Acta Polytechnica
https://doi.org/10.14311/AP.2022.62.0118
Acta Polytechnica 62(1):118–156, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence
Published by the Czech Technical University in Prague
HOW TO UNDERSTAND THE STRUCTURE OF BETA FUNCTIONS
IN SIX-DERIVATIVE QUANTUM GRAVITY?
Lesław Rachwał
Universidade Federal de Juiz de Fora, Departamento de Física–Instituto de Ciências Exatas,
33036-900, Juiz de Fora, MG, Brazil
correspondence: grzerach@gmail.com
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 → +∞ 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 gen-
eral 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 de-
scribed by the symmetric tensor field gµν being the
metric tensor of the pseudo-Riemannian smooth differ-
ential 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 densi-
tized Lagrangian
√
|g|R we call as Einstein-Hilbert
action. The QG models based on it were originally
studied in [1–3]. 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 struc-
ture 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 modi-
fications 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 lev-
els with the benefits of getting new and deeper in-
sights in the theory of relativistic gravitational field.
Our framework on the classical level can be summa-
rized 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 relativis-
tic 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 ten-
sor 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 prob-
lems. This happens already on the classical level when
they supposed to describe the modified dynamics of
the gravitational field (metric field gµν (x) living on
the spacetime manifold). These successes and prob-
lems get amplified even more on the quantum level.
The pros for HD theories give strong motivations
why to consider seriously these modifications of Ein-
stein’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.
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vol. 62 no. 1/2022 Beta Functions in Six-Derivative Quantum Gravity
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 expla-
nations 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 respec-
tively. 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 classi-
cal 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 covari-
ance. 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 gµν and contravariant gµν ), Riemann
curvature tensor Rµνρσ and also of covariant deriva-
tives ∇µ acting on these Riemann tensors1. Initially
this may look as presumably unnecessary complica-
tion 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 com-
ponents of the metric tensor field in Einstein’s gravity.
However, on the cosmological and galactic scales some
gravitational models with higher derivatives give suc-
cesses 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 gravi-
ties. First, still on the classical level, within the class
of higher-derivative gravitational theories, there are
models that are the first, which besides relativistic
1We do not need to consider covariant derivatives on the
metric tensor because of the metricity condition, ∇µgνρ = 0.
symmetries, enjoy also invariance under conformal
symmetry understood in the GR framework. Prop-
erly this is called as Weyl symmetry of the rescaling
of the covariant metric tensor, according to the law:
gµν → Ω2gµν with Ω = Ω(x) being an arbitrary scalar
parameter of these transformations. To understand
better this fact, one may first recall that the metric
tensor gµν 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 gravita-
tional theory the classical action must be precisely
with d derivatives on the metric. (One sees due to the
requirement of general covariance that this considera-
tion of conformal gravitational theories makes sense
only in even dimensions d of spacetime.) Another in-
teresting 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
□R is trivially a total derivative term, so cannot be
used.) Therefore for d-dimensional conformal gravi-
tational theories (d > 2) we inevitably must consider
HD metric theories. The conformally invariant gravi-
tational 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 gravita-
tional theories in dimensions d > 2 come instead from
quantum considerations. After all, it is not so surpris-
ing 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 con-
sistent 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 en-
ergy 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 con-
straining the possibilities for quantum gravitational
theories. It is more constraining than it was origi-
nally thought of. Moreover, not all macroscopic, so
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Lesław Rachwał Acta Polytechnica
long wavelength limit, classical theories are with these
quantum correspondence features, only those which
emerge as classical limits of consistent quantum grav-
ity 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 consider-
ations of gravitationally “charged” matter fields and
their mutual interactions consistent with quantum me-
chanics. In this way we could make gravity dynamical
and quantum with a proper form of graviton’s propa-
gation. 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 in-
troduced 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 gravita-
tional fields with HD form of the graviton’s propa-
gators 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 the-
ory to some energetic quantum matter fields moving
under the influence of classical initially non-dynamical
gravitational background field. (The background grav-
itational 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 expecta-
tion 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 gen-
erated. This is so even if the original matter theory
is with two-derivative actions (like for example stan-
dard 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 back-
grounds. In such a way we generate the dynamics
to the gravitational field due to quantum matter in-
teractions 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
Sdiv =
∫
d4x
√
|g|
(
αC C
2 + αRR2
)
, (1)
so we see that counterterms of the GR-covariant form
of C 2 and R2 are being generated. (In the equation
above, the R2 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 = Cµνρσ C µνρσ . Collectively, we will denote
these curvatures as R2, so R2 = R2, C2.) 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 stan-
dard two-derivative Einstein-Hilbert action, but still
the above results persist. One notices that in these
two counterterms C 2 and R2 one has four derivatives
acting on a metric tensor, so these are theories of
a general higher-derivative type, differently from orig-
inally intended E-H gravitational theory whose action
is just based on the Ricci scalar R. These C 2 and R2
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 considera-
tions higher derivatives are inevitable.
We also remark here that in the special case, where
the matter theory is classically conformally invari-
ant 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 covari-
ant counterterm C 2 is generated, while the coefficient
αR = 0 in (1). This is due to the fact that the quanti-
zation 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 R2 counterterm is not needed but
still the action of a quantum consistent coupled con-
formal 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
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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 deriva-
tives in the gravitational action was first considered by
Stelle in [7] and some exact classical solutions of such
a theory were analyzed in [8–10]. In d = 4 spacetime
dimensions, the minimal number of derivatives is ex-
actly 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 dy-
namics (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 pos-
sible 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) per-
turbatively 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 the-
ories – 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 real-
ized. 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 [15–18]. 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 diver-
gences 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 quan-
tum level induce the higher-derivative terms in the
gravitational actions, although these further argu-
ments are all related to the original one from DeWitt
and Utiyama. One can, for example, consider inte-
grating 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 exam-
ple popularized by ’t Hooft [19–21], 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 mass-
less fields are integrated out in the path integral.
In this way we can discover the quantum consis-
tent dynamics of the gravitational field even if we did
not know that such quantum fields mediating gravi-
tational interactions between particles existed in the
first place. The graviton becomes a propagating parti-
cle 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 addi-
tional 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 renor-
malizable. 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 di-
mensional 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 spe-
cial 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 sym-
metry 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 ac-
tion, 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
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Lesław Rachwał Acta Polytechnica
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 simi-
larity 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 gravita-
tional 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 ampli-
fied 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 Dµ (covari-
ant both with respect to Yang-Mills internal group
G and with respect to gravitational field). For such
an object, one can define the curvature Fµν that is
decomposed in its respective sectors into the gauge
field strength Fµν and the Riemann gravitational ten-
sor Rµνρσ . But the most natural thing to do here is
to consider symmetric action constructed with such
a total curvature of the derivative Dµ and then the
generalized 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 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 the-
ory, 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 grav-
itational fields entirely characterized by the metric
tensor of gravitational spacetime. Secondly, we spec-
ify the set of symmetries (invariance group) of our
theory. Again, in our setup these are, in general, in-
variances under general coordinate transformations
also known as diffeomorphism symmetries of gravita-
tional theories. In this sense, we also restrict the set
of possible theories from general models considered
in the gauge treatment of gravity, when the transla-
tion group or full Poincaré groups are gauged. Then
finally following Landau we define the theory by spec-
ifying 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 (con-
tracted in various combinations). Specifying now, to
the case motivated above, we shall use and study be-
low 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
SHD, being the functional of the classical metric field
SHD[gµν ], the following object
Z =
∫
Dgµν exp(iSHD), (2)
where in the functional integral above we must be
more careful than just on the formal level in defining
properly the integration measure Dgµν . 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 clas-
sify in four dimensions all gravitational configurations
(all gravitational pseudo-Riemannian manifolds) over
which we should integrate above. The functional in-
tegral, 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 symme-
tries 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 gµν ) to the classical
external current J . And also for other theoretical rea-
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sons, 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, gµν = ḡµν + hµν , where the background
classical metric is denoted by ḡµν and metric pertur-
bations by hµν . By computing variational derivatives
of the partition function Z[J ] with respect to the clas-
sical current J one gets higher n-point functions with
the accuracy of the full quantum level. One can com-
pute 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 quan-
tum 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 k6 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 fluctu-
ations 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 R2 as in [7]) leads to the following
equality
∆ + d∂ = 4, (3)
where ∆ is superficial degree of divergence of any
Feynman graph G, d∂ 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 ⩾ 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)2. 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
∆ + d∂ = 6 − 2L. (4)
The above formula can be also rewritten as a useful
inequality (bound on the superficial degree ∆):
∆ ⩽ 6 − 2L = 4 − 2(L − 1), (5)
since d∂ ⩾ 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 ∆ < 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 pro-
portional to the cosmological constant Λ 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 Λ
2We remind for completeness that the Gauss-Bonnet scalar
term GB = E4 = R2µνρσ − 4R2µν + R2 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.
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Lesław Rachwał Acta Polytechnica
(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∂ = 4 and ∆ = 0. The last infor-
mation signifies that they are universal logarithmic
divergences. Their names originate from the fact that
they arise when the ultraviolet cutoff ΛUV 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 the-
ory 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 func-
tions 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 Λ
or R) this is not true. The theory is four-loop finite,
while the beta functions of R2, C 2 and GB terms
are one-loop exact. All these miracles are only pos-
sible to happen in very special super-renormalizable
model since we have six derivatives in the gravita-
tional propagator around flat spacetime. This number
is bigger than the minimal for a renormalizable and
scale-invariant QG theory in d = 4 spacetime dimen-
sions 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 de-
cide to study the quantum theory described by the
following classical Lagrangian,
L = ωC Cµνρσ□C µνρσ + ωRR□R
+ θC C 2 + θRR2 + θGBGB + ωκR + ωΛ. (6)
From this Lagrangian we construct the action of our
HD quantum gravitational model, here with six deriva-
tives as the leading number of derivatives in the UV
regime, by the formula
SHD =
∫
d4x
√
|g|L. (7)
Above by Cµνρσ we denote the Weyl tensor (con-
structed from the Riemann Rµνρσ , Ricci tensor Rµν
and Ricci scalar R and with coefficients suitable for
d = 4 case). Moreover, by 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 curva-
tures,
GB = E4 = R2µνρσ − 4R
2
µν + R
2. (8)
Similarly, we can write for the “square” of the Weyl
tensor in d = 4
C 2 = C 2µνρσ = Cµνρσ C
µνρσ = R2µνρσ − 2R
2
µν +
1
3
R2.
(9)
Finally, to denote the box operator we use the symbol
□ with the definition □ = gµν ∇µ∇ν , which is a GR-
covariant analogue of the d’Alembertian operator ∂2
known from the flat spacetime.
It is important to emphasize here that the La-
grangian (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 □
(which is constructed using the GR-covariant deriva-
tive ∇µ). These two terms exhaust all other pos-
sibilities 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 cyclic-
ity 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 prop-
agators. 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 general-
ized 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
√
|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 ωκ
and ωΛ respectively. The first one is related to the
Newton gravitational constant GN , while the last one
ωΛ to the value of the physical cosmological constant
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vol. 62 no. 1/2022 Beta Functions in Six-Derivative Quantum Gravity
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 the-
ory (6). In order to have a non-degenerate classi-
cal action and the well-defined Hessian operator of
the second variational derivative, one needs to re-
quire that both coefficients of the UV-leading terms,
namely ωC and ωR, 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 re-
ported 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 grav-
itational 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 ωC = 0, θC ≠ 0 and ωR ̸= 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 deriva-
tives in both special and also in generic theories (with
ωC ≠ 0 and ωR ̸= 0). For another special version of
the model, with ωC ̸= 0 and ωR = 0, the situation is
quite opposite regarding the spectrum, but the nega-
tive 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 ana-
lyzed 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 αC , αR and αGB 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□C) to the coefficient in front of the corresponding
term with two Ricci scalars (i.e. R□R), so only on the
ratio ωC /ωR also to be analyzed later at length here.
These coefficients of UV divergences αC , αR and αGB
do not depend on θC , θR, θGB, ωκ nor on ωΛ. 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 αC , αR and αGB. For example, the
terms with coefficients θC or θR 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
SHD =
∫
d
4
x
√
|g| (ωC Cµνρσ□C µνρσ + ωRR□R) . (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 deriva-
tives 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 for-
mula could be analytically extended). All three cases
of: first minimal renormalizable theory, and the mod-
els 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 deriva-
tives 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 dis-
continuity 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 com-
pletely 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
125
Lesław Rachwał Acta Polytechnica
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 na-
ture 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 num-
ber of derivatives 2n must be continuous in the pa-
rameter of the number of derivatives 2n, which could
be analytically extended to the whole complex plane
from the even integer values 2n ⩾ 10, which it origi-
nally 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 ⩾ 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 super-
ficial 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 kUV for loop integration momenta for
the degree ∆ vanishing, and power-law type for the
degree ∆ positive. This sharp distinction between
what is convergent and what is divergent (based on
the non-negativity of the degree of divergences ∆ 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 func-
tions 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 re-
search 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 di-
vergences, when one goes from six- to four-derivative
gravitational theories. So, in other words, when one
reduces 3-loop super-renormalizability to just renor-
malizability. 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 de-
gree ∆ < 0), to the situation when at this loop level
still UV divergences are present (since their degree
∆ is zero for logarithmic UV divergences in the cut-
off). 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 R3) to the La-
grangian in (6). These terms again will come with
the coefficients of the highest energy dimensionality,
equal to the dimensionality of the coefficients ωC and
ωR. 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
LR3 = s1R3 + s2RRµν Rµν + s3Rµν RµρRνρ
+ s4RRµνρσ Rµνρσ + s5Rµν Rρσ Rµρνσ
+ s6Rµνρσ Rµν κλRρσκλ . (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
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vol. 62 no. 1/2022 Beta Functions in Six-Derivative Quantum Gravity
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 sR3 and
are to be added to the original Lagrangian in (6) of
six-derivative theories, where the UV-leading terms
were of the type ωR□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 Ĥ) from the full action. The contributions
from cubic killers to it will be of the form of at least
sR, when counted in powers of generalized curvature
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 −
1
2
z2 + . . . . (12)
Hence we need to take maybe up to the square of
the contribution sR 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 R2 in the UV-divergent part of the effec-
tive action. Hence the contribution of the cubic killer
in curvatures would produce addition to the covari-
ant terms with UV divergences of the general type
f (s)R2, where the yet unknown functions f (s) can be
polynomials up to the second order in the coefficients
si of these killers. Now, requiring the total beta func-
tions vanish (for complete UV-finiteness) we need in
general to solve the system of the quadratic equations
in the coefficients si. 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 si coefficients to be real
for the definiteness of the action (for example in the
Euclidean case of the signature of the metric). There-
fore this issue requires a more detailed mathematical
analysis, but the preliminary results based on [26, 29]
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
si.
One can compare the situation here with cubic
killers to the more known situation where the quartic
killers are used to obtain UV-finiteness. Unfortu-
nately, 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. Quar-
tic 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 R4) is always linear in d = 4 to UV
divergences proportional to R2 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 theo-
ries [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) [36–38], 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 diver-
gences 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
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Lesław Rachwał Acta Polytechnica
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 gen-
uine 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 the-
oretical 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 perturba-
tive 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 ac-
count, although its expression is not gauge-invariant
and depends on the gauge fixing. These nice prop-
erties 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 quan-
tum 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 com-
putation 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. Un-
fortunately, 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 com-
bination of the two- and three-point functions of the
quantum theory, while the wave function renormaliza-
tion 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 quan-
tum corrections. Hence, here at the one-loop level
there is no quantum renormalization of the graviton’s
wave function and UV divergences related to interac-
tions are derived solely from propagation of free modes
(here of graviton fields) around the flat spacetime and
corrected (dressed) at the first quantum loop. Effec-
tive 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 θC (t), θR(t) and
θGB(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 gravi-
tational models [27, 41–43]. 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 diver-
gence linear in the scalar curvature R requires really
big efforts and was achieved only recently in our col-
laboration in [44]. In the present work, we comment
on the next step, and we show the results of the calcu-
lations of the simply looking one-loop UV divergences
for the four-derivative sector in the six-derivative min-
imal gravity model. In our result, we have now full
answers to the beta functions for the Weyl-squared
C 2, Ricci scalar-squared R2 and the Gauss-Bonnet
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 R3. 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 [30, 45, 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 func-
tion coefficients for the four-derivative gravitational
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vol. 62 no. 1/2022 Beta Functions in Six-Derivative Quantum Gravity
couplings, namely θC , θR and θGB, extracted as the
coefficients of the UV-divergent part of the effective
action in (1). Without loss of generality, the calcu-
lation 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
R2, 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 deriva-
tives 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 [26, 44] as considered here. In
what follows, we can skip a great part of the explana-
tions. 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 back-
ground field method, which is defined by the following
splitting of the metric
gµν −→ ḡµν + hµν (13)
to the background ḡµν and the quantum fluctuation
parts given by the spin-2 symmetric tensor hµν .
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 α, β and γ parameters. First, we choose the
parameter β in the harmonic background gauge-fixing
condition χµ, according to
χµ = ∇λhλµ − β ∇µh, h = hν ν , (14)
in the most simple “minimal” form, as will be indicated
below. The same concerns the parameters α and γ.
Finally, we select a general form of the weighting
operator, Ĉ = C̃ µν , which is defined by the formula
below:
Ĉ = C̃ µν = −
1
α
(
gµν□2 + (γ − 1)∇µ□∇ν
)
. (15)
This together with the gauge-fixing condition, that is
χµ, defines the gauge-fixing action [41] in the following
form,
Sgf =
∫
d4x
√
|g| χµ C̃ µν χν . (16)
The action of the complex Faddeev-Popov (FP) ghost
fields (respectively C̄ µ and Cµ) has in turn the form
Sgh =
∫
d4x
√
|g| C̄ µMµν Cν , (17)
where the bilinear part between the anti-ghost C̄ µ and
ghost fields Cµ, the so called FP-matrix M̂ , depends
differentially on χµ gauge-fixing conditions and also
on the contracted form of the generator of gauge
transformations R̂,
M̂ = Mµν =
δχµ
δgαβ
Rαβ
ν = δµν □+∇
ν ∇µ−2β∇µ∇ν .
(18)
In the above equation by the matrix-valued operator
Rαβ
ν we mean the generator of infinitesimal diffeo-
morphism (local gauge) transformations in any metric
theory of gravity.
Since as proven and explained at the end of sec-
tion 1.3, our final results for UV divergences are here
completely universal and they are independent of any
parameter used to regularize, compute and renormal-
ize 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 ap-
proach, a contribution from a subset of divergent
129
Lesław Rachwał Acta Polytechnica
diagrams may not be absorbed by a gauge-covariant
counterterm: F 2 in gauge theories, or R2 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 prop-
erty 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 R2 terms), but its front coefficient
may depend on these gauge parameters α, β, γ, 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 inde-
pendently and they are a good check of the correctness
of the calculation.
Unfortunately, it seems that using the BV com-
putational methods even in the intermediate results
for traces of separate matrix-valued differential opera-
tors (like Ĥ, M̂ and Ĉ), we see already both gauge-
independence and gauge-fixing parameters indepen-
dence 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 con-
cerning the choice of the gauge-fixing parameters α,
β and γ. The bilinear form of the action is defined
from the second variational derivative (giving rise to
the Hessian operator Ĥ)
Ĥ = H µν,ρσ =
1√
|g|
δ2 (S + Sgf )
δhµν δhρσ
= H µν,ρσlead + O(∇
4),
(19)
where the first term H µν,ρσlead contains six-derivative
terms, which are leading in the UV regime. By O(∇4)
we denote the rest of the bilinear form, with four or
less derivatives and with higher powers of gravitational
curvatures R. The energy dimension of this expression
is compensated by the powers of curvature tensor R
and its covariant derivatives, hence in this case, we can
also denote O(∇4) = O(R). The corresponding full
expression for the Hessian operator Ĥ is very bulky,
and we will not include it here.
The highest derivative part (leading in the UV
regime) of the Ĥ operator, after adding the gauge-
fixing term (16) that we have selected, has the form
H
µν,ρσ
lead =
[
ωC δ
µν,ρσ +
( β2γ
α
−
ωC
3
+2ωR
)
gµν gρσ
]
□3
+
( ωC
3
− 2ωR −
βγ
α
)(
gρσ ∇µ∇ν + gµν ∇ρ∇σ
)
□2
+
( 1
α
gµρ − 2ωC gµρ
)
∇ν ∇σ□2
+
( 2ωC
3
+ 2ωR +
γ − 1
α
)
∇µ∇ν ∇ρ∇σ□. (20)
In this expression, we do not mark explicitly the sym-
metrization in and between the pairs of indices (µ, ν)
and (ρ, σ) for the sake of brevity.
To make the UV-leading part of the Hessian opera-
tor H µν,ρσlead minimal, one has to choose the following
values for the gauge-fixing parameters [44]:
α =
1
2ωC
, β =
ωC − 6ωR
4ωC − 6ωR
, γ =
2ωC − 3ωR
3ωC
. (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 ωC → 0
and also when ωC = 32 ωR. While the first one is
clearly understandable, because then we are losing
one term ωC C□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 R2 and C 2 terms).
To explain this other spurious degeneracy one rather
goes to the Ricci basis of writing terms (with R2 and
R2µν = Rµν Rµν 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 R2µν leads
to the pathology in the case of ωC = 32 ωR and also
formal divergence of the β 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 R2µν in the Ricci basis does not
have any sensible and crucial meaning – this is only
a spurious intermediate dependence on (4ωC −6ωR)−1.
Contrary, the singular dependence on ωC coefficient
is very crucial and will be analyzed at length here.
Actually, to verify that the denominators with (4ωC −
6ωR)−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],
Γ̄(1) =
i
2
Tr ln Ĥ − iTr ln M̂ −
i
2
Tr ln Ĉ. (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
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vol. 62 no. 1/2022 Beta Functions in Six-Derivative Quantum Gravity
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 α, β and γ. However, only the
final results, so the weighted sum as in (22) is prop-
erly 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 con-
sists 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 compu-
tation presented here can be based on the analysis of
the scalar case given by the action
S =
∫
d4x
(
−
1
2
ϕ□ϕ −
λ
4!
ϕ4
)
. (23)
From this action one reads the second variational
derivative operator (also known as the Hessian) given
by the formula
H =
δ2S
δϕ2
= −□ −
λ
2
ϕ2. (24)
Next, one needs to compute the following functional
trace Tr ln H to get the UV-divergent part of the one-
loop effective action
Tr ln H = Tr ln
(
−□ −
λ
2
ϕ2
)
= Tr ln
(
−□
(
1 +
λ
2
ϕ2□−1
))
= Tr ln (−□) + Tr ln
(
1 +
λ
2
ϕ2□−1
)
. (25)
In the above expression, one concentrates on the sec-
ond part which contains the λ coupling. One expands
the logarithm, as in (12), in the second trace to the
second order in λ. This yields
Tr ln
(
1 +
λ
2
ϕ2□−1
)
= Tr
(
λ
2
ϕ2□−1
)
−
1
2
Tr
(
λ
2
ϕ2□−1
)2
+ . . . (26)
and one picks up from it only the expression quadratic
in λ and quartic in the background scalar field ϕ, which
is also formally quadratic in the inverse box operator
□−1, that is the part
Tr ln H ⊃ −
1
2
λ2
4
ϕ4Tr □−2 = −
λ2
8
ϕ4Tr □−2. (27)
Precisely this expression is relevant for the UV di-
vergence proportional to the quartic interaction term
− λ4! ϕ
4 in the original scalar field action (23). Noticing
that the functional trace of the □−2 scalar operator
in d = 4 is given by
Tr □−2 = i
ln L2
(4π)2
, (28)
where L is a dimensionless UV-cutoff parameter re-
lated to the ΛUV dimensionful momentum UV-cutoff
and the renormalization scale µ via ΛUV = Lµ, one
finds for the UV-divergent and interesting us part of
the one-loop effective action here
Γ(1)div =
i
2
Tr ln H ⊃
∫
d4x
ln L2
(4π)2
λ2
16
ϕ4. (29)
Now, one can compare this to the original action
terms in (23) describing quartic interactions of the
scalar fields ϕ: −
∫
d4x λ24 ϕ
4. The counterterm action
(to absorb UV divergences) is opposite to Γdiv 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 Γct we expect terms
Γct = −Γdiv = −
1
2
ln L2
24
∫
d4xβλϕ
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 − 14! = −
1
24 ).
From this one reads that (identifying that effectively
ln L2 → 1 for comparison)
−
1
48
βλ = −
λ2
16(4π)2
(31)
and finally that
βλ =
3λ2
(4π)2
, (32)
which is a standard result for the one-loop beta func-
tion of the quartic coupling λ in 14! λϕ
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 deriva-
tion of the explicit results for beta functions of the the-
ory. The reader, who wants to see some samples can
consult more explicit similar calculations as presented
in references [26, 30, 44]. In particular, the appendix
of [30] compares two approaches to the computation of
UV divergences in gauge theory (simpler than gravity
131
Lesław Rachwał Acta Polytechnica
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 di-
vergences of the six-derivative gravitational theory
are
Γ(1)R,Cdiv = −
ln L2
2(4π)2
∫
d4x
√
|g|
{( 397
40
+
2x
9
)
C 2
+
1387
180
GB −
7
36
R2
}
(33)
for the case of six-derivative pure QG model in d = 4
spacetime dimensions and
Γ(1)R,Cdiv = −
ln L2
2(4π)2
∫
d
4
x
√
|g|
{
−
133
20
C
2
+
196
45
GB +
(
−
5
2
x
−2
4−der +
5
2
x
−1
4−der −
5
36
)
R
2
}
(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 co-
variant cut-off regulator L [27], which stays in the
following relations to the dimensional regularization
parameter ϵ [27, 43],
ln L2 ≡ ln
Λ2UV
µ2
=
1
ϵ
=
1
2 − ω
=
2
4 − n
, (35)
where we denoted by n the generalized dimensionality
of spacetime in the DIMREG scheme of regularization
(additionally ΛUV is the dimensionful UV cutoff en-
ergy parameter and µ 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 =
ωC
ωR
, (36)
while for Stelle four-derivative theory in (34) we use
analogously but now with the theta couplings instead
of omegas, namely
x4−der =
θC
θR
. (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 Lren is obtained start-
ing from the classical Lagrangian written in terms of
the renormalized coupling constants and then adding
the counterterms to subtract the divergences,
Lren = L(αb(t)) = L
(
Zαi(t)αi(t)
)
= L(αi(t))+Lct
= L(αi(t)) + (ZC − 1) θC (t) C 2 + (ZR − 1) θR(t) R2
+ (ZGB − 1) θGB(t) GB, (38)
where we have that Lct = −Ldiv and αi(t) =
{θC (t), θR(t), θGB(t)}. Above we denoted by αb(t) the
RG running bare values of coupling parameters, by Lct
and Ldiv the counterterm and divergent Lagrangians
respectively, by Zαi (t) renormalization constants for
all dimensionless couplings and finally by αi(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 dimen-
sional regularization,
Γ(1)ct = −Γ
(1)
div =
1
2ϵ
1
(4π)2
∫
d
4
x
√
|g|
{( 397
40
+
2x
9
)
C
2
−
7
36
R
2 +
1387
180
GB
}
≡
1
2ϵ
1
(4π)2
∫
d
4
x
√
|g|
{
βC C
2 + βRR2 + βGBGB
}
.
(39)
Comparing the last two formulas we can identify the
beta functions and finally get the renormalization
group equations for the six derivative theory,
βC = µ
dθC
dµ
=
1
(4π)2
(
397
40
+
2x
9
)
, (40)
βR = µ
dθR
dµ
= −
1
(4π)2
7
36
, (41)
βGB = µ
dθGB
dµ
=
1
(4π)2
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 µ
µ0
to the renormalization scale µ, where µ0 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 char-
acteristic 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.
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vol. 62 no. 1/2022 Beta Functions in Six-Derivative Quantum Gravity
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 calcula-
tions 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 compu-
tation of UV divergences, which was the main task of
the work presented here.
(1.) First, the validity of the expression for the Hes-
sian operator from the classical action with six
derivatives was verified in the following way. The
covariant divergence of the second variational deriva-
tive operator (Hessian) with respect to gravita-
tional fluctuations hµν , from each GR-covariant
term Sgrav,i in the gravitational action must be
separately zero, namely
∇µ
(
δ2Sgrav,i
δhµν δhρσ
)
= 0 + O
(
∇kRl, k + 2l > 4
)
, (43)
where SHD =
∑
i Sgrav,i. This formula was explic-
itly 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 R.
(2.) The computation of the functional trace of the log-
arithm of the gauge weighting operator Ĉ, namely
of Tr ln Ĉ was checked using three methods. Since
the Ĉ 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 ad-
ditional verifications of our partial results for this
trace. Our three methods consist basically of trans-
forming 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 op-
erator Ŷ 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
Ĉ from the right by Ŷ one time, in the second
method we multiplied by Ŷ from the left also once,
and in the final third method we used the explicitly
symmetric form of multiplication ŶĈŶ . (This last
form of multiplication is presumably very important
for the manifest self-adjointness property of the
resulting 8-derivative differential operator ŶĈŶ .)
For these operatorial multiplications, Ŷ 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 (ŶĈ
and ĈŶ respectively) were six-derivative ones, while
in the last one with double multiplication from both
sides, ŶĈŶ was an eight-derivative matrix-valued
differential operator. At the end, all three described
above methods of computation of Tr ln Ĉ 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: R2, R2µν , and R2µνρσ ).
(3.) Similarly, the computation of Tr ln Ŷ for the two-
derivative operator Ŷ was verified using three anal-
ogous methods. We used multiplication from both
sides by the operator  and also the symmetric form
of multiplication ÂŶÂ, where Ŷ is a two-derivative
operator, whose functional trace of the logarithm we
searched here. Above, Â was another two-derivative
non-minimal spin-one vector gauge (massless) oper-
ator, whose trace of the logarithm is well known [41]
and can be easily found. Again, for the final results
for Tr ln Ŷ all three methods presented here agree
to the order of terms quadratic in curvatures R.
(4.) In total divergent part Γdiv of the quantum ef-
fective action, we checked a complete cancellation
of terms with poles in y = 2ωC − 3ωR variable,
namely all terms with 1
y
and 1
y2
in denominators
(originating from the expression for the gauge-fixing
parameter β in (21)) completely cancel out. This
is not a trivial cancellation between the results of
the following traces: Tr ln Ĥ and Tr ln M̂ .
(5.) Finally, using the same code written in Mathe-
matica [47] a similar computation in four-derivative
gravitational theory (Stelle theory in four dimen-
sions) was repeated. We easily were able to re-
produce 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 x4−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
133
Lesław Rachwał Acta Polytechnica
four and six derivatives). We study in detail the sit-
uation 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 ob-
tained by continuous limit procedures.
First, we discuss the situation with a possible de-
generacy of the kinetic operator of the theory acting
between quantum metric fluctuations hµν on the level
of the quadratized action. If the action of a theory in
the UV regime has the following UV-leading terms
Sgrav =
∫
d
d
x
√
|g|
(
ωC,N C□
N
C + ωR,N R□N R
)
, (44)
with ωC,N ≠ 0 and also ωR,N ̸= 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 ωC,N ̸= 0 and ωR,N ̸= 0, but also
when ωR,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 ωC,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 addi-
tion of the gauge-fixing functional here is necessary
since without it the kinetic operator (Hessian) is auto-
matically 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,
Sgrav =
∫
d4x
√
|g|
(
ωC,N C□
N C + ωR,N R□N R
+ ωC,N −1C□N −1C + ωR,N −1R□N −1R
+ωC,N −2C□N −2C + ωR,N −2R□N −2R
)
, (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 follow-
ing convention specially suited for six-derivative grav-
itational theories, so in the case when N = 1. We will
call coupling coefficients in front of the leading terms
as respective omega coefficients (like ωC = ωC,1 and
analogously ωR = ωR,1), while the coefficients of the
subleading terms with four derivatives we will denote
as theta coefficients (like θC = ωC,0 and analogously
θR = ωR,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 cos-
mological constant type of UV divergence. We denote
this coefficient as ω−1 = ωR,−1 and it is in front of
the Ricci scalar term in the original classical action
of the theory (6). Simply this coefficient ω−1 is re-
lated to the value of the 4-dimensional gravitational
Newton’s constant GN .
The expressions for the RG running of the cosmo-
logical constant and the Newton’s constant in [26, 44]
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:
ωR,N −1
ωR,N
,
ωC,N −1
ωC,N
,
ωR,N −2
ωR,N
,
ωC,N −2
ωC,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
Cµνρσ and Ricci scalars R the only fractions, which
appear in such subleading UV divergences are “diag-
onal” and do not mix terms from the spin-2 (Weyl)
sector with terms from the spin-0 (Ricci scalar) sec-
tor. If in any of the above fractions, we take the
limits: ωR,N −1 → 0, ωC,N −1 → 0, ωR,N −2 → 0, or
ωC,N −2 → 0, then the corresponding fractions and
also related UV divergences (and resulting beta func-
tions in question) simply vanish, provided that the
coefficients in their denominators ωR,N and ωC,N are
non-zero.
On the other hand, if ωC,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 situa-
tion 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 ωR,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 addi-
tional kind of gauge fixing has to be used, which fixes
the value also of the trace of the metric fluctuations
h = gµν hµν and this last one in the theory with N = 0
resembles the conformal gauge fixing of the trace.
If ωR,N = 0 and additionally ωR,N −1 or ωR,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 renormaliza-
tion scheme and the renormalizability of such a theory
is likely lost. But if the model is with ωR,N = 0 and
at the same time ωR,N −1 = ωR,N −2 = 0, then the con-
tributions of corresponding fractions in (46) are van-
ishing, because the limits ωR,N −1 → 0 or ωR,N −2 → 0
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vol. 62 no. 1/2022 Beta Functions in Six-Derivative Quantum Gravity
must be taken as the first respectively. Only after
this, the final limiting procedure ωR,N → 0 should be
performed. Therefore, in this limiting situation, the
proper sequence of limits on respective fractions is as
follows:
lim
ωR,N →0
(
lim
ωR,N −1→0
ωR,N −1
ωR,N
)
= 0 (47)
and
lim
ωR,N →0
(
lim
ωR,N −2→0
ωR,N −2
ωR,N
)
= 0 . (48)
In this case, there are no contributions to the beta
functions from these fractions, so the R2 sector does
not contribute anything to the mentioned UV diver-
gences, 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 ωC,N = 0 and at the same time
ωC,N −1 = ωC,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 (R2 sector) survives and the theory is likely non-
renormalizable. Then we expect contributions to UV
divergences only from terms in the R2 sector.
Regarding the divergences proportional to expres-
sions quadratic in curvatures (R2, C 2, and the Gauss-
Bonnet term GB), we have found the following generic
structure in four-derivative gravity [42]:
A−2
x24−der
+
A−1
x4−der
+ A0, (49)
where in this case of four-derivative gravity the fun-
damental ratio of the theory is defined as
x4−der =
ωC,0
ωR,0
=
θC
θR
. (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 R2, C 2 and GB terms respectively). The explicit
numerical values are given in the formula (34). One
observes negative powers of the ratio x4−der in (49)
and in (34), implying also negative powers of the cou-
pling θC in the final results for these UV divergences.
This result signifies that the theory with θC = 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 θC → 0
of the above formula in (49) do not exist. Such theories
with θC = 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 R2 Starobinsky
theory as well as to theories in the R2 sector with
addition of the Einstein-Hilbert R or the cosmological
constant ωΛ terms.
On the other side, the limit θR → 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 A0
for (49) for each of the UV divergences in this case.
But this is an incorrect answer since for pure four-
derivative gravity with θR = 0 in the Weyl basis of
terms, we have an enhancement of the symmetry in
the model, beyond the case where θR was non-zero.
In this situation, the theory enjoys also conformal
symmetry and a more specialized and delicate compu-
tation 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 con-
formal gravity, there are also some subleading terms
from the almost “pure” R2 sector, that is ω−1 ̸= 0
or when we allow for non-vanishing cosmological con-
stant term ωΛ ̸= 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 con-
formal transformations of gravitational gauge poten-
tials (not the conformal Weyl gauge potentials bµ)
are without derivatives. At the end, when the more
sophisticated method is employed, the eventual result
is different than A0 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 A0 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 ωΛ term. We conclude
that in the four-derivative theory, the two possible
extreme cases of θR = 0 or θC = 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
B0 + B1x, (51)
with new values for the constants B0 and B1. The
explicit numerical values are given in our formula (33)
with the results. We also remark that the values of the
constant terms B0 are different than the values of A0
in the previous four-derivative gravity case. Moreover,
the numerical coefficients B0 and B1 are different
for different types of UV divergences of the effective
action (R2, C 2 and 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 =
ωC,1
ωR,1
=
ωC
ωR
. (52)
135
Lesław Rachwał Acta Polytechnica
We emphasize that in such a case, we cannot con-
tinuously take the limit ωC → 0. Although, naively
this would mean the limit x → 0, the result just B0
from (51) would be incorrect. This is because in this
case we cannot trust the method of the computation.
When ωC = 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 ωR → 0 is clearly impossible too, because it
gives divergent results. However, in this case (ωR = 0)
and on the contrary to the previous case (ωC = 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 analysis3.
We conclude that in the case of six-derivative gravity,
both cases ωR = 0 or ωC = 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 grav-
itational 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 1
ωR
or 1
ωC
dependence (corre-
sponding to positive or negative powers of the funda-
mental ratio x = ωC
ωR
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
ωC , ωR and x coefficients and we will not distinguish
and not change it to the special notation originally
adequate only to Stelle quadratic theory (with θC , θR
and x4−der). We hope that this will not lead to any
confusion.
We emphasize, that when we have one of the two
terms missing – with ωR or ωC front couplings – in
the leading in UV part of the action of the model, then
3We 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 ωC ̸= 0 and ωR ̸= 0.
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 ωR = 0 or ωC = 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 θC = 0 is highly problematic. The same
regards taking the limit θC → 0, because then the
pure R2 theory can be fully gauge-fixed. And for ex-
ample, 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 loga-
rithms of Ĥ and of Ĉ operators both contain singular
expressions 1
θC
, and there is no final cancellation be-
tween 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 x4−der.
The results in quadratic Stelle gravity, when we set
θR = 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
θR = 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 Ĥ and Ĉ are different
than their limiting versions under θR → 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 θR ̸= 0.
If we start with the theory with θC = 0 from the be-
ginning, 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
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vol. 62 no. 1/2022 Beta Functions in Six-Derivative Quantum Gravity
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 R2 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 θC = 0 case) one could choose
the Ĉ matrix-valued differential operator different
from the standard minimal prescription and choose
different values for the γ gauge-fixing parameter. In
the standard minimal choice for the gauge-fixing pa-
rameters and in this model, the Ĉ matrix contains
an irregular part in θC coupling ( 1θC pole), because
of the dependence of γ on θC . 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 θC ≠ 0. However,
knowing that in the case with θC = 0, this procedure
is anyhow unsuccessful, we have the freedom to choose
the value of γ 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 ωR = 0 and N = 1 is
not conformally invariant in d = 4 dimensions. And
the above problems with the gauge fixing of the Hes-
sian operator Ĥ and non-minimality of it in the case
ωC = 0 still persist. This is because here for six-
derivative gravitational theories the box operator □
acting between two gravitational curvatures is only
a spectator from the point of view of the UV-leading
part of the Ĥ operator (with the highest number of
derivatives and with the zeroth powers in gravitational
curvatures) or from the point of view of flat space-
time kinetic operator and flat spacetime graviton’s
propagator. The box operator in momentum space
gives only one additional factor of −k2 to the kinetic
operator and additional suppression by −k−2 to the
propagator. The Hessian in the six-derivative theory
with ωR = 0 must possess the same definitional issues
as the one in the four-derivative theory (with θR = 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 ∂2 is −k2 in
Fourier space). So then, if we know that the Hessian
Ĥ is almost well-defined for the conformal gravity case
(up to the need for additional gauge fixing of the con-
formal symmetry), then the same will be true for the
Hessian in the six-derivative theory with the ωR = 0
condition in d = 4 spacetime dimensions, although
then the theory ceases to be conformal anymore. In
conformal gravity in d = 4, when θR = 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, set-
ting ωR = 0 does not create any problem for the
form of neither Ĥ nor Ĉ operators. Only the final
results for the beta functions show 1
ωR
poles as this
was manifest from the results in [26]. In turn, in
six-derivative theories, the limit ωC → 0 seems regu-
lar, but it is questionable that now we can trust the
results of this limit. In the pure R□R theory, we
expect to get some discontinuous results for the beta
functions not obtainable by the limit ωC → 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 ωC → 0 on the ki-
netic operator from the generic case ωC ̸= 0 produces
a Hessian Ĥ 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 ωC → 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 Ĥ of the theory between
graviton fluctuations. The insertions of box operators,
like any power or functions of the box operator □, are
only the immaterial differences between the cases of
four- and six-derivative theories here. These opera-
tors 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 R = 0. Using formula (20)
with solutions for gauge-fixing parameters as in (21),
one finds in the generic case ωC ̸= 0 and ωR ̸= 0, that
the kinetic operator (leading part of the Hessian) is
indeed minimal and of the form
H
µν,ρσ
lead =
ωC
2
□ (gµρgνσ + gµσ gνρ)
− ωC
ωC − 6ωR
4ωC − 6ωR
□gµν gρσ . (53)
In the above formula, one does not see any singularity
when ωC is vanishing (one saw ω−1C divergences in
the expressions for α and γ parameters in (21)), but
in this case the above treatment was not justified.
When ωC = 0, one can solve the system for gauge-
fixing parameters for β and γ′ = γ
α
and assume that
formally 1
α
= 0 and 1
γ
= 0, but in the ratio γ
α
the limit
is finite. One then finds that β = 1 and γ′ = −2ωR
and after substitution to the original Hessian, one
gets that its leading part explicitly vanishes. The
same one gets by plugging the naive limit ωC → 0
in (53). One also sees from the explicit solutions
in (21) and resulting general expression for γ′ (i.e.
γ′ = 43 ωC − 2ωR) that by plugging ωC = 0 one finds
again that β = 1 and γ′ = −2ωR as derived exactly
above. The highest derivative level of the gravitational
action is then completely gauge-fixed.
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Lesław Rachwał Acta Polytechnica
In the opposite case, when ωR = 0, the leading part
of the Hessian does not vanish, but it is degenerate
and in the form
H
µν,ρσ
lead =
ωC
2
□ (gµρgνσ + gµσ gνρ) −
ωC
4
□gµν gρσ , (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 τ h□3h
with a new (fourth) gauge-fixing parameter τ and
where the trace of metric fluctuations h = hµµ is
used, removes the degeneracy provided that τ ̸= 0
is selected. Then the kinetic operator takes the form
H
µν,ρσ
lead =
ωC
2
□ (gµρgνσ + gµσ gνρ)
+
(
τ −
ωC
4
)
□gµν gρσ . (55)
Moreover, for any non-zero value of τ the Hessian
is still a minimal operator. For τ ̸= 0 the inverse
exists and also the propagator can be defined around
flat spacetime. The only question is whether the fi-
nal results are τ -independent since this is a spurious
gauge-fixing parameter. The reason for such indepen-
dence is obvious in the four-derivative case, since τ
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 τ parameter drops out
from final results as it should for them to be physical
and τ gauge choice independent.
In four-derivative gravitational theory, one can see
the dependence on the x4−der ratio only in the coeffi-
cient of the R2 counterterm. This dependence is with
the general schematic form A−2x−24−der + A−1x
−1
4−der +
A0x
0
4−der like in (49) and in (34). We remark that
for other counterterms (namely for C 2 and GB in
this Weyl basis), the coefficients of UV divergences
are numbers completely independent of x4−der. One
could try to explain here this quadratic dependence
in the inverse ratio x−14−der in front of the R
2 countert-
erm 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 de-
grees 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 enumer-
ate all of them and their strength of interactions in
cubic vertices when they interact with background
gravitational potentials. Therefore, this task of ex-
plaining 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 depen-
dence on the x4−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 B1x1 + B0x0 like in (51) with
non-negative powers of the x ratio. The other coun-
terterms R2 and GB are with constant coefficients
(only B0 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 R2, R2µν , and
R2µνρσ 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 gravi-
tational theory, which makes the C 2 sector of UV
divergences completely finite. This value is exactly
x = − 357380 = −44.6625. The R
2 sector of UV diver-
gences 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 x4−der,
which made contrary the R2 sector UV-finite, were
two and they were x4−der = 3(3 ±
√
7) (their numeri-
cal approximations are as follows: x4−der,− ≈ 1.0627
and x4−der,+ ≈ 16.937) as solutions of some non-
degenerate quadratic algebraic equation. Again, con-
trary 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 ωC = 0 and ωR = 0. In six-derivative
model or when we have even more derivatives, su-
perficially these two couplings and their roles for the
computation of UV divergences may look symmetric.
This is however not true due to the different impact of
these two conditions on the form of the kinetic opera-
tor Ĥ. In the case when ωR = 0, the Hessian operator
still exists, while for ωC = 0 we lose its form. This
observation has profound implications as we explain
below. First, it is a fact that both these conditions
lead to badly non-renormalizable theories, in which
the flat spacetime propagator cannot be simply de-
fined. Moreover, if N > 0 in none of these two reduced
models we have an enhancement of symmetries and
none of them has anything to do with conformal grav-
ity models, which are present only for N = 0 and
ωR = 0 case, despite that in constructions of these
six-derivative models we might use only terms with
Weyl tensor. (However, here we use the term C□C,
where it is known that the GR-covariant box operator
□ is not conformal.)
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Our explanation of the x-dependence is as follows.
First, in the generic model with N > 0, since it hap-
pens that it is the N = 0 scale-invariant gravitational
model which is here the exceptional one. For six-
derivative theory (or any one with N > 0) the two
reduced models with conditions that ωC = 0 or ωR = 0
respectively are not renormalizable and likely even at
the one-loop level higher types of divergences (besides
C 2 and R2 from (1)) will be generated. From this we
expect that there must be some problems with UV
divergences obtained from naive power counting ar-
guments here. The problems must show up somehow
in the final numerical values for divergent terms or
in the intermediate steps of the process of computing
these divergences. These problems then signal that we
are working with non-renormalizable theory, which do
not have a good control over perturbative divergences
showing up at the one-loop level.
First, in the case ωC = 0, we see that the problems
are already there with the definition of the kinetic op-
erator (Hessian) between quantum metric fluctuation
fields. This implies that further processing with this
operator is ill-defined, we cannot trust it and even
if it would give us some final results for divergences,
then they are not reliable at all since the theory is
non-renormalizable. But we already found here the
instance of the problem, which makes our final limit-
ing results (in the ωC → 0 limit) not trustable. This
means that from the expression in (51), we do not
expect any additional obstacles, like 1
ωC
poles since
the price for non-renormalizability was already paid
and we have already met dangerous problems, which
signal the incorrectness of the naive limit ωC → 0.
This should already take away our trust in the limit
ωC → 0 of expressions for UV divergences in (33).
Then this line of thought in the case ωC = 0 does
not put any constraints at all on the final form of the
x-dependence in (51) since these results like in (51)
in the limit ωC → 0 will anyway be likely incorrect.
Now, in the other case with N > 0 and ωR = 0, we
do not have the problem with the definition of the
Hessian Ĥ. Formally, we can process it till the end of
taking the functional trace of the logarithm and adding
contributions from Tr ln M̂ and Tr ln Ĉ. But somehow,
we must find the occurrence of the problem, because
the theory is non-renormalizable! So the only place in
which the problem may sit is in the final x-dependence
of the results for UV divergences. These results should
be ill-defined, when the limit ωR → 0 is attempted.
And this implies that they must be with poles in the
ωR coefficient, so they must be instead with positive
powers of the x ratio of the theory. Hence, we conclude
that the x-dependence must be linear or quadratic,
but always with positive powers of the ratio x. This
is now confirmed by explicit form as in (33) for UV
divergences of six-derivative theory. The problems
with renormalizability of the pure theory C□C show
up in the last possible moment in the procedure for
obtaining the result, namely when one wants to take
the limit ωR → 0 or equivalently x → ∞. This is the
generic situation for any super-renormalizable theory
and for any N > 0. There are still some mysterious
things here, like why the dependence is only linear in
x and why only for the C 2 type of UV counterterm,
while two other counterterms R2 and GB are numbers
completely independent of x. Right now we cannot
provide satisfactory mathematical explanations for
these facts.
Using this argumentation in the theory models with
N = 1, we get an explanation for the x-dependence in
formula for UV divergences in (51). The logical chain
for the explanation should be as follows. Firstly, in the
pure theory C□C, one concludes that the problems of
non-renormalizability shows only in the final results as
impossibility to take the limit ωR → 0 or equivalently
x → ∞ of formula in (51) for divergences of the model.
Hence the dependence must be on non-negative powers
of the ratio x in formula (51), as it is clearly confirmed
by explicit inspection of this formula. This settles
the issue of the structure of exact beta functions for
N = 1 models (and also for higher N ⩾ 1 cases
too). Now, the same formula is a starting point for
an attempt to take the other limit x → 0 of the
also non-renormalizable model of the type R□R. But
in such a model we have already found a source of
the problem caused by non-renormalizability earlier,
that it is here connected with the impossibility to
properly define non-degenerate Hessian operator in
the model. But this limiting case of x → 0 must
follow the same structure as already established in
formula (51). Simply, theoretically speaking, there
is no need to see more instances of problems due to
non-renormalizability in the R□R model. Hence, the
first explanation based on the C□C model is sufficient
and the results in the model R□R must be consistent
with it. Moreover, from just the analysis of the case
x → ∞, we have concluded what is the structure in
a generic renormalizable case, when we have both
ωR ̸= 0 and ωC ̸= 0 (so x ̸= 0 and x ̸= ∞). This
structure is beautifully confirmed by the formula (51)
or (33) explicitly for the generic case.
As emphasized above, it is in turn the N = 0 case,
which is extraordinary and it changes the pattern of
x4−der-dependencies described above. This all can be
traced back to the fact that for N = 0 we have the
possibility of reducing the generic HD scale-invariant
model to the conformal one, when the full conformal
symmetry is preserved (at least on the classical level
of the theory). This happens, when one takes the iso-
lated case of θR = 0 and θC ̸= 0 for the four-derivative
theory (for positive-definiteness we may also assume
that θC > 0). This case is discontinuous and cannot
be taken as the naive limit x4−der → ∞ of the for-
mula (49) for the R2 type of UV divergences, which
would leave us effectively only with the A0 coefficient.
It is well known that the conformal gravity model is
renormalizable one (at least to the one-loop level),
contrary to the case of the theory C□C, which was
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Lesław Rachwał Acta Polytechnica
discussed above. This means that we shall not find any
source of the problem when computing and getting
results for UV divergences of this C 2 model. We do
not find problems with the Hessian or the propagator
provided we also gauge-fix the conformal local symme-
try of Weyl conformal gravity. We shall not find the
problem with the final expression of UV divergences,
so there we shall not expect poles with θR coefficient.
But some x4−der-dependence up to the quadratic order
could be present (this is due to the one-loop character
of the computation here; one can understand this eas-
ily from contributing Feynman diagrams). So then, we
conclude that this dependence may be only in positive
powers of the inverse ratio, namely of x−14−der = θR/θC .
This is again confirmed in the formula (49) and (34),
where we indeed find the quadratic dependence but
in the inverse ratio x−14−der.
Simply, the final results for the generic case
x4−der ̸= 0 and x4−der ̸= ∞ cannot depend on pos-
itive powers of x4−der since then the limit of con-
formal gravity in d = 4 (i.e. x4−der → ∞) would
produce divergent results, but we know that Weyl
gravity is renormalizable with a good control over
one-loop UV divergences. However, this does not
mean that the results for conformal gravity are con-
tinuous and obtainable from the generic ones in (34)
by taking the limit x4−der → ∞ there. We admit the
fact that the coefficients there may show some finite
discontinuities. However, both in the true and naive
x4−der → ∞ limiting forms they must be finite – we
only exclude the case when they would be divergent
in the x4−der → ∞ limit. In this way, the results in
renormalizable 4-dimensional conformal gravity for
UV divergences may be expressed via finite numbers
multiplying just one common overall divergence (like
1/ϵ parameter in dimensional regularization (DIM-
REG) scheme). The theory is renormalizable and
there are no new divergences inside coefficients of es-
tablished form of UV divergences in generic HD Stelle
theory in d = 4 spacetime dimensions, as in (1).
The significant difference between the cases of N =
0 and N ⩾ 1 is that in the former case the theory with
θR = 0 is conformal on the classical tree-level as well
as on the first quantum loop, since we know that Weyl
conformal quantum gravity is one-loop renormalizable.
This is why the pattern of the x-dependence in these
two cases is diametrically different. In both these cases
of N = 0 and N ̸= 0, one can derive the general
structure of beta functions in generic HD theory with
any finite value of the fundamental ratio x (x ̸= 0
and x ̸= ∞) by just analyzing the limit x → ∞ (or
respectively x4−der → ∞) and its divergences which
should or should not appear there respectively for the
cases of N ̸= 0 or N = 0.
The inverse quadratic dependence on the ratio
x4−der in the case of four-derivative Stelle theory can
be easily understood as well. It is up to the quadratic
order and the same dependence we would expect in the
case of six-derivative gravitational theory in the C 2
sector of UV divergences. However, there as a surprise
we find only up to linear dependence on the fundamen-
tal ratio x and only in one distinguished sector of C 2
divergences. In general, we can have up to quadratic
dependence on x in six-derivative models or on x−14−der
in the Stelle gravity case in d = 4 spacetime dimen-
sions. The UV divergences of some renormalizable
HD gravity models in d = 4 spacetime dimensions are
all at most quadratic in the general gravitational cur-
vature (schematically they are R2). Hence they can
be all read from the one-loop perturbative quantum
corrections to the two-point graviton Green function,
so equivalently from the quantum dressed graviton’s
propagator around flat spacetime background. We
remind that here there is no quantum divergent renor-
malization of the graviton wave function. Moreover,
higher orders in graviton fields (appearing in interac-
tion vertices) are completely determined here due to
the gauge invariance (diffeomorphism) present in any
QG model, so we can concentrate below only on these
two-point Green functions.
As it is known from diagrammatics, here the con-
tributing Feynman diagrams may have either one prop-
agator (topology of the bubble attached to the line)
or two propagators (sunset diagrams) at the one-loop
order and for corrections to the two-point function. In
the most difficult case, there are here two perturbative
propagators. Since in our higher-derivative theory we
have two leading terms shaping the UV form of the
graviton’s propagator, namely the terms ωC C□C and
ωRR□R, then the corresponding propagator may be
either with the front coefficient ω−1C or ω
−1
R respec-
tively as the leading term. To change between the
two expansions (in ωC or in ωR) one needs to use
one power of the ratio x. Since we have two such
propagators in the one-loop diagrams considered here,
then dependence is up to the quadratic power in x.
Sometimes we need to change back from ωC to ωR
as the leading coefficient of the tree-level propagator,
and then we need to multiply by inverse powers of
the ratio x. The quadratic dependence is what we
can have here in the most complicated case, which is
actually realized in Stelle generic theory with both
θC ̸= 0 and θR ̸= 0. (The argumentation above can be
repeated very similarly for quadratic gravity in d = 4
forgetting about one power of box operator □ and
changing corresponding omega coefficients to theta
coefficients and x to x4−der.) Apparently, in the case
of six-derivative gravitational theories there is some,
for the moment, unexplained cancellation, and we see
there only the dependence up to the first power on
the ratio x of that theory.
One should acknowledge here the speciality of the
case of d = 4 and one-loop type of computation. For
higher loop orders the powers of the x ratio may ap-
pear higher in the final expressions for UV divergences
of the theory. Similarly, if one goes to higher dimen-
sional QG models, then even in renormalizable models
at the one-loop level, one needs to compute higher
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n-point Green function. This is because in even di-
mension d one needs in renormalizable theory not
only to renormalize terms of the type R□(d−4)/2R
but also others with more curvatures (and correspond-
ingly less powers of covariant derivatives) down to the
term of the type Rd/2, where we do not find covariant
derivatives acting on curvature at all. In the middle,
the general terms can be schematically parametrized
as ∇d−2iRi for i = 2, . . . , d2 – all these terms have
the energy dimensionality equal to the dimensionality
of spacetime d. For the last term of the type Rd/2
one needs to look at the quantum dressed n = d2 -
point function at the one-loop order. In conclusion,
in higher dimensions one should consider not only
two-point functions with one-loop diagrams with the
two topologies described above, but up to quantum
dressed d2 -point functions. And even for one-loop per-
turbative level these additional diagrams may have
more complicated topology meaning more vertices and
more propagators and this means that also powers
of the ratio x or x−1 respectively will be higher and
higher. They are expected to be up to the upper
bound given by the maximal power exponent equal to
d
2 – this can be derived from the expression of quan-
tum dressed d2 -point function, which is built exactly
with d2 propagators joining precisely
d
2 3-leg the same
perturbative vertices. Then the topology of such a
diagram is this one of the main one-loop ring and d2
external legs attached to it, with each one separately
and each one emanating from one single 3-leg vertex.
Again the situation at the one-loop and in d = 4 is
quite special and simple since the ratio x appears
here only up to the maximal power exponent given
by d2 = 2.
As a side result, one also sees that the situation
in four-derivative model with the condition θC = 0
is somehow “doubly” bad. First, the Hessian is not
well-defined to start with and this takes away our trust
in this type of computation. Moreover, if we would
attempt to take the limit θC → 0 (or equivalently
x4−der → 0) in the final result like in (49), then we
get a second problem since such limit gives infinite re-
sults. This means that we somehow doubly confirmed
the problem with the perturbative and multiplicative
renormalizability of such a model. It is not that the
two instances of the problem support each other – they
appear somehow independently and are not related,
nor they cancel out. Above, we have seen that in
the six-derivative (or general N > 0) case, they could
occur completely independently for two completely
different types of non-renormalizable theories (with
the conditions of ωR = 0 or ωC = 0 respectively).
Here, we see that since conformal gravity at one-loop
must be without any problem of this type (no prob-
lem with the Hessian and no problem with getting
infinite results of the limits x4−der → ∞), then the
occurrence of these two problems at the same time
must happen in badly non-renormalizable model with
θC = 0 condition. In other words, since conformal
gravity is a safe model, then the model with θC = 0
must suffer twice since all these two problems must
inevitably appear in one model or the other, if some
extreme special cases like θR = 0 or θC = 0 are being
considered.
Again, we remark that in generic quadratic gravity
model we see up to quadratic dependence on the
inverse ratio x−14−der, but a precise location where this
dependence shows up is still not amenable for an easy
explanation. We do not know why this happens in
the R2 sector only, while the C 2 and GB sectors are
free from any x4−der-dependence. But at least the
dependence on the inverse ratio x−14−der, rather than
on its original form x4−der = θCθR , in the exceptional
case of N = 0 can be explained by the miraculous one-
loop perturbative renormalizability of the conformal
gravity model in d = 4.
4.3. Case of conformal gravity
Here we continue the discussion of related issues, but
now in the framework of conformal gravity, so within
the model with the reduced HD action with N = 0
and formally with θR = 0. There are various mo-
tivations for conformal gravity in d = 4 spacetime
dimensions [13, 14]. As it is well known the reason
for the multiplicative renormalizability of such a re-
duced model, when we have from the beginning that
θR = 0 is the presence of conformality – conformal
symmetry both on the tree level and also on the level
of the first loop. Unfortunately, the story with con-
formal gravity in d = 4 is even more complicated
than what we argued above. First, already at the
one-loop level one discovers the presence of conformal
anomaly, which is typically thought as not so harmful
on the first loop level. However, it heralds the soon
breaking of conformal symmetry like for example via
the appearance of the R2 counterterm at the two-
loop level. Such term as a counterterm is not fully
invariant under local conformal transformations – it
is only invariant under so called restricted conformal
transformations that is such transformations whose
parameters satisfy the source-free background GR-
covariant d’Alembert equation (□Ω = 0) on a general
spacetime. Hence the R2 term is still scale-invariant
but it breaks full conformal symmetry of the quantum
conformal gravity. It seems the only way out of this
conformal anomaly problem is to include and couple
to conformal gravity specific matter sector to cancel
in effect the anomaly. This is, for example, done in
N = 4 conformal supergravity coupled to two copies
of N = 4 super-Yang-Mills theory, first proposed by
Fradkin and Tseytlin [49]. In such a coupled supergrav-
ity model, we have vanishing beta functions, implying
complete UV-finiteness and conformality present also
on the quantum level. This is conformal symmetry in
the local version (not a rigid one) with Weyl conformal
transformations in the gravitational setup and on the
quantum field theory level.
141
Lesław Rachwał Acta Polytechnica
If not in the framework of N = 4 Fradkin and
Tseytlin supergravity, the conformal anomaly of local
conformal symmetry signals breaking of conformal
symmetry, while scale-invariance (global part of it)
still may remain intact. In the long run, besides the
presence of non-conformal R2 counterterm, this break-
ing will put conformal Ward identities in question
and also the constraining power of the quantum con-
formality in question too. It will not constrain any
more the detailed form of gravitational correlation
functions of the quantum theory. The conformal sym-
metry will not be there and it will not protect the
spectrum from the emergence of some spurious ghost
states in it. This last thing will endanger the pertur-
bative unitarity of the theory (and we do not speak
here about the danger of unitarity breaking due to
the HD nature of conformal gravity). Without the
power of quantum conformal symmetry, we may have
unwanted states in the spectrum corresponding to
the states from generic Stelle gravity, and not from
the tree-level spectrum of conformal gravity, so we
can see mismatch in counting number of degrees of
freedom and also in their characters, namely whether
they are spin-1 or spin-0, ghosts or healthy particles,
etc.
Moreover, in pure conformal gravity described by
the action simply C 2 without any supergravitational
extension, we notice the somehow nomenclature prob-
lem with the presence of quantum conformality. Even
barring the issue of conformal anomaly, the general
pure gravitational theory has non-vanishing beta func-
tions, so there is no UV-finiteness there. This implies
that there is an RG running and scale-dependence of
couplings and of various correlators on the renormal-
ization energy scale. Hence already at the one-loop
level one could say that scale-invariance is broken,
which implies violation of conformal symmetry too.
However, one can live with this semantic difference
provided that there are no disastrous consequences
of the conformal anomaly. One can adopt the point
of view that the theory at the one-loop level is still
good provided that the UV-divergent action is con-
formally invariant too, that is when one has only
conformally invariant UV counterterms. (Although in
the strict meaning having them implies non-vanishing
beta function, RG running, loss of UV-finiteness and
of scale-invariance.) In our case the conformally in-
variant counterterms are only of the type C 2 and GB,
so if the R2 counterterm is not present at the one-loop
level, then we can speak about this preserved quantum
conformality in the second sense. It happens this is
exactly the situation we meet for quantum conformal
gravity in d = 4 at the one-loop level.
In order to see quantum conformality of one-loop
level conformal gravity in d = 4 described by the
action, one first naively may try to take the limit
x → +∞ from the expression for the R2 sector of UV
divergences from formula in (34). One would end up
with the results, just a constant A0, which is generally
not zero. The whole story is again more subtle, since
the limits in this case are again not continuous, al-
though as we advocated above they are luckily also not
divergent, when we want to send θR → 0. In the end,
we have only a finite discrepancy in numbers, which
can be easily explained. As emphasized above in this
case of the special reduced model we have the enhance-
ment of symmetries and this new emergent conformal
symmetry in the local version must be gauge-fixed
too. This means that the kinetic operator needs to be
modified and some new conformal gauge-fixing func-
tional must be added to it for the consistency of the
generalized Faddeev-Popov quantization prescription
of theories with local gauge symmetries. This means
that we will also have a new conformal gauge-fixing
parameter (the fourth one), which can be suitably ad-
justed to provide again the minimality of the Hessian
operator. Although, of course, the whole details of
the covariant quantization procedure for conformal
gravity are more delicate and more subtle, here we
can just take a shortcut and pinpoint the main points
of attention. When computing UV divergences using
generally covariant methods like BV trace technology
and functional traces of logarithms of operators, one
also necessarily needs to add here the contribution
of the third conformal ghosts, which are scalars from
the point of view of Lorentz symmetry but they come
with anti-commuting statistics. They are needed here
because the conformal gravity theory is a natural HD
theory and then third ghosts are necessary for covari-
ant treatment of any gauge symmetry in the local
form. It is true that for conformal local symmetry we
do not need FP ghost fields (because of the reasons
elucidated above), but we need a new third conformal
ghost, which is moreover independent from the third
ghost of diffeomorphism symmetry. Each symmetry
with local realization comes with its own set of third
ghosts, when the theory is with higher derivatives.
It is also well known that classically conformal fields
(like massless gauge fields of electromagnetism and
also of Yang-Mills theory) give contribution to diver-
gences which is conformally invariant counterterm,
so only of the type C 2 or GB terms. This can be
understood easily as a kind of conformal version of
the DeWitt-Utiyama argument used before. Hence, if
the scalars of the anti-commuting type that we have
to subtract were conformally coupled, then they will
not contribute anything to the R2 type of the coun-
terterm. But we see from the formula in (34) that
the A0 coefficient there is non-zero, so only this one
survives after the limit x → ∞ is taken. To cancel
the R2 counterterm is crucial for the hypothesized
conformal invariance of the conformal gravity also on
the first loop quantum level. And this must be done
by explicitly non-conformal fields with non-conformal
contributions to divergences. They cannot be massless
gauge fields, but they can be minimally, so not confor-
mally coupled scalar fields. Here for the consistency of
the whole formalism of the FP covariant quantization
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of conformal gravity, this role is played by the one
real conformal third ghost with the kinetic operator
□2.
The contribution of the third conformal ghosts is
what we actually need to complete the whole pro-
cess of the computation of the UV divergences in the
conformal gravity model. We need them for the over-
all consistency since in this covariant framework we
cannot a posteriori check the presence of all gauge
invariances. Here we assume that on the first quan-
tum loop level, the conformal gravity model enjoys
fully diffeomorphism as well as conformal symmetry.
The terms given in the covariant BV framework of
computation all satisfy these requirements, so only we
must be careful to take all these contributions into ac-
count. The contribution of the third conformal ghosts
is like that of two real scalars coupled minimally (but
not conformally to one’s surprise) to the background
gravitational metric, but of the ghost type. Indeed
this means that we have to subtract the contribution
of two scalars, which is of course, UV-divergent but
after extracting the overall divergence there is only
a finite number. This is the number that when sub-
tracted now matches with number obtained after the
naive limit x → ∞ of the generic results from (49).
We explain that we need to subtract two real scalars
each one coming with the standard two-derivative
GR-covariant box operator as the kinetic operator
since in HD conformal gravity the operator between
third conformal ghosts is of the □2 type as for the
four-derivative theory. The limit to conformal gravity
is discontinuous, but only in this sense that one has
to take out also contribution of real scalar fields min-
imally coupled to gravitation. The first part of the
limiting procedure, namely x → ∞ is only a partial
step and to complete the whole limiting procedure
one must also deal properly with conformal symmetry.
This applies not only to the coefficients in front of the
R2 term, where we see the mysterious but explainable
x-dependence, but also to other coefficients in front
of terms like C 2 and GB terms. Of course, for the
last two terms the limits x → ∞ do not change any-
thing, but the contribution of third conformal ghosts
makes impact and change the numerical results, which
are luckily still finite in conformal gravity. The co-
efficients in front of the R2 and GB counterterms
are also finite in generic four-derivative gravity (cf.
with (34)), however by these types of arguments with
conformal gravity we cannot at present understand
why the x-dependence happens only in front of the R2
counterterm. Of course, in conformal gravity model,
there is not any x-dependence at all.
At the end, when one accounts for all these nu-
merical contributions, one indeed finds that at the
one-loop level in conformal gravity, the coefficient of
the R2 term is vanishing, so the quantum conformal-
ity is present in the second sense. And we have only
conformally invariant counterterms in pure confor-
mal gravity without any conformal matter, but there
is still interesting RG flow of couplings there. This
also signifies that there is conformally invariant but
non-trivial divergent part of the effective action with
finite numerical constant coefficients, when the overall
divergence is extracted. These finite coefficients arise
in the two-step process. First as the limit x → ∞ of
a generic HD gravity and then by subtraction of UV-
divergent contributions of two real scalars minimally
coupled to gravitational field. Since these last contri-
butions are known to be finite numbers multiplying
the overall UV divergence, then this implies that the
limit x → ∞ of the generic expression in (49) must
also give finite numbers. This explains why in pertur-
batively one-loop renormalizable model of conformal
gravity in d = 4 there are standard UV divergences,
although this is a reduced model with θR = 0 and
N = 0 case. So the x-dependence in (34) must be
as emphasized above that is with inverse powers of
the fundamental ratio x of the theory and in accord
with what was schematically displayed in formula (49).
Hopefully now the dissimilarities between the cases
with N = 0 and N > 0 are more clear.
In short, we think that the only sensible reason, why
we see completely different behaviour when going from
N = 0 to N = 1 class of theories is that the theory
with ωR = 0 and N = 1 ceases to be conformally
invariant in d = 4. In a different vein, the degeneracy
of the kinetic operator Ĥ in the ωC = 0 cases, for
both N = 0 and N = 1, remains always the same.
This proves again and again that the case of con-
formal gravity is very special among all HD theories,
in d = 4 among all theories quadratic in gravitational
curvatures. One can also study the phenomenological
applications of the Weyl conformal gravity models to
the evaporation process of black holes [50, 51] and also
use the technology of RG flows (and also functional
RG flows) in the quantum model of conformal grav-
ity to derive some interesting consequences for the
cosmology (like for example for the presence of dark
components of the universe in [52–54]). Finally, the
conformal symmetry realized fully on the classical level
(and as we have seen also to the first loop level and
perhaps also beyond) is instrumental in solving the
issue with spacetime singularities [34, 55, 56], which
are otherwise ubiquitous problems in any other model
of generally covariant gravitational physics (both on
the classical and quantum level). To resolve all singu-
larities one must be sure that the conformality (Weyl
symmetry) is present also on the full quantum level
(and it is not anomalous there), so the resolution of
singularities from the classical level (by some compen-
sating conformal transformations) is not immediately
destroyed by some dangerous non-conformal quantum
fluctuations and corrections.
4.4. More on limiting cases
Here again we analyze closer the situation with various
limits, when some coefficients in the gravitational
action (45) disappear.
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Lesław Rachwał Acta Polytechnica
In a generic six-derivative theory, the trace of the
logarithm of the FP ghosts kinetic operator M̂ and
of the standard minimal Ĉ matrix are regular in the
limit ωR → 0, but not in the limit ωC → 0. For
the Ĉ matrix this is understandable, because the γ
parameter contains factor ω−1C in the minimal gauge.
However, for the FP ghosts kinetic operator M̂ , this
dependence was unexpected, because in the explicit
definition of the M̂ operator there was never any
singularity in ωC . Moreover, this singularity is even
quadratic in ωC coefficient.
We also emphasize that in the general six-derivative
theory the trace of the logarithm of the Hessian op-
erator Ĥ is irregular both in the limits ωR → 0 and
ωC → 0 separately. It seems that in the total sum of
contributions to the beta functions of the theory the
singularity in ωC cancels completely between Tr ln Ĥ,
Tr ln Ĉ, and Tr ln M̂ , while the poles in ωR remain
and this is what is seen as a dependence of the final
results on the non-negative powers of the fundamental
ratio x. For the definition of the Ĥ operator, if the
limit ωR → 0 is taken, nothing bad is seen. This
may be a partial surprise. Of course, when the limit
ωC → 0 is taken, then this operator does vanish on
flat spacetime, so then its degeneracy is clearly visible.
The situation with limits (θR → 0 or θC → 0) in
four-derivative theory we see as follows. The func-
tional trace Tr ln Ĉ is regular in both limits. It ac-
tually does not depend on any gauge-fixing param-
eters here, despite that in its formal definition we
used the γ parameter, which shows the 1
θC
singularity.
The situation with the FP operator M̂ is the same
as previously, because the operator is identical as in
the six-derivative theory case. The operator Ĥ shows
the problem with its definition only when the limit
θC → 0 is considered. The same is true for its trace
of the functional logarithm, which shows singularity
in θC coupling coefficient up to the quadratic order.
In this case and in the total sum of all contributions,
we see only 1
θC
singularity to the quadratic order.
However, here the limit θR → 0 is not continuous
either, because the theory reaches a critical point in
the theory space with enhanced symmetry for θR = 0
(conformal enhancement of local symmetries) as it was
explained in subsection 4.3.
Let us also comment on what special happens in the
computation of UV divergences for quadratic theory
from the perspective of problems that we have initially
encountered in six-derivative theory for the same com-
putation. First, we established, in the middle steps
of our computation for the results published in [26],
that in the traces Tr ln Ĥ and Tr ln Ĉ in Stelle grav-
ity there are no any dangerous 1
y
= 12ωC −3ωR poles
(cf. [42]). The cancellations happen separately within
each trace. Second thing is that we found that the
trace Tr ln Ĉ surprisingly completely does not depend
on the gauge-fixing parameter γ, which was needed
and used in the initial definition of the Ĉ operator
in (15). Finally, one can notice that the addition of
the Gauss-Bonnet term in d = 4 spacetime dimension,
does not change anything for R2, R2µν , and R2µνρσ di-
vergences (as it was expected), because its variation
is a topological term in d = 4.
In this last part, we use the schematic notation for
various gravitational theories, when we do not write,
for simplicity, the coupling coefficients in front of var-
ious terms since they are not the most important for
the considerations here. In the case of six-derivative
theories, it is impossible to obtain the results for the
cases with ωC = 0 or ωR = 0 by any limiting proce-
dures of the corresponding results obtained for the
general six-derivative theory with ωR ̸= 0 and ωC ̸= 0.
These reduced theories have different bilinear parts,
with degenerate forms of the kinetic operator and our
calculation methods break down here. Similarly, one
can calculate the beta functions in a theory with R2
only and this was done indirectly many times. One
can also calculate UV divergences in C□C + R2 the-
ory or in an analogous R□R + C 2 theory, but this is
actually not easy to do. But then we cannot easily ex-
tract these results from our general calculation done in
C□C + R□R six-derivative theory. The simple reason
is that all these theories have different amount and
characteristics of degrees of freedom and the transition
from one to another at quantum level is complicated
(and to some extent unknown).
Moreover, we remark, that the results for beta func-
tions in models C□C + R2 (or C□C + R2 + C 2) or
in R□R + C 2 (or R□R + C 2 + R2) could be obtained
by though different computations than what we have
done here. We summarize that the six-derivative grav-
itational theory to be renormalizable must contain
both terms of the type C□C and R□R. Then the ki-
netic operator (Hessian) between gravitational fluctu-
ations and the graviton’s propagator are well-defined.
In all other models, there is not a balance between
the number of derivatives in the vertices of the theory
and in all gauge-invariant pieces of the propagator,
so the theory behaves badly regarding perturbative
UV divergences at higher loops. This does not mean
that the computation of UV divergences at one-loop
level is forbidden, just only that usually these are not
all divergences in the theory, they may not be the
UV-leading ones anymore or the theory does not have
decent control over all of them.
For the strictly non-renormalizable theory with
the leading in the UV term C□C we can have addi-
tions of various subleading terms which do not change
the fact of non-renormalizability. We can add terms
(separately or in conjunction) of the following types:
ωΛ (cosmological constant term), R (E-H term), R2
(Starobinsky’s term), C 2 (Weyl square term). The
UV-leading part of the Hessian still is defined as it con-
tains six-derivative differential operator understood
on flat spacetime and between tensorial fluctuations,
so derived from the terms quadratic in curvatures.
The Hessian is non-degenerate. (It has to be non-
degenerate here, because the GR-covariant box oper-
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ator is here only a spectator, and the Hessian must
be “almost” non-degenerate for the case of conformal
gravity with the action C 2.) The flat spacetime prop-
agator can be defined only if we make addition of
ωΛ, or R or R2 terms – this is because of the prob-
lematic part of it proportional to the projector P (0)
which must for the consistency of the inverting pro-
cedure for the whole propagator be non-zero. This
scalar part (spin-0 part) is sourced from any scalar
term or from the cosmological constant term. If only
the C 2 term is added, then the propagator still is
ill-defined. Still these additions do not change the
fact that the theory is non-renormalizable, if there is
not an accompanying six-derivative term of the form
R□R. As for the final results for UV divergences in
these extended models, naively one would think that
there are no additional UV divergences proportional
to terms with four derivatives of the metric (namely to
terms R2, C 2 and GB), because of the limit ωR → 0
and the dependence on the x ratio in (33) in the linear
way. We would naively think that divergences with
R2 and GB terms are the same as in (33). The only
problematic one could be this proportional to the C 2
term since the limit gives already divergent results
(so “doubly” divergent) – this would mean that the
coefficient of the C 2 divergence is further divergent
and renormalization of just C 2 does not absorb ev-
erything at the one-loop level. Since the model is
non-renormalizable we cannot trust this computation
and these limits at the end, especially if they give
divergent results. But this probably means that we
cannot sensibly define the C 2 counterterm needed for
the UV renormalization in these theories. In a sense
an attempt of adding ωΛ or R, or R2 terms to reg-
ularize the theory C□C + C 2, or even the simplest
one, just C□C, is unsuccessful so we perhaps still
cannot trust there in the final results just given by
two B0 coefficients of UV divergences proportional to
terms R2 and GB, while the C 2 divergences are never
well-defined in this class of models.
Instead, in the case of the reduced model with
R□R UV-leading action, one may keep some hope
that the results for the C 2 counterterms will be finite
at the end, but maybe still discontinuous, despite the
non-renormalizability of the model with R□R action
(plus possible lower derivative additions to regularize
it as it was mentioned above). Maybe in this re-
duced models the results of the projection procedure
of the UV-divergent functional of the effective action
of the theory onto the sector with only C 2 terms will
result here in giving sense to pure C 2 divergences in
this limiting model. (Here we may try to resort to
some projection procedure for the functional with UV
divergences since in these non-renormalizable models,
one may expect to find more divergences than just of
the form of C 2 and R2 as presented initially in (1).
There could exist new UV divergences, which contain
even more than four derivatives on the background
metric tensor, even in d = 4 case.) But the final finite
value may be discontinuous and may not be obtain-
able by the naive limit x → 0 of the expression for the
divergent term in the C 2 sector of UV divergences, so
it may not be just B0 there. This remark about pos-
sible discontinuities may apply also to coefficients in
front of divergent terms of the type R2 and GB. They
may still end up with some finite definite values for
this model, but probably they are not the same as the
coefficients B0 of these terms from (51), so we proba-
bly will be able to see here another discontinuities in
taking the naive limit x → 0.
These above results about discontinuities and
negative consequences due to the overall non-
renormalizability of the two considered above reduced
models, are also enforced by the analysis of power
counting of UV divergences. One can try to perform
the “worst case scenario” analysis of one-loop inte-
grals and the results show complete lack of control
over perturbative UV divergences in such reduced
models. This is even worse that in the case of off-
shell E-H gravity considered in d = 4 dimensions,
which is known to be one-loop off-shell perturbatively
non-renormalizable theory. In the latter case the su-
perficial divergence of the divergence ∆ is bounded at
the one-loop level (L = 1) in formula (4) by the value
4. In general, at the arbitrary loop order we have the
formula for power counting reads then
∆ + d∂ = 4 + 2(L − 1), (56)
which if we concentrate on logarithmic UV divergences
only (with ∆ = 0), we get that at the one-loop level
for all Green functions we need counterterms with
up to d∂ = 4 partial derivatives on the metric tensor.
At the two-loop level we instead need to absorb the
divergence term with d∂ = 6 partial derivatives as this
was famously derived by Goroff and Sagnotti [17, 18].
The counterterm that they have found was of the form
of the C 3 GR-covariant term and its perturbative
coefficient at the two-loop order does not vanish, and
this implies that the whole UV-divergent term does
not vanish even in the on-shell situation. But still we
know which counterterms to expect at the given loop
order and the absorption of UV divergences works for
all divergent Green functions of the QG model.
The situation in the reduced models of the type
C□C or R□R in the leading UV terms is much worse
even at the one-loop level from naive power counting
there. One sees that different GR-covariant countert-
erms are needed to absorb divergences in different
divergent Green functions of the quantum model at
the one-loop level, so the counting does not stop at the
two-point function level. We think that despite these
tremendous difficulties, one still can compute the di-
vergent parts of the effective action and the actual
computations are very tedious and still possible. This
is provided that one can invert the propagator, so one
has some non-vanishing parts in both gauge-invariant
parts of it with the spin-0 and spin-2 projectors. So,
145
Lesław Rachwał Acta Polytechnica
it is at present practically impossible to do compu-
tation in the pure models C□C or R□R only. We
know that they give contributions in momentum space
proportional to k6 in the spin-2 and spin-0 parts of
the propagator respectively, while other parts are not
touched. In order to regularize the theory and to
give sense to the perturbative propagator around flat
spacetime, one has to add the regulator terms as this
was mentioned above. Let us assume that they give
contributions to the other sector of the spin projectors
in the graviton’s propagator of the form k−m, where
m is some integer and m < 6, they may likely also
give additional subleading contributions to the main
respective part of the propagator which was there with
six derivatives in the UV regime correspondingly to
the spin-2 sector in C□C theory and to the spin-0 sec-
tor in R□R model. The values of m are respectively:
m = 0 for the cosmological constant addition (it still
regularizes the propagator but very, very weakly),
m = 2 when E-H term is added (it contributes both
to the spin-0 and spin-2 parts), m = 4 when either R2
or C 2 terms are added (they contribute exclusively in
their respective sectors).
Since now after the regularization of the graviton’s
propagator, its behaviour is still very unbalanced in
the UV regime between different components, then
one sees the following results of the analysis of UV
divergences at the one-loop order. First, the general
gravitational vertex is still with six derivatives, while
the propagator is k−6 in the best (most suppressed) be-
haviour and k−m is the worst behaviour in the other
components. For the most dangerous situation we
have to assume that the overall behaviour of the prop-
agator is in the worst case, so with the UV scaling of
the form k−m. Then the relation between the number
of derivatives in a general gravitational vertex and in
the propagator is broken and this is a reason for very
bad behaviour with UV divergences here. Such rela-
tion is typically present even in non-renormalizable
models, like in E-H gravity. The lack of this relation
means that now the result for d∂ of any Feynman
graph G depends on the number of external graviton
lines ng emanating from the one-loop diagram. Pre-
viously in the analysis of power counting there was
never any dependence on this ng parameter. This is
a source for problems even bigger when one increases
ng . For definiteness we can assume that ng > 1 since
here we will not be interested in vacuum or tadpole
diagrams and quantum corrections to them. Now,
for a general diagram G with ng external graviton
lines, the worst situation from the point of view of
UV divergences is for the following topology of the
diagram, namely there is one loop of gravitons (so
called “ring of gravitons”) in the middle with ng 3-leg
vertices joined by ng propagators. In the case when
we concentrate on logarithmic divergences D = 0, we
get the following results for the quantity d∂ which
tells us how many derivatives we must have in the
corresponding counterterm to absorb the divergence,
d∂ = 4 + ng (6 − m) (57)
for the graph contributing one-loop quantum correc-
tions to the ng -point Green function. One sees that
this d∂ grows without a bound even at the one-loop
level, when ng grows, so in principle to renormalize the
theory at the one-loop level one would need already
infinitely many GR-covariant terms, if one does not
bound the number of external legs of Green functions
that must be considered here. A few words about
explanation of numbers appearing in the formula (57).
The 4 there is the number of spacetime dimensions
(integration over all momenta components at the one-
loop level), while the (6 − m) factor comes from the
difference between the highest number of derivatives
in the vertex, i.e. 6 compensated by the worst be-
haviour in some propagator components given in the
UV by k−m only.
Moreover, there are precisely ng segments of the
structure propagator joined with 3-leg vertex to create
a big loop. This behaviour signals complete lack of
control over perturbative one-loop divergences even
at the one-loop level. Moreover, they have to be
absorbed in the schematic terms of the type
∇4+4ng −ng m+2iRng −i, (58)
for the index i running over integer values in the range
i = 0, 1, 2, . . . ng − 2, where we only mentioned the
total number of covariant derivatives not specifying
how they act on these general gravitational curvatures.
This is because this is an expression for the quantum
dressed one-loop Green function with ng graviton legs
on the flat spacetime, so terms with more curvatures
than ng will not contribute to absorb these divergence
of flat Green ng -point function. We only mentioned
here the really the worst situation, where the diver-
gence may be finally absorbable not only by the high-
est curvature terms of the type ∇4+4ng −ng mRng but
also for terms with smaller number of curvatures (up
to R2 terms and in the precise form R□
1
2 ng (6−m)R).
We neglect writing counterterms here which are total
derivatives and which are of the cosmological constant
type. These are then the needed counterterms off-shell
at the one-loop level in such general reduced model.
To make it more concrete, we will analyze the cases
of m = 0, 2 and 4 with special attention here in these
badly non-renormalizable models for some small num-
bers of legs of quantum dressed Green functions. We
have that at m = 0 to absorb UV divergences from
the 2-point function we need generic counterterms of
the form: R□j R with the exponent j running over
values j = 0, 1, 2, 3, 4, 5, 6, while to renormalize a three-
point function one needs previous terms and possibly
new terms of the type ∇j R3 with j = 0, . . . , 16 and for
four-point functions new terms are of the type ∇j R4
with j = 0, . . . , 20, etc. for higher Green functions (for
ng -leg correlators one needs j up to jmax = 4ng + 4).
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When we regularize by adding the E-H term with
m = 2 the situation is slightly better, but then to
absorb UV divergences from the 2-point function we
need generic counterterms of the form: R□j R with
the exponent j running over values j = 0, 1, 2, 3, 4,
while to renormalize a three-point function one needs
previous terms and possibly new terms of the type
∇j R3 with j = 0, . . . , 10 and for four-point functions
new terms are of the type ∇j R4 with j = 0, . . . , 12,
etc. for higher Green functions (for ng -leg correlators
one needs j up to jmax = 2ng +4). Finally, we can add
terms of the type R2 and C 2 for the regularization pur-
poses. In such final case to be considered here one has
that to absorb UV divergences from the 2-point func-
tion we need generic counterterms of the form: R□j R
with the exponent j running over values j = 0, 1, 2,
while to renormalize a three-point function one needs
previous terms and possibly new terms of the type
∇j R3 with j = 0, . . . , 4 and for four-point functions
new terms are of the type ∇j R4 with j = 0, . . . , 4,
etc. for higher Green functions (for ng -leg correlators
one needs j up to jmax = 4 here independently on the
number of legs ng ). Still in this last case one sees that
one needs infinitely many counterterms to renormalize
the theory at the one-loop level, although the index j
of added covariant derivative is bounded by the values
4, still one needs more terms with more powers of
gravitational curvatures R.
This shows how badly non-renormalizable are these
models already at the one-loop level and that any
control over perturbative UV divergence is likely lost,
when the number of external legs is not bounded
here from above. These reduced models are examples
of theories when the dimensionality and the num-
ber of derivatives one can extract from the vertices
and propagators of the theory differ very much. Pre-
viously in quantum gravity models these two num-
bers were identical which leads to good properties of
control over perturbative divergences (renormalizabil-
ity, super-renormalizability and even UV-finiteness).
With these reduced models we are on the other bad
extreme of vast possibilities of QG models. But see-
ing them explicitly proves to us how precious is the
renormalizability property and why we strongly need
them in HD models of QG, in particular how we need
super-renormalizability in six-derivative QG models.
The arguments above convince us to think that
there is no hope to get convergent results for the front
coefficient coming with the C 2 counterterm in the
UV-divergent part of the effective action in the consid-
ered here model with the only presence of the C□C
as the leading one in the UV. This may signify that
here there exists another UV-divergent term (perhaps
of the structure like C□nC), which contains more
derivatives, and this could be a reason why the coeffi-
cient in front of the C 2 term is itself a divergent one.
The presence of such new needed counterterms with
more derivatives can be motivated by the analysis of
power counting of UV divergences in this reduced un-
balanced model, which is also presented below. Even
if the higher C□nC type of UV divergence is prop-
erly extracted and taken care of, then we can still be
unable to properly define and see as convergent the di-
vergence proportional to the four-derivative term C 2.
Even such a projection of the UV-divergent functional
of the theory onto the sector with only C 2 terms will
not help here in giving sense to pure C 2 divergences
in this limiting model. However, this remark does not
need necessarily to apply to coefficients in front of
divergent terms of the type R2 and GB. They may
still end up with some finite definite values for this
model, but probably they are not the same as the
coefficients B0 of these terms from (51), so we could
be able to see here another discontinuity in taking the
limit x → +∞.
On the other hand, for the strictly non-
renormalizable theory with the leading in the UV
term R□R we can have additions of similar various
subleading terms which do not change the fact of non-
renormalizability. We can add terms (separately or in
conjunction) of the following types: ωΛ, R, R2, or C 2.
The UV-leading part of the Hessian still is not well-
defined as it should contain six-derivative differential
operator understood on flat spacetime and between
tensorial fluctuations, while from the term R□R we
get only operator between traces of metric fluctua-
tions h = ηµν hµν (between spin-0 parts), so derived
from the terms quadratic in curvatures present in the
UV regime. Probably the degeneracy of the Hessian
operator can be easily lifted out, if we add one of the
ωΛ, or R or C 2 terms. If only the R2 term is added,
then the Hessian still is degenerate. Similarly, the flat
spacetime propagator can be defined only if we make
addition of ωΛ, or R or C 2 terms – this is because of
the problematic part of it proportional to the projector
P (2) which must for the consistency of the inverting
procedure for the whole propagator be non-zero. This
tensorial part (spin-2 part) is sourced exclusively from
any GR-invariant term built out with Weyl tensors
in adopted here Weyl basis of terms or from the E-H
term or from the cosmological constant term. If only
the R2 term is added, then the propagator still is
ill-defined. Still these additions do not change the
fact that the theory is formally non-renormalizable, if
there is not an accompanying six-derivative term of
the form C□C.
As for the final results for UV divergences in these
extended models, naively one would think that there
are no new UV divergences proportional to terms with
four derivatives of the metric (namely to terms R2,
C 2 and GB), because of the limit ωC → 0 and the de-
pendence on the x ratio in (33) in the linear way. We
would naively think that divergences with R2 and GB
terms are the same as in (33). The only problematic
one could be this proportional to the C 2 term since
the limit gives already constant result, namely the B0
coefficient. Since the model is non-renormalizable we
cannot trust this computation and these limits at the
147
Lesław Rachwał Acta Polytechnica
end, even if they give here convergent results. But
this probably means that we cannot sensibly define
the C 2 counterterm needed for the renormalization
of these theories. When we include additions to the
action, which remove the degeneracy of the flat space-
time graviton’s propagator, then at least the pertur-
bative computation using Feynman diagrams can be
attempted in such a theory. Although of course, in
this case the different parts of the propagator have
different UV scalings, so the situation for one-loop
integrals is a bit unbalanced and there is not a stable
control over perturbative UV divergences, when for
example one goes to the higher loop orders. Proba-
bly new counterterms (with even higher number of
derivatives) will be at need here. Making additions
of some subleading terms from the point of view of
the UV regime, may help in defining the unbalanced
perturbative propagator, but still one expects (due
to energy dimension considerations) that these addi-
tions do not at all influence the quantitative form of
UV divergences with four-derivative terms, so these
ones which are leading in the UV. These additions
are needed here only quantitatively and on the formal
level to let the computation being done for example
using Feynman diagrams with some mathematically
existing expressions for the graviton’s propagator. In
a sense adding ωΛ or R, or C 2 terms regularizes the
theory R□R + R2 or even the simplest one, just R□R,
so we can perhaps trust there in the final results just
given by all three B0 coefficients of UV divergences
proportional to terms R2, C 2 and GB. This can be
motivated by the observation that here the limits
ωC → 0 or θC → 0 respectively do not enhance any
symmetry of the model in question, so they can be
naively and safely taken. But we agree that this case
requires a special detailed and careful computation to
prove this conjectural behaviours.
In the general case of badly non-renormalizable the-
ories, with both ωC = 0 or ωR = 0, one trusts more the
computation using Feynman diagrams and around flat
spacetime than of the fully GR-covariant BV method
of computation. For the former one only needs to be
able to define properly the propagator – all physical
sectors of it, and for this purpose one can regularize
the theory by adding the term ωκR, which is a dy-
namical one with the smallest number of derivatives
and for which flat spacetime is an on-shell vacuum
background. (In this way, we exclude adding the
cosmological constant term ωΛ, which would require
adding some source and the flat spacetime propagator
could not be considered anymore in vacuum there.)
Then in such regulated non-renormalizable theory one
can get results around flat spacetime and in Fourier
momentum space, and then at the end one can take
the limit ωκ → 0. The results for some UV divergences
in these non-renormalizable models must be viewed as
projected since higher-derivative (like 6-derivative and
even higher) infinities may be present as well. These
last results must coincide with the ones we obtained
in (33), when the proper limits of ωC → 0 or ωR → 0
are taken. We notice that adding the E-H term, which
is always a good regulator, changes the dynamics very
insignificantly for these higher-derivative models and
the results from Feynman diagram computations can
be always derived. Instead for the leading in the UV
regime part of the Hessian operator, which is a crucial
element for the BV method of computation, addition
of just ωκR does not help too much and the operator
is still degenerate since it is required that all its sec-
tors are with six-derivative differential operators: are
non-vanishing and non-degenerate there.
We propose the following procedure for the deriva-
tion of correct limiting cases analyzed here. First
the theory with ωC ̸= 0, ωR ̸= 0 and ωκ ̸= 0 is
analyzed using Feynman diagram approach. The re-
sults for UV divergences must be identical to the ones
found in (33) using the BV technique. They do not
show any singularity at this moment. In Feynman
diagram computation one can take the limit ωC →
or ωR → 0, while the propagator and perturbative
vertices are still well-defined. In these circumstances
we have that still ωκ ̸= 0. We admit that the the-
ory loses now its renormalizability properties, but we
just want to project the UV-divergent action onto the
terms with the structure of three GR-covariant terms
C 2, R2 and GB. For this the method of Feynman
graph computation is still suitable since it only re-
quires the well-defined propagator, but it can work
even in badly non-renormalizable theories. This is
like taking the naive limits of ωC → or ωR → 0 in the
results from (33) respectively, regardless which way
they were obtained. One justifies the step of taking
these limits by recalling that we are still working in
the Feynman diagram approach, and not with the BV
technique. Finally, one sends ωκ → 0 hoping that this
does not produce any finite discontinuity in the results
for four-derivative UV divergences. This is justified by
dimensional analysis arguments provided also earlier
in this article and by the fact that except in the case
of conformal gravity theory in d = 4, there is no any
enhancement of local symmetries in the limit ωκ → 0.
Then one gets the sense for the limits considered in
these sections.
This analysis concludes the part with special lim-
iting cases of extended six-derivative theories, where
one of the coefficients ωR or ωC is to be set to zero.
Probably the same considerations of some limits can
be repeated very similarly (with the exception of the
conformal gravity case) for the Stelle quadratic gravity
models, but we will skip this analysis here since it can
be found in the literature.
5. Stability of HD theories
Above we have seen that HD theories of gravitation
are inevitable due to quantum considerations. They
also come with a lot of benefits that we have discussed
at length before like super-renormalizability and the
possibility for UV-finiteness. However, it is also well
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vol. 62 no. 1/2022 Beta Functions in Six-Derivative Quantum Gravity
known that they have their own drawbacks and prob-
lems. One of the most crucial one is the issue of
unitarity of the scattering (S-matrix) in perturbative
framework. This is of course in the situation when we
can discuss the scattering problems, so when we can
define asymptotic states in interacting gravitational
background, so when the gravitational spacetime is
asymptotically flat. In more generality, the related
issue is of quantum stability of the theory.
In general, in literature about general HD theories
there exist various proposals for solutions of these
perennial problems. We can mention here a few of
them: P T -symmetric quantum theory, Anselmi-Piva
fakeon prescription, non-local HD theories, benign
ghosts as proposed by Smilga, etc. Below we will
try to describe some of their methods and show that
the problems with unitarity or with the stability of
the quantum theory can be successfully solved. We
also provide arguments thanks to Mannheim [57, 58]
that the gravitational coupled theory must be without
problems of this type, if the original matter theory
was completely consistent.
We first express the view that the stability of the
quantum theory is fundamental, while the classical
theory may emerge from it only in some properly de-
fined limits. Hence we should care more about the
full even non-linear stability on the quantum level and
some instabilities on the classical level may be just
artifacts of using classical theory which cannot be de-
fined by itself without any reference to the original
fundamental quantum theory. An attempt to under-
stand the stability entirely in classical terms may be
doomed to clearly fail since forgetting about the quan-
tum origin may be here detrimental for the limiting
process. If the quantum theory is stable and unitarity
is preserved, then this is the only thing we should
require since the world is in its nature quantum and
physically we know that it is true that ℏ = 1 in proper
units, rather than ℏ → 0, so the classical limit may
be only some kind of illusion. If there are problems
with classical stability analysis like this done origi-
nally by Ostrogradsky, then this may only mean that
the classical theory obtained this way neglects some
important features that were relevant on the quantum
level for the full quantum stability of the system.
First, in Anselmi-Piva prescription one solves com-
pletely the unitarity issue for HD theories by invoking
fakeon prescription to take properly into account the
contribution of particles which in the spectrum are
related to higher derivatives theories and which typ-
ically are considered as dangerous for the unitarity
of the theory. The presence of a particle with nega-
tive residue called a ghost at the classical level makes
the theory not unitary in its original quantization
based on the standard Feynman prescription [7] of
encircling the poles for the loop integrals. A new quan-
tum prescription, as recently introduced by Anselmi
and Piva [59–61] was based on the earlier works by
Cutkosky, Landshoff, Olive, and Polkinghorne [62].
The former authors invented a procedure for the Lee-
Wick theories [63, 64], which allow them to tame the
effects typically associated to the presence of ghosts in
the Stelle’s theory. In this picture, the ghost problem
(also known as unitarity problem) is solved conse-
quently at any perturbative order in the loop expan-
sion [61] done for the loop integrals which need to be
computed in any QFT, if one requires higher order
accuracy.
At the classical level, the ghost particle (or what
Anselmi and Piva define as “fakeon”, because this par-
ticle understood as a quantum state can only appear
as a virtual particle and inside perturbative loops) is
removed from the perturbative spectrum of the the-
ory. This is done by solving the classical equations of
motion for the fakeon field by the mean of a very spe-
cific combination of advanced plus retarded Green’s
functions and by fixing to zero the homogeneous so-
lution of resulting field equations [65, 66]. This is
then equivalent to removing the complex ghosts in
the quantum theory from the spectrum of asymptotic
quantum states by hand. However, this choice and
this removal decision is fully preserved and protected
by quantum corrections, hence it does not invalidate
the unitarity of the S-matrix at higher loop orders.
Such prescription of how to treat virtual particles
arising due to HD nature of the theories is very general
and can be applied to both real and complex ghosts,
and also to normal particles, if one wishes to. (Every
particle can be made fake, so without observable ef-
fects on the unitarity of the theory.) In particular, this
prescription is crucial in order to make perturbatively
unitary the theory proposed by Modesto and Shapiro
in [67, 68] which comes under the name of “Lee-Wick
quantum gravity”. The latter class of theories is based
on the general gravitational higher-derivative actions
as proposed by Asorey, Lopez, and Shapiro [12]. In
this range of theories, we can safely state to have
a class of super-renormalizable or UV-finite and uni-
tary higher-derivative theories of QG. In order to
guarantee tree-level unitarity, the theory in [67, 68]
has been constructed in such a way that it shows
up only complex conjugate poles in the graviton’s
propagator, besides the standard spin-2 pole typically
associated with the normal massless graviton particle
with two polarizations. Afterwards, the new prescrip-
tion by Anselmi and Piva [61] guarantees the unitarity
of this theory at any perturbative order in the loop
expansion.
We also emphasize that the Stelle’s quadratic theory
in gravitational curvatures [7] with the Anselmi-Piva
prescription is the only strictly renormalizable theory
of gravity in d = 4 spacetime dimensions, while the
theories proposed in [67, 68] are from a large (in
principle infinite) class of super-renormalizable or UV-
finite models for quantum gravity.
Next, in the other approach pioneered by Bender
and Mannheim to higher-derivative theories and to
non-symmetric and non-Hermitian quantum mechan-
149
Lesław Rachwał Acta Polytechnica
ics [69, 70], one exploits the power of non- Hermi-
tian P T -symmetric quantum gravity. Here, the basic
idea is that the gravitational Hamiltonian in such
theories (if it can be well-defined), is not a Hermi-
tian operator on the properly defined Hilbert space
of quantum states, rather it is only P T -symmetric
Hamiltonian. Then some eigenstates of such a Hamil-
tonian may correspond to non-stationary solutions of
the original classical wave equations. They would in-
deed correspond in the standard classical treatment to
the Ostrogradsky instabilities. The famous example
are cosmological run-away solutions or asymptotically
non-flat gravitational potentials for the black hole so-
lutions. The problem of ghosts manifests itself already
on the classical level of equations of motion, where one
studies the linear perturbations and its evolution in
time. For unstable theories, the perturbations growth
is without a bound in time. But in some special solu-
tions, like for example present in models of conformal
gravity, these instabilities are clearly avoided and then
one can speak that ghosts are benign in opposition to
them being malign in destroying the unitarity of the
theory. Such benign ghosts [71, 72] are then innocent
for the issues of perturbative stability.
In the P T -symmetric approach to HD theories at
the beginning, one cannot determine the Hilbert space
by looking at the c-number propagators of quantum
fields. In this case, one has to from the start quan-
tize the theory and construct from the scratch the
Hilbert space, which is different than the original naive
construction based on the extension of the one used
normally for example for two-derivative QFT’s. With
this new Hilbert space and with the non-Hermitian
(but P T -symmetric) Hamiltonian the theory revealed
to be quantum-mechanically stable. This is dictated
by the construction of the new Hilbert space and the
structure of the Hamiltonian operator. In that case
the procedure of taking the classical limit, results
in the definition of the theory in one of the Stokes
wedges and in such a region the Hamiltonian is not
real-definite and the corresponding classical Hamil-
tonian is not a Hermitian operator. Therefore, the
whole discussion of Ostrogradsky analysis is correct as
far as the theory with real functions and real-valued
Hamiltonians is concerned, but it is not correct for
the theory which corresponds to the quantum the-
ory which was earlier proven to be stable quantum-
mechanically. The whole issue is transmitted and now
there is not any problem with unitarity or classical
stability of the theory, but one has to be very careful
in attempts to define the classical limiting theory.
We also repeat here arguments proposed by
Mannheim about stability of the resulting gravitation-
matter coupled theory [57, 58]. First we take some
matter two-derivative model (like for example stan-
dard model of particle physics, where we have various
scalars, fermions and spin-1 gauge bosons). This the-
ory as considered on flat Minkowski background is
well known to be unitary so it gives S-matrix of inter-
actions with these properties. The model can be said
that it is also stable on the quantum level. Now, we
want to couple it to gravity, or in other words put it on
gravitational spacetime with non-trivial background
in such a way that the mutual interactions between
gravitational sector and matter sector are consistent.
This, in particular, implies that the phenomena of
back-reaction of matter species on geometry are not
to be neglected. The crucial assumption here is that
this procedure of coupling to gravity is well behaved
and for example, it will not destroy the unitarity prop-
erties present in the matter sector. We know that
the theory in the matter sector is stable and also
its coupling to geometry should be stable on the full
quantum level. After all, this is just simple coupling
procedure (could be minimal coupling) to provide
mutual consistent interactions with the background
configurations of the gravitational field.
Next, on the quantum level described, for example,
by functional path integral, we can decide to com-
pletely integrate out matter species still staying on
the general gravitational background. As emphasized
in section 1, such procedure in d = 4 spacetime dimen-
sions generate effective quantum gravitational dynam-
ics of background fields with higher derivatives, pre-
cisely in this case there are terms of the type C 2 and
R2 (the latter term is absent when the matter theory
is classically conformally invariant). In other words,
the resulting functional of the quantum partition func-
tion of the total coupled model is a functional of only
background gravitational fields. This last reduced or
“effective” functional is given by the functional integral
over quantum fluctuations of gravitational field of the
theory given classically by the action with these HD
above terms. Let us recall now what we have done,
namely we have simply integrated out all quantum
matter fields, which is an identity transformation for
the functional integral representation of the partition
function Z of the quantum coupled theory. Since
this transformation does not change anything, then
also the resulting theory of gravitational background
must necessarily possess the same features as the orig-
inal coupled theory we started with. Since the first
theory was unitary, then also the last one theory of
pure gravity but with higher-derivative terms must be
unitary too. We emphasized that both theories give
the same numerical values of the partition function Z
understood here as the functional of the background
spacetime metric. In the first theory the integration
variables under functional integrals are quantum mat-
ter fields, while in the second case we are dealing with
pure gravity so we need to integrate over quantum
fluctuations of the gravitational fields. In the last case
the model, which gives the integrand of the functional
integral is given by the classical action SHD, so it con-
tains necessarily higher derivatives of the gravitational
metric field.
There also exist possibilities that ghosts or classical
instabilities one sees on the classical level thanks to
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Ostrogradsky analysis disappear. This may happen if
for example, some very specific (or fine-tuned) initial
or boundary conditions are used for solving non-linear
higher-derivative classical equations of motion of the
theory. It is not excluded as proven by Smilga that
some instabilities may go away if one analyzes such
special situations.
Various cures have been proposed in the literature
for dealing with the ghosts-tachyon issue: Lee-Wick
prescription [63, 64], fakeons [61, 65, 66, 73], non-
perturbative numerical methods [71, 72, 74–78], ghost
instabilities [79–81], non- Hermitian P T -symmetric
quantum gravity based on P T -symmetric quantum
mechanics [69, 70], etc (see also [82–87]). One might
even entertain the idea that unitarity in quantum
gravity is not a fundamental concept. So far, there
is no a consensus in the community which solutions is
the correct one. The unfortunate prevalent viewpoint
is that none of the proposed solutions solves conclu-
sively and completely the problem. And it seems that
sadly the solutions proposed in the literature are not
compatible and are unrelated to each other.
Therefore all the arguments given above should con-
vince the reader that the HD (gravitational) theories
are stable on the full quantum level. In particular,
this means that for situations in which we can define
asymptotic states (like for asymptotically flat space-
times) the scattering matrix between fluctuations of
the gravitational field is unitary on the quantum level
and both perturbatively and non-perturbatively.
6. Conclusions
In this contribution, we have discussed the HD grav-
itational theories, in particular six-derivative gravi-
tational theories. First, we motivated them by em-
phasizing their various advantages as for the models
of consistent Quantum Gravities. We showed that
six-derivative theories are even better behaved on
the quantum level than just four-derivative theories,
although the latter ones are very useful regarding
scale- and conformal invariance of gravitational mod-
els. Moreover, the models with four-derivative ac-
tions serve as good starting examples of HD theo-
ries and they are reference points for consideration
of six- and higher order gravitational theories. We
first tried to explain the dependence of the beta func-
tions in six-derivative theories by drawing analogies
exactly to these prototype theories of Stelle gravity.
We also emphasize that only in six-derivative grav-
itational models we have the very nice features of
super-renormalizability and the narrow but still vi-
able option for complete UV-finiteness. This is why
we think super-renormalizable six-derivative theories
have better control over perturbative UV divergences
and give us a good model of QG, where this last issue
with perturbative divergences is finally fully under our
control and theoretical understanding.
In the main part of this paper, we analyzed the struc-
ture of perturbative one-loop beta functions in six-
derivative gravity for couplings in front of terms con-
taining precisely four-derivative in the UV-divergent
part of the effective actions. These terms can be con-
sidered as scale-invariant term since couplings in front
of them are all dimensionless in d = 4 spacetime di-
mensions. Our calculation for these divergences was
done originally in the Euclidean signature using the
so-called Barvinsky-Vilkovisky trace technology. How-
ever, the results are the same also in the Minkowskian
signature independently which prescription one uses
to rotate back to the physical relativistic Lorentz sig-
nature case, whether this is standard Wick rotation, or
the one using Anselmi-Piva prescription using fakeons.
This is because they are the leading divergences in the
UV regime, and hence they do not completely depend
how the rotation procedure is done from Euclidean to
Minkowskian and how for example the contributions
of arcs on the complex plane is taken into account
since the last ones give subleading contributions to
the UV-divergent integrals. Moreover, the calcula-
tions of beta functions that we presented in this paper
has very nice and important features of being renor-
malization scheme-independent since they are done
at the one-loop, but the expressions we get for them
are valid universally. These are exact beta functions
since they do not receive any perturbative correc-
tions at the higher loop orders since the six-derivative
gravitational theory is super-renormalizable in d = 4.
Another part of good properties of the beta functions
obtained here are the complete gauge independence
and also independence on the gauge-fixing parame-
ters one can use in the definition of the gauge-fixing
functional. These last two properties are very impor-
tant since in general gravitational theory we have the
access to perturbative computation only after intro-
ducing some spurious element to the formalism which
are related to gauge freedoms (in this case these are
diffeomorphism symmetries). We modify the original
theory (from the canonical formalism) by adding vari-
ous additional fields and various spurious nonphysical
(gauge) polarizations of mediating gauge bosons (in
our case of gravitons) in order also to preserve rela-
tivistic invariances. These are redundancies that have
to be eliminated when at the end one wants to com-
pute some physical observables. Therefore, it is very
reassuring that our final results are completely insen-
sitive to these gauge-driven modifications of original
theories.
Our beta functions being exact and with a lot of
nice other properties, constitute one significant part
of the accessible observables in the QG model with
six-derivative actions. Their computation is a nice
theoretical exercise, which of course from the sense of
algebraic and analytic methods used in mathematical
physics has its own sake of interest. However, as
we emphasized above these final results for the beta
functions may have also meaning as true physical
observables in the model of six-derivative QG theories.
151
Lesław Rachwał Acta Polytechnica
We described in greater detail the analysis of the
structure of beta functions in this model. First we
used arguments of energy dimensionality and the de-
pendence of couplings on the dimensionless funda-
mental ratio of the theory x. Next, we tried to draw
a comparison between the structure of 4-derivative
gravitational Stelle theory and six-derivative theory in
d = 4 dimensions. We showed the dependence on the
parameters x is quite opposite in two cases. The case
with four-derivative theory is exceptional because the
model without any R term in the action (and also
without the cosmological constant term) enjoys en-
hanced symmetry and then the quantum conformal
gravity is renormalizable at the one-loop order, so
then it is a special case of a sensible quantum physical
theory (up to conformal anomaly problems discussed
earlier). We also remark that in the cases of x → 0
and x → ∞ the generic six-derivative theories are
badly non-renormalizable. This was the source of the
problem with attempts to obtain sensible answers in
these two limits. Non-renormalizability problem must
show in some place in the middle or at the end of the
computation to warn us that at the end we cannot
trust in the final results for the beta functions in these
cases. In these two cases this problem showed indeed
in two different places and the logical consequences
of this were strongly constraining the possible form
of the rational x-dependence of these results. Thanks
to these considerations we were able finally to un-
derstand whether the positive or inverse powers of
the ratio x must appear in the final results for beta
functions in question. Of course, we admit that this
analysis is a posteriori since we first derived the results
for the divergences and only later tried to understand
the reasons behind these results. But eventually we
were able to find a satisfactory explanation.
And there are a few of additional spin-offs of the pre-
sented argumentation. First, we can make predictions
about the structure of beta functions in 8-derivative
and also of other higher-derivative gravitational theo-
ries with the number of derivatives in the action which
is bigger than 4 and 6 (analyzed in this paper). We
conjecture that the structure should be very similar to
what we have seen already in the generic case with six
derivatives, so only positive powers of the correspond-
ing fundamental ratio x of the theory, and probably
only in the sector with C 2 type of UV divergences.
Another good side effect is that we provide first (to
our knowledge) theoretical explanation of the struc-
ture of beta functions as seen in four-derivative case
of Stelle theory in d = 4 spacetime dimensions. It
is not only that the theory with C 2 action is excep-
tional in d = 4 dimensions; we also “explained” these
differences based on an extension of the theory to
include higher-derivative terms like with 6-derivative
and quantify to which level the theory with C 2 action
is special and how this reflects on the structure of
its one-loop beta functions. We remark that in Stelle
gravity (or even in its subcase model with conformal
symmetry based on the C 2 action), there are con-
tributions to beta functions originating from higher
perturbative loops since the super-renormalizability
argument based on power counting analysis does not
apply here. Our partial explanation of the structure
of the one-loop beta functions in Stelle theory in d = 4
uses a general philosophy that to “explain” some nu-
merical results in theoretical physics, one perhaps
has to generalize the original setup and in this new
extended framework looks for simplifying principles,
which by reduction to some special cases show explic-
itly how special are these cases not only qualitatively
but also quantitatively and what this reduction pro-
cedure implies on the numbers one gets as the results
of the reduction. For example, one typically extend
the original framework from d = 4 spacetime dimen-
sion fixed condition to more general situation with
arbitrary d and then draw the general conclusion as
a function of d based on some general simple prin-
ciples. Then finally, the case of d = 4 is recovered
as a particular value one gets when the function is
evaluated for d = 4. And this should explain its spe-
ciality. In our case, we extended the four-derivative
theory by adding terms with six derivatives and in
this way we were able to study a more generic situ-
ation. This was in order to understand and explain
the structure of divergences in the special reduced
case of conformal gravity in d = 4 and of still generic
four-dimensional Stelle theory. We think that this
is a good theoretical explanation which sheds some
light on the so far mysterious issue of the structure
of beta functions. One can also see this as another
advantage of why it is worth to study generalizations
of higher-derivative gravitational actions to include
terms with even more higher number of derivatives,
like 6-derivative, 8-derivative actions, etc.
Finally, here we can comment on the issue of exper-
imental bounds on the values of the ratio x. Since it
appears in six-derivative gravitational theory the con-
straints on its possible values are very weak. Slightly
stronger constraints apply now for the corresponding
value of the ratio in four-derivative Stelle gravitational
theory in d = 4 case. Since the main reason for higher-
derivative modifications of gravity comes because of
consistency of the coupled quantum theory, then one
would expect that the stringent bounds would come
from experimental measurement in the real domain
of true quantum gravity. Of course, right now this is
very, very far, if possible at all, future for experimental
gravitational physics. This is all due to smallness of
gravitational couplings characterized by GN propor-
tional inversely to the Planck mass MP ∼ 1019 GeV.
And in the quantum domain of elementary particle
physics this scale is bigger than any energy scale of in-
teractions between elementary quanta of matter. This
implies that also quantum gravitational interactions
are very weak in strength. Hence the only experimen-
tal/observational bounds we have on the coefficients in
front of higher-derivative terms come from the classi-
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vol. 62 no. 1/2022 Beta Functions in Six-Derivative Quantum Gravity
cal/astrophysical domain of gravitational physics and
they are still very weak. To probe the values of the
coefficients in front of six-derivative terms, one would
have to really perform a gravitational experiment to
the increased level of accuracy between elementary
particles in the full quantum domain, which is now
completely unfeasible. Hence, we have to be satisfied
with already existing very weak bounds, but this lets
us to freely consider theoretical generic situation with
arbitrary values of the ratio x since maybe only (very
far) future experiments can force us theoreticians to
consider some more restricted subset or interval for
the values of the x ratio as consistent with observed
situation in the Nature. For the moment it is reason-
able to consider and explore theoretically all possible
range of values for the x ratio and also both possi-
ble signs. (Only the case with x = 0 is excluded as
a non-renormalizable theory that we have discussed
before.)
In the last section of this contribution, we com-
mented on the important issue of stability of higher-
derivative theories. We touched both the classical and
quantum levels, while the former should not be under-
stood as a standalone level on which we can initially
(before supposed quantization) define the classical the-
ory of the relativistic gravitational field. We followed
the philosophy that the quantum theory is more fun-
damental and it is a starting point to consider various
limits, if it is properly quantized (in a sense that the
quantum partition function is consistently defined,
regardless of how we get there to its form, no matter
which formal quantization procedure we have been fol-
lowing). One of such possible limit is the classical limit
where the field expectation values are large compared
to characteristic values as found in the microworld of
elementary particles. And also the occupation number
for bosonic states are large number (of the order of
Avogadro number for example). Then we could speak
about coherent states which could define well classical
limit of the theory. Such procedure has to be followed
in order to define HD classical gravitational theory.
We emphasized that quantum theory is the basis and
classical theory is the derived concept, not vice versa.
On the quantum level we shortly discussed various ap-
proaches present in the literature to solve the problems
with unwanted ghost-like particle states. They were
classified in two groups: theories with P T -symmetric
Hamiltonian and theories with Anselmi-Piva prescrip-
tion instead of the Feynman prescription to take into
account contributions of the poles of the ghost but
without spoiling the unitarity issue. On the quantum
level, we considered mainly the issue with unitarity
of the scattering matrix since this seems the most
problematic one. The violation of unitarity would
signal the problem with conservation of the probabil-
ity of quantum processes. Something that we cannot
allow to happen in quantum-mechanical framework
for the isolated quantum system (non-interacting with
the noisy decohering and dissipative or some thermal
environment). Of course, such an analysis was tailor-
made for the cases of gravitational backgrounds on
which we can define properly the scattering process.
In general, the scattering processes are not every-
thing we can talk about in quantum field theories even
for on-shell quantities. The analysis of some on-shell
dressed Green functions may also show some prob-
lems with quantum stability of the system. Therefore,
we briefly also described the results of the stability
analysis, both on the classical and quantum level and
to the various loop accuracy in QG models. This
analysis is in principle applicable to the case of any
gravitational background, more general than the one
coming with the requirement of asymptotic flatness.
We also mentioned that in some cases of classical field
theory the analysis of classical exact solutions shows
that the very special and tuned solutions are without
classical instabilities and they are well-defined for any
time starting with very special initial or boundary
conditions. For example, here we can mention the
case of so-called benign ghosts of higher-derivative
gravitational theories as proposed by Smilga some
time ago. This should prove to the reader that we
are dealing with the theories which besides a very
interesting structure of perturbative beta functions,
are also amenable to solve the stability and unitarity
issues in these theories, both on the quantum as well
as on the classical level. With some special care we
can exert control and HD gravitational theories are
stable quantum-mechanically and this is what matters
fundamentally.
Acknowledgements
We would like to thank I. L. Shapiro, L. Modesto and
A. Pinzul for initial comments and encouragement about
this work. L. R. thanks the Department of Physics of the
Federal University of Juiz de Fora for kind hospitality and
FAPEMIG for a technical support. Finally, we would like
to express our gratitude to the organizers of the “Alge-
braic and analytic methods in physics” – AAMP XVIII
conference for accepting our talk proposal and for creating
a stimulating environment for scientific online discussions.
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Acta Polytechnica 62(1):118–156, 2022
1 Introduction and motivation
1.1 Motivations for and introduction to six-derivative gravitational theories
1.2 Addition of killer operators
1.3 Universality of the results
2 Brief description of the technique for computing UV divergences
2.1 Example of the BV method of computation for the scalar case
2.2 Results in six-derivative gravity
3 Some theoretical checks of the results (33)
4 Structure of beta functions in six-derivative quantum gravity
4.1 Limiting cases
4.2 Dependence of the final results on the fundamental ratio x
4.3 Case of conformal gravity
4.4 More on limiting cases
5 Stability of HD theories
6 Conclusions
Acknowledgements
References