
tensor as

$S_{{\rm conf}}=\\!\int\\!d^{4}x\sqrt{|g|}\alpha_{C}C^{2},$ (9)

where the square of the Weyl tensor has the following expansion in known
quadratic in curvatures invariants

$C^{2}=C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-2R_{\mu\nu}R^{\mu\nu}+\frac{1}{3}R^{2},$
(10)

which shows a big resemblance to the formula (6) regarding the numerical
coefficients. This is easily explained by the tracelessness of the Weyl tensor
in any pair of its indices. To check that the action (9) is conformally
invariant in $d=4$ spacetime dimension, one must also recall a conformal
transformation law for the volume element, namely

$\sqrt{|g|}\to\sqrt{|g^{\prime}|}=\Omega^{4}\sqrt{|g|},$ (11)

while the coordinate volume differentials collected in $d^{4}x$ do not
transform (that is $d^{4}x\to d^{4}x$) since the coordinates are not changed
under the transformation ($x^{\mu}\to x^{\mu}$) and all conformal
transformations are induced from the way how the metric changes in (2).

One notices that the scalar invariant $C^{2}$ transforms co-covariantly, that
is we have

$C^{2}\to\Omega^{-4}C^{2}$ (12)

and only its properly densitized version

$\sqrt{|g|}C^{2}\to\sqrt{|g|}C^{2}$ (13)

is conformal invariant. However, this last one is not a scalar invariant from
the perspective of GR, since it is a density. Therefore in the local framework
we are in conflict of finding local conformal invariants which are also
scalars from the point of view of diffeomorphisms. This observation has some
important consequences for the study of black hole solutions and the question
of the fate of singularities there. Probably, one should look at some non-
local invariants which are integrated versions of the densitized scalar
invariant in (13), i.e. at the values of the action integral (9) over some
small region of spacetimes.

The action integral is a scalar and it is conformally invariant, but as a main
feature of the action functionals it is obtained via integration, so that is
why it must be highly non-local (or strongly non-local). This action in (9)
describes completely the classical conformal Weyl gravity in $d=4$. This
theory is both diffeomorphically (relativistic coordinate changes) and
conformally invariant (changes of the conformal factors of the metric). This
is a unique theory in $d=4$ since there are no other local conformal
invariants (besides $\sqrt{|g|}C^{2}$ in (13)) possible to be constructed
here. Additionally, the theory is described by only one conformal coupling
constant $\alpha_{C}$. In this way this theory is a very similar to Maxwell or
Yang-Mills action, which is also quadratic in gauge field strengths and which
also comes with one dimensionless coupling constant. The fact that
$\alpha_{C}$ is dimensionless and unique constrains the choice of the theory
even more in the gravitational setting. Compare this situation with Einstein-
Hilbert with cosmological constant which already comes with two coupling
parameters. Models with higher derivatives and quadratic in curvatures contain
even more such coupling parameters. (The general Stelle theory has four such
parameters appearing in the Lagrangian $\lambda_{{\rm
cc}}+\kappa_{N}R+\alpha_{R}R^{2}+\alpha_{C}C^{2}$). The uniqueness of the
theory is a very elegant and economic property to select a correct
gravitational theory. For definiteness in Euclidean setup we require that the
coupling $\alpha_{C}$ is given by a real number and strictly positive.

One should examine the advantages of classical Weyl gravity as a theory of
relativistic gravitational field. First, it is noticed that the energy-
momentum tensor which is a source of gravity must be here traceless to comply
with the tracelessness of the tensor of equations of motion (EOM) coming from
the action (9) required by conformal invariance of both matter (source) and
gravity sides of the coupled EOM. Explicitly, the tensor of EOM for Weyl
conformal gravity has the form

$B^{\mu\nu}=\frac{1}{\alpha_{C}}\frac{1}{\sqrt{|g|}}\frac{\delta S_{{\rm
conf}}}{\delta
g_{\mu\nu}}=2\nabla_{\kappa}\nabla_{\lambda}C^{\mu\kappa\nu\lambda}+C^{\mu\kappa\nu\lambda}R_{\kappa\lambda}$
(14)

and it bears the name of Bach tensor. Generally, in $d=4$ the Bach tensor is
symmetric ($B^{\mu\nu}=B^{\nu\mu}$), automatically traceless
($g_{\mu\nu}B^{\mu\nu}$), divergence-free ($\nabla_{\mu}B^{\mu\nu}=0$) and it
originates from the conformal action $S_{{\rm conf}}$ as the variational
derivative of the action integral in (9). The field equations in Weyl gravity
read

$B^{\mu\nu}=\frac{1}{2}\alpha_{c}T^{\mu\nu}$ (15)

This form of EOM puts a stringent condition on the form of a possible energy-
momentum tensor of the matter source because it has to be traceless, so that
only massless or scale-invariant matter on their equations of motion can give
rise to it. This condition is satisfied in the deep UV (regime of high
energies), where all the matter becomes “effectively” massless and conformal
symmetry is restored. Since the effects of relativistic gravity are not well
investigated and understood in the regime of very high energies and also of
high matter densities, then it is more practical to look for tests of CWG in
the setup of vacuum solutions, where there is no matter at all
($T^{\mu\nu}=0$).

Here, for empty space solutions, we can perform and verify three basic tests
of gravitational theory which originally gave impetus to the Einstein
gravitational theories. Those were the perihelion of Mercury, the light
deflection by the Sun and the redshift of spectral lines due to gravitational
effects. To this one could also add the time delay of radar echoes and
precessions of gyroscopes. All these tests were carried out in empty space and
they gave a verification of Einstein’s theory. But they are also equally valid
and satisfied in CWG since they are based in particular on static and
spherically symmetric exact vacuum solutions in the form of the Schwarzschild
metric. This metric is also an exact vacuum solution in CWG. A bolder
statement is that all exact vacuum solutions of Einstein gravitational theory
(without cosmological constant) are also exact vacuum solutions of classical
Weyl gravity. For this, one have to notice that in the expansion in (10) only
the Ricci tensor appears, hence the EOM of CWG are quadratic in Ricci tensors
or in Ricci scalars (and not of Riemann tensor), and hence they vanish
identically on Ricci-flat spacetime which are known to be gravitational vacuum
in original Einsteinian theory. We conclude that metrics describing vacuum
spacetimes like Schwarzschild, Kerr, etc are also solutions there. This
implies that CWG reproduces all precise tests of relativistic gravitation like
Einstein theory in the vacuum.

But CWG contains more solutions that just Einsteinian theory. And this is one
of its powers. For example, the solutions found by Mannheim and Kazanas for
static and spherically symmetric situations in conformal gravitational vacuum
[32] are a family of solutions with more parameters than in Einsteinian
theory. But they can serve as a solution of “dark matter” or missing matter
problems in galaxies [16, 17, 18]. These solutions describe accurately more
than 200 galactic rotation curves with just one fittable parameter and mimic
the behavior of the gravitational potential produced by the conjectured dark
matter halo. However, in CWG dark matter is not necessary at all and the power
of classical conformal gravity is to explain flat rotation curves as exact
vacuum solutions of the theory. Moreover, as emphasized at the beginning of
the previous section conformal Weyl gravity provides also an explanation for
“almost” scale-invariant power spectrum of scalar cosmological fluctuations
therefore explaining one of the issues in current cosmology. Other problems in
cosmology where conformal gravity helps is the problem of cosmological
constant, which is not needed in conformal cosmology any more, and the problem
of flatness. According to [33, 34] conformal gravity also solves the problem
related to inflation and dark energy problems (vacuum energy) since all
cosmological manifolds are conformally flat. This fact also simplifies the
analysis of the cosmological fluctuations both in the scalar as well as in the
tensorial sectors.

Classical CWG provides also a natural and very robust resolution of the
problem of spacetime singularities, which are probably the most serious
inconsistencies of classical relativistic gravitational physics. These
singularities signal the breakdown of classical relativistic physics where the
theory inherently sets its own limits of validity. One would say that the fate
of classical singularities of the gravitational field is to be eventually
solved somehow by a hypothetical theory of quantum gravity. However, we think
that for the full resolution of them, one has to use the power of additional
new local symmetry in the gravitational framework. And this symmetry is a
dynamical symmetry related to arbitrary conformal rescalings of the metric of
the spacetime (2). By allowing for conformal rescalings one resolves the
problem of classical GR singularities (which put in danger exact solutions of
the gravitational theory). For example, the Big Bang or Big Crunch
singularities in the cosmological framework, or black hole singularities
(Schwarzschild, Kerr, and other metrics, which are singular at the origin of
the black hole) [35, 36, 37]. The idea is to relegate the “observable” general
relativistic singularity into a (conformal) gauge-dependent unobservable
conformal factor $\Omega^{2}$. In this way something which was for example the
curvature singularity (like measured by the Kretschmann invariant scalar
$R_{\mu\nu\rho\sigma}^{2}$) in Einsteinian theory, now is not anymore a good
conformal invariant to look at. By performing a change of the conformal gauge
according to (2), one shows that such an invariant scalar from GR loses its
unambiguous meaning and its actual value changes.

When one analyzes the conformal invariants (there are very few of them), then
one sees that the singularities can be completely resolved and that the
conformal equivalence class of metrics is not singular, similarly like the
coordinate “singularity” at the horizon of Schwarzschild metric is not present
in the equivalence class of metrics related by coordinate transformations. The
point is that there exist metrics (describing physical gravitational
spacetimes), which are not singular in these classes, compared to one
particular example metric which was singular but the singularity was assessed
by looking at “wrong” invariants, that is scalars which are not invariant
under the full group of symmetries of the theory in question. Then the choice
of non-singular metrics should be a preferred one, and the one of original
singular metric should be discarded as driven by a singular conformal gauge-
dependent scale factor $\Omega^{2}$ of the metric. Since the choice of any
metric in the equivalence class is possible (and should be done to perform
actual physical computations), then the one not giving rise to singularity is
completely equivalent to others, but about the resolution of singularities one
decides by choosing a most convenient one from the physical purposes. And that
is why one choose Kruskal-Szekeres coordinates for example over original
Schwarzschild coordinates to cover the whole black hole horizon region. One
concludes that there is not a true singularity at the location of the horizon,
it was only a coordinate gauge choice singularity. In complete analogy for the
conformal resolution of singularities, one derives that the conformally
related metrics which do not develop singularities are better and physically
more convenient than the original metrics in Einsteinian theory. One concludes
that there is not a true singularity with a conformally invariant meaning
here, it was only an original conformal gauge choice that was singular and led
to not “conformally” invariant notion of curvature singularities like in
Einsteinian GR. This again shows the interplay and similarities between
relativistic and conformal ways of treatment of some things - a theme which is
constantly recurring in this original contribution.

We also believe that the true resolution of classical singularities still
should be achieved on the classical level of the theory. Even if the quantum
effects play an important role for this issue, they all are captured by the
quantum effective action. The true quantum-corrected solutions of the theory
are understood as classical solutions to the quantum effective action, so the
same problem returns to the classical level. Our take on is to assume that
additional new symmetry in the gravitational relativistic dynamics comes with
the rescue to these annoying problems of ubiquitous and inevitable classical
singularities of gravitational spacetimes.

Another virtue of CWG is that this approach to relativistic dynamics of the
gravitational field is fully Machian. This realizes completely the old ideas
of Mach, which originally inspired Einstein in his construction of the
gravitational theory. However, as it is well known in Einsteinian gravity the
dynamics is only partially Machian, as this is obvious from Einstein equations
in vacuum, which depend on the matter distribution only locally. However, in
CWG gravitational field at a particular location is determined completely by
the conformal matter distribution everywhere else in the Universe (in Machian
sense, although of course the EOM of CWG are local despite that they are with
four derivatives on the metric tensor). This particular feature is seen also
from the explicit form of Mannheim-Kazanas solutions for which the particular
slope of the galactic rotation curve is determined from cosmological reasons,
so the matter density in the whole universe influences the gravitational
potential inside galaxies. In particular, Mannheim-Kazanas solutions provide
one exact solution to the problem of embedding the local galactic gravity
solution in a bigger scale picture of cosmological expansion of the Universe
as a whole. The fully Machian character of CWG is obviously related to the
preservation of only causal structure of the gravitational spacetimes under
conformal transformations. And only the last structure of the spacetime
manifold retains its meaning in full CWG.

Conformal symmetry is quite often seen on the classical level of the
relativistic dynamics. For example, classical electromagnetic and Yang-Mills
(YM) (non-Abelian) gauge theories, and the theory of massless Dirac fermions
in $d=4$ spacetime dimensions, are perfectly conformally invariant. They enjoy
spacetimes symmetries from the full $15$-parameter conformal group containing
as its subgroup the $10$-element usual Poincaré group of standard relativistic
symmetries. Moreover, a scalar field conformally coupled to gravity in $d=4$
with the non-minimal coupling of the form

$S_{{\rm int}}=\\!\int\\!d^{4}x\sqrt{|g|}\frac{1}{6}R\phi^{2}$ (16)

represents another classically conformal model with matter. One can also ask
what are the corresponding consequences of this fact on the quantum level. For
this one has to discuss the quantum Weyl gravity QWG.

##  4 Quantum Weyl Gravity

There are various reasons for quantum Weyl gravity. More clear conceptually is
to couple first quantum matter fields to fixed but non-trivial background
geometry. This setup of quantum fields in curved spacetime is well known and
it has very profound consequences also for the construction of resulting
quantum gravitational theory, where the gravity is taken as dynamical and the
dynamics of the related field of the graviton is fully quantum. One can induce
gravity from quantum matter fluctuations [38]. In this way one can find a
correct quantum gravity theory, the same like this was originally done with YM
theory when at the beginning only fermions transforming non-trivially under
non-Abelian symmetry groups were known.

The idea is the following. It is the embodiment of the original deWitt-Utiyama
argument [39]. When the matter fields are put in a non-trivial classical
external non-dynamical gravitational field, then as quantum fields described
by QFT on non-trivial backgrounds they produce various UV divergences, both in
the matter and in the gravitational (geometric sector). The correct quantum
gravitational theory should absorb all the divergences of the gravitational
character in its counterterms. For this the original action of the
gravitational theory should contain the corresponding terms. And here the
results of the computation about these UV divergences is as follows. When one
uses the conformal models described above as matter models on curved
spacetimes, then in $d=4$ spacetime dimensions the only non-trivial
gravitational counterterm which is generated by quantum corrections induced
from the matter side is of the conformal character, that is, it is precisely

$S_{{\rm div}}=\\!\int\\!d^{4}x\sqrt{|g|}C^{2}.$ (17)

Hence all matter-generated UV-divergences will be completely absorbed in the
gravitational counterterm coming from the action of WG. This means that QWG
allows for the quantum coupling of conformal matter to gravity in a way
preserving conformal symmetry. The main message is that the conformal symmetry
of classical matter models is not destroyed by quantum corrections when
coupled to conformal gravity. Moreover, gravity enhances this coupling
providing the whole uniform treatment of the classical system, which is
described in a fully conformally invariant way.

One can introduce QWG completely from the scratch along similar lines of
thoughts. For this, one needs to consider the path integral on curved fixed
spacetime backgrounds, and to integrate out the quantum fields of conformal
matter models (like electromagnetism, YM or SM Dirac fermions before the
electroweak symmetry breaking). As proved by ’t Hooft and Veltman [40, 41],
this generates the very specific action for quantum gravitational theories.
Since the conformal symmetry was originally in the matter model, then the
careful quantization procedure cannot break it and it must be inherited by the
resulting induced gravitational theory. This must be the theory described by
the $C^{2}$ action. Hence QWG is inevitable, if one uses conformal matter
models and if the conformal symmetry is treated with sufficient care. If one
did not know that dynamical gravity is out there (like one did not know about
the dynamical theory of the gluonic field related to non-Abelian gauge
symmetries), then one would discover that theory (dynamical gravity) by
integrating out example massless Dirac fermions. And this dynamical quantum
gravity must be described by QWG theory. There is also a related line of
argumentation on how to rederive Einsteinian theory starting from Weyl gravity
in $d=4$ (due to Maldacena and others, see e.g. [42]), but we will not discuss
this direction here, since it goes against the main spirit of our
presentation, although we admit that there are striking similarities between
the two theories (apparent for example on the level of exact vacuum solutions
discussed at length above). It is also interesting with this line of thought
to study processes of black hole evaporation within classical conformal
gravity as it was done in [43].

Having discussed motivation for Weyl gravity as the theory of quantum gravity,
now we analyze properties of QWG understood as the QFT of gravitational
interactions with conformal symmetry. This last element is additional and
requires a special care. This is because the conformal symmetry is in the
local version here. It is not a global symmetry as it was for example in
highly (super)-symmetric ${\cal N}=4$ matter models. Since it is in the gauged
version, then this additional symmetry of local character must be properly
taken care of during the formal process of quantization of gravity. The
framework of Faddeev-Popov quantization allows for this since it parallels the
treatment of diffeomorphism symmetries, which are from their nature local
transformations. On the other side, the conformal symmetry is here gauged
(localized) and there are some subtle differences compared to the other case
of gauge symmetries of relativistic gravitation (understood as the local gauge
theory of the Poincaré group). We refer the reader to a special treatment of
local gauge symmetries in this case. Of course, on the quantum level one must
secure that these local symmetries are not anomalous. Otherwise, the whole
program of quantization would miserably fail. The issue of conformal anomaly
is very important and we discuss some comments about it at the end of this
contribution.

For covariant quantization the best theoretical framework is that one with
functional path integrals. Since in gauge theory we have a local symmetry and
also degeneracy of the kinetic operator on the level of path integral, then
this symmetry must be constrained to define properly the functional integral.
One typically does a gauge fixing. Here this should be done separately for
diffeomorphisms and for conformal symmetry of WG. The framework lets all this
to be done in a covariant manner. Next, one has to add appropriate Faddeev-
Popov quantum ghost fields. We remark that these ghost fields are good and
useful and they have nothing in common with the Boulware-Deser ghosts of Weyl
gravity (bad and malevolent ghosts). They have to be specific to the
diffeomorphism part of the symmetry group, as well as to the conformal group.
With such conformal ghosts new Feynman rules need to be added to the
perturbative rules of the quantum gravitational theory turning it therefore
into QWG theory. Moreover, finally we can unambiguously find the propagator of
the QWG graviton field when some gauge choice is made. Of course, this
propagator will depend on the gauge fixing parameters used to kill the
additional infinite gauge freedom on the level of path integral.

The properties of QWG on the perturbative level of analysis in QFT are very
interesting. This quantum system gives rise to a renormalizable model of
quantum gravitation. The perturbative UV-divergences are fully under control
and they are all absorbable in the original covariant terms of the theory,
namely the known and seen before counterterm $\sqrt{|g|}C^{2}$ of the original
action of the theory. QWG has a good control not only over divergences
generated from the matter side, it also constrains the divergences generated
by quantum graviton loops. The counterterm $R^{2}$ is not generated (at least
on the level of first quantum loop). The renormalizability of this model of
quantum conformal gravity is a very remarkable feature, and it persists even
after coupling to conformal matter models (which in $d=4$ are characterized by
dimensionless coupling constants). The QWG in its pure gravitational sector
has one unique coupling constant $\alpha_{C}$. Of course, this coupling
exhibits RG running due to the fact that the corresponding $\beta$-function as
read from the UV-divergences of the theory is non-vanishing. However,
asymptotically in the UV the control over this running of $\alpha_{C}$ is
gained since the QWG is asymptotically free in deep UV regime. In this way,
this model resembles very much the situation met in QCD, where the theory also
contains only one positive unique coupling which in the UV limit is
asymptotically free.

One knows that QCD as a YM theory of the non-Abelian group of $SU(3)$ is
defined without any mass scale and the QCD coupling parameter is dimensionless
although it runs under RG. However, in QCD one sees in the IR region
spontaneous creation of mass scales, in the form of generation of
$\Lambda_{{\rm QCD}}$ or effectively the pion mass $m_{\pi}$ (this last
particle mediates effectively the strong interactions between protons and
neutrons at very low energies). This effect is entirely due to the phenomenon
known as dimensional transmutation. Similar effects will take lead in QWG,
where in the low energy regime we should see a spontaneous generation of mass
scales (like the Planck scale), which describe effectively quantum
gravitational interactions at sufficiently low energies.

The fact that the QWG is asymptotically free in the ultraviolet regime can be
also seen as a particular case of a special situation when the fixed point
(FP) of RG is met in the UV. And the theory in the UV reaches some CFT
describing conformal gravitational interactions also on the quantum level. The
quest for such CFT theory with local conformal symmetry and with gravitational
interactions is very restricting. Especially, the constraint that in the UV
gravitational CFT there is no conformal anomaly fixes the final UV theory
quite uniquely. The theory in the UV has to be an ${\cal N}=4$ conformal
supergravity as discovered by Fradkin and Tseytlin with four copies of ${\cal
N}=4$ super-Yang-Mills theories [44]. In such a coupled model, there are no
UV-divergences and the quantum model is completely UV-finite. This framework
of supergravity is very constrained and the total symmetry algebra has many
generators, hence the divergences are constrained to vanish. Moreover, the
conformal current, the energy-momentum tensor and all other supersymmetric
currents find themselves in the same supersymmetry multiplet and that is why
they have to be simultaneously conserved, which implies the absence of any
violation of conformal symmetry, the preservation of the quantum structure of
CFT, the independence of any mass scale and UV-finiteness. Only with extended
supersymmetries it is possible to relate the conservation of the energy-
momentum tensor $\nabla_{\mu}T^{\mu\nu}=0$ to the conservation of conformal
current $\nabla_{\mu}j^{\mu}=0$ and hence to the situation with unbroken scale
and conformal invariance. In this very constrained framework, the relativistic
dynamics is very special, and for example there is no dynamics related to RG
flows. Still constraints put a big amount of control on possible places from
which divergences could pop out requiring regularization. In such a
constrained theory it is still possible to perform a very well defined quantum
computation, which is the main result of this paper. For this it is enough to
use powers of the relativistic symmetries and of conformal symmetries in the
QFT theoretical framework.

The model due to Fradkin and Tseytlin realized fully the idea of quantum
conformality with gravitational interactions. In this way the conformality of
${\cal N}=4$ SYM theory is extended away from flat spacetime to a general
curved background. It is very reassuring that when ${\cal N}=4$ SYM theory is
put on curved background then even the conformally invariant gravitational
counterterm $\sqrt{|g|}C^{2}$ is not generated (of course in the gauge sector
the $F^{2}$ counterterm is not generated to the flat spacetime UV-finiteness
of this gauge theory model). Only the coupling of the two theories
(supergravity and super-Yang-Mills) provides at the end the theory with no
conformal (gauged) anomaly on the quantum level. This theory is with quantum
scale invariance promoted to quantum conformal invariance. There are no beta
functions, no running couplings, no RG flows, only anomalous dimensions of
quantum operators. The conformal Ward identities are clearly preserved and
they constrain highly the Green functions. And as we discussed this above,
this theory sits all the time at the UV FP of RG, however the breaking of
conformal symmetry by adding some deformation operator can be achieved. This
will result with some non-trivial RG flows. For some very particular
situation, one can view this as a spontaneous conformal symmetry breaking
(induced and related to supersymmetry breaking as well) and one can hope that
when this symmetry is broken in a soft way, then there is a remnant of some
information in conformal Ward identities of the theory even away from the UV
FP, when the theory was described by a gravitational CFT.

Another virtue of QWG in Euclidean framework is that the Euclidean theory
defined by action functional with $C^{2}$ as Lagrangian density is bounded
from below. This implies that the partition function of the corresponding
statistical mechanical model is completely positive-define and well-defined,
does not need any regularization, and the functional integrals are
mathematically very well-defined. Contrary to Euclidean Einsteinian
gravitational theory in QWG there is no conformal instability problem, of
course because here we do not have a conformal mode with negative sign. In QWG
we have only propagating modes which are invariant under symmetry properties
of the theory, so the conformal mode, (trace of the metric tensor
fluctuations) does not show up in the perturbative spectrum. One can analyze
with all available methods from statistical mechanics and field theory the
quantum partition function $Z$ here, which is positive-definite and describes
the quantum statistical partition function. It is one of the very first
consistent model of field theory which in the context of quantum statistical
mechanics gives statistical and stochastic properties of Euclidean field
theory. Especially, since in Euclidean QWG this is a field theory with
propagating metric degrees of freedom describing differential geometry of
$4$-dimensional manifolds (surfaces). It could be said as an analogue of
Euclidean gravity - quantum statistical mechanical theory of differential
manifolds. Since the metric structure does not matter here, only the conformal
properties, and angles as mentioned in the introductions, this is really a
study of (scaleless) shape of manifolds. This theory can be viewed as a first
consistent model of quantum shape dynamics.

As emphasized a lot previously QWG is a system with a very peculiar
relativistic dynamics. Diffeomorphism symmetry is fully realized (symmetry
under the change of coordinate system used to describe physics). Moreover, it
is also conformal symmetry that it is gauged here (appears in the local
version). And this is a symmetry under an arbitrary changes of the conformal
factor used to measure distances. This puts a lot of constraints on the
ensuing relativistic dynamics of the gravitational field. One could say from
the historical perspective, that this is a first modified gravitational theory
(introduced by Weyl in 1918), although it was not called like this neither by
Weyl, nor by Einstein \- his opponent. His theory was clearly different than
Einsteinian gravity (introduced in 1918) because it embodies more symmetries
and the resulting dynamics is with higher derivatives, particularly in $d=4$
this is with four derivatives, while Einstein theory gives a dynamics with two
derivatives only. One can see this difference also on the level of the spectra
of two theories. For example, in QWG as analyzed on the tree-level the
physical degrees of freedom around any background are only transverse
traceless (TT) gravitons as this is obvious from the York decomposition (to be
explained in detail below). And we do not have for example a problematic
conformal mode of Einsteinian gravity.

Moreover, QWG is different than any other higher derivative (HD) gravitational
theory containing four derivatives in $d=4$ spacetime dimensions. Such a
general Stelle theory is characterized by two coupling parameters and two
covariant terms in the gravitational Lagrangian, namely
$\alpha_{R}R^{2}+\alpha_{C}C^{2}$. However, QWG is different and not a similar
to a general quadratic Stelle theory. The spectra are different since only in
QWG we have only TT gravitons present. One cannot also take in a simple
continuous way the limit $\alpha_{R}$ in the results of the general Stelle
theory. This is another incarnation of the famous Veltman discontinuities.
This time it is to be analyzed in HD gravitational theories. It is not like
original problem with the mass gap of gauge theories and counting of
propagating degrees of freedom there ($2$ in massless vs $3$ in massive gauge
theories). Here the discontinuity has to do not only with transition from
massive to massless excitations, but also with a drastic change of the
character of fluctuations in the two theories. For example, they fill
different representations of the Lorentz group which is used to classify
particle excitations around flat spacetime. So the particles are in different
tensorial representation. For example, in QWG on the tree-level we have 2
massless tensors and a massless vector. This last particle is not seen in any
other spectrum of ordinary HD gravitational theories in $d=4$. This vector
could be viewed as a remnant and the signal of the remaining (vectorial)
conformal symmetry on the level of the action of the theory. One should see
that in general the Veltman discontinuity means that in the discontinuous
limiting theory there exists an enhancement of local symmetries. The theory
with $\alpha_{R}=0$ is very special and highly symmetric, higher than any
ordinary HD gravitational theories. And this is of course, due to the presence
of local conformal symmetry there.

##  5 Quantum computations

We would like to present here an example of a quantum computation done in the
framework of QWG. This computation has to be done in the framework of quantum
field theory (QFT) of gravitational (and conformal) interactions. For this we
will use the formalism of Faddeev-Popov quantization where special care should
be exerted in the treatment of local conformal symmetry on the quantum level.
We do not wish that this symmetry is broken by the mere quantization process.
This quantum computation (at the one-loop level) will be performed within the
background-independent formalism and for this we will use the background field
method (BFM). Moreover, as our background we will choose a general Einstein
spacetime (ES) background, characterized by the condition for Ricci tensor

$R_{\mu\nu}=\Lambda g_{\mu\nu},$ (18)

where $\Lambda$ is an arbitrary value constant parameter, that is we have
$\Lambda={\rm const}$ ($\nabla_{\mu}\Lambda=0$). The values and the form of
the Weyl tensor $C_{\mu\nu\rho\sigma}$ can be arbitrary on general ES and only
in this way the generic ES differs from a maximally symmetric spacetimes (MSS)
characterized by the parameter $\Lambda$. Selecting a general ES background we
get in one stroke the result of the previous computations on two different
backgrounds: namely on Ricci-flat background and MSS. They were considered in
[45].

We will perform the computation considering only the physical degrees of
freedom which are present as dynamical degrees of freedom in QWG. As it is
well known due to diffeomorphism symmetry and due to the conformal symmetry
they are transverse and traceless (TT) gravitons respectively. Of course, a
quantum graviton is described as a fluctuation of the metric tensor of the
spacetime, so these fluctuations can be considered as a symmetric rank-2
tensor.

Our computation is an RG-improved one-loop level computation of the RG running
of couplings in QWG. In this calculation we take into account the special role
of non-perturbative phenomena (and this is why this is an RG-improvement)
encoded in the threshold phenomena (predominantly) related to mass thresholds
of quantum massive modes. Additionally we also study the impact of the
graviton’s anomalous dimension on these results. These are two non-
perturbative effects that improve over the standard quantum one-loop
computation which was achieved in 1980’s. In this sense, we can say that our
computation is a $1.5$-loop one since we still acknowledge that the effects of
two perturbative loops will give a more accurate description of quantum
phenomena. We start our computation with the one-loop partition function of
QWG on ES and make it more sensitive to quantum effects by working with it in
the non-perturbative Wetterich equation describing the exact RG flows. This
means that our one-loop beta functions are functionally improved in the sense
of using functional Renormalization Group (FRG) approach.

The novelty of this approach is that for the first time we analyze the general
situation on ES, and not separately on MSS [46] or Ricci-flat backgrounds.
This approach follows [47]. After analyzing the RG flow mainly in the deep IR
regime we will search for FP’s of RG (defined by condition that beta functions
vanish $\beta=0$). Finally, we discuss the physical applications and
interpretations of these results about the IR behavior of QWG. This conclusion
will open a new window for cosmology.

Before we embark on showing details of the computation, we should explain why
it is sensible to perform such quantum computation with running couplings and
beta function in QWG. We emphasized before that if the conformality is
preserved on the quantum level, then the theory must be at a FP of RG, and
there are no beta functions and no RG flow. For consistency such a regime of
the theory must be also conformal anomaly-free. This implies that this theory
has to be (very probably) uniquely the superconformal gravity constructed by
Fradkin and Tseytlin [44]. We believe such theory describes the conformal
gravitational physics in the deep UV regime, where all the symmetries
(including supersymmetry in local version) are restored. Therefore we find
more interesting to look at the deep UV behavior of the QWG where we can find
some non-trivial FP of RG describing probably new infrared physics. What
happens in the IR regime is the soft breaking of conformal symmetry so that
only the global part of the full $15$-dimensional conformal group is broken.
In particular, the generator related to global scale-invariance, i.e. to the
dilatation transformations on the physical system, is broken. This breaking in
a spontaneous way is a deformation of the original UV conformal theory by
giving a vev to some operator, which is globally scale-invariant but not
invariant under full conformal group.

This is what may happen in QWG described by pure gravitational action (without
local supersymmetry) when analyzed consequently at one-loop and two-loop
perturbative levels of the theory. In the UV this theory meets a UV FP of RG,
where the scale-invariance is regained. Also conformal anomaly vanishes there
(due to participation of other fields in the supermultiplet). When one lowers
the energy scale and decouples the superpartners, one sees that the quantum
operator $R^{2}$ acquires a non-vanishing vev due to contribution at the two-
loop order. This is also related to the fact that the beta function
$\beta_{R^{2}}$ of the coupling in front of the $R^{2}$ term appears on the
second perturbative loop order, while it was not present there at the first
loop order. In such a theory we start a non-trivial RG running, which when
approaching the IR limit, generates mass scales (like the Planck scale) and
the symmetries of the theory are changed. Likewise, symmetries in QCD are
changed when we start describing strong interactions via interactions with
massive pions.) This happens because of the effect of dimensional
transmutation occurring in this theory. The quantum theory which describes
these phenomena towards IR regime of the spectrum is QWG with non-vanishing
beta functions of couplings and running coupling parameters. Hence, it is
useful to describe quantum RG effects in such theory described by the
quantized version of Weyl gravity, when we still had the conformal symmetry on
the classical level, and when the quantization procedure did not break this
and was done very delicately. One sees that if sufficient care is exerted then
conformal Ward identities are still satisfied like in softly spontaneously
broken local gauge theories.

The issue of conformal anomaly (CA) does not prevent sensible computations in
such a theory since the computation is done towards the IR regime. One notices
that the CA is really an UV problem or in a different disguise a problem with
the UV-completion of the theory. In other words, for a good definition of the
theory one must find a UV FP with good conformal properties. We already have
discussed such a candidate theory, namely the ${\cal N}=4$ conformal
supergravity (CSG). Moreover, the CA at one-loop, for example, is completely
expressed via the perturbative beta functions of the theory. And it is well
known that these beta functions are related to the UV-divergences of the
theory, so to the UV limits of the theory. Hence physics in the IR does not
matter for this. If we assume that the UV-completion is done by CSG, then the
problem with CA is gone since this last one probes the UV physics. We know
that in IR the conformal symmetry has to be broken anyway, but this will not
influence at all the securement of CA issue in the UV regime. With this
understanding in mind, we can still reasonably well and logically correct
describe the situation with running in the IR. In order to have the power of
the remnants of the conformal symmetry expressed on the quantum level via
conformal Ward identities, the conformal symmetry towards the IR regime has to
be broken but only in a softly way as described schematically above. Of
course, this violation of conformality has to be analyzed better and in more
details, and we leave it for future investigations. In this way, we can still
sensibly talk about RG flow in QWG in the IR regime, where the softly violated
conformal Ward identities still constrain the physics in a sense that the
quantized theory which takes into account these phenomena in the IR regime is
the quantized version of Weyl gravity, and for example not a generic higher
derivative (quadratic in curvatures) gravitational theory. This is the
perspective which gives sense to the computation that we present below. This
is exactly a quantum computation in the IR regime of the QWG theory (non-
supersymmetric).

In order to proceed with the quantum computation we need a few technical
details. First in order to get rid of ambiguities related to gauge degrees of
freedom and other non-physical degrees of freedom in QWG and also explicit
addition of Faddeev-Popov diffeomorphic and conformal ghosts, we use the York
decomposition of the metric fluctuations. Working in BFM, we start with
writing a general expansion of the full quantum metric $\tilde{g}_{\mu\nu}$

$\tilde{g}_{\mu\nu}=g_{\mu\nu}+h_{\mu\nu},$ (19)

where the fluctuations fields $h_{\mu\nu}$ are general and the background
metric $g_{\mu\nu}$ is a general ES metric. We perform the York decomposition
of the gravitational fluctuations in two steps. First, we remove the spurious
trace degree of freedom by writing that

$h_{\mu\nu}=\bar{h}_{\mu\nu}+\frac{1}{4}g_{\mu\nu}\phi\quad{\rm
with}\quad\phi=h_{\mu}^{\mu}=g^{\mu\nu}h_{\mu\nu}$ (20)

and with the tracelessness condition on the new fluctuations

$g^{\mu\nu}\bar{h}_{\mu\nu}=\bar{h}_{\mu}^{\mu}=0.$ (21)

In the second step of York decomposition we remove the transverse part from
the traceless metric fluctuations $\bar{h}_{\mu\nu}$. This is achieved by
recalling the following formula

$\bar{h}_{\mu\nu}=\bar{h}_{\mu\nu}^{\perp}+\nabla_{\mu}\eta_{\nu}^{\perp}+\nabla_{\nu}\eta_{\mu}^{\perp}+\nabla_{\mu}\nabla_{\nu}\sigma-\frac{1}{4}g_{\mu\nu}\square\sigma,$
(22)

where the transversality conditions on new metric fluctuations
$\bar{h}_{\mu\nu}^{\perp}$ and on the vector field $\eta_{\mu}^{\perp}$ read

$\nabla^{\mu}\bar{h}_{\mu\nu}^{\perp}=0\quad{\rm
and}\quad\nabla^{\mu}\eta_{\mu}^{\perp}=0.$ (23)

Moreover, still the metric fluctuation field
$\bar{h}_{\mu\nu}^{\perp}=h_{\mu\nu}^{TT}$ retains its traceless character

$g^{\mu\nu}\bar{h}_{\mu\nu}^{\perp}=\bar{h}_{\mu}^{\perp\mu}=0$ (24)

and this implies that for convenience we can also redefine the trace scalar
field $\phi\to\bar{\phi}=\phi-\square\sigma$, where $\sigma$ is another scalar
field. We notice that the final York decomposition reads

$h_{\mu\nu}=\bar{h}_{\mu\nu}^{\perp}+\nabla_{\mu}\eta_{\nu}^{\perp}+\nabla_{\nu}\eta_{\mu}^{\perp}+\nabla_{\mu}\nabla_{\nu}\sigma-\frac{1}{4}g_{\mu\nu}\bar{\phi}.$
(25)

The purpose of this decomposition is its main result namely the irreducible
part of the fluctuation, which is in the traceless transverse tensor
$\bar{h}_{\mu\nu}^{\perp}=h_{\mu\nu}^{TT}$. One can see that in the QWG the
physical field which propagates around any background metric $g_{\mu\nu}$ (it
does not even have to be an ES metric) is only a traceless transverse
graviton. This is a physical field of QWG, where all redundancies due to the
local gauge symmetries of the formalism are removed. This feature is due to
diffeomorphism (transversality) and due to conformality (tracelessness). One
can see it explicitly when writing the second variation of the Weyl action
functional (9) around any on-shell background (a Bach-flat background). In
such an on-shell situation the spurious quantum fields $\eta_{\mu}^{\perp}$,
$\sigma$ and $\bar{\phi}$ completely decouple and they do not show up in the
expression for the second variation of the action integral. On a completely
general background there appear terms with fields $\eta_{\mu}^{\perp}$,
$\sigma$ and $\bar{\phi}$ but they are all proportional to the Bach tensor
$B^{\mu\nu}$, so they correctly vanish when EOM in (14) are used. But below we
consider only on-shell backgrounds (all ES are automatically Bach-flat in
$d=4$ spacetime dimensions).

Next, we discuss the most general expansion of terms quadratic in generalized
curvatures in $d=4$ dimensions. We consider here only local scalar invariants
of dimension four. This implies that in the action they come only with
dimensionless coupling coefficients, which is important for the property of
soft breaking of conformal symmetry. We also analyze their properties under
global and local scale transformations and also their first variations.

The first invariant is known already to us conformal scalar invariant $C^{2}$
whose expansion in other terms in curvature reads

$C^{2}=R_{\mu\nu\rho\sigma}^{2}-2R_{\mu\nu}^{2}+\frac{1}{3}R^{2}=2R_{\mu\nu}^{2}-\frac{2}{3}R^{2},$
(26)

where in the last line we used the integration by parts under spacetime volume
integral and the Gauss-Bonnet topological relation. As we discussed at length
in the introduction, this invariant when properly densitized is perfectly
conformally invariant, which we express as the following statement below. Its
infinitesimal local conformal variation vanishes identically, that is we have

$\delta_{c}\left(\sqrt{|g|}C^{2}\right)=0.$ (27)

Of course, this is also true for finite conformal transformation which is also
a basis for CWG.

The next curvature scalar invariant that we can consider here is a Gauss-
Bonnet invariant (also known as topological Euler invariant). Its expansion
reads explicitly

${\rm GB}=E=R_{\mu\nu\rho\sigma}^{2}-4R_{\mu\nu}^{2}+R^{2}.$ (28)

In four dimensions one can call it topological since it is related by Gauss-
Bonnet theorem to the topological (completely metric-independent)
characteristics of the spacetime, known as Euler topological invariant or
Euler characteristics $\chi_{4}$. This also implies that any its variation
under the integral integrates to zero, that is we have

$\delta\left(\int d^{4}x\sqrt{|g|}{\rm GB}\right)=0.$ (29)

In particular, this means that the variation is zero also under conformal
transformations, both infinitesimal and finite, local and global.

The last invariant we can call as a Starobinsky invariant since first was
invented by him in 1980 [48]. It is simply the square of the Ricci curvature
scalar $R^{2}$. Besides it being globally scale-invariant (in $d=4$) related
to the fact that the coefficient $\alpha_{R}$ is dimensionless, it also shows
some more extended form of global conformal invariance. Namely, it is
invariant with respect to restricted local conformal invariance. The general
local conformal transformations are parametrized by arbitrary scalar field
$\Omega=\Omega(x)$ as in (2). However, the Starobinsky invariant $R^{2}$ does
not change under conformal transformations if the parameter $\Omega$ satisfies
the GR-covariant d’Alembertian (wave) equation, namely if

$\square\Omega(x)=0.$ (30)

This is less than general local conformal transformations with arbitrary
$\Omega$, but still it is more than just global scale-invariance or no
invariance at all. Therefore usage of this operator for the breaking of the
conformal symmetry in $d=4$ dimensions in the softest possible way may be a
preferable solution to get a soft RG flow in QWG.

Now, we characterize the background ES manifolds. They are more general than
MSS and Ricci-flat spacetimes, but they include both these cases as special
subsets. To describe a general ES, one needs to specify the value of the
$\Lambda$ parameter in (18) and also the form of the Weyl tensor
$C_{\mu\nu\rho\sigma}$ consistent with all symmetries (so in results 10
algebraically independent components in $d=4$). As a consequence of defining
condition of ES, we derive the following curvature relations

$R=4\Lambda,\quad{\rm GB}=R_{\mu\nu\rho\sigma}^{2}$ (31)

and since the expression for the Riemann tensor on a general ES is given by

$R_{\mu\nu\rho\sigma}=\frac{\Lambda}{3}\left(g_{\mu\rho}g_{\nu\sigma}-g_{\mu\sigma}g_{\nu\rho}\right)+C_{\mu\nu\rho\sigma},$
(32)

which is up to the Weyl tensor the same like on a general MSS with $\Lambda$
parameter, we derive also relations for various curvature square invariants

$C^{2}=R_{\mu\nu\rho\sigma}^{2}-\frac{8}{3}\Lambda^{2}\quad{\rm and}\quad{\rm
GB}=C^{2}+\frac{8}{3}\Lambda^{2}.$ (33)

We remark that in $d=4$ all ES are vacuum solutions to CWG, i.e. they satisfy
Bach EOM (14). This is an important consequence for the evaluation of the
partition function on on-shell backgrounds and also for the possibility to use
WKB approximation to describe one-loop quantum physics. It has also
implications for the form of the second variation on the considered here
backgrounds. In what follows we will only consider classical on-shell
backgrounds. However, quantum perturbations do not have to satisfy classical
EOM of the theory since QFT effects are typically off-shell.

The next step consists of writing the expression for the one-loop partition
function in QWG on ES in terms of functional determinants of some differential
operators considered on this background. We first start with writing the
second variation of the conformal action of classical conformal Weyl gravity.
It reads

$\delta^{2}S_{{\rm
conf}}=\\!\int\\!d^{4}x\sqrt{|g|}h^{TT\mu\nu}\left(\hat{\square}-\frac{2}{3}\Lambda\hat{1}+2\hat{C}\right)\left(\hat{\square}-\frac{4}{3}\Lambda\hat{1}+2\hat{C}\right)h_{\mu\nu}^{TT},$
(34)

where we used York decomposition of fluctuations (25) and we commuted and
integrated by parts covariant derivatives under the volume spacetime integral.
We also exploited the definition of the Weyl tensor as a matrix $\hat{C}$
whose action on the graviton fluctuations is the following

$\left(\hat{C}h\right)_{\mu\nu}=C_{\mu\alpha\nu\beta}h^{\alpha\beta}.$ (35)

Finally, we define the identity matrix $\hat{1}$ in the space of rank-2
symmetric tensor fluctuations by the formula

$\left(\hat{1}\right)^{\mu\nu}{}_{\alpha\beta}=\delta_{\alpha\beta}^{\mu\nu}=\delta_{(\alpha}^{\mu}\delta_{\beta)}^{\nu}=\frac{1}{2}\left(\delta_{\alpha}^{\mu}\delta_{\beta}^{\nu}+\delta_{\beta}^{\mu}\delta_{\alpha}^{\nu}\right).$
(36)

The formula (34) was proven by an explicit computation where we also used
various non-trivial identities coming from taking variational derivatives of
the GB term in (28). One notices that for ES being on-shell background there
is no participation of spurious gauge-redundant fields of transverse vectors
$\eta_{\mu}^{\perp}$ and two scalars $\sigma$ and $\bar{\phi}$. Moreover, one
sees that in the kernel of the expression in (34) between traceless and
transverse fluctuations $h_{\mu\nu}^{TT}$ there is a matrix-differential
operator, which has a direct factorization in a product of two single two-
derivative operators (shifted by some constant factors proportional to
$\Lambda$ and also by some matrix of the Weyl tensor $\hat{C}$. It is
remarkable that there is not a mixing term $\Lambda\hat{C}$ which could be
allowed by dimensional reasons, and moreover such a term separately would not
be seen by taking a limit to MSS ($\hat{C}\to 0$) or to Ricci-flat spacetime
($\Lambda\to 0$). This lets the factorization of the kernel to be successful
here. One can think additionally of this kernel as the simplest generalization
of the kernels that appear separately in the expressions for the second
variations on MSS and on Ricci-flat spacetimes. In the last case, the second
variation gets simplified even to a quadratic form

$\delta^{2}S_{{\rm
conf}}=\\!\int\\!d^{4}x\sqrt{|g|}h^{TT\mu\nu}\left(\hat{\square}+2\hat{C}\right)^{2}h_{\mu\nu}^{TT}.$
(37)

One could naively derive the expression for the partition function from the
result in (34). The result would be

$\tilde{\tilde{Z}}_{{\rm
1-loop}}^{2}=\frac{1}{\det_{2TT}\left(\hat{\square}-\frac{2}{3}\Lambda\hat{1}+2\hat{C}\right)\det_{2TT}\left(\hat{\square}-\frac{4}{3}\Lambda\hat{1}+2\hat{C}\right)}.$
(38)

But since in the theory we have local symmetries then the true expression for
the one-loop partition function is more complicated. We have to take care of
the Jacobian of the path integral variable transformation, of fixing all the
local gauge symmetries, and also of adding Faddeev-Popov determinant (both for
diffeomorphism and conformal part of the local symmetry group here in QWG). At
the end, one has to take care of possible zero modes on ES and carefully
exclude them from the computation of determinants of the differential
operators here.

When one uses only physical and gauge-invariant degrees of freedom (only TT
gravitons), then there is no need to additionally care about fixing all the
local gauge symmetries, and also about the addition of Faddeev-Popov
determinants. This is the advantage of using real degrees of freedom which do
not come with any redundancy related to local symmetries. However, one still
has to treat delicately the Jacobian of the change of integration variables
and the issue of zero modes. The first problem of the functional change of the
integration variables under field theory path integral from original
unconstrained spin-2 field $h_{\mu\nu}$ to a set of constrained fields
$\left(h_{\mu\nu}^{TT},\eta_{\mu}^{\perp},\sigma,\bar{\phi}\right)$ is easily
done even on a general spacetime background. For this issue, one can easily
understand that the situation on a general ES is the same like on a general
MSS with the same $\Lambda$ since the terms in the Jacobian depend only on
contractions of the Riemann tensor, and not on the Riemann tensor itself. And
one can understand from the formula (32) that as for Ricci tensors and Ricci
scalars there is no any difference between MSS and ES. Therefore, following
what was done in the analysis of Jacobian on MSS we can write

$\tilde{Z}_{{\rm
1-loop}}^{2}=\frac{\det_{1T}\left(\hat{\square}+\Lambda\hat{1}\right)\det_{0}\left(\hat{\square}+\frac{4}{3}\Lambda\hat{1}\right)}{\det_{2TT}\left(\hat{\square}-\frac{2}{3}\Lambda\hat{1}+2\hat{C}\right)\det_{2TT}\left(\hat{\square}-\frac{4}{3}\Lambda\hat{1}+2\hat{C}\right)}.$
(39)

The above expression is closer to the final one for the one-loop partition
function on the general ES, but not yet the final one. We observe that the
factors in the numerator originate only from the analysis on MSS background.
However, they cannot be “corrected” by the addition of the matrix of the Weyl
tensor $\hat{C}$ since this matrix does not makes sense between spin-1 or
spin-0 fluctuations, which are used when we take determinants of the operators
in the numerators. These determinants in the subspaces of spin-1 and spin-0
fluctuations respectively originate completely from the treatment of the
Jacobian of the field variables transformation (to the TT gravitational
fluctuations).

The last problem is related to the presence of zero modes of the differential
operators which appear in (39) on a general ES background. In full generality,
such a problem is very complicated to be tackled analytically. What we need
actually, is a much more modest answer to the problem of rewriting the
determinants of operators in the above formula. As it is well known zero modes
appear mostly due to constraints on operators that they have to act only in
the subspace of transverse fluctuations (both spin-1 and spin-2) since these
last constraints are of the differential character. The constraint on the
operator to act on traceless rank-2 tensor fluctuations $\bar{h}_{\mu\nu}$
does not create any problem for this since the last one is an algebraic
condition. Therefore it is desirable to change the determinants of operators
acting in subspaces of transverse fluctuations to determinants of operators
acting in extended subspaces, where the condition of transversality is not
imposed anymore. In our candidate expression for the one-loop partition
function on the ES background, we have to change determinants in the space of
spin-2 TT graviton fluctuations and for spin-1 transverse vector fluctuations.

In general, formulas relating determinants in subspace of transverse and
extended fluctuations are derivable around any background. For this purpose
one must consider the following expressions

$\bar{h}^{\mu\nu}\square\bar{h}_{\mu\nu}\quad{\rm
and}\quad\eta^{\mu}\square\eta_{\mu}$ (40)

under the volume spacetime integrals and expand them using the York
decomposition for both traceless spin-2 fluctuations $\bar{h}_{\mu\nu}$ and
for general unconstrained spin-1 fluctuations $\eta_{\mu}$ separately. One has
to commute derivatives, integrate them by parts, exploit various tracelessness
and transversality conditions as stipulated in formulas (23) and (24). At the
end one has to use various curvature relations valid on a general ES from (18)
and (32).

One sees that the relation between determinants of operators for transverse
and unconstrained spin-1 fluctuations does depend only on the Ricci tensor of
the background spacetime. Hence in this place we again find no difference
between MSS and ES case. On MSS background spacetime, the final formula reads

$\det{}_{1T}\left(\hat{\square}+X\hat{1}\right)=\frac{\det_{1}\left(\hat{\square}+X\hat{1}\right)}{\det_{0}\left(\hat{\square}+X\hat{1}-\Lambda\hat{1}\right)}$
(41)

and this formula allows to perform change from the subspace (1T) of transverse
spin-1 fluctuations to the space of unconstrained spin-1 fluctuations. The
correction term is proportional to the functional determinant of some scalar
box operator, which receives also an additional shift by the term proportional
to the $\Lambda$ parameter (which is the same for both ES and MSS
backgrounds).

The situation with the sector of spin-2 TT fluctuations is more complicated.
The expressions for the expansion of $\bar{h}^{\mu\nu}\square\bar{h}_{\mu\nu}$
using the York decomposition on a generic ES background contains terms
proportional to the powers of the $\Lambda$ parameter as well as terms
proportional to the matrix of Weyl tensor $\hat{C}$. However, the terms from
the last group come without derivatives, they are only of the endomorphic
character without any remnant differential operator. Again, one does not note
any mixing terms (proportional for example to $\Lambda\hat{C}$), which would
be invisible in MSS or in Ricci-flat limits. This signifies that the
contribution to the exclusion of zero modes for differential operators is done
separately from MSS backgrounds (parametrized by the value of the $\Lambda$
parameter) and from Ricci-flat backgrounds (parametrized by arbitrary Weyl
tensor $\hat{C}$). Since in the last case the resulting terms in
$\bar{h}^{\mu\nu}\square\bar{h}_{\mu\nu}$ proportional to $\hat{C}$ are not of
the differential characters, then one concludes that there is no contribution
that has to be subtracted from the determinants of the operators on Ricci-flat
background due to zero modes. The same happens on general ES and one can
safely forget about the contributions with Weyl tensor $\hat{C}$ there, and
only concentrate on the contribution that really have to be subtracted, namely
those which are proportional to the $\Lambda$ parameter of ES. For this last
aspect the computation is identical to the general case of MSS. One ends up
with the following formula

$\det{}_{2TT}\left(\hat{\square}+X\hat{1}\right)=\frac{\det_{2T}\left(\hat{\square}+X\hat{1}\right)}{\det{}_{1T}\left(\hat{\square}+X\hat{1}-\frac{5}{3}\Lambda\hat{1}\right)\det_{0}\left(\hat{\square}+X\hat{1}-\frac{8}{3}\Lambda\hat{1}\right)}$
(42)

and this formula allows to perform change from the subspace (2TT) of traceless
transverse spin-2 gravitational fluctuations to the space of traceless spin-2
fluctuations. There are two correction terms. They are proportional to the
functional determinant of the differential operator acting on transverse
spin-1 fluctuations and to the functional determinants of some scalar box
operator, which receives also an additional shift by the term proportional to
the $\Lambda$ parameter on ES. In accordance with the previous remarks, one
sees in general that such correction terms could not have a term with the
matrix of the Weyl tensor $\hat{C}$ because such matrix cannot act in the
subspace of spin-1 or spin-0 fluctuations due to simple algebraic reasons
(there is not a sufficient number of indices to contract with Weyl to produce
an endomorphism of the vector or scalar bundle).

Finally, one gets the expressions for the one-loop partition around a general
ES in the following form

$Z_{{\rm
1-loop}}^{2}=\frac{\det_{1}^{2}\left(\hat{\square}+\Lambda\hat{1}\right)\det_{1}\left(\hat{\square}+\frac{1}{3}\Lambda\hat{1}\right)\det_{0}\left(\hat{\square}+\frac{4}{3}\Lambda\hat{1}\right)}{\det_{2T}\left(\hat{\square}-\frac{2}{3}\Lambda\hat{1}+2\hat{C}\right)\det_{2T}\left(\hat{\square}-\frac{4}{3}\Lambda\hat{1}+2\hat{C}\right)\det_{0}\left(\hat{\square}+2\Lambda\hat{1}\right)}.$
(43)

To get this expression we applied (41) and (42) to (39) and finally once again
(41) to the intermediate result. In this form, this expression resembles very
much the one obtained as the final one on MSS backgrounds. The only difference
is the addition of two terms with the matrix of the Weyl tensor $\hat{C}$ in
the denominator, when this algebraically makes sense so only for spin-2
fluctuations. This is the final correct expression for the one-loop partition
function of QWG theory on ES background. Its correctness was verified by
performing the explicit computation of the $b_{0}$, $b_{2}$ and $b_{4}$
coefficients related to UV divergences of the QWG theory. For example, for the
$b_{4}$ coefficient, they reproduce the famous results by Fradkin and Tseytlin
of divergences at one-loop level proportional to $\Lambda^{2}$ and $C^{2}$ in
one stroke.

One can study various limiting situations of the one-loop partition function
we have just computed in (43). First, one can easily take the limit
$\hat{C}\to 0$ to reduce to MSS case. This is only a small cosmetic
modification since the terms with matrix of the Weyl tensor $\hat{C}$ will be
gone, but the structure of the partition function $Z$ is exactly the same in
MSS like in (43). The limit $\Lambda\to 0$ brings us back the situation on
general Ricci-flat spacetime. Then one sees that the form gets reduced to the
following

$Z_{{\rm 1-loop},{\rm Ricci-
flat}}^{2}=\frac{\det_{1}^{3}\left(\hat{\square}\right)}{\det_{2T}^{2}\left(\hat{\square}+2\hat{C}\right)}.$
(44)

The last partition function can be extended to contain only fully
unconstrained fields of fluctuations (in particular of spin-2 fluctuations)
and then it reads

$Z_{{\rm 1-loop},{\rm Ricci-
flat}}^{2}=\frac{\det_{1}^{3}\left(\hat{\square}\right)\det_{0}^{2}\left(\hat{\square}\right)}{\det_{2}^{2}\left(\hat{\square}+2\hat{C}\right)}.$
(45)

The similar operation of extension can be also performed on the level of the
one-loop partition function on a general ES background. Namely, we can write

$Z_{{\rm
1-loop}}^{2}=\frac{\det_{1}^{2}\left(\hat{\square}+\Lambda\hat{1}\right)\det_{1}\left(\hat{\square}+\frac{1}{3}\Lambda\hat{1}\right)\det_{0}\left(\hat{\square}-\frac{2}{3}\Lambda\hat{1}\right)\det_{0}\left(\hat{\square}-\frac{4}{3}\Lambda\hat{1}\right)\det_{0}\left(\hat{\square}+\frac{4}{3}\Lambda\hat{1}\right)}{\det_{2}\left(\hat{\square}-\frac{2}{3}\Lambda\hat{1}+2\hat{C}\right)\det_{2}\left(\hat{\square}-\frac{4}{3}\Lambda\hat{1}+2\hat{C}\right)\det_{0}\left(\hat{\square}+2\Lambda\hat{1}\right)}.$
(46)

The two formulas in (43) and in (46) are completely equivalent due to the
following extension formula

$\det{}_{2T}\left(\hat{\square}+X\hat{1}+2\hat{C}\right)=\frac{\det{}_{2}\left(\hat{\square}+X\hat{1}+2\hat{C}\right)}{\det{}_{0}\left(\hat{\square}+X\hat{1}\right)}.$
(47)

This last formula has nothing to do with zero modes and this extension of the
operator to fully unconstrained spin-2 representation from the traceless one.
In this form this is valid on any spacetime. One notices that due to algebraic
reasons, the matrix of the Weyl tensor $\hat{C}$ cannot appear in the
denominator of the expression above.

The next step in the computation is the application of the functional RG flow
equation due to Wetterich [49, 50]. In a schematic form for a two-derivative
theory defined by the following kinetic part of the action in arbitrary
fluctuation field $\phi$ in whatever representation

$S_{{\rm kin}}=\int
d^{4}x\sqrt{|g|}\phi\left(\hat{\square}+a\hat{1}\right)\phi$ (48)

the Wetterich equation takes the form

$k\partial_{k}\Gamma_{k}=\frac{1}{2}{\rm
Tr}\left(\frac{k\partial_{k}R_{k}\hat{1}}{\hat{\square}+R_{k}\hat{1}+a\hat{1}}\right),$
(49)

where $\Gamma_{k}$ denotes the scale-dependent effective (average) action for
all fields in the theory, $a$ is a general constant shift of the differential
operator, and the cutoff kernel $R_{k}(z)$ is a function of the leading part
in derivatives of the differential operator $z=\hat{\square}$ and also of the
RG energy scale $k$. The IR-cutoff kernel $R_{k}(z)$ leads to a suppression of
the contribution of modes with small eigenvalues of the covariant Laplacian
operator $\hat{\square}$ (namely these for which we have $-\hat{\square}\ll
k^{2}$ in momentum space representation), while the factor
$k\partial_{k}R_{k}\hat{1}$ in the numerator of the Wetterich equation removes
contributions from large eigenvalues $-\hat{\square}\gg k^{2}$. In this way,
the loop integrals to be done as functional traces in Wetterich equation are
both IR- and UV-finite. The functional traces in (49) are traces both in the
spacetime integration sense as well as in the normal traces over all internal
indices the fields may carry with themselves. In our case, this will be traces
over Lorentz indices on metric fluctuations $h_{\mu\nu}$ (or on traceless
fluctuations $\bar{h}_{\mu\nu}$) and on vector field fluctuations. Obviously,
for scalars such internal traces are identical to multiplication by unity.

Following the developments of the Wetterich equation for the constrained
fields and for the quantum theory read from the expression of its one-loop
partition function understood as product of various functional determinants
(raised to various powers) of simple two-derivative operators $\hat{\square}$
possibly shifted by some constant endomorphic terms $a\hat{1}$, the resulting
reduced Wetterich equation looks as follows

$k\partial_{k}\Gamma_{k}=\frac{1}{2}\sum_{i}\sum_{j}\pm{\rm
Tr}_{\phi_{i}}\left(\frac{\left(k\partial_{k}R_{k}-\eta
R_{k}\right)\hat{1}}{\hat{\square}+R_{k}\hat{1}+a_{i,j}\hat{1}}\right)\\!,$
(50)

where we have denoted the anomalous dimension (identical for all fluctuation
fields) as $\eta$. Additionally, the $\pm$ signs depend on what was in the
exponent on the corresponding term in the one-loop partition function (43) or
(46). In other words, this decides whether the general factor of the type
$\left(\hat{\square}+R_{k}\hat{1}+a_{i,j}\hat{1}\right)$ was originally in the
denominator or the numerator of the partition function, respectively and with
which power. The functional traces were separately considered in each subspace
of fluctuation fields $\phi_{i}$, and in each subsector of fluctuations we
allowed for different factors of the type
$\left(\hat{\square}+R_{k}\hat{1}+a_{i,j}\hat{1}\right)$ with different shifts
$a_{i,j}$ (the index $j$ here is internal to each subsector of fluctuations,
while the index $i$ counts different subspaces with different representation
of fields $\phi_{i}$). The correctness of the above steps that lead to the
above FRG flow equation in a reduced form (50) is, for example, verified by an
independent check of the coefficients for one-loop perturbative beta functions
of dimensionless coupling parameters in QWG.

One should write the Wetterich equation in a reduced form in an explicit way
on a general ES background using the two form of the one-loop partition
function. Here are the results of these expansions. From formula in (43) one
gets

$k\partial_{k}\Gamma_{k}=\frac{1}{2}{\rm
Tr}_{2T}\left(\frac{\left(k\partial_{k}R_{k}-\eta
R_{k}\right)\hat{1}}{\hat{\square}-\frac{2}{3}\Lambda\hat{1}+2\hat{C}}\right)+\frac{1}{2}{\rm
Tr}_{2T}\left(\frac{\left(k\partial_{k}R_{k}-\eta
R_{k}\right)\hat{1}}{\hat{\square}-\frac{4}{3}\Lambda\hat{1}+2\hat{C}}\right)+\frac{1}{2}{\rm
Tr}_{0}\left(\frac{\left(k\partial_{k}R_{k}-\eta
R_{k}\right)\hat{1}}{\hat{\square}+2\Lambda\hat{1}}\right)-$ $-{\rm
Tr}_{1}\left(\frac{\left(k\partial_{k}R_{k}-\eta
R_{k}\right)\hat{1}}{\hat{\square}+\Lambda\hat{1}}\right)-\frac{1}{2}{\rm
Tr}_{1}\left(\frac{\left(k\partial_{k}R_{k}-\eta
R_{k}\right)\hat{1}}{\hat{\square}+\frac{1}{3}\Lambda\hat{1}}\right)-\frac{1}{2}{\rm
Tr}_{0}\left(\frac{\left(k\partial_{k}R_{k}-\eta
R_{k}\right)\hat{1}}{\hat{\square}+\frac{4}{3}\Lambda\hat{1}}\right),$ (51)

while when we use the alternative (but equivalent) expression from (46) one
finds instead a slightly more complicated formula which reads

$k\partial_{k}\Gamma_{k}=\frac{1}{2}{\rm
Tr}_{2T}\left(\frac{\left(k\partial_{k}R_{k}-\eta
R_{k}\right)\hat{1}}{\hat{\square}-\frac{2}{3}\Lambda\hat{1}+2\hat{C}}\right)+\frac{1}{2}{\rm
Tr}_{2T}\left(\frac{\left(k\partial_{k}R_{k}-\eta
R_{k}\right)\hat{1}}{\hat{\square}-\frac{4}{3}\Lambda\hat{1}+2\hat{C}}\right)+\frac{1}{2}{\rm
Tr}_{0}\left(\frac{\left(k\partial_{k}R_{k}-\eta
R_{k}\right)\hat{1}}{\hat{\square}+2\Lambda\hat{1}}\right)-$ $-{\rm
Tr}_{1}\left(\frac{\left(k\partial_{k}R_{k}-\eta
R_{k}\right)\hat{1}}{\hat{\square}+\Lambda\hat{1}}\right)-\frac{1}{2}{\rm
Tr}_{1}\left(\frac{\left(k\partial_{k}R_{k}-\eta
R_{k}\right)\hat{1}}{\hat{\square}+\frac{1}{3}\Lambda\hat{1}}\right)-\frac{1}{2}{\rm
Tr}_{0}\left(\frac{\left(k\partial_{k}R_{k}-\eta
R_{k}\right)\hat{1}}{\hat{\square}+\frac{4}{3}\Lambda\hat{1}}\right)-$
$-\frac{1}{2}{\rm Tr}_{0}\left(\frac{\left(k\partial_{k}R_{k}-\eta
R_{k}\right)\hat{1}}{\hat{\square}-\frac{2}{3}\Lambda\hat{1}}\right)-\frac{1}{2}{\rm
Tr}_{0}\left(\frac{\left(k\partial_{k}R_{k}-\eta
R_{k}\right)\hat{1}}{\hat{\square}-\frac{4}{3}\Lambda\hat{1}}\right).$ (52)

In the next step, we need to write and simplify the form of the beta
functional (LHS of the Wetterich equation) on a general ES background. This is
of course, related to the choice of the action ansatz for the running
effective action $\Gamma_{k}$ and to our abilities to read various beta
functions from it.

In general, we are interested only in running of dimensionless couplings since
their RG flow appears first when the conformal symmetry is broken softly and
spontaneously. In our $4$-dimensional setup such couplings appear in front of
three quadratic invariants in curvatures which we defined above and which are
consistent with diffeomorphism symmetry of the quantum theory (since conformal
symmetry is already softly violated by the mere presence of RG flow for
dimensionless couplings). Therefore the general beta functional of the theory
may have the form

$k\partial_{k}\Gamma_{k}=\\!\int\\!d^{4}x\sqrt{|g|}\left(\beta_{R}R^{2}+\beta_{C}C^{2}+\beta_{{\rm
GB}}{\rm GB}\right).$ (53)

In the above formula we defined that the general running effective action had
the form

$\Gamma_{k}=\\!\int\\!d^{4}x\sqrt{|g|}\left(\alpha_{R}(k)R^{2}+\alpha_{C}(k)C^{2}+\alpha_{{\rm
GB}}(k){\rm GB}\right)$ (54)

and that the beta functions of the running couplings parameter
$\alpha_{i}=\alpha_{i}(k)$ are defined as logarithmic derivative of them with
respect to RG scale $k$ that is the following formula holds

$\beta_{i}=k\frac{d}{dk}\alpha_{i}(k).$ (55)

We notice that as it is common in some FRG approaches the $k$-dependence for
the one-loop RG-improved effective action $\Gamma_{k}$ sits only in the
running couplings, while the diffeomorphically invariant terms there are not
touched at all by the derivative with respect to RG scale $k$ needed when
defining non-perturbatively the beta function$\beta_{i}$. Moreover, this is
not the end of our choice of the general ansatz on ES. It is well known that
perturbatively in QWG the beta function of the Starobinsky $R^{2}$ term is
highly suppressed compared to other beta functions $\beta_{C}$ and
$\beta_{{\rm GB}}$ (that is we have parametrically that
$\beta_{R}\ll\beta_{C},\,\beta_{{\rm GB}}$). This hierarchy of the system of
beta functions is already seen on the level of first loop. There holds that
$\beta_{R}=0$ exactly to one-loop level. If one considers higher loops then
the expression for $\beta_{R}$ is smaller because it contains higher powers of
the perturbative expansion parameter which for us here is assumed to be small.
These parameters are the couplings of the original theory. Following this
important observation we may neglect in our computation the presence of the
$\beta_{R}$ function both on the perturbative and also on non-perturbative
levels. Therefore our reduced beta functional of the theory describing running
effective action $\Gamma_{k}$ has the form

$k\partial_{k}\Gamma_{k}=\\!\int\\!d^{4}x\sqrt{|g|}\left(\beta_{C}C^{2}+\beta_{{\rm
GB}}{\rm GB}\right)$ (56)

derived from

$\Gamma_{k}=\\!\int\\!d^{4}x\sqrt{|g|}\left(\alpha_{C}(k)C^{2}+\alpha_{{\rm
GB}}(k){\rm GB}\right),$ (57)

while we unanimously agree that on the RHS of the Wetterich flow equation we
have the partition function coming exactly from the QWG action functional
described by the truncation ansatz $\alpha_{C}C^{2}$ in agreement with (9). On
the RHS of the FRG flow equation the full $k$-dependence is brought about by
IR cutoff kernels $R_{k}$ and we decide not to include any additional
$k$-dependence in the Weyl coupling $\alpha_{C}$ on this RHS (a source side
which is a reason for all effects of RG running of the LHS of the RG flow
equation).

The beta functional (56) evaluated on the generic ES background gives

$\left.\beta_{C}C^{2}+\beta_{{\rm GB}}{\rm GB}\right|_{{\rm
ES}}=\beta_{C}C^{2}+\beta_{{\rm
GB}}\left(C^{2}+\frac{8}{3}\Lambda^{2}\right)=\left(\beta_{C}+\beta_{{\rm
GB}}\right)C^{2}+\frac{8}{3}\beta_{{\rm GB}}\Lambda^{2}$ (58)

Therefore on a general ES manifold we can read in principle independently two
beta functions $\beta_{C}$ and $\beta_{{\rm GB}}$. This is possible due to an
extraction of terms proportional to the invariant quantities $C^{2}$ and
$\Lambda^{2}$ respectively from the RHS of the FRG flow equation in (49).
Again, we emphasize that the advantage of using ES as background spaces is
that we read these two beta functions from one expression evaluated on one
single background, hence there is no any issue related to changing the
background and to mutual interrelations and interdependences between them. Of
course, such a background should be understood only as a theoretical tool and
a trick which is very helpful in extracting the beta functions of the quantum
theory. One should not think that our theory has to be considered always on ES
background. For consistency of the quantum theory the QWG in the form we
present here can be considered on general Bach-flat backgrounds.

In order to proceed with the quantum computation of beta functions in the one-
loop RG improved scheme due to non-perturbative FRG phenomena, one has to make
a choice for the explicit cutoff kernel function $R_{k}(z)$. We follow the
standard choice given by the Litim cutoff

$R_{k}=R_{k}(z)=\left(k^{2}-z\right)\theta\left(k^{2}-z\right),$ (59)

where $\theta(x)$ denotes standard Heaviside theta step function. For such a
choice of the cutoff kernel we find that the functional traces (the spacetime
integration part of them) of some general function of the general at most two-
derivative operator $\hat{{\cal O}}$ is given by

${\rm Tr}f\left(\hat{{\cal O}}\right)=B_{4}\left(\hat{{\cal
O}}\right)Q_{0}[f],$ (60)

where the $B_{4}$ coefficients of any differential operator are well known in
QFT and in the theory of operators. The $Q_{0}$ functional of the function $f$
can be simplified in our case of interest, if we recall what is the generic
form of the function of the operator $f=f(z)$ that we usually meet in our
consideration related to the Wetterich equation. This form reads

$f(z)=\frac{k\partial_{k}R_{k}(z)}{z+R_{k}(z)+a_{i,j}}$ (61)

for a generic two-derivative factor of the type
$\left(\hat{\square}+R_{k}\hat{1}+a_{i,j}\hat{1}\right)$. In our case we also
accept the identification that $\hat{{\cal O}}=\hat{\square}$ as the leading
in the number of derivatives part of the operator
$\hat{\square}+a_{i,j}\hat{1}$. Then in such circumstances the expression for
$Q_{0}$ functional simplifies to

$Q_{0}[f]=Q_{0}\left[\frac{k\partial_{k}R_{k}(z)}{z+R_{k}(z)+a_{i,j}}\right]=(2-\eta)\left(1+\frac{a_{i,j}}{k^{2}}\right)^{-1}.$
(62)

Using such a form of IR-decoupling we are sure that we correctly take care of
the threshold effects related to massive modes of the excitations present
around a given background. This decoupling is explicitly realized by the
second factor in the above formula, while the first factor describes the
corrections due to the inclusion of the anomalous dimension for the quantum
fields. Here $\eta$ denotes the uniform anomalous dimension for the graviton
field, since all TT graviton fields, vector fields and scalars must have it
the same (due to the requirements of original conformal symmetry of the
initial theory and due to always unbroken diffeomorphism symmetry).

With the formulas (60) and (62) at hand, one can easily compute the ensuing
functional traces in the FRG flow equation. First, we write it for the compact
form of the one-loop partition function on a general ES given in (51). Then
the flow simplifies to

$\frac{2}{2-\eta}k\partial_{k}\Gamma_{k}=\left(1-\frac{\frac{2}{3}\Lambda}{k^{2}}\right)^{-1}B_{4}\left(\hat{\square}_{2T}-\frac{2}{3}\Lambda\hat{1}_{2T}+2\hat{C}\right)+\left(1-\frac{\frac{4}{3}\Lambda}{k^{2}}\right)^{-1}B_{4}\left(\hat{\square}_{2T}-\frac{4}{3}\Lambda\hat{1}_{2T}+2\hat{C}\right)+$
$+\left(1+\frac{2\Lambda}{k^{2}}\right)^{-1}B_{4}\left(\hat{\square}_{0}+2\Lambda\hat{1}_{0}\right)-\left(1+\frac{\frac{4}{3}\Lambda}{k^{2}}\right)^{-1}B_{4}\left(\hat{\square}_{0}+\frac{4}{3}\Lambda\hat{1}_{0}\right)-$
$-2\left(1+\frac{\Lambda}{k^{2}}\right)^{-1}B_{4}\left(\hat{\square}_{1}+\Lambda\hat{1}_{0}\right)-\left(1+\frac{\frac{1}{3}\Lambda}{k^{2}}\right)^{-1}B_{4}\left(\hat{\square}_{1}+\frac{1}{3}\Lambda\hat{1}_{0}\right).$
(63)

In the case of the alternative equation for the one-loop partition function
given in (52) we instead find

$\frac{2}{2-\eta}k\partial_{k}\Gamma_{k}=\left(1-\frac{\frac{2}{3}\Lambda}{k^{2}}\right)^{-1}B_{4}\left(\hat{\square}_{2}-\frac{2}{3}\Lambda\hat{1}_{2}+2\hat{C}\right)+\left(1-\frac{\frac{4}{3}\Lambda}{k^{2}}\right)^{-1}B_{4}\left(\hat{\square}_{2}-\frac{4}{3}\Lambda\hat{1}_{2}+2\hat{C}\right)+$
$+\left(1+\frac{2\Lambda}{k^{2}}\right)^{-1}B_{4}\left(\hat{\square}_{0}+2\Lambda\hat{1}_{0}\right)-\left(1+\frac{\frac{4}{3}\Lambda}{k^{2}}\right)^{-1}B_{4}\left(\hat{\square}_{0}+\frac{4}{3}\Lambda\hat{1}_{0}\right)-$
$-\left(1-\frac{\frac{2}{3}\Lambda}{k^{2}}\right)^{-1}B_{4}\left(\hat{\square}_{0}-\frac{2}{3}\Lambda\hat{1}_{0}\right)-\left(1-\frac{\frac{4}{3}\Lambda}{k^{2}}\right)^{-1}B_{4}\left(\hat{\square}_{0}-\frac{4}{3}\Lambda\hat{1}_{0}\right)-$
$-2\left(1+\frac{\Lambda}{k^{2}}\right)^{-1}B_{4}\left(\hat{\square}_{1}+\Lambda\hat{1}_{0}\right)-\left(1+\frac{\frac{1}{3}\Lambda}{k^{2}}\right)^{-1}B_{4}\left(\hat{\square}_{1}+\frac{1}{3}\Lambda\hat{1}_{0}\right).$
(64)

However, already on this level one sees that these two forms of the FRG flows
are equivalent because of the following formula

$B_{4}\left(\hat{\square}_{2T}-X\hat{1}_{2T}+2\hat{C}\right)=B_{4}\left(\hat{\square}_{2}-X\hat{1}_{2}+2\hat{C}\right)-B_{4}\left(\hat{\square}_{0}-X\hat{1}_{0}\right)$
(65)

for any constant $X$. Therefore for future considerations, we will consider
only the explicit RG flow given by the formula (63). In the formulas above one
explicitly sees the presence of massive threshold functions of the general
type

$\left(1+\frac{a_{i,j}}{k^{2}}\right)^{-1}.$ (66)

As we will see in a moment they are very important for the consideration of
the RG flows towards the deep IR regime in QWG. To eventually simplify the
expression for the RG flow and read the non-perturbative expressions for the
beta functions with the threshold phenomena included, one needs to evaluate
the $B_{4}$ coefficients related to perturbative UV-divergences of the theory.
Precisely they are related to logarithmic dimensionless UV-divergences in
front of the terms proportional to $C^{2}$ and ${\rm GB}$ term in the quantum
theory.

Explicit computations on the general ES give us the results for these $B_{4}$
coefficients of various operators

$B_{4}\left(\hat{\square}_{2T}-\frac{2}{3}\Lambda\hat{1}_{2T}+2\hat{C}\right)=\frac{21}{20}C^{2}-\frac{7}{5}\Lambda^{2}$
(67)
$B_{4}\left(\hat{\square}_{2T}-\frac{4}{3}\Lambda\hat{1}_{2T}+2\hat{C}\right)=\frac{21}{20}C^{2}+\frac{3}{5}\Lambda^{2}$
(68)
$B_{4}\left(\hat{\square}_{1}+\Lambda\hat{1}_{0}\right)=-\frac{11}{180}C^{2}+\frac{716}{135}\Lambda^{2}$
(69)
$B_{4}\left(\hat{\square}_{1}+\frac{1}{3}\Lambda\hat{1}_{0}\right)=-\frac{11}{180}C^{2}+\frac{236}{135}\Lambda^{2}$
(70)
$B_{4}\left(\hat{\square}_{0}+2\Lambda\hat{1}_{0}\right)=\frac{1}{180}C^{2}+\frac{479}{135}\Lambda^{2}$
(71)
$B_{4}\left(\hat{\square}_{0}+\frac{4}{3}\Lambda\hat{1}_{0}\right)=\frac{1}{180}C^{2}+\frac{269}{135}\Lambda^{2}$
(72)

We conveniently above wrote these formulas in terms of $C^{2}$ and
$\Lambda^{2}$invariant to be prepared for the extraction of two beta functions
$\beta_{C}$ and $\beta_{{\rm GB}}$. Once again one notices that in these
formulas there are no interference terms between $\Lambda$ and matrix
$\hat{C}$ terms. Of course, they are not possible due to algebraic reasons
here, because the final expression for the $B_{4}$ coefficient must be a
scalar and the matrix $\hat{C}$ is completely traceless. On the other hand
this result signifies that the contributions from the MSS part and from the
Ricci-flat part to the general ES are independent of each other and there is
no any mixing between them and also that they can be computed separately.

Now, we read the beta functions exploiting together formulas (63), (56), (58)
and a set of formulas from (67) to (72). We therefore find that the following
equation holds

$\frac{2}{2-\eta}\left(\left(\beta_{C}+\beta_{{\rm
GB}}\right)C^{2}+\frac{8}{3}\beta_{{\rm
GB}}\Lambda^{2}\right)=\left(1-\frac{\frac{2}{3}\Lambda}{k^{2}}\right)^{-1}\left(\frac{21}{20}C^{2}-\frac{7}{5}\Lambda^{2}\right)+\left(1-\frac{\frac{4}{3}\Lambda}{k^{2}}\right)^{-1}\left(\frac{21}{20}C^{2}+\frac{3}{5}\Lambda^{2}\right)+$
$+\left(1+\frac{2\Lambda}{k^{2}}\right)^{-1}\left(\frac{1}{180}C^{2}+\frac{479}{135}\Lambda^{2}\right)-\left(1+\frac{\frac{4}{3}\Lambda}{k^{2}}\right)^{-1}\left(\frac{1}{180}C^{2}+\frac{269}{135}\Lambda^{2}\right)-$
$-2\left(1+\frac{\Lambda}{k^{2}}\right)^{-1}\left(-\frac{11}{180}C^{2}+\frac{716}{135}\Lambda^{2}\right)-\left(1+\frac{\frac{1}{3}\Lambda}{k^{2}}\right)^{-1}\left(-\frac{11}{180}C^{2}+\frac{236}{135}\Lambda^{2}\right).$
(73)

By comparing both sides of this equation and treating the two invariants
$C^{2}$ and $\Lambda^{2}$ as exclusive and mutually independent, we arrive to
the two equations

$\frac{2}{2-\eta}\left(\frac{8}{3}\beta_{{\rm
GB}}\Lambda^{2}\right)=\left(1-\frac{\frac{2}{3}\Lambda}{k^{2}}\right)^{-1}\left(-\frac{7}{5}\Lambda^{2}\right)+\left(1-\frac{\frac{4}{3}\Lambda}{k^{2}}\right)^{-1}\left(\frac{3}{5}\Lambda^{2}\right)+$
$+\left(1+\frac{2\Lambda}{k^{2}}\right)^{-1}\left(\frac{479}{135}\Lambda^{2}\right)-\left(1+\frac{\frac{4}{3}\Lambda}{k^{2}}\right)^{-1}\left(\frac{269}{135}\Lambda^{2}\right)-$
$-2\left(1+\frac{\Lambda}{k^{2}}\right)^{-1}\left(\frac{716}{135}\Lambda^{2}\right)-\left(1+\frac{\frac{1}{3}\Lambda}{k^{2}}\right)^{-1}\left(\frac{236}{135}\Lambda^{2}\right)$
(74)

and

$\frac{2}{2-\eta}\left(\beta_{C}+\beta_{{\rm
GB}}\right)C^{2}=2\times\left(\frac{21}{20}C^{2}\right)+\left(\frac{1}{180}C^{2}\right)-\left(\frac{1}{180}C^{2}\right)-3\times\left(-\frac{11}{180}C^{2}\right).$
(75)

These two equations represent respectively the RG flows as considered
separately on MSS and Ricci-flat backgrounds. In order to obtain the equation
in (75) we also took a zeroth order series coefficient in expansion in the
$\Lambda$ parameter, since this parameter is non-vanishing only on MSS. In
this way we prove that the results as obtained on a general ES completely
reproduce the previous results obtained separately on MSS and Ricci-flat
backgrounds. This also confirms that the original method with using of two
distinct backgrounds is consistent and does not lead to any contradiction.
These two backgrounds are mutually exclusive since the MSS can be Ricci-flat
only if this is a flat spacetime, however this observation regarding the
physical background does not lead to any inconsistency here. We used these two
backgrounds as a trick to get two combinations of beta functions but none
should understand that we consider a QWG theory put simultaneously on these
two backgrounds. The results we get there are consistent with each other.
Moreover, they are also consistent with the general considerations that we
make on ES, when the background used again as a mathematical trick is a single
one. However, we do not find any problem in taking respective limits
$\Lambda\to 0$ or $\hat{C}\to 0$ and we recover previous results from Ricci-
flat and MSS respectively. Moreover, there are no mixing terms
$\Lambda\hat{C}$ which would invalidate the independent analysis of MSS and
Ricci-flat to get beta functions $\beta_{C}$ and $\beta_{{\rm GB}}$
separately. This verifies a general philosophy that the quantum beta functions
of the theory (in a given scheme) are universal and one can use arbitrary
backgrounds to evaluate them. Before we used MSS and Ricci-flat because of the
reasons of convenience. Now, we have proven that the same consistent results
are also obtained on the general ES.

By doing algebraic simplifications in the formula one finally arrives to the
system

$\frac{2}{2-\eta}\beta_{{\rm
GB}}\Lambda^{2}=\left(1-\frac{\frac{2}{3}\Lambda}{k^{2}}\right)^{-1}\left(-\frac{21}{40}\Lambda^{2}\right)+\left(1-\frac{\frac{4}{3}\Lambda}{k^{2}}\right)^{-1}\left(\frac{9}{40}\Lambda^{2}\right)+\left(1+\frac{2\Lambda}{k^{2}}\right)^{-1}\left(\frac{479}{360}\Lambda^{2}\right)-$
$-\left(1+\frac{\frac{4}{3}\Lambda}{k^{2}}\right)^{-1}\left(\frac{269}{360}\Lambda^{2}\right)-\left(1+\frac{\Lambda}{k^{2}}\right)^{-1}\left(\frac{179}{45}\Lambda^{2}\right)-\left(1+\frac{\frac{1}{3}\Lambda}{k^{2}}\right)^{-1}\left(\frac{59}{90}\Lambda^{2}\right)$
(76)

and

$\frac{2}{2-\eta}\left(\beta_{C}+\beta_{{\rm
GB}}\right)C^{2}=\frac{137}{60}C^{2}.$ (77)

From this system one solves for the beta functions $\beta_{{\rm GB}}$ and
$\beta_{C}$ in the following way

$\beta_{{\rm
GB}}=\frac{1}{2}(2-\eta)\left[-\frac{21}{40}\left(1-\frac{\frac{2}{3}\Lambda}{k^{2}}\right)^{-1}+\frac{9}{40}\left(1-\frac{\frac{4}{3}\Lambda}{k^{2}}\right)^{-1}+\frac{479}{360}\left(1+\frac{2\Lambda}{k^{2}}\right)^{-1}-\right.$
$\left.-\frac{269}{360}\left(1+\frac{\frac{4}{3}\Lambda}{k^{2}}\right)^{-1}-\frac{179}{45}\left(1+\frac{\Lambda}{k^{2}}\right)^{-1}-\frac{59}{90}\left(1+\frac{\frac{1}{3}\Lambda}{k^{2}}\right)^{-1}\right]$
(78)

and

$\beta_{C}=\frac{1}{2}(2-\eta)\left[\frac{137}{60}+\frac{21}{40}\left(1-\frac{\frac{2}{3}\Lambda}{k^{2}}\right)^{-1}-\frac{9}{40}\left(1-\frac{\frac{4}{3}\Lambda}{k^{2}}\right)^{-1}-\frac{479}{360}\left(1+\frac{2\Lambda}{k^{2}}\right)^{-1}+\right.$
$\left.+\frac{269}{360}\left(1+\frac{\frac{4}{3}\Lambda}{k^{2}}\right)^{-1}+\frac{179}{45}\left(1+\frac{\Lambda}{k^{2}}\right)^{-1}+\frac{59}{90}\left(1+\frac{\frac{1}{3}\Lambda}{k^{2}}\right)^{-1}\right]$
(79)

Now, everything depends on the expression for the anomalous dimension $\eta$
of the TT graviton field. Using its expression motivated by perturbative one-
loop considerations, we get that

$\eta=-\frac{1}{\omega_{C}}k\frac{d}{dk}\omega_{C}(k)$ (80)

where the coupling $\omega_{C}$ is related to the original Weyl coupling by
the inverse relation (i.e. $\omega_{C}=\alpha_{C}^{-1}$). This extension to
include a non-trivial anomalous dimension is here for free since it does not
change anything related to the structure of the two beta functions that we
have obtained.

To check the consistency of these results for the beta functions $\beta_{C}$
and $\beta_{{\rm GB}}$ one can obtain their asymptotic values in the UV
asymptotic regimes of energies. This means that we should take the limit
$k\to+\infty$ together with $\eta\to 0$ since this anomalous dimension is a
quantum effect of higher order compared to the perturbative one-loop running
that the true running asymptotes to in the deep UV regime. These limits result
in the following expressions

$\beta_{{\rm GB}}=-\frac{87}{20}$ (81) $\beta_{C}+\beta_{{\rm
GB}}=\frac{137}{60}$ (82)

which are solved by

$\beta_{C}=\frac{199}{30}\quad{\rm and}\quad\beta_{{\rm GB}}=-\frac{87}{20}.$
(83)

These two results perfectly agree with the one-loop perturbative results as
obtained earlier by Fradkin and Tseytlin.

Having obtained the results for the non-perturbatively improved beta functions
with the inclusion of the threshold effects of massive modes and of anomalous
dimension of the graviton $\eta,$ we can now discuss the implications for the
RG flows and its topology in QWG. The situation in the UV regime is quite
trivial and boring since in QWG both couplings reach a trivial UV FP of RG.
The coupling $\beta_{{\rm GB}}$ is induced by quantum corrections to the
action of QWG theory in (9), but as we have seen in (29) it non-zero value
does not lead to any violation of conformal symmetry even in the local
version. This means that the perturbative couplings that should be defined as
$\left(\sqrt{\omega_{C}}\right)^{-1}=\sqrt{\alpha_{C}}$ and
$\left(\sqrt{-\omega_{{\rm GB}}}\right)^{-1}=\sqrt{-\alpha_{{\rm GB}}}$ reach
an asymptotically free FP in the UV, when the FP values of these couplings
both vanish. (The coupling of the Gauss-Bonnet term is negative since its beta
function $\beta_{{\rm GB}}<0$ is also negative and this is why we took the
definitions with the change of the sign under the square roots.) We conclude
that in the UV the quantum theory of Weyl gravity meets a Gaussian FP with
trivial scaling of couplings and operators. This behavior is in agreement with
that in the UV regime the QWG theory should meet a conformally invariant FP
where this model is probably merged to a broader theory such as CSG in which
the UV FP with fully realized quantum conformality is present. In this UV-
complete theory there is no running and the value of the Weyl coupling can be
arbitrary (even non-perturbatively big), but still with any value of
$\alpha_{C}$ the conformal symmetry is explicitly present and not violated by
quantum effects.

Instead, the situation towards the IR regime of the energy spectrum is more
interesting. One looks in that regime for non-trivial new FP of RG flows. They
are defined by the simultaneous condition that $\beta_{C}=0$ and $\beta_{{\rm
GB}}=0$. One can see that for this system of equations the presence of the
anomalous dimension $\eta$ in the expressions for the beta functions (78) and
(79) is completely spurious and this higher order quantum effect can be safely
neglected here. The results we have found are really curious since for some
finite value of the energy (in the IR regime) we see with a very good
precision of 2% a non-trivial point where the RG flow for the two beta
functions $\beta_{C}$ and $\beta_{{\rm GB}}$ stops completely for a moment (of
energies, or a while in RG time $t=\log\frac{k}{k_{0}}$). This confluence of
two FP’s for respectively $\alpha_{C}$ and $\alpha_{{\rm GB}}$ couplings is a
very stunning feature that is present only in QWG. Other higher derivative
quadratic gravities do not enjoy such a behavior in the IR regime. When one
uses a rescaled dimensionless energy RG scale variable, here conveniently
defined as

$\kappa=\frac{k}{\sqrt{|\Lambda|}},$ (84)

then one can numerically determine the location of these FP’s for both of the
couplings separately. First, in the case of $\Lambda>0$ we have a very good
agreement and the FP for the $\alpha_{C}$ coupling is found around
$\kappa_{C}\approx 1.17709$ and for the $\alpha_{{\rm GB}}$ coupling is found
around $\kappa_{{\rm GB}}\approx 1.19163$. For the case of $\Lambda<0$ we
still find a good agreement and the particular values for energy locations of
FP’s are $\kappa_{C}\approx 1.49722$ and $\kappa_{{\rm GB}}\approx 1.52128$
for the couplings $\alpha_{C}$ and $\alpha_{{\rm GB}}$ respectively.

Now, we provide some interpretation to this special point (happening almost at
the same energy scales for both couplings) at some finite energy scale. It
cannot be a true IR FP of RG flow since this happens at finite energy scale,
so the flow may momentarily stop, turn back but generally must continue, when
we decrease the value of the RG time $t$ more towards the deeper IR regime. We
decided to call such a special point of RG flow as turning point (TP) in
distinction to a true IR FP. One can plot the beta functions of the two
couplings analyzed here. One sees that the curves for RG beta functions cross
zero line almost at the same energy scale. Moreover, one expects that by the
inclusion of higher order phenomena (like by going to the two-loop level) one
will improve the accuracy of matching such that in a truly non-perturbative
theory the two curves cross zero exactly at the same RG time. We can call this
position as a $t_{{\rm TP}}$. The estimates that we have made above suggest
that its value is around $1.2$ and definitely it is for values bigger than
$1$. We emphasize once again that this TP is not a true FP of RG that still
has to be searched for in the deeper IR regime. Instead, for this turning
point there exists a holographic interpretation that we describe in short
below.

When one wants to describe the IR regime, one leaves the turning point and
continue the RG flow still towards the IR regime. Here we can employ the
perturbation calculus in value of the couplings that we met at the TP. We
remind that only the beta functions of couplings vanish there, while this is
not the case for the coupling themselves. We denote a general coupling there
by $\omega_{*}$ at the location of TP. We therefore later perform
perturbations in the small $\omega_{*}$ parameter treating the TP as a
starting point for this perturbation and assuming that $\omega_{*}\ll 1$. The
consequences of this are the following. We found a non-trivial IR FP at $k=0$
so in the true IR regime. This IR FP of RG is completely stable in a sense
that the dimensions of two operators $C^{2}$ and ${\rm GB}$ near it are finite
negative and bounded. Their actual values are equal to each other and given by
$\theta=-\frac{1}{3}$. This implies that they are relevant operators near this
IR FP. They describe the RG flow of the two originally dimensionless couplings
$\alpha_{C}$ and $\alpha_{{\rm GB}}$ of the theory. One sees that thanks to
the quantum effects the operator and couplings acquire some anomalous but
controllable dimensions and in the IR regime they are not anymore
dimensionless (and the corresponding operators $C^{2}$ and ${\rm GB}$ are not
of energy dimension $4$ in $d=4$). This means that the perturbation by
addition of these two operators to the CFT of IR FP of the theory is
completely stable and does not destabilize the physics which in the deep IR
regime is controlled and dictated by IR FP of the theory. Moreover, one finds
the following relation between the couplings $\omega_{*}$ defined at the TP
and the ensuing couplings $\omega_{**}$ at the true IR FP,

$\omega_{**}=\omega_{*}+\frac{9}{2}\kappa\beta^{\prime}(\kappa),$ (85)

where $\kappa$ is the schematic notation for the location of the TP, and also
the derivatives of the beta functions of the corresponding couplings are
evaluated at the TP location. This formula is valid for both types of
couplings $\omega_{C}$ and $\omega_{{\rm GB}}$. Additionally, one finds that
the couplings $\omega_{*}$ are not constrained at all and the couplings
$\omega_{**}$ are only forced to stay in relation (85), where the couplings
$\omega_{*}$ are completely arbitrary. This means that also resulting true IR
FP values of the couplings $\omega_{**}$ can be both in the perturbative as
well as in some other non-perturbative regimes.

##  6 Interpretation and further discussion of the results

Firstly, we would like to discuss the possible interpretation of our findings
about the TP of RG flow that occurs at some finite location in energy scales
(around $\kappa=1.2$, so for $k=1.2\sqrt{\Lambda}$ for $\Lambda>0$). The whole
fact that this is a turning point of RG corresponds to the existence of a
multi-branch holographic RG flow. A possibly correct analysis of such an IR
behavior is provided by the AdS/CFT correspondence, which describes the
geometrized understanding of RG flows in various QFT’s ( see [51] and
references within). The geometrization comes by promoting RG flows to some
gravitational spacetime (characterized by some metric tensor) in some higher
dimensional spacetimes which resembles the deformations of exact AdS
spacetime. In our case, this extended completely artificial spacetime is
5-dimensional and has nothing to do with real physical gravitational spacetime
considered as exact or background solution in CWG. The geometrization of the
RG flow is only a useful theoretical tool to visualize various properties of
the otherwise complicated RG flows in quantum field theories [52]. As the name
suggests only in the case of FP of RG (when we know that they are described by
some CFT’s) we are sure that the resulting gravitationally dual 5-dimensional
bulk is exactly identical to some patch of the anti-de Sitter spacetime. The
departure from the FP is encoded on the dual side of the theory by some radial
geometry that differs from the AdS. Therefore in the case of the RG running of
couplings we need to speak about the modified AdS spacetime used for
geometrization of RG flow. This bulk spacetime can be also understood as a
gravitational dual in 5-dimension to the field theory which was considered in
$d=4$ dimensions of real physical spacetime. When the running takes place only
as a function of one variable, like in our case, where $k$ is its argument,
then in the 5-dimensional bulk we can use a metric of 5-dimensional spacetime
which has a very special metric structure. Namely this could be a conformally
flat metric (in a 5-dimensional sense) with only a non-trivial scale factor
and in this way this analogue of the FLRW spacetime used very frequently in
description of physical 4-dimensional cosmology. This is the framework and the
metric ansatz with which we will try to describe the geometrization of the
interesting RG flow phenomena that we have found towards the IR regime in QWG.
In particular, the situation with TP of RG is well described by bounce
cosmological solutions.

One recalls that this TP happens at some finite energy scales $\kappa$. They
in general, correspond to some finite radial locations in the asymptotically
AdS spacetime, where we geometrize the RG flow. Of course, due to the fact
that both in the UV regime as well as in the IR we find a FP of RG in QWG, the
gravitational dual spacetime must be asymptotically (that is for zero and
infinite values of the radial locations) approaching some 5-dimensional AdS
types of spacetimes. Precisely, in the dual gravitational spacetime, the IR FP
corresponds to infinite radial location $\rho\to+\infty$ and UV FP is in turn
located at the origin of the AdS spacetime, that is for $\rho\to 0$. In
general, there exists a relation between the energy scale in RG flow $k$ and
the radial coordinate $\rho$ of the dual AdS, which reads $\rho\sim k^{-1}$,
which is true up to a constant of proportionality. The particular spacelike
location of the TP of RG flow is dual to 4-dimensional surface (so co-
dimension 1 in 5-dimensional bulk spacetime) embedded in AdS-like
5-dimensional geometry, which is located at particular constant value of the
radial coordinate $\rho={\rm const}$ of the AdS-like geometry. From this setup
one easily understands that the true IR FP of RG flow must corresponds to the
conformal boundary of AdS spacetime which is asymptotically located at
infinite values of the radial coordinate $\rho$. Therefore the TP cannot be a
true IR FP as we have also found earlier.

One can consider the gravitational cosmological spacetimes describing the
bounce situation. A particular useful example is the case of a FLRW spacetime
with a specific form of the scale factor $a(t)$. In cosmology if the scale
factor is not an injective function but a continuous function of the
cosmological time $t$, then there must exist a bouncing point. This moment of
the cosmological time corresponds to the physical bounce like in the motion of
a material point in the given gravitational potential, when it bounces from
the floor. This means that the scale factor just after the bounce grows in