[L. Rachwał]Lesław<EMAIL_ADDRESS>
myDepartamento de Física—Instituto de Ciências Exatas, Universidade Federal de
Juiz de Fora,
33036-900, Juiz de Fora, MG, Brazil secondspeaker at AAMP XVIII, online
session

# How to understand the Structure of Beta Functions in Six-derivative Quantum
Gravity?

###### Abstract

We extensively motivate the studies of higher-derivative gravities, and in
particular we emphasize which new quantum features theories with six
derivatives in their definitions possess. Next, we discuss the mathematical
structure of the exact on the full quantum level beta functions obtained
previously for three couplings in front of generally covariant terms with four
derivatives (Weyl tensor squared, Ricci scalar squared and the Gauss-Bonnet
scalar) in minimal six-derivative quantum gravity in $d=4$ spacetime
dimensions. The fundamental role here is played by the ratio $x$ of the
coupling in front of the term with Weyl tensors to the coupling in front of
the term with Ricci scalars in the original action. We draw a relation between
the polynomial dependence on $x$ and the absence/presence of enhanced
conformal symmetry and renormalizability in the models where formally
$x\to+\infty$ in the case of four- and six-derivative theories respectively.

###### keywords:

Quantum Gravity, Higher derivatives, Beta functions, UV-finiteness, Conformal
symmetry

##  1 Introduction and motivation

Six-derivative Quantum Gravity (QG) is a model of quantum dynamics of
relativistic gravitational field with higher derivatives. It is a special case
of general higher-derivative (HD) models which are very particular
modifications of Einsteinian gravitational theory. The last one is based on
the theory with up to two derivatives (an addition of the cosmological
constant term brings terms with no derivatives on the metric field at all) and
it is simply based on the action composed of the Ricci curvature scalar $R$
understood as the function of the spacetime metric. In this setup, we consider
that gravitational field is completely described by the symmetric tensor field
$g_{\mu\nu}$ being the metric tensor of the pseudo-Riemannian smooth
differential manifold of a physical spacetime. In Einstein’s theory the scalar
$R$ contains precisely two ordinary (partial) derivatives of the metric. The
action obtained by integrating over the spacetime volume the densitized
Lagrangian $\sqrt{|g|}R$ we call as Einstein-Hilbert action. The QG models
based on it were originally studied in [1, 2, 3]. Below we consider
modifications of two-derivative gravitational theory, where the number of
derivatives on the metric is higher than just two.

It must be remarked, however, that the kinematical framework of general
relativity (GR) (like metric structure of the spacetime manifold, the form of
Christoffel coefficients, the motion of probe particles, or geodesic and fluid
dynamics equations) remains intact for these modifications. Therefore these
higher-derivative (HD) models of gravitational field are still consistent with
the physical basis of GR, the only difference is that their dynamics – the
dynamics of the gravitational field – is described by classical equations of
motion with higher-derivative character. Thence these modifications of
standard Einsteinian gravitational theory are still in the set of generally
relativistic models of the dynamics of the gravitational field. They could be
considered both on the classical and quantum levels with the benefits of
getting new and deeper insights in the theory of relativistic gravitational
field. Our framework on the classical level can be summarized by saying that
we work within the set of metric theories of gravity, where the metric and
only the metric tensor characterizes fully the configurations of the
gravitational fields which are here represented by pseudo-Riemannian
differential manifolds of relativistic four-dimensional continuous spacetimes.
Therefore, in this work we neglect other classical modifications of GR, like
by adding torsion, non-metricity, other geometric elements or other scalars,
vectors or tensor fields. This choice of the dynamical variables for the
relativistic gravitational field bears impact both on the classical dynamics
as well as on the quantum theory.

Theories with higher derivatives come naturally both with advantages and with
some theoretical problems. This happens already on the classical level when
they supposed to describe the modified dynamics of the gravitational field
(metric field $g_{\mu\nu}(x)$ living on the spacetime manifold). These
successes and problems get amplified even more on the quantum level. The pros
for HD theories give strong motivations why to consider seriously these
modifications of Einstein’s gravitation. We will briefly discuss various
possibilities of how to resolve the problems of higher-derivative field
dynamics in one of the last sections of this contribution, while here we will
consider more the motivations.

On the classical level, the set of HD gravitational theories can be viewed as
one of the many possible modifications of two-derivative gravitational theory.
It is true that now observations, mainly in cosmology and on the intergalactic
scales, point to some possible failures of Einsteinian theory of gravity or to
our lack in understanding the proper nature of the sources of gravity in these
respective situations. There are various views possible on this situation and
its explanations by gravitational theories. In the first view, researchers say
that Einstein-Hilbert theory is still fine, but we need to add locally new
exotic (meaning coming with some non-standard properties) matter source. Since
we do not know what these sources for energy-momentum tensor (EMT) of matter
are built out of (for example – from which quantum fields of particle physics
as understood nowadays), we call the missing sources as dark energy and dark
matter respectively. Contrary to this approach, in the other, the
gravitational source is standard, that is we describe what we really see in
the galaxies and in the universe, without any “dark” components, but the
gravitational theory should be modified. In this second path, the internal
dynamics of the gravitational field is changed and that is why it reacts
differently to the same classical visible EMT source of standard matter. One
of the promising options is to add higher derivatives of the metric on the
classical level, but in such a way to still preserve the local Lorentz
symmetry of the dynamics that is to be safe with respect to the general
covariance. Hence all HD terms in the action of the theory must come from
generally densitized scalars which are HD analogs of the Ricci scalar. They
can be in full generality built as contractions of the metric tensors (both
covariant $g_{\mu\nu}$ and contravariant $g^{\mu\nu}$), Riemann curvature
tensor $R_{\mu\nu\\!\rho\sigma}$ and also of covariant derivatives
$\nabla_{\mu}$ acting on these Riemann tensors111We do not need to consider
covariant derivatives on the metric tensor because of the metricity condition,
$\nabla_{\mu}g_{\nu\\!\rho}=0$.. Initially this may look as presumably
unnecessary complication since classical equations of motion (EOM) with higher
derivatives of the gravitational field are even more complicated than already
a coupled system of non-linear partial differential equations for the
components of the metric tensor field in Einstein’s gravity. However, on the
cosmological and galactic scales some gravitational models with higher
derivatives give successes in explaining: the problem of dark matter halos,
flat galactic rotation curves, cosmological dark energy (late-time exponential
expansion of the universe) and also primordial inflation without a necessity
of having the actual inflaton field. These are amongst all of observational
pieces of evidence that can be taken for HD models.

Since our work is theoretical we provide below some conceptual and consistency
arguments for HD gravities. First, still on the classical level, within the
class of higher-derivative gravitational theories, there are models that are
the first, which besides relativistic symmetries, enjoy also invariance under
conformal symmetry understood in the GR framework. Properly this is called as
Weyl symmetry of the rescaling of the covariant metric tensor, according to
the law: $g_{\mu\nu}\to\Omega^{2}g_{\mu\nu}$ with $\Omega=\Omega(x)$ being an
arbitrary scalar parameter of these transformations. To understand better this
fact, one may first recall that the metric tensor $g_{\mu\nu}$ is taken here
as a dimensionless quantity and all energy dimensions are brought only by
partial derivatives acting on it. Next, the prerequisite for full conformal
symmetry is scale-invariance of the classical action, so the absence of any
dimensionful parameter in the definition of the theory. From these facts, one
derives that in four spacetime dimensions ($d=4$) the gravitational conformal
models must possess terms with precisely four derivatives acting on the
metric. In general, in $d$ dimensions, for conformal gravitational theory the
classical action must be precisely with $d$ derivatives on the metric. (One
sees due to the requirement of general covariance that this consideration of
conformal gravitational theories makes sense only in even dimensions $d$ of
spacetime.) Another interesting observation, is that the gravitational theory
with Einstein-Hilbert action is classically conformally invariant only in two-
dimensional framework. For $4$-dimensional scale-invariant gravitational
theory one must use a combination of the squares of the Riemann tensors and
various contractions thereof. (The term $\square R$ is trivially a total
derivative term, so cannot be used.) Therefore for $d$-dimensional conformal
gravitational theories ($d>2$) we inevitably must consider HD metric theories.
The conformally invariant gravitational dynamics is very special both on the
classical level and also on the quantum level as we will see in the next
sections.

The main arguments for higher-derivative gravitational theories in dimensions
$d>2$ come instead from quantum considerations. After all, it is not so
surprising that it is the quantum coupling between quantum field theory (QFT)
of matter fields and quantum (or semi-classical) gravity or self-interactions
within pure quantum gravity that dictates what should be a consistent quantum
theory of gravitational interactions. Our initial guess (actually Einstein’s
one) might not be the best one when quantum effects are taken fully into
account. Since it is the classical theory that is emergent from the more
fundamental quantum one working not only in the microworld, but at all energy
scales (equivalent to various distances), then the underlying fundamental
quantum theory must necessarily be mathematically consistent, while some
different classical theories may not possess the same strong feature. Already
here we turn the reader’s attention to the fact that the purely mathematical
requirement of the consistency on the quantum level of gravitational self-
interactions is very strongly constraining the possibilities for quantum
gravitational theories. It is more constraining than it was originally thought
of. Moreover, not all macroscopic, so long wavelength limit, classical
theories are with these quantum correspondence features, only those which
emerge as classical limits of consistent quantum gravity theories. Following
this path, at the end, we must also correct our classical gravitational
theory, and likely it will not be Einsteinian gravity any more.

From a different side, we know that matter fields are quantum, they interact
and they are energetic, so they are “charged” under gravitation since energy-
momentum content is what the gravity couples to. If we did not know nothing
about gravity, then we could discover something about it from quantum
considerations of gravitationally “charged” matter fields and their mutual
interactions consistent with quantum mechanics. In this way we could make
gravity dynamical and quantum with a proper form of graviton’s propagation.
Actually, it is the quantum consideration that makes the gauge bosons
mediating the interactions between quantum charged particles dynamical. These
gauge bosons are emanations or quantum realizations of classical dynamical
gauge fields that must be introduced in the classical dynamics of matter
fields or particles for the overall consistency. Below we will present a few
detailed arguments why we need HD gravities in $d>2$ giving rise to dynamical
gravitational fields with HD form of the graviton’s propagators in the quantum
domain. They are all related and in a sense all touch upon the issue of
coupling of a potential unknown dynamical quantum gravity theory to some
energetic quantum matter fields moving under the influence of classical
initially non-dynamical gravitational background field. (The background
gravitational field does not have to be static, stationary or completely time-
independent, what we only require here is that it is not a dynamical one.)
These last classical fields can be understood as frozen expectation values of
some dynamical quantum gravitational fields. As one can imagine for this
process of quantum balancing of interactions the issue of back-reaction of
quantum matter fields on the classical non-dynamical geometry is essential.

Firstly, we recall the argument of DeWitt and Utiyama [4]. Due to quantum
matter loops some UV divergences in the gravitational sector are generated.
This is so even if the original matter theory is with two-derivative actions
(like for example standard model of particle physics). The reasons for these
divergences are pictorially Feynman diagrams with quantum matter fields
running in the perturbative loops, while the graviton lines are only external
lines of the diagrams since they constitute classical backgrounds. In such a
way we generate the dynamics to the gravitational field due to quantum matter
interactions with gravity, so due to the back-reaction phenomena. If the
latter was neglected we would have only the impact of classical gravitational
field on the motion and interactions of quantum matter particles. We can be
very concrete here, namely for example in $d=4$ spacetime dimensions, the
dynamical action that is generated for gravity takes the form

$S_{\rm
div}=\\!\int\\!d^{4}x\sqrt{|g|}\left(\alpha_{C}C^{2}+\alpha_{R}R^{2}\right),$
(1)

so we see that counterterms of the GR-covariant form of $C^{2}$ and $R^{2}$
are being generated. (In the equation above, the $R^{2}$ and $C^{2}$ terms
denote respectively the square of the Ricci scalar and of the Weyl tensor,
where the indices are contracted in the natural order, i.e.
$C^{2}=C_{\mu\nu\\!\rho\sigma}C^{\mu\nu\\!\rho\sigma}$. Collectively, we will
denote these curvatures as ${\cal R}^{2}$, so ${\cal R}^{2}=R^{2},C^{2}$.)
This is true no matter what was our intention of what was the dynamical theory
of the gravitational field. We might have thought that this was described by
the standard two-derivative Einstein-Hilbert action, but still the above
results persist. One notices that in these two counterterms $C^{2}$ and
$R^{2}$ one has four derivatives acting on a metric tensor, so these are
theories of a general higher-derivative type, differently from originally
intended E-H gravitational theory whose action is just based on the Ricci
scalar $R$. These $C^{2}$ and $R^{2}$ terms appear in the divergent part of
the dynamically induced action for the gravitational fields. We must be able
to absorb these divergences to have a consistent quantum theory of the
gravitational field coupled to the quantum matter fields present here on such
curved (gravitational) backgrounds [5]. This implies that in the dynamics of
the gravitational field we must have exactly these terms with higher
derivatives as in (1). Finally, we can even abstract and forget about matter
species and consider only pure gravitational quantum theory. The consistency
of self-interactions there on the quantum level puts the same restriction on
the form of the action of the theory. In such a situation in the language of
Feynman diagrams, one considers also loops with quantum gravitons running
inside. These graphs induce the same form of UV divergences as in (1). Then in
such a model, we must still consider the dynamics of the quantum gravitational
field with higher derivatives. Hence, from quantum considerations higher
derivatives are inevitable.

We also remark here that in the special case, where the matter theory is
classically conformally invariant with respect to classical gravitational
background field (the examples are: massless fermion, massless Klein-Gordon
scalar field conformally coupled to the geometry, electrodynamic field or non-
Abelian Yang-Mills field in $d=4$), then only the conformally covariant
counterterm $C^{2}$ is generated, while the coefficient $\alpha_{R}=0$ in (1).
This is due to the fact that the quantization procedure preserves conformal
symmetries of the original classical theory coupled to the non-trivial
spacetime background. Such argument can be called as a conformal version of
the original DeWitt-Utiyama argument. Then the $R^{2}$ counterterm is not
needed but still the action of a quantum consistent coupled conformal system
requires the higher-derivative dynamics in the gravitational sector [6]. Here
this is clearly the gravitational dynamics only in the spin-$2$ sector of
metric fluctuations, which is contained entirely in the (conformal) $C^{2}$
sector of the generic four-derivative theory presented in (1).

An intriguing possibility for having higher derivatives in the gravitational
action was first considered by Stelle in [7] and some exact classical
solutions of such a theory were analyzed in [8, 9, 10]. In $d=4$ spacetime
dimensions, the minimal number of derivatives is exactly four, the same as the
number of dimensions [11]. This reasoning coincides with the one presented
earlier that we need to have in even number of dimensions $d$, precisely $d$
derivatives in the gravitational action to have first scale-invariant model of
gravitational dynamics (later possible to be promoted to enjoy also the full
conformal invariance). However, as proven by Asorey, Lopez and Shapiro in
[12], there are also possible theories with even higher number of derivatives,
and they still have good properties on the quantum level and when coupled to
quantum matter fields. Similarly, in the literature there are various known
motivations for conformal gravity in $d=4$ spacetime dimensions, one can
consult representatives in [13, 14].

Secondly, we emphasize that to have a minimal (in a sense with the smallest
number of derivatives) perturbatively renormalizable model of QG in dimensions
$d$, one also has to consider actions with precisely $d$ derivatives. The
actions with smaller number are not scale-invariant and have problems on the
quantum level to control all perturbative UV divergences, and not all of them
are absorbable in the counterterms coming from the original classical actions
of the theories – such models with less than $d$ derivatives are not
multiplicatively renormalizable. The first case for renormalizability is when
the action contains all generic terms with arbitrary coupling coefficients
with $d$ (partial) derivatives on the metric for $d$ dimensions. The special
cases when some coefficients and some coupling parameters vanish may lead to
restricted situations in which full renormalizability is not realized. We
discuss such special limiting cases in further sections of this paper. The
argument with the first renormalizable theory is a very similar in type to the
quantum induced action from matter fields, but this time the particles which
run in the perturbative loops of Feynman diagrams are quantum gravitons
themselves. So this argument about renormalizability applies to pure quantum
gravity cases. Unfortunately, the original Einstein-Hilbert action for QG
model is not renormalizable (at least not perturbatively) in $d=4$ dimensions
[15, 16, 17, 18]. The problems show up when one goes off-shell, couples some
matter, or goes to the two-loop order, while at the first loop order with pure
E-H gravitational action on-shell all UV divergences could be successfully
absorbed [15] on Einstein vacuum backgrounds (so on Ricci-flat
configurations). Actually, in such a case in vacuum configurations the theory
at the one-loop level is completely UV-finite.

There are also other ways how one can on the quantum level induce the higher-
derivative terms in the gravitational actions, although these further
arguments are all related to the original one from DeWitt and Utiyama. One
can, for example, consider integrating out completely quantum matter species
on the level of functional integral which represents all accessible
information of the quantum theory. In the situation, when these matter species
are coupled to some background gravitational field, then the resulting
partition function $Z$ is a functional of the background gravitational field.
Not surprisingly, this functional is of the higher-derivative nature in terms
of number of derivatives of the fundamental metric field, if we work in the
dimension $d>2$. This reasoning was for example popularized by ’t Hooft [19,
20, 21], especially since in $d=4$ it can give rise to another motivations for
conformal gravity as a quantum consistent model of conformal and gravitational
interactions, when massless fields are integrated out in the path integral.

In this way we can discover the quantum consistent dynamics of the
gravitational field even if we did not know that such quantum fields mediating
gravitational interactions between particles existed in the first place. The
graviton becomes a propagating particle and with higher-derivative form of the
propagator, which translates in momentum space to the enhanced suppression of
the fall-off of the propagator for large momenta in the UV regime. This is due
to the additional higher powers of propagating momentum in the perturbative
expression for the graviton’s propagator. This enhanced UV decaying form of
the propagator is what makes the UV divergences under perturbative control and
what makes the theory at the end renormalizable. Besides a few (finite number
of) controlled UV divergences the theory is convergent and gives finite
perturbative answers to many questions one can pose about the quantum dynamics
of the gravitational field, also in models coupled consistently to quantum
matter fields.

Another way is to consider the theory of Einsteinian gravity and corrections
to it coming from higher dimensional theories. One should already understood
from the discussion above, that E-H action is a good quantum action for the QG
model only in the special $2$-dimensional case. There in $d=2$ QG is very
special renormalizable and finite theory, but without dynamical content
resembling anything what is known from four dimensions (like for example the
existence of gravitational waves, graviton spin-2 particles, etc.). This is
again due to infinite power of conformal symmetry in $d=2$ case. Instead, if
one considers higher dimensions like $6$, $8$, etc. and then compactifies them
to common $4$-dimensional case, one finds that even if in the higher
dimensions one had to deal with the two-derivative theory based on the
Einstein-Hilbert action, then in the reduced case in four dimensions, one
again finds effective (dimensionally reduced) action with four derivatives.
These types of arguments were recently invoked by Maldacena [22] in order to
study higher-derivative (and conformal) gravities from the point of view of
higher dimensions, when the process of integration out of quantum modes
already took place and one derives a new dynamics for the gravitational field
based on some compactification arguments.

All this above shows that many arguments from even various different
directions lead to the studies of higher-derivative gravitational theories in
dimensions of spacetime $d>2$. Therefore, it is very natural to quantize such
four-derivative theories (like it was first done by Stelle) and treat them as
a starting point for discussion of QG models in $d=4$ case. At the end, one
can also come back and try to solve for exact solutions of these higher-
derivative gravitational theories on the classical level, although due to
increased level of non-linearities this is a very difficult task [23].

Yet another argument is based on apparent similarity and symmetry seen in the
action of quadratic gravity and action for a general Yang-Mills theory. Both
these actions are quadratic in the corresponding field strengths (or
curvatures). They are curvatures respectively in the external spacetime for
the gravitational field and in the internal space for gauge degrees of
freedom. The Einstein-Hilbert action is therefore _not similar_ to the $F^{2}$
action of Yang-Mills theory and the system of Einstein-Maxwell or Einstein-
Yang-Mills theory does not look symmetric since the number of curvatures in
two sectors is not properly balanced. Of course, this lack of balance is later
even amplified to the problematic level by quantum corrections and the
presence of unbalanced UV divergences (non-renormalizability!). Still, already
on the classical level, one sees some dichotomy, especially when one tries to
define a common total covariant derivative $D_{\mu}$ (covariant both with
respect to Yang-Mills internal group $G$ and with respect to gravitational
field). For such an object, one can define the curvature ${\cal F}_{\mu\nu}$
that is decomposed in its respective sectors into the gauge field strength
$F_{\mu\nu}$ and the Riemann gravitational tensor $R_{\mu\nu\\!\rho\sigma}$.
But the most natural thing to do here is to consider symmetric action
constructed with such a total curvature of the derivative $D_{\mu}$ and then
the generalized ${\cal F}^{2}$ is the first consistent option to include both
dynamics of the non-Abelian gauge field and also of the gravitational field.
As we have seen this choice is also stable quantum-mechanically [24] since
there are no corrections that would destabilize it and the only quantum
corrections present they support this ${\cal F}^{2}$ structure of the theory,
even if this was not there from the beginning. We emphasize that this was
inevitably the higher-derivative structure for the dynamics of the quantum
relativistic gravitational field studied here.

###  1.1 Motivations for and introduction to six-derivative gravitational
theories

Now, we would like to summarize here on what is the general procedure to
define the gravitational theory, both on the classical as well as on the
quantum level. First, we decide what our theory is of – which fields are
dynamical there. In our case these are gravitational fields entirely
characterized by the metric tensor of gravitational spacetime. Secondly, we
specify the set of symmetries (invariance group) of our theory. Again, in our
setup these are, in general, invariances under general coordinate
transformations also known as diffeomorphism symmetries of gravitational
theories. In this sense, we also restrict the set of possible theories from
general models considered in the gauge treatment of gravity, when the
translation group or full Poincaré groups are gauged. Then finally following
Landau we define the theory by specifying its dynamical action functional. In
our case for a classical level, this is a GR-invariant scalar obtained by
integrating some GR-densitized scalar Lagrangian over the full 4-dimensional
continuum (spacetime). As emphasized above, for theoretical consistency, we
must use Lagrangians (actions) which contain higher (partial) derivatives of
the metric tensor, when the Lagrangian is completely expanded to a form where
ordinary derivatives act on the metric tensors (contracted in various
combinations). Specifying now, to the case motivated above, we shall use and
study below the theories defined by classical action functionals which contain
precisely six derivatives of the metric tensor field.

In order to define the theory on the quantum level, we use the standard
functional integral representation of the partition function (also known as
the vacuum transition amplitude) of the quantum theory. That is we construct,
having the classical action functional $S_{\rm HD}$, being the functional of
the classical metric field $S_{\rm HD}[g_{\mu\nu}]$, the following object

$Z=\\!\int\\!{\cal D}g_{\mu\nu}\exp(iS_{\rm HD}),$ (2)

where in the functional integral above we must be more careful than just on
the formal level in defining properly the integration measure ${\cal
D}g_{\mu\nu}$. For example, we should sum over all backgrounds and also over
all topologies of the classical background gravitational field. One can hope
that it is also possible to classify in four dimensions all gravitational
configurations (all gravitational pseudo-Riemannian manifolds) over which we
should integrate above. The functional integral, if properly defined, is the
basis for quantum theory. One can even promote the point of view that by
giving the functional $Z$ one _defines_ the quantum theory even without
reference to any classical action $S$. However, it is difficult a priori to
propose generating functionals $Z$, which are consistent with all symmetries
of the theory (especially gauge invariances) and such that they possess
sensible macroscopic (classical) limits. For practical purposes of evaluating
various correlation functions between quantum fields and their fluctuations,
one modifies this functional $Z$ by adding a coupling of the quantum field
(here this role of the integration variable is played by $g_{\mu\nu}$) to the
classical external current $J$. And also for other theoretical reasons, one
can compute this functional in background field method, where the functional
integration is over fluctuation fields, while the classical action functional
is decomposed into background and parts quadratic, cubic and of higher order
in quantum fluctuation fields. For this one defines that the full metric is
decomposed as follows, $g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu}$, where the
background classical metric is denoted by $\bar{g}_{\mu\nu}$ and metric
perturbations by $h_{\mu\nu}$. By computing variational derivatives of the
partition function $Z[J]$ with respect to the classical current $J$ one gets
higher $n$-point functions with the accuracy of the full quantum level. One
can compute them both perturbatively (in loop expansion) or non-
perturbatively, and also on trivial backgrounds or in background field method.
Finally, for spacetimes which asymptotically reach Riemann-flatness, from on-
shell quantum Green functions dressed by wave functions of external classical
states, one derives quantum matrix elements of scattering processes. Only in
such conditions one can define general scattering problem in quantum
gravitational theory.

In this article, we want to analyze the quantum gravitational model with six
derivatives in the action. That the theory is with six derivatives can be
seen, because of two related reasons. First, one can derive the classical
equations of motion based on such an action. Then one will see that the number
of partial derivatives acting on a metric tensor in a general term of such
tensor of equations of motion is at most 6 in our model. Or similarly, one can
compute the tree-level graviton’s propagator for example around flat Minkowski
background. And then one notices that some components of this propagator are
suppressed in the UV regime by the power $k^{6}$ in Fourier space, when $k$ is
the propagating momentum of the quantum mode. Actually, for this last check
one does not even have to invert and compute the propagator, one can perform a
very much the same analysis on the level of the kinetic operator between
gravitational fluctuations around some background (of course the flat
background is here the easiest one). Later in the main text of this article,
we discuss how to overcome the problems in defining the propagator in some
special cases, but the situation with the terms of the kinetic operator is
almost always well-defined and one can read the six-derivative character of
the theory easily from there.

We have seen in the previous section that the four-derivative gravitational
theories in $d=4$ spacetime dimensions are scale-invariant (can be conformally
invariant) on the classical level and that they are also first minimal
renormalizable models of dynamical QG. This last assertion is proved by the
power counting analysis. We will show below that it is possible to further
extend the theory in such a way that the control over divergences is
strengthened even more and this is again based on the analysis of the
superficial degrees of divergences of any graph and also on the energy
dimensionality arguments. In this way we will also explain why we can call
generic six-derivative gravitational theories in $d=4$ as perturbatively
super-renormalizable theories.

The power counting analysis in the case of four-derivative Stelle gravity as
in (1) (quadratic gravity of the schematic type ${\cal R}^{2}$ as in [7])
leads to the following equality

$\Delta+d_{\partial}=4,$ (3)

where $\Delta$ is superficial degree of divergence of any Feynman graph $G$,
$d_{\partial}$ is the number of derivatives of the metric on the external
lines of the diagram $G$, and for future use we define $L$ as the number of
loop order. For tree-level (classical level) we have $L=0$, while for
concreteness we shall assume $L\geqslant 1$. This theory is simply
renormalizable since the needed GR-covariant counterterms (to absorb
perturbative UV divergences) have the same form as the original action in
(1)222We remind for completeness that the Gauss-Bonnet scalar term ${\rm
GB}=E_{4}=R_{\mu\nu\\!\rho\sigma}^{2}-4R_{\mu\nu}^{2}+R^{2}$ is a topological
term, that is its variation in four spacetime dimensions leads to total
derivative terms contributing nothing to classical EOM and also to quantum
perturbation theory. It may however contribute non-perturbatively when the
topology changes are expected. But for the sake of computing UV divergences we
might simply neglect the presence of this term both in the original action as
well as in the resulting one-loop UV-divergent part of the effective action..
In general local perturbatively renormalizable HD model of QG in $d=4$, the
divergences at any loop order must take the form as in (1) with a potential
addition of the topological Gauss-Bonnet term.

The change in the formula (3), when the six-derivative terms are leading in
the UV regime, is as follows

$\Delta+d_{\partial}=6-2L.$ (4)

The above formula can be also rewritten as a useful inequality (bound on the
superficial degree $\Delta$):

$\Delta\leqslant 6-2L=4-2(L-1),$ (5)

since $d_{\partial}\geqslant 0$. From this one sees an interesting feature
that while in the case of four-derivative Stelle theory the bound was
independent on the number of loops $L$, for the case of six derivatives (and
higher too) the bound is tighter for higher number of loops. This is the basis
for super-renormalizability properties. In particular, in the case of six-
derivative theories there are no any loop divergences at the level of fourth
loop, since for $L=4$ we find that $\Delta<0$, so all graphs are UV-
convergent. We also emphasize that a super-renormalizable model is still
renormalizable, but at the same time it is more special since infinities in
the former do not show up at arbitrary loop order $L$, which is instead the
case for merely renormalizable models. From the formula (4) at the $L=3$ loop
level the possible UV divergences are only of the form proportional to the
cosmological constant $\Lambda$ parameter, so completely without any partial
derivatives acting on the metric tensor. Similarly for the case of $L=2$, we
have that divergences can be proportional to the $\Lambda$ (with no
derivatives) and also to the first power of the Ricci scalar $R$ of the
manifold (with two derivatives on the metric, when it is expanded). In what
follows we will not concentrate on these types of subleading in the UV
divergences and our main attention in this paper will be placed on the four-
derivative divergences as present in the action (1). Up to the presence of the
Gauss-Bonnet term they are the same as induced from quantum matter loops.
These types of divergences are only generated at the one-loop level since for
them we must have $d_{\partial}=4$ and $\Delta=0$. The last information
signifies that they are universal logarithmic divergences. Their names
originate from the fact that they arise when the ultraviolet cutoff
$\Lambda_{\rm UV}$ is used to cut the one-loop integrations over momenta of
modes running in the loop in the upper limits.

The analysis of power counting implies that the theory has divergences only at
the first, second and third loop order and starting from the fourth loop level
it is completely UV-finite model of QG. Moreover, based on the above
argumentation, the beta functions that we report below (in front of GR-
covariant terms with four derivatives in the divergent effective action)
receive contributions only at the one-loop level and higher orders (like two-
and three-loop) do not have any impact on them. This means that the beta
functions that we are interested in and that we computed at the one-loop level
are all valid to all loop orders, hence our results for them are truly exact.
They do not receive any perturbative contributions from higher loops. For
other terms in the divergent action (like $\Lambda$ or $R$) this is not true.
The theory is four-loop finite, while the beta functions of $R^{2}$, $C^{2}$
and ${\rm GB}$ terms are one-loop exact. All these miracles are only possible
to happen in very special super-renormalizable model since we have six
derivatives in the gravitational propagator around flat spacetime. This number
is bigger than the minimal for a renormalizable and scale-invariant QG theory
in $d=4$ spacetime dimensions and this is the origin of the facts above since
we have a higher momentum suppression in the graviton’s propagator.

According to what we have stated before, we decide to study the quantum theory
described by the following classical Lagrangian,

${\cal L}=\omega_{C}C_{\mu\nu\\!\rho\sigma}\square
C^{\mu\nu\\!\rho\sigma}+\omega_{R}R\square R\\\
+\theta_{C}C^{2}+\theta_{R}R^{2}+\theta_{\rm GB}{\rm
GB}+\omega_{\kappa}R+\omega_{\Lambda}.$ (6)

From this Lagrangian we construct the action of our HD quantum gravitational
model, here with six derivatives as the leading number of derivatives in the
UV regime, by the formula

$S_{\rm HD}=\\!\int\\!d^{4}x\sqrt{|g|}{\cal L}.$ (7)

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 $d=4$ case). Moreover, by ${\rm
GB}$ we mean the Euler term which gives rise to Euler characteristic of the
spacetime after integrating over the whole manifold. Its integrand is given by
the term also known as the Gauss-Bonnet term and it has the following
expansion in other terms quadratic in the gravitational curvatures,

${\rm GB}=E_{4}=R_{\mu\nu\\!\rho\sigma}^{2}-4R_{\mu\nu}^{2}+R^{2}.$ (8)

Similarly, we can write for the “square” of the Weyl tensor in $d=4$

$C^{2}=C_{\mu\nu\\!\rho\sigma}^{2}=C_{\mu\nu\\!\rho\sigma}C^{\mu\nu\\!\rho\sigma}=R_{\mu\nu\\!\rho\sigma}^{2}-2R_{\mu\nu}^{2}+\frac{1}{3}R^{2}.$
(9)

Finally, to denote the box operator we use the symbol $\square$ with the
definition $\square=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$, which is a GR-
covariant analogue of the d’Alembertian operator $\partial^{2}$ known from the
flat spacetime.

It is important to emphasize here that the Lagrangian (6) describes the most
general six-derivative theory describing the propagation of gravitational
fluctuations on flat spacetime. For this purpose it is important to include
all terms that are quadratic in gravitational curvature. As it is obvious from
the construction of the Lagrangian in (6) for six-derivative model we have to
include terms which are quadratic in the Weyl tensor or Ricci scalar and they
contain precisely one power of the covariant box operator $\square$ (which is
constructed using the GR-covariant derivative $\nabla_{\mu}$). These two terms
exhaust all other possibilities since other terms which are quadratic and
contain two covariant derivatives can be reduced to the two above exploiting
various symmetry properties of the curvature Riemann tensor as well as
cyclicity and Bianchi identities. Moreover, 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. Other bases are possible as well
but then they distort and entangle various contributions of various terms to
these propagators. We also remark that the addition of the Gauss-Bonnet term
is possible here (but it is a total derivative in $d=4$); one could also add a
generalized Gauss-Bonnet term, which is an analogue of the formula in (8),
where the GR-covariant box operator in the first power is inserted in the
middle of each of the tensorial terms there, which are quadratic in
curvatures. Eventually, there is no contribution of the generalized Gauss-
Bonnet term in any dimension to the flat spacetime graviton propagator, so for
this purpose we do not need to add such term to the Lagrangian as it was
written in (6).

In what follows we employ the pseudo-Euclidean notations and by $\sqrt{|g|}$
we will denote the square root of the absolute value of the metric determinant
(always real in our conventions). The two most subleading terms in the
Lagrangian (6) are with couplings $\omega_{\kappa}$ and $\omega_{\Lambda}$
respectively. The first one is related to the Newton gravitational constant
$G_{N}$, while the last one $\omega_{\Lambda}$ to the value of the physical
cosmological constant parameter. The QG model with the Lagrangian (6) 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.

We would like to already emphasize here, that there are two remarkable special
limiting cases in the theory (6). In order to have a non-degenerate classical
action and the well-defined Hessian operator of the second variational
derivative, one needs to require that both coefficients of the UV-leading
terms, namely $\omega_{C}$ and $\omega_{R}$, should be non-zero. Only in this
case the theory is renormalizable, moreover only in this case it also has nice
additional features like super-renormalizability and that the fourth and
higher perturbative loop contributions are completely finite. We want to say
that the quantum calculations reported in the next section correspond only to
this kind of well balanced model with both Weyl tensor and Ricci scalar
squared terms and one power of the GR-covariant box operator inserted in the
middle. (This is in order to have a six-derivative action, but also with terms
that are precisely quadratic in gravitational curvatures.) In principle, there
exist also models with non-balanced situations and dichotomy between different
sectors of fluctuations. For example, in the special case of $\omega_{C}=0$,
$\theta_{C}\neq 0$ and $\omega_{R}\neq 0$, the theory has the propagating
spin-two mode with four derivatives and the propagating spin-zero mode with
six derivatives in the perturbative spectrum around flat spacetime. This has
to be contrasted with the fact that interaction vertices have always six
derivatives in both special and also in generic theories (with $\omega_{C}\neq
0$ and $\omega_{R}\neq 0$). For another special version of the model, with
$\omega_{C}\neq 0$ and $\omega_{R}=0$, the situation is quite opposite
regarding the spectrum, but the negative conclusions are the same. According
to the power counting arguments from [6, 25] and also from (3) in both special
cases the theories are unfortunately non-renormalizable. (We also discuss in
greater details the power counting for these two special limiting models in
section 4.4.) Hence one should be very careful in performing computations in
such cases and in trusting the results of limits there. These cases will be
analyzed in more details in the next sections as it will be revealed that they
are crucial for understanding the issue of the structure of perturbative
divergences both in the four-derivative as well as also in six-derivative QG
models in $d=4$.

The other consequences of the formula for power counting as presented in (4)
is that the subleading in the UV terms of the original action in (6) do not at
all contribute to the four-derivative terms leading in the UV regime of the
divergences in (1). That is we have that the coefficients $\alpha_{C}$,
$\alpha_{R}$ and $\alpha_{\rm GB}$ in (1) depend only on the ratio of the
coefficient in front of the term with Weyl tensors and box inserted in the
middle (i.e. $C\square C$) to the coefficient in front of the corresponding
term with two Ricci scalars (i.e. $R\square R$), so only on the ratio
$\omega_{C}/\omega_{R}$ also to be analyzed later at length here. These
coefficients of UV divergences $\alpha_{C}$, $\alpha_{R}$ and $\alpha_{\rm
GB}$ do not depend on $\theta_{C}$, $\theta_{R}$, $\theta_{\rm GB}$,
$\omega_{\kappa}$ nor on $\omega_{\Lambda}$. This is due to the energy
dimensionality considerations of other UV-subleading terms in the action in
(6). Only the terms having the same energy dimensionality as the leading in
the UV regime (shaping the UV form of the perturbative propagator) may
contribute to the leading form of UV divergences, which in the divergent
action (1) are represented by dimensionless numbers (in $d=4$) such as
$\alpha_{C}$, $\alpha_{R}$ and $\alpha_{\rm GB}$. For example, the terms with
coefficients $\theta_{C}$ or $\theta_{R}$ have different energy dimensions and
cannot appear there. This pertinent observation lets us for our computation to
use just the reduced action, where we write only the terms that are important
for the UV divergences we want to analyze in this paper. This action takes
explicitly the following form

$S_{\rm
HD}=\\!\int\\!d^{4}x\sqrt{|g|}\left(\omega_{C}C_{\mu\nu\\!\rho\sigma}\square
C^{\mu\nu\\!\rho\sigma}+\omega_{R}R\square R\right).$ (10)

We want to just remark here that the results in the theory with six-derivative
gravitational action are discontinuous to the results one obtains for the
similar type of computations in four-derivative Stelle quadratic QG models,
which are usually analyzed in $d=4$ as the first and the most promising models
of higher-derivative QG. This discontinuity is based on the known fact (both
for HD gauge and gravitational theories) that the cases with two and four more
derivatives in the action of respective gauge fields (metric fields in
gravity) than in the minimal renormalizable model are discontinuous and
exceptional, while the general formula exists starting from action with six
derivatives more in its definition (and then this formula could be
analytically extended). All three cases of: first minimal renormalizable
theory, and the models with two or four derivatives more are special and
cannot be obtained by any limiting procedure from the general results which
hold for higher-derivative regulated actions, which contain six or more
derivatives than in the minimal renormalizable model. For the case of QG in
$d=4$ in the minimal model we have obviously four derivatives. Of course, this
discontinuity is related to the different type of enhanced renormalizability
properties of the models in question. As we have already explained above the
gravitational model with six derivatives in $d=4$ is the first super-
renormalizable model of QG, where from the fourth loop on the perturbative UV
divergences are completely absent. The case of Stelle theory gives just the
renormalizable theory, where the divergences are present at any loop order
(they are always the same divergences, always absorbable in the same set of
counterterms since the theory is renormalizable). One sees the discontinuity
already in the behaviour of UV divergences as done in the analysis of power
counting. When the number of derivatives is increased in steps (by two), then
the level of loops when one does not see divergences at all decreases but in
some discontinuous jumps. And for example for the QG theory with ten or more
derivatives the UV divergences are only at the one-loop level. (For
gravitational theories with $8$ derivatives the last level which is divergent
is the second loop.)

There exist also analytic formulas, which combine the results for UV
divergences for the cases of theories with four or more derivatives more
compared to the minimal renormalizable model with four derivatives in $d=4$.
Again one sees from such formulas, that the correct results for the minimal
renormalizable model and the one with six derivatives are discontinuous. Then
the case with 8-derivative gravitational theory is the first one for which the
analytic formulas hold true. However, this has apparently nothing to do with
the strengthened super-renormalizability properties at some loop level as it
was emphasized above.

The six-derivative gravitational theory is therefore 3-loop super-
renormalizable since the 3-loop level is the last one, when one needs to
absorb infinities and renormalize anew the theory. These jumps from 3-loop
super-renormalizability to 2-loop and finally to one-loop super-
renormalizability are from their nature discontinuous and hence also the
results for divergences inherit this discontinuity. For theories with ten or
more derivatives we have one-loop super-renormalizability and the results for
even higher number of derivatives $2n$ must be continuous in the parameter of
the number of derivatives $2n$, which could be analytically extended to the
whole complex plane from the even integer values $2n\geqslant 10$, which it
originally had. In this analytically extended picture, the cases with eight,
six and four derivatives are special isolated points, which are discontinuous
and cannot be obtained from the general analytic formula valid for any
$n\geqslant 5$. The origin of this is again in power counting of divergences,
when some integrals over loop momenta are said to be convergent, when the
superficial degree of divergence is smaller than zero, and when this is non-
negative, then one meets non-trivial UV divergences. These infinities are
logarithmic in the UV cutoff $k_{\rm UV}$ for loop integration momenta for the
degree $\Delta$ vanishing, and power-law type for the degree $\Delta$
positive. This sharp distinction between what is convergent and what is
divergent (based on the non-negativity of the degree of divergences $\Delta$
of any diagram) introduces the discontinuity, which is the main source of the
problems here.

In this contribution, we mainly discuss and analyze the results which were
first obtained in our recent publication [26]. 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
[27] applied 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 [26] we read the beta functions of running dimensionless
gravitational couplings. The results for them in six-derivative gravitational
theory in $d=4$ spacetime dimensions were the main results there. They are
also described in section 2 here. Instead, in the present contribution, we
decided to include an extended discussion of the theoretical checks done on
these results in section 3. However, the main novel contribution is in section
4, where we present the analysis of the structure of these obtained results
for the beta functions. Our main goal here is to show an argumentation that
provides an explanation why the structure of the beta function is unique and
why it depends in this particular form on the ratio $x$ (to be defined later
in the main text in (36)). These comments were not initially included in the
main research article [26] and they constitute the main new development of the
present paper.

We remind to the reader that in this paper, in particular, we will spend some
time on attempts to explain the discontinuity of such results for UV
divergences, when one goes from six- to four-derivative gravitational
theories. So, in other words, when one reduces 3-loop super-renormalizability
to just renormalizability. Or equivalently, when the situation at the fourth
perturbative loop gets modified from not having divergences at all, because
all loop integrations give convergent results (with negative superficial
degree $\Delta<0$), to the situation when at this loop level still UV
divergences are present (since their degree $\Delta$ is zero for logarithmic
UV divergences in the cutoff). This clearly sharp contrast in the sign of the
superficial degree of divergences is one of the reasons, why the discontinuity
between the cases of six- and four-derivative gravitational theories in $d=4$
persists.

###  1.2 Addition of killer operators

As a matter of fact, we can also add other terms (cubic in gravitational
curvatures ${\cal R}^{3}$) to the Lagrangian in (6). These terms again will
come with the coefficients 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. The general form of them is given by the following list of six GR-
covariant terms

${\cal L}_{{\cal
R}^{3}}=s_{1}R^{3}+s_{2}RR_{\mu\nu}R^{\mu\nu}+s_{3}R_{\mu\nu}R^{\mu}{}_{\rho}R^{\nu\\!\rho}\\\
+s_{4}RR_{\mu\nu\\!\rho\sigma}R^{\mu\nu\\!\rho\sigma}+s_{5}R_{\mu\nu}R_{\rho\sigma}R^{\mu\\!\rho\nu\sigma}\\\
+s_{6}R_{\mu\nu\\!\rho\sigma}R^{\mu\nu}{}_{\kappa\lambda}R^{\rho\sigma\kappa\lambda}\,.$
(11)

Actually, these terms can be very essential for making the gravitational
theory with six-derivative actions completely UV-finite. However, for
renormalizability or super-renormalizability properties 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. In the analysis below
we did not take their contributions into account and made already a
technically demanding computation in a simplest minimal model with six-
derivative actions. The set of terms in (11) is complete in $d=4$ for all what
regards terms cubic in gravitational curvatures. This non-trivial statement is
due to various identities as proven in [28].

These cubic terms are also sometimes called “killers” of the beta functions
since they may have profound effects on the form of the beta functions of all
terms in the theory. This is roughly very simple to explain. These killer
terms are generally of the type $s{\cal R}^{3}$ and are to be added to the
original Lagrangian in (6) of six-derivative theories, where the UV-leading
terms were of the type $\omega{\cal R}\square{\cal R}$. It is well known that
to extract UV divergences at the one-loop level one has 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 of at least $s{\cal
R}$, when counted in powers of generalized curvature ${\cal R}$. Next, when
computing the trace of the functional logarithm of the Hessian operator for
the form of the one-loop UV-divergent effective action one uses the expansion
of the logarithm in a series according to

$\ln(1+z)=z-\frac{1}{2}z^{2}+\ldots.$ (12)

Hence we need to take maybe 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.
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. Hence the
contribution of the cubic killer in curvatures would produce addition to the
covariant terms with UV divergences of the general type $f(s){\cal R}^{2}$,
where the 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 general to solve
the system of the quadratic equations in the coefficients $s_{i}$. The only
obstacle for finding coefficients of the killers can be that some solutions of
this system reveal to be complex numbers, not real, but we need to require all
$s_{i}$ coefficients to be real for the definiteness of the action (for
example in the Euclidean case of the signature of the metric). Therefore this
issue requires a more detailed mathematical analysis, but the preliminary
results based on [29, 26] show that in most of the cases the UV-finiteness is
possible and easily can be achieved by adding the cubic killer operators from
(11) with real coefficients $s_{i}$.

One can compare the situation here with cubic killers to the more known
situation where the quartic killers are used to obtain UV-finiteness.
Unfortunately, such quartic killers cannot be added to the six-derivative
gravitational theory from (6) since they would have too many partial
derivatives and would destroy the renormalizability of the model. Quartic
killers can be included in theories with at least 8 derivatives. Such approach
seems to be preferred one since the contribution of quartic killers (of the
type schematically as ${\cal R}^{4}$) is always linear in $d=4$ to UV
divergences proportional to ${\cal R}^{2}$ schematically. And to solve linear
system of equations with linear coefficients is always doable and one always
finds solutions and they are always real. This approach was successfully
applied to gravity theories in [29], to gauge theories in [30], to the
theories on de Sitter and anti-de Sitter backgrounds [31] and also in general
non-local theories [32]. One could show that the UV-finiteness may be an
universal feature of quantum field-theoretical interactions in nature [33].
Moreover, this feature of the absence of perturbative UV divergences is
related to the quantum conformality as advocated in [34, 35].

###  1.3 Universality of the results

Finally, one of the most important features of the expression for the UV-
divergent part of the effective action in the six-derivative gravitational
theories is its complete independence of any parameter used in the
computation. This parameter can be gauge-fixing parameter, or it can appear in
gauge choice, or in details of some renormalization scheme, etc. This bold
fact of complete universality of the results for the effective action was
proven by the theorem by Kallosh, Tarasov and Tyutin (KTT) [36, 37, 38],
applied here to the six-derivative QG theories. The theorem expresses the
difference between two effective actions of the same theory but computed using
different set of external parameters. Basically, this difference is
proportional to the off-shell tensor of classical equation of motion of the
original theory. And this difference disappears on-shell. However, in our
computation we want to exploit the case when the effective action and various
Green functions are computed from it understood as the off-shell functional.

But in super-renormalizable theories there is still some advantage of using
this theorem, namely for this one notices the difference in number of
derivatives on the metric tensor between the original action and Lagrangian of
the theory as it is in the form (10) (and resulting from it classical EOM) and
between the same counting of derivatives done in the divergent part of the
effective action. We remind the reader that in the former case we have six
derivatives on the metric, while in the latter we count up to four
derivatives. This mismatch together with the theorem of KTT implies that the
difference between the two UV-divergent parts of the effective actions (only
for these parts of the effective actions) computed using two different schemes
or methods must vanish in super-renormalizable QG theories with six-derivative
actions for whatever change of the external parameters that are used for the
computation of these UV-divergent functionals. This means that our results for
divergences are completely universal and cannot depend on any parameter. Hence
we derive the conclusion that our found divergences do not depend on the
gauge-fixing parameters, gauge choices nor on other parametrization
ambiguities. We remark that this situation is much better than for example in
E-H gravity, where the dependence on a gauge is quite strong, or even in
Stelle four-derivative theory, where four-derivative UV-divergent terms also
show up some ambiguous dependence on gauge parameters off-shell. Here we are
completely safe from such problems and such cumbersome ambiguities.

In this way such beta functions are piece of genuine observable quantity that
can be defined in super-renormalizable models of QG. They are universal,
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 observable in QG models. Therefore all these nice features gives us
even more push towards analyzing the structure of such physical quantities and
to understand this based on some theoretical considerations. This is what we
are trying to attempt in this contribution.

Another important feature is that in theories with higher derivatives in their
defining classical action, on the full quantum level there is no need for
perturbative renormalization of the graviton‘s wave function. This is also
contrary to the case of two-derivative theory, when one has to take this
phenomena into account, although its expression is not gauge-invariant and
depends on the gauge fixing. These nice properties of no need for wave
function renormalization can be easily understood in the Batalin-Vilkovisky
formalism for quantization of gauge theories (or in general theories with
differential constraints) [39, 40]. This important feature is also shared by
other, for example, four-derivative QG models. Since the wave function of the
graviton does not receive any quantum correction, then one can derive the form
of the beta functions for couplings just from reading the UV divergences of
the dressed two-point functions with two external graviton lines. We can
simplify our computation drastically since for this kind of one-loop
computation we do not have to bother ourselves with the three- or higher
$n$-point function to independently determine the wave function
renormalization. Unfortunately, the latter is the case, for example, for
standard gauge theory (Yang-Mills model) or for E-H gravity, where the
renormalization of the coupling constant of interactions has to be read from
the combination of the two- and three-point functions of the quantum theory,
while the wave function renormalization of gauge fields or graviton field
respectively can be just read from quantum dressed two-point Green function.
For the case of six-derivative theories, just from the two-point function we
can read everything about the renormalization of the coupling parameters of
gravitons’ interactions. Additionally, we have that on the first quantum loop
level we do not need to study effective interaction vertices dressed by
quantum corrections. Hence, here at the one-loop level there is no quantum
renormalization of the graviton’s wave function and UV divergences related to
interactions are derived solely from propagation of free modes (here of
graviton fields) around the flat spacetime and corrected (dressed) at the
first quantum loop. Effective vertices of interactions between gravitons do
not matter for this, but that situation may be changed at higher loop orders.
At the one-loop level this is a great simplification for our algorithm of
derivation of the covariant form of UV divergences since we just need to
extract them from the expression for one-loop perturbative two-point
correlators of the theory, both in cases of four- and six-derivative QG
models.

All these nice features of the six-derivative QG model makes it further worth
studying as an example of non-trivial RG flows in QG. Here we have exactness
of one-loop expressions for running $\theta_{C}(t)$, $\theta_{R}(t)$ and
$\theta_{\rm GB}(t)$ coupling parameters in (6), together with super-
renormalizability. This is one of the most powerful and beautiful features of
the super-renormalizable QG theory analyzed here. Therefore, this model gives
us a good and promising theoretical laboratory for studying RG flows in
general quantum gravitational theories understood in the field-theoretical
framework.

We remark that from a technical point of view, the one-loop calculations in
super-renormalizable models of QG are more difficult when compared to the ones
done in the four-derivative just renormalizable gravitational models [41, 27,
42, 43]. The level of complexity of such calculations depends strongly on the
number of derivatives in the classical action of the model as well as on the
type of one-loop counterterms one is looking for. The counterterm for the
cosmological constant is actually very easy to obtain and this was done
already in [12]. Next, the derivation of the divergence linear in the scalar
curvature $R$ requires really big efforts and was achieved only recently in
our collaboration in [44]. In the present work, we comment on the next step,
and we show the results of the calculations of the simply looking one-loop UV
divergences for the four-derivative sector in the six-derivative minimal
gravity model. In our result, we have now full answers to the beta functions
for the Weyl-squared $C^{2}$, Ricci scalar-squared $R^{2}$ and the Gauss-
Bonnet ${\rm GB}$ scalar terms. The calculation is really tedious and
cumbersome and it was done for the simplest possible six-derivative QG theory
without cubic terms in the classical action, which here would be third powers
of the generalized curvature tensor ${\cal R}^{3}$. Even in this simplest
minimal case, the intermediate expressions are too large for the explicit
presentation here, hence they will be mostly omitted. Similar computations in
four-, six- and general higher-derivative gauge theory were also performed in
[45, 30, 46].

As it was already mentioned above, the derivation of zero- and two-derivative
ultraviolet divergences has been previously done in Refs. [12] and [44]. Below
we will show the results for the complete set of beta functions for the theory
(10). This we will achieve by deriving the exact and computed at one-loop beta
function coefficients for the four-derivative gravitational couplings, namely
$\theta_{C}$, $\theta_{R}$ and $\theta_{\rm GB}$, extracted as the
coefficients of the UV-divergent part of the effective action in (1). Without
loss of generality, the calculation will be performed in the reduced model
(10), so without terms subleading in the number of partial derivatives acting
on the metric tensor after the proper expansion here. (We will not need to
include terms like $R^{2}$, $C^{2}$ or even $R$ in (10).) This is clearly
explained by the arguments from dimensional analysis since the divergences
with four derivatives of the metric, in (1), are of our biggest interest here.
Moreover, numerical coefficients of those subleading terms cannot in any way
combine with coefficients of propagators (shaped in the UV regime by the
leading terms with six derivatives in the action (10)) to form dimensionless
ratios in front of terms in (1) in $d=4$ spacetime dimensions.

##  2 Brief description of the technique for computing UV divergences

An essential part of the calculations is pretty much the same as usually done
in any higher-derivative QG model, especially in the renormalizable or super-
renormalizable models [44, 26] as considered here. In what follows, we can
skip a great part of the explanations. We will focus on the calculation of the
fourth derivative terms of the divergent part of the effective action.

First, to perform pure computation we use the background field method, which
is defined by the following splitting of the metric

$g_{\mu\nu}\longrightarrow{\bar{g}}_{\mu\nu}+h_{\mu\nu}$ (13)

to the background $\bar{g}_{\mu\nu}$ and the quantum fluctuation parts given
by the spin-2 symmetric tensor $h_{\mu\nu}$.

The next step is to define the gauge-fixing condition. Since our theory with
six derivatives still possesses gauge invariance due to diffeomorphism
symmetry we have to fix the gauge to make the graviton propagator non-
degenerate. For this we will make some choice of the gauge-fixing parameters,
here represented by numerical $\alpha$, $\beta$ and $\gamma$ parameters.
First, we choose the parameter $\beta$ in the harmonic background gauge-fixing
condition $\chi_{\mu}$, according to

$\chi_{\mu}=\nabla^{\lambda}h_{\lambda\mu}-\beta\,\nabla_{\mu}h,\qquad
h=h^{\nu}{}_{\nu}\,,$ (14)

in the most simple “minimal” form, as will be indicated below. The same
concerns the parameters $\alpha$ and $\gamma$. Finally, we select a general
form of the weighting operator, $\,\hat{C}=\tilde{C}^{\mu\nu}$, which is
defined by the formula below:

${\hat{C}}=\tilde{C}^{\mu\nu}\,=\,\,-\frac{1}{\alpha}\left(g^{\mu\nu}\Box^{2}+(\gamma-1)\nabla^{\mu}\Box\nabla^{\nu}\right).$
(15)

This together with the gauge-fixing condition, that is $\chi^{\mu}$, defines
the gauge-fixing action [41] in the following form,

$S_{{\rm
gf}}=\\!\int\\!d^{4}x\sqrt{|g|}\,\,\chi_{\mu}\,\tilde{C}^{\mu\nu}\,\chi_{\nu}.$
(16)

The action of the complex Faddeev-Popov (FP) ghost fields (respectively
$\bar{C}^{\mu}$ and $C_{\mu}$) has in turn the form

$S_{{\rm
gh}}\,=\,\\!\int\\!d^{4}x\sqrt{|g|}\,\,\bar{C}^{\mu}M_{\mu}{}^{\nu}C_{\nu},$
(17)

where the bilinear part between the anti-ghost $\bar{C}^{\mu}$ and ghost
fields $C_{\mu}$, the so called FP-matrix $\hat{M}$, depends differentially on
$\chi_{\mu}$ gauge-fixing conditions and also on the contracted form of the
generator of gauge transformations $\hat{R}$,

${\hat{M}}\,=\,M_{\mu}{}^{\nu}\,=\,\frac{\delta\chi_{\mu}}{\delta
g_{\alpha\beta}}\,R_{\alpha\beta}\,^{\nu}\,=\,\delta_{\nu}^{\mu}\Box+\nabla^{\nu}\nabla_{\mu}-2\beta\nabla_{\mu}\nabla^{\nu}.$
(18)

In the above equation by the matrix-valued operator $R_{\alpha\beta}\,^{\nu}$
we mean the generator of infinitesimal diffeomorphism (local gauge)
transformations in any metric theory of gravity.

Since as proven and explained at the end of section 1.3, our final results for
UV divergences are here completely universal and they are independent of any
parameter used to regularize, compute and renormalize the effective action of
the theory, then we can take the following philosophy at work here. We choose
some specific gauge choice in order to simplify our calculation, but then we
are sure that the final results will be still correct, if obtained
consistently within this computation done in a particular gauge choice. It is
true that intermediate steps of the computation may be different in different
gauges, but the final results must be unique and it does not matter which way
we arrive to them. We think we could choose one of the simplest path to reach
this goal. A posteriori this method is justified, but the middle steps of the
processing of the Hessian operator will not have any invariant objective
physical meaning. These are just steps in the calculational procedure in some
selected gauge.

One knows that in such a case, for example, for a formalism due to Barvinsky-
Vilkovisky (BV) [27] of functional traces of differential operators applied in
the background field method framework, all intermediate results are manifestly
gauge-independent. Then still such partial contributions (any of them)
separately do not have any sensible physical meaning, although such results
are gauge-independent and look superficially physical – any physical meaning
cannot be properly associated to them, if all these terms are not taken in
total and only in the final sum. On the contrary, if the computation is
performed using Feynman diagrams, momentum integrals and around flat
spacetime, then the intermediate results are not gauge-invariant, as it is
well known for partial contributions of some graphs, and only in the final sum
they acquire such features of gauge-independence.

We also need to distinguish here two different features. Some partial results
may be still gauge-dependent and their form may not show up gauge symmetry
(for example, using Feynman diagram approach, a contribution from a subset of
divergent diagrams may not be absorbed by a gauge-covariant counterterm:
$F^{2}$ in gauge theories, or ${\cal R}^{2}$ in gravity in $d=4$). This
feature should be however regained when the final results are obtained. This
is actually a good check of the computation. But another property is
independence of the gauge-fixing parameters, which are spurious non-physical
parameters. At the same time, a counterterm might be gauge-covariant (built
with $F^{2}$ or ${\cal R}^{2}$ terms), but its front coefficient may depend on
these gauge parameters $\alpha$, $\beta$, $\gamma$, etc. This should not
happen for the final results and they should be both gauge-covariant (so
gauge-independent or gauge-invariant) and also gauge-fixing parameters
independent. These two necessary properties, to call the result physical, must
be realized completely independently and they are a good check of the
correctness of the calculation.

Unfortunately, it seems that using the BV computational methods even in the
intermediate results for traces of separate matrix-valued differential
operators (like $\hat{H}$, $\hat{M}$ and $\hat{C}$), we see already both
gauge-independence and gauge-fixing parameters independence provided that such
parameters were not used in the definition of these operators. Only in some
cases, the total result is only gauge-fixing parameter independent. This means
that within this formalism of computation this check is not very valuable and
one basically has to be very careful to get the correct results at the end.
Instead, we perform a bunch of other rigorous checks of our results as it is
mentioned, for example, in section 3.

Finally, let us here give briefly a few details concerning the choice of the
gauge-fixing parameters $\alpha$, $\beta$ and $\gamma$. The bilinear form of
the action is defined from the second variational derivative (giving rise to
the Hessian operator $\hat{H}$)

${\hat{H}}=H^{\mu\nu,\rho\sigma}=\frac{1}{\sqrt{|g|}}\,\,\frac{\delta^{2}\left(S+S_{{\rm
gf}}\right)}{\delta h_{\mu\nu}\,\delta h_{\rho\sigma}}\,=\,H_{{\rm
lead}}^{\mu\nu,\rho\sigma}\,+\,{\cal O}(\nabla^{4}),$ (19)

where the first term $H_{{\rm lead}}^{\mu\nu,\rho\sigma}$ contains six-
derivative terms, which are leading in the UV regime. By $\,{\cal
O}(\nabla^{4})\,$ we denote the rest of the bilinear form, with four or less
derivatives and with higher powers of gravitational curvatures $\cal R$. The
energy dimension of this expression is compensated by the powers of curvature
tensor $\cal R$ and its covariant derivatives, hence in this case, we can also
denote $\,{\cal O}(\nabla^{4})\,=\,O({\cal R})$. The corresponding full
expression for the Hessian operator $\hat{H}$ is very bulky, and we will not
include it here.

The highest derivative part (leading in the UV regime) of the ${\hat{H}}$
operator, after adding the gauge-fixing term (16) that we have selected, has
the form

$H_{{\rm
lead}}^{\mu\nu,\rho\sigma}=\Big{[}\omega_{C}\,\delta^{\mu\nu,\rho\sigma}\,+\,\Big{(}\frac{\beta^{2}\gamma}{\alpha}-\frac{\omega_{C}}{3}+2\omega_{R}\Big{)}g^{\mu\nu}g^{\rho\sigma}\Big{]}\square^{3}\\\
+\Big{(}\frac{\omega_{C}}{3}-2\omega_{R}-\frac{\beta\gamma}{\alpha}\Big{)}\Big{(}g^{\rho\sigma}\nabla^{\mu}\nabla^{\nu}+g^{\mu\nu}\nabla^{\rho}\nabla^{\sigma}\Big{)}\square^{2}\\\
+\Big{(}\frac{1}{\alpha}g^{\mu\rho}-2{\omega_{C}}g^{\mu\rho}\Big{)}\nabla^{\nu}\nabla^{\sigma}\square^{2}\\\
+\Big{(}\frac{2\omega_{C}}{3}+2\omega_{R}+\frac{\gamma-1}{\alpha}\Big{)}\nabla^{\mu}\nabla^{\nu}\nabla^{\rho}\nabla^{\sigma}\square.$
(20)

In this expression, we do not mark explicitly the symmetrization in and
between the pairs of indices $(\mu,\nu)$ and $(\rho,\sigma)$ for the sake of
brevity.

To make the UV-leading part of the Hessian operator $H_{{\rm
lead}}^{\mu\nu,\rho\sigma}$ minimal, one has to choose the following values
for the gauge-fixing parameters [44]:

$\alpha=\frac{1}{2\omega_{C}},\qquad\beta=\frac{\omega_{C}-6\omega_{R}}{4\omega_{C}-6\omega_{R}},\qquad\gamma=\frac{2\omega_{C}-3\omega_{R}}{3\omega_{C}}.$
(21)

We previously explained that this choice does not affect the values and the
form of one-loop divergences in super-renormalizable QG. Thus, we assume it as
the most simple option.

One notices that the expressions for gauge-fixing parameters in (21) are
singular in the limit $\omega_{C}\to 0$ and also when
$\omega_{C}=\frac{3}{2}\omega_{R}$. While the first one is clearly
understandable, because then we are losing one term $\omega_{C}C\square C$ in
the action (10) and the theory is degenerate and non-generic, the second
condition is not easily understandable in the Weyl basis of writing terms in
the action in (10) (with $R^{2}$ and $C^{2}$ terms). To explain this other
spurious degeneracy one rather goes to the Ricci basis of writing terms (with
$R^{2}$ and $R^{2}_{\mu\nu}=R_{\mu\nu}R^{\mu\nu}$ elements and also properly
generalized to the six-derivative models by inserting one power of the box
operator in the middle). There one sees that the absence of the coefficient in
front of the $R^{2}_{\mu\nu}$ leads to the pathology in the case of
$\omega_{C}=\frac{3}{2}\omega_{R}$ and also formal divergence of the $\beta$
gauge-fixing parameter. We remark that in the final results there is no any
trace of this denominator and this divergence, hence the condition for non-
vanishing of the coefficient in front of the covariant term $R^{2}_{\mu\nu}$
in the Ricci basis does not have any sensible and crucial meaning – this is
only a spurious intermediate dependence on $(4\omega_{C}-6\omega_{R})^{-1}$.
Contrary, the singular dependence on $\omega_{C}$ coefficient is very crucial
and will be analyzed at length here. Actually, to verify that the denominators
with $(4\omega_{C}-6\omega_{R})^{-1}$ completely cancel out in the final
results is a powerful check of our method of computation.

Now we can collect all the necessary elements to write down the general
formula for the UV-divergent part of the one-loop contribution to the
effective action of the theory [41],

${\bar{\Gamma}}^{(1)}\,=\,\frac{i}{2}{\rm Tr}\ln{\hat{H}}\,-\,i{\rm
Tr}\ln{\hat{M}}\,-\,\frac{i}{2}{\rm Tr}\ln{\hat{C}}.$ (22)

The calculation of the divergent parts of the first two expressions in (22) is
very standard. One uses for this the technique of the generalized Schwinger-
DeWitt method [27], which was first introduced by Barvinsky and Vilkovisky.
For this reason we shall skip most of the standard technical details here. We
use the Barvinsky-Vilkovisky trace technology related to the covariant heat
kernel methods together with methods of dimensional regularization (DIMREG) to
evaluate the functional traces present in (22) and to have under control the
general covariance of the final results. Due to this we cannot check it
because all three contributions in (22) gives results which look covariant and
sensible. We remind the reader that here we work with the minimal gauge choice
and in general all three terms separately will show the gauge dependence and
also spurious dependence on gauge-fixing parameters $\alpha$, $\beta$ and
$\gamma$. However, only the final results, so the weighted sum as in (22) is
properly gauge-independent and gauge-fixing independent and gives rise to a
physical observable of the beta functional of the theory at the one-loop
level.

The computational method that we adopt here consists basically of using the
Barvinsky-Vilkovisky trace technology 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 are obtained in
background field method and from UV divergences in [26] we read the beta
functions of running gravitational couplings. We also present here below an
illustrative scalar example of the techniques by which these results were
obtained.

###  2.1 Example of the BV method of computation for the scalar case

The simplest example to use the technique of computation presented here can be
based on the analysis of the scalar case given by the action

$S=\\!\int\\!d^{4}x\left(-\frac{1}{2}\phi\square\phi-\frac{\lambda}{4!}\phi^{4}\right).$
(23)

From this action one reads the second variational derivative operator (also
known as the Hessian) given by the formula

$H=\frac{\delta^{2}S}{\delta\phi^{2}}=-\square-\frac{\lambda}{2}\phi^{2}.$
(24)

Next, one needs to compute the following functional trace ${\rm Tr}\ln H$ to
get the UV-divergent part of the one-loop effective action

${\rm Tr}\ln H={\rm Tr}\ln\left(-\square-\frac{\lambda}{2}\phi^{2}\right)\\\
={\rm
Tr}\ln\left(-\square\left(1+\frac{\lambda}{2}\phi^{2}\square^{-1}\right)\right)\\\
={\rm Tr}\ln\left(-\square\right)+{\rm
Tr}\ln\left(1+\frac{\lambda}{2}\phi^{2}\square^{-1}\right).$ (25)

In the above expression, one concentrates on the second part which contains
the $\lambda$ coupling. One expands the logarithm, as in (12), in the second
trace to the second order in $\lambda$. This yields

${\rm Tr}\ln\left(1+\frac{\lambda}{2}\phi^{2}\square^{-1}\right)={\rm
Tr}\left(\frac{\lambda}{2}\phi^{2}\square^{-1}\right)\\\ -\frac{1}{2}{\rm
Tr}\left(\frac{\lambda}{2}\phi^{2}\square^{-1}\right)^{2}+\ldots$ (26)

and one picks up from it only the expression quadratic in $\lambda$ and
quartic in the background scalar field $\phi$, which is also formally
quadratic in the inverse box operator $\square^{-1}$, that is the part

${\rm Tr}\ln H\supset-\frac{1}{2}\frac{\lambda^{2}}{4}\phi^{4}{\rm
Tr}\,\square^{-2}=-\frac{\lambda^{2}}{8}\phi^{4}{\rm Tr}\,\square^{-2}.$ (27)

Precisely this expression is relevant for the UV divergence proportional to
the quartic interaction term $-\frac{\lambda}{4!}\phi^{4}$ in the original
scalar field action (23). Noticing that the functional trace of the
$\square^{-2}$ scalar operator in $d=4$ is given by

${\rm Tr}\,\square^{-2}=i\frac{\ln L^{2}}{(4\pi)^{2}},$ (28)

where $L$ is a dimensionless UV-cutoff parameter related to the $\Lambda_{{\rm
UV}}$ dimensionful momentum UV-cutoff and the renormalization scale $\mu$ via
$\Lambda_{{\rm UV}}=L\mu$, one finds for the UV-divergent and interesting us
part of the one-loop effective action here

$\Gamma_{{\rm div}}^{(1)}=\frac{i}{2}{\rm Tr}\ln
H\supset\\!\int\\!d^{4}x\frac{\ln
L^{2}}{(4\pi)^{2}}\frac{\lambda^{2}}{16}\phi^{4}.$ (29)

Now, one can compare this to the original action terms in (23) describing
quartic interactions of the scalar fields $\phi$:
$-\\!\int\\!d^{4}x\frac{\lambda}{24}\phi^{4}$. The counterterm action (to
absorb UV divergences) is opposite to $\Gamma_{{\rm div}}$ and the form of the
terms in the counterterm action is expressed via perturbative beta functions
of the theory. That is in the counterterm action $\Gamma_{{\rm ct}}$ we expect
terms

$\Gamma_{{\rm ct}}=-\Gamma_{{\rm div}}=-\frac{1}{2}\frac{\ln
L^{2}}{24}\\!\int\\!d^{4}x\beta_{\lambda}\phi^{4}$ (30)

with the front coefficient exactly identical to the one half of the one in
front of the quartic interactions in the original action in (23) (being equal
to $-\frac{1}{4!}=-\frac{1}{24}$). From this one reads that (identifying that
effectively $\ln L^{2}\to 1$ for comparison)

$-\frac{1}{48}\beta_{\lambda}=-\frac{\lambda^{2}}{16(4\pi)^{2}}$ (31)

and finally that

$\beta_{\lambda}=\frac{3\lambda^{2}}{(4\pi)^{2}},$ (32)

which is a standard result for the one-loop beta function of the quartic
coupling $\lambda$ in $\frac{1}{4!}\lambda\phi^{4}$ scalar theory in $d=4$
spacetime dimensions.

One sees that even in the simplest framework, the details of such a
computation are quite cumbersome, and we decide not to include in this
manuscript other more sophisticated illustrative examples of such derivation
of the explicit results for beta functions of the theory. The reader, who
wants to see some samples can consult more explicit similar calculations as
presented in references [26, 44, 30]. In particular, the appendix of [30]
compares two approaches to the computation of UV divergences in gauge theory
(simpler than gravity but with non-Abelian gauge symmetry) – using BV heat
kernel technique and using standard Feynman diagram computation using graphs
and Feynman rules around flat space and in Fourier momentum space.

###  2.2 Results in six-derivative gravity

The final results for this computation of all UV divergences of the six-
derivative gravitational theory are

$\Gamma^{(1)R,C}_{\rm div}=-\frac{\ln
L^{2}}{2(4\pi)^{2}}\\!\int\\!d^{4}x\sqrt{|g|}\left\\{\Big{(}\frac{397}{40}+\frac{2x}{9}\Big{)}C^{2}\right.\\\
\left.+\frac{1387}{180}{\rm GB}-\frac{7}{36}R^{2}\right\\}\qquad$ (33)

for the case of six-derivative pure QG model in $d=4$ spacetime dimensions and

$\Gamma^{(1)R,C}_{\rm div}=-\frac{\ln
L^{2}}{2(4\pi)^{2}}\\!\int\\!d^{4}x\sqrt{|g|}\left\\{-\frac{133}{20}C^{2}\right.\\\
\left.+\frac{196}{45}{\rm GB}+\left(-\frac{5}{2}x_{\rm
4-der}^{-2}+\frac{5}{2}x_{\rm
4-der}^{-1}-\frac{5}{36}\right)R^{2}\right\\}\qquad$ (34)

for the case of four-derivative pure Stelle quadratic model of QG to the one-
loop accuracy. This last result was first reported in [42]. The result in six-
derivative gravity is freshly new [26]. Here we define the covariant cut-off
regulator $L$ [27], which stays in the following relations to the dimensional
regularization parameter $\epsilon$ [43, 27],

$\ln L^{2}\equiv\ln\frac{\Lambda_{\rm
UV}^{2}}{\mu^{2}}\,=\,\frac{1}{\epsilon}=\frac{1}{2-\omega}=\frac{2}{4-n}\,,$
(35)

where we denoted by $n$ the generalized dimensionality of spacetime in the
DIMREG scheme of regularization (additionally $\Lambda_{\rm UV}$ is the
dimensionful UV cutoff energy parameter and $\mu$ is the quantum
renormalization scale). Moreover, to write compactly our finite results for
the six-derivative theory we used the definition of the fundamental ratio of
the theory $x$ as

$x=\frac{\omega_{C}}{\omega_{R}},$ (36)

while for Stelle four-derivative theory in (34) we use analogously but now
with the theta couplings instead of omegas, namely

$x_{\rm 4-der}=\frac{\theta_{C}}{\theta_{R}}.$ (37)

It is worth to describe briefly here also the passage from UV divergences of
the theory at the one-loop level to the perturbative one-loop beta functions
of relevant dimensionless couplings. Using the divergent contribution to the
quantum effective action, derived previously, we can define the beta functions
of the theory. Let us first fix some definitions.

The renormalized Lagrangian $\mathcal{L}_{\rm ren}$ is obtained starting from
the classical Lagrangian written in terms of the renormalized coupling
constants and then adding the counterterms to subtract the divergences,

$\mathcal{L}_{\rm
ren}=\mathcal{L}(\alpha_{b}(t))\,=\,\mathcal{L}\left(Z_{\alpha_{i}(t)}\alpha_{i}(t)\right)\,=\,\mathcal{L}(\alpha_{i}(t))+\mathcal{L}_{\rm
ct}\\\
=\mathcal{L}(\alpha_{i}(t))+\left(Z_{C}-1\right)\theta_{C}(t)\,C^{2}+\left(Z_{R}-1\right)\theta_{R}(t)\,R^{2}\\\
+\left(Z_{\rm GB}-1\right)\theta_{\rm GB}(t)\,{\rm GB},$ (38)

where we have that $\mathcal{L}_{\rm ct}=-\mathcal{L}_{\rm div}$ and
$\alpha_{i}(t)=\\{\theta_{C}(t),\theta_{R}(t),\theta_{\rm GB}(t)\\}.$ Above we
denoted by $\alpha_{b}(t)$ the RG running bare values of coupling parameters,
by ${\cal L}_{\rm ct}$ and ${\cal L}_{\rm div}$ the counterterm and divergent
Lagrangians respectively, by $Z_{\alpha_{i}}(t)$ renormalization constants for
all dimensionless couplings and finally by $\alpha_{i}(t)$ these running
couplings. Here and above we neglect writing terms which are UV-divergent but
subleading in the number of derivatives in the UV regime. From (38), the full
counterterm action reads, already in dimensional regularization,

$\Gamma^{(1)}_{\rm ct}=-\Gamma^{(1)}_{\rm
div}=\frac{1}{2\epsilon}\frac{1}{(4\pi)^{2}}\\!\int\\!d^{4}x\sqrt{|g|}\bigg{\\{}\Big{(}\frac{397}{40}+\frac{2x}{9}\Big{)}C^{2}\\\
-\frac{7}{36}R^{2}+\frac{1387}{180}{\rm GB}\bigg{\\}}\\\
\equiv\frac{1}{2\epsilon}\frac{1}{(4\pi)^{2}}\\!\int\\!d^{4}x\sqrt{|g|}\left\\{\beta_{C}C^{2}+\beta_{R}R^{2}+\beta_{\rm
GB}{\rm GB}\right\\}\,.$ (39)

Comparing the last two formulas we can identify the beta functions and finally
get the renormalization group equations for the six derivative theory,

$\beta_{C}=\mu\frac{d\theta_{C}}{d\mu}=\frac{1}{(4\pi)^{2}}\left(\frac{397}{40}+\frac{2x}{9}\right),$
(40)
$\beta_{R}=\mu\frac{d\theta_{R}}{d\mu}=-\frac{1}{(4\pi)^{2}}\,\frac{7}{36}\,,$
(41) $\beta_{\rm GB}=\mu\frac{d\theta_{\rm
GB}}{d\mu}=\frac{1}{(4\pi)^{2}}\,\frac{1387}{180}\,,$ (42)

The three lines above constitute the main results of this work. Their
structure, mainly the $x$-dependence is the main topic of discussion in the
next sections. Above we denoted by $t$ the so called logarithmic RG time
parameter related in the following way: $t=\log\frac{\mu}{\mu_{0}}$ to the
renormalization scale $\mu$, where $\mu_{0}$ is some reference energy scale.

As we will show below the differences between the cases of four-derivative
theory and six-derivative one are significant and the dependence on the ratio
$x$ is with quite opposite pattern and in completely different sectors of
ultraviolet divergences of the two respective theories. In the main part of
this contribution we will try an attempt to explain the mentioned difference,
which is now clearly noticeable, using some general principles and arguments
about renormalizability of the quantum models. We will also study some
limiting cases of the non-finite (infinite or zero) values of the $x$
parameter and motivate that in such cases the QG model is non-renormalizable
and this leads to characteristic patterns in the structure of beta functions
mentioned above for six-derivative theories. This is also why we can call the
ratio $x$ as the fundamental parameter of the gravitational theory.

##  3 Some theoretical checks of the results (33)

Let us say that regardless of the simplicity of the final formulas with the
final result in [26], the intermediate calculations were quite big and this is
why we cannot present these intermediate steps here. This was not only because
of the size of the algebraic expressions, where we used Mathematica for help
with symbolic algebra manipulations, but also due to the complexity of all the
steps of the computation starting from the quadratic expansions of the action
of six-derivative classical theory. The ultimate validity of the calculations
has been checked in several different ways. This is also briefly described
below.

The following checks were performed to ensure the correctness of the
intermediate results of the computation of UV divergences, which was the main
task of the work presented here.

  1. 1.

First, the validity of the expression for the Hessian operator from the
classical action with six derivatives was verified in the following way. The
covariant divergence of the second variational derivative operator (Hessian)
with respect to gravitational fluctuations $h_{\mu\nu}$, from each GR-
covariant term $S_{{\rm grav},i}$ in the gravitational action must be
separately zero, namely

$\nabla_{\mu}\left(\frac{\delta^{2}S_{{\rm grav},i}}{\delta h_{\mu\nu}\delta
h_{\rho\sigma}}\right)=0+O\left(\nabla^{k}{\cal R}^{l},k+2l>4\right),$ (43)

where $S_{{\rm HD}}=\sum_{i}S_{{\rm grav},i}$. This formula was explicitly
checked for each term in the action in (10) to the order quadratic in
curvatures and up to total of four covariant derivatives acting on the general
gravitational curvature $\cal R$.

  2. 2.

The computation of the functional trace of the logarithm of the gauge
weighting operator $\hat{C}$, namely of ${\rm Tr}\ln\hat{C}$ was checked using
three methods. Since the $\hat{C}$ operator is a non-minimal four-derivative
differential operator and matrix-valued (so with vector indices), then the
computation of its trace of the logarithm is a bit troublesome. One has to be
more careful here. Therefore, we performed additional verifications of our
partial results for this trace. Our three methods consist basically of
transforming the problem to computing the same trace of logarithm but of new
operators (with higher number of derivatives). Next, by selecting some
adjustable parameters present in the construction of these new operators,
these morphed operators could be put into a minimal form and easily traced
(under the functional logarithm operation) using standard methods and
prescriptions of Barvinsky-Vilkovisky trace technology [27]. This construction
of new operators was achieved by an operatorial multiplication by some two-
derivative spin-one operator $\hat{Y}$ containing one free adjustable
parameter. For details one can look up the section III of [26].

In the first variant of the method, we multiplied $\hat{C}$ from the right by
$\hat{Y}$ one time, in the second method we multiplied by $\hat{Y}$ from the
left also once, and in the final third method we used the explicitly symmetric
form of multiplication $\hat{Y}\\!\hat{C}\hat{Y}$. (This last form of
multiplication is presumably very important for the manifest self-adjointness
property of the resulting 8-derivative differential operator
$\hat{Y}\\!\hat{C}\hat{Y}$.) For these operatorial multiplications, $\hat{Y}$
was a two-derivative operator, whose trace of the logarithm is known and can
be easily verified. (This was also checked independently below.) We emphasize
that in the first two methods the resulting operators ($\hat{Y}\\!\hat{C}$ and
$\hat{C}\hat{Y}$ respectively) were six-derivative ones, while in the last one
with double multiplication from both sides, $\hat{Y}\\!\hat{C}\hat{Y}$ was an
eight-derivative matrix-valued differential operator. At the end, all three
described above methods of computation of ${\rm Tr}\ln\hat{C}$ agree for terms
quadratic in curvatures. These terms are only important for us here since they
appear in the form of UV divergences of the theory (and are composed from GR-
invariants: $R^{2}$, $R_{\mu\nu}^{2}$, and $R_{\mu\nu\\!\rho\sigma}^{2}$).

  3. 3.

Similarly, the computation of ${\rm Tr}\ln\hat{Y}$ for the two-derivative
operator $\hat{Y}$ was verified using three analogous methods. We used
multiplication from both sides by the operator $\hat{A}$ and also the
symmetric form of multiplication $\hat{A}\hat{Y}\\!\\!\hat{A}$, where
$\hat{Y}$ is a two-derivative operator, whose functional trace of the
logarithm we searched here. Above, $\hat{A}$ was another two-derivative non-
minimal spin-one vector gauge (massless) operator, whose trace of the
logarithm is well known [41] and can be easily found. Again, for the final
results for ${\rm Tr}\ln\hat{Y}$ all three methods presented here agree to the
order of terms quadratic in curvatures $\cal R$.

  4. 4.

In total divergent part $\Gamma_{\rm div}$ of the quantum effective action, we
checked a complete cancellation of terms with poles in
$y=2\omega_{C}-3\omega_{R}$ variable, namely all terms with $\frac{1}{y}$ and
$\frac{1}{y^{2}}$ in denominators (originating from the expression for the
gauge-fixing parameter $\beta$ in (21)) completely cancel out. This is not a
trivial cancellation between the results of the following traces: ${\rm
Tr}\ln\hat{H}$ and ${\rm Tr}\ln\hat{M}$.

  5. 5.

Finally, using the same code written in Mathematica [47] a similar computation
in four-derivative gravitational theory (Stelle theory in four dimensions) was
repeated. We easily were able to reproduce all results about one-loop UV
divergences there [42]. To our satisfaction, we found a complete agreement for
all the coefficients and the same non-trivial dependence on the parameter
$x_{\rm 4-der}$, which was already defined above for Stelle gravity. This was
the final check.

##  4 Structure of beta functions in six-derivative quantum gravity

###  4.1 Limiting cases

In this subsection, we discuss various limiting cases of higher-derivative
gravitational theories (both with four and six derivatives). We study in
detail the situation when some of the coefficients of action terms in the Weyl
basis tend to zero. We comment whether in such cases our method of computation
is still valid and whether the final results for UV divergences are correct in
that cases and whether they could be obtained by continuous limit procedures.

First, we discuss the situation with a possible degeneracy of the kinetic
operator of the theory acting between quantum metric fluctuations $h_{\mu\nu}$
on the level of the quadratized action. If the action of a theory in the UV
regime has the following UV-leading terms

$S_{{\rm
grav}}=\\!\int\\!d^{d}x\sqrt{|g|}\left(\omega_{C,N}C\square^{N}C+\omega_{R,N}R\square^{N}R\right),$
(44)

with $\omega_{C,N}\neq 0$ and also $\omega_{R,N}\neq 0$ and after adding the
proper gauge-fixing functional, then the kinetic operator can be defined, so
in these circumstances it is not degenerate. Then it constitutes the
operatorial kernel of the part of the action which is quadratic in the
fluctuation fields. It can be well-defined not only for the cases when
$\omega_{C,N}\neq 0$ and $\omega_{R,N}\neq 0$, but also when $\omega_{R,N}=0$.
This last assertion one can check by explicit inspection, but due to the
length of the resulting expression we decided not to include such a bulky
formula here. However, in the case $\omega_{C,N}=0$, a special procedure must
be used to define the theory of perturbations and to extract UV divergences of
the model. We remark that in the last case the theory is non-renormalizable.
We also emphasize that the addition of the gauge-fixing functional here is
necessary since without it the kinetic operator (Hessian) is automatically
degenerate as the result of gauge invariance of the theory (here in
gravitational setup represented by the diffeomorphism gauge symmetry).

In general, as emphasized in [26], in four spacetime dimensions, the general
UV divergences depend only on the coefficients appearing in the following UV-
leading part of the gravitational HD action,

$S_{{\rm
grav}}=\\!\int\\!d^{4}x\sqrt{|g|}\left(\omega_{C,N}C\square^{N}C+\omega_{R,N}R\square^{N}R\right.\\\
+\omega_{C,N-1}C\square^{N-1}C+\omega_{R,N-1}R\square^{N-1}R\\\
\left.+\omega_{C,N-2}C\square^{N-2}C+\omega_{R,N-2}R\square^{N-2}R\right),$
(45)

where the last two lines contain subleading terms in the UV regime. However,
they are the most relevant for the divergences proportional to the Ricci
curvature scalar and also to the cosmological constant term [44]. Below for
notational convenience, we adopt the following convention specially suited for
six-derivative gravitational theories, so in the case when $N=1$. We will call
coupling coefficients in front of the leading terms as respective omega
coefficients (like $\omega_{C}=\omega_{C,1}$ and analogously
$\omega_{R}=\omega_{R,1}$), while the coefficients of the subleading terms
with four derivatives we will denote as theta coefficients (like
$\theta_{C}=\omega_{C,0}$ and analogously $\theta_{R}=\omega_{R,0}$).
Eventually, for the most subleading terms with subindex values of $(N-2)$
equal formally to $-1$ here, we have just one term contributing to the
cosmological constant type of UV divergence. We denote this coefficient as
$\omega_{-1}=\omega_{R,-1}$ and it is in front of the Ricci scalar term in the
original classical action of the theory (6). Simply this coefficient
$\omega_{-1}$ is related to the value of the 4-dimensional gravitational
Newton’s constant $G_{N}$.

The expressions for the RG running of the cosmological constant and the
Newton’s constant in [44, 26] contain various fractions of parameters of the
theory appearing in the action (45). Still, for a generic value of the integer
$N$, giving roughly the half of the order of higher derivatives in the model,
we have the following schematic structure of these fractions:

$\frac{\omega_{R,N-1}}{\omega_{R,N}}\,,\quad\frac{\omega_{C,N-1}}{\omega_{C,N}}\,,\quad\frac{\omega_{R,N-2}}{\omega_{R,N}}\,,\quad\frac{\omega_{C,N-2}}{\omega_{C,N}}\,.$
(46)

The structure of the UV divergences and of these fractions can be easily
understood from the energy dimensionality arguments. We notice that in the
Weyl basis with terms in (45) written with Weyl tensors
$C_{\mu\nu\\!\rho\sigma}$ and Ricci scalars $R$ the only fractions, which
appear in such subleading UV divergences are “diagonal” and do not mix terms
from the spin-2 (Weyl) sector with terms from the spin-0 (Ricci scalar)
sector. If in any of the above fractions, we take the limits:
$\omega_{R,N-1}\to 0$, $\omega_{C,N-1}\to 0$, $\omega_{R,N-2}\to 0$, or
$\omega_{C,N-2}\to 0$, then the corresponding fractions and also related UV
divergences (and resulting beta functions in question) simply vanish, provided
that the coefficients in their denominators $\omega_{R,N}$ and $\omega_{C,N}$
are non-zero.

On the other hand, if $\omega_{C,N}=0$, then we cannot rely on this limiting
procedure. In this case, on the level of the quadratized action the operator
between quantum fluctuations is degenerate even after adding the gauge-fixing
terms. This means that in this situation a special procedure has to be used to
extract the UV divergences of the model. This is possible, but we will not
discuss it here.

It is worth to notice that in turn, if $\omega_{R,N}=0$, then the kinetic
operator for small fluctuations and after adding the gauge fixing is still
well-defined, as emphasized also above. For this case a special additional
kind of gauge fixing has to be used, which fixes the value also of the trace
of the metric fluctuations $h=g^{\mu\nu}h_{\mu\nu}$ and this last one in the
theory with $N=0$ resembles the conformal gauge fixing of the trace.

If $\omega_{R,N}=0$ and additionally $\omega_{R,N-1}$ or $\omega_{R,N-2}$ are
non-zero, then the corresponding beta functions for the cosmological constant
and Newton’s gravitational constant are indeed infinite and ill-defined as
viewed naively from the expressions in (46). This situation could be
understood as that there is an additional new divergence not absorbed in the
adopted renormalization scheme and the renormalizability of such a theory is
likely lost. But if the model is with $\omega_{R,N}=0$ and at the same time
$\omega_{R,N-1}=\omega_{R,N-2}=0$, then the contributions of corresponding
fractions in (46) are vanishing, because the limits $\omega_{R,N-1}\to 0$ or
$\omega_{R,N-2}\to 0$ must be taken as the first respectively. Only after
this, the final limiting procedure $\omega_{R,N}\to 0$ should be performed.
Therefore, in this limiting situation, the proper sequence of limits on
respective fractions is as follows:

$\lim_{\omega_{R,N}\to 0}\left(\lim_{\omega_{R,N-1}\to
0}\frac{\omega_{R,N-1}}{\omega_{R,N}}\right)=0$ (47)

and

$\lim_{\omega_{R,N}\to 0}\left(\lim_{\omega_{R,N-2}\to
0}\frac{\omega_{R,N-2}}{\omega_{R,N}}\right)=0\,.$ (48)

In this case, there are no contributions to the beta functions from these
fractions, so the $R^{2}$ sector does not contribute anything to the mentioned
UV divergences, while it is expected that the terms in the $C^{2}$ sector make
some impact on beta functions.

However, the similar procedure cannot be applied in the sector with Weyl
square terms ($C^{2}$ sector), so to the model with $\omega_{C,N}=0$ and at
the same time $\omega_{C,N-1}=\omega_{C,N-2}=0$ since these cases have to be
treated specially and separately. In the last case, after the limit, only the
pure sector with Ricci scalar square terms ($R^{2}$ sector) survives and the
theory is likely non-renormalizable. Then we expect contributions to UV
divergences only from terms in the $R^{2}$ sector.

Regarding the divergences proportional to expressions quadratic in curvatures
($R^{2}$, $C^{2}$, and the Gauss-Bonnet term ${\rm GB}$), we have found the
following generic structure in four-derivative gravity [42]:

$\frac{A_{-2}}{x^{2}_{\rm 4-der}}+\frac{A_{-1}}{x_{\rm 4-der}}+A_{0},$ (49)

where in this case of four-derivative gravity the fundamental ratio of the
theory is defined as

$x_{\rm
4-der}=\frac{\omega_{C,0}}{\omega_{R,0}}=\frac{\theta_{C}}{\theta_{R}}.$ (50)

The numerical coefficients $A_{-2}$, $A_{-1}$ and $A_{-0}$ are different for
different types of UV divergences (here they are given by terms with four
derivatives, namely by $R^{2}$, $C^{2}$ and $\rm GB$ terms respectively). The
explicit numerical values are given in the formula (34). One observes negative
powers of the ratio $x_{\rm 4-der}$ in (49) and in (34), implying also
negative powers of the coupling $\theta_{C}$ in the final results for these UV
divergences. This result signifies that the theory with $\theta_{C}=0$ should
be treated separately and then we do not have well-defined kinetic operator in
a standard scheme of computation. The naive results with the limit
$\theta_{C}\to 0$ of the above formula in (49) do not exist. Such theories
with $\theta_{C}=0$ entail complete absence of gravitational terms in the
$C^{2}$ sector. They are again very special and perturbatively non-
renormalizable models. The above remarks apply both to pure $R^{2}$
Starobinsky theory as well as to theories in the $R^{2}$ sector with addition
of the Einstein-Hilbert $R$ or the cosmological constant $\omega_{\Lambda}$
terms.

On the other side, the limit $\theta_{R}\to 0$ in pure $C^{2}$ gravity seems
not to produce any problem with the degeneracy of the kinetic operator, nor
with the final expression (49). The naive answer would be just $A_{0}$ for
(49) for each of the UV divergences in this case. But this is an _incorrect_
answer since for pure four-derivative gravity with $\theta_{R}=0$ in the Weyl
basis of terms, we have an enhancement of the symmetry in the model, beyond
the case where $\theta_{R}$ was non-zero. In this situation, the theory enjoys
also conformal symmetry and a more specialized and delicate computation must
be performed to cover this case. This is the case of four-dimensional
conformal (Weyl) gravity. (We decided for simplicity not to analyze here the
cases when besides the $C^{2}$ action for four-dimensional conformal gravity,
there are also some subleading terms from the almost “pure” $R^{2}$ sector,
that is $\omega_{-1}\neq 0$ or when we allow for non-vanishing cosmological
constant term $\omega_{\Lambda}\neq 0$ – these terms in the action would cause
breaking of classical conformality.)

The computation in this case should reflect the fact that also the conformal
symmetry should be gauge-fixed. We remark that the conformal symmetry does not
require dynamical FP ghosts, because the conformal transformations of
gravitational gauge potentials (not the conformal Weyl gauge potentials
$b_{\mu}$) are without derivatives. At the end, when the more sophisticated
method is employed, the eventual result is different than $A_{0}$ for each
type out of three types of four-derivative UV-divergent GR-invariant terms in
the quantum effective action of the model. The strict result $A_{0}$ is still
correct only for theories in which conformality is violated by inclusion of
other non-conformal terms like the Einstein-Hilbert $R$ term or the
cosmological constant $\omega_{\Lambda}$ term. We conclude that in the four-
derivative theory, the two possible extreme cases of $\theta_{R}=0$ or
$\theta_{C}=0$ are not covered by the general formula (49). But in each of
these cases the reasons for this omission are different. In both these cases
the separate more adapted methods of computation of UV divergences have to be
used.

In the case of six-derivative theory studied in [26], we have the following
structure of UV divergences quadratic in gravitational curvatures

$B_{0}+B_{1}x,$ (51)

with new values for the constants $B_{0}$ and $B_{1}$. The explicit numerical
values are given in our formula (33) with the results. We also remark that the
values of the constant terms $B_{0}$ are different than the values of $A_{0}$
in the previous four-derivative gravity case. Moreover, the numerical
coefficients $B_{0}$ and $B_{1}$ are different for different types of UV
divergences of the effective action ($R^{2}$, $C^{2}$ and $\rm GB$ terms
respectively). When the leading dynamics in the UV regime is governed by the
theory with six derivatives, then the fundamental ratio $x$ we define as

$x=\frac{\omega_{C,1}}{\omega_{R,1}}=\frac{\omega_{C}}{\omega_{R}}.$ (52)

We emphasize that in such a case, we cannot continuously take the limit
$\omega_{C}\to 0$. Although, naively this would mean the limit $x\to 0$, the
result just $B_{0}$ from (51) would be incorrect. This is because in this case
we cannot trust the method of the computation. When $\omega_{C}=0$ the kinetic
operator is degenerate (the same as it was in the four-derivative gravity
case) and needs non-standard treatment, that we will not discuss here.

Moreover, looking at the last formula (51), the other limit $\omega_{R}\to 0$
is clearly impossible too, because it gives divergent results. However, in
this case ($\omega_{R}=0$) and on the contrary to the previous case
($\omega_{C}=0$), we could trust the computation at least on the level of the
kinetic operator (Hessian) and its subsequent computation of the functional
trace of the logarithm of. In this case, the final divergent results in (51)
signify that the theory likely is non-renormalizable and that there are new UV
divergences besides those ones derived from naive power counting analysis333We
remark that the generic power counting analysis of UV divergences in six-
derivative quantum gravity, as presented in section 1.1, applies only in cases
when $\omega_{C}\neq 0$ and $\omega_{R}\neq 0$.. We conclude that in the case
of six-derivative gravity, both cases $\omega_{R}=0$ or $\omega_{C}=0$ require
special treatment and the type of formula like in (51) or (33) does not apply
there and the limiting cases are not continuous. More discussion of these
limits is contained also in the further subsection 4.4.

###  4.2 Dependence of the final results on the fundamental ratio $x$

Here we just want to understand the $x$-dependence in the result for the beta
functions in six-derivative gravitational theory. We first try to analyze the
situation for simpler theory (with four derivatives), prepare the ground for
the theory with six derivatives, and then eventually draw some comparison
between the two. We look for singular $\frac{1}{\omega_{R}}$ or
$\frac{1}{\omega_{C}}$ dependence (corresponding to positive or negative
powers of the fundamental ratio $x=\frac{\omega_{C}}{\omega_{R}}$
respectively) in functional traces of the fundamental operators defining the
dynamics of quantum perturbations important to the one-loop perturbative
level. We note that the two definitions for the ratio $x$ in (50) and in (52)
respectively for four- and six-derivative gravities are compatible with each
other and the proper use of them (with theta or omega couplings) is obvious in
the specific contexts they are used in. Below, when we will refer to features
shared by both four- and six-derivative gravitational theories, we will use
common notation with general $\omega_{C}$, $\omega_{R}$ and $x$ coefficients
and we will not distinguish and not change it to the special notation
originally adequate only to Stelle quadratic theory (with $\theta_{C}$,
$\theta_{R}$ and $x_{\rm 4-der}$). We hope that this will not lead to any
confusion.

We emphasize, that when we have one of the two terms missing – with
$\omega_{R}$ or $\omega_{C}$ front couplings – in the leading in UV part of
the action of the model, then the theory is badly non-renormalizable and
degenerate. For example, one cannot define even at the tree-level the flat
spacetime graviton propagator since the parts proportional to $P^{(0)}$ or
$P^{(2)}$ projectors do not exist in cases when $\omega_{R}=0$ or
$\omega_{C}=0$ respectively. However, there we can still use the Barvinsky-
Vilkovisky (BV) trace technology to compute the new UV divergences. The fact
that they are not possible to be absorbed in counterterms of the original
theory is another story related to the non-renormalizability of the model that
we will not discuss further here. We think that, for example, using the BV
technique one can fast compute UV divergences in Einstein-Hilbert (E-H) theory
in $d=4$ (which is a non-renormalizable model) and this method still gives a
definite result (besides that these divergences are gauge-fixing dependent and
valid only for one gauge choice). Moreover, using the BV traces machinery and
the minimal form of the kinetic operator is essential to get final results for
the unique effective action (as introduced by Barvinsky [27, 48]), also in
perturbatively non-renormalizable models.

In quadratic gravity (four-derivative theory) in $d=4$, setting $\theta_{C}=0$
is highly problematic. The same regards taking the limit $\theta_{C}\to 0$,
because then the pure $R^{2}$ theory can be fully gauge-fixed. And for
example, this means that on flat spacetime background, the kinetic operator
vanishes, perturbative modes are not dynamical and there is no graviton
propagator. Using the standard technique of the one-loop effective action one
sees that the traces of the functional logarithms of $\hat{H}$ and of
$\hat{C}$ operators both contain singular expressions $\frac{1}{\theta_{C}}$,
and there is no final cancellation between them. In this case of four-
derivative gravity, in final results for UV divergences, we really see inverse
powers of the fundamental ratio of the theory $x_{\rm 4-der}$.

The results in quadratic Stelle gravity, when we set $\theta_{R}=0$, are not
continuous either. Because in this case the local gauge symmetry of the theory
is enhanced. We have also conformal symmetry there. The model is identical to
the Weyl gravity in $d=4$ described by the action $C^{2}$. As emphasized in
[41], this case of $\theta_{R}=0$ has to be treated specially. Also, in this
model, the conformal symmetry has to be gauge-fixed and in this special case
the operators $\hat{H}$ and $\hat{C}$ are different than their limiting
versions under $\theta_{R}\to 0$ limit from the generic four-derivative theory
case. Hence also the results for the beta functions are different than the
limits of the corresponding beta functions in the situation with
$\theta_{R}\neq 0$.

If we start with the theory with $\theta_{C}=0$ from the beginning, then there
are serious problems with the kinetic operator. We checked that it cannot be
put by standard fixing of the gauge to the minimal form with four-derivative
leading operator. Moreover, as the result of this process one of the typical
gauge-fixing parameters remains undetermined. Here one can try to compute the
trace of the logarithm of the Hessian using the method proposed in [44]
consisting of multiplying by some two-derivative non-minimal operator and
getting a six-derivative operator, whose trace can be easily found. But it is
hard to believe that one has any chance to get a non-singular answer for all
the beta functions in pure $R^{2}$ theory since it is known that this theory
is non-renormalizable (because it lacks the $C^{2}$ counterterm in the bare
action).

Actually, here (for the $\theta_{C}=0$ case) one could choose the $\hat{C}$
matrix-valued differential operator different from the standard minimal
prescription and choose different values for the $\gamma$ gauge-fixing
parameter. In the standard minimal choice for the gauge-fixing parameters and
in this model, the $\hat{C}$ matrix contains an irregular part in $\theta_{C}$
coupling ($\frac{1}{\theta_{C}}$ pole), because of the dependence of $\gamma$
on $\theta_{C}$. This last dependence originates from the conditions forced on
gauge-fixing parameters in order to put the kinetic operator in the minimal
form, in the standard case $\theta_{C}\neq 0$. However, knowing that in the
case with $\theta_{C}=0$, this procedure is anyhow unsuccessful, we have the
freedom to choose the value of $\gamma$ different than the standard one and at
our wish.

In principle, similar considerations can be repeated verbatim for the case of
six-derivative theory (with $N=1$ power exponent on the box operator in the
defining the theory action in (45)). But we remark here that the theory with
$\omega_{R}=0$ and $N=1$ is _not_ conformally invariant in $d=4$ dimensions.
And the above problems with the gauge fixing of the Hessian operator $\hat{H}$
and non-minimality of it in the case $\omega_{C}=0$ still persist. This is
because here for six-derivative gravitational theories the box operator
$\square$ acting between two gravitational curvatures is only a spectator from
the point of view of the UV-leading part of the $\hat{H}$ operator (with the
highest number of derivatives and with the zeroth powers in gravitational
curvatures) or from the point of view of flat spacetime kinetic operator and
flat spacetime graviton’s propagator. The box operator in momentum space gives
only one additional factor of $-k^{2}$ to the kinetic operator and additional
suppression by $-k^{-2}$ to the propagator. The Hessian in the six-derivative
theory with $\omega_{R}=0$ must possess the same definitional issues as the
one in the four-derivative theory (with $\theta_{R}=0$), because for the
kinetic terms box operator again plays only the role of the spectator. Hence
the difference on this level between four- and six-derivative theories is only
in some overall multiplicative coefficient (like flat spacetime d’Alembertian
operator $\partial^{2}$ is $-k^{2}$ in Fourier space). So then, if we know
that the Hessian $\hat{H}$ is almost well-defined for the conformal gravity
case (up to the need for additional gauge fixing of the conformal symmetry),
then the same will be true for the Hessian in the six-derivative theory with
the $\omega_{R}=0$ condition in $d=4$ spacetime dimensions, although then the
theory ceases to be conformal anymore. In conformal gravity in $d=4$, when
$\theta_{R}=0$, we have almost well-defined Hessian, because we know that it
gives rise to a good renormalizable theory at least to the one-loop
perturbative level of computations.

Now, also in the case of six-derivative theories, setting $\omega_{R}=0$ does
not create any problem for the form of neither $\hat{H}$ nor $\hat{C}$
operators. Only the final results for the beta functions show
$\frac{1}{\omega_{R}}$ poles as this was manifest from the results in [26]. In
turn, in six-derivative theories, the limit $\omega_{C}\to 0$ seems regular,
but it is questionable that now we can trust the results of this limit. In the
pure $R\square R$ theory, we expect to get some discontinuous results for the
beta functions not obtainable by the limit $\omega_{C}\to 0$ since this model
is non-renormalizable. In this model, there is still an open problem that one
cannot make the kinetic operator of fluctuations a minimal 6-derivative one.
Furthermore, taking the limit $\omega_{C}\to 0$ on the kinetic operator from
the generic case $\omega_{C}\neq 0$ produces a Hessian $\hat{H}$ that vanishes
on flat spacetime. Hence it seems that in this case the intermediate steps of
the process of computing the divergent part of the effective action are not
well-defined, while the final result is amenable to taking the limit
$\omega_{C}\to 0$, but exactly because of this former reason, we should not
trust these apparently continuously looking limits.

One should analyze deeper the form of the leading in the number of derivatives
(and also in the UV regime) part of the kinetic operator $\hat{H}$ of the
theory between graviton fluctuations. The insertions of box operators, like
any power or functions of the box operator $\square$, are only the immaterial
differences between the cases of four- and six-derivative theories here. These
operators are only spectators for getting the leading part of the Hessian,
which is with the highest number of derivatives and also considered on flat
spacetime, so with the condition that ${\cal R}=0$. Using formula (20) with
solutions for gauge-fixing parameters as in (21), one finds in the generic
case $\omega_{C}\neq 0$ and $\omega_{R}\neq 0$, that the kinetic operator
(leading part of the Hessian) is indeed minimal and of the form

$H_{{\rm
lead}}^{\mu\nu,\rho\sigma}=\frac{\omega_{C}}{2}\square\left(g^{\mu\rho}g^{\nu\sigma}+g^{\mu\sigma}g^{\nu\rho}\right)\\\
-\omega_{C}\frac{\omega_{C}-6\omega_{R}}{4\omega_{C}-6\omega_{R}}\square
g^{\mu\nu}g^{\rho\sigma}\,.$ (53)

In the above formula, one does not see any singularity when $\omega_{C}$ is
vanishing (one saw $\omega_{C}^{-1}$ divergences in the expressions for
$\alpha$ and $\gamma$ parameters in (21)), but in this case the above
treatment was not justified. When $\omega_{C}=0$, one can solve the system for
gauge-fixing parameters for $\beta$ and
$\gamma^{\prime}=\frac{\gamma}{\alpha}$ and assume that formally
$\frac{1}{\alpha}=0$ and $\frac{1}{\gamma}=0$, but in the ratio
$\frac{\gamma}{\alpha}$ the limit is finite. One then finds that $\beta=1$ and
$\gamma^{\prime}=-2\omega_{R}$ and after substitution to the original Hessian,
one gets that its leading part explicitly vanishes. The same one gets by
plugging the naive limit $\omega_{C}\to 0$ in (53). One also sees from the
explicit solutions in (21) and resulting general expression for
$\gamma^{\prime}$ (i.e. $\gamma^{\prime}=\frac{4}{3}\omega_{C}-2\omega_{R}$)
that by plugging $\omega_{C}=0$ one finds again that $\beta=1$ and
$\gamma^{\prime}=-2\omega_{R}$ as derived exactly above. The highest
derivative level of the gravitational action is then completely gauge-fixed.

In the opposite case, when $\omega_{R}=0$, the leading part of the Hessian
does not vanish, but it is degenerate and in the form

$H_{{\rm
lead}}^{\mu\nu,\rho\sigma}=\frac{\omega_{C}}{2}\square\left(g^{\mu\rho}g^{\nu\sigma}+g^{\mu\sigma}g^{\nu\rho}\right)-\frac{\omega_{C}}{4}\square
g^{\mu\nu}g^{\rho\sigma},$ (54)

because this operator does not possess a well-defined inverse, precisely in
$d=4$ dimensions. An addition of a new conformal-like type of gauge-fixing
here $\tau h\square^{3}h$ with a new (fourth) gauge-fixing parameter $\tau$
and where the trace of metric fluctuations $h=h^{\mu}{}_{\mu}$ is used,
removes the degeneracy provided that $\tau\neq 0$ is selected. Then the
kinetic operator takes the form

$H_{{\rm
lead}}^{\mu\nu,\rho\sigma}=\frac{\omega_{C}}{2}\square\left(g^{\mu\rho}g^{\nu\sigma}+g^{\mu\sigma}g^{\nu\rho}\right)\\\
+\left(\tau-\frac{\omega_{C}}{4}\right)\square g^{\mu\nu}g^{\rho\sigma}.$ (55)

Moreover, for any non-zero value of $\tau$ the Hessian is still a minimal
operator. For $\tau\neq 0$ the inverse exists and also the propagator can be
defined around flat spacetime. The only question is whether the final results
are $\tau$-independent since this is a spurious gauge-fixing parameter. The
reason for such independence is obvious in the four-derivative case, since
$\tau$ is a gauge-fixing parameter for conformal symmetry (conformal gauge-
fixing parameter, so this is then in such circumstances a symmetry argument).
But in the case of six-derivative model in $d=4$, the reasoning with conformal
symmetry is not adequate since this model is not conformal anymore. Only the
explicit computation may show that $\tau$ parameter drops out from final
results as it should for them to be physical and $\tau$ gauge choice
independent.

In four-derivative gravitational theory, one can see the dependence on the
$x_{\rm 4-der}$ ratio only in the coefficient of the $R^{2}$ counterterm. This
dependence is with the general schematic form $A_{-2}x_{\rm
4-der}^{-2}+A_{-1}x_{\rm 4-der}^{-1}+A_{0}x_{\rm 4-der}^{0}$ like in (49) and
in (34). We remark that for other counterterms (namely for $C^{2}$ and ${\rm
GB}$ in this Weyl basis), the coefficients of UV divergences are numbers
completely independent of $x_{\rm 4-der}$. One could try to explain here this
quadratic dependence in the inverse ratio $x_{\rm 4-der}^{-1}$ in front of the
$R^{2}$ counterterm in a spirit similar to the argumentation presented in
[44], where we counted active degrees of freedom contributing to the
corresponding beta functions of the theory. It is well known by the examples
of beta functions in QED coupled to some charged matter and in Yang-Mills
theory, that the beta function at the one-loop level expresses weighted
counting of degrees of freedom and their charges in interactions with gauge
bosons in question (minimal couplings in three-leg vertices are enough to be
considered here due the gauge symmetry). The similar counting could be
attempted here, but in gravity, especially in HD gravity, there is a plenty of
other gravitational degrees of freedom, so this is quite a difficult task to
enumerate all of them and their strength of interactions in cubic vertices
when they interact with background gravitational potentials. Therefore, this
task of explaining $x$-dependence and numbers present in the expressions for
all the beta functions both in four- and six-derivative theories, now seems to
be too ambitious and we leave it for some further future considerations.

Instead, we comment briefly on the general dependence on the $x_{\rm 4-der}$
ratio in four-derivative theory and compare this with six-derivative theory.
In the case of $N=1$ (six-derivative gravitational theory), it was found as a
main result in [26], that the dependence on $x$ is only in front of the
$C^{2}$ counterterm and this is a linear dependence $B_{1}x^{1}+B_{0}x^{0}$
like in (51) with non-negative powers of the $x$ ratio. The other counterterms
$R^{2}$ and ${\rm GB}$ are with constant coefficients (only $B_{0}$ terms
present) (cf. with (33)). If the other than the Weyl basis is used to write
counterterms, then the $x$-dependence is linear in coefficients in front of
each basis term (like in the basis with $R^{2}$, $R^{2}_{\mu\nu}$, and
$R^{2}_{\mu\nu\rho\sigma}$ terms). These explicit dissimilarities between
$N=0$ and $N=1$ models certainly require deeper investigations.

It is interesting also to analyze a special value of the fundamental ratio $x$
of the six-derivative gravitational theory, which makes the $C^{2}$ sector of
UV divergences completely finite. This value is exactly
$x=-\frac{3573}{80}=-44.6625$. The $R^{2}$ sector of UV divergences cannot be
made finite this way. We remind for comparison that in the case of quadratic
gravity with four derivatives in $d=4$, the special values for $x_{\rm
4-der}$, which made _contrary_ the $R^{2}$ sector UV-finite, were two and they
were $x_{\rm 4-der}=3(3\pm\sqrt{7})$ (their numerical approximations are as
follows: $x_{\rm 4-der,}{}_{-}\approx 1.0627$ and $x_{\rm 4-der,}{}_{+}\approx
16.937$) as solutions of some non-degenerate quadratic algebraic equation.
Again, contrary to the previous case with six derivatives, here the
divergences in the $C^{2}$ sector cannot be made vanish.

Now, we discuss the differences between the two extreme cases $\omega_{C}=0$
and $\omega_{R}=0$. In six-derivative model or when we have even more
derivatives, superficially these two couplings and their roles for the