computation of UV divergences may look symmetric. This is however not true due
to the different impact of these two conditions on the form of the kinetic
operator $\hat{H}$. In the case when $\omega_{R}=0$, the Hessian operator
still exists, while for $\omega_{C}=0$ we lose its form. This observation has
profound implications as we explain below. First, it is a fact that both these
conditions lead to badly non-renormalizable theories, in which the flat
spacetime propagator cannot be simply defined. Moreover, if $N>0$ in none of
these two reduced models we have an enhancement of symmetries and none of them
has anything to do with conformal gravity models, which are present only for
$N=0$ and $\omega_{R}=0$ case, despite that in constructions of these six-
derivative models we might use only terms with Weyl tensor. (However, here we
use the term $C\square C$, where it is known that the GR-covariant box
operator $\square$ is not conformal.)

Our explanation of the $x$-dependence is as follows. First, in the generic
model with $N>0$, since it happens that it is the $N=0$ scale-invariant
gravitational model which is here the exceptional one. For six-derivative
theory (or any one with $N>0$) the two reduced models with conditions that
$\omega_{C}=0$ or $\omega_{R}=0$ respectively are not renormalizable and
likely even at the one-loop level higher types of divergences (besides $C^{2}$
and $R^{2}$ from (1)) will be generated. From this we expect that there must
be some problems with UV divergences obtained from naive power counting
arguments here. The problems must show up somehow in the final numerical
values for divergent terms or in the intermediate steps of the process of
computing these divergences. These problems then signal that we are working
with non-renormalizable theory, which do not have a good control over
perturbative divergences showing up at the one-loop level.

First, in the case $\omega_{C}=0$, we see that the problems are already there
with the definition of the kinetic operator (Hessian) between quantum metric
fluctuation fields. This implies that further processing with this operator is
ill-defined, we cannot trust it and even if it would give us some final
results for divergences, then they are not reliable at all since the theory is
non-renormalizable. But we already found here the instance of the problem,
which makes our final limiting results (in the $\omega_{C}\to 0$ limit) not
trustable. This means that from the expression in (51), we do not expect any
additional obstacles, like $\frac{1}{\omega_{C}}$ poles since the price for
non-renormalizability was already paid and we have already met dangerous
problems, which signal the incorrectness of the naive limit $\omega_{C}\to 0$.
This should already take away our trust in the limit $\omega_{C}\to 0$ of
expressions for UV divergences in (33). Then this line of thought in the case
$\omega_{C}=0$ does not put any constraints at all on the final form of the
$x$-dependence in (51) since these results like in (51) in the limit
$\omega_{C}\to 0$ will anyway be likely incorrect.

Now, in the other case with $N>0$ and $\omega_{R}=0$, we do not have the
problem with the definition of the Hessian ${\hat{H}}$. Formally, we can
process it till the end of taking the functional trace of the logarithm and
adding contributions from ${\rm Tr}\ln\hat{M}$ and ${\rm Tr}\ln\hat{C}$. But
somehow, we must find the occurrence of the problem, because the theory is
non-renormalizable! So the only place in which the problem may sit is in the
final $x$-dependence of the results for UV divergences. These results should
be ill-defined, when the limit $\omega_{R}\to 0$ is attempted. And this
implies that they must be with poles in the $\omega_{R}$ coefficient, so they
must be instead with positive powers of the $x$ ratio of the theory. Hence, we
conclude that the $x$-dependence must be linear or quadratic, but always with
positive powers of the ratio $x$. This is now confirmed by explicit form as in
(33) for UV divergences of six-derivative theory. The problems with
renormalizability of the pure theory $C\square C$ show up in the last possible
moment in the procedure for obtaining the result, namely when one wants to
take the limit $\omega_{R}\to 0$ or equivalently $x\to\infty$. This is the
generic situation for any super-renormalizable theory and for any $N>0$. There
are still some mysterious things here, like why the dependence is only linear
in $x$ and why only for the $C^{2}$ type of UV counterterm, while two other
counterterms $R^{2}$ and $\rm GB$ are numbers completely independent of $x$.
Right now we cannot provide satisfactory mathematical explanations for these
facts.

Using this argumentation in the theory models with $N=1$, we get an
explanation for the $x$-dependence in formula for UV divergences in (51). The
logical chain for the explanation should be as follows. Firstly, in the pure
theory $C\square C$, one concludes that the problems of non-renormalizability
shows only in the final results as impossibility to take the limit
$\omega_{R}\to 0$ or equivalently $x\to\infty$ of formula in (51) for
divergences of the model. Hence the dependence must be on non-negative powers
of the ratio $x$ in formula (51), as it is clearly confirmed by explicit
inspection of this formula. This settles the issue of the structure of exact
beta functions for $N=1$ models (and also for higher $N\geqslant 1$ cases
too). Now, the same formula is a starting point for an attempt to take the
other limit $x\to 0$ of the also non-renormalizable model of the type
$R\square R$. But in such a model we have already found a source of the
problem caused by non-renormalizability earlier, that it is here connected
with the impossibility to properly define non-degenerate Hessian operator in
the model. But this limiting case of $x\to 0$ must follow the same structure
as already established in formula (51). Simply, theoretically speaking, there
is no need to see more instances of problems due to non-renormalizability in
the $R\square R$ model. Hence, the first explanation based on the $C\square C$
model is sufficient and the results in the model $R\square R$ must be
consistent with it. Moreover, from just the analysis of the case $x\to\infty$,
we have concluded what is the structure in a generic renormalizable case, when
we have both $\omega_{R}\neq 0$ and $\omega_{C}\neq 0$ (so $x\neq 0$ and
$x\neq\infty$). This structure is beautifully confirmed by the formula (51) or
(33) explicitly for the generic case.

As emphasized above, it is in turn the $N=0$ case, which is extraordinary and
it changes the pattern of $x_{\rm 4-der}$-dependencies described above. This
all can be traced back to the fact that for $N=0$ we have the possibility of
reducing the generic HD scale-invariant model to the conformal one, when the
full conformal symmetry is preserved (at least on the classical level of the
theory). This happens, when one takes the isolated case of $\theta_{R}=0$ and
$\theta_{C}\neq 0$ for the four-derivative theory (for positive-definiteness
we may also assume that $\theta_{C}>0$). This case is discontinuous and cannot
be taken as the naive limit $x_{\rm 4-der}\to\infty$ of the formula (49) for
the $R^{2}$ type of UV divergences, which would leave us effectively only with
the $A_{0}$ coefficient. It is well known that the conformal gravity model is
renormalizable one (at least to the one-loop level), contrary to the case of
the theory $C\square C$, which was discussed above. This means that we shall
not find any source of the problem when computing and getting results for UV
divergences of this $C^{2}$ model. We do not find problems with the Hessian or
the propagator provided we also gauge-fix the conformal local symmetry of Weyl
conformal gravity. We shall not find the problem with the final expression of
UV divergences, so there we shall not expect poles with $\theta_{R}$
coefficient. But some $x_{\rm 4-der}$-dependence up to the quadratic order
could be present (this is due to the one-loop character of the computation
here; one can understand this easily from contributing Feynman diagrams). So
then, we conclude that this dependence may be only in positive powers of the
inverse ratio, namely of $x_{\rm 4-der}^{-1}=\theta_{R}/\theta_{C}$. This is
again confirmed in the formula (49) and (34), where we indeed find the
quadratic dependence but in the inverse ratio $x_{\rm 4-der}^{-1}$.

Simply, the final results for the generic case $x_{\rm 4-der}\neq 0$ and
$x_{\rm 4-der}\neq\infty$ cannot depend on positive powers of $x_{\rm 4-der}$
since then the limit of conformal gravity in $d=4$ (i.e. $x_{\rm
4-der}\to\infty$) would produce divergent results, but we know that Weyl
gravity is renormalizable with a good control over one-loop UV divergences.
However, this does not mean that the results for conformal gravity are
continuous and obtainable from the generic ones in (34) by taking the limit
$x_{\rm 4-der}\to\infty$ there. We admit the fact that the coefficients there
may show some finite discontinuities. However, both in the true and naive
$x_{\rm 4-der}\to\infty$ limiting forms they must be finite – we only exclude
the case when they would be divergent in the $x_{\rm 4-der}\to\infty$ limit.
In this way, the results in renormalizable 4-dimensional conformal gravity for
UV divergences may be expressed via finite numbers multiplying just one common
overall divergence (like $1/\epsilon$ parameter in dimensional regularization
(DIMREG) scheme). The theory is renormalizable and there are no new
divergences inside coefficients of established form of UV divergences in
generic HD Stelle theory in $d=4$ spacetime dimensions, as in (1).

The significant difference between the cases of $N=0$ and $N\geqslant 1$ is
that in the former case the theory with $\theta_{R}=0$ is conformal on the
classical tree-level as well as on the first quantum loop, since we know that
Weyl conformal quantum gravity is one-loop renormalizable. This is why the
pattern of the $x$-dependence in these two cases is diametrically different.
In both these cases of $N=0$ and $N\neq 0$, one can derive the general
structure of beta functions in generic HD theory with any finite value of the
fundamental ratio $x$ ($x\neq 0$ and $x\neq\infty$) by just analyzing the
limit $x\to\infty$ (or respectively $x_{\rm 4-der}\to\infty$) and its
divergences which should or should not appear there respectively for the cases
of $N\neq 0$ or $N=0$.

The inverse quadratic dependence on the ratio $x_{\rm 4-der}$ in the case of
four-derivative Stelle theory can be easily understood as well. It is up to
the quadratic order and the same dependence we would expect in the case of
six-derivative gravitational theory in the $C^{2}$ sector of UV divergences.
However, there as a surprise we find only up to linear dependence on the
fundamental ratio $x$ and only in one distinguished sector of $C^{2}$
divergences. In general, we can have up to quadratic dependence on $x$ in six-
derivative models or on $x_{\rm 4-der}^{-1}$ in the Stelle gravity case in
$d=4$ spacetime dimensions. The UV divergences of some renormalizable HD
gravity models in $d=4$ spacetime dimensions are all at most quadratic in the
general gravitational curvature (schematically they are ${\cal R}^{2}$). Hence
they can be all read from the one-loop perturbative quantum corrections to the
two-point graviton Green function, so equivalently from the quantum dressed
graviton’s propagator around flat spacetime background. We remind that here
there is no quantum divergent renormalization of the graviton wave function.
Moreover, higher orders in graviton fields (appearing in interaction vertices)
are completely determined here due to the gauge invariance (diffeomorphism)
present in any QG model, so we can concentrate below only on these two-point
Green functions.

As it is known from diagrammatics, here the contributing Feynman diagrams may
have either one propagator (topology of the bubble attached to the line) or
two propagators (sunset diagrams) at the one-loop order and for corrections to
the two-point function. In the most difficult case, there are here two
perturbative propagators. Since in our higher-derivative theory we have two
leading terms shaping the UV form of the graviton’s propagator, namely the
terms $\omega_{C}C\square C$ and $\omega_{R}R\square R$, then the
corresponding propagator may be either with the front coefficient
$\omega_{C}^{-1}$ or $\omega_{R}^{-1}$ respectively as the leading term. To
change between the two expansions (in $\omega_{C}$ or in $\omega_{R}$) one
needs to use one power of the ratio $x$. Since we have two such propagators in
the one-loop diagrams considered here, then dependence is up to the quadratic
power in $x$. Sometimes we need to change back from $\omega_{C}$ to
$\omega_{R}$ as the leading coefficient of the tree-level propagator, and then
we need to multiply by inverse powers of the ratio $x$. The quadratic
dependence is what we can have here in the most complicated case, which is
actually realized in Stelle generic theory with both $\theta_{C}\neq 0$ and
$\theta_{R}\neq 0$. (The argumentation above can be repeated very similarly
for quadratic gravity in $d=4$ forgetting about one power of box operator
$\square$ and changing corresponding omega coefficients to theta coefficients
and $x$ to $x_{\rm 4-der}$.) Apparently, in the case of six-derivative
gravitational theories there is some, for the moment, unexplained
cancellation, and we see there only the dependence up to the first power on
the ratio $x$ of that theory.

One should acknowledge here the speciality of the case of $d=4$ and one-loop
type of computation. For higher loop orders the powers of the $x$ ratio may
appear higher in the final expressions for UV divergences of the theory.
Similarly, if one goes to higher dimensional QG models, then even in
renormalizable models at the one-loop level, one needs to compute higher
$n$-point Green function. This is because in even dimension $d$ one needs in
renormalizable theory not only to renormalize terms of the type ${\cal
R}\square^{(d-4)/2}{\cal R}$ but also others with more curvatures (and
correspondingly less powers of covariant derivatives) down to the term of the
type ${\cal R}^{d/2}$, where we do not find covariant derivatives acting on
curvature at all. In the middle, the general terms can be schematically
parametrized as $\nabla^{d-2i}{\cal R}^{i}$ for $i=2,\ldots,\frac{d}{2}$ – all
these terms have the energy dimensionality equal to the dimensionality of
spacetime $d$. For the last term of the type ${\cal R}^{d/2}$ one needs to
look at the quantum dressed $n=\frac{d}{2}$-point function at the one-loop
order. In conclusion, in higher dimensions one should consider not only two-
point functions with one-loop diagrams with the two topologies described
above, but up to quantum dressed $\frac{d}{2}$-point functions. And even for
one-loop perturbative level these additional diagrams may have more
complicated topology meaning more vertices and more propagators and this means
that also powers of the ratio $x$ or $x^{-1}$ respectively will be higher and
higher. They are expected to be up to the upper bound given by the maximal
power exponent equal to $\frac{d}{2}$ – this can be derived from the
expression of quantum dressed $\frac{d}{2}$-point function, which is built
exactly with $\frac{d}{2}$ propagators joining precisely $\frac{d}{2}$ 3-leg
the same perturbative vertices. Then the topology of such a diagram is this
one of the main one-loop ring and $\frac{d}{2}$ external legs attached to it,
with each one separately and each one emanating from one single $3$-leg
vertex. Again the situation at the one-loop and in $d=4$ is quite special and
simple since the ratio $x$ appears here only up to the maximal power exponent
given by $\frac{d}{2}=2$.

As a side result, one also sees that the situation in four-derivative model
with the condition $\theta_{C}=0$ is somehow “doubly” bad. First, the Hessian
is not well-defined to start with and this takes away our trust in this type
of computation. Moreover, if we would attempt to take the limit $\theta_{C}\to
0$ (or equivalently $x_{\rm 4-der}\to 0$) in the final result like in (49),
then we get a second problem since such limit gives infinite results. This
means that we somehow doubly confirmed the problem with the perturbative and
multiplicative renormalizability of such a model. It is not that the two
instances of the problem support each other – they appear somehow
independently and are not related, nor they cancel out. Above, we have seen
that in the six-derivative (or general $N>0$) case, they could occur
completely independently for two completely different types of non-
renormalizable theories (with the conditions of $\omega_{R}=0$ or
$\omega_{C}=0$ respectively). Here, we see that since conformal gravity at
one-loop must be without any problem of this type (no problem with the Hessian
and no problem with getting infinite results of the limits $x_{\rm
4-der}\to\infty$), then the occurrence of these two problems at the same time
must happen in badly non-renormalizable model with $\theta_{C}=0$ condition.
In other words, since conformal gravity is a safe model, then the model with
$\theta_{C}=0$ must suffer twice since all these two problems must inevitably
appear in one model or the other, if some extreme special cases like
$\theta_{R}=0$ or $\theta_{C}=0$ are being considered.

Again, we remark that in generic quadratic gravity model we see up to
quadratic dependence on the inverse ratio $x_{\rm 4-der}^{-1}$, but a precise
location where this dependence shows up is still not amenable for an easy
explanation. We do not know why this happens in the $R^{2}$ sector only, while
the $C^{2}$ and $\rm GB$ sectors are free from any $x_{\rm 4-der}$-dependence.
But at least the dependence on the inverse ratio $x_{\rm 4-der}^{-1}$, rather
than on its original form $x_{\rm 4-der}=\frac{\theta_{C}}{\theta_{R}}$, in
the exceptional case of $N=0$ can be explained by the miraculous one-loop
perturbative renormalizability of the conformal gravity model in $d=4$.

###  4.3 Case of Conformal Gravity

Here we continue the discussion of related issues, but now in the framework of
conformal gravity, so within the model with the reduced HD action with $N=0$
and formally with $\theta_{R}=0$. There are various motivations for conformal
gravity in $d=4$ spacetime dimensions [13, 14]. As it is well known the reason
for the multiplicative renormalizability of such a reduced model, when we have
from the beginning that $\theta_{R}=0$ is the presence of conformality -
conformal symmetry both on the tree level and also on the level of the first
loop. Unfortunately, the story with conformal gravity in $d=4$ is even more
complicated than what we argued above. First, already at the one-loop level
one discovers the presence of conformal anomaly, which is typically thought as
not so harmful on the first loop level. However, it heralds the soon breaking
of conformal symmetry like for example via the appearance of the $R^{2}$
counterterm at the two-loop level. Such term as a counterterm is not fully
invariant under local conformal transformations - it is only invariant under
so called restricted conformal transformations that is such transformations
whose parameters satisfy the source-free background GR-covariant d’Alembert
equation ($\square\Omega=0$) on a general spacetime. Hence the $R^{2}$ term is
still scale-invariant but it breaks full conformal symmetry of the quantum
conformal gravity. It seems the only way out of this conformal anomaly problem
is to include and couple to conformal gravity specific matter sector to cancel
in effect the anomaly. This is, for example, done in ${\cal N}=4$ conformal
supergravity coupled to two copies of ${\cal N}=4$ super-Yang-Mills theory,
first proposed by Fradkin and Tseytlin [49]. In such a coupled supergravity
model, we have vanishing beta functions, implying complete UV-finiteness and
conformality present also on the quantum level. This is conformal symmetry in
the local version (not a rigid one) with Weyl conformal transformations in the
gravitational setup and on the quantum field theory level.

If not in the framework of ${\cal N}=4$ Fradkin and Tseytlin supergravity, the
conformal anomaly of local conformal symmetry signals breaking of conformal
symmetry, while scale-invariance (global part of it) still may remain intact.
In the long run, besides the presence of non-conformal $R^{2}$ counterterm,
this breaking will put conformal Ward identities in question and also the
constraining power of the quantum conformality in question too. It will not
constrain any more the detailed form of gravitational correlation functions of
the quantum theory. The conformal symmetry will not be there and it will not
protect the spectrum from the emergence of some spurious ghost states in it.
This last thing will endanger the perturbative unitarity of the theory (and we
do not speak here about the danger of unitarity breaking due to the HD nature
of conformal gravity). Without the power of quantum conformal symmetry, we may
have unwanted states in the spectrum corresponding to the states from generic
Stelle gravity, and not from the tree-level spectrum of conformal gravity, so
we can see mismatch in counting number of degrees of freedom and also in their
characters, namely whether they are spin-1 or spin-0, ghosts or healthy
particles, etc.

Moreover, in pure conformal gravity described by the action simply $C^{2}$
without any supergravitational extension, we notice the somehow nomenclature
problem with the presence of quantum conformality. Even barring the issue of
conformal anomaly, the general pure gravitational theory has non-vanishing
beta functions, so there is no UV-finiteness there. This implies that there is
an RG running and scale-dependence of couplings and of various correlators on
the renormalization energy scale. Hence already at the one-loop level one
could say that scale-invariance is broken, which implies violation of
conformal symmetry too. However, one can live with this semantic difference
provided that there are no disastrous consequences of the conformal anomaly.
One can adopt the point of view that the theory at the one-loop level is still
good provided that the UV-divergent action is conformally invariant too, that
is when one has only conformally invariant UV counterterms. (Although in the
strict meaning having them implies non-vanishing beta function, RG running,
loss of UV-finiteness and of scale-invariance.) In our case the conformally
invariant counterterms are only of the type $C^{2}$ and $\rm GB$, so if the
$R^{2}$ counterterm is not present at the one-loop level, then we can speak
about this preserved quantum conformality in the second sense. It happens this
is exactly the situation we meet for quantum conformal gravity in $d=4$ at the
one-loop level.

In order to see quantum conformality of one-loop level conformal gravity in
$d=4$ described by the action, one first naively may try to take the limit
$x\to+\infty$ from the expression for the $R^{2}$ sector of UV divergences
from formula in (34). One would end up with the results, just a constant
$A_{0}$, which is generally not zero. The whole story is again more subtle,
since the limits in this case are again not continuous, although as we
advocated above they are luckily also not divergent, when we want to send
$\theta_{R}\to 0$. In the end, we have only a finite discrepancy in numbers,
which can be easily explained. As emphasized above in this case of the special
reduced model we have the enhancement of symmetries and this new emergent
conformal symmetry in the local version must be gauge-fixed too. This means
that the kinetic operator needs to be modified and some new conformal gauge-
fixing functional must be added to it for the consistency of the generalized
Faddeev-Popov quantization prescription of theories with local gauge
symmetries. This means that we will also have a new conformal gauge-fixing
parameter (the fourth one), which can be suitably adjusted to provide again
the minimality of the Hessian operator. Although, of course, the whole details
of the covariant quantization procedure for conformal gravity are more
delicate and more subtle, here we can just take a shortcut and pinpoint the
main points of attention. When computing UV divergences using generally
covariant methods like BV trace technology and functional traces of logarithms
of operators, one also necessarily needs to add here the contribution of the
third conformal ghosts, which are scalars from the point of view of Lorentz
symmetry but they come with anti-commuting statistics. They are needed here
because the conformal gravity theory is a natural HD theory and then third
ghosts are necessary for covariant treatment of any gauge symmetry in the
local form. It is true that for conformal local symmetry we do not need FP
ghost fields (because of the reasons elucidated above), but we need a new
third conformal ghost, which is moreover independent from the third ghost of
diffeomorphism symmetry. Each symmetry with local realization comes with its
own set of third ghosts, when the theory is with higher derivatives. It is
also well known that classically conformal fields (like massless gauge fields
of electromagnetism and also of Yang-Mills theory) give contribution to
divergences which is conformally invariant counterterm, so only of the type
$C^{2}$ or $\rm GB$ terms. This can be understood easily as a kind of
conformal version of the DeWitt-Utiyama argument used before. Hence, if the
scalars of the anti-commuting type that we have to subtract were conformally
coupled, then they will not contribute anything to the $R^{2}$ type of the
counterterm. But we see from the formula in (34) that the $A_{0}$ coefficient
there is non-zero, so only this one survives after the limit $x\to\infty$ is
taken. To cancel the $R^{2}$ counterterm is crucial for the hypothesized
conformal invariance of the conformal gravity also on the first loop quantum
level. And this must be done by explicitly non-conformal fields with non-
conformal contributions to divergences. They cannot be massless gauge fields,
but they can be minimally, so not conformally coupled scalar fields. Here for
the consistency of the whole formalism of the FP covariant quantization of
conformal gravity, this role is played by the one real conformal third ghost
with the kinetic operator $\square^{2}$.

The contribution of the third conformal ghosts is what we actually need to
complete the whole process of the computation of the UV divergences in the
conformal gravity model. We need them for the overall consistency since in
this covariant framework we cannot a posteriori check the presence of all
gauge invariances. Here we assume that on the first quantum loop level, the
conformal gravity model enjoys fully diffeomorphism as well as conformal
symmetry. The terms given in the covariant BV framework of computation all
satisfy these requirements, so only we must be careful to take all these
contributions into account. The contribution of the third conformal ghosts is
like that of two real scalars coupled minimally (but not conformally to one’s
surprise) to the background gravitational metric, but of the ghost type.
Indeed this means that we have to subtract the contribution of two scalars,
which is of course, UV-divergent but after extracting the overall divergence
there is only a finite number. This is the number that when subtracted now
matches with number obtained after the naive limit $x\to\infty$ of the generic
results from (49). We explain that we need to subtract two real scalars each
one coming with the standard two-derivative GR-covariant box operator as the
kinetic operator since in HD conformal gravity the operator between third
conformal ghosts is of the $\square^{2}$ type as for the four-derivative
theory. The limit to conformal gravity is discontinuous, but only in this
sense that one has to take out also contribution of real scalar fields
minimally coupled to gravitation. The first part of the limiting procedure,
namely $x\to\infty$ is only a partial step and to complete the whole limiting
procedure one must also deal properly with conformal symmetry. This applies
not only to the coefficients in front of the $R^{2}$ term, where we see the
mysterious but explainable $x$-dependence, but also to other coefficients in
front of terms like $C^{2}$ and $\rm GB$ terms. Of course, for the last two
terms the limits $x\to\infty$ do not change anything, but the contribution of
third conformal ghosts makes impact and change the numerical results, which
are luckily still finite in conformal gravity. The coefficients in front of
the $R^{2}$ and $\rm GB$ counterterms are also finite in generic four-
derivative gravity (cf. with (34)), however by these types of arguments with
conformal gravity we cannot at present understand why the $x$-dependence
happens only in front of the $R^{2}$ counterterm. Of course, in conformal
gravity model, there is not any $x$-dependence at all.

At the end, when one accounts for all these numerical contributions, one
indeed finds that at the one-loop level in conformal gravity, the coefficient
of the $R^{2}$ term is vanishing, so the quantum conformality is present in
the second sense. And we have only conformally invariant counterterms in pure
conformal gravity without any conformal matter, but there is still interesting
RG flow of couplings there. This also signifies that there is conformally
invariant but non-trivial divergent part of the effective action with finite
numerical constant coefficients, when the overall divergence is extracted.
These finite coefficients arise in the two-step process. First as the limit
$x\to\infty$ of a generic HD gravity and then by subtraction of UV-divergent
contributions of two real scalars minimally coupled to gravitational field.
Since these last contributions are known to be finite numbers multiplying the
overall UV divergence, then this implies that the limit $x\to\infty$ of the
generic expression in (49) must also give finite numbers. This explains why in
perturbatively one-loop renormalizable model of conformal gravity in $d=4$
there are standard UV divergences, although this is a reduced model with
$\theta_{R}=0$ and $N=0$ case. So the $x$-dependence in (34) must be as
emphasized above that is with inverse powers of the fundamental ratio $x$ of
the theory and in accord with what was schematically displayed in formula
(49). Hopefully now the dissimilarities between the cases with $N=0$ and $N>0$
are more clear.

In short, we think that the only sensible reason, why we see completely
different behaviour when going from $N=0$ to $N=1$ class of theories is that
the theory with $\omega_{R}=0$ and $N=1$ ceases to be conformally invariant in
$d=4$. In a different vein, the degeneracy of the kinetic operator $\hat{H}$
in the $\omega_{C}=0$ cases, for both $N=0$ and $N=1$, remains always the
same.

This proves again and again that the case of conformal gravity is very special
among all HD theories, in $d=4$ among all theories quadratic in gravitational
curvatures. One can also study the phenomenological applications of the Weyl
conformal gravity models to the evaporation process of black holes [50, 51]
and also use the technology of RG flows (and also functional RG flows) in the
quantum model of conformal gravity to derive some interesting consequences for
the cosmology (like for example for the presence of dark components of the
universe in [52, 53, 54]). Finally, the conformal symmetry realized fully on
the classical level (and as we have seen also to the first loop level and
perhaps also beyond) is instrumental in solving the issue with spacetime
singularities [55, 34, 56], which are otherwise ubiquitous problems in any
other model of generally covariant gravitational physics (both on the
classical and quantum level). To resolve all singularities one must be sure
that the conformality (Weyl symmetry) is present also on the full quantum
level (and it is not anomalous there), so the resolution of singularities from
the classical level (by some compensating conformal transformations) is not
immediately destroyed by some dangerous non-conformal quantum fluctuations and
corrections.

###  4.4 More on limiting cases

Here again we analyze closer the situation with various limits, when some
coefficients in the gravitational action (45) disappear.

In a generic six-derivative theory, the trace of the logarithm of the FP
ghosts kinetic operator $\hat{M}$ and of the standard minimal $\hat{C}$ matrix
are regular in the limit $\omega_{R}\to 0$, but not in the limit
$\omega_{C}\to 0$. For the $\hat{C}$ matrix this is understandable, because
the $\gamma$ parameter contains factor $\omega_{C}^{-1}$ in the minimal gauge.
However, for the FP ghosts kinetic operator $\hat{M}$, this dependence was
unexpected, because in the explicit definition of the $\hat{M}$ operator there
was never any singularity in $\omega_{C}$. Moreover, this singularity is even
quadratic in $\omega_{C}$ coefficient.

We also emphasize that in the general six-derivative theory the trace of the
logarithm of the Hessian operator $\hat{H}$ is irregular both in the limits
$\omega_{R}\to 0$ and $\omega_{C}\to 0$ separately. It seems that in the total
sum of contributions to the beta functions of the theory the singularity in
$\omega_{C}$ cancels completely between ${\rm Tr}\ln\hat{H}$, ${\rm
Tr}\ln\hat{C}$, and ${\rm Tr}\ln\hat{M}$, while the poles in $\omega_{R}$
remain and this is what is seen as a dependence of the final results on the
non-negative powers of the fundamental ratio $x$. For the definition of the
$\hat{H}$ operator, if the limit $\omega_{R}\to 0$ is taken, nothing bad is
seen. This may be a partial surprise. Of course, when the limit $\omega_{C}\to
0$ is taken, then this operator does vanish on flat spacetime, so then its
degeneracy is clearly visible.

The situation with limits ($\theta_{R}\to 0$ or $\theta_{C}\to 0$) in four-
derivative theory we see as follows. The functional trace ${\rm Tr}\ln\hat{C}$
is regular in both limits. It actually does not depend on any gauge-fixing
parameters here, despite that in its formal definition we used the $\gamma$
parameter, which shows the $\frac{1}{\theta_{C}}$ singularity. The situation
with the FP operator $\hat{M}$ is the same as previously, because the operator
is identical as in the six-derivative theory case. The operator $\hat{H}$
shows the problem with its definition only when the limit $\theta_{C}\to 0$ is
considered. The same is true for its trace of the functional logarithm, which
shows singularity in ${\theta_{C}}$ coupling coefficient up to the quadratic
order. In this case and in the total sum of all contributions, we see only
$\frac{1}{\theta_{C}}$ singularity to the quadratic order. However, here the
limit $\theta_{R}\to 0$ is not continuous either, because the theory reaches a
critical point in the theory space with enhanced symmetry for $\theta_{R}=0$
(conformal enhancement of local symmetries) as it was explained in subsection
4.3.

Let us also comment on what special happens in the computation of UV
divergences for quadratic theory from the perspective of problems that we have
initially encountered in six-derivative theory for the same computation.
First, we established, in the middle steps of our computation for the results
published in [26], that in the traces ${\rm Tr}\ln\hat{H}$ and ${\rm
Tr}\ln\hat{C}$ in Stelle gravity there are no any dangerous
$\frac{1}{y}=\frac{1}{2\omega_{C}-3\omega_{R}}$ poles (cf. [42]). The
cancellations happen separately within each trace. Second thing is that we
found that the trace ${\rm Tr}\ln\hat{C}$ surprisingly completely does not
depend on the gauge-fixing parameter $\gamma$, which was needed and used in
the initial definition of the $\hat{C}$ operator in (15). Finally, one can
notice that the addition of the Gauss-Bonnet term in $d=4$ spacetime
dimension, does not change anything for $R^{2}$, $R_{\mu\nu}^{2}$, and
$R_{\mu\nu\\!\rho\sigma}^{2}$ divergences (as it was expected), because its
variation is a topological term in $d=4$.

In this last part, we use the schematic notation for various gravitational
theories, when we do not write, for simplicity, the coupling coefficients in
front of various terms since they are not the most important for the
considerations here. In the case of six-derivative theories, it is impossible
to obtain the results for the cases with $\omega_{C}=0$ or $\omega_{R}=0$ by
any limiting procedures of the corresponding results obtained for the general
six-derivative theory with $\omega_{R}\neq 0$ and $\omega_{C}\neq 0$. These
reduced theories have different bilinear parts, with degenerate forms of the
kinetic operator and our calculation methods break down here. Similarly, one
can calculate the beta functions in a theory with $R^{2}$ only and this was
done indirectly many times. One can also calculate UV divergences in $C\square
C+R^{2}$ theory or in an analogous $R\square R+C^{2}$ theory, but this is
actually not easy to do. But then we cannot easily extract these results from
our general calculation done in $C\square C+R\square R$ six-derivative theory.
The simple reason is that all these theories have different amount and
characteristics of degrees of freedom and the transition from one to another
at quantum level is complicated (and to some extent unknown).

Moreover, we remark, that the results for beta functions in models $C\square
C+R^{2}$ (or $C\square C+R^{2}+C^{2}$) or in $R\square R+C^{2}$ (or $R\square
R+C^{2}+R^{2}$) could be obtained by though different computations than what
we have done here. We summarize that the six-derivative gravitational theory
to be renormalizable must contain both terms of the type $C\square C$ and
$R\square R$. Then the kinetic operator (Hessian) between gravitational
fluctuations and the graviton’s propagator are well-defined. In all other
models, there is not a balance between the number of derivatives in the
vertices of the theory and in all gauge-invariant pieces of the propagator, so
the theory behaves badly regarding perturbative UV divergences at higher
loops. This does not mean that the computation of UV divergences at one-loop
level is forbidden, just only that usually these are not all divergences in
the theory, they may not be the UV-leading ones anymore or the theory does not
have decent control over all of them.

For the strictly non-renormalizable theory with the leading in the UV term
$C\square C$ we can have additions of various subleading terms which do not
change the fact of non-renormalizability. We can add terms (separately or in
conjunction) of the following types: $\omega_{\Lambda}$ (cosmological constant
term), $R$ (E-H term), $R^{2}$ (Starobinsky’s term), $C^{2}$ (Weyl square
term). The UV-leading part of the Hessian still is defined as it contains six-
derivative differential operator understood on flat spacetime and between
tensorial fluctuations, so derived from the terms quadratic in curvatures. The
Hessian is non-degenerate. (It has to be non-degenerate here, because the GR-
covariant box operator is here only a spectator, and the Hessian must be
“almost” non-degenerate for the case of conformal gravity with the action
$C^{2}$.) The flat spacetime propagator can be defined only if we make
addition of $\omega_{\Lambda}$, or $R$ or $R^{2}$ terms – this is because of
the problematic part of it proportional to the projector $P^{(0)}$ which must
for the consistency of the inverting procedure for the whole propagator be
non-zero. This scalar part (spin-0 part) is sourced from any scalar term or
from the cosmological constant term. If only the $C^{2}$ term is added, then
the propagator still is ill-defined. Still these additions do not change the
fact that the theory is non-renormalizable, if there is not an accompanying
six-derivative term of the form $R\square R$. As for the final results for UV
divergences in these extended models, naively one would think that there are
no additional UV divergences proportional to terms with four derivatives of
the metric (namely to terms $R^{2}$, $C^{2}$ and $\rm GB$), because of the
limit $\omega_{R}\to 0$ and the dependence on the $x$ ratio in (33) in the
linear way. We would naively think that divergences with $R^{2}$ and $\rm GB$
terms are the same as in (33). The only problematic one could be this
proportional to the $C^{2}$ term since the limit gives already divergent
results (so “doubly” divergent) – this would mean that the coefficient of the
$C^{2}$ divergence is further divergent and renormalization of just $C^{2}$
does not absorb everything at the one-loop level. Since the model is non-
renormalizable we cannot trust this computation and these limits at the end,
especially if they give divergent results. But this probably means that we
cannot sensibly define the $C^{2}$ counterterm needed for the UV
renormalization in these theories. In a sense an attempt of adding
$\omega_{\Lambda}$ or $R$, or $R^{2}$ terms to regularize the theory $C\square
C+C^{2}$, or even the simplest one, just $C\square C$, is unsuccessful so we
perhaps still cannot trust there in the final results just given by two
$B_{0}$ coefficients of UV divergences proportional to terms $R^{2}$ and $\rm
GB$, while the $C^{2}$ divergences are never well-defined in this class of
models.

Instead, in the case of the reduced model with $R\square R$ UV-leading action,
one may keep some hope that the results for the $C^{2}$ counterterms will be
finite at the end, but maybe still discontinuous, despite the non-
renormalizability of the model with $R\square R$ action (plus possible lower
derivative additions to regularize it as it was mentioned above). Maybe in
this reduced models the results of the projection procedure of the UV-
divergent functional of the effective action of the theory onto the sector
with only $C^{2}$ terms will result here in giving sense to pure $C^{2}$
divergences in this limiting model. (Here we may try to resort to some
projection procedure for the functional with UV divergences since in these
non-renormalizable models, one may expect to find more divergences than just
of the form of $C^{2}$ and $R^{2}$ as presented initially in (1). There could
exist new UV divergences, which contain even more than four derivatives on the
background metric tensor, even in $d=4$ case.) But the final finite value may
be discontinuous and may not be obtainable by the naive limit $x\to 0$ of the
expression for the divergent term in the $C^{2}$ sector of UV divergences, so
it may not be just $B_{0}$ there. This remark about possible discontinuities
may apply also to coefficients in front of divergent terms of the type $R^{2}$
and ${\rm GB}$. They may still end up with some finite definite values for
this model, but probably they are not the same as the coefficients $B_{0}$ of
these terms from (51), so we probably will be able to see here another
discontinuities in taking the naive limit $x\to 0$.

These above results about discontinuities and negative consequences due to the
overall non-renormalizability of the two considered above reduced models, are
also enforced by the analysis of power counting of UV divergences. One can try
to perform the “worst case scenario” analysis of one-loop integrals and the
results show complete lack of control over perturbative UV divergences in such
reduced models. This is even worse that in the case of off-shell E-H gravity
considered in $d=4$ dimensions, which is known to be one-loop off-shell
perturbatively non-renormalizable theory. In the latter case the superficial
divergence of the divergence $\Delta$ is bounded at the one-loop level ($L=1$)
in formula (4) by the value $4$. In general, at the arbitrary loop order we
have the formula for power counting reads then

$\Delta+d_{\partial}=4+2(L-1),$ (56)

which if we concentrate on logarithmic UV divergences only (with $\Delta=0$),
we get that at the one-loop level for all Green functions we need counterterms
with up to $d_{\partial}=4$ partial derivatives on the metric tensor. At the
two-loop level we instead need to absorb the divergence term with
$d_{\partial}=6$ partial derivatives as this was famously derived by Goroff
and Sagnotti [17, 18]. The counterterm that they have found was of the form of
the $C^{3}$ GR-covariant term and its perturbative coefficient at the two-loop
order does not vanish, and this implies that the whole UV-divergent term does
not vanish even in the on-shell situation. But still we know which
counterterms to expect at the given loop order and the absorption of UV
divergences works for all divergent Green functions of the QG model.

The situation in the reduced models of the type $C\square C$ or $R\square R$
in the leading UV terms is much worse even at the one-loop level from naive
power counting there. One sees that different GR-covariant counterterms are
needed to absorb divergences in different divergent Green functions of the
quantum model at the one-loop level, so the counting does not stop at the two-
point function level. We think that despite these tremendous difficulties, one
still can compute the divergent parts of the effective action and the actual
computations are very tedious and still possible. This is provided that one
can invert the propagator, so one has some non-vanishing parts in both gauge-
invariant parts of it with the spin-0 and spin-2 projectors. So, it is at
present practically impossible to do computation in the pure models $C\square
C$ or $R\square R$ only. We know that they give contributions in momentum
space proportional to $k^{6}$ in the spin-2 and spin-0 parts of the propagator
respectively, while other parts are not touched. In order to regularize the
theory and to give sense to the perturbative propagator around flat spacetime,
one has to add the regulator terms as this was mentioned above. Let us assume
that they give contributions to the other sector of the spin projectors in the
graviton’s propagator of the form $k^{-m}$, where $m$ is some integer and
$m<6$, they may likely also give additional subleading contributions to the
main respective part of the propagator which was there with six derivatives in
the UV regime correspondingly to the spin-2 sector in $C\square C$ theory and
to the spin-0 sector in $R\square R$ model. The values of $m$ are
respectively: $m=0$ for the cosmological constant addition (it still
regularizes the propagator but very, very weakly), $m=2$ when E-H term is
added (it contributes both to the spin-0 and spin-2 parts), $m=4$ when either
$R^{2}$ or $C^{2}$ terms are added (they contribute exclusively in their
respective sectors).

Since now after the regularization of the graviton’s propagator, its behaviour
is still very unbalanced in the UV regime between different components, then
one sees the following results of the analysis of UV divergences at the one-
loop order. First, the general gravitational vertex is still with six
derivatives, while the propagator is $k^{-6}$ in the best (most suppressed)
behaviour and $k^{-m}$ is the worst behaviour in the other components. For the
most dangerous situation we have to assume that the overall behaviour of the
propagator is in the worst case, so with the UV scaling of the form $k^{-m}$.
Then the relation between the number of derivatives in a general gravitational
vertex and in the propagator is broken and this is a reason for very bad
behaviour with UV divergences here. Such relation is typically present even in
non-renormalizable models, like in E-H gravity. The lack of this relation
means that now the result for $d_{\partial}$ of any Feynman graph $G$ depends
on the number of external graviton lines $n_{g}$ emanating from the one-loop
diagram. Previously in the analysis of power counting there was never any
dependence on this $n_{g}$ parameter. This is a source for problems even
bigger when one increases $n_{g}$. For definiteness we can assume that
$n_{g}>1$ since here we will not be interested in vacuum or tadpole diagrams
and quantum corrections to them. Now, for a general diagram $G$ with $n_{g}$
external graviton lines, the worst situation from the point of view of UV
divergences is for the following topology of the diagram, namely there is one
loop of gravitons (so called “ring of gravitons”) in the middle with $n_{g}$
3-leg vertices joined by $n_{g}$ propagators. In the case when we concentrate
on logarithmic divergences $D=0$, we get the following results for the
quantity $d_{\partial}$ which tells us how many derivatives we must have in
the corresponding counterterm to absorb the divergence,

$d_{\partial}=4+n_{g}(6-m)$ (57)

for the graph contributing one-loop quantum corrections to the $n_{g}$-point
Green function. One sees that this $d_{\partial}$ grows without a bound even
at the one-loop level, when $n_{g}$ grows, so in principle to renormalize the
theory at the one-loop level one would need already infinitely many GR-
covariant terms, if one does not bound the number of external legs of Green
functions that must be considered here. A few words about explanation of
numbers appearing in the formula (57). The $4$ there is the number of
spacetime dimensions (integration over all momenta components at the one-loop
level), while the $(6-m)$ factor comes from the difference between the highest
number of derivatives in the vertex, i.e. $6$ compensated by the worst
behaviour in some propagator components given in the UV by $k^{-m}$ only.

Moreover, there are precisely $n_{g}$ segments of the structure propagator
joined with 3-leg vertex to create a big loop. This behaviour signals complete
lack of control over perturbative one-loop divergences even at the one-loop
level. Moreover, they have to be absorbed in the schematic terms of the type

$\nabla^{4+4n_{g}-n_{g}m+2i}{\cal R}^{n_{g}-i},$ (58)

for the index $i$ running over integer values in the range $i=0,1,2,\ldots
n_{g}-2$, where we only mentioned the total number of covariant derivatives
not specifying how they act on these general gravitational curvatures. This is
because this is an expression for the quantum dressed one-loop Green function
with $n_{g}$ graviton legs on the flat spacetime, so terms with more
curvatures than $n_{g}$ will not contribute to absorb these divergence of flat
Green $n_{g}$-point function. We only mentioned here the really the worst
situation, where the divergence may be finally absorbable not only by the
highest curvature terms of the type $\nabla^{4+4n_{g}-n_{g}m}{\cal R}^{n_{g}}$
but also for terms with smaller number of curvatures (up to ${\cal R}^{2}$
terms and in the precise form ${\cal R}\square^{\frac{1}{2}n_{g}(6-m)}{\cal
R}$). We neglect writing counterterms here which are total derivatives and
which are of the cosmological constant type. These are then the needed
counterterms off-shell at the one-loop level in such general reduced model.

To make it more concrete, we will analyze the cases of $m=0,2$ and $4$ with
special attention here in these badly non-renormalizable models for some small
numbers of legs of quantum dressed Green functions. We have that at $m=0$ to
absorb UV divergences from the $2$-point function we need generic counterterms
of the form: ${\cal R}\square^{j}{\cal R}$ with the exponent $j$ running over
values $j=0,1,2,3,4,5,6$, while to renormalize a three-point function one
needs previous terms and possibly new terms of the type $\nabla^{j}{\cal
R}^{3}$ with $j=0,\ldots,16$ and for four-point functions new terms are of the
type $\nabla^{j}{\cal R}^{4}$ with $j=0,\ldots,20$, etc. for higher Green
functions (for $n_{g}$-leg correlators one needs $j$ up to $j_{\rm
max}=4n_{g}+4$). When we regularize by adding the E-H term with $m=2$ the
situation is slightly better, but then to absorb UV divergences from the
$2$-point function we need generic counterterms of the form: ${\cal
R}\square^{j}{\cal R}$ with the exponent $j$ running over values
$j=0,1,2,3,4$, while to renormalize a three-point function one needs previous
terms and possibly new terms of the type $\nabla^{j}{\cal R}^{3}$ with
$j=0,\ldots,10$ and for four-point functions new terms are of the type
$\nabla^{j}{\cal R}^{4}$ with $j=0,\ldots,12$, etc. for higher Green functions
(for $n_{g}$-leg correlators one needs $j$ up to $j_{\rm max}=2n_{g}+4$).
Finally, we can add terms of the type $R^{2}$ and $C^{2}$ for the
regularization purposes. In such final case to be considered here one has that
to absorb UV divergences from the $2$-point function we need generic
counterterms of the form: ${\cal R}\square^{j}{\cal R}$ with the exponent $j$
running over values $j=0,1,2$, while to renormalize a three-point function one
needs previous terms and possibly new terms of the type $\nabla^{j}{\cal
R}^{3}$ with $j=0,\ldots,4$ and for four-point functions new terms are of the
type $\nabla^{j}{\cal R}^{4}$ with $j=0,\ldots,4$, etc. for higher Green
functions (for $n_{g}$-leg correlators one needs $j$ up to $j_{\rm max}=4$
here independently on the number of legs $n_{g}$). Still in this last case one
sees that one needs infinitely many counterterms to renormalize the theory at
the one-loop level, although the index $j$ of added covariant derivative is
bounded by the values $4$, still one needs more terms with more powers of
gravitational curvatures $\cal R$.

This shows how badly non-renormalizable are these models already at the one-
loop level and that any control over perturbative UV divergence is likely
lost, when the number of external legs is not bounded here from above. These
reduced models are examples of theories when the dimensionality and the number
of derivatives one can extract from the vertices and propagators of the theory
differ very much. Previously in quantum gravity models these two numbers were
identical which leads to good properties of control over perturbative
divergences (renormalizability, super-renormalizability and even UV-
finiteness). With these reduced models we are on the other bad extreme of vast
possibilities of QG models. But seeing them explicitly proves to us how
precious is the renormalizability property and why we strongly need them in HD
models of QG, in particular how we need super-renormalizability in six-
derivative QG models.

The arguments above convince us to think that there is no hope to get
convergent results for the front coefficient coming with the $C^{2}$
counterterm in the UV-divergent part of the effective action in the considered
here model with the only presence of the $C\square C$ as the leading one in
the UV. This may signify that here there exists another UV-divergent term
(perhaps of the structure like $C\square^{n}C$), which contains more
derivatives, and this could be a reason why the coefficient in front of the
$C^{2}$ term is itself a divergent one. The presence of such new needed
counterterms with more derivatives can be motivated by the analysis of power
counting of UV divergences in this reduced unbalanced model, which is also
presented below. Even if the higher $C\square^{n}C$ type of UV divergence is
properly extracted and taken care of, then we can still be unable to properly
define and see as convergent the divergence proportional to the four-
derivative term $C^{2}$. Even such a projection of the UV-divergent functional
of the theory onto the sector with only $C^{2}$ terms will not help here in
giving sense to pure $C^{2}$ divergences in this limiting model. However, this
remark does not need necessarily to apply to coefficients in front of
divergent terms of the type $R^{2}$ and ${\rm GB}$. They may still end up with
some finite definite values for this model, but probably they are not the same
as the coefficients $B_{0}$ of these terms from (51), so we could be able to
see here another discontinuity in taking the limit $x\to+\infty$.

On the other hand, for the strictly non-renormalizable theory with the leading
in the UV term $R\square R$ we can have additions of similar various
subleading terms which do not change the fact of non-renormalizability. We can
add terms (separately or in conjunction) of the following types:
$\omega_{\Lambda}$, $R$, $R^{2}$, or $C^{2}$. The UV-leading part of the
Hessian still is not well-defined as it should contain six-derivative
differential operator understood on flat spacetime and between tensorial
fluctuations, while from the term $R\square R$ we get only operator between
traces of metric fluctuations $h=\eta^{\mu\nu}h_{\mu\nu}$ (between spin-0
parts), so derived from the terms quadratic in curvatures present in the UV
regime. Probably the degeneracy of the Hessian operator can be easily lifted
out, if we add one of the $\omega_{\Lambda}$, or $R$ or $C^{2}$ terms. If only
the $R^{2}$ term is added, then the Hessian still is degenerate. Similarly,
the flat spacetime propagator can be defined only if we make addition of
$\omega_{\Lambda}$, or $R$ or $C^{2}$ terms – this is because of the
problematic part of it proportional to the projector $P^{(2)}$ which must for
the consistency of the inverting procedure for the whole propagator be non-
zero. This tensorial part (spin-2 part) is sourced exclusively from any GR-
invariant term built out with Weyl tensors in adopted here Weyl basis of terms
or from the E-H term or from the cosmological constant term. If only the
$R^{2}$ term is added, then the propagator still is ill-defined. Still these
additions do not change the fact that the theory is formally non-
renormalizable, if there is not an accompanying six-derivative term of the
form $C\square C$.

As for the final results for UV divergences in these extended models, naively
one would think that there are no new UV divergences proportional to terms
with four derivatives of the metric (namely to terms $R^{2}$, $C^{2}$ and $\rm
GB$), because of the limit $\omega_{C}\to 0$ and the dependence on the $x$
ratio in (33) in the linear way. We would naively think that divergences with
$R^{2}$ and $\rm GB$ terms are the same as in (33). The only problematic one
could be this proportional to the $C^{2}$ term since the limit gives already
constant result, namely the $B_{0}$ coefficient. Since the model is non-
renormalizable we cannot trust this computation and these limits at the end,
even if they give here convergent results. But this probably means that we
cannot sensibly define the $C^{2}$ counterterm needed for the renormalization
of these theories. When we include additions to the action, which remove the
degeneracy of the flat spacetime graviton’s propagator, then at least the
perturbative computation using Feynman diagrams can be attempted in such a
theory. Although of course, in this case the different parts of the propagator
have different UV scalings, so the situation for one-loop integrals is a bit
unbalanced and there is not a stable control over perturbative UV divergences,
when for example one goes to the higher loop orders. Probably new counterterms
(with even higher number of derivatives) will be at need here. Making
additions of some subleading terms from the point of view of the UV regime,
may help in defining the unbalanced perturbative propagator, but still one
expects (due to energy dimension considerations) that these additions do not
at all influence the quantitative form of UV divergences with four-derivative
terms, so these ones which are leading in the UV. These additions are needed
here only quantitatively and on the formal level to let the computation being
done for example using Feynman diagrams with some mathematically existing
expressions for the graviton’s propagator. In a sense adding
$\omega_{\Lambda}$ or $R$, or $C^{2}$ terms regularizes the theory $R\square
R+R^{2}$ or even the simplest one, just $R\square R$, so we can perhaps trust
there in the final results just given by all three $B_{0}$ coefficients of UV
divergences proportional to terms $R^{2}$, $C^{2}$ and $\rm GB$. This can be
motivated by the observation that here the limits $\omega_{C}\to 0$ or
$\theta_{C}\to 0$ respectively do not enhance any symmetry of the model in
question, so they can be naively and safely taken. But we agree that this case
requires a special detailed and careful computation to prove this conjectural
behaviours.

In the general case of badly non-renormalizable theories, with both
$\omega_{C}=0$ or $\omega_{R}=0$, one trusts more the computation using
Feynman diagrams and around flat spacetime than of the fully GR-covariant BV
method of computation. For the former one only needs to be able to define
properly the propagator – all physical sectors of it, and for this purpose one
can regularize the theory by adding the term $\omega_{\kappa}R$, which is a
dynamical one with the smallest number of derivatives and for which flat
spacetime is an on-shell vacuum background. (In this way, we exclude adding
the cosmological constant term $\omega_{\Lambda}$, which would require adding
some source and the flat spacetime propagator could not be considered anymore
in vacuum there.) Then in such regulated non-renormalizable theory one can get
results around flat spacetime and in Fourier momentum space, and then at the
end one can take the limit $\omega_{\kappa}\to 0$. The results for some UV
divergences in these non-renormalizable models must be viewed as projected
since higher-derivative (like 6-derivative and even higher) infinities may be
present as well. These last results must coincide with the ones we obtained in
(33), when the proper limits of $\omega_{C}\to 0$ or $\omega_{R}\to 0$ are
taken. We notice that adding the E-H term, which is always a good regulator,
changes the dynamics very insignificantly for these higher-derivative models
and the results from Feynman diagram computations can be always derived.
Instead for the leading in the UV regime part of the Hessian operator, which
is a crucial element for the BV method of computation, addition of just
$\omega_{\kappa}R$ does not help too much and the operator is still degenerate
since it is required that all its sectors are with six-derivative differential
operators: are non-vanishing and non-degenerate there.

We propose the following procedure for the derivation of correct limiting
cases analyzed here. First the theory with $\omega_{C}\neq 0$, $\omega_{R}\neq
0$ and $\omega_{\kappa}\neq 0$ is analyzed using Feynman diagram approach. The
results for UV divergences must be identical to the ones found in (33) using
the BV technique. They do not show any singularity at this moment. In Feynman
diagram computation one can take the limit $\omega_{C}\to$ or $\omega_{R}\to
0$, while the propagator and perturbative vertices are still well-defined. In
these circumstances we have that still $\omega_{\kappa}\neq 0$. We admit that
the theory loses now its renormalizability properties, but we just want to
project the UV-divergent action onto the terms with the structure of three GR-
covariant terms $C^{2}$, $R^{2}$ and $\rm GB$. For this the method of Feynman
graph computation is still suitable since it only requires the well-defined
propagator, but it can work even in badly non-renormalizable theories. This is
like taking the naive limits of $\omega_{C}\to$ or $\omega_{R}\to 0$ in the
results from (33) respectively, regardless which way they were obtained. One
justifies the step of taking these limits by recalling that we are still
working in the Feynman diagram approach, and not with the BV technique.
Finally, one sends $\omega_{\kappa}\to 0$ hoping that this does not produce
any finite discontinuity in the results for four-derivative UV divergences.
This is justified by dimensional analysis arguments provided also earlier in
this article and by the fact that except in the case of conformal gravity
theory in $d=4$, there is no any enhancement of local symmetries in the limit
$\omega_{\kappa}\to 0$. Then one gets the sense for the limits considered in
these sections.

This analysis concludes the part with special limiting cases of extended six-
derivative theories, where one of the coefficients $\omega_{R}$ or
$\omega_{C}$ is to be set to zero. Probably the same considerations of some
limits can be repeated very similarly (with the exception of the conformal
gravity case) for the Stelle quadratic gravity models, but we will skip this
analysis here since it can be found in the literature.

##  5 Stability of HD theories

Above we have seen that HD theories of gravitation are inevitable due to
quantum considerations. They also come with a lot of benefits that we have
discussed at length before like super-renormalizability and the possibility
for UV-finiteness. However, it is also well known that they have their own
drawbacks and problems. One of the most crucial one is the issue of unitarity
of the scattering ($S$-matrix) in perturbative framework. This is of course in
the situation when we can discuss the scattering problems, so when we can
define asymptotic states in interacting gravitational background, so when the
gravitational spacetime is asymptotically flat. In more generality, the
related issue is of quantum stability of the theory.

In general, in literature about general HD theories there exist various
proposals for solutions of these perennial problems. We can mention here a few
of them: $PT$-symmetric quantum theory, Anselmi-Piva fakeon prescription, non-
local HD theories, benign ghosts as proposed by Smilga, etc. Below we will try
to describe some of their methods and show that the problems with unitarity or
with the stability of the quantum theory can be successfully solved. We also
provide arguments thanks to Mannheim [57, 58] that the gravitational coupled
theory must be without problems of this type, if the original matter theory
was completely consistent.

We first express the view that the stability of the quantum theory is
fundamental, while the classical theory may emerge from it only in some
properly defined limits. Hence we should care more about the full even non-
linear stability on the quantum level and some instabilities on the classical
level may be just artifacts of using classical theory which cannot be defined
by itself without any reference to the original fundamental quantum theory. An
attempt to understand the stability entirely in classical terms may be doomed
to clearly fail since forgetting about the quantum origin may be here
detrimental for the limiting process. If the quantum theory is stable and
unitarity is preserved, then this is the only thing we should require since
the world is in its nature quantum and physically we know that it is true that
$\hbar=1$ in proper units, rather than $\hbar\to 0$, so the classical limit
may be only some kind of illusion. If there are problems with classical
stability analysis like this done originally by Ostrogradsky, then this may
only mean that the classical theory obtained this way neglects some important
features that were relevant on the quantum level for the full quantum
stability of the system.

First, in Anselmi-Piva prescription one solves completely the unitarity issue
for HD theories by invoking fakeon prescription to take properly into account
the contribution of particles which in the spectrum are related to higher
derivatives theories and which typically are considered as dangerous for the
unitarity of the theory. The presence of a particle with negative residue
called a ghost at the classical level makes the theory not unitary in its
original quantization based on the standard Feynman prescription [7] of
encircling the poles for the loop integrals. A new quantum prescription, as
recently introduced by Anselmi and Piva [59, 60, 61] was based on the earlier
works by Cutkosky, Landshoff, Olive, and Polkinghorne [62]. The former authors
invented a procedure for the Lee-Wick theories [63, 64], which allow them to
tame the effects typically associated to the presence of ghosts in the
Stelle’s theory. In this picture, the ghost problem (also known as unitarity
problem) is solved consequently at any perturbative order in the loop
expansion [61] done for the loop integrals which need to be computed in any
QFT, if one requires higher order accuracy.

At the classical level, the ghost particle (or what Anselmi and Piva define as
“fakeon”, because this particle understood as a quantum state can only appear
as a virtual particle and inside perturbative loops) is removed from the
perturbative spectrum of the theory. This is done by solving the classical
equations of motion for the fakeon field by the mean of a very specific
combination of advanced plus retarded Green’s functions and by fixing to zero
the homogeneous solution of resulting field equations [65, 66]. This is then
equivalent to removing the complex ghosts in the quantum theory from the
spectrum of asymptotic quantum states by hand. However, this choice and this
removal decision is fully preserved and protected by quantum corrections,
hence it does not invalidate the unitarity of the $S$-matrix at higher loop
orders.

Such prescription of how to treat virtual particles arising due to HD nature
of the theories is very general and can be applied to both real and complex
ghosts, and also to normal particles, if one wishes to. (Every particle can be
made fake, so without observable effects on the unitarity of the theory.) In
particular, this prescription is crucial in order to make perturbatively
unitary the theory proposed by Modesto and Shapiro in [67, 68] which comes
under the name of “Lee-Wick quantum gravity”. The latter class of theories is
based on the general gravitational higher-derivative actions as proposed by
Asorey, Lopez, and Shapiro [12]. In this range of theories, we can safely
state to have a class of super-renormalizable or UV-finite and unitary higher-
derivative theories of QG. In order to guarantee tree-level unitarity, the
theory in [67, 68] has been constructed in such a way that it shows up only
complex conjugate poles in the graviton’s propagator, besides the standard
spin-2 pole typically associated with the normal massless graviton particle
with two polarizations. Afterwards, the new prescription by Anselmi and Piva
[61] guarantees the unitarity of this theory at any perturbative order in the
loop expansion.

We also emphasize that the Stelle’s quadratic theory in gravitational
curvatures [7] with the Anselmi-Piva prescription is the only strictly
renormalizable theory of gravity in $d=4$ spacetime dimensions, while the
theories proposed in [67, 68] are from a large (in principle infinite) class
of super-renormalizable or UV-finite models for quantum gravity.

Next, in the other approach pioneered by Bender and Mannheim to higher-
derivative theories and to non-symmetric and non-Hermitian quantum mechanics
[69, 70], one exploits the power of non- Hermitian $PT$-symmetric quantum
gravity. Here, the basic idea is that the gravitational Hamiltonian in such
theories (if it can be well-defined), is not a Hermitian operator on the
properly defined Hilbert space of quantum states, rather it is only
$PT$-symmetric Hamiltonian. Then some eigenstates of such a Hamiltonian may
correspond to non-stationary solutions of the original classical wave
equations. They would indeed correspond in the standard classical treatment to
the Ostrogradsky instabilities. The famous example are cosmological run-away
solutions or asymptotically non-flat gravitational potentials for the black
hole solutions. The problem of ghosts manifests itself already on the
classical level of equations of motion, where one studies the linear
perturbations and its evolution in time. For unstable theories, the
perturbations growth is without a bound in time. But in some special
solutions, like for example present in models of conformal gravity, these
instabilities are clearly avoided and then one can speak that ghosts are
benign in opposition to them being malign in destroying the unitarity of the
theory. Such benign ghosts [71, 72] are then innocent for the issues of
perturbative stability.

In the $PT$-symmetric approach to HD theories at the beginning, one cannot
determine the Hilbert space by looking at the $c$-number propagators of
quantum fields. In this case, one has to from the start quantize the theory
and construct from the scratch the Hilbert space, which is different than the
original naive construction based on the extension of the one used normally
for example for two-derivative QFT’s. With this new Hilbert space and with the
non-Hermitian (but $PT$-symmetric) Hamiltonian the theory revealed to be
quantum-mechanically stable. This is dictated by the construction of the new
Hilbert space and the structure of the Hamiltonian operator. In that case the
procedure of taking the classical limit, results in the definition of the
theory in one of the Stokes wedges and in such a region the Hamiltonian is not
real-definite and the corresponding classical Hamiltonian is not a Hermitian
operator. Therefore, the whole discussion of Ostrogradsky analysis is correct
as far as the theory with real functions and real-valued Hamiltonians is
concerned, but it is not correct for the theory which corresponds to the
quantum theory which was earlier proven to be stable quantum-mechanically. The
whole issue is transmitted and now there is not any problem with unitarity or
classical stability of the theory, but one has to be very careful in attempts
to define the classical limiting theory.

We also repeat here arguments proposed by Mannheim about stability of the
resulting gravitation-matter coupled theory [57, 58]. First we take some
matter two-derivative model (like for example standard model of particle
physics, where we have various scalars, fermions and spin-1 gauge bosons).
This theory as considered on flat Minkowski background is well known to be
unitary so it gives $S$-matrix of interactions with these properties. The
model can be said that it is also stable on the quantum level. Now, we want to
couple it to gravity, or in other words put it on gravitational spacetime with
non-trivial background in such a way that the mutual interactions between
gravitational sector and matter sector are consistent. This, in particular,
implies that the phenomena of back-reaction of matter species on geometry are
not to be neglected. The crucial assumption here is that this procedure of
coupling to gravity is well behaved and for example, it will not destroy the
unitarity properties present in the matter sector. We know that the theory in
the matter sector is stable and also its coupling to geometry should be stable
on the full quantum level. After all, this is just simple coupling procedure
(could be minimal coupling) to provide mutual consistent interactions with the
background configurations of the gravitational field.

Next, on the quantum level described, for example, by functional path
integral, we can decide to completely integrate out matter species still
staying on the general gravitational background. As emphasized in section 1,
such procedure in $d=4$ spacetime dimensions generate effective quantum
gravitational dynamics of background fields with higher derivatives, precisely
in this case there are terms of the type $C^{2}$ and $R^{2}$ (the latter term
is absent when the matter theory is classically conformally invariant). In
other words, the resulting functional of the quantum partition function of the
total coupled model is a functional of only background gravitational fields.
This last reduced or “effective” functional is given by the functional
integral over quantum fluctuations of gravitational field of the theory given
classically by the action with these HD above terms. Let us recall now what we
have done, namely we have simply integrated out all quantum matter fields,
which is an identity transformation for the functional integral representation
of the partition function $Z$ of the quantum coupled theory. Since this
transformation does not change anything, then also the resulting theory of
gravitational background must necessarily possess the same features as the
original coupled theory we started with. Since the first theory was unitary,
then also the last one theory of pure gravity but with higher-derivative terms
must be unitary too. We emphasized that both theories give the same numerical
values of the partition function $Z$ understood here as the functional of the
background spacetime metric. In the first theory the integration variables
under functional integrals are quantum matter fields, while in the second case
we are dealing with pure gravity so we need to integrate over quantum
fluctuations of the gravitational fields. In the last case the model, which
gives the integrand of the functional integral is given by the classical
action $S_{\rm HD}$, so it contains necessarily higher derivatives of the
gravitational metric field.

There also exist possibilities that ghosts or classical instabilities one sees
on the classical level thanks to Ostrogradsky analysis disappear. This may
happen if for example, some very specific (or fine-tuned) initial or boundary
conditions are used for solving non-linear higher-derivative classical
equations of motion of the theory. It is not excluded as proven by Smilga that
some instabilities may go away if one analyzes such special situations.

Various cures have been proposed in the literature for dealing with the
ghosts-tachyon issue: Lee-Wick prescription [63, 64], fakeons [61, 65, 73,
66], non-perturbative numerical methods [74, 72, 71, 75, 76, 77, 78], ghost
instabilities [79, 80, 81], non- Hermitian $PT$-symmetric quantum gravity
based on $PT$-symmetric quantum mechanics [69, 70], etc (see also [82, 83, 84,
85, 86, 87]). One might even entertain the idea that unitarity in quantum
gravity is not a fundamental concept. So far, there is no a consensus in the
community which solutions is the correct one. The unfortunate prevalent
viewpoint is that none of the proposed solutions solves conclusively and
completely the problem. And it seems that sadly the solutions proposed in the
literature are not compatible and are unrelated to each other.

Therefore all the arguments given above should convince the reader that the HD
(gravitational) theories are stable on the full quantum level. In particular,
this means that for situations in which we can define asymptotic states (like
for asymptotically flat spacetimes) the scattering matrix between fluctuations
of the gravitational field is unitary on the quantum level and both
perturbatively and non-perturbatively.

##  6 Conclusions

In this contribution, we have discussed the HD gravitational theories, in
particular six-derivative gravitational theories. First, we motivated them by
emphasizing their various advantages as for the models of consistent Quantum
Gravities. We showed that six-derivative theories are even better behaved on
the quantum level than just four-derivative theories, although the latter ones
are very useful regarding scale- and conformal invariance of gravitational
models. Moreover, the models with four-derivative actions serve as good
starting examples of HD theories and they are reference points for
consideration of six- and higher order gravitational theories. We first tried
to explain the dependence of the beta functions in six-derivative theories by
drawing analogies exactly to these prototype theories of Stelle gravity. We
also emphasize that only in six-derivative gravitational models we have the
very nice features of super-renormalizability and the narrow but still viable
option for complete UV-finiteness. This is why we think super-renormalizable
six-derivative theories have better control over perturbative UV divergences
and give us a good model of QG, where this last issue with perturbative
divergences is finally fully under our control and theoretical understanding.

In the main part of this paper, we analyzed the structure of perturbative one-
loop beta functions in six-derivative gravity for couplings in front of terms
containing precisely four-derivative in the UV-divergent part of the effective
actions. These terms can be considered as scale-invariant term since couplings
in front of them are all dimensionless in $d=4$ spacetime dimensions. Our
calculation for these divergences was done originally in the Euclidean
signature using the so-called Barvinsky-Vilkovisky trace technology. However,
the results are the same also in the Minkowskian signature independently which
prescription one uses to rotate back to the physical relativistic Lorentz
signature case, whether this is standard Wick rotation, or the one using
Anselmi-Piva prescription using fakeons. This is because they are the leading
divergences in the UV regime, and hence they do not completely depend how the
rotation procedure is done from Euclidean to Minkowskian and how for example
the contributions of arcs on the complex plane is taken into account since the
last ones give subleading contributions to the UV-divergent integrals.
Moreover, the calculations of beta functions that we presented in this paper
has very nice and important features of being renormalization scheme-
independent since they are done at the one-loop, but the expressions we get
for them are valid universally. These are exact beta functions since they do
not receive any perturbative corrections at the higher loop orders since the
six-derivative gravitational theory is super-renormalizable in $d=4$. Another
part of good properties of the beta functions obtained here are the complete
gauge independence and also independence on the gauge-fixing parameters one
can use in the definition of the gauge-fixing functional. These last two
properties are very important since in general gravitational theory we have
the access to perturbative computation only after introducing some spurious
element to the formalism which are related to gauge freedoms (in this case
these are diffeomorphism symmetries). We modify the original theory (from the
canonical formalism) by adding various additional fields and various spurious
nonphysical (gauge) polarizations of mediating gauge bosons (in our case of
gravitons) in order also to preserve relativistic invariances. These are
redundancies that have to be eliminated when at the end one wants to compute
some physical observables. Therefore, it is very reassuring that our final
results are completely insensitive to these gauge-driven modifications of
original theories.

Our beta functions being exact and with a lot of nice other properties,
constitute one significant part of the accessible observables in the QG model
with six-derivative actions. Their computation is a nice theoretical exercise,
which of course from the sense of algebraic and analytic methods used in
mathematical physics has its own sake of interest. However, as we emphasized
above these final results for the beta functions may have also meaning as true
physical observables in the model of six-derivative QG theories.

We described in greater detail the analysis of the structure of beta functions
in this model. First we used arguments of energy dimensionality and the
dependence of couplings on the dimensionless fundamental ratio of the theory
$x$. Next, we tried to draw a comparison between the structure of 4-derivative
gravitational Stelle theory and six-derivative theory in $d=4$ dimensions. We
showed the dependence on the parameters $x$ is quite opposite in two cases.
The case with four-derivative theory is exceptional because the model without
any $R$ term in the action (and also without the cosmological constant term)
enjoys enhanced symmetry and then the quantum conformal gravity is
renormalizable at the one-loop order, so then it is a special case of a
sensible quantum physical theory (up to conformal anomaly problems discussed
earlier). We also remark that in the cases of $x\to 0$ and $x\to\infty$ the
generic six-derivative theories are badly non-renormalizable. This was the
source of the problem with attempts to obtain sensible answers in these two
limits. Non-renormalizability problem must show in some place in the middle or
at the end of the computation to warn us that at the end we cannot trust in
the final results for the beta functions in these cases. In these two cases
this problem showed indeed in two different places and the logical
consequences of this were strongly constraining the possible form of the
rational $x$-dependence of these results. Thanks to these considerations we
were able finally to understand whether the positive or inverse powers of the
ratio $x$ must appear in the final results for beta functions in question. Of
course, we admit that this analysis is a posteriori since we first derived the
results for the divergences and only later tried to understand the reasons
behind these results. But eventually we were able to find a satisfactory
explanation.

And there are a few of additional spin-offs of the presented argumentation.
First, we can make predictions about the structure of beta functions in
8-derivative and also of other higher-derivative gravitational theories with
the number of derivatives in the action which is bigger than 4 and 6 (analyzed
in this paper). We conjecture that the structure should be very similar to
what we have seen already in the generic case with six derivatives, so only
positive powers of the corresponding fundamental ratio $x$ of the theory, and
probably only in the sector with $C^{2}$ type of UV divergences. Another good
side effect is that we provide first (to our knowledge) theoretical
explanation of the structure of beta functions as seen in four-derivative case
of Stelle theory in $d=4$ spacetime dimensions. It is not only that the theory
with $C^{2}$ action is exceptional in $d=4$ dimensions; we also “explained”
these differences based on an extension of the theory to include higher-
derivative terms like with 6-derivative and quantify to which level the theory
with $C^{2}$ action is special and how this reflects on the structure of its
one-loop beta functions. We remark that in Stelle gravity (or even in its
subcase model with conformal symmetry based on the $C^{2}$ action), there are
contributions to beta functions originating from higher perturbative loops
since the super-renormalizability argument based on power counting analysis
does not apply here. Our partial explanation of the structure of the one-loop
beta functions in Stelle theory in $d=4$ uses a general philosophy that to
“explain” some numerical results in theoretical physics, one perhaps has to
generalize the original setup and in this new extended framework looks for
simplifying principles, which by reduction to some special cases show
explicitly how special are these cases not only qualitatively but also
quantitatively and what this reduction procedure implies on the numbers one
gets as the results of the reduction. For example, one typically extend the
original framework from $d=4$ spacetime dimension fixed condition to more
general situation with arbitrary $d$ and then draw the general conclusion as a
function of $d$ based on some general simple principles. Then finally, the
case of $d=4$ is recovered as a particular value one gets when the function is
evaluated for $d=4$. And this should explain its speciality. In our case, we
extended the four-derivative theory by adding terms with six derivatives and
in this way we were able to study a more generic situation. This was in order
to understand and explain the structure of divergences in the special reduced
case of conformal gravity in $d=4$ and of still generic four-dimensional
Stelle theory. We think that this is a good theoretical explanation which
sheds some light on the so far mysterious issue of the structure of beta
functions. One can also see this as another advantage of why it is worth to
study generalizations of higher-derivative gravitational actions to include
terms with even more higher number of derivatives, like 6-derivative,
8-derivative actions, etc.

Finally, here we can comment on the issue of experimental bounds on the values
of the ratio $x$. Since it appears in six-derivative gravitational theory the
constraints on its possible values are very weak. Slightly stronger
constraints apply now for the corresponding value of the ratio in four-
derivative Stelle gravitational theory in $d=4$ case. Since the main reason
for higher-derivative modifications of gravity comes because of consistency of
the coupled quantum theory, then one would expect that the stringent bounds
would come from experimental measurement in the real domain of true quantum
gravity. Of course, right now this is very, very far, if possible at all,
future for experimental gravitational physics. This is all due to smallness of
gravitational couplings characterized by $G_{N}$ proportional inversely to the
Planck mass $M_{P}\sim 10^{19}\,{\rm GeV}$. And in the quantum domain of
elementary particle physics this scale is bigger than any energy scale of
interactions between elementary quanta of matter. This implies that also
quantum gravitational interactions are very weak in strength. Hence the only
experimental/observational bounds we have on the coefficients in front of
higher-derivative terms come from the classical/astrophysical domain of
gravitational physics and they are still very weak. To probe the values of the
coefficients in front of six-derivative terms, one would have to really
perform a gravitational experiment to the increased level of accuracy between
elementary particles in the full quantum domain, which is now completely
unfeasible. Hence, we have to be satisfied with already existing very weak
bounds, but this lets us to freely consider theoretical generic situation with
arbitrary values of the ratio $x$ since maybe only (very far) future
experiments can force us theoreticians to consider some more restricted subset
or interval for the values of the $x$ ratio as consistent with observed
situation in the Nature. For the moment it is reasonable to consider and
explore theoretically all possible range of values for the $x$ ratio and also
both possible signs. (Only the case with $x=0$ is excluded as a non-
renormalizable theory that we have discussed before.)

In the last section of this contribution, we commented on the important issue
of stability of higher-derivative theories. We touched both the classical and
quantum levels, while the former should not be understood as a standalone
level on which we can initially (before supposed quantization) define the
classical theory of the relativistic gravitational field. We followed the
philosophy that the quantum theory is more fundamental and it is a starting
point to consider various limits, if it is properly quantized (in a sense that
the quantum partition function is consistently defined, regardless of how we
get there to its form, no matter which formal quantization procedure we have
been following). One of such possible limit is the classical limit where the
field expectation values are large compared to characteristic values as found
in the microworld of elementary particles. And also the occupation number for
bosonic states are large number (of the order of Avogadro number for example).
Then we could speak about coherent states which could define well classical
limit of the theory. Such procedure has to be followed in order to define HD
classical gravitational theory. We emphasized that quantum theory is the basis
and classical theory is the derived concept, not vice versa. On the quantum
level we shortly discussed various approaches present in the literature to
solve the problems with unwanted ghost-like particle states. They were
classified in two groups: theories with $PT$-symmetric Hamiltonian and
theories with Anselmi-Piva prescription instead of the Feynman prescription to
take into account contributions of the poles of the ghost but without spoiling
the unitarity issue. On the quantum level, we considered mainly the issue with
unitarity of the scattering matrix since this seems the most problematic one.
The violation of unitarity would signal the problem with conservation of the
probability of quantum processes. Something that we cannot allow to happen in
quantum-mechanical framework for the isolated quantum system (non-interacting
with the noisy decohering and dissipative or some thermal environment). Of
course, such an analysis was tailor-made for the cases of gravitational
backgrounds on which we can define properly the scattering process.

In general, the scattering processes are not everything we can talk about in
quantum field theories even for on-shell quantities. The analysis of some on-
shell dressed Green functions may also show some problems with quantum
stability of the system. Therefore, we briefly also described the results of
the stability analysis, both on the classical and quantum level and to the
various loop accuracy in QG models. This analysis is in principle applicable
to the case of any gravitational background, more general than the one coming
with the requirement of asymptotic flatness. We also mentioned that in some
cases of classical field theory the analysis of classical exact solutions
shows that the very special and tuned solutions are without classical
instabilities and they are well-defined for any time starting with very
special initial or boundary conditions. For example, here we can mention the
case of so-called benign ghosts of higher-derivative gravitational theories as
proposed by Smilga some time ago. This should prove to the reader that we are
dealing with the theories which besides a very interesting structure of
perturbative beta functions, are also amenable to solve the stability and
unitarity issues in these theories, both on the quantum as well as on the
classical level. With some special care we can exert control and HD
gravitational theories are stable quantum-mechanically and this is what
matters fundamentally.

###### Acknowledgements.

We would like to thank I. L. Shapiro, L. Modesto and A. Pinzul for initial
comments and encouragement about this work. 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 “Algebraic and analytic methods in physics”
– AAMP XVIII conference for accepting our talk proposal and for creating a
stimulating environment for scientific online discussions.

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