###### Abstract

We present and discuss well known conditions for ultraviolet finiteness and
asymptotic safety. The requirements for complete absence of ultraviolet
divergences in quantum field theories and existence of a non-trivial fixed
point for renormalization group flow in the ultraviolet regime are compared
based on the example of a six-derivative quantum gravitational theory in $d=4$
spacetime dimensions. In this model, it is possible for the first time to have
fully UV-finite quantum theory without adding matter or special symmetry, but
by inclusion of additional terms cubic in curvatures. We comment on
similarities and some apparent differences between the two approaches, but we
show that they are both compatible to each other. Finally, we motivate the
claim that actually asymptotic safety needs UV-finite models for providing
explicit form of the ultraviolet limit of Wilsonian effective actions
describing special situations at fixed points.

###### keywords:

quantum gravity; quantum field theory; super-renormalizable gravity; higher-
derivative gravity; UV-finite gravity; asymptotic safety; six-derivative
gravity

1 1 0 Academic Editor: Mariusz P. Dąbrowski, Adam Balcerzak, Vincenzo Salzano
and Gonzalo J. Olmo Received: 31 January 2022; Accepted: 05 April 2022;
Published: 08 April 2022 Ultraviolet Finiteness or Asymptotic Safety in Higher
Derivative Gravitational Theories Lesław Rachwał ${}^{1,\,{{\dagger}}}$ Leslaw
Rachwal

##  1 Introduction

The behaviour of quantum gravity (QG) models at very high energies, so in the
ultraviolet (UV) regime, was always of big interest for research in
theoretical gravity. Quantum considerations in the perturbative framework
force us to use modified gravity models as viable and consistent models for QG
dynamics. This is because of the well known fact of quantum non-
renormalizability of the theory based on the Einstein–Hilbert gravitational
action. In this note we want to compare the methodology and conditions between
ultraviolet-finite and asymptotically safe theories. We will consider them on
a particular example of higher derivative gravities in $d=4$ spacetime
dimensions. To be definite, the model with six derivatives in the
gravitational dynamics will be for these purposes studied here. After the
considerate analysis, we decided that the $6$-derivative gravitational theory,
being in the general class of higher derivative local gravitational models,
better shows the differences between the two approaches for UV-completion of
gravitational theories. Namely, we believe it does this better than
$4$-derivative Stelle quadratic theory, which in $d=4$ is simply
perturbatively renormalizable model Stelle:1976gc . In a pure gravity case,
this last model is not completely without perturbative UV divergences (is not
UV-finite), but it can reach a non-trivial fixed point of the renormalization
group (RG) flow in the UV regime (can give rise to the asymptotically safe
model as found in Codello:2006in ). The longer discussion of this issue is
presented in Section 2.1, where we point out that the conformal window is not
possible for pure Stelle gravity. However, the case of pure six-derivative
gravity offers for us such a possibility, probably for the first time. Pure
refers to a gravitational theory which is a quantum theory of only metric
degrees of freedom, without addition of matter fields or additional symmetry
(like in supergravity cases). Such a modified gravity model with six
derivatives in its defining classical action, first studied on the quantum
level in Rachwal:2021bgb , is a good and interesting theoretical laboratory
for investigations of various perturbative and also non-perturbative RG flows.

In the framework of quantum field theories (QFT) of fundamental interactions,
one of the biggest theoretical problems is the proper form of UV-completion of
theories which work well at some low energy regimes. One could simply view
them as effective theories valid in these energy ranges. As such, they must
work well there. The interesting conceptual question is what happens with
these theories when the energy scale is increased. There are various possible
options here—one positive is that the model in question can be renormalizable
(perturbatively or non-perturbatively). However, this is not the end of the
story in the UV regime since even some renormalizable theories may meet some
problems on the way towards the UV regime. A possible danger is for example,
the presence of the Landau pole for running coupling parameters. Then, such
models are not UV-complete. A first possible resolution is to require that the
models are not only renormalizable but that they are also asymptotically free
(AF) in the UV regime and that the couplings tend to vanishing values in the
high energy limits. This scenario is realized, for example in the UV-
consistent QCD—theory of strong interactions. We remark however, that in such
a model the UV divergences are still present but that they are fully under
control and hence are not anymore dangerous and that the couplings do not run
to divergent values at any finite (or at infinite) energy scales.

The renormalizability brings the additional control over divergences that one
encounters by doing higher order (loop) computations in the QFT framework
within the perturbative calculus very frequently used for considerations of
quantum corrections. Unfortunately, this is not the case for Einsteinian
gravity in $d=4$. In the latter theory the control over perturbative
infinities is lost when the situation is considered off-shell, at the two-loop
or higher order of accuracy, or when the matter fields are added to the
quantum system.

One of the way out for the problem of infinities in QG, is to consider higher-
derivative theories in $d=4$ starting first with quadratic gravity of Stelle.
Then, the renormalizability is regained, but the price is the higher-
derivative nature of classical and quantum dynamics with various
instabilities. Another option as proposed by Weinberg in 1979 Weinberg:1980gg
is to solve the problem with perturbative UV divergences by demanding an
existence of a non-trivial fixed point (FP) Smolin:1981rm  of RG flow in the
UV regime. This last scenario is inherently non-perturbative in values of
coupling parameters since at the FP the running couplings must attain some
finite non-zero values. In this way the situation differs from the trivial or
Gaussian FP, where the FP values are vanishing. We also note that in AS the
problem of higher derivative instabilities is probably solved by the arguments
which were presented in Platania:2020knd , which show that the Boulware–Deser
ghosts are most probably fictitious particles. Contrary to the solution
proposed by Weinberg of the problem of UV divergences, usually in the
perturbative framework, another approach is proposed. Namely one seeks to find
a theory so special and tuned that all UV divergences disappear that there are
no issues with them and all the perturbative loop integrals are perfectly
finite regarding the UV limits of integration.

The above two approaches for definitely solving the problem with UV infinities
have a lot in common. There are however some subtle but important differences
that we want to discuss and oppose in this article. We believe that the
analysis of such dissimilarities will be beneficial for bringing together the
research directions and for finding a unified description of QFT of QG which
is without any problem at arbitrary high energy scales. For this goal, we will
present the results of the analysis for the promising model of six-derivative
gravitational theories, which can be made completely without infinities in the
UV regime.

In full generality, a consistent model of QFT of fundamental interactions,
like in the case of QG for metric fluctuations $h_{\mu\nu}$ for the quantum
graviton fields, must be defined as an interpolating theory between two FPs of
RG (or in the special case, like of QED, we must invoke some threshold
phenomena in IR to decouple massive charged modes). This relates to the
necessity of having the possibility to define the interpolating $c$-function
as this was emphasized in Anselmi:1997am ; Anselmi:1997ys  for any well-
defined QFT model. One of the two FPs may be in the infrared (IR) regime and
we will not discuss here the infrared safety notion related to it. The second
FP must be in the UV regime and its presence there is required for UV-
completion of the QFT model.

It is well known that QFT valid at all energy scales (without any other form
of UV-completion like by string theory or by some quantum theory of some other
non-local objects) should be defined as an interpolating QFT between two
FPs—between IR and UV regimes. These two endpoints FP of RG are musts,
otherwise for finite or infinitely large (or infinitely small) energies the RG
flow in the theory reaches a dangerous Landau pole, and this means that the
description of the quantum system in terms of continuous field theory of
point-like particles fails, or at least that the perturbativity fails and one
has to resort to study some non-perturbative effects here. For the question of
UV consistency the IR problems do not matter and the IR FP is unnecessary. The
safe situation for the IR regime of the theory can be obtained, for example,
if threshold phenomena decouple all interacting modes (by giving them mass)
and only free (Abelian) massless modes remain, then there is no need for IR FP
of interactions since there are no IR interactions. There exist possible UV-
completions by a quantum theory of something else and these are not
relativistic fields and we will not discuss them here. However, here and in
the QFT approach to QG, the field-theoretical description of the quantum
system is assumed to be valid at all energy scales. Hence QFT of fundamental
interactions must be defined with the sense at all energy scales and the two
FPs must exist (some of them may be of course trivial Gaussian FP describing a
free non-interacting theory).

Next, these two FPs may be found in QFT of completely different sets of active
degrees of freedom from the point of view of UV and IR FPs, respectively. So
we must allow for the possibility of transmutation and of changing the nature
of degrees of freedom of the theory in the course of RG flow. Still, we
require them to be quantum fields with point-like nature of excitations around
the vacuum. We should also allow for the impact of the important effects due
to threshold phenomena when some degrees of freedom acquire mass, they
decouple, and are effectively integrated out on the level of functional
integral. In such a case, they are absent from the spectrum at lower energies,
towards the IR limit. The vacuum for these theories (both in the UV and IR
regimes) is always the same since it has to be defined non-perturbatively and
exactly as the stable state with minimal value of energy. This definition is
independent of the energy of quantum fluctuations, regardless whether they are
at UV FP (infinitely large energies) or at the IR FP (infinitely small
energies). All this is needed for the consistent description of the scheme of
QFT valid at all energies and with the RG flow in it which interpolates
between the two FPs.

In the next two subsections we briefly discuss the main ideas behind the
theoretical notions of UV-finiteness in quantum field theories and also behind
the asymptotic safety (AS). The second section will be devoted for a short
presentation of the six-derivative gravitational theories in $d=4$. We will
also quote the main results for exact beta functions of perturbative couplings
there. In the main section of this paper, we analyze the similarities and
differences between AS and UV-finite scenarios based on the example six-
derivative theory. Finally we draw our conclusions and comment on a
possibility of unified framework and how UV-finiteness should be incorporated
in the UV safety scenario.

###  1.1 UV-Finiteness

In the first case, one typically relies heavily on the perturbative approach.
It is not so surprising since the first place where one meets dangerous UV
divergences is during the expansion of quantum predictions of general QFTs in
small coupling parameters. The ultraviolet divergences are the most ubiquitous
theoretical phenomena appearing in almost any QFT models. Contrarily, if the
QFT model is very special, then these infinities may be absent. Such
situations typically call for an enhancement of symmetries to full quantum
conformal symmetry. However, here we want to speak about perturbative UV
divergences. If they are absent in all orders of perturbative calculus, then
we can say that the theory is UV-finite. Since, for example, at the one-loop
level, these UV divergences are related to the perturbative beta functions of
the couplings of the theory, then an equivalent reformulation of the
conditions for UV-finiteness reads

$\beta_{i}=0,$ (1)

where the index $i$ counts all the couplings $g_{i}$ of the theory. For this
the general theory we write in the form defined by the action
$S=\sum_{i}g_{i}{\cal O}_{i}$, where ${\cal O}_{i}$ lists all classical terms
built out of fields and derivatives of the model.

The condition in (1) is not typically satisfied for any value of the
perturbative couplings $g_{i}$ of the model. The situation must be very fine-
tuned to have $\beta_{i}=0$ to all loop orders (and also non-perturbatively).
This requirement must also hold at any energy scale since the running energy
scale $E$ is never an explicit argument of the equation in (1). In the
language of RG flows such a special theory sits at the FP of RG all the RG
time and never leaves it. This is obvious since RG running phenomena arise
from the ability to absorb all UV-divergent quantities in bare non-observable
parameters of the model. In the case of UV-finite theories, there is no RG
running and the theory is scale-invariant on the full quantum level. Likely
this last symmetry can be promoted to the full quantum conformal invariance.

There are few known examples of theories that they are completely UV-finite in
the standard relativistic QFT framework. Enhanced symmetry helps in finding
such models, but this is not a necessary ingredient. We can mention here an
example of ${\cal N}=4$ super-Yang-Mills theory Namazie:1982br  and also of
${\cal N}=4$ supergravity due to Fradkin and Tseytlin Fradkin:1985am . These
models require the highest possible for consistency level of supersymmetry in
their formulations. However, in other gauge-scalar-Yukawa models as found by
Litim and Sannino in Codello:2016muj ; Litim:2014uca , the UV-finiteness is
with a less degree of symmetry.

Generally, it is very difficult to find QFT models completely without any
perturbative UV divergences at any loop orders and such UV-finite theories are
very special. Of course, they do not show any problem in the UV regime and
they are definitely UV-complete quantum models of field theory. Strictly
speaking UV-finiteness should also hold non-perturbatively and one should
avoid in such models also possible UV divergences that one could find in some
non-perturbative approaches to QFT. We envisage therefore the possibility that
infinities and divergences can be also non-perturbative in some schemes. The
condition for complete absence of UV divergences is typically a non-linear
condition in all couplings of the theory and first it is difficult to find
one. Secondly it is even more difficult to solve it for couplings and in this
way define completely the position of the FP where the RG flow effectively
stops.

In some matter theories (without gravity) UV-finiteness may be viewed as some
additional condition on the quantum dynamics. It gives rise to enhanced
conformal symmetry on the quantum level, but in matter models as we have seen
for QCD this is not essential. The problem with UV-completion still may be
solved even if the UV divergences are present even in the asymptotically very
high energies. This only means that one has to perform renormalization of
these divergences also at infinite energies.

However, in the gravitational setup UV-finiteness of some QG theory is a very
desirable feature since only with it consequently promoted to full conformal
invariance in the general-relativistic (GR) setup, we can deal with the
problem of singularities of the classical gravitational field. As emphasized
in Bambi:2016wdn ; Modesto:2016max ; Rachwal:2018gwu  full conformal symmetry
on the quantum level is a key ingredient for the successful resolution of
singularities. One of the prerequisite for this is that the quantum theory
should be completely without divergences and conformal symmetry should not be
broken at very high energies. Of course, for the issue of singularities we
need only to study the UV regime of the theory. The important question here is
how quantum corrections help or destabilize the solution of the problem with
these pathologies in classical theory. An example of the theory which solves
these issues on the classical level is a conformal gravity (Weyl gravity) in
$d=4$ spacetime dimensions. For the stabilization of these resolutions one has
to ensure that quantum effects do not destroy this conformal symmetry of the
classical theory, or in other words, that conformal symmetry is also preserved
in the quantum theory. This is achieved when there is UV-finiteness and there
are no divergences and the quantum effective action also enjoys the important
feature of being conformally symmetric. Hence UV-finiteness is a must for
consistent QG model if it has to deal also with spacetime singularities.

On the other side, one has to recover the Einsteinian GR in the IR limit as it
was explained above. However, the problem of infinities is in the opposite UV
regime. The situation at the highest energies should be very symmetric and one
can view that the IR phase is a spontaneously broken phase, where the
conformal symmetry is not present in the vacuum state of the theory. From the
UV-consistent theory we require the complete cancellation of infinities and
reaching the UV conformal phase. Moreover, this conformal phase does not
extend for all energies, especially for the low energies (or even Planckian
energies). In the IR the breaking of quantum conformality is expected
Jizba:2020hre  and is welcomed there to provide a consistent description of IR
low-energy gravitational phenomena, like gravitational scattering amplitudes,
cosmological perturbations Jizba:2014taa , gravitational threshold phenomena
Jizba:2019oaf , etc.

###  1.2 Asymptotic Safety

Asymptotic safety (AS) is an approach that in turn generalizes the notion of
asymptotic freedom for all running coupling parameters Niedermaier:2006wt . It
describes this both in the perturbative as well as in non-perturbative
attitude towards quantum physics.

The idea of Asymptotic Safety scenarios came from Weinberg and it was proposed
in 1977 Weinberg:1980gg . This was in close analogue to asymptotic freedom
(AF), which makes strong interactions not so strong in the UV regime due to
non-trivial RG flow of Yang–Mills coupling. The idea of asymptotic safety was
that many problems of Einstein quantum gravity are also solved in a similar
way due to non-trivial RG flow of coupling parameters in the gravitational
theory. One of the problem is the famous non-renormalizability of
Einstein–Hilbert (E-H) gravity (with or without cosmological constant) and in
$d=4$ spacetime dimensions, off-shell, at one-loop level and coupled to some
mater species, or also in pure gravity at the second loop level as proved by
explicit computation of UV divergences by Goroff and Sagnotti Goroff:1985sz ;
Goroff:1985th . The other issues concern the unitarity bound on the scattering
amplitudes of graviton-graviton interactions, which is apparently broken in
the theory based just on E-H action at energies of the scattering gravitons
being of the order of the Planck scale. Although it is well known that the E-H
gravity is unitary two-derivative theory these two problems deal really with
the UV regime of the quantum gravitational theory, where the simple E-H theory
is untenable on the full quantum level.

The idea of Weinberg to solve the non-renormalizability issue and the growth
without bound of the quantum scattering amplitudes was to invoke the situation
that in the UV regime a non-trivial fixed point (FP) is met of the RG flow of
running coupling parameters in the gravitational action, which is typically
taken as different from the one given by the E-H truncation action. The values
of the couplings are non-zero and can be also not very small, so the existence
of such a FP is truly a non-perturbative issue to settle which one probably
needs to use methods of non-perturbative field theory and RG flows there. One
still can hope that such a non-Gaussian FP can be reached by methods of
perturbation theory in some various small deformation parameters (not
necessarily like the coupling)—they could be dimension of spacetime away from
$d=4$ or similar ones. The existence of non-trivial FP apparently solves the
problem of non-renormalizability since now in the UV regime the theory is
without divergences, without RG flows, since the FP is met. This is also often
called as non-perturbative renormalizability. Although for example, the issue
of non-perturbative form of UV divergences of the theory and whether they are
present here or not is only rarely considered. In turn, there may exist also
non-perturbative divergences of the QFT models and the issue of their
reabsorption in some models of field theory is well posed too. Speaking maybe
too conservatively, one could say that in this last sense one has non-
perturbative renormalizability, if even non-perturbative divergences are
possible to be absorbed in the counterterms of the form identical to the terms
present in the original action of the model. After all, the notion of the
effective action of any QFT model is not restricted only to the perturbative
calculus, and also its divergent part has the meaning independently of using
it. Then the issue of renormalizability of these non-perturbative UV-
divergences is sensible too and this would constitute the core part of the
conservative definition. Of course, then the UV-completion by non-perturbative
UV FP of RG flow for all couplings is conceptually something else and this
requires the departure from the original conservative definition of what was
renormalizability. To support more modern point of view we can remark that
Wilson and others Wilson:1973jj  have formally shown that the existence of a
non-trivial UV FP (with non-perturbative values of couplings) guarantees
renormalizability of the theory in the first sense (possibility to absorb all
perturbative UV-divergences of the model in question computed via loop
integrals in counterterms of the original action). Moreover, the number of
parameters needed to reabsorb such divergences equals the number of relevant
directions of the RG flow near that UV FP.

One shall not forget the reasons why we have RG flow in the QFT at the first
place. Historically first classical models of field theories were with
constant coupling parameters. Then one could have thought of models with
spacetime-dependent coupling parameters as unnecessary complication since they
would typically break homogeneity of the spacetime. Originally coupling
parameters in field theory were constants. The great discovery and also the
great puzzle of interacting QFT in the perturbative framework came with
infinities. They were unwanted, but theoreticians had to learn how to live
with them and to deal with them in such a way to bring the predictions back to
finite values. This was finally achieved in the advent of renormalization of
infinities in QFT. The divergences were reabsorbed in counterterms of the
theory. This was possible in any renormalizable QFT model. However, there was
also a price to pay for this possibility—the couplings could not be anymore
constant, they would have to show dependence on the arbitrary renormalization
energy scale $\mu$. In this way they could be viewed as dependent on the fifth
external dimension being the energy. However, this is fully consistent with
relativistic symmetries of the underlying spacetime, if the extended
5-dimensional background spacetime is maximally symmetric and negatively
curved. The dependence on the energy as the dependence on additional
dimensions can be further developed in the framework of AdS/CFT correspondence
and its generalizations. Coming back to the perturbative framework, the reason
for RG flow is traced back to the presence of UV divergences. If the latter
were not present, then we would have only finite renormalization of couplings
and not an ensuing RG flow. Hence all the story with perturbative RG flows
started because of infinities. One can very carefully isolate them from the
functional of the quantum effective action of the theory. There they could be
both of the perturbative and also non-perturbative character. Hence the notion
of renormalizability and its further generalizations should primarily deal
with these divergences and the possibility of their reabsorption in the
original terms of the theory named here as counterterms. Additionally, this
story of UV divergences can be finally closed too, if the theory is completely
UV-finite. Therefore UV-finiteness is a very natural generalizations to the
notion of (perturbative) renormalizability.

It is however a fact that at the non-trivial (non-Gaussian FP), the theory by
definition does not show any RG effects. The issue with non-renormalizability
is more tricky here, since one cannot say then whether the perturbative or
non-perturbative UV divergences are still there or not. As a matter of fact
about divergences presence one derives by looking at the asymptotic behaviour
of various integrals, so then a special limiting procedure must be performed
to accomplish this task. Instead, the situation with an UV FP of RG is that
one is sitting at the FP already and for all the time, and then one cannot
change the energy scale and cannot perform any asymptotic limit to see the
behaviour when the FP is reached. It is true that what is really important is
not that whether the FP exists or not but rather how it reached there in the
UV regime. We can concentrate just on the leading asymptotics with the most
relevant behaviour. One could derive whether the perturbative (or also non-
perturbative) UV divergences are there so whether the problem with non-
renormalizability is ameliorated or not. For example, from the paper by Kogut
and Wilson Wilson:1973jj , it was established that the FP with a finite number
of IR-relevant directions gives rise to renormalizability in the perturbative
scheme.

As mentioned above, in the AS programme for UV-completion of some interacting
QFT models, one is more interested in the way how the FP is reached in the UV
than just the location of this FP in the parameter space (moreover, the last
one is not invariant under arbitrary coupling redefinitions, hence it does not
possess any observable physical meaning). To find a fixed point of RG, one
studies the solution in the parameter space of the equation
$\beta_{i}=\frac{dg_{i}(t)}{dt}=0$, where $t$ is the variable of the
logarithmic RG time. To quantify the way FP is reached, one must study higher
derivatives of the flow of the couplings $g_{i}(t)$, for example the second
derivatives $\beta^{\prime}_{i}=\frac{d^{2}g_{i}(t)}{dt^{2}}$ and evaluate
them at the FP, so with the condition that $\beta_{i}=0$. However, from the
theoretical point of view much better quantities are the following derivatives
put in the matrix form $M_{ij}=\frac{\partial\beta_{i}}{\partial g_{j}(t)}$,
which do not show now any explicit $t$-dependence and they are autonomous in
the system of all running couplings $g_{i}=g_{i}(t)$. The eigenvalues of such
a matrix (so called stability matrix) of derivatives are related to the so
called critical exponents of the theory at the FP. This issue is studied
extensively in all FRG papers, since these critical exponents are thought to
be observable physical quantities as we know, for example, from the theory of
critical phenomena applied to the condensed matter physics. Close to the FP
one has non-trivial scaling of couplings in a form of various power laws that
are parametrized by these critical exponents. The computation of these
critical exponents is one of the main task of the theoretical description of
the situation at the FP.

Moreover, as it was explicit in the original Weinberg’s construction besides
the existence of a non-trivial FP in UV to make the theory safe (from the
problems of the UV regime), one must also ensure that the dimensionality of
the critical surface on which such a FP lays is finite. Such a surface is
typically parametrized by the relevant directions from the point of view of
the FP in question, which is located in the UV regime. Then this
classification of the ir/relevant directions is sensitive which FP this is –
in the UV or in the IR we speak about. To each relevant direction correspond
an operator in the action of a theory that when it is included it drives the
RG flow away from the FP. Simply unmodified theory is at the FP, while the
deformation by adding some operator breaks the scale invariance and the RG
flow starts taking us away from the FP when the energy scale is lowered
compared to the formally infinite abstract theoretical energy scale of the UV
FP. The deformation action is of the general form

$S_{\rm def}=\int d^{4}x\sqrt{|g|}g_{i}{\cal O}_{i},$ (2)

where we summed over all deformation parameters and their coupling parameters
$g_{i}$. Each of this coupling should reach a finite FP value in the UV, and
away from the FP we should see its non-trivial RG running.

Additionally, if the operators are (classically) marginal from the point of
view of the non-trivial UV FP, then the running coupling at least in the
vicinity of such FP should be logarithmic with the energy scale, but the value
it attains in the limit of infinite energies should be finite and typically
non-zero (for a non-Gaussian FP). There are known examples of the studies how
such non-trivial FPs can be reached when going towards the UV regime, with
explicit both perturbative (at the one-loop level) and non-perturbative forms
of the running of the coupling parameters.

The second requirement from Weinberg about finite number of relevant
directions is very important for the predictability of the theory since the
theory with FP existing but with infinitely many ways how it can be reached is
not useful physically since this vast infinite freedom of choices for
infinitely many relevant couplings does not constrain at all the physics at
some finite intermediate energy scale, which is neither in the UV, nor in the
IR. Of course, we could also add an arbitrary (possibly even infinite) number
of irrelevant deformations from the point of view of UV FP with some
irrelevant coupling parameters. The scaling of such irrelevant couplings would
be $g\sim g^{*}+ck^{\alpha}+O(k^{\alpha-1})$, with $\alpha>0$. For relevant
couplings (in my terminology) one would have that $\alpha<0$. We remark that
since relevant couplings from the UV FP perspective run to fixed values
$g^{*}$, when $k\to+\infty$, so towards the UV FP, they do not change or
modify anything regarding the question of how the UV FP is reached. This is
why they are irrelevant from the point of view of the UV physics. (However,
they can be relevant from the point of view of IR, where also their scaling
exponent $\alpha$ may change to something else like $\alpha^{\prime}$ possibly
being or positive or negative.) As for the issue with irrelevant couplings
with $\alpha>0$ the situation with them must be definitely under full control
and they are not unconstrained as relevant couplings were at the UV FP. For
$\alpha>0$, one has that the running in the UV leads to a blow-up
$g\to+\infty$, unless $c=0$. In order to avoid this and reach the FP, one has
to lie on the UV critical surface, so the initial condition is such that the
constant $c=0$ for all irrelevant deformations (in terminology with
$\alpha>0$) and thus at the level of the effective action one would have all
such operators multiplying the corresponding (generally non-zero) fixed-point
values of the couplings $g=g^{*}$. This also means that all possible operators
consistent with symmetries and particle content of the theory must be kept in
the action (unless by chance we have that $g^{*}=0$). However, in the case of
six-derivative gravitational theory we also add here a requirement from
effective field theory that the number of derivatives in them is not bigger
than 6.

Finally, we could also add some marginal operators with marginal logarithmic
running RG flow of couplings, but here as usual with marginal deformations,
the situation needs further attention (and for example going to the higher
order in the perturbation theory) to finally pinpoint the issue whether they
are marginally relevant or irrelevant operators due to higher order
corrections. They then could fall into one of the two categories described
above.

The conditions for asymptotic safety (AS) should solve the problems with
perturbative non-renormalizability of some theories (like quantum gravity) and
also should tame the growth of some scattering amplitudes beyond the unitarity
bound and also up to infinity for infinite energies. The operational
definitions of couplings due to Weinberg is to read them from some observed
physical process, when it is sure that they are physical observables and they
are well defined and measurable, at least in principle. Then the physical
requirement of AS means that the physical scattering amplitudes or cross
sections tend to some finite values for the energy of scattering particles
tending to infinity. The theory is then indeed safe regarding the possible
infinities present in the observables of the theory. However, this very
operational scheme is very difficult to realize in practice for theoretical
computations and for finding evidence for the existence or not of the fixed
point and for the characteristic of the critical surface, on which it lays.
One rather uses the theoretical and abstract scheme of computation with
abstract energy scale and computes correlations in terms of abstract, non-
physical coupling parameters of the theory. For example, in AS approach to the
UV-completion, this is done right away using the vertex expansion
Pawlowski:2020qer  and references therein. Only later, in standard
perturbation calculus approach, one has to physically dress all the quantities
computed theoretically like Green functions (on-shell), scattering amplitudes
Christiansen:2017cxa ; Knorr:2019atm ; Denz:2016qks ; scattering  and
couplings. The usage of this scheme in principle leads to some other important
questions. We have to keep in mind that AS makes the situation under control
in the deep UV regime, asymptotically at the point where the energies are
formally infinite. However, in practical aspects of quantum gravitational
physics we will be more interested in situation at some intermediate energies,
where the energies are of the order of Planck energy or beyond (trans-
Planckian energies) and make the situation safe there regarding the scattering
amplitudes. The RG flow from the UV to the IR has also been studied in many
papers in the framework of AS and in functional renormalization group (FRG),
e.g., in Gubitosi:2019ncx  (pure gravity and in the quadratic approximation)
and also in Eichhorn:2019ybe  (coupled gravity-matter systems).

The main conditions for AS are formulated in such a way that they touch upon
the issue of existence of a non-trivial fixed point (FP) in the UV regime,
which must also lay on a finite-dimensional critical surface parameterized
locally in the vicinity of this FP by a finite number of relevant directions.
All these relevant directions correspond to the deformations that could be
added to the original action of the theory at the UV FP, so to the UV action.
Then one violates the scale-invariance present in the FP and causes a non-
trivial RG flow down with energies. The relevant operators with respect this
UV FP parametrize locally the tangent space to the critical manifold where the
touch point is precisely the FP in question. Besides these conditions imposed
in the UV regime of the theory, for the physical consistency and
phenomenological viability the RG flow in the whole parameter space down with
energies must also possess some further special features when it is considered
on the phase diagram. We will not consider them in more details here. We just
want to remark that the scenario for AS or UV-finiteness regards really the
situation at infinite energies, where the theoretical problems are the most
severe. However, one knows that to compare with QG phenomenology one must
really study the RG flow phenomena at physical energy scales, like around the
Planck scale. The conditions in the UV regime should leave some imprint and
when one runs down with energies, then for example the effective action
functional should be constrained because of this very particular asymptotics
in the UV regime.

#### AS for Dimensionless Couplings and Dimensional Transmutation for
Dimensionful Ones

Finally, when looking for pieces of evidence for AS scenario in the UV regime,
we need to discuss the beta functions $\beta_{i}$ of which couplings $g_{i}$
we will seriously consider for finding a non-trivial UV FP of RG. In the
perturbative calculus, it is well known that only logarithmic divergences in
the formal UV-cutoff (as put in momentum space) are universal and unambiguous
in any models of QFT. The infamous power-law like UV divergences (cf. their
description in Anselmi_reno ) are not in such a class since they can be easily
reabsorbed and they do not require and they do not cause any RG flow. (After
all, we remind the reader that the true reason for the RG flow of coupling
parameters, which were originally thought of to be always constant in the
classical physics, is that there are non-absorbable in the field redefinition
logarithmic UV divergences that need counterterms and renormalization and
hence also introduction of bare and running coupling parameters $g_{i}(t)$. In
the last expression we denote by $t$ the logarithmic RG-time which is defined
as $t=\ln({k}/{k_{0}})$ with some reference energy scale $k_{0}$ and with the
running energy scale $k$.) The power-law UV divergences are ambiguous and for
example they depend on the scheme of regularization used to find them, hence
the information contained in them can never be physical and will never have
any influence on true observables of theoretical models in QFT.

Due to the above reasons, we only study the AS conditions for couplings whose
RG flow is determined in the perturbative approach from logarithmic UV
divergences. If the theory is only with higher derivatives and does not
contain in their original defining action any subleading terms with the
smaller number of derivatives, then the couplings with the above description
are simply dimensionless gravitational constants. For example, in $d=4$ these
are couplings in front of GR-covariant terms of dimension four, so in front of
$R^{2}$ and $C^{2}$ terms, where $R$ denotes the Ricci scalar and $C$ the Weyl
tensor. Their RG flow is read from logarithmic UV divergences. We will only
concentrate on them in the QG models without subleading terms. The six-
derivative gravitational theory that we will present results of in the next
section is precisely of this type. The running of other prominent
gravitational couplings (like of the Newton’s constant $G_{N}$ or of the
cosmological constant $\lambda$) in the minimal six-derivative gravitational
models is absent, while when one adds also some subleading terms in the number
of derivatives, like $R^{2}$, $C^{2}$ or the E-H $R$ term, then the RG flow
for them is back and it should be then read exclusively from logarithmic UV
divergences now present also in such an extended model. The computations of
this type were first presented in Asorey:1996hz ; Modesto:2017hzl .

A corresponding FRG study of the system with six derivatives has been done
recently in Knorr:2021lll  to the one-loop perturbative level, extending at
the same time the original work on Stelle gravity from Julve:1978xn ;
Fradkin:1981iu ; Avramidi:1985ki  and later from Codello:2006in  to the case
of $d=6$ spacetime dimensions. In this new setup much more non-trivial FPs
were found. This can be viewed also an extension of our work with six-
derivative quantum gravitational dynamics to the case of six dimensions, where
the latter theory is a minimal renormalizable gravitational model. However, as
it is obvious from counting of constraints $\beta_{i}=0$ for all 10 couplings
of such a model in $d=6$ and comparing this with the number of active
couplings of the theory on which one-loop divergences are dependent, such a
minimal model cannot be UV-finite, although of course there is much more UV
FPs found in the perturbative AS framework. The reason is the presence of the
topological term (an analogue of the Gauss-Bonnet/Euler term from four
dimension) which does not give any impact for one-loop UV divergences. However
the UV divergence proportional to this term and the corresponding beta
function for this coupling has to be made vanish in fully UV-finite model.
Hence we see that we have by one too many equations than the number of
independent couplings (the value of the topological coupling is arbitrary—it
is completely unconstrained from the conditions of perturbative UV-finiteness)
and such an algebraic system is generically without solutions. One can resort
here to getting partial UV-finiteness only in some subsectors of couplings,
similarly like this was done in the case of quadratic gravity in
Codello:2006in . It is true that in the framework of the paper Knorr:2021lll ,
the cubic terms like ${\cal R}^{3}$ are present; however, they do not play the
role of cubic killers since couplings in front of them in the minimal
renormalizable model (which must have six derivatives in $d=6$) do exhibit RG
running (perturbative even at the one-loop level). This in clear contradiction
to the six-derivative model as considered here, where the couplings in front
of cubic killers operators Rachwal:2021bgb  do not run. Therefore, if one
would like to make the presented model in six dimensions completely UV-finite,
then one should first promote the theory to super-renormalizable by including
8-derivative or even higher 10-derivative kinetic terms in the action. In the
next step, one could use quartic killers (of the type ${\cal R}^{4}$) to kill
all the beta functions, although then the algebraic system of equations is
non-linear. If instead the theory would be with 10 derivatives, in $d=6$, then
addition of quintic killer terms would definitely solve the conditions for UV-
finiteness. (This is analogous to the construction of 8-derivative QG UV-
finite model in $d=4$ as presented in Modesto:2014lga ; Modesto:2015lna ;
Modesto:2017sdr .) Additionally, one can add some matter or (super-)symmetries
to make the model of Knorr:2021lll  completely UV-finite and with the
vanishing anomaly coefficient $b_{6}=0$. Such conclusions can be extended to
any minimal renormalizable theory containing terms with up to $d=2n$
derivatives of the metric in $d$ spacetime dimensions. In particular, this
explains the difference between full AS and only partial UV-finiteness of the
Stelle quadratic gravity in $d=4$ as analyzed in Codello:2006in .

As we emphasized above in the six-derivative gravitational model studied in
this article, we will not have any logarithmic perturbative UV divergences
proportional to the terms whose front coupling coefficients play the role of
the Newton’s gravitational constant or of the cosmological constant.
Therefore, we will not discuss the findings related to the Reuter’s FP of
gravitational theory giving the first evidence for asymptotic safety scenario
in gravity Reuter:1996cp . Actually, we know that these results give a
theoretical solution to the theoretical problems of non-renormalizability of
the E-H action in $d=4$ spacetime dimensions Lauscher:2001ya . However, we
think that perhaps there is no any observable physics at intermediate energy
scales behind them and they are only purely theoretical developments. First
and foremost we believe that the true quantum gravitational theory should be
with higher derivatives in their defining actions. The corresponding strong
arguments given by DeWitt and Utiyama are already quite old Utiyama:1962sn .
Moreover, these results as obtained in Gies:2015tca  are scheme-dependent and
also gauge- and gauge-fixing dependent, hence they cannot have any true gauge-
invariant physical meaning. Moreover, they are read from ambiguous power-law
like divergences in the perturbative calculus. There they are read for
dimensionful couplings and this typically creates more problems with their
universality in the framework of RG flows.

Instead of looking for something like the Reuter’s FPs for dimensionless
Newton’s constant and dimensionless cosmological constant Dou:1997fg , we
would prefer to look for completely unambiguous situation of the RG flow of
the couplings $g_{R}$ in front of the $R^{2}$ and $g_{C}$ in front of the
$C^{2}$ terms, respectively. Their RG flows is gauge-independent, universal
and unambiguous in six-derivative gravitational theory as we explained in the
next section of this contribution. In this way we will look for analogous of
FP as first found by Fradkin and Tseytlin for conformal supergravity four-
dimensional quantum models Fradkin:1985am .

Here, we remark that in a general situation the RG running is not physical and
is not observable quantity. Additionally, the location of the FP is not
physical, but the critical exponents are observable. However, first at the
one-loop level the running of dimensionless coupling parameters is physical
and observable and it has to do with the counting of perturbative degrees of
freedom of the theory (both real and virtual in a specific sense). Moreover,
the running at the one-loop perturbative level around the Gaussian FP is
completely universal since any coupling redefinition, which preserves the
perturbativity of couplings cannot change the results for these beta
functions. In a more special situation it is possible to give an unambiguous
physical meaning to the running of couplings, if the symmetry is involved. For
example, the NSVZ exact beta functions are universal and observable since they
have a very beautiful geometrical interpretation. As explained in shif1 ;
shif2  they are related to the counting of zero modes, the dimension of the
corresponding moduli spaces, and also they express the number of the
instantonic zero modes Fradkin:1983qk , which, in turn, is related to the
number of nontrivially realized and hidden symmetries in the model. They have
the geometric nature implying that the one-loop exact expressions for beta
functions of Kähler sigma models have universal observable meaning despite
that their expressions are fully non-perturbative. Being observable
quantities, they are also renormalization-group invariant.

The similar situation also happens in the case of six-derivative gravitational
theories, where due to the enhanced renormalizability properties (so called
super-renormalizability as explained in Section 2), the exact expressions for
beta functions, which happen to coincide with one-loop expressions, have also
true physical meaning. There is a hidden symmetry behind this which lets
everyone to give them a new geometrical interpretation. Only in such situation
we can give a sensible meaning to the one-loop expressions for six-derivative
gravitational theories and for the FPs of RG found there. Here, these
quantities play the role of observables, because already in them one can show
theoretically that all scheme dependencies mentioned above drop out, as it
should be. Precisely in this situation, these beta functions, RG flows and the
position of the FPs may have an invariant physical and geometrical meaning.
The analogous conditions occur for the case of FPs of RG found first by
Fradkin and Tseytlin in Fradkin:1985am  in the model of 4-dimensional
conformal supergravity. There when one concentrates on only dimensionless
coupling constant parameters (in front of the square of the Weyl tensor
$C^{2}$), one derives the same universal conclusions about the RG flow, FP
structure, critical exponents, etc.

One of the most disappointing features of RG flows for made dimensionless
Newton’s constant $G_{N}$ (or cosmological constant $\lambda$) is their gauge
dependence of special FP values Falkenberg:1996bq . Additionally, also the
expressions for anomalous dimensions of various operators show some spurious
dependence. These are some unacceptable statements in any gauge theory since
they cannot simply lead to any definite observable physical consequences being
gauge-variant (it does not matter whether this dependence is strong or only
light, any dependence is wrong and incorrect). Such ideas to work with
physical effects of gauge-dependent quantities clearly contradict from the
definition the postulate of gauge invariance for relativistic field theories.
After all, in such gauge models, one were able to treat gauge invariances in a
relativistic manner but the price was to introduce a lot of gauge redundancy
baggage. This was of course under the proviso, that this baggage will be
completely unobservable and will not have any impact on physical effects of
the theory. The relativistic formalism of QFT was rescued by the condition
that everything gauge-dependent will never be observable. Or in more clear
words in the formalism we have added some spurious degrees of freedom, only
needed to save and marry Lorentz symmetry and gauge invariance. However, they
do not carry with themselves any physically relevant information. Now, we
think that deriving physical consequences (like scattering amplitudes) from
explicitly gauge-dependent detailed description of the purported UV FPs is
wrong since the latter have to be scheme-independent. This task can be then at
best viewed as an incorrectly set exercise and this will not have any impact
on physics at all. One of the way to solve this issue with explicit gauge
dependence is to use some special formalism like presented in Lavrov:2012xz ;
Barra:2019rhz ; Lavrov:2019agp . This is why in this paper we will not comment
any more on the Reuter type of FPs, which were the original basis for the AS
program in QG research Reuter:1996cp .

We add that it is natural to expect that the FP values of couplings in general
are not physical, in particular, couplings can always be redefined or non-
linearly rescaled, and thus it is completely normal that FP values depend on
the scheme. This also happens in condensed matter physics. At the same time,
these dependencies must disappear in physical observables (i.e., in scattering
amplitudes like computed in Draper:2020knh ; scattering ) and thus there these
dependencies are not an issue. The existence of FPs must be an universal
statement (i.e., gauge- and scheme-independent), while their precise location
in the parameter space is not. In correctly computed scattering amplitudes (as
in Draper:2020knh ) there should not be seen any dependence on the gauge-
fixing, scheme or the method of calculations. This is actually a good check
for the formalism in which they are computed. In standard QFT, the amplitudes
are obtained from gauge-dependent Green functions. For example, in the
perturbative approach the independence and universality of the former ones is
secured by famous Weinberg theorem about scattering on-shell amplitudes.

Eventually, we comment that in our theories (in particular in six-derivative
QG) some values of the Newton’s constant and the cosmological constant must be
attained in the IR regime. This is likely due to the mechanism of dimensional
transmutation, when the RG flow is run towards lower energies, and the model
transforms into something else—into some IR description of gravitational
physics and there it is not any more six-derivative QG theory. Of course, in
such a regime the conditions for AS or for UV-finiteness do not hold any more
and the RG flow has already taken place. The effective IR description can be
probably accurate if one assumes that this is just E-H gravity plus some small
corrections. We explain that the values of $G_{N}$ and of $\lambda$ that are
reached in the IR regime must probably arise due to the phenomena of
dimensional transmutation, since these are for the dimensionful gravitational
couplings. In this way our theories resemble very much the theory of strong
interactions—QCD, where in the UV regime they are well described by the Yang-
Mills theories together with some matter species and characterized completely
by only one dimensionless Yang-Mills coupling parameter $g_{\rm YM}$. While in
the IR regime of the QCD, the famous $\Lambda_{\rm QCD}$ scale arises and this
is a dimensionful quantity of mass dimension and the theoretical most accepted
mechanism for its generation is this mentioned above dimensional
transmutation. We think that the same mechanism is at work also for
dimensionful gravitational constants ($G_{N}$ and $\lambda$) and for this the
issue of an existence of UV FP is probably not very relevant. Instead, in this
paper we concentrate on dimensionless couplings $g_{R}$ and $g_{C}$ and for
them we do not need to invoke any special mechanism in the IR regime.

##  2 Six-Derivative Gravitational Model

In this section we present briefly the main aspects of the considerations on
the quantum level of the recently proposed model of six-derivative in the
gravitational action in $d=4$ spacetime dimensions. For more details we refer
the interested reader to the review paper in Rachwal:2021bgb . Such six-
derivative model of QG is a simple next order generalization of the original
model as presented for Stelle in 1977. We consider it here on the quantum
level since then it has some additional benefits in comparison to the general
higher-derivative models of gravity. Roughly speaking regarding perturbative
UV divergences this model behaves even better than the original Stelle
gravity.

The Lagrangian of the six-derivative gravitational model reads

${\cal L}=\omega_{C}C_{\mu\nu\rho\sigma}\square
C^{\mu\nu\rho\sigma}+\omega_{R}R\square R+{\cal L}_{K}.$ (3)

From this Lagrangian we construct the action of our higher-derivative QG
model, here with six derivatives as the leading number of derivatives in the
UV regime, by the formula

$S_{\rm HD}=\\!\int\\!d^{4}x\sqrt{|g|}{\cal L}.$ (4)

Above by $C_{\mu\nu\rho\sigma}$ we denote the Weyl tensor (constructed from
the Riemann $R_{\mu\nu\rho\sigma}$, Ricci tensor $R_{\mu\nu}$ and Ricci scalar
$R$ and with coefficients suitable for the case of $d=4$). Similarly we can
write for the “square” scalar of the Weyl tensor

$C^{2}=C_{\mu\nu\rho\sigma}^{2}=C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}=R_{\mu\nu\rho\sigma}^{2}-2R_{\mu\nu}^{2}+\frac{1}{3}R^{2}.$
(5)

Finally, to denote the box operator we use the symbol $\square$ with the
following definition, i.e., $\square=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$ which
is a GR-covariant analogue of the d’Alembertian operator known from the flat
spacetime case.

It is important to emphasize here that the Lagrangian (3) describes quite
general six-derivative theory describing the propagation of gravitational
fluctuations around flat spacetime. For this purpose it is important to
include terms that are quadratic in gravitational curvature. As it is obvious
from the construction of the Lagrangian in (3) for six-derivative model we
have to include terms which are quadratic in the Weyl tensor or Ricci scalar
and they must contain precisely one power of the covariant box operator
$\square=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$ (which is constructed using the
GR-covariant derivative $\nabla_{\mu}$). These two terms exhaust all other
possibilities since all other terms which are quadratic and contain two
covariant derivatives can be reduced to the two above. This is achieved by
exploiting various symmetry properties of the curvature Riemann tensor as well
as cyclicity and Bianchi identities. The basis with Weyl tensors and Ricci
scalars is the most convenient when one wants to study the form of the
propagator of graviton around flat spacetime. The QG model with the Lagrangian
(3) is definitely the simplest one that describes the most general form of the
graviton propagator around flat spacetime, in four spacetime dimensions and
for the theory with six derivatives.

The special part of the Lagrangian in (3) contain terms that are cubic in
curvatures. The explicit form of them in $d=4$ after using all of symmetries
and identities reads,

$\displaystyle{\cal L}_{K}$ $\displaystyle=$ $\displaystyle
s_{1}R^{3}+s_{2}RR_{\mu\nu}R^{\mu\nu}+s_{3}R_{\mu\nu}R^{\mu}{}_{\rho}R^{\nu\rho}+s_{4}RR_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$
(6)
$\displaystyle+s_{5}R_{\mu\nu}R_{\rho\sigma}R^{\mu\rho\nu\sigma}+s_{6}R_{\mu\nu\rho\sigma}R^{\mu\nu}{}_{\kappa\lambda}R^{\rho\sigma\kappa\lambda}\,.$

These terms are essential for killing the beta functions hence the common name
for them—killers. As one can see they are all constructed as terms cubic in
gravitational curvatures, while here six various contractions are possible.
The coefficients $s_{i}$ will be determined in a moment from the condition of
all-loop (or exact) UV-finiteness.

One can make the following additional remark here. Looking at the kinetic
terms in the Lagrangian (3), one can see that this type of construction bears
great similarity to other higher-order derivative models, for example Hořava
gravity (and Einstein–Aether theory to a lesser extent Jacobson:2010mx ;
Haghani ). These models were analyzed, for example, in Pinzul:2010ct ;
Lopes:2015bra ; Pinzul:2016dwy , where a special attention was put on their
coupling to matter sectors. Moreover, some higher-derivative anisotropic
operators were constructed and discussed in Mamiya:2013wqa .

The six-derivative model, as presented here, is to some extent similar to the
original higher-derivative model of Stelle quadratic gravity in $d=4$
Stelle:1976gc . As emphasized before, higher derivatives are inevitably
generated from matter quantum loops Utiyama:1962sn . However, compared to the
case of quadratic gravity, the model described by the Lagrangian in (3),
possesses also some significant differences due to the “higher than minimal
number of higher derivatives”. For renormalizable models in $d=4$, the
dynamics with four derivatives is enough to achieve full renormalizability. By
increasing the number of derivatives, the theory is still renormalizable.
However, as we will describe in details below, the theory with six and more
derivatives is super-renormalizable. Hence it possesses more distinct features
of better convergence of loop integrals in the UV limits. Finally, this leads
to the first UV-finite model of pure quantum gravity in $d=4$. As we
emphasized in the introduction such a goal was not possible to be reached in
the minimal model of quadratic gravity with four derivatives in $d=4$. In
other words, we can term the theory from (3) as described by “higher than
higher-derivative” quantum gravitational model.

Before presenting the final results for UV divergences of this theory, we
briefly discuss the quantum properties of the theory based on the Lagrangian
(3). First, from the form of perturbative propagator around flat spacetime
$\eta_{\mu\nu}$ for the graviton field, so for the fluctuations of the metric
$h_{\mu\nu}=g_{\mu\nu}-\eta_{\mu\nu}$, we see that in the UV regime there is a
suppression of the form $k^{-6}$. This is higher fall-off of the propagator
than in standard Einsteinian theory and also than in Stelle quadratic theory
with four derivatives, where it is like $k^{-4}$. This already implies that
the theory has better control over perturbative UV divergences.

From the more precise analysis of power counting of UV divergences one sees a
few important features here of the six-derivative model. First, this theory is
renormalizable Buchbinder:2021wzv  and the bound on the dimensionality of
counterterms and of the superficial degree of divergences is $\Delta\leqslant
4$. For a situation at the generic loop order $L$, the following bound on
$\Delta$ is found: $\Delta\leqslant 4-2(L-1)$ Rachwal:2021bgb . This signifies
that when higher loops are analyzed ($L>1$), then there is more and more
restricted form of possible GR-covariant UV divergences of the model. In what
follows below, we will only analyze for simplicity counterterms built out with
terms containing precisely four derivatives of the background metric tensor.
The other counterterms with smaller number of derivatives receive
contributions also from higher loops (up to the order $L=4$ when the quantum
theory is completely UV-finite).

From the bound $\Delta\leqslant 4$ we derive that UV divergences at any loop
level $L$ can at most contain four derivatives of the metric. Since we read
the beta functions of running couplings in the perturbative framework from
corresponding UV divergences of the model, then we immediately conclude that
couplings $\omega_{C}$ and $\omega_{R}$ in front of terms with six derivatives
in the Lagrangian in (3) never show any RG running behaviour. Hence these are
constant non-running coupling parameters of the theory. Therefore, this
implies that the situation with RG flow in this quantum model is simpler (than
in Stelle theory for example) and is not so involved, despite that in the
original definition of the model in 3 we have significantly more couplings. In
this way, we reduce the non-linearity of the system of equations for the RG
evolution of couplings and one can solve it hierarchically starting with the
couplings in front of terms with six derivatives (these are constants). Next
are couplings for the terms with four derivatives and after solving for them
we can read unambiguously the RG running of the effective Newton’s constant
$G_{N}$ (as described in Modesto:2017hzl ). Finally, knowing the RG flow for
effective $G_{N}$ one derives and solves for the RG equation for the running
cosmological constant $\Lambda$ (as described in Asorey:1996hz ). We remark
that in the minimal super-renormalizable model as described by the Lagrangian
3, there are no subleading terms with four, two or zero derivatives on the
metric and hence to the one-loop order in this model we do not have any
unambiguous running of the couplings $G_{N}$ and $\Lambda$. Therefore, below
we focus on UV divergences and related RG flows only for couplings
$\theta_{C}$, $\theta_{R}$ and $\theta_{\rm GB}$ in front of terms in the
divergent action with precisely four derivatives of the metric.

When one considers only terms in the UV-divergent action with dimensionless
couplings (for which the condition of AS has gauge-invariant and independent
meanings), then the form of the one-loop divergent action reads, following
Buchbinder:1992rb

$S_{\rm
div}=-\frac{1}{\epsilon}\\!\int\\!d^{4}x\sqrt{|g|}\left(\beta_{R}R^{2}+\beta_{C}C^{2}+\beta_{\rm
GB}{\rm GB}\right),$ (7)

where by ${\rm GB}$ we denote the famous Gauss-Bonnet term in $d=4$
dimensions. In the above formula $\epsilon$ is a regulator (related to the one
often used in DIMREG scheme) that when it is sent to zero the infinities are
seen. Second advantage is that the theory is super-renormalizable meaning only
first few loops (a finite number of them) gives divergent result. However,
more importantly, when one analyzes from the power counting formula for
$\Delta$ the number of derivatives necessary in counterterms, then for terms
of dimension $4$ (like these in the action in (7)) one finds that only one-
loop quantum corrections to the divergences contribute here. This means that
for the terms $R^{2}$, $C^{2}$ and for $\rm GB$ only contributions at the one-
loop level must be taken. Hence by knowing the results at the first loop we
know the effect of performing all loop resummations. The results are therefore
one-loop exact. The quantum one-loop super-renormalizability that we meet in
this model is encouraging us and we can also easily made the theory’s control
over perturbative UV divergences complete – that is by playing with the
additions to the Lagrangian in (6) we can satisfy the conditions $\beta_{i}$
for all $i$ (i.e., $i=R,C,{\rm GB}$). Moreover, we do not need to study here
higher loops!

We finally remark that the whole notion of super-renormalizability
(differently than just ordinary renormalizability) is inherently related to
the perturbative development of the quantum model and perturbative loop
integrals evaluated for parametrically small values of the coupling constants.
Hence for it is almost practically impossible to find a non-perturbative
meaning like the AS is.

Other advantages for studying the divergences with dimensionless couplings in
front in $d=4$ dimensions, are that in this super-renormalizable model, the
expressions for one-loop ultraviolet divergences are completely gauge-
invariant, independent and also they do not depend on any gauge-fixing
parameters, nor of parametrization ambiguities for definition of quantum
fields (whether it is $h_{\mu\nu}$ or $h^{\mu\nu}$ or some other combination
taken as the basic quantum variable) or details of the scheme chosen for
renormalization or regularization of the quantum theory Kallosh:1978wt .
Finally these leading UV divergences containing four partial derivatives on
the metric tensors are completely independent on some subleading terms in the
number of partial derivatives that can be added to the action in (3). This is
again a consequence of power counting and of arguments of the character of
dimensional analysis.

In this way such beta functions for dimensionless gravitational couplings are
pieces of genuine observable quantities that can be defined in super-
renormalizable models of QG. They are universal, unambiguous, independent of
spurious parameters needed to define the gauge theory with local symmetries,
and moreover they are exact, but still being computed at the one-loop level in
perturbation calculus. They are clearly very good candidates for the good
physical observable in QG models. Therefore all these nice features gives us
even more encouragement towards analyzing the structure of such physical
quantities, to understand this based on some theoretical considerations and to
use this QG model for the comparative description between AS and UV-
finiteness.

All these above nice features of the six-derivative QG model makes it further
worth studying as an example of an exact and non-trivial RG flows in QG. Here
we have exactness of one-loop expressions for running $\theta_{R}(t)$,
$\theta_{C}(t)$ and $\theta_{\rm GB}(t)$ coupling parameters together with
super-renormalizability. This exact character of UV divergences is one of the
most powerful and beautiful features of the super-renormalizable QG model
here. Therefore this model gives us a very promising theoretical laboratory
for studying RG flows in general quantum gravitational theories understood in
the quantum field-theoretical framework.

The computation of the UV divergences in the model (3) was done recently in
Rachwal:2021bgb . The details of the methods used to obtain them were
presented to some extent in this recent article. The method consists basically
of using the Barvinsky-Vilkovisky trace technology Barvinsky:1985an  to
compute functional traces of differential operators giving the expression for
the UV-divergent parts of the effective action at the one-loop level. The main
results were obtained in background field method and from UV divergences in
Rachwal:2021bgb  we read the beta functions of running gravitational
couplings. The results for them in six-derivative gravitational theory in
$d=4$ spacetime dimensions were the main achievements there.

###  2.1 Killer Terms

Now, we discuss in greater details the role of the cubic killer terms in the
action (3) contained collectively in ${\cal L}_{K}$. These terms again will
come with front coefficients $s_{i}$ (for $i=1,\ldots,6$) of the highest
energy dimensionality, equal to the dimensionality of the coefficients
$\omega_{C}$ and $\omega_{R}$. Hence they could contribute to the leading
four-derivative terms with UV divergences of the theory as written in (7). The
general form of them was given in the list in (6).

Actually, these terms are essential for making the gravitational theory with
six-derivative actions completely UV-finite. However, for renormalizability or
super-renormalizability properties of the model in (3), these terms are not
necessary, e.g., they do not make impact on the renormalizability of the
theory and therefore should be regarded as non-minimal ones. The set of terms
in (6) is complete in $d=4$ for everything what regards terms cubic in
gravitational curvatures. This non-trivial statement is due to various
identities in $d=4$ as proven in Mistry:2020iex .

These cubic terms are also sometimes called “killers” of the beta functions
since they have profound effects on the form of the beta functions of all
dimensionless gravitational couplings in the theory. This is very easy to
explain. These terms are generally of the type $s{\cal R}^{3}$ and they are
added to the original Lagrangian of six-derivative theories, where the main
terms were of the type $\omega{\cal R}\square{\cal R}$ (important for flat
spacetime graviton’s propagator). It is well known that to get UV divergences
at the one-loop level one has first to compute the second variational
derivative operator (Hessian) $\hat{H}$ from the full action. The
contributions from cubic killers to it will be of the form $s{\cal R}$ or
higher in powers of curvatures. The second step is to derive the logarithm of
the Hessian and subsequently the functional trace of such a matrix-valued
differential operator (i.e., ${\rm Tr}\ln\hat{H}$). Next, when computing the
trace of the functional logarithm one uses the known expansion of the
logarithm in a Taylor series according to

$\ln(1+z)=z-\frac{1}{2}z^{2}+\ldots.$ (8)

Hence one needs to take for sure up to the square of the contribution $s{\cal
R}$ to the Hessian from the cubic killer term. The third power would be too
much because of the following observation—we must remember that we are looking
for terms of the general type ${\cal R}^{2}$ in the UV-divergent part of the
effective action at the one-loop level. Hence the contribution to $S_{\rm
div}$ of the cubic killer would produce the addition of the following general
type $f(s){\cal R}^{2}$, where yet unknown functions $f(s)$ can be polynomials
up to the second order in the coefficients $s_{i}$ of these killers. Now,
requiring the total beta functions vanish (for complete UV-finiteness) we need
in full generality to solve the system of some quadratic equations in the
coefficients $s_{i}$. The only potential problem for finding coefficients of
the killers can be that some solutions of this system reveal to be complex
numbers, rather than real. However, we need to require that all $s_{i}$
coefficients to be real for the definiteness of the classical action (for
example in the Euclidean case of the signature of the metric). Therefore this
issue requires a subtle and more detailed mathematical analysis, but the
preliminary results based on Modesto:2011kw ; Modesto:2014lga ;
Rachwal:2021bgb  show that in most of the cases the UV-finiteness is possible
and easily can be achieved by adding some cubic killer operators from (6) with
real coefficients $s_{i}$.

Below for definiteness we provide a sample result for these coefficients
$s_{i}$ which make the full QG model from (3) completely UV-finite. One can
choose the following values for the coefficients $s_{i}$ and a relation
$\omega_{R}=\omega_{C}$,

$s_{1}=s_{2}=0,\quad
s_{3}=5,\quad\frac{s_{4}}{\omega_{C}}\approx-0.847625,\quad\frac{s_{5}}{\omega_{C}}\approx
2.1177,\quad\frac{s_{6}}{\omega_{C}}\approx-9.83078.$ (9)

This choice is not by any means unique and one should scan a space of
parameters for other numerical solutions. The figures above are given also by
exact formulas from solving some resulting cubic equations, and using Cardano
formulas, but we will not need them here. One finds that real coefficients are
quite generic solutions here to the system of three quadratic algebraic
equations coming from requirements that $\beta_{R}=\beta_{C}=\beta_{\rm GB}=0$
in (7).

One can compare the situation here with cubic killers to the more known
situation where the quartic killers are used to obtain UV-finiteness of
gravitational theories. The second approach seems to be preferred one since
the contribution of quartic killers (of the type schematically as ${\cal
R}^{4}$) in $d=4$ to UV divergences is always linear and proportional to
${\cal R}^{2}$ schematically. To solve a linear system of equations with
linear coefficients, one always finds solutions and they are always real. This
approach was successfully applied to QG theories in Modesto:2014lga , to gauge
theories in Modesto:2015foa , to the theories on de Sitter and anti-de Sitter
backgrounds Koshelev:2017ebj  and also in general non-local theories
Modesto:2017sdr . One shows that the UV-finiteness may be an universal feature
of quantum field-theoretical interactions in nature Modesto:2015lna .
Moreover, this feature of the absence of perturbative UV divergences is
related to the quantum conformality as advocated in Modesto:2016max ;
Modesto:2017xha .

For swift comparison with other results we give also the expressions for
perturbative one-loop UV divergences in $d=4$ for standard quadratic Stelle
gravity Julve:1978xn ; Fradkin:1981iu ; Avramidi:1985ki . They are shown below

$\displaystyle\Gamma^{(1)}_{\rm
div}=-\frac{1}{2\epsilon(4\pi)^{2}}\\!\int\\!d^{4}x\sqrt{|g|}\left\\{-\frac{133}{20}C^{2}+\left(-\frac{5}{2}x^{\prime
2}+\frac{5}{2}x^{\prime}-\frac{5}{36}\right)R^{2}+\frac{196}{45}{\rm
GB}\right\\}\quad\quad\quad$ (10) $\displaystyle{\rm with}\quad
x^{\prime}=\frac{\theta_{R}}{\theta_{C}},$

where in the original action of Stelle gravity the RG running couplings are
$\theta_{R}$ and $\theta_{C}$ according to

$S_{\rm
Stelle}=\\!\int\\!d^{4}x\sqrt{|g|}\left(\theta_{R}R^{2}+\theta_{C}C^{2}\right).$
(11)

One sees that for this one-loop situation one can have without matter or
additional symmetry arguments, only the restricted UV-finiteness in the
$R^{2}$ sector of UV divergences. This is obtained for some special value of
the ratio $x^{\prime}=3(3\pm\sqrt{7})$ which approximate to
$x^{\prime}_{-}\approx 1.06275$ or $x^{\prime}_{+}\approx 16.9373$. This is
regarding the generic situation in quadratic four-derivative gravity. Such
situation was also studied for the first time for the sake of conditions for
AS in Codello:2006in .

The authors of Codello:2006in , in their paper indeed found AS for all
couplings although these couplings were redefined and an overall rescaling of
the total divergent action by the inverse Weyl coupling was performed.
However, the point is the following. Even in the case of the model reaching UV
AS as analyzed in Codello:2006in  the divergent part of the effective action
was non-zero, or in other words the coefficient of the trace anomaly, so
called $b_{4}$ coefficient in $d=4$ spacetime dimensions is non-vanishing. We
must admit that the results of Codello:2006in  fully agree with the one-loop
computation obtained earlier by Fradkin, Tseytlin and independently by
Avramidi in Fradkin:1981iu ; Avramidi:1985ki . The most clear interpretation
of these results is that for none of the values of the $x^{\prime}$ ratio, all
independent terms in (2.1) (so $C^{2}$, $R^{2}$ and $\rm GB$ terms) can be
cancelled out and hence this divergent part of the effective action is always
non-zero. Hence the pure quadratic theory is never fully UV-finite, and only
in the sector of $R^{2}$ divergences UV-finiteness is possible.

One would find it useful for pedagogical purposes to compare the situation in
pure Yang–Mills theory (YM) in $d=4$, where in the UV one meets AF, which is a
subcase of AS. However, the one-loop divergent effective action in this model
is independent on the coupling and it is non-zero, hence the theory is not
finite, although it reaches a (trivial) UV FP of RG flow. In such a model also
the trace anomaly does not vanish ($b_{4}\neq 0$) if it is coupled to the
external metric. We think that the similar situation happens for the coupling
in front of the Weyl tensor $C^{2}$ in Stelle gravity since its FP value also
shows AF in this subsector of couplings. Therefore, in pure quadratic gravity,
there is no conformal window possible for the model, while the AS in the UV
regime is still an option. One of the important point of the present paper is
that the vanishing of the full trace anomaly, divergent part of the effective
action and consequently the existence of the conformal window for the pure
gravitational model is achieved for the first time in the theory with six
derivatives, which is analyzed here. One then can understand that this special
UV-finite model sitting at the conformal window is a subcase of perturbatively
realized AS models of quantum gravity.

In six-derivative gravity one has quite a similar situation but with some
significant differences. For definiteness we consider here the situation
completely without killer operators in (6). Then the running couplings are
$\theta_{R}(t)$, $\theta_{C}(t)$ and $\theta_{\rm GB}(t)$, but they themselves
do not source the RG flow since the last one comes only from non-running
$\omega_{R}$ and $\omega_{C}$. In such a situation the exact one-loop results
for UV divergences in 6-derivative gravitational theory in $d=4$ reads as
follows

$\Gamma_{\rm
div}=-\frac{1}{2\epsilon(4\pi)^{2}}\\!\int\\!d^{4}x\sqrt{|g|}\left\\{\Big{(}\frac{2x}{9}+\frac{397}{40}\Big{)}C^{2}-\frac{7}{36}R^{2}+\frac{1387}{180}{\rm
GB}\right\\},$ (12)

where in this case the fundamental ratio of the theory is defined as
$x=\frac{\omega_{C}}{\omega_{R}}$. In this case, only the $C^{2}$ sector of UV
divergences can be made finite and this partial UV-finiteness is achieved for
the value $x_{*}=-\frac{3573}{80}=-44.6625$ of the fundamental ratio. An
attempt to discuss, understand and analyze the structure of these UV
divergences (in comparison with divergences as presented in (2.1) for Stelle
gravity) an interested reader may find in Rachwal:2022hln .

In the generic situation, the two couplings $\theta_{R}(t)$ and $\theta_{\rm
GB}(t)$ run in such a way that in the deep UV regime asymptotic freedom is
achieved in them. Their values tend for $t\to+\infty$ to infinity,
respectively to $-\infty$ for $\theta_{R}$ and to $+\infty$ for $\theta_{\rm
GB}$, so the initial conditions for the RG flows so we avoid Landau pole in
must be such that $\theta_{R}(t=0)<0$ and that $\theta_{\rm GB}(t=0)>0$. For
the case of the $\theta_{C}(t)$ coupling the situation is more complicated. If
the value of $x=x_{*}$, then this is an example of UV-finiteness and AS in
this sector of $C^{2}$ divergences and this value of the non-running coupling
$\theta_{C}$ is kept fixed at all energies, not only in the UV regime. We are
then sitting at the non-trivial FP of RG. If non-running $x$ is smaller or
bigger than $x_{*}$, then there is a non-trivial running for $\theta_{C}(t)$
and the conditions for AF in the UV regime are similar to the stipulated
above. If $x<x_{*}$, then $\beta_{C}<0$ and we must require an initial
condition that $\theta_{C}(t=0)<0$, while in the opposite case if $x>x_{*}$,
then respectively $\beta_{C}>0$ and the initial condition that
$\theta_{C}(t=0)>0$ ensures that we meet AF in the deep UV regime without any
dangerous Landau pole on our way towards it.

##  3 UV-Finiteness Versus Asymptotic Safety in Six-Derivative Gravitational
Model

Here we discuss and compare asymptotic safety approach to HD QG, which gives a
lot of success for quantum field theory of gravitational interactions treated
both on the perturbative as well as on the non-perturbative level. We would
like to compare this approach with the other one offering full UV-finiteness
of any Green function or scattering amplitude. This last case is realized by
first making the theory with six-derivative (so super-renormalizable) and
later by adding some special operators cubic in gravitational curvature. One
then ends up with completely finite theories (without UV divergences). We
mentioned already above that in the original Stelle theory we can only make
the $R^{2}$ sector of UV perturbative divergences at the one-loop level
finite. Contrary, in the pure six-derivative theory (without terms cubic in
curvatures) it is possible by adjusting values of the fundamental ratio $x$ of
the theory to make instead the $C^{2}$ sector of UV divergences completely
finite.

In this section, we compare the conditions for UV-finiteness and for
asymptotic safety based on the particular example of a six-derivative QG model
as presented in the previous Section 2. We also discuss the common practices
that are used to extract some physical information from these two approaches
towards the UV-completion of QG models. At first sight, there is a lot of
similarities between these two research directions Percacci:2017fkn . However,
as we will point out also below there are some distinct differences, but this
does not exclude the possibility for the unification and merging of the two
approaches towards consistent QFT of quantum gravitational interactions.

First similarity is that in both approaches, one studies the same condition,
namely

$\beta_{i}=0.$ (13)

And this is the starting common point of two approaches. Here, we want to
emphasize the differences. In models which are UV-finite, this condition of
vanishing beta functions should hold for all running couplings of the theory
and at all energy scales. This requirement may look as much stronger, but this
is a true condition for the quantum model to look the same at any energy
scale, hence there are no divergences possibly spoiling the conformal
invariance of the quantum theory.

Instead, in the AS paradigm, the condition (13) is to be put only in the UV
regime of the RG flow. Only in the limits of infinite energies, we shall find
a situation safe with UV divergences and only then we should reach a non-
trivial FP, this is the asymptotic condition. Moreover, while in the approach
with UV-finiteness, we require complete absence of all UV divergences, here
for AS we may consider some sectors of couplings, in which there is a non-
trivial FP, but some other couplings may run towards AF in the UV regime, and
hence in the last case the UV divergences are still present (as also explained
in full details below). We described previously that one can think of UV-
finiteness as the special case of the AS, where the equation (13) for all
couplings must hold at all energy scales and not as a special autonomous
condition in the parameter space. Therefore, the condition for vanishing of
conformal anomaly are more special than just for the existence of the non-
trivial FP of RG flow. Generically, the AS describes a special point in the
parameter space, when the condition (13) is solved for the running coupling
parameters. (Actually at the FP they do not run anymore since they take their
fixed valued there.) In different vein by closely analyzing the perturbative
asymptotic safety models as introduced by Litim and Sannino in Litim:2014uca ,
one sees that the requirement there for perturbative AS is equivalent to the
vanishing of UV divergences. Hence in such a situation the UV-finiteness for
some sectors of couplings is a prerequisite for fully developed AS scenario.
In the last perturbative model, where the AS is guaranteed, the UV-finiteness
condition is put on the UV action since at lower energies the action may not
be at the conformal window. It reaches the conformality only for
asymptotically high energies due to the RG flow of dimensionless couplings.
The consistency condition for the good behaviour in the infinite limit of
energies (AS) is translated into the perturbative language as the condition
for absence of ultraviolet divergences of perturbative loop integrals in the
limiting theory described by the UV action. This also shows the
complementarity between these two approaches to the problems of UV-completion
and the UV limits of RG flows.

We remind here that in an example of AF theories, which are in a sense a
trivial subset of AS theories, the perturbative UV divergences are present
even in the UV regime of very high energies (the $b_{4}$ coefficient of the
trace anomaly never vanishes since it is constant as computed to the one-loop
level), while the perturbative couplings do run to zero values as it is
required consistently with asymptotic freedom in the UV—the example of the QCD
model is very paradigmatic here. To be more explicit when one writes
schematically the YM action as $\alpha_{F}^{-1}F^{2}$, the perturbative
coupling is $\alpha_{F}$, that is the loop expansion is in powers of this
$\alpha_{F}$. One has at the tree-level the action proportional to the
starting power $\alpha_{F}^{-1}$. Then at the first loop level, one finds that
$\Gamma^{(1)}\sim\alpha_{F}^{0}$ – is independent on the couplings and finally
starting from two loops we see that for example $\Gamma^{(2)}\sim\alpha_{F}$
and in full generality $\Gamma^{(L)}\sim\alpha_{F}^{L-1}$. This confirms that
again in pure YM theory the divergences are still there at the one-loop level
and even in the situation when the trivial Gaussian FP is reached in the UV,
and the coupling is zero, that is we have that when $k\to+\infty$ then
$\alpha_{F}(k)\to 0$, they are still present there and for example cause the
non-vanishing of the trace anomaly. This is true even at the Gaussian FP where
the couplings are formally zero. At the Gaussian FP we have that
$\beta_{\alpha_{F}}=0$, however for the inverse coupling
$\omega_{F}=\alpha_{F}^{-1}$ needed to write the classical action in the form
$\omega_{F}F^{2}$ there is no condition that $\beta_{\omega_{F}}=0$ at the FP.
Instead, this last beta function is there constant ($\beta_{\omega_{F}}={\rm
const}$), because it is constant at all energy scales to the one-loop
accuracy.

Therefore to refine our comparison between AS and UV-finiteness, we can say
that they are very compatible to each other, except the AF sectors of the
couplings of the theory, where the UV divergences generically still remain. In
the last sectors the conditions for AF, despite the unambiguous presence of
divergences as explained above, are still consistent with AS although they are
not consistent with UV-finiteness. This is the only known exception so far.

Moreover, we remark that when there is a FP (free or safe, or both) there will
be generically plenty of trajectories approaching this point. Only few special
trajectories will reach the FP. Other trajectories will stay some RG time in
the vicinity of the FP and eventually they will escape (they are not UV-
complete) and they correspond to inconsistent theories, for example whose UV
divergence cannot be cured. If these diverging trajectories did not exist
(i.e., if the fixed point is fully UV attractive), the theory would have an
infinite (for infinite couplings in the truncation) or unbounded number of
relevant directions and thus it would not be predictive (and renormalizable),
although it formally reaches a UV FP. Of course, this situation is not
satisfactory. On the diverging trajectories (not reaching the UV FP) the
running of couplings under RG flow may lead to somewhere else (extreme
regions) in the parameter space (like an asymptotic Landau pole for some
couplings at infinite energies), where the theory is ill defined as well.
Therefore also this situation should be definitely excluded from
considerations of AS.

Therefore, here we would say that UV-finiteness is more restrictive condition
because the FP is for all couplings and for all energy scales. One could even
say that the theory without renormalization is easier to solve exactly, since
what matters is mainly the classical level of such a model and the conformal
symmetry helps with it. An important essence of the AS program is that the
emphasis should be put also on studies of the way how the FP is approached in
the UV regime, not only the mere existence of the FP at the infinite energies.
One could even risk a statement that UV-finiteness is needed for asymptotic
safety at the close vicinity of the FP in order to have a full control over
perturbative divergences in the scheme of expansion in small coupling
parameters. The RG flow from the UV to the IR has also been studied in many
papers in the framework of AS and in functional renormalization group (FRG),
e.g., in Gubitosi:2019ncx  (pure gravity and in the quadratic approximation)
and also in Eichhorn:2019ybe  (coupled gravity-matter systems). One can also
explicitly integrate out the RG flow and compute the scale-dependent effective
action $\Gamma_{k}$ as this was done for example in Codello:2015oqa ;
Ohta:2020bsc . One of the goal of such a following of the flow towards the IR
limit was to get the following limit $\Gamma_{k\to 0}=\Gamma$, which gives the
definition of the standard (quantum) effective action of the theory.

The second set of differences stems from the fact that as it was obvious from
the example of quadratic Stelle gravity understood on the quantum level, it is
impossible to have a fully UV-finite theory there, while in some sectors of
couplings the AS is possible. At the end such a model of pure QG with only
dimensionless coupling constants does not entail the possibility in $d=4$ to
ameliorate the problem with divergences of QFT. In AS program, the models with
dimensionless couplings in gravity are usually considered, however, as we
pointed out on the example of six-derivative gravity from the previous
section, the addition of higher order operators in gravitational curvatures is
essential for the success of the construction of the UV-finite QG theories. To
our knowledge the consistent addition of cubic or quartic operators in
gravitational curvatures for the AS program in the framework of super-
renormalizable QG models and the better control of the FPs in the UV regime
was never studied before in the literature. Our papers (Rachwal:2021bgb ;
Rachwal:2022hln ) instead suggest that these terms are crucial for having
better control of perturbative UV divergences. In QG models, which are super-
renormalizable only divergences at the one-loop level matters and one possible
way to kill them is to add these terms higher in curvatures. This is one
solution in pure QG models, where the matter fields are not added.

Here one should also mention that there are studies of conditions for AS in
models with additions of terms quartic and cubic in curvatures (see e.g.,
Knorr:2021lll ; Knorr:2021slg ), and also in the $f(R)$ approximation, where
$R$ is the Ricci scalar of the spacetime and $f(x)$ is an arbitrary function.
In the last truncation the beta functions have been impressively computed up
to the order $R^{70}$ Falls:2018ylp . These studies add significantly to our
understanding of situation with AS when cubic and quartic terms are included.
The inclusion of terms of the type $C\square C$ and $R\square R$ together with
terms cubic in curvatures $O({\cal R}^{3})$ was done here for the first time.
We remind that the first group of terms (being quadratic in gravitational
curvatures) have the impact on the improved scaling of the propagator in the
UV regime and it is entirely responsible for the super-renormalizability
property of the six-derivative gravitational theory. The addition of terms
precisely cubic in gravitational curvatures (and with no covariant derivatives
acting between them) is motivated by effective field theory (EFT)
considerations since all such terms contain precisely 6 (partial) derivatives
on the metric tensor after expansion around flat spacetime and for the quantum
field $h_{\mu\nu}$. In EFT all terms with given number of derivatives must be
potentially included for theoretical completion of the framework. The
expansion in number of derivatives is also counted by powers of the high
energy scale $M$, so this is a kind of derivative expansion or an expansion in
the inverse powers of the big mass parameter $M$. Finally, we remark that the
terms with higher powers like $R^{4}$ or up to $R^{70}$ are not needed to be
explicitly added here, if one works to the order of six derivatives. Moreover,
the basis of all terms as presented in (6) is complete in $d=4$ spacetime
dimensions since terms of all tensorial structure are present there (up to
various identities holding in $d=4$).

Another option is to add some matter fields (or supersymmetric multiplets of
fields) in such a way that even pure QG model with dimensionless coupling
parameters is UV-finite. The example here is mentioned before ${\cal N}=4$
conformal supergravity which is this highly supersymmetric analogue of scale-
invariant Weyl gravity in $d=4$. In such a case without killer operators and
without higher order terms, but in coupled QG model, one can find complete
absence of UV divergences and presumably at the same time also conditions for
AS in the UV regime. This model due to Fradkin and Tseytlin is probably one of
the most promising for solving problems of QG and for having at the same time
AS in the UV regime and UV-finiteness, together with local conformal symmetry
and supersymmetry of matter and gravitational interactions. In general,
however, classically scale-invariant QG models, may enjoy partial AS in some
sector of couplings, but they are never fully UV-finite if the theory is in
such a pure form. On this example one sees very well the differences between
AS and UV-finiteness approaches to the problems of divergences in QFT models
of QG.

One more difference can come from the observation that in AS scenario for QG,
the FP values for the couplings are very special and they are grouped in
isolated points on the parameter space (defined by coordinates being coupling
parameters of the theory). Therefore the situation to find a FP is fine-tuned,
and as the name suggests, these are points (so $0$-dimensional objects) in
higher-dimensional parameter space of the theory. In contrary, the approach
with UV-finite theory restricts couplings to lay on some one-dimensional
curves being submanifolds in the bigger parameter space. Or they could be even
higher dimensional submanifolds of FPs in super-renormalizable theories. From
this point of view, one could say that the situation with UV-finite theory is
less restrictive. However, another point of view is that having a fixed point
or a line fixed points (or even a plane of them) does not make a difference,
they are all solutions to the defining equation $\beta_{i}=0$. Instead, one
could consider the situation in UV-finite theories as more fine-tuned since
there the action of the model is restricted to have a few operators only, put
there by hand, in order to have UV-finiteness. As we remarked previously, the
UV-finite models can be viewed as special cases to the more broad class of UV-
complete AS models.

The standard illustrative example one can recall here is given by the
situation in ${\cal N}=4$ super-Yang–Mills theory, where if the coupling space
is parametrized by Yang-Mills gauge coupling $g$, Yukawa coupling $y$ and the
quartic self-interactions of scalars $\lambda$, then the ${\cal N}=4$ theory
is obtained by putting two constraints on $g$, $y$ and $\lambda$. Such theory
is fully conformal on the quantum level analyzed around flat spacetime
background. There are then no UV divergences and the theory sits all the time
at a FP. If the value of the coupling is non-zero, then this is a non-trivial
FP. As one can see by counting constraints, the condition for UV-finiteness in
this case is a one-dimensional curve, which can be parametrized for example by
the value of the Yang–Mills coupling $g$. The other couplings are then
functions of it. The non-trivial FP is met for any value of the $g$ coupling,
so this is strictly speaking a line of FP of RG and moving on this line does
not cost any energy and there is no any RG flow within it.

A similar situation one also meets in the QG model with six derivatives. There
the complete UV-finiteness (absence of the $R^{2}$, $C^{2}$ and the ${\rm GB}$
divergences) is obtained when the ratios $\frac{s}{\omega_{C}}$, as presented
in (9), are fixed, but the values of the coupling $\omega_{C}$ can be for this
arbitrary, provided that this is not zero. We also find a line of FPs here and
the condition for UV-finiteness does not determine uniquely the value of $s$,
neither of $\omega_{C}$, only their ratio is fixed for the condition of UV-
finiteness. Here we denote by $s$ a generic coupling in front of the killer
operators from (6) admitting that in our case of six-derivative QG in $d=4$
there are six different non-running parameters $s_{i}$.

This last situation situation is reminiscent of what happens in AS paradigm to
QG models, when one can rescale by the overall coupling the whole action. Then
the couplings which appear there are basically the ratios and the condition
for AS is insensitive to the overall coupling. If it was reintroduced there,
then the condition for true AS would again define for us not just isolated
points, but a line of them on the parameter space. However, we remark that for
full UV-finiteness one cannot perform such procedure since if one rescales by
the overall divergence of the action, then one loses all other divergences
since relative coefficients between divergent terms of the effective action
stay typically in finite mutual relations.

Finally, we must mention that for proper framing theoretically AS in the UV
regime must embody the case of UV-finiteness. This is because of the presence
of another issue within the AS program. This issue which must be taken into
account is that the presence of a non-trivial UV FP must be necessarily
accompanied by the UV-finiteness of the functional of the Wilsonian effective
action when the last one is to be computed at the formally infinite energy
scales. Otherwise, the whole construction of AS theory in the UV will not work
and the UV divergences will still inevitably show up. Thus, AS would not be
UV-safe. This means that it must exist an UV action functional that is an UV
endpoint of the RG trajectory as followed by Wilsonian effective action. One
typically knows that the UV effective action is a very complicated object
since it may describe a theory with a lot of derivatives, or even a non-local
one. But just for its consistency such a functional must be UV-finite at the
infinite energy scale, in the opposite case the UV divergences are still
there.

The precise logical construction here is as follows. The UV limit of the
Wilsonian effective action must exist as an action describing fully the UV
physics, if this is understood as effective field theory. On the other hand,
it must also exist as a well defined endpoint of the RG flow at energies
$E_{\infty}$. Such action is typically call as the UV action. Sometimes people
also call it as a bare or classical action of the theory, although in
principle it has no relations to the classical action which was presumably
there before the quantization process. In AS program it is often assumed that
the UV action exists but its explicit construction is almost never showed. It
is incorrect to think that it is the simplest action of the local theory in
the UV limit. For consistent removal of divergences such local actions must
come with very special properties. One can treat this UV action as a basis for
quantum theory and ask the question of what happens in theoretically “even
higher energies” $E>E_{\infty}$. For consistency of the whole AS program, the
UV divergences for “even higher energies” $E>E_{\infty}$ should be absent.
(Here we assume the following sequence of limits, first $E\to+\infty$ and only
later $E_{\infty}\to+\infty$).

Hence, we conclude that necessarily the functional for the UV action must be
itself UV-finite when it is understood as the basis for some QFT. Otherwise we
run back into a problem of pathological infinities. Moreover, if one is able
to provide an explicit construction for the UV-finite action, then one can
give a satisfactory question to the old question in studies of RG flows—of
what is the explicit form for the UV action for a consistent QFT which reaches
a non-trivial UV FP. Previously, the existence of such actions was only
hypothesized—now we can provide explicit examples of local UV action
functionals, also in the gravitational frameworks.

Unfortunately, we remark that the issue of the proper construction of a true
UV-finite UV Wilsonian effective action in the theories that reach AS in the
UV regime is typically neglected in the literature. Maybe because one has not
had good examples of UV-finite quantum gravitational theories. They are not
easily constructible using Einstein–Hilbert action or even Stelle gravity
actions in $d=4$ spacetime dimensions. As we emphasized earlier for this issue
one must include _higher derivatives_ than four-derivative actions or add some
very special matter content which probably needs to be related also by some
symmetry to the pure gravitational sector.

Complete UV-finiteness is not possible there without adding some special
matter content (like in HD supergravity with ${\cal N}=4$ due to Fradkin and
Tseytlin). Instead the case of six-derivative theories is the first one in
which one obtains for a model with enhanced symmetry all beta functions
vanishing, exactly, which here is equivalent to the situation at the one-loop
due to super-renormalizability properties. In this way one can find the
example of a HD gravitational theory which completely describes the situation
at the UV fixed point of the RG flow. Hence one gives rise to a more detailed
description of the AS program for QG by specifying an explicit example of the
Wilsonian effective action that is there at the UV FP. One now understands
that for the AS to work at the very high energies it is not enough that a
hypothetical FP exists, the critical surface is finite dimensional, but also
that there exists an action which describes the physics precisely at the UV FP
and which is completely without UV divergences.

After all, we remark that the title of the seminal paper by Weinberg
Weinberg:1980gg  was about UV divergences in gravitational theories. Of
course, this paper started the whole revolution related to the AS program. The
theoretical problem considered there was exactly with UV divergences and how
to solve them to get the quantum theory UV-finite and UV-safe and as Weinberg
coined it—asymptotically safe meaning safe from UV divergences (so finite) at
asymptotically high energies. This is in close similarity to the origin of the
terminology for AF theories, which are free for asymptotically high energies,
so in the deep UV regime. We think that it is worth, following Weinberg, to
come back, to unify and to appreciate again UV-finiteness considerations for
asymptotically safe models of QFT since this is a necessary ingredient for the
latter ones.

In this section, we discussed and compared the asymptotic safety approach to
HD QG with the other one offering full UV-finiteness of any Green function or
scattering amplitude. This last case was realized by first making the theory
with six-derivative (so super-renormalizable) and later by adding some cubic
killer operators to end up with completely finite theories (without UV
divergences). We mentioned that in the original Stelle theory we can only make
the $R^{2}$ sector of UV perturbative divergences at the one-loop level
finite, and complete UV-finiteness is not possible there without adding some
special matter content (like in HD supergravity with ${\cal N}=4$ due to
Fradkin and Tseytlin). Instead the case of six-derivative theories is the
first one in which one obtains for a model with enhanced symmetry all beta
functions vanishing, exactly, which here is equivalent to the situation at the
one-loop due to super-renormalizability properties. In this way one can find
the example of a HD gravitational theory which completely describes the
situation at the UV fixed point of the RG flow.

##  4 Conclusions

In this paper, we described the two approaches towards a consistent
formulations of quantum field theories of gravitational interactions valid at
arbitrary high energy scales. We discussed both UV-finiteness and also
Asymptotic Safety as apparently different ways of achieving UV-completion of
gravity without necessity of invoking string theory or any other non-local
extensions of standard physics. They are not rivals to each other. On the
contrary, they possess a lot of common features and only superficial
differences that upon closer inspection and analysis show the two different
sides of the same coin. We showed that these differences can be understood and
explained and that the mathematical basis for the two approaches is very
similar. It is however a fact that typically UV-finiteness is considered for
theories which are perturbative in coupling parameters, while AS is inherently
rather a consequence of non-perturbative phenomena since FP values of
couplings are not needed to be assumed to be small. This points to the problem
that methodologies used to study these two approaches are usually different
like perturbative vs. non-perturbative physics.

We still think that these two approaches can be unified and that they need
each other. For example, we emphasized above that although the non-
perturbative formulation of AS does not assume the smallness of any coupling
parameter, it must be consistent with the results obtained in the perturbative
framework where this smallness is assumed and exploited. One particular and
pertinent finding here is that the UV action functional, which describes the
physics at the UV FP of RG flow, which stays at the core of AS paradigm, must
itself entail the model UV-finite, when it is considered in standard
perturbation theory. This implies that for consistency and for explicitness
the AS program must contain the results already known from the perturbative
approach to the UV-finite theories. Although as we remarked before UV-
finiteness may also regard the non-perturbative divergences. This is why we
think we can properly call the AS scenario not merely as non-perturbative
renormalizability, but as a non-perturbative UV-finiteness in some theories.
It is obvious that the full non-perturbative results (in AS) cannot go in
contradiction with the perturbative results (from UV-finite theories). This
last statement is true generally.

Let us summarize here the main ideas of this paper. The goal was exactly to
compare the two approaches (UV-finiteness and AS) to perennial problems of
quantum gravity with the conclusion that they are very similar and should not
be viewed as rivals to each other. In this article, we wanted to point out
similarities and the potential differences may be traced back only to the
differences between perturbative and non-perturbative approaches to QFT. We
think the honest conclusion now is that the class of perturbatively UV-finite
theories is a subset of the theories which are generally asymptotically safe,
both in non-perturbative or as well as in perturbative framework. Simply UV-
finiteness is a more restrictive requirement than just AS since the former is
related to the vanishing of trace anomaly coefficient $b_{4}$ at all energy
scales, and not only to the situation at an ultraviolet FP of RG reached at
infinite energies.

In the course of our investigations, we first provided examples where these
relations between AS and UV-finite theories can be studied both perturbatively
and non-perturbatively. We pointed to the scalar-gauge-Yukawa theories, in
which perturbative analysis shows that they can be with guaranteed AS and also
with perturbative UV-finiteness. It was more difficult to find similar
examples in the case of QG models. For this one necessarily needs to study
higher-derivative modified models of gravity. We remarked that in pure gravity
models, in $d=4$ these are gravitational models due to Stelle with second
powers of gravitational curvatures and with precisely four derivatives in the
action, UV-finiteness in all sectors is impossible. This is in a close
similarity to the case of pure non-Abelian gauge gauge theory in $d=4$, which
is known that cannot be finite at the quantum level. In the last case one has
to add charged matter, and for higher loop finiteness the matter must be of
both fermionic and scalar types and matter interactions must consist of Yukawa
scalar-fermion interactions as well as also of quartic scalar interactions.
The examples of gauge UV-finite theories are here highly supersymmetric ${\cal
N}=4$ super-Yang–Mills theories and also less symmetric scalar-gauge-fermion
models of Litim and Sannino. The same considerations for gravity are way more
complicated.

For the case of QG, in the quest for UV-finite theories one very
(super-)symmetric option was to consider conformal supergravity with ${\cal
N}=4$ as the analogue of the super-Yang–Mills theory. Besides this very
symmetric option we decided to study pure gravitational models but with higher
than usual number of higher derivatives. The considerations of six-derivative
quantum gravitational models comes here with advantages that they were super-
renormalizable and UV-finiteness was achieved to all loop orders. Our
expressions for the beta functions were exact since there are no divergences
at higher loop level, moreover they were exact, scheme and gauge parameter and
gauge fixing independent. Based on such expressions after adding killers
(cubic terms in gravitational curvatures) one could easily get an UV-finite
theory at the one-loop perturbative level. However, this result we extended to
all perturbative loops by the power of the super-renormalizability argument
since divergences were expected and checked to be present only at the first
loop level. On this example of some explicit UV-finite models of six-
derivative QG we were able to discuss various differences and also
similarities between the AS and UV-finiteness approach to the role of
perturbative and non-perturbative UV divergences in the theory. This was the
main purpose of studying here such an example of UV-finite pure six-derivative
QG theory.

Here, we analyzed and constructed all gravitational terms in the action with
up to six derivatives on the metric. Of course, one can take models with more
derivatives (with a finite number of them giving rise to local higher
derivatives theories) or even fully non-local models (with a formally infinite
number of derivatives) Knorr:2019atm ; Modesto:2017sdr . We just restricted to
6-derivative models since in this hierarchy these are the simplest and they
are the first after Stelle quadratic theory in $d=4$ spacetime dimensions. We
showed that what was not possible for quadratic gravity is possible for six-
derivative model and then the $b_{4}$ coefficient vanishes, there is no trace
anomaly on the quantum level and the theory sits at the conformal window. As
one can see from the the formula (9), there is also a significant freedom (as
opposed to fine-tuning situations) to satisfy the conditions for UV-finiteness
since for example the values of the coefficients of $\omega_{C}$ and $s_{i}$
are not determined separately, and only their ratio is specified for the
conditions that all three beta functions of dimensionless couplings are zero
here. We emphasized previously in the text that this is one example of UV-
finite theory, but actually there are an infinite amount of them. The
preliminary analysis shows that the allowed region for UV-finiteness spans a
3-dimensional compact region in a 7-dimensional parameter space of all
couplings of the Lagrangian in (3) (one coupling is removed here due to the
overall rescaling of the action by $\omega_{C}\neq 0$ coefficient, for
example). However, the analytic description of this region is a difficult
algebraic task. Hence one could say that the conditions for UV-finiteness are
quite generic ones and these infinity of models mean that they are not fine-
tuned. Similar advantages we also find in models, which are AS in the UV
regime.

In some standard matter theories, like QCD for example, just perturbative
renormalizability or AS is enough to have a well defined matter model, which
is quantum consistent. However, as we repeated many times above, in quantum
gravity we may look for UV-finiteness because of the problem with spacetime
singularities of classical gravitational theory. Moreover, for renormalizable
models one can verify experimentally the RG running of couplings, while in the
UV conformal gravitational phase such phenomena could not be seen, even in
theory. We think this is one of the distinguishing features of these models.
Besides this, in mere renormalizable models, the couplings take different
values than in the non-UV-finite models, so we expect that predictions for
some observables will be different. Additionally, there is a vast literature
on the implications of conformal windows for models of particle physics
phenomenology (e.g., Lombardo:2014pda ; Ryttov:2007sr ; Orlando:2020yii  and
also in Antipin:2013exa ; Antipin:2017ebo ), so there are definitely clear
observational signatures here.

One could also pose a question of how to recover the standard two-derivative
Einsteinian gravity (popularly known as the first example of a generally
relativistic theory) from this six-derivative model, which is moreover
selected to be UV-finite after conditions in (9). The answer is very simple
since one can freely add terms subleading with the number of derivatives to
the Lagrangian in (3). In particular, one can add there the Einstein–Hilbert
term, Stelle quadratic terms $R^{2}$ and $C^{2}$ or even a cosmological
constant term. One can supplement the conditions for UV-finiteness since then
two new beta functions are there to cancel (of the cosmological constant and
of the Newton’s coupling) according to the results from Asorey:1996hz ;
Modesto:2017hzl . In this new situation one again may find full finiteness in
the UV regime. On the other hand, when one goes towards IR the terms with
smaller number of derivatives become more and more important and effectively
the action of the theory takes the form of Einstein–Hilbert action (possibly
with the cosmological constant term). In this way the known IR gravitational
physics is fully recovered here.

Instead, the quest for UV-finite theories is a completely UV question. Then in
such regime the theory may be completely different than the theory, which
remains in the IR regime. As we know and expect (based on Utiyama:1962sn , for
example) in the UV regime higher derivatives are inevitable and then the UV
action necessarily differs from the two-derivative Einstein–Hilbert action,
which is safely recovered in the IR regime of the theory. We already
emphasized in the introduction to this paper that the quantum conformality in
the UV is needed for the consistent resolution of spacetime singularities111
Another options for a consistent resolution of GR spacetime singularities are
provided by some non-local models or when one adds ${\cal R}^{3}$ operators
(and higher) since they are sufficient to solve black hole singularities
problem even in the presence of the Einstein–Hilbert part of the action (see,
e.g., Holdom:2002xy ; Knorr:2022kqp )., which is first done on the classical
level of conformal Weyl $C^{2}$ gravity and later this is not destabilized by
quantum corrections, if the conformal symmetry is fully realized in the UV
regime. It is exactly that regime, where the theory should reach the
description of point-like singularities which are so ubiquitous problems for
classical Einsteinian general relativity theory. This could be viewed as one
of the most pressing reasons why the classical gravity is not a complete
theory and why it has to be quantized right away. Moreover, the theory in the
UV regime must be changed and it cannot be a two-derivative gravitational
theory anymore. We find that that theory which solves the problem of short
distance singularities and at the same time of UV divergences must incorporate
conformal symmetry on the quantum level. Hence, one of the necessary
conditions is that the trace anomaly completely vanishes $b_{4}=0$ and one
cannot bypass this condition by any overall rescalings. This gives rise
naturally also to UV-finite theories. The examples of conformal anomaly-free
theories were already well studied in the past Fradkin:1983tg ; however, in
this work we extended the discussion for pure six-derivative gravitational
theories (without matter species), where such UV-finiteness was possible for
the first time.

We comment that the conformal symmetry is probably the last one to be found in
the UV regime of particle physics. This is because probing it in principle
requires access to infinitely high energies, or at least arbitrarily high. In
the UV conformal gravitational phase all scales should be practically
identical, so this is practically the same situation as at the FP of RG flow.
The condition for cancellation of conformal anomalies is very powerful since
then it implies that the theory is completely without divergences. We can
eventually quote the famous words of late Paul Dirac about QFT. He believed
that in a fully consistent quantum theory of the world all infinities present
in QFT (like in QED on which he worked extensively) should be absent. This we
achieved in UV-finite models of quantum gravity.

We admit that the situation at the UV FP, so formally at infinite energies,
provided that it exists and that some UV action at it is UV-finite, is not the
end of the story with QG. This only solves the pending and long standing
theoretical problem with infinities within QG models. Of course, the last one
due to conformal enhanced symmetry at the FP is also related to the issue of
resolution of spacetime singularities. However, these two theoretical problems
and their successful solutions within the QFT framework do not address the
phenomenological consequences of QG models at some finite energy scales, for
example around the Planck energy scales ($M_{\rm Pl}=10^{19}\,{\rm GeV}$).
However, the philosophy that we have in mind is the following. After solving
all theoretical problems in the UV for QG, we finally could have a consistent
theoretical model for QG as a QFT and within this we should run down in
energies and check for some observable predictions of the theory. Then, we can
make relations of these QG models to QG phenomenology, physics of the SM of
particle physics coupled to gravity, unifications of gravity with other
interactions, cosmology, black hole physics, sub- and trans-Planckian
gravitational physics.

For this ambitious goal one has to leave the UV FP, and start a deformation of
the UV-finite UV action functional for the gravitational theory. The conformal
symmetry and scale-invariance which were the situation at the FP must at lower
energies be somehow broken (there are various ways of doing this, cf.
Jizba:2020hre ; Jizba:2014taa ) and the ensuing RG flow will start
immediately. The model will not be anymore at infinite energies, will not be
at the UV FP, the running effective Wilsonian action Codello:2008vh ;
Codello:2015oqa ; Ohta:2020bsc  will be different than the action of UV-finite
theory, the couplings will start to run down with the energy and they will not
be anymore equal to their FP values. The RG flow will be back non-trivial, but
the conformal perturbation theory (CPT) will allow for its studies around the
non-trivial UV FP. Following such RG flows within the framework of CPT or FRG
one should be able to reach physical energy scales like the mentioned above
physical Planck scale $M_{\rm Pl}$. For this one can use the methods of
Wilsonian (RG-running) effective actions $\Gamma_{k}$ (as done for example in
Codello:2015oqa ; Ohta:2020bsc ). There, at lower energies, one will not be in
the AS regime, and because of this, the UV AS scenario will not be at work
since one will not be there in the formally deep UV regime. (If one looks at
the face value at the Wilsonian effective action, then the quantum theory
based on it likely will not be UV-finite). However, the form of the
interpolating (effective) action at these energies will have imprinted on
itself the conditions that the theory satisfied in the deep UV regime (AS and
UV-finiteness) and then this means that such effective theory at the energies
$E\sim M_{\rm Pl}$ will be very special and unique. We also remark that the
full quantum effective action $\Gamma$ describes the theory across all
energies. For large energies, the couplings or form-factors Knorr:2019atm
reach the AS/UV-finite regime, for small energies one should recover GR, and
for intermediate energies (e.g., inflationary scale or Planck scale) one can
describe the crossover or a smooth transition between the two phases of the
theory. We emphasize that at intermediate energies the two theories described
by $\Gamma_{k}$, there with $k\sim M_{\rm Pl}$, are still consistent, since
their UV-completion is achieved independently in the UV regime and the
corresponding theories at higher energies are well defined. The predictions of
such theories at intermediate gravitational energies for observable
gravitational physics will be of our next great interest.

This is the final important link of the theory to the reality and to the
observational physics. We plan to work on this in the future since the
theoretical framework for QFT of gravity within the UV-finite or AS is
consistent and established now. Moreover, these two approaches seem to be very
promising in solving the theoretical problems of QG within well tested field-
theoretical framework. We are also in possession of explicit formulations of
QG models that satisfy the above conditions both perturbatively and non-
perturbatively. The link to QG phenomenology will create a possibility and a
necessity for some final tests or verification of the theoretical ideas for
quantum modified gravity that we advocated in this work.

This research received no external funding.

###### Acknowledgements.

We would like to thank I.L.S., L.M. and A.P. 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. We also would like to acknowledge critical suggestions for
improvements of the manuscript from an anonymous referee in the Universe
journal. We believe with them the part of the description of asymptotic safety
in gravity is now much more clear and honest. Finally, we would like to
express our gratitude to the organizers of the ALTECOSMOFUN’21 conference
(Alternative Gravities and Fundamental Cosmology) for accepting our talk
proposal and for creating a stimulating environment for scientific online
discussions. The author declares no conflict of interest. References

## References

  * (1) Stelle, K.S. Renormalization of Higher Derivative Quantum Gravity. _Phys. Rev. D_ 1977, _16_ , 953–969.
  * (2) Codello, A.; Percacci, R. Fixed points of higher derivative gravity. _Phys. Rev. Lett._ 2006, 97, 221301.
  * (3) Rachwal, L.; Modesto, L.; Pinzul, A.; Shapiro, I.L. Renormalization group in six-derivative quantum gravity. _Phys. Rev. D_ 2021, _104_ , 085018.
  * (4) Weinberg, S. Ultraviolet Divergences in Quantum Theories of Gravitation. In _General Relativity: An Einstein Centenary Survey_ ; Hawking, S.W., Israel, W., Eds.; Cambridge University Press: Cambridge, UK; Chapter 16, pp. 790–831.
  * (5) Smolin, L. A Fixed Point for Quantum Gravity. _Nucl. Phys. B_ 1982, _208_ , 439–466.
  * (6) Platania, A.; Wetterich, C. Non-perturbative unitarity and fictitious ghosts in quantum gravity. _Phys. Lett. B_ 2020, _811_ , 135911.
  * (7) Anselmi, D.; Erlich, J.; Freedman, D.Z.; Johansen, A.A. Positivity constraints on anomalies in supersymmetric gauge theories. _Phys. Rev. D_ 1998, _57_ , 7570–7588.
  * (8) Anselmi, D.; Freedman, D.Z.; Grisaru, M.T.; Johansen, A.A. Nonperturbative formulas for central functions of supersymmetric gauge theories. _Nucl. Phys. B_ 1998, _526_ , 543–571.
  * (9) Namazie, M.A.; Salam, A.; Strathdee, J.A. Finiteness of Broken $N=4$ Superyang-mills Theory. _Phys. Rev. D_ 1983, _28_ , 1481.
  * (10) Fradkin, E.S.; Tseytlin, A.A. Conformal Supergravity. _Phys. Rept._ 1985, _119_ , 233–362.
  * (11) Codello, A.; Langæble, K.; Litim, D.F.; Sannino, F. Conformal Gauge-Yukawa Theories away From Four Dimensions. _J. High Energy Phys._ 2016, _7_ , 118.
  * (12) Litim, D.F.; Sannino, F. Asymptotic safety guaranteed. _J. High Energy Phys._ 2014, _12_ , 178.
  * (13) Bambi, C.; Modesto, L.; Rachwał, L. Spacetime completeness of non-singular black holes in conformal gravity. _J. Cosmol. Astropart. Phys._ 2017, _05_ , 003.
  * (14) Modesto, L.; Rachwal, L. Finite Conformal Quantum Gravity and Nonsingular Spacetimes. _arXiv_ 2016, arXiv:1605.04173.
  * (15) Rachwał, L. Conformal Symmetry in Field Theory and in Quantum Gravity. _Universe_ 2018, _4_ , 125.
  * (16) Jizba, P.; Giaccari, S.G.; Kňap, J. Dark side of Weyl gravity. _Universe_ 2020, _6_ , 123.
  * (17) Jizba, P.; Kleinert, H.; Scardigli, F. Inflationary cosmology from quantum Conformal Gravity. _Eur. Phys. J. C_ 2015, _75_ , 245.
  * (18) Jizba, P.; Kňap, J. Infrared behavior of Weyl Gravity: Functional Renormalization Group approach. _Phys. Rev. D_ 2020, _101_ , 044050.
  * (19) Niedermaier, M.; Reuter, M. The Asymptotic Safety Scenario in Quantum Gravity. _Living Rev. Rel._ 2006, _9_ , 5–173.
  * (20) Goroff, M.H.; Sagnotti, A. The Ultraviolet Behavior of Einstein Gravity. _Nucl. Phys. B_ 1986, _266_ , 709–736.
  * (21) Goroff, M.H.; Sagnotti, A. Quantum Gravity at Two Loops. _Phys. Lett. B_ 1985, _160_ , 81–86.
  * (22) Wilson, K.G.; Kogut, J.B.; The Renormalization group and the epsilon expansion. _Phys. Rept._ 1974, _12_ , 75–199.
  * (23) Pawlowski, J.M.; Reichert, M. Quantum Gravity: A Fluctuating Point of View. _Front. Phys._ 2021, _8_ , 527.
  * (24) Christiansen, N.; Litim, D.F.; Pawlowski, J.M.; Reichert, M. Asymptotic safety of gravity with matter. _Phys. Rev. D_ 2018, _97_ , 106012.
  * (25) Knorr, B.; Ripken, C.; Saueressig, F. Form Factors in Asymptotic Safety: Conceptual ideas and computational toolbox. _Class. Quant. Grav._ 2019, _36_ , 234001.
  * (26) Denz, T.; Pawlowski, J.M.; Reichert, M. Towards apparent convergence in asymptotically safe quantum gravity. _Eur. Phys. J. C_ 2018, _78_ , 336.
  * (27) Dona, P.; Giaccari, S.; Modesto, L.; Rachwal, L.; Zhu, Y. Scattering amplitudes in super-renormalizable gravity. _J. High Energy Phys._ 2015, _8_ , 038.
  * (28) Gubitosi, G.; Ripken, C.; Saueressig, F. Scales and hierachies in asymptotically safe quantum gravity: A review. _Found. Phys._ 2019, _49_ , 972–990.
  * (29) Eichhorn, A.; Platania, A.; Schiffer, M. Lorentz invariance violations in the interplay of quantum gravity with matter. _Phys. Rev. D_ 2020, _102_ , 026007.