Let us take, in particular, the example of conformal gravitational theory in
$d=4$ spacetime dimensions. The action for this theory was introduced in
Section 2 in (16). One may think that this is a very special case of a generic
four-derivative theory which is scale-invariant precisely in $d=4$ dimensions.
The action of such a HD gravitational theory can be conveniently written as

$S_{{\rm
HD}}=\\!\int\\!d^{4}x\sqrt{|g|}\left(\alpha_{R^{2}}R^{2}+\alpha_{C^{2}}C^{2}\right),$
(25)

where we have deliberately chosen the basis with curvature invariants $R^{2}$
and $C^{2}$, which make explicitly manifest the conformal covariance of the
part with the $C^{2}$ term there999In principle, one could also add here a
different term written as the Gauss–Bonnet density, but it is topological in
$d=4$ and its variation is a total derivative, hence vanishes under spacetime
volume integral.. One would think (naively) that to specify to a conformal
gravitational theory in $d=4,$ one has to simply take a limit
$\alpha_{R^{2}}\to 0$ from the generic HD theory in (25).

One notices that for generic HD gravitational theories such as in (25), there
are no problems or subtleties with the FP quantization procedure of the
existing diffeomorphism local symmetry, which can successfully be brought to
the very end here, and, for example, the propagator around flat spacetime of
graviton or covariant functional traces of the Hessian operator understood
around general background can be computed. In general, the method of FP
quantization of higher-derivative gravitational theories (with local
diffeomorphism symmetries) works for the cases when both $\alpha_{R^{2}}\neq
0$ and $\alpha_{C^{2}}\neq 0$. One may think that the results of the same
procedure in the case when the limit $\alpha_{R^{2}}\to 0$ is taken will be
smooth and continuous ones.

However, this is not the case and one sees some strong discontinuities in the
details of this procedure as well as in the final results obtained within the
QFT framework. One can see this discontinuity already in the perturbative
spectrum of the two theories (with $\alpha_{R^{2}}\neq 0$ and with
$\alpha_{R^{2}}=0$) when analyzed around flat spacetime. As it is well known
in general higher-derivative gravitational theories with four derivatives, one
typically has eight degrees of freedom, where also the Einstein–Hilbert term
$\kappa R$ with the Newton’s constant is included into the action in (25).
These results are derived either from the canonical formalism of treatment of
theories with gauge symmetries and performed by counting constraints Kluson ,
or by taking a covariant perturbative approach and looking at the spectrum of
modes which solve linearized EOM of the theory. A similar procedure can be
also applied there to Weyl conformal gravity. However, there are more
subtleties due to new constraints due to local conformal symmetry in the
canonical framework and due to the masslessness character of all fluctuations
considered in the perturbation theory (some poles of the propagator are double
and need an additional resolution). But, using some tricks with well-designed
limiting procedures Riegert ; Ferrara:2020zef , one obtains from both the two
methods the consistent result, namely six degrees of freedom in conformal
gravity in $d=4$. This last number of degrees of freedom in conformal gravity
is unambiguous. As one can see, this result is clearly discontinuous from the
other one (eight degrees of freedom) obtained for any non-zero value of the
coupling $\alpha_{R^{2}}$ and for $G_{N}\neq 0$. Hence, the limit
$\alpha_{R^{2}}\to 0$ will also not change anything and we inevitably find a
discontinuity.

###  4.3 Counting of Degrees of Freedom

The full story of counting active degrees of freedom in two theories (generic
HD gravity in (25) and Weyl conformal gravity in $d=4$) is, however, more
complicated and more twisted. For scale-invariant HD theory, one should never
include the $\kappa$ coupling related to the Newton’s constant multiplying the
Ricci scalar $R$ term in the Einstein-Hilbert action. This is obvious from the
fact that this dimensionful parameter gives rise to masses of perturbative
modes, which are, however, strictly forbidden in theory without scales. On the
other side, the $\kappa$ coupling is technically needed to resolve the double
poles of the graviton propagator, otherwise it is problematic both in generic
HD theory and in conformal Weyl gravity too. The standard particle
interpretation is only available for modes whose propagation after simple
fraction decomposition is described by simple poles in the $k^{2}$ variable.
To resolve the multiple poles, one tries to modify the theory and introduce a
spurious massive parameter $M$ (breaking conformal and even scale invariance).
At the end of the computation, one should, as the last limit, take $M\to 0$
hoping that for physical observables this limit will be unambiguous and
possible to define. In this way, the problem with multiple poles can be
resolved and physical quantities can be determined.

If instead one naively counts possible number of degrees of freedom in
massless theory (HD theory in (25) with $G_{N}=0$), then one may have at most
six degrees of freedom ($2\times 2=4$ from two massless poles of the spin-2
part of the propagator and $2\times 1=2$ from two massless poles of the spin-0
part). This final number is in clear disagreement with the number obtained
using other methods, also when $\kappa=0$ (such as by counting dynamical
constraints or computing one-loop functional determinants in scale-invariant
HD theory). However, as one of the advantages of the previous method, we
notice that we do not anymore play with the mass to resolve the double poles
and we do not inadvertently violate the scale invariance by intrusion of mass
to the definition of the theory and perturbative states around flat spacetime.

Still, assuming that in generic scale-invariant HD theory in (25) we have
eight perturbative degrees of freedom, it is difficult in a standard way to
assign these degrees of freedom to particular spin states. This is because we
do not have mass in the theory, so we cannot have massive spin-2 excitations
either, and this forbids us from having $2+5=7$ degrees of freedom in the
spin-2 sector. In the massless spin-2 sector with two modes, we can have at
most four degrees of freedom, while in the spin-0 sector we can have at most
only two degrees of freedom when we have two poles. This does not sum up to
eight. Instead, the maximal number of degrees of freedom we can obtain in this
way is six. This different way of counting shows that there may not be such
strong discontinuity when one takes the limit $\alpha_{R^{2}}\to 0$ from
generic massless HD gravity to a special conformal Weyl gravity in $d=4$
spacetime dimensions. In actuality, one may think that since in this case of
counting, the number of active degrees of freedom is continuous $6\to 6$, then
the limit $\alpha_{R^{2}}\to 0$ is not so dangerous and signifies the mere
different accounting or reshuffling of degrees of freedom. One could be
tempted to say that in generic scaleless HD gravity in $d=4$, we have six
degrees of freedom distributed as two spin-2 massless particles, each with two
polarizations, and two spin-0 massless particles. When the transition to
conformal gravity happens, the two scalar degrees of freedom (from spin-0)
merge into one massless spin-1 vector particle with two helicities. It is
clear from this counting that the number of degrees of freedom is conserved in
this limiting procedure. They just obtain different specification and
distribution in conformal gravity into slightly different irreducible
representations of Lorentz group describing massless particle perturbative
excitations around flat spacetime. The emerging massless vector spin-1 gauge
particle is reminiscent of the Weyl gauge photon, but here, its role in the
gravitational context is completely different.

Due to additional constraints and symmetries of the conformal gravitational
theory, one also makes visible the discontinuity in the structure of the two
theories. New symmetries are present in the special theory with
$\alpha_{R^{2}}=0$. The discontinuity in the number of degrees of freedom
signals that something is very special with the theory characterized by the
absence of the $R^{2}$ term in (25). The symmetries are enhanced in this case,
the limit is not continuous, and this is not similar to any other generic HD
gravitational theory. This point in the theory space (actually a line
parametrized by any value of the coupling $\alpha_{C^{2}}>0$) is very special
and it is isolated from its neighborhood; hence, any limiting procedures do
not give any continuous results here. Moreover, one can view that the
conformal gravity is a critical or extremal point in the theory space, not
reachable by any sequence of generic HD theories asymptoting to it. We already
know the reason for such a behavior, since the symmetries are enhanced and the
conformal symmetry of rescaling transformations appears there in the local
form.

As is known, the counting of degrees of freedom can be also performed in
canonical (Hamiltonian) formalism for gauge field theories. This formalism
Henneaux:1992ig  is also very useful as a starting point for canonical
quantization of theories with degeneracy of the Lagrangian, so when not all of
the generalized velocities can be re-expressed via generalized momenta, this
generically happens due to the presence of symmetries (in local form). The
freedom related to gauge symmetries typically indicates that there is more
than one set of canonical variables that corresponds to the same physical
state. This choice can be realized by adding some gauge-fixing conditions.
General gauge theory can be viewed as a mechanical theory with constraints,
since not all canonical momenta are non-zero during the spacetime evolution of
fields. For example, these constraints are analogues of the Gauss laws for the
electric part of the field strength tensors. The constraint is of the first
class if its Poisson bracket with every other constraints vanishes on the
constraint surface, but not necessarily in the full phase space. Then one
could say that they vanish weakly. If Poisson bracket of a constraint with at
least one other constraint does not vanish even weakly, then they are of the
second or higher class. Typically in gauge theories, first-class constraints
play the role of generators of local gauge symmetries. The same interpretation
cannot be applied to the second-class constraints since they do not preserve
constraints even on the constraint manifold. They are non-physical and they
have to be treated differently to generators of gauge symmetries. The second
class constraints can be set strongly to zero (so valid in the whole phase
space and not only on the constraint surface) either before or after the
evaluation of a Dirac bracket, which here substitutes for the Poisson bracket
for the considerations of the extended Hamiltonian function. They just become
identities which must be satisfied during the field evolution, expressing some
canonical variables in terms of others.

In the case of conformal gravity, setting second-class constraints strongly
equal to zero can actually be used to eliminate some of the canonical
dynamical variables of the formalism. In this way, one can completely select
the gauge by eliminating all first-class constraints and ending up only with
second-class constraints. Following this approach, one arrives at the
following counting of physical degrees of freedom (understood as field-
theoretical degrees of freedom, contrary to mechanical ones) $N_{\rm dof}$ in
generic gauge field theory:

$N_{\rm dof}=N_{\rm can}=N_{\rm tot,can}-N_{1}-N_{2}-N_{\rm gfix}=N_{\rm
tot,can}-2N_{1}-N_{2},$ (26)

where by $N_{\rm can}$ we denote the number of independent canonical variables
in the formalism. Moreover, $N_{\rm tot,can}$ stands for total number of
canonical variables (possibly mutually dependent) and $N_{1}$ and $N_{2}$ are
for the number of first- and second-class constraints, respectively. Finally,
by $N_{\rm gfix}$ we mean the number of gauge-fixing conditions added to
unambiguously pinpoint the dynamics of the system in question. This counting
in the case of Weyl gravity is as follows Irakleidou:2018qxn : the physical
degrees of freedom are $N_{\rm dof}=12-3-0-3=12-6=6$ (according to (26)). The
interpretation of these resulting six physical degrees is as follows: two of
these describe a massless graviton, such as in the case of two-derivative
general relativity, and the remaining four can be used for a description of a
“partially massless” graviton, as in Deser:1983mm ; Deser:2001pe . Following
appendix C.3 of Irakleidou:2018qxn , one can see that due to the higher-
derivative nature of the conformal gravity, two additional primary constraints
are actually of the second class; however, one can set them strongly to zero,
and they are effectively eliminated from the Lagrangian of the conformal
gravity. In other words, these second-class constraints can be solved
everywhere in the phase space. This also implies that instead of the extended
Dirac bracket, one can use, for the extended Hamiltonian, simple Poisson
bracket here without too much complication.

In these above two paragraphs, the conformal gravity theory was formulated in
the Hamiltonian formalism. The original work can be found in
Irakleidou:2014vla . Earlier, a Hamiltonian formulation of generic higher-
derivative theories in $d=4$ spacetime dimensions, including conformal
gravity, was considered in Buchbinder:1992rb ; Kluson .

The general fact of discontinuity is an emanation of a quite ubiquitous
phenomena which was first discovered by Veltman and van Dam in 1970
vanDam:1970vg . When some parameters of the theory reach critical values, then
the limits are discontinuous and the resulting theory can be very special. Of
course, originally Veltman discovered these discontinuities in massive vector
theories when taking the mass parameter (which from the nature of things is
dimensionful) to zero, which changes the character of symmetries present in
the theory, and local gauge symmetry arises. This new symmetry and new
dynamical situation is not possible to predict, being always within massive
vector models. Of course, there we also have the discontinuities in the number
of degrees of freedom, namely, three in the massive case and two in the
massless situation with enhanced gauge symmetries. One may think that it is
the gauge symmetry which kills the propagation of one longitudinal mode of the
massive vector gauge boson. In the case when $m^{2}=0$, the dynamics are
different and there is an enhancement of symmetries as well. The perturbative
spectrum is different and also the quantum correlation functions differ. One
also understands well the reason behind this case, because then local gauge
symmetry is born. However, one could view sending the mass parameter to zero
as a very crude procedure. Similar considerations in the gravitational setup
were also given to transition: massive gravity into massless gravity, first by
Veltman, van Dam, and Zakharov. The discontinuities in that case bear their
names. The reasoning is exactly the same as in the case of massive and
massless gauge theories analyzed already by Veltman. In the gravitational
context, the local enhanced symmetry of the massless gravitation here kills
three “longitudinal” states of the previous massive spin-2 graviton, morphing
it into a massless spin-2 graviton with only two polarizations.

We emphasize that in the case of various limits of HD theories, and in
particular of the limit $\alpha_{R^{2}}\to 0$, the situation is more delicate
since the coupling parameter here is dimensionless. Hence, its limit is not as
rough as for a dimensionful parameter. One actually may expect that these kind
of limits are better behaved and are continuous. This is still not the case
for the transition to conformal gravity from generic HD gravitational theory,
since the number of degrees of freedom changes abruptly, that is, this
symmetry, when present, modifies the perturbative spectrum of active degrees
of freedom. This could be viewed as a more subtle form of the Veltman
discontinuity; however, it is still a discontinuity. We see that, in
principle, it is possible to distinguish two types of Veltman-like
discontinuities. When the transition from scale-invariant HD gravity to
conformal gravity occurs, we see a softened version of discontinuity. The more
strong is when we modify the HD gravity from (25) by adding the dimensionful
parameter, as in with the Einstein–Hilbert term. The situation with both these
limits is not so clear because one time they are treated as technical tricks,
but the other time they change the counting of degrees of freedom and
interfere with symmetries of the models in question. One still sees that the
limit to conformal gravity $\alpha_{R^{2}}\to 0$ is very special and delicate
and should be approached with great care, as the last one in the computation
procedure, where the order of different limiting procedures matters.

This unpleasant situation with the problems of definition of spectrum of
perturbative modes around flat spacetime happens also for conformal gravities
in higher dimensions of spacetime, and here, the case of the $C^{2}$ gravity
from $d=4$ is not an exception. In general, even in dimensions $d$ of
spacetime, the Lagrangian of conformal gravity may be found in the schematic
form

$\alpha_{C^{2}}C\square^{(d-4)/2}C+O\left(C^{3}\right),$ (27)

while the Lagrangian of scale-invariant respective HD and quadratic (in
curvatures) gravity there has the similar form

$\alpha_{C^{2}}C\square^{(d-4)/2}C+\alpha_{R^{2}}R\square^{(d-4)/2}R.$ (28)

However, the same problems with limits (such as $\alpha_{R^{2}}\to 0$),
discontinuities, disagreements in number of active perturbative degrees of
freedom, and resolutions of poles in the propagators are met in this higher-
dimensional setting with only modifications of different counting of degrees
of freedom in different dimensions.

All the arguments above again and again confirm that the theories with local
conformal symmetry have to be treated specially and no naive limiting
procedures can reproduce their results. Since the symmetry is the mathematical
embodiment of the beauty of nature, this simply means that theories with local
conformal symmetries are discontinuous from any other scale-invariant HD
theories in (25). This fact of discontinuity is true, but it makes the
conformal gravitational theories even more beautiful.

Let us repeat that the conformal gravity is special and is not obtainable by a
continuous limit from a general Stelle theory, when $\alpha_{R^{2}}\to 0$.
There is an enhancement of symmetries in this critical point, where the
$R^{2}$ term is not present for Stelle four-derivative gravitational theory.
This is an enhancement to full local conformal symmetry on the classical
level, and on the first loop quantum level (up to the issue of conformal
anomaly). This theory with enhanced symmetry should be treated specially and
differently to generic HD theories. First, the FP method for fixing conformal
gauge symmetry (with proper conformal gauge-fixing and additional conformal
sector of ghosts fields) should be used in conformal gravitational models. In
such a way, the quantization procedure is performed as conformally covariant
as possible and with due care given to local conformal symmetry. Finally,
conformally invariant regularization procedure should be employed to define
quantum theory, especially to take care of perturbative UV-divergences. If all
these requirements are satisfied then we can announce an achievement of a
construction of quantum theory with local conformal symmetry conserved in
field theory models. The physical predictions in such quantum theories
preserve the underlying local conformal symmetry as much as is possible and
dictated by the quantum nature of the model. The last is shaped and controlled
exclusively by nature and completely independently of what regards the
formalism used by theoreticians to describe the nature and to perform the
artificial acts of quantizations or regularizations within given models.

##  5 Fradkin–Tseytlin Approach to Pure Conformal Gravity in $d=4$

In the context of a limiting procedure from generic HD gravitational theories
in (25), one must notice the paper by Fradkin and Tseytlin from 1982 FT1 .
They worked in the four-dimensional setup and analyzed perturbative one-loop
UV-divergences of some HD theories generically described by the action in
(25). Their computation is fully well-defined for the case when
$\alpha_{R^{2}}\neq 0$ and $\alpha_{C^{2}}\neq 0$. To perform the covariant
quantization process they use the general FP method for diffeomorphisms, as
elucidated at length before. Here, this method was applied to higher-
derivative gravitational theories. For extracting UV-divergences, they use
general diffeomorphism, preserving regularization of QFT of the HD
gravitational model. Their computation gives correct results in the generic
case when $\alpha_{R^{2}}\neq 0$ and $\alpha_{C^{2}}\neq 0$. However, as one
of the additional developments, they wanted also to consider the conformal
Weyl gravity as a special case of HD theories and also as the limit of HD
theories, when $\alpha_{R^{2}}\to 0$. Of course, the authors were aware of the
discontinuity present in such a limit. However, they also knew that the model,
in the case when precisely $\alpha_{R^{2}}=0$, has enhanced symmetries, and
this new symmetry, on the classical level before quantization, was the local
conformal symmetry of conformal Weyl gravity.

###  5.1 The Limit $\alpha_{R^{2}}\to 0$

The strategy that authors of FT1 ; FT2  followed was the following. Knowing
that the conformal symmetry was present at the classical level of the theory
with $\alpha_{R^{2}}=0$, they also expected it to be present in some unbroken
form on the quantum level, here analyzed to the level of one-loop accuracy.
This assumption allowed them, as we will explain later, to use the power of
conformal symmetry at the end of their computation for clearing the situation
with counterterms of the UV-divergent one-loop effective action. As a matter
of fact, these counterterms in $d=4$ are all globally scale-invariant for
generic HD gravitational theories. They are, namely, of three possible forms
here; $R^{2}$, $C^{2}$, and $E={\rm GB}$, where the last one is a Gauss–Bonnet
scalar term.

As analyzed before, the Weyl squared term, given explicitly by

$C^{2}=C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-2R_{\mu\nu}R^{\mu\nu}+\frac{1}{3}R^{2},$
(29)

when properly densitized to the form $\sqrt{|g|}C^{2}$, is a conformal
invariant. That is, we have for infinitesimal conformal transformations
(denoted below by $\delta_{c}$), which only matter here, that

$\delta_{c}\left(\sqrt{|g|}C^{2}\right)=0.$ (30)

On the other hand, the last invariant is topological and is known as the
celebrated Gauss–Bonnet scalar (or as an Euler density $\sqrt{|g|}E$ in a
densitized form). Its expansion, in other terms quadratic in standard
gravitational curvatures, reads,

${\rm
GB}=E=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}.$
(31)

Since it is a topological term (depending only on the boundary of spacetime),
then any of its bulk variation carried out in the volume of spacetime (in
particular the conformal infinitesimal one $\delta_{c}$) vanishes, that is, we
have

$\delta\left(\\!\int\\!d^{4}x\sqrt{|g|}{\rm GB}\right)=0$ (32)

as an identity of variational four-dimensional calculus.

Finally, the last invariant with respect to diffeomorphisms (scalar) out of
the group of three local invariants in curvature in quantum gravity, that is,
quadratic in the curvature, is simply the square of the Ricci scalar $R^{2}$.
Here, one notices an interesting thing pertaining to conformal transformations
of this invariant which is quadratic in gravitational curvatures (similar to
Riemann tensor, Ricci tensor, and Ricci scalar). This term is globally
conformally invariant in $d=4$, i.e., it is scale-invariant because its energy
dimension as the operator is ${\rm dim}\left(R^{2}\right)=4$. The coupling
coefficient in front of it in the action in (25), this $\alpha_{R^{2}}$, is of
course a dimensionless parameter in $d=4$ spacetime dimensions. This fact is
common also for all the other invariants considered above (in (29) and (31)),
and this happens precisely because all these invariants contain four
derivatives on the metric tensor and this number coincides here with the
dimension of spacetime. Hence, the $R^{2}$ term, $C^{2}$ term in (29), and the
${\rm GB}$ term in (31) are properly scale-invariant terms when densitized on
the level of action functional. As we have seen above for the last two, we
have upgrade of these symmetry invariance properties to the full invariance
under local conformal symmetry understood in the GR framework.

For the first term $R^{2}$, however, the situation is different and its
symmetry invariance obtains some promotion but not to the full conformal
invariance. Namely, the $R^{2}$ term is invariant in $d=4$ dimensions under
so-called restricted conformal transformations, that is, the transformations
from (9), satisfying the additional condition that

$\square\Omega(x)=0,$ (33)

which is a kind of GR-covariant wave equation (GR-covariant d’Alembert
equation) for the parameter of the conformal transformations $\Omega(x)$. This
means that this parameter $\Omega=\Omega(x)$ can be spacetime-dependent here
(already this is more than a case of invariance under global transformations
when we have that $\Omega={\rm const}),$ but this dependence cannot be
arbitrary here. It must satisfy an analogue of the wave equation, which in
Euclidean signature would require the $\Omega$ field again to be constant.
Instead, in the Minkowskian signature, this wave equation for $\Omega(x)$ does
have some non-trivial spacetime-dependent solutions. This signifies that the
condition of restricted conformal invariance is more general than global scale
invariance in GR with constant parameter $\Omega={\rm const}$, but still it is
less than a general invariance under fully local conformal symmetry with
completely arbitrary spacetime dependence in $\Omega(x)$.

One can easily see this fact by recalling the following properties of
conformal transformations of the Ricci scalar. For full generality, we write
them in arbitrary dimensions $d$. We have explicitly, that under local
conformal transformations,

$R\to
R^{\prime}=\Omega^{-2}\left(R-2(d-1)\square\ln\Omega-(d-2)(d-1)\nabla_{\mu}\ln\Omega\nabla^{\mu}\ln\Omega\right)$
(34)

and then using the Leibniz property of the box operator: $\square
f^{2}=2f\square f+2\nabla^{\mu}f\nabla_{\mu}f$, and the fact that
$\nabla_{\mu}\ln\Omega=\Omega^{-1}\nabla_{\mu}\Omega$, we rewrite above as

$R\to
R^{\prime}=\Omega^{-2}\left(R-2(d-1)\Omega^{-1}\square\Omega-(d-4)(d-1)\Omega^{-2}\nabla_{\mu}\Omega\nabla^{\mu}\Omega\right).$
(35)

Now, exactly in $d=4$ (and only in $d=4$, neglecting the degenerate case of
$d=1$ where $R=0$ automatically), we see that the last term in the bracket in
the above formula disappears, and we are left with the following simple local
conformal transformation law in $d=4$, that

$R\to R^{\prime}=\Omega^{-2}\left(R-6\Omega^{-1}\square\Omega\right),$ (36)

which indeed shows that for restricted form of conformal transformations
satisfying that $\square\Omega=0$, we obtain that the Ricci scalar transforms
fully covariantly under such local conformal rescalings. In actuality, one
sees that this is the special property of $d=4$ spacetime dimensions, since in
other dimensions even restricted conformal transformations do not make Ricci
scalar transform conformally covariantly because of the second term in (35)
multiplied by the factor $(d-4)$ and by a mixed term $(\nabla\Omega)^{2}$.
From this, one easily concludes the transformation law

$\sqrt{|g|}R^{2}\to\left(\sqrt{|g|}R^{2}\right)^{\prime}=\sqrt{|g|}\left(R-6\Omega^{-1}\square\Omega\right)^{2},$
(37)

which shows that, indeed, for restricted conformal transformations, the
densitized term $\sqrt{|g|}R^{2}$ is (partial) conformal invariant.

###  5.2 Final Conformal Transformation

Equipped with the above knowledge, we can discuss the trick and the method
used by Fradkin and Tseytlin to obtain, a bit heuristically, the UV-
divergences of conformal Weyl gravity in $d=4$ dimensions. The computation for
the $\alpha_{R^{2}}\neq 0$ case showed that, generally, the $R^{2}$ term is
present in UV-divergences, and this term is not conformally invariant as a
counterterm, when we mean full arbitrary conformal transformations. Therefore,
its presence violates the assumed conformal symmetry at the one-loop level. It
would be desirable for consistency of the theory to make it vanish. We explain
below what was performed to achieve this.

In an HD theory from (25), the one-loop UV-divergent part of the effective
action when the limit $\alpha_{R^{2}}\to 0$ is taken explicitly reads,

$\Gamma_{{\rm div}}^{{\rm
1-loop}}=-\frac{1}{(d-4)(4\pi)^{2}}\\!\int\\!d^{4}x\sqrt{|g|}\left(\frac{133}{20}C^{2}+\frac{5}{36}R^{2}-\frac{196}{45}{\rm
GB}\right),$ (38)

where we effectively used the dimensional regularization scheme (DIMREG), and
whole divergences are written as singular expressions in the deviation from
$d=4$ spacetime dimensions. As obvious from above, the $R^{2}$ term in this
naive procedure is generated at the one-loop level; therefore, conformal
symmetry seems to be broken by quantum corrections.

Assuming, in contrary and _a posteriori_ , that the local conformal symmetry
was present also on the level of the first quantum loop, the authors decided
to, after the computation, take the limit $\alpha_{R^{2}}\to 0$ in a naive way
and eventually performed a compensating additional conformal transformation,
as in (9). They simply wanted to exploit the freedom of making arbitrary post-
computation conformal transformations on the results of the non-conformally
covariant calculation. The parameter of such a conformal transformation
$\Omega(x)$ could be chosen at one’s will. The authors chose such $\Omega(x)$
that $R^{\prime}=0$ in (36), that is, the result of final conformal
transformation completely nullifies the presence of the square of the Ricci
scalar $R^{2}$ counterterm. Exactly this term before was spoiling the
conformal invariance of the full set of counterterms. In this way, and somehow
by hand manipulations, the non-fully conformal counterterm $R^{2}$ is
completely cancelled out, so the problem with the desired and originally-not-
present conformal invariance of the UV-divergent action in conformal gravity
at one-loop is solved.

For consistency, the following question then arose: Is it possible to find
solutions for the spacetime dependence of the $\Omega(x)$ parameter of
conformal transformations in (9), such that indeed we find that in the result
$R^{\prime}=0$? The solution of the related equation in $d=4$,

$R-6\Omega^{-1}\square\Omega=0$ (39)

or to an equivalent equation,

$\left(\square-\frac{R}{6}\right)\Omega=0$ (40)

exists and it is given for $\Omega$ by

$\Omega=\Omega(x)=1+\left(\square-\frac{R}{6}\right)^{-1}\frac{R}{6}.$ (41)

In the last formula, we performed some formal manipulations with treatment of
the operator $\square-\frac{1}{6}R$ as completely algebraic, although of
course it is a differential operator, and taking its inverse requires some
additional care. One can convince oneself that the solution for $\Omega$ in
(41) satisfies the equation (40), when one notices that
$(\square-\frac{1}{6}R)$ when acting on $1$ gives $-\frac{1}{6}R$ as the
result of the operatorial linearity, and moreover
$\left(\square-\frac{1}{6}R\right)^{-1}$ is treated here as the operatorial
inverse of the operator in the last parentheses. In particular, to define the
inverse of such shifted GR-box covariant operator by possibly spacetime-
dependent Ricci scalar, understood here as an arbitrary scalar field $R(x)$ on
a curved manifold, one must resort to operating with Green functions of the
operator since only then can we sensibly speak about the inverse on the
general curved backgrounds. Moreover, the last formula in (41) is valid only
when the metric $g_{\mu\nu}$, whose Ricci scalar $R=R\left(g_{\mu\nu}\right)$
appears there, is asymptotically flat.

Therefore, one concludes that the solution for $\Omega$ in $d=4$ always exists
for any form of the spacetime-dependent Ricci scalar $R=R(x).$ However, the
price is the resulting strong non-locality in the expression for the
$\Omega=\Omega(x)$ in (41), which is the parameter of the conformal
transformation that has to be performed at the end to bring back the conformal
invariance to the theory on the level of one loop. This is how the authors of
FT1  recover, at the end, conformal invariance in the model under studies on
the quantum level. It is well understood that this procedure looks very _ad
hoc_. This is so because the formalism they used was suitable only for dealing
with diffeomorphisms as local symmetries, and the regularization that they
used was only preserving this last symmetry. The care was not exerted to
properly treat the local conformal symmetry and to gauge-fix it and to add
appropriate FP determinant and FP ghosts for conformal symmetry. At the end,
the authors performed quite arbitrarily-looking conformal transformations
somehow to cure their results obtained in the limit $\alpha_{R^{2}}\to 0$. For
this, they had to know _a priori_ , but not _a posteriori_ , that conformal
symmetry survives on the quantum level. Somehow, they knew it and understood
why the conformal symmetry is still present on the level of one-loop
counterterms and they worked out an ingenious solution and procedure to
improve on their non-conformally covariantly-looking results. Eventually, they
were able to show that, indeed, the conformal invariance was preserved on the
level of one-loop divergent effective action.

As emphasized already a few times, it seems that the authors of FT1  assumed
what they wanted to prove by explicit computation, that is, that the conformal
symmetry is not violated on the first-loop quantum level. _A posteriori_ ,
they proved that their initial assumption was indeed correct, but in this way
they entered into some kind of logical vicious circle. Somehow, we need an
argument that does not rely on the presence of conformal invariance before it
is explicitly checked that it is there. This argument is given in the form of
analysis of the conformal anomaly at the one-loop level, that the same authors
also discussed in later publications. We will discuss it also below. Here, we
want to remark that all the considerations of the presence or absence of the
conformal symmetry were considered so far only on the level of the divergent
effective action, and the issue of conformal anomaly (CA) and its form were
completely neglected for these aspects. So far, we have just analyzed whether
the counterterms are conformally invariant, while, as emphasized earlier, the
mere existence of them breaks full conformal symmetry due to presence of non-
vanishing beta functions for couplings, and thus also of non-trivial RG flow
and of non-trivial scale dependence of various correlation functions.

The results from the limit $\alpha_{R^{2}}\to 0$ did naively contain the
$R^{2}$ term, which is dangerous for full local conformal symmetry, but these
were only incomplete results since additional care had to be given to them. By
performing final conformal transformations, Fradkin and Tseytlin were able to
reduce to zero such non-conformal counterterms; however, the parameter of such
a transformation was expressed as highly non-local function in (41). They
could still perform the conformal transformations on the form of counterterms
as the last operation, because the symmetry they acknowledged earlier had to
be present also on the quantum level since it was present also on the
classical level before the quantization procedure. Implicitly, they assumed
that the quantization act does not destroy conformal symmetry. They just used
the freedom to select a conformal gauge, although one has a strong impression
that their approach could be more clear since they knew only _a posteriori_
that local conformal symmetry must have been present on the level of the first
quantum loop in this theory. The fact that the solution for $\Omega$ contains
this apparent non-locality in local Weyl conformal gravity in four-dimensional
QFT framework could be quite disappointing, and one could wonder whether it is
possible to perform all such procedures better and in a more clean way.
Moreover, the parameter $\Omega$ must be always spacetime-dependent since
constant values do not change the form of the Ricci scalar $R$ according to
(36), because we always have that $\square\Omega=0$ for $\Omega={\rm const}$.

When one is finished with the dangerous conformal symmetry-violating $R^{2}$
term in UV-divergences, then this is not the end of the story because, as
authors say, the contribution of two scalar degrees of freedom has to be
subtracted from the results in (38) without $R^{2}$ term there. These are
assumed to be two minimally coupled to gravity scalar modes on a general
background. At the end, the form of UV-divergences in Weyl conformal gravity
in $d=4$ spacetime dimensions regularized in the DIMREG scheme reads,

$\Gamma_{{\rm div}}^{{\rm
1-loop}}=-\frac{1}{(d-4)(4\pi)^{2}}\\!\int\\!d^{4}x\sqrt{|g|}\left(\frac{199}{30}C^{2}-\frac{87}{20}{\rm
GB}\right).$ (42)

The results in the formula above should be understood as final ones. No
additional conformal transformation or subtraction of some contributions
should be performed on them.

One also wonders whether the final conformal transformation nullifying the
$R^{2}$ term can be reversed. It is easily acceptable that one obtains as the
result of conformal transformation that $R\neq 0\to R^{\prime}=0$. However,
here is visible apparent irreversibility of the conformal transformation
because one does not obtain anything from $R^{\prime}=0$ as the starting
point, since zero transforms to zero under homogeneous transformations. Of
course, this is only apparent since by inverse conformal transformations which
are achieved with $\Omega^{-1}$ parameter, one obtains from $R^{\prime}=0$
back to the case $R^{\prime\prime}=R\neq 0$, or one can even perform a new
transformation with $\Omega\neq 1$, and then from $R^{\prime}=0$ one will find
oneself in the spacetime where $R^{\prime\prime}\neq 0$. One, therefore, has a
strange impression that the power of conformal transformations in a theory
with conformal symmetry on the quantum level was only used instrumentally by
authors of FT1  to reach the assumed goals and, for example, after doing this,
one is forbidden to use it anymore, or to use it for reversing its effects. Of
course, such a situation with understanding of the symmetry in the theory is
not satisfactory, and one asks for a natural formalism which makes such
transformations obvious, and all of them are allowed to be performed provided
the symmetry is realized there on any level of the theory.

In other words, we can say that the $R^{2}$ term in UV-divergences of the
theory is conformal gauge-dependent, that is, by a choice of some conformal
gauge one can eliminate it but _a priori_ it is there. We choose a conformal
gauge by performing arbitrary conformal transformation in the theory. Is the
$R^{2}$ term there or not?—This statement is not universal and it is actually
gauge-dependent in conformal gravity. The value of the $R^{2}$ beta function
is not universal either. Therefore, in the description preserving conformal
symmetry we shall not speak at all about this term, and the issue related to
its presence or non-existence is not a physical issue, it is just another
example of the gauge artifact statement, here due to conformal gauge.

If one looks at the difference between formulas (42) and (38) (the last one
obtained as the limit in the case $\alpha_{R^{2}}=0$), one sees that the
correcting contribution of two scalars is essential and that this together
with the resolution of the problem of the $R^{2}$ counterterm signifies why
the simple naive limit $\alpha_{R^{2}}\to 0$ does not work for the case of
quantum conformal gravity. These are all post-computational procedures that
one has to perform after the naive limit $\alpha_{R^{2}}\to 0$ is taken. They
constitute the theoretical reasons (or the problems within the formalisms of
simple QFT of gravitational interactions with only diffeomorphisms symmetries)
why the limits in question are discontinuous and why the case of conformal
gravity is very special one.

###  5.3 Subtraction of Two Scalars

There could be various attempts to explain the role of the mysterious two
scalar degrees of freedom that have to be subtracted for the last corrections
to the theory. The possible questions are why they have to be subtracted, not
added, why they are two, and why scalar degrees of freedom, and not, for
example, one massless vector (spin-1) or tensor (spin-2), each of the last two
coming with two helicity states as well. Finally, one could ask, accepting
that they are scalars, why they have to be minimally coupled to a curved
gravitational background, and not conformally coupled to gravity. The last
option looks to be the most reasonable one, especially in the framework of
conformal gravity, where conformal symmetry of the whole physical system
(gravity + matter) on the quantum level is the thing we care so much about
here.

A possible explanation of the problem with two scalars may come from analyzing
the spectrum. Exactly two degrees of freedom are the difference between eight
and six perturbative degrees of freedom (as analyzed around flat spacetime
background), respectively, for generic HD gravitational theories and for Weyl
conformal gravity. Hence, if one wants to obtain six degrees of freedom one
would have to subtract two degrees of freedom from all degrees of freedom
present in HD gravity with both $C^{2}$ and $R^{2}$ terms. The arithmetic here
agrees. However, there could be some problems with this interpretation. First,
as we discussed before, this way of counting degrees of freedom is not
completely unambiguous and one may be tempted to assume number of six degrees
of freedom also in generic HD gravitational theory without the
Einstein–Hilbert term added. Secondly, when one looks at the spectrum of
generic HD theory, one does not see that there are two more scalar degrees of
freedom compared to Weyl gravity. One sees only one physical scalar degree of
freedom in generic HD theory (with standard counting to give $8=2+5+1$ degrees
of freedom), while in conformal gravity, there are no scalars at all101010In
actuality, to twist the story even more, we could add that proponents of
counting of six degrees of freedom in massless scale-invariant HD gravity (but
not in conformal gravity!) would see that there indeed are two perturbative
scalar degrees of freedom (with spin-0).. The actual reshuffling of degrees of
freedom (from eight to six) is intricately more complicated: first, one must
subtract one scalar d.o.f. and then later divide five d.o.f.s from massive
spin-2 field into two helicities of massless spin-2 and another two helicity
states of massless spin-1 (fields present in conformal gravity), and finally
one has to remove the last remaining degree of freedom. This game with various
Lorentz representations of perturbative degrees of freedom would suggest to
try something with one scalar, massless vector or massive spin-2 mode.
However, it is incorrect to add or subtract here, for example, the
contribution to one-loop divergences of traces of a vector (spin-1) and the
final results work only when precisely two scalars minimally coupled are
subtracted. Hence, this reinterpretation of subtracting contribution of two
scalars can be, at most, only viewed as a heuristic tentative explanation.

On the other hand, one could also come up with a different idea that these two
scalars are conformal ghosts. This would explain why, at the end, Fradkin and
Tseytlin had to take them into account when limiting to Weyl gravity (with
enhanced conformal symmetry). These fields obviously were not present in
generic HD gravities, but when one moves to conformal theory, one has to
gauge-fix this symmetry and also supply proper ghosts (depending on the
character of interactions, they have to be either FP ghosts of new local
symmetry or third ghosts if HD terms are present, or both). It is true that
the authors of FT1  performed an additional simple conformal gauge-fixing,
requiring vanishing of the trace $h=0$. This performs the first part of the
job of treating with due care the eminent conformal symmetry here. However,
the next part is to deal with respective ghost fields of conformal symmetry,
and, moreover, what is reassuring here is that since they are of Grassmannian
nature (as all quantization ghosts are), then to add their contribution means
effectively to subtract the contribution of “normal” particles. Hence, this is
consistent with the idea that to reach conformal gravity, one has to add
contribution of conformal ghosts, or, in a sense, subtract contribution of
normal particles.

As it could be explained thoroughly in the next publications on this matter,
these conformal ghost fields are here in four-dimensional conformal gravity of
the type of third ghosts exclusively, namely, they are not FP conformal
ghosts. However, what is more important here is that they are of scalar
character, and they do not carry any Lorentz index. Moreover, as will be made
clear in the next section, they are just minimally (covariantly, similar to in
GR) coupled to the background and not conformally coupled, since such is the
feature of the FP quantization procedure. This fits very well with the
description of subtraction of two scalars; however, again, there could be some
problems with such an interpretation. First, ghosts used during the covariant
quantization procedure (that is, FP ghosts or third ghosts) are typically not
counted for active perturbative degrees of freedom around a given background,
hence they would not entail any change in the number of degrees of freedom
between the two theories (limiting HD gravity and conformal gravity). This is
in clear contradiction with the counting suggesting that the change is $8\to
6$, while it is also in perfect agreement with other counting, where we have
advocated before that there is no change and we have $6\to 6$. One cannot also
say that potential additional two degrees of freedom, as present in the
standard counting of degrees of freedom of HD gravity, are these two conformal
ghost modes (counted negatively). The ambiguity and the potential problem with
this interpretation remain.

However, after all, the role of such ghost contributions (both whether these
are FP or third ghosts) is not to change the number of perturbative active
(physical to some extent) degrees of freedom, but to provide the correct
expression for the partition function. For example, from such an expression,
one could, at the one-loop level when expressed via determinants of some
differential operators, obtain the one-loop partition function and, finally,
read from it the number of degrees of freedom. This number typically agrees
with the number obtained via counting of constraints and other similar
canonical methods (and, for example, it gives eight degrees of freedom in
scale-invariant HD gravity). However, on the level of such an expression
written using determinants in various subspaces, it is not evident to
understand the origin of various factors and functional traces there. As
emphasized and, for example, explained in the case of conformal gravity, one
can find there some various auxiliary elements of the theory used for
quantization and for writing path integrals, such as contributions from
Jacobians of change of field variables under path integrals, from gauge-fixing
functionals, and also from FP and third ghost determinants. One does not find
there contributions clearly assigned to modes appearing in the perturbative
spectrum of the theory around flat spacetime, such as no clear origin with
massless spin-2 or spin-1 fields. Eventually, from one-loop partition
function, one finds the correct total number of degrees of freedom (including
also these spin-2 and spin-1 massless states in conformal gravity). However,
eventually, there is no sign that this counting manifestly takes FP ghosts or
Jacobians into account. It only has to take them in intermediate levels of
computation for the whole consistency of the procedure.

To answer the question of why there are two scalars, one has to recall a few
facts. First, it is well known that the third ghost is just one single real
field. This would already look like a contradiction to our counting, since we
stated that there are two scalar contributions that need to be subtracted, but
secondly, one has to pay enough attention to the fact that the mentioned
contributions of minimally coupled real scalars are understood as coming from
the following two-derivative scalar action coupled to curved spacetime
background:

$S_{{\rm sc}}=\frac{1}{2}\int
d^{4}x\sqrt{|g|}g^{\mu\nu}\left(\partial_{\mu}\phi\right)\left(\partial_{\nu}\phi\right)=-\frac{1}{2}\int
d^{4}x\sqrt{|g|}\phi\square\phi,$ (43)

where we do not write any scalar potential term $V$ (containing mass terms or
self-interactions) since it is well known that the terms in $V$ do not at all
contribute to divergence proportional to $R^{2}$ and ${\rm GB}$ term. This is
undoubtedly true at the one-loop order and for dimension-four counterterms in
the effective action, which could receive contributions only from
dimensionless couplings, but, for example, here they do not receive any
contribution from quartic self-interactions of scalars. Only the form of the
kinetic operator (precisely, how many derivatives there are) and the
multiplicities of scalars (how many scalars there are) matter for this
computation. There is a slight dependence of the contribution to the $C^{2}$
counterterm on the non-minimal coupling $\xi$ with gravity (that is, with a
possible non-minimal term $\xi R\phi^{2}$, which is so-called coupling to a
scalar curvature of spacetime), but these are all neglected for our purposes.
One could say that, at the end, our model is very poor, since it describes
here two massless scalar fields with no interactions, no self-interactions,
and even without mass parameters, just coupled simplistically and minimally to
the background geometry. Even the normalization of the scalar field here is
not an issue for UV-divergences, but here we stick to the canonical one.

One has to compare the above with the contribution from the $G_{c}$ matrix. To
make the story short, the suitable operator in $d=4$ is of the form
$G_{c}=\square^{2}$ and this acts between two real scalar third ghosts for
conformal symmetry $b_{c}$. Hence, due to fractionalization property of
functional traces of a logarithm of such an operator as $G_{c}$, one sees that
on the level of the trace of the logarithm, its contribution precisely agrees
with two contributions from one scalar box $\square$ in a simple scalar field
representation in the last part of the formula (43). This explains the
presence of two scalars. In more generality, one could say that, in general,
even dimensionality of spacetime $d$, the number of simple two-derivative
scalars, each with action in (43), the contribution of which have to be
subtracted, has to be equal to half of the dimension $\frac{1}{2}d$. This
assertion can place our theoretical explanation in verification, for example,
in six-dimensional conformal gravity models.

Within this second interpretation, we have a confirmation for why these two
contributions have to be subtracted, but not added, and why they are scalar
degrees of freedom, and not massless vectors or symmetric rank-2 tensors. We
also see why they are minimally coupled to GR-covariant background and why
they are two scalars. However, as remarked before, the whole existence or need
for third ghosts (both of diffeomorphism symmetry or of local conformal
symmetry) depends on the choice of the background. As we emphasized previously
with some clever choice of partially covariant weighting functional $G_{c}$,
their contribution can be safely eliminated and they can be forgotten on the
quantum level. Such is, for example, the situation around flat spacetime,
when, in the gauge-fixing functional $G_{c}$, one uses only partial
derivatives, and from them builds scalar d’Alembertian operator
$\partial^{2}$, and on the contrary, one does not use background-covariant
derivatives from GR to construct a proper GR-covariant box operator
$\square=\nabla^{2}$. Then, in such circumstances, there are no conformal
third ghosts at all and so their contribution also does not exist and then
this interpretation breaks down.

One could try to rescue this idea that the two scalars are third ghosts for
conformal symmetry by claiming that for GR-covariant results of one-loop
divergences obtained within some covariant framework, such as within the
method of covariant Barvinsky–Vilkovisky traces, one ought to use only the
form of the weighting functional $G_{c}$, which is GR-covariant, so it must be
built using the covariant d’Alembertian operator $\square$. Otherwise, one
could be afraid of losing the general covariance of the results at the end,
but they are guaranteed to be provided such that in the whole process of
computation of them there are no steps which explicitly and manifestly violate
the general covariance. Using only partially covariant $G_{c}$ functional (or
in a sense forgetting about the third ghosts contributions) is not in line
with such a formalism, and the results obtained by this oversightare, of
course, incorrect. Hence, one can conclude that third ghosts must be
remembered in all GR-covariant formalisms of computation, while one can still
safely forget about them around flat spacetime and when working with Feynman
diagrams.

However, here one must remember that the original computation due to Fradkin
and Tseytlin was performed just in manifestly covariant framework and using
background field method (BFM), when they sum various contributions and they do
not use non-gauge-covariant results from some partial resummation of Feynman
diagrams. It is well known that the two methods at the end should agree for
final results for UV-divergences, which are proved to be expressed via gauge-
invariant and gauge-fixing-independent counterterms. One must remark that in
the covariant BFM formalism, as the Barvinsky–Vilkovisky (BV) trace technology
is, the partial results, although looking generally covariant (so in this
sense they are better looking than results of some subset of Feynman diagram
contributions, which are not gauge-invariant), are without any sense if
considered separately. Only their total sum has a clear physical meaning.
Therefore, one cannot say with a good meaning, for example, what the generally
covariantly written and unambiguous contribution of FP ghosts (or third
ghosts) is to total divergences. Namely, it is true that they contribute and
one usually should not forget about them, but their contribution alone is not
physical. One should not be deluded by intermediate results, which look
covariant, and would invite an interpretation of such partial results.
Therefore, one must remember third ghosts for local conformal symmetry, but,
again, their contribution alone is not very meaningful.

It is still a very interesting fact that their contribution (with all the
provisos mentioned above) precisely agrees with the discontinuity between
conformal gravity and a four-dimensional HD gravity after the limit
$\alpha_{R^{2}}\to 0$, thus realizing the subtraction of two scalar degrees of
freedom minimally coupled to background geometry. It remains yet to be seen if
such a discontinuity has any physical meaning, or it is just an artifact of
our clumsy way of approaching conformal gravity from generic HD gravity.
Therefore, the interpretation of two scalars as two third ghosts for conformal
symmetry in Weyl gravity is quite a plausible one, since it defends itself
quite well, and certainly it is worthy studying further. For this, one should
also better understand the issues related to conformal third ghosts.

Instead, here we summarize what is known about the contribution of two scalars
to UV-divergences. This does not presuppose that these scalar fields are real
or physical or whether they really live on a curved spacetime background.
Nonetheless, their contribution can be singled out (with all the provisos
above). Namely, two minimally coupled scalar fields (each with one degree of
freedom and with a two-derivative action from (43)) on a curved general
background contribute twice the following contribution to UV-divergences,

$\Gamma_{{\rm div},\,{\rm sc}}^{{\rm
1-loop}}=-\frac{1}{(d-4)(4\pi)^{2}}\\!\int\\!d^{4}x\sqrt{|g|}\left(\frac{1}{120}C^{2}+\frac{1}{72}R^{2}-\frac{1}{360}{\rm
GB}+\frac{1}{30}\square R\right).$ (44)

The last term above, proportional to $\square R$, was written just for
definiteness and we do not use it in the final main computation. The
contribution of two scalars in (42) can also be studied in the $R^{2}$ sector
of UV-divergences, although the final result is non-universal and it leads to
a conformal gauge-dependent coefficient $\frac{1}{9}R^{2}$, which was
justifiably neglected when writing the final formula (42).

###  5.4 Relation to Conformal Anomaly

When one speaks about UV-divergences at the one-loop level, one should not
forget to relate them to the celebrated conformal anomaly. Basically, on the
quantum level where there are UV-infinities and also resulting beta functions
of couplings, there is an RG flow, and Green functions do not show perfect
scale invariance. They depend on energy scale, and this dependence is dictated
by RG phenomena. This means that inevitably in such situation, the conformal
symmetry of the matter model (global group, as we discussed in Section 2.1) is
broken and we do not have a globally conformally invariant model on flat
spacetime. This is also confirmed by analyzing conformal anomaly of global
conformal symmetry in the setup of a curved geometric background (as explained
in Section 2.2). Then, the global group of scale transformations from GR in
some matter models (which were globally conformally invariant) may be broken
on the quantum level when one couples these matter theories to a non-trivial
gravitational background. As for global anomaly of conformal symmetry, this is
not a big deal in matter theories, where conformality on the classical level
might be just a lucky coincidence, and then one does not view the effects of
this anomaly as disastrous for the original matter theory. Simply, on the
classical (tree-) level before the quantization, the conformal symmetry was
present, but just after quantization of matter fields and matter symmetries
(which could be also local internal symmetries) due to quantum effects, this
symmetry is not there anymore and it does not constrain quantum dynamics
anymore. One loses some power and simplicity of the theory but then the
quantum physics enters and all correlations have to be tediously computed
according to rules of QFT (in distinction from rules of CFT). One cannot use
the rigidity of constraints placed on Green functions due to CFT models
anymore, since one is out of their realm.

Technically speaking, the conformal anomaly (CA) understood as an anomaly for
global conformal symmetry is equivalent to a trace anomaly. The last one
signifies that the trace of the energy-momentum tensor of the matter theory
does not vanish, although the theory on the classical level possessed scale
invariance or even conformal invariance (before this was coupled to gravity).
These last two requirements on the classical level imply that the classical
EMT111111The definition of the energy-momentum tensor (EMT) that one uses here
is the unambiguous Hilbert definition as the variational derivative of the
action with respect to metric fluctuations and properly de-densitized. is
trace-free ($T=0$) upon using EOM from the matter sector, and this is
completely tantamount to invariance of the theory under infinitesimal
conformal transformations understood in the framework of GR (as they were
studied in Section 2.2). Therefore, we end up in the situation that on the
quantum level, due to physical quantum effects (but not due to an act of some
uncareful quantization), one finds that the trace of EMT on-shell is not zero
anymore. For this, one has to use the quantum matter equation of motion
obtained from the full quantum effective action restricted to the matter
sector. One can specify the situation at the perturbative one-loop and then
one correctly expects that this violation of tracelessness condition (thus,
violation of scale invariance) is proportional to beta functions of the
theory, which in turn are proportional to UV-divergences. Such a simplistic
identification is no longer true at higher loop orders, but we can restrict
all of the below analysis to this case only.

To summarize, there is an anomaly because some condition known from the
classical theory (here, that $T=0$) is not present on the quantum level of the
theory anymore. If some symmetry is on the classical and was expected to be a
symmetry of the theory at all conditions, then its lack on the quantum level
is a true anomaly in a physical sense, and this tells us that maybe we had a
wrong expectation about the theory, and that theory with this symmetry is
simply anomalous and very sensitive to the quantum effects, or that the
original matter theory with conformal symmetry was not in a symmetric enough
situation to protect the conformal symmetry from violation at the quantum
level. Since, in the matter models, this conformal symmetry is not in a gauged
form, there actually is not a big problem with such an anomaly. One has to
simply accept that symmetries on the quantum level of the model are in the
poorer form, and, for example, conformal symmetry is not realized in the
quantum dynamics.

Still, in the matter models, but coupled to an external geometry, one expects
that the lack of conformal symmetry on the quantum level manifests itself as
the presence of non-trivial counterterms. Here, we can concentrate on UV-
divergences in the gravitational sector, but one can also consider them in the
matter part. For what we are going to say, this last restriction will be
completely sufficient. Therefore, we can define that conformal anomaly ${\cal
A}_{c}$ is exactly proportional to the $b_{4}$ coefficient of UV-divergences
of the theory at the one-loop of accuracy. Moreover, one can view it as proper
trace of the quantum EMT defined from the effective action, that is, we have

$T=g^{\mu\nu}T_{\mu\nu}=-\frac{2}{\sqrt{|g|}}g^{\mu\nu}\frac{\delta\Gamma_{{\rm
div}}}{\delta g_{\mu\nu}}=\frac{1}{(4\pi)^{2}}b_{4}.$ (45)

It is important to realize that in the above expression, the divergent part of
the effective action is taken in full generality as off-shell, so without
using quantum (or even classical) EOM. For matter models, the divergent
effective action there $\Gamma_{{\rm div}}$ is with all contributions when the
matter fields are taken quantum (so they run in the loops of Feynman
diagrams), while the gravity is treated as a classical external field to which
the quantum matter system is coupled (so graviton lines can be only external
lines of the diagrams). Moreover, in the above formula (45), a regulator which
was used to isolate UV-divergences was already extracted, so the expression
for $b_{4}$ (and also equivalently for $T$) is finite and not UV-divergent.

The $b_{4}$ coefficient of divergences is, of course, special to $d=4$
dimensions because then counterterms have the energy dimension equal to the
dimensionality of spacetime and couplings in front of them are dimensionless,
so naively we already have classical scale invariance. Its relation to the
divergent part of the one-loop effective action (part with classical scale
invariance of the action) reads,

$\Gamma_{{\rm div}}=\frac{1}{(d-4)(4\pi)^{2}}\int d^{4}x\sqrt{|g|}b_{4}.$ (46)

In the case of only gravitational counterterms that we are interested in, this
coefficient is related to the beta functions of couplings in the following
way:

$b_{4}=\beta_{C}C^{2}+\beta_{R}R^{2}+\beta_{{\rm GB}}{\rm GB}+\beta_{\square
R}\square R.$ (47)

One notices a few interesting facts about this expression for anomaly. First,
the coefficient $\beta_{R}$ is not universal since it is conformal gauge-
dependent (for example, in the background conformal gauge, its value is zero).
Secondly, the beta function of the total derivative term $\beta_{\square R}$
is known to be ambiguous on the level of the trace of EMT, but not on the
level of the $b_{4}$ coefficient. In the results we can forget about these
last two terms and consider only universal contributions to two beta
functions: $\beta_{C}$ and $\beta_{{\rm GB}}$. Once again, we remind the
reader that the presence of CA is not problematic for matter models, where
this symmetry does not show up in the gauged form on the quantum level. One
really sees, unambiguously, the presence of CA when, off-shell, the expression
for it is not proportional to matter EOM.

One can see the consequences of CA for the quantum matter theory not only on
the level of ultraviolet divergent expressions in correlation functions, but
also in finite parts of these correlators on the quantum level computed to a
given accuracy. For example, due to RG flow phenomena and within RG formalism
(which basically tells in the simplest way that physical results should be
independent on the arbitrary renormalization energy scale), one finds that the
finite terms in the semi-classical general off-shell action for gravity (when
quantum matter fields are integrated over) take the following form:

$\Gamma_{{\rm fin}}\supset\int
d^{4}x\sqrt{|g|}\left(\beta_{C}C\log\frac{\square}{\mu^{2}}C+\beta_{{\rm
GB}}{\rm GB}_{\log\square/\mu^{2}}\right),$ (48)

where, for simplicity, we wrote only contributions which are universal on the
level of the trace of EMT $T$ related to the $b_{4}$ coefficient. The non-
local insertion of the logarithm of the GR-covariant box operator is both
present between two Weyl tensors and also between Gauss–Bonnet invariant. The
last one explicitly rewritten takes the form

${\rm
GB}_{\log\square/\mu^{2}}=R_{\mu\nu\rho\sigma}\log\frac{\square}{\mu^{2}}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}\log\frac{\square}{\mu^{2}}R^{\mu\nu}+R\log\frac{\square}{\mu^{2}}R,$
(49)

which is in the direct analogy with the formula (31). One sees that these
contributions in (48) explicitly break conformal symmetry, despite that on the
level of divergent action, so without non-local logarithms of the box
operator, they entail completely conformally invariant counterterms. This
shows that in such a setting, the conformal symmetry is explicitly broken and
it is no longer there on the quantum level. As one confirms in (48), the
expressions vanish when the corresponding beta functions vanish. Only there we
are sure of quantum conformality present in some matter models. In the
expressions in (48), we see highly non-local functions, namely, of the
logarithm of the covariant box operator. Their origin is due to RG invariance
of the total effective action when the running of couplings is also taken into
account and where $\mu$ is a physical scale of renormalization which cancels
with the $\mu$-dependence of running coupling parameters for any physical
observable computed within the given model.

Thus far, we have only discussed some issues related to conformal anomaly when
gravity was the external dynamical classical field. When we use the same
procedure towards quantum gravity when it is both dynamical and quantum, and,
for example, propagates inside quantum loops (where there is finally
graviton’s propagator), and gravitons can be on internal lines of some Feynman
diagrams, then in conformal gravity (so, the theory with local conformal
symmetry), one has to be very careful since now this symmetry is in the gauged
form. If one sees an anomaly, this might be a sign of a just creation of
pathological, sick theory. This issue of how CA in conformal gravity may
create disastrous effects was discussed at length in various previous works
confan ; confan2 ; confan3 ; confreview . We will also discuss these issues
and potential resolution of them at full length elsewhere.

Now, if the conformal symmetry is in the gauged (local) form, then one can
adapt the discussion which was presented above to the total system composed of
matter and gravity when both components are quantum and dynamical. The
conformal symmetry was the defining local symmetry on the classical level of
gravitational theory, hence the issue with potential conformal anomaly for it
is very important and crucial to resolve satisfactorily. Moreover, local
conformal symmetry is instrumental in defining the spectrum of the theory, for
example, around flat spacetime, so any change with this would ruin the
spectrum and classification of irreducible Lorentz representations of modes
there. It would also destroy the counting of perturbative degrees of freedom
of the theory. All these problems show in other disguise as violation of
unitarity, not due to the HD character of gravitational theory, but due to
destroying one of the local gauge symmetries, here in this case of local
conformal symmetry. We will not comment on these perennial issues here, but we
will refer to the vast literature on this topic.

In the case of a total gravitational system where matter is coupled to
gravity, one has to analyze the total EMT of the full system. As is well
known, it is composed of two pieces—gravitational and matter parts. One also
notices that following Weinberg’s definition, the total EMT is zero on-shell
as a consequence of invariance of the total action (gravitational part and
coupled matter part) under GCT. Thus, on-shell we do not find any problem with
CA in the total system. This remark applies both to the classical and quantum
level, since in the last case, one can, for example, compute the effective EMT
from the full quantum effective action functional, and for this argument to
work, the gravitational theory or matter part alone do not have to be even
classically conformally symmetric. This is indeed a very robust argument valid
in any diffeomorphism invariant theory, which also preserves this symmetry on
the quantum level (so, under the condition that there is no quantum anomaly of
diffeomorphism symmetry). In any such theories on-shell, we find that the
trace anomaly $T$ vanishes. However, as usual in QFT, the situation off-shell
is also very important, and, actually, in these circumstances one can again
relate this trace $T={\cal A}_{c}$ to the coefficient $b_{4}$ of divergences
according to the last formulas (45)–(47). These formulas, also in the more
general case, with quantum gravitational part remain valid. Of course, the
respective beta functions now receive contributions from both quantum
gravitational interactions and the quantum matter sector.

In the total system interacting gravitationally, the issue of detecting
physical consequences from the expressions for $b_{4}$ is more complicated
than just a case of simple matter theory. First, one knows that even on a
classical level there exist some matter models for which the trace $T$ does
not vanish classically off-shell but only using matter EOM. Now, with gravity,
we have this twist where $T=0$ on-shell as a kind of tautology. One must
definitely analyze the situation quantum off-shell, but even if one finds
there that $T\neq 0$, this does not necessarily imply that there is a
conformal anomaly in the total system. Sometimes one finds for the total $T$
off-shell an expression which does vanish, then one is sure of the absence of
CA, or in the gravitational context, if the expression for it is proportional
neither to the gravitational EOM nor to the EOM from the matter sector. These
EOM should be derived from the respective quantum EOM originated from quantum
effective action. This is a conformal Noether identity for the total system
(matter + gravity).

One could also in parallel analyze the situation in six dimensions since then
the space for all terms in conformal gravity is bigger. It is interesting
viewing the situation with divergences in six-dimensional conformal gravity of
the type $C\square C+\ldots$ which still possesses the propagator around flat
spacetime. We expect only three conformally invariant counterterms, $C\square
C+\ldots$, $\left(C^{3}\right)_{1}$, and $\left(C^{3}\right)_{2}$, out of 10
possible terms in the action with six dimension operators nieuwe ;
Mistry:2020iex . There, we also expect that the conformal anomaly ${\cal
A}_{c}$ is described in a conformally covariant way such that ${\cal
A}_{c}=b_{6}$ and that only three terms, $C\square C+\ldots$,
$\left(C^{3}\right)_{1}$, and $\left(C^{3}\right)_{2}$, out of 10 possible
terms in the action with six dimension operators appear there. What about the
remaining seven terms? If all this formalism is correct they should be made to
vanish by just one conformal transformation in six dimensions with just one
parameter $\Omega(x)$. This is quite improbable in a general situation to
remove six counterterms (although they are related) by just one conformal
transformation, but it should be true in a general case when the reasoning
with conformal anomaly is correct. The UV-divergences after conformal
transformations should be described by conformally invariant terms only in
${\cal A}_{c}$, according to the general wisdom that terms in conformal
anomalies are written in a way that preserves conformal symmetry (despite that
they are in the anomaly heralding the disastrous breaking of this symmetry).

###  5.5 Summary of the Method

The results of all these above procedures, although maybe not deeply
justified, revealed to be true for Weyl conformal gravity, as it was checked
after using other methods of computation. The coefficient of the $R^{2}$
divergence indeed vanishes at the level of the first loop, and the two other
coefficients were correctly obtained in the limit $\alpha_{R^{2}}\to 0$: for
$C^{2}$ and for the Gauss–Bonnet divergences. This procedure was justified to
be used and to be a correct one. The potential general problem could be that
only diffeomorphism symmetry was treated with due care since quantization and
regularization procedures were performed in a way to preserve it. The
conformal symmetry was restored by hand using final conformal transformations,
where one actually admits that this symmetry was always there, even on the
quantum level, but maybe in its hidden form. Of course, one could act much
better and treat it well from the beginning, and then at the end, one would
not have to perform correcting conformal transformation. However, even if we
want to perform this rectification procedure, there would be a clear
justification that this is possible since it would be obvious in a more
manifest formalism that the conformal symmetry is present on the quantum
level. One would not have to perform these compensating final transformations
with parameters which are apparently strongly non-local functions of spacetime
points.

Finally, this procedure, as employed confidently by the authors, recovers
local conformal symmetry on the quantum level, although there were some hidden
assumptions and not _a priori_ justified steps in its rederivation. One could
ask how such procedure can be systematized, if at all, which transformations
we should perform at the end, and whether this works also to higher loop
level, or only at the level of the first loop. One may wish to work all the
time from the start within the formalism that preserves conformal symmetry in
the framework of covariant quantization of theories with local gauge
symmetries. This is exactly what we plan to explain in the future sequel
article of this paper.

##  6 Conclusions

In this introductory review article, we tried to present and extensively
motivate the need for a new quantization procedure. This method should be
designed to specially treat local conformal symmetry when this is present in
gravitational interactions introduced in the same way as Weyl did in 1918.
Local conformal symmetry is very special, as we emphasized in the introduction
to this paper and also in the motivations.

We discuss how to view it as a gauge symmetry both on the flat spacetime and
also on curved spacetime framework. The latter is generally best suited to
study fields and their transformations, as in the proper GR framework. Keeping
this in mind, we later viewed conformal symmetry as a particular example of
the general gauge theories. However, compared to other gauge symmetries, in
the minimal version, conformal symmetry comes with some differences. It does
not come with its own potential field, but it just uses the metric, somehow
borrowing it from the diffeomorphism group. This relation is another emanation
of the fact that the full conformal group is not a direct product of the
Poincaré group and the proper conformal factor (which consists of five
generators in $d=4$ spacetime dimensions). Moreover, we emphasize that the
transformation law for the metric field, which could be treated similar to a
charged matter for conformal symmetry, does not contain any differential
operators, and this is just an algebraic operation of multiplication. In this
way, the conformal gauge theory differs from any other local gauge theory with
standard potential fields, wherein the generators of infinitesimal gauge
transformations are generically differential operators.

This last observation has profound implications for the quantization program
of conformal symmetry in conformal gravity. Namely, this implies that in
simple conformal gauges, we do not need to supply Faddeev–Popov ghost fields
for conformal symmetry, since they can always be algebraically eliminated.
However, one always has to additionally fix the gauge, remembering the local
conformal symmetry with its special gauge-fixing conditions. The simplest
choice is to place some constraints on the trace of the metric fluctuation
field $h=g^{\mu\nu}h_{\mu\nu}$. The easiest choice is to just place some
algebraic condition, the more complicated conformal gauge choice is to place
some differential condition here Strominger:1984zy . Irrespective of which
choice is selected on a general background for the consistency of the whole
quantization approach, one must add to the spectrum one real minimally
gravitationally coupled scalar field, which here plays the role of the third
ghost field for conformal symmetry. This third ghost comes naturally with a
four-derivative kinetic term. Equivalently, one can add two real scalar fields
minimally coupled and enjoying two-derivative dynamics. The need for this
additional third ghost scalar field is because of the higher-derivative nature
of conformal gravity in $d=4$ (actually, this is true in any $d>2$). We also
emphasize that these conformal third ghosts are different to the usual third
ghost fields related to diffeomorphisms, which are also present in general
theories (not necessarily conformally invariant) with higher derivatives on a
general background. Since these third ghost fields couple only to the
gravitational background, then if one studies Feynman rules around flat
spacetime (gravitational vacuum configuration), then one can safely forget
about them. They are needed for propagation of modes around non-trivial
spacetime configurations or for necessarily and inherently background-
independent approaches to QG, such as, for example, in Barvinsky–Vilkovisky
(BV) BVreport  trace technology, which is a generalization of the
Schwinger–DeWitt covariant methods of computing the effective action at the
one-loop level around general curved backgrounds.

These conformal third ghost fields also played an essential role in our
explanation for the one-loop results correctly obtained for the first time by
Fradkin and Tseytlin regarding the UV divergences in Weyl gravity FT1 . Their
method, although looking a bit ad hoc, was correct, and we found a
justification for why this was so. From the generic results that one obtains
in Stelle quadratic theory with two independent terms, $\omega_{R}R^{2}$ and
$\omega_{C}C^{2}$, one first takes the limit $\omega_{R}\to 0$, obtaining some
finite results. However, they are not the final ones because of the mentioned
Veltman-like discontinuities in conformal gravity. Their cure is to subtract
the contribution of two real scalars minimally coupled. They worked in the
fully covariant framework of BV methods, hence it was natural for them to work
on a general curved background. The explanation why these are two scalar,
real, and minimally coupled conformal ghosts was found by identifying them as
the third conformal ghost fields, which are new fields in the here-presented
proper conformal quantization approach to conformal gravity, and they have to
be included on general backgrounds. This proves that our framework of
quantization specialized to the cases of local conformal symmetry in the
gravitational setup already brings correct and valid results.

Having constructed a detailed framework for quantization which has breaking of
conformal symmetry under full control, we can now use this approach to write
with full confidence the Feynman rules of the perturbative theory that we can
use for computation, for example, at the two-loop level. Such computations are
very interesting, though very difficult too, in pure conformal gravity to
check the consistency of conformal symmetry on this loop level. They are also
very important for the further developments in FT conformal supergravity
theories. As primary areas of application of our results, we mentioned the two
above. In pure conformal gravity in $d=4$, we could investigate the situation
at higher loop orders and decide the fate of conformal symmetry on the quantum
level, the presence of conformal anomaly, and dangerous $R^{2}$ terms. In this
model, we need to exert special care to check whether the conformal symmetry
is finally dramatically violated by quantum corrections and not by the mere
quantization method or procedure. For example, having the well-defined
formalism of Feynman rules in conformal gravity around a flat background, one
could extend the computation of scattering amplitudes from tree-level
scattering  (where it coincides with the results of Einstein–Hilbert gravity)
to higher loop orders, the results of which will surely be very interesting
regarding the unitarity bound of gravitons’ scattering.

Instead, in the framework of ${\cal N}=4$ conformal FT supergravity, we know
that the symmetry is fully preserved on the quantum loop level. Therefore, one
is required and tempted here to work in the formalism, which preserves this
symmetry on the quantum level, or at least which performs the gauge-fixing of
this symmetry in a fully controllable way. Similarly to how one uses
superfield formalism and supersymmetric gauge-fixing of local symmetry to
preserve this symmetry in local supergravity models, in the framework of FT
supergravity, we shall implement the same with the conformal part of the full
superconformal algebra of symmetries. As it is well known, to perform a real
quantum computation at some loop orders, one really has to fix the gauge for
all local symmetries, otherwise one meets a degeneracy problem of the kinetic
terms for small fluctuations around the flat spacetime, and then the
perturbative propagator cannot be defined. To avoid these unfortunate
circumstances, one adds gauge-fixings and breaks all local symmetries, but in
the controllable manner. This is all quantified by the presence of gauge-
fixing parameters. In the final results, for example, for $S$-matrix elements,
one expects complete independence of these spurious gauge-fixing parameters.
Then, this means that the symmetry is restored in these final results and the
effects of intentional breaking are fully under control. When one applies our
results regarding the correct quantization method, then the computation using,
for example, Feynman diagrams and around flat spacetime can be brought to the
very end. Moreover, one is then sure that the conformal symmetry is not
touched and has its powerful impact and the constraining power on the final
results. This is what is expected in the highly symmetric framework of ${\cal
N}=4$ FT conformal supergravity and regarding perturbative results in this
theory.

Of course, there are also other possible applications of the framework for
conformal quantization that we have presented here besides the two mentioned
above. One can use it also in the BV framework at the one-loop level to
perform honest and fully controlled computation of UV divergences in conformal
gravity without any ad hoc unjustified methods or tricks. We have already
performed such a computation and these results fully confirmed the ones
obtained using standard non-conformally covariant techniques. We will report
about the details of them elsewhere.

This research received no external funding.

###### Acknowledgements.

We would like to thank I.L. Shapiro, P. Jizba, and L. Modesto for initial
comments and encouragement regarding this work. Especially useful comments
regarding the issues with conformal anomaly on the quantum level we owe to S.
Giaccari. L.R. thanks the Department of Physics of the Federal University of
Juiz de Fora for kind hospitality and FAPEMIG for a technical support.
Finally, we would like to express our gratitude to the organizers of the
TQTG’21 conference (“The Quantum & The Gravity”) for accepting our talk
proposal and for creating a stimulating environment for scientific online
discussions. The author declares no conflicts of interest. References

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