# Generalization of conformal Hamada operators

Lesław Rachwał<EMAIL_ADDRESS>Públio Rwany B. R. do Vale
<EMAIL_ADDRESS>Departamento de Física - Instituto de Ciências Exatas,
Universidade Federal de Juiz de Fora, 33036-900, Juiz de Fora, MG, Brazil

###### Abstract

The six-derivative conformal scalar operator was originally found by Hamada in
its critical dimension of spacetime, $d=6$. We generalize this construction to
arbitrary dimensions $d$ by adding new terms cubic in gravitational curvatures
and by changing its coefficients of expansion in various curvature terms. The
consequences of global scale-invariance and of infinitesimal local conformal
transformations are derived for the form of this generalized operator. The
system of linear equations for coefficients is solved giving explicitly the
conformal Hamada operator in any $d$. Some singularities in construction for
dimensions $d=2$ and $d=4$ are noticed. We also prove a general theorem that a
scalar conformal operator with $n$ derivatives in $d=n-2$ dimensions is
impossible to construct. Finally, we compare our explicit construction with
the one that uses conformal covariant derivatives and conformal curvature
tensors. We present new results for operators built with different orders of
conformal covariant derivatives.

##  I Introduction

The theory of General Relativity (GR) is a successful relativistic theory of
gravitation that has given us a new understanding of space and time, since
then renamed as spacetime, and gravity in this setup can be interpreted as the
curvature of the latter. This general Einstein’s theory showed us that the
spacetime itself may be viewed as having many various unusual characteristics
and also possessing specific sets of symmetries, different from the symmetries
present on the flat space (Euclidean space or Minkowskian spacetime), which
was earlier usually considered as the background arena for all non-
gravitational physics. We are talking here about a collection of aspects and
properties of spacetimes which show that they are more flexible and dynamical
than previously thought of. Therefore, we can say in that theoretical ensemble
that “gravity” is a byproduct of the presence of such a tangible and elastic
spacetime, which by itself is a prominent object in General Relativity.

A fundamental feature of this new understanding of space and time is known as
diffeomorphism invariance, which can be interpreted simply as “an invariance
of the laws of physics between distinct coordinate systems” , which is also
known as covariance of these physical laws with respect to general coordinate
transformations. When the latter ones are assumed to be differentiable
mappings between different coordinatizations of the same manifold, then we
speak about diffeomorphisms. The symmetry with respect to arbitrary
diffeomorphisms (arbitrary changes of coordinate systems and reference frames)
is typically taken as the only symmetry of Einsteinian gravitational theory.
However, this symmetry is not the maximal one of a possible consistent
relativistic gravitational theory. The diffeomorphism symmetry of gravity can
be extended in many ways. One of the directions of such extensions is to
invoke another very important symmetry in GR, that is the conformal symmetry.
This is a symmetry and invariance (of a possible gravitational theory and
coupled to it matter theory) with respect to so called conformal (or Weyl)
transformations on the metric structure on the differential manifold (and
corresponding rescalings of matter fields present additionally there). We will
emphasize about such symmetry below.

The conformal symmetry has a significant role in physics mainly due to its
applications that cover several areas, such as condensed matter, statistical
mechanics, high energy physics, cosmology and quantum field theory. This
theoretical tool was first introduced and studied by Hermann Weyl [1, 2]. The
conformal (Weyl) transformation of the metric tensor

$g_{\alpha\beta}\rightarrow e^{2\Omega\left(x\right)}g_{\alpha\beta}$ (1)

produces another metric with respect to diffeomorphisms but the metric being
within the same conformal class (equivalence class with respect to conformal
rescalings). Above, the conformal parameter is represented by
$\Omega\left(x\right)$. The conformal rescaling of the metric as in (1) in
this form is valid for any spacetime of any dimension $d$. The transformations
in (1) scale the non-conformally invariant infinitesimal line element from GR
according to

$ds^{2}\to e^{2\Omega(x)}ds^{2},$ (2)

while the coordinates $x^{\mu}$ and their differentials $dx^{\mu}$ remain
untouched under such transformations. Among such changes in physical scales to
measure length distances and time intervals, there are not only global
transformations, where the conformal parameter is constant, i.e.,
$\Omega\left(x\right)\rightarrow\Omega$ (so called scale transformations), but
also local Weyl transformations111In this case the parameter $\Omega(x)$ may
depend on both the location in space and in time.. A theory invariant with
respect to this transformation is known as conformally invariant (Weyl
symmetric) or a theory with Weyl invariance, or simply with Weyl symmetry. A
special class of theories, which preserve conformal symmetry on the full
quantum level is given by a set of conformal field theories (CFT’s in short)
[3]. Such CFT’s arose originally in some lower dimensional field theory models
of statistical mechanics [4, 5]. Being the subset of general quantum field
theories (QFT) they come however with their own specific set of methods and
rules owing to the presence of fully realized conformal symmetry, which
strongly constrains the resulting quantum dynamics. Additionally the formalism
of general and abstract CFT’s makes heavy use of mathematical physics in the
form of conformal algebras and conformal operators (both primary, secondary
and also of higher orders) and also of conformal correlation functions. The
construction of primary operators of the CFT algebras as conformally covariant
operators is the central issue of any CFT, being conceptually quite different
from the main problem of a general QFT. The former is the problem that some
aspects of it for higher derivative scalar operators coupled to non-trivial
background geometry we want to discuss in this article.

It is understood in spacetime physics that upon the transformations as
proposed by Weyl in (1), only angles remain invariant, while sizes, magnitudes
and scalar products between vectors could change. Hence, due to these
properties and their consistency with relativity, the invariance with respect
to conformal transformations could be an additional symmetry fully consistent
with the theory of relativistic gravitation [6]. In original Einstein’s
theory, the dynamics of the gravitational field does not enjoy conformal
symmetry. Here, we would like to consider a possibility of adding the
requirements of local conformal symmetry of the gravitational action in a
slightly different theory. Such a symmetry is fully realized and embodied in
the gravitational theory known as Weyl gravity (conformal gravity) [1, 2, 7,
8, 9, 10], which in $d=4$ spacetime dimensions is based on the square of the
Weyl tensor in the gravitational action, in opposition to the mere presence of
the Ricci scalar in the action of Einstein’s gravity. This is why conformal
gravity can be seen as the first example of a different (or modified)
classical gravitation, but still being in agreement with main concepts and
ideas of General Relativity. Simply, in Weyl gravity the group of local
gravitational gauge symmetries is bigger than just of diffeomorphisms, because
it is extended by local conformal group factor.

However, Weyl’s gravitational theory was forgotten for a while due to the
great success of Einstein’s theory of gravitation, which only includes
symmetries in relation to the diffeomorphism transformations. Along Einstein’s
theory’s apogee, Weyl’s theory was left out, for being considered as a non-
physical theory. Fortunately, as the time passed by, Weyl’s theory returned to
the theoretical physics mainstream [11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23]. The return of conformal symmetry and conformal methods [24, 25,
26, 27] occurred precisely in the general theory of relativity, later also
enriched in quantum field theory, where it became so much appreciated [11, 28,
29]. This curious fact happened mainly because of the applications of
conformal methods (but not so much of Weyl’s theory) in gravity, high energy
physics and field theory; some of these applications exerted crucial roles in
the development of renormalization methods and its importance to the
renormalization group (RG) flows [30, 31] both in quantum field theories and
also in models of statistical physics. Other recent developments in conformal
gravity were done in [32, 33, 34], for black hole physics in [35, 36], and
also generally for the use of conformal methods to resolve spacetime
singularities in [37, 38].

Related to this usage of methods of conformal symmetry there were also
significant studies on conformal anomalies, which analysis of related quantum
effects can be found in [39, 40, 118, 41, 42, 43, 44], and other applications
that follow on them are discussed in [45, 46, 6]. In particular, the conformal
anomaly-generated gravitational effective action in curved spacetime [47, 48],
is of considerable interest due to numerous physical applications in black
hole physics, cosmology, supersymmetry, string theory and statistical
mechanics (see [49] for the review) and also because of the hope to use such
an action as an insight for the full theory of quantum gravity [50, 51, 40,
42, 52]. The effective action for the gravitationally coupled conformal scalar
field [53, 54, 55, 56] is a nonlocal functional but all the nonlocalities are
related with the Green functions of a local conformal operator acting on
scalar fields. Previously, the cases of the second order and the fourth order
conformal operators were investigated for this purpose in the literature [57,
58, 59]. The first time two-derivative scalar operator with conformal coupling
to background geometry was considered in the works of Penrose in $d=4$
spacetime dimensions [60, 61] and later in a case of generalized dimensions by
Ørsted [62]. In turn, the four-derivative scalar conformal operator was
constructed by Fradkin and Tseytlin in their search for supergravitational
theories in $d=4$ [63] and then immediately generalized by Paneitz in 1982
[64], and also independently by Eastwood and Singer [65]. As remarked later by
Branson [64] this, what is now called Paneitz operator, played a significant
role in various applications in differential geometry, not only conformal
differential geometry [66], in $d=4$ and also in higher dimensions [67] and
both in Euclidean [68] and in Minkowskian [69] signature of the metric. This
significance of conformal scalar operators in differential geometry and
theoretical physics is visible at the present moment too.

A great example about the applications of conformal transformations on
operators with higher derivatives can be seen in [70], which shows us briefly
that such operators are useful to investigate quantum gravitational effects.
There are also other applications as well: with an ability to provide
consistent renormalization of quantum divergences [71], an opportunity to use
group renormalization methods to study asymptotic behavior of couplings on
curved backgrounds [72], and finally with a possibility of building candidate
Grand Unified Theories (GUT) models of particle physics with higher derivative
operators [16].

Conformal symmetry and conformal methods are also ubiquitously found in
condensed matter physics [73, 74], where, for example, conformal field
theories are used to solve interaction problems in many-body physics [3]. The
formalism of CFT’s offers a possibility to describe a large set of quantum
field theories reaching a non-Gaussian fixed point (FP) of renormalization
group flows that can be moreover solved exactly around such FP’s. However, in
condensed matter not all of the tools of general CFT’s are often exploited
since the general and abstract methods of CFT were developed in a slightly
different context (of purely mathematical models of 2-dimensional field
theories, cf. [75]). In [74], for example, a well known Tomonaga theory in the
Kac-Moody CFT language is used to show how the properties of Fermi surfaces
can be understood on the purely algebraic CFT base.

In general, CFT’s are used to describe the situation of RG flows in generic
QFT’s with running coupling parameters near a fixed point of renormalization
group flow. There, where the RG flow stops one surely meets scale invariance
which can be easily promoted to full conformal invariance on the quantum
interacting level [76]. In such circumstances, the reason for quantum RG
running of couplings is absent, namely there are also no perturbative UV
divergences and the theory is completely UV-finite. In general, it is quite
difficult to find models of such finite interacting theories when the gauge or
gravitational interactions are included consistently on the quantum level, but
one can consult the examples found in [77, 78, 79, 37, 80, 6]. As one can see
there the absence of UV divergences is closely related to the quantum
conformality and the theory which sits at the FP of RG can be cast in the form
of a CFT (possibly on a curved gravitational background like studied in this
paper). When the FP is non-trivial and the theory around it is still with non-
vanishing interactions, so this is a non-Gaussian FP, simply due to the mere
fact of its existence, the theory is said to be asymptotically safe since it
does not have anymore problems with UV divergences [81]. The values of
couplings are non-zero at such FP, but still the theory is finite and without
problems in the UV regime, that is it can be defined at arbitrarily high
energy scales. In other words, conformality is reached asymptotically in the
UV.

One can also see that the correlation functions of quantum fields in the
vicinity of FP show the features of complete scale invariance. A convenient
description of these theoretical phenomena is also in terms of anomalous
dimensions of relevant operators defined near the FP. They physically provide
the tools for the analysis of critical exponents in the broad area of physics
of critical phenomena usually studied in the IR regimes [82], while in QFT
models of fundamental interactions these are usually anomalous dimensions of
operators defined in the vicinity of UV FP’s of RG flows. Moreover, it is
known in statistical physics that the particular microscopic (UV) details of
the models are irrelevant for the situation near IR FP’s, which are used to
describe critical phenomena as studied in condensed matter since some
different physical models might belong to the same universality classes each
one characterized by one and the same IR FP, where all the correlation lengths
diverge. Similarly, in the QFT domain of applicability different QFT models
may be in the same UV universality class from the point of view of the UV FP
of quantum RG flows [83, 84], regardless of their distinguishing infrared
details (like mass deformation, etc.). There in QFT, the fact that correlation
lengths diverge is a consequence of the presence of conformal symmetry in the
UV regime.

When the CFT description of the theory near UV FP is obtained (with central
charges of CFT algebras and with the set of primary conformal operators of
these algebras, their commutation relations, their spectra, and also their
anomalous scaling dimensions or conformal weights, so in total with all CFT
data), then one can construct a new perturbation calculus around the UV FP, so
called conformal perturbation theory [85]. In it one deforms the conformal
symmetry by adding new deformation operators or by giving vacuum expectation
values to some non-conformally invariant quantum fields, and then one also
allows for perturbation of the conformal structure and an RG running of new
couplings in front of these new deformation operators. These operators have to
be relevant deformations from the point of view of the UV FP, must be built
out of conformal primary operators of the CFT, and moreover are required to be
such that the RG flow is back generated when one goes down with energies away
from the UV FP. However, to achieve this formidable task, first one needs to
have a full description of the situation at the UV FP, including also the
description of all primary conformal operators. This is why the study of
higher derivative scalar conformal operators222By coupling matter fields to
dynamical quantum background geometry one sees that higher derivative
gravitational terms are naturally generated [86] in spacetime dimensions
$d>2$, hence one also has to include higher derivatives when coupling scalars
to gravity. as the basic ingredients for the construction of UV FP CFT
algebras containing operators with higher derivatives coupled to background
geometry is an essential initial task in this ambitious programme.

The six-derivative scalar operator with conformal properties was originally
found for physical applications related to conformal anomaly in $d=6$
spacetime dimensions in 2001 by Hamada in [87]. In this work, Hamada analyzed
how the integrability conditions for conformal anomalies constrain the
possible form of the effective action for an even-dimensional differential
operators. His analysis first showed that these integrability conditions are
satisfied for an action that is constructed using a four-derivative scalar
conformal operator in $d=4$ spacetime dimensions (the Paneitz operator). The
general form of this operator makes manifest diffeomorphism invariance,
renormalization and regularization scheme-independence. Additionally its
coefficients are chosen such that the anomaly vanishes and the action of the
higher-derivative scalar field coupled to background gravitational field
(background geometry) is fully conformally invariant. These results were
crucial for further developments in [88, 89]. In [87] Hamada later generalized
these arguments to a new six-derivative scalar operator considered exclusively
in its critical dimension, namely in $d=6$. This is the reason why in this
paper we will call a six-derivative scalar conformal operator and acting on
scalars by the name of Hamada operator.

###  I.1 Scope and the plan of the article

In this work we will study the conformal properties of scalar operators with
six derivatives in general dimensions $d$ of spacetime. Our results will be
applicable both in the Euclidean (space) and also in the Minkowskian domain
(spacetime) of the metric signature. The first case besides the obvious
applications in conformal Euclidean differential geometry of curved manifolds
will also be useful in condensed matter theory to describe conformal behaviour
of some higher-derivative scalar quasi-particle excitations coupled to
geometries of effectively higher-dimensional spaces. We expect this situation
to be present in the close vicinity of the RG fixed points, so where the
critical phenomena occur. Our results can be applied, for example, to phonon
excitations propagating on a curved graphene 2-dimensional sheets moving and
deforming themselves in time [90] or to some analogue (Euclidean) gravity
models with the presence of scalars and of Weyl symmetry [91, 92]. In the
Minkowskian case of the signature, such a conformal scalar operator gives
natural answers to a problem of conformal coupling to gravity for a scalar
field whose kinetic operator on the flat spacetime contains higher
derivatives, in our case precisely six derivatives.

We will propose a general form of conformal scalar operators in general
dimensions $d$ with arbitrary a priori undetermined coefficients. The
tensorial structure of the terms in the expansion of the operators will be
kept fixed, while we will search for values of these numerical coefficients by
demanding full conformal covariance of these operators when acting on a scalar
field with properly assigned conformal weight under Weyl transformations in
the GR setting. Of course, to check the conditions for conformality of
operators, we will need to investigate their transformations under conformal
rescalings (1) since they will be built from metric tensors, covariant
derivatives and gravitational curvature tensors, which are all functions of
the metric as a basic atomic building element, which transforms as in (1). We
will especially make use of a method of infinitesimal conformal variations
because it is well known that one has to check the conformal invariance only
to the first linearized order in the conformal parameters. In this procedure,
we consider conformal parameters (parameters $\Omega(x)$ of conformal
transformations in (1)) to have values very close to zero, so the difference
of $\exp(2\Omega(x))$ from $1$ is perturbative and very small. This lets us to
consider variations as linear operators and hence the problem of finding a
generalized form of conformal operators with higher derivatives in $d$
dimensions will be reduced to an algebraic one of solving some linear system
of equations for coefficients of expansion of operators in some basis of GR-
invariant terms.

Although the second and fourth order in derivatives scalar conformal operators
in their generalized forms in any dimension $d$ (Ørsted and Paneitz operators
respectively) were already reported in the physics literature [62, 64, 63,
43], we will analyze them again. This is because they have interesting
features that cannot be left behind, for example, their singular dependence in
coefficients of expansion in GR scalar terms on the number of dimensions $d$
and the presence of some critical dimensions $d_{{\rm crit}}$ related to the
order of derivatives. In particular, we will emphasize a known and clearly
visible fact that in even dimension of space(time) smaller than $d_{{\rm
crit}}$ such respective operators are not constructible. Moreover, these
conformal operators are useful building blocks in order to construct a
generalized form of the Hamada operator (which we call an operator with six
derivatives acting on scalars and transforming covariantly under general
conformal transformations) in any dimension $d$.

This article is organized as follows: in section II, we present a brief review
of the main peculiarities and properties of conformal transformations as
standardly used in GR. We also complement this section by a short review there
of the Ørsted and Paneitz conformal operators analyzed in any dimension of
spacetime. In the next section (section III), we discuss the Hamada operator,
first in $d=6$, following the original work of Hamada. In the same section, we
also intend to clarify the need for the presence of some additional terms
(higher in curvatures) in the construction of generalizations for both the
Paneitz (away from $d=4$) and Hamada operators (away from $d=6$). This step is
crucial in correctly understanding the listing of all basis elements for the
construction of the Hamada operator in general dimensions $d$, which we
undertake in the next sections. Namely, in section IV, we present a general
method of derivation, based on the complete basis of GR-invariant terms but
with undetermined coefficients, first concentrating on global conformal
transformations and their consequences and later on local infinitesimal
transformations. Finally, at the end of this section, we present all
coefficients of the generalized Hamada operator $\Delta_{6}$ in arbitrary
dimension $d$ of spacetime. This gives us the main result of the paper. In the
next main section V, we reconsider the original Hamada operator in $d=6$ and
we point out the singularities in its construction in dimensions $d=2$ and
$d=4$. We also indicate there what are the possible dimensions $d$ in which
the operators like $\Delta_{4}$ and $\Delta_{6}$ (generalized Paneitz and
Hamada respectively) become conformally covariant and in what dimensions the
coefficients of expansion of these operators have irremovable singularities.
Furthermore, to strengthen these conclusions, we generalize these observations
and in the last part of section V we prove a theorem that a general operator
$\Delta_{n}$ cannot be constructed in dimension $d=n-2$. In the second part of
the paper, first we discuss thoroughly the construction of conformal covariant
derivative and corresponding covariant box operator, this is in section VI. We
apply it to find the generalization of the Hamada operator written with
conformal curvatures and derivatives and for this the expression due to Wunsch
is much shorter. As a part with completely _new_ results we also present
section VII when we use a new construction method with different orders of
conformal derivatives. Finally in section VIII we compare all three methods
for finding generalization of conformal Hamada operators. In section IX we
draw our conclusions. We also there give some historical remarks about Hamada
operators.

##  II Conformal transformations

In this short section, we discuss our conventions about conformal
transformations of the covariant metric tensor of spacetime and its derived
quantities. We also postulate the conformally covariant transformations laws
for scalar fields present in the theory. This finally lets us to show the
simplest conformally covariant scalar operators when they act on corresponding
scalar fields with proper conformal weights, namely we will show here the
Ørsted and Paneitz operators. Here, we keep the dimension $d$ of spacetime and
the signature of the metric in it completely arbitrary.

To set our convention we specify the conformal transformations on the
covariant metric tensor as

$g_{\mu\nu}\to g^{\prime}_{\mu\nu}=e^{2\Omega}g_{\mu\nu},$ (3)

where $\Omega=\Omega(x)$ is a local (spacetime dependent) parameter of
conformal transformations [93] in the GR framework for performing conformal
(or Weyl) transformations. Under these transformations the infinitesimal
length element of the physical spacetime transforms as

$ds^{2}\to ds^{\prime 2}=e^{2\Omega}ds^{2},$ (4)

so the spacetime distances are not anymore invariant. However, the angles
remain conformally invariant. It is required that the conformal parameter
function $\Omega(x)$ is everywhere real-valued, hence the exponential factor
$\exp(2\Omega(x))$ in (3) is always positive-definite. This is important since
then under general conformal transformations the signature of the metric
tensor of space(time) remains invariant and also the metric never becomes
degenerate. From this basic conformal transformation law in (3), one derives
the corresponding transformations for the other objects like the contravariant
(inverse metric), metric density, Levi-Civita covariant derivatives and
finally various curvature tensors (Riemann tensor, Ricci tensor and Ricci
scalar). We will not need explicit formulas for the conformally transformed
versions of these objects, but one can consult, for example, the textbook [93]
as a useful reference for them. We just here for future convenience notice the
conformal transformation law of the fully covariant Weyl tensor (tensor of
conformal curvature) with all four spacetime indices covariant, which in
arbitrary dimension $d$ of spacetime (where the Weyl tensor can be defined, so
in $d\geqslant 3$) reads,

$C_{\mu\nu\\!\rho\sigma}\to
C_{\mu\nu\\!\rho\sigma}^{\prime}=e^{2\Omega}C_{\mu\nu\\!\rho\sigma}.$ (5)

From the above formula, one explicitly sees that this tensor transforms
covariantly (with a specific conformal weight that can be assigned to it)
under general conformal transformations. This is a notable exception among
other ordinary curvature tensors as considered in differential geometry. Hence
the role of Weyl tensor for conformal transformations and for conformal
methods in geometry is very significant. One can even consider a conformally
invariant GR-tensor
$C_{\mu\nu\\!\rho}{}^{\sigma}=g^{\sigma\kappa}C_{\mu\nu\\!\rho\kappa}$ and
this example proves that the position of indices (whether they are covariant
or contravariant) on a general tensor in GR matters for an assignment of
proper conformal weight, if such definition is possible.

When the scalar field $\phi=\phi(x)$ has assigned conformal weight
$w\in\mathbb{R}$, then it changes according to

$\phi\to\phi^{\prime}=e^{w\Omega}\phi.$ (6)

The spacetime dependence of the scalar field $\phi(x)$ is not changed by
conformal transformations, so for simplicity further we will omit writing it
explicitly. The conformal weight $w$ of the scalar field is constant in a
given model provided the dimensionality $d$ of the spacetime is fixed.
Actually, as we will show by explicit computation from reasons of simple
global scale invariance of the scalar models, this weight can only depend as a
simple rational function on the general dimension $d$ of spacetime (it is a
polynomial of the degree 1 in $d$ parameter). Moreover, we call the case when
$w\neq 0$ as conformally covariant scalar field, while only in the special
case of $w=0$ we can speak of conformally invariant (or inert) scalar field.
As we will see in section V, this special case for conformally inert scalars
is on the boundary of the regime where we cannot define conformal scalar
operators acting on such scalars. Finally, here we also emphasize that for
different scalar models (characterized by a different number of derivatives in
the kinetic term on the flat spacetime background), but still considered
within the spacetime of the same fixed dimensionality $d$, the conformal
weights for the scalars are different since a priori these are different and
unrelated scalar fields, and for example, their energy dimensions are also
different.

One sees that the covariant metric can be assigned a weight $w=2$, while its
inverse (the contravariant metric) must then have $w=-2$. This must be in such
a way that the Kronecker delta tensor (the metric with mixed position of
indices) $\delta^{\mu}_{\nu}$ remains completely invariant, with $w=0$, and
absolute object of any geometry. One also notices the transformations of the
relativistic spacetime density measure

$\sqrt{|\,\mbox{det}\,g_{\mu\nu}|}\underset{{\rm
df}}{=}\sqrt{|g|}\to\sqrt{|g|}^{\prime}=e^{d\Omega}\sqrt{|g|}$ (7)

in spacetime of dimension $d$, while the integration volume element in
coordinates $d^{d}x$ remains conformally invariant. This means that the
density measure (being the square root of the absolute value of the
determinant of the covariant metric tensor $g_{\mu\nu}$) $\sqrt{|g|}$ may be
assigned a conformal weight $w=d$.

In the infinitesimal form (for $\Omega\ll 1$), these conformal transformations
of the basic building blocks of geometrical GR-covariant tensors and covariant
differential operators read,

$g_{\mu\nu}\to g^{\prime}_{\mu\nu}=(1+2\Omega)g_{\mu\nu},$ (8) $g^{\mu\nu}\to
g^{\prime\mu\nu}=(1-2\Omega)g^{\mu\nu},$ (9)
$\sqrt{|g|}\to\sqrt{|g|}^{\prime}=\left(1+d\Omega\right)\sqrt{|g|},$ (10)
$C_{\mu\nu\\!\rho\sigma}^{\prime}=(1+2\Omega)C_{\mu\nu\\!\rho\sigma},$ (11)
$\phi\to\phi^{\prime}=(1+w\Omega)\phi.$ (12)

The transformation rules for curvature tensors or other covariant differential
operators are more complicated – they might contain also derivatives of the
parameter $\Omega=\Omega(x)$ and we will not need to write them explicitly
here. The formulas in (8) to (12) express the forms of infinitesimal conformal
variations of the objects on the left hand sides of these equations. In later
sections, we will make an extensive use of the formalism of conformal
variations of various tensors and differential operators of geometry since for
conformal invariance or covariance we need to check only to the linearized
level in the small parameter $\Omega(x)$ and hence only infinitesimal
conformal variations settle the issues with conformal symmetry and good
transformation properties under its corresponding transformations.

###  II.1 Conformal operators and conformal actions

Below, we will present the final results for the construction of two- and
four-derivative scalar conformal operators acting on scalar fields present on
the spacetime manifold and coupled to it in the conformal way. We remark that
conformal operators $\Delta$ when they are conformally covariant and when they
act on conformally covariant scalar fields $\phi$ with properly assigned
weights, give rise to another scalars $\Delta\phi$, which for conformality
must also transform as conformally covariant objects. Here, the crucial thing
is that the weight $w^{\prime}$ for the new composed object $\Delta\phi$ will
be almost always necessarily different than the original weight $w$ of the
scalar field $\phi$. Therefore, the general action of conformally covariant
differential operator on some conformal scalars is characterized and adjusted
by specifying two constant numbers: $w$ and $w^{\prime}$ for any type of
conformally covariant differential operator. One could say that only in the
rare case, when $w^{\prime}=w$, we can tell that the differential operator is
conformally invariant since then it does not add anything to the weight of the
scalar field it acts upon on the right. In general, the weight of the
differential operator $w_{\Delta}$ could be defined (following [41, 118]) as
the difference

$w_{\Delta}=w^{\prime}-w.$ (13)

This definition works also for endomorphic operators, that is such ones that
they do not have any differential character and they do not contain any
derivatives, nor partial nor covariant ones in their definitions, and in that
case, it coincides with the standard definition of the natural conformal
weight of a conformally covariant tensor. One assumes judiciously here that in
the case of a product of two conformally covariant objects, let them be
tensors, scalars or differential operators, the product is also an object that
transforms covariantly under the conformal rescalings and that its conformal
weight is equal to the sum of weights. This additivity of conformal weights is
a natural property like the additivity of charge for $U(1)$ gauge
transformations since also conformal transformations have some formal
similarities to the Abelian one-parameter gauge transformations. One can prove
this additive relation between conformal weights of tensors in a product by
exploiting the linearity and the Leibniz rule for the infinitesimal conformal
variations of products. For the case of action of differential operators on
some tensors, which could be viewed as a “product” only by going to suitable
momentum space, we adopt this rule and actually based on this we define the
weights of various conformally covariant differential operators as the
differences like $w^{\prime}-w$ as used above.

Having at our disposal a conformally covariant operator with the conformal
weight $w_{\Delta}$ defined above and acting on precisely one tensorial field
on the right, one can construct also a conformally invariant action
functional. One then treats the conformal operator as the kinetic term for the
tensorial field coupled non-trivially to a background geometry. This way of
coupling to the external gravitational field is non-minimal, but it is also
conformal from the definition of the conformally covariant operator. In order
to finish the explicit construction of the action, one uses the conformally
covariant object schematically denoted as $\Delta\phi$, which is linear in the
scalar field $\phi$ and couple it on the left hand side with the conjugate
tensor field. For the uncharged scalar field case, that we consider in this
work, the conjugation means here no operations, so we just multiply by the
same scalar field evaluated at the same spacetime point $\phi=\phi(x)$. In the
more general case of various tensorial representations of various internal
symmetry groups, one has to conjugate this tensorial field on the left in the
charge conjugation sense, internal indices sense, and also use the tensors
with the complementary positions of all spacetime and Lorentz indices present
on the field. Or in other words, one has to exploit the form of the kinetic
term for the field $\phi$ and naturally quadratic in this tensorial field, as
it is done in a standard manner in any QFT model with these tensorial fields.
The Lagrangian built in such a way is already a conformally covariant GR
scalar. When it is properly densitized by adjoining the factor of
$\sqrt{|g|}$, then one gets a conformal invariant and a GR scalar density.
Finally, after integrating over the whole spacetime points of the manifold
(with the conformally invariant coordinate integration element $d^{d}x$) this
yields an action functional which also enjoys the same properties of being
fully conformally invariant and also of being a nonlocal scalar action
functional from the point of view of GR. Such action functional written in the
schematic form as

$S=\\!\int\\!d^{d}x\sqrt{|g|}\phi\Delta\phi$ (14)

can be chosen as well as the basis for studies of conformal invariance
properties of the field theory model and of the differential operator in
question. This is what we are going to use in some further parts of this
paper. Hence one can study various conformally covariant differential
operators or related to them conformally invariant action functionals. The
fact that the operator $\Delta$ is conformally covariant and that it gives
rise to the action (14) implies that it also possesses many nice and
interesting features. We can already mention one of them here since this
operator automatically has to be self-adjoint with respect to the conjugation
and the operation of integrating by parts under the volume spacetime integral
in (14). This also means that its action on the right scalar field $\phi$
(canonically) must give the same results as its conjugate action on the left
scalar field $\phi$ in (14) for the case of scalar fields, when conjugation
boils down to no operations at all. This operator $\Delta$ arises also
naturally as the second variational derivative of the action $S$ in (14) with
respect to scalar fields $\phi$, here without the need of additional self-
adjoining procedure (such latter operation is needed for ordinary operators
not being conformally covariant, for example). Only the self-adjointness
property in such a form allows us to have both conformally covariant operator
$\Delta$ and conformally invariant action $S$, quadratic in scalar fields
$\phi$ and with the same operator $\Delta$ in it (as in (14)), at the same
time.

It is also possible, in principle, to consider operators that involve non-
linear actions on conformally covariant tensorial fields as their arguments,
they even do not have to be the same tensorial fields. Such complication is
actually quite easy to tackle on the level of differential geometry
considerations. On the level of the corresponding QFT models, this is related
to the non-trivial interactions between various conformal fields of the model
and not only with non-trivial interactions of these fields with gravitational
background geometry. We will comment briefly on these possibilities in section
V for the case of scalar fields. However, here one has to take into account
that from the perspective of QFT the addition of such non-trivial interactions
is possible but then the weights of the scalar fields participating in such
interactions must be determined firstly and solely from the kinetic term. The
considerations of the conformal invariance of the action as constructed in
(14) with only the kinetic (quadratic) term for the scalar field $\phi$ are
the most relevant here since the kinetic term is the most important one and
this is because it contains the smallest possible number of powers of fields,
when we also neglect tadpole diagrams.

In the light of what we have said above, in a consistent conformally symmetric
theory coupled to background gravitational field, it is possible to have only
one unique even number of derivatives in the kinetic term on the flat
spacetime background, for example, for the scalar field, at the same time. So,
for example, it is forbidden to have simultaneously, in one model, Ørsted and
Paneitz operators for one and the same scalar field $\phi$. This is already
due to scale invariance requirement, so from the considerations of the energy
dimensions of the scalar fields. The same restriction will apply to non-
linearities and basically only non-linearities containing less derivatives
than in the leading kinetic term of the model will be allowed. One does not
allow here cases with negative or rational, fractional powers of the covariant
derivatives and also cases of the same pathologies of exponents for scalar
fields – this is due to requirement of locality of the Lagrangian. It is
perhaps possible to add some non-linear interaction terms with derivatives,
but we have not yet checked this option. Still, one can very easily add non-
linearities containing no derivatives at all and corresponding to the very
specific powers of the scalar fields or equivalently to the form of local
conformal interaction vertices with no momentum dependence in the language of
QFT. Then in the last case there is no issue with their generalization to the
curved spacetime background since the latter is easily achieved by just
supplying the $\sqrt{|g|}$ factor for conformality of the resulting Lagrangian
density, because that part of the interaction Lagrangian does not contain
derivatives at all and all the fields present in the interaction vertex
transform conformally covariantly. One is sure that on the flat spacetime
background the non-linear interaction terms with derivatives exist and are
there scale-invariant, however the question is really whether there exists a
suitable generalization of such terms to the case of non-trivial gravitational
backgrounds. If this is possible, we can add such terms as well to the final
expression for the conformally invariant action functional of the theory,
which will now include also non-trivial but conformal self-interaction terms
of the scalar field $\phi$, possibly also with some number of background
covariant derivatives acting between these fields.

However, as it should be clear from the preceding analysis, this issue with
non-linearities is in no relation to the main issue of this paper, namely of
constructing the kinetic operators so such that they act precisely on one
power of the scalar field $\phi$ on the right. This is because for conformal
covariance of the latter operator one has to analyze only operatorial terms
which act linearly and on one scalar field $\phi$, and non-linearities here do
not play any role. Simply, the propagator of quantum modes for the field
$\phi$ on a curved background, and in particular on flat Minkowskian
background, depends only and exclusively on terms which are quadratic in the
field $\phi$ on the level of Lagrangian and higher non-linearities do not
matter for this issue. This is what also conformal coupling to background
geometry repeats here, namely the structure of the kinetic term describing
propagation of modes of the field $\phi$ and this is what needs to be only
checked here for conformal covariance properties of the latter operator. When
one couples the kinetic term describing the propagation of modes of the fields
$\phi$ to background geometry, one can resort to the construction of a new
conformally covariant operator which only describes a propagation on this non-
trivial background and consistently with the requirements of conformal
symmetry and at the same time the issue of higher interactions or self-
interactions of the scalar field $\phi$ may be for this purpose completely
omitted. In other words, one does not need to include such non-linearities for
the main parts of this paper, where we show explicitly the construction for
conformal kinetic operators acting between two scalar fields on the level of
the action functional, like in (14).

The examples of the kinetic scalar conformal operators when coupled to
external geometry give us explicit results for the conformal coupling to
background geometry procedure. This is not a minimal gravitational coupling,
and so it may not exist for any operator that one can consider on the level of
flat spacetime theory, without intervening gravity. The question whether such
conformal covariantization from flat spacetime to a curved background exists
and whether it is successful in all interesting us cases is the main
scientific question that we want to address here. Having discussed general
issues of the construction of conformal scalar operators in the form of
suitable kinetic operators coupling the scalar fields in a conformal way to
background geometry, now we can show the details of the construction of Ørsted
and Paneitz operator. We achieve this in the two subsections below. We adopt
the notation here that the operator with $n$ derivatives in the leading
kinetic term, which is a term which comes with no gravitational curvatures (so
equivalently as it is considered on the flat spacetime background), is denoted
by the symbol $\Delta_{n}$.

###  II.2 Ørsted operator $\Delta_{2}$

Two-derivative scalar conformal operator333Note that this operator is a two-
derivative Klein-Gordon kinetic operator (acting on the scalar field) non-
minimally coupled to a Ricci scalar $R$ of the background geometry and without
the mass since a constant mass term would generally break the conformal
symmetry. (But actually here one can treat effectively the Ricci scalar $R$ as
a possibly spacetime-dependent mass term for the scalar field $\phi$). This
constitutes the case of standard conformal coupling of a scalar field to
background gravitational field. $\Delta_{2}$

$\Delta_{2}=\square-\frac{d-2}{4(d-1)}R$ (15)

was first studied by Penrose in $d=4$ and by Ørsted in general dimensions $d$
and gives the scalar action

$S=\\!\int\\!d^{d}x\sqrt{|g|}\phi\Delta_{2}\phi$ (16)

conformally invariant if the conformal weight of the scalar field is given by

$w=\frac{2-d}{2}.$ (17)

Such a two-derivative operator in (15) can be defined in any spacetime
dimension $d>1$. It can be easily found here that $w_{\Delta_{2}}=-2$, so in
the result $w^{\prime}=-\frac{d+2}{2}$, which is the weight of the scalar
$\Delta_{2}\phi$. The action in (16) is conformally invariant (has effectively
$w=0$) because of the compensating total conformal weight of the metric
density $\sqrt{|g|}$ as in (10). The action in (16) describes just a free
standard scalar field without self-interactions and only coupled in a
conformal way to the background geometry. This action consists only of the
two-derivative kinetic term for scalars, which is the leading term in the
number of derivatives. Moreover, only this term survives in a limiting case of
flat spacetime backgrounds giving rise to a standard scalar flat space
propagator with two derivatives in momentum space.

The action in (16) and in $d=4$ familiar spacetime dimensions contains the
famous non-minimal coupling $\xi R\phi^{2}$ of the scalar field $\phi$ to the
background geometry represented here by the Ricci scalar $R$. In $d=4$, this
coupling still gives rise to a renormalizable quantum theory of scalar matter
coupled to quantum gravity (since the coefficient $\xi$ is dimensionless in
$d=4$). The coupling of this type is naturally generated when one puts
massless scalar matter (with two-derivative kinetic operator) from flat
spacetime background on a non-trivial gravitational configuration. This is an
additional term of interactions between scalars and gravitation that has to be
added by the token of conformal version of the DeWitt-Utiyama arguments for
higher derivatives [86]. So this coupling term is still renormalizable, but
non-minimal and required by the consistency of the coupled quantum matter and
gravitational theory. In $d=4$, one finds a special value of the coefficient
$\xi_{d=4}=-\frac{1}{6}$. For other values $\xi\neq-\frac{1}{6}$ the theory is
not conformally invariant. Actually, for non-conformal values of the $\xi$
coefficient in $d=4$, the theory is still scale-invariant (under global Weyl
transformations and under restricted conformal transformations satisfying
$\square\Omega(x)=0$). Moreover, the coupling of the form $\xi R\phi^{2}$ is
the most general one, that one can add to the two-derivative scalar theory,
but still being consistent on the quantum level with renormalizability of the
coupled theory. Other types of higher couplings of the scalar field $\phi$
containing for example, other gravitational curvature tensors or other powers
of $R$, necessarily come with higher-derivative character, so with the
benefits and drawbacks of higher-derivative models of gravitational dynamics.

One notices that for $d=2$ the term proportional to the Ricci scalar curvature
$R$ in (15) drops down. This is again a special thing of the case of $d=2$
space(time) dimensions, which is related to special extended conformal
symmetry on the 2-dimensional complex plane (the algebra of symmetry
generators is in such a case infinite-dimensional), to the speciality of a
2-dimensional quantum gravity based on a two-derivative Einstein-Hilbert
action, and also to the whole idea of construction of a conformal
2-dimensional Minkowskian worldsheet spacetimes as used in string theories. In
such a case, one can say that the background GR-covariant box operator (the
generalization from the flat spacetime of the d’Alembertian operator
$\partial^{2}$, namely $\square=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$) when
acting on massless, dimensionless and conformally inert scalar field $\phi$
with $w=0$ in $d=2$, is perfectly conformally covariant with the conformal
weight $w_{\square}=-2$.

Another interesting thing is that for the action of the Ørsted operator in
(15) on a scalar field $\phi$, the conformal weight $w$ of the scalar field in
interesting and physically sensible dimensions, namely for $d\geqslant 2$, is
fixed and completely determined by the dimension of spacetime and it is also
always non-positive, reaching zero only in the case of $d=2$. As we will show
later this conformal weight is closely related to the energy dimensionality of
the scalar field with the same kinetic term on the flat spacetime, so in our
case this is just the $\partial^{2}$ operator and this is canonically
normalized standard massless scalar field on flat spacetime with two-
derivative action. Another known speciality of the $d=2$ case (both for
Euclidean and Minkowskian signature of the metric) is that there such a scalar
field is dimensionless. From this we derive that its conformal weight vanishes
too, so this is a case for a conformally invariant scalar field $\phi$.

###  II.3 Paneitz operator $\Delta_{4}$

Four-derivative scalar conformal operator $\Delta_{4}$ was originally found
independently – first by Fradkin and Tseytlin and later by Paneitz and
Eastwood and Singer. Its form in $d=4$ is the following444Our notation
regarding differential operators acting on non-operatorial objects is the most
compact one and does not use additional parentheses. That is the sequence of
differential operators (product of operators) only acts on the first tensorial
object it finds on its right and finishes its derivative action there. For
example, by $\nabla_{\mu}R\nabla^{\mu}$ we mean
$\left(\nabla_{\mu}R\right)\nabla^{\mu}$. Such a form of writing all expanded
formulas in this paper can be achieved by extensive use of Leibniz rule. If
the operator acts on a product of tensors, then we put them in one common
parenthesis.

$\Delta_{4}=\square^{2}+\frac{1}{3}\nabla_{\mu}R\nabla^{\mu}+2R_{\mu\nu}\nabla^{\mu}\nabla^{\nu}-\frac{2}{3}R\square.$
(18)

We first observe that in $d=4$, terms quadratic in curvatures do not appear in
the construction of this operator. Moreover, there the conformal weight of the
scalar field must be chosen as $w=0$. This coincides with the fact known from
simple dimensional analysis that the scalar field in $d=4$ dimensions with
four-derivative kinetic term must be dimensionless (in energy units), when,
for example, analyzed on the flat spacetime background.

The Paneitz operator was later generalized by Paneitz to arbitrary number of
dimensions of spacetime $d$ (except the case of $d=2$ where such simple
generalization described below does not work). Its form reads

$\Delta_{4}=\square^{2}-\frac{d-6}{2(d-1)}\nabla_{\mu}R\nabla^{\mu}+\frac{4}{d-2}R_{\mu\nu}\nabla^{\mu}\nabla^{\nu}-\frac{(d-2)^{2}+4}{2(d-1)(d-2)}R\square+$
$+\frac{(d-4)\left(d^{3}-4d^{2}+16d-16\right)}{16(d-1)^{2}(d-2)^{2}}R^{2}-\frac{d-4}{(d-2)^{2}}R_{\mu\nu}R^{\mu\nu}-\frac{d-4}{4(d-1)}\square
R.$ (19)

The conformal weight of the scalar field must then be given by

$w=\frac{4-d}{2},$ (20)

which is negative for $d>4$ and positive for $d<4$. One notices immediately
that terms quadratic in gravitational curvatures or containing four energy
dimensions in the background quantities (namely $R^{2}$,
$R_{\mu\nu}R^{\mu\nu}$ and $\square R$) come with coefficients which are
proportional to the factor $(d-4)$. This confirms that in $d=4$ they are not
necessary in the construction of the $\Delta_{4}$ operator. One also sees that
various coefficients in front of various terms contain the singularity
$\frac{1}{d-2}$, hence they cannot be defined as a simple limit in $d=2$
spacetime dimensions. We will come back to this issue later. Some coefficients
explode which means that Paneitz operator cannot be constructed in this way in
$d=2$. This happens when the dimension $d$ of spacetime (being an even integer
number) is lower than the critical dimension given by the index on the
$\Delta$ operator. Namely $\Delta_{n=4}=\Delta_{4}$ is a Paneitz operator,
$\Delta_{n=2}=\Delta_{2}$ is conformal two-derivative operator, etc. This is
also related to the negativity of the conformal weight $w$ of the scalar field
and vanishing weight happens precisely in the critical dimension. As we will
see below, in the case when the weight is positive and equals to one, we are
able to prove nonexistence of the dimensionally generalized $\Delta_{n}$ for
$d=n-2$ as a mathematical theorem.

One can, of course, freely change the basis for writing terms quadratic in
curvature in the Paneitz operator. For example, squares of the Riemann tensor
are possible there too $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$. However, we
notice one peculiar fact, that in dimensions $d>3$, one can add for free and
with an arbitrary coefficient a conformally invariant operator whose structure
makes evident its conformal properties. This operator is the square of the
Weyl tensor $C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}\underset{\rm
df}{=}C^{2}$ and it acts as endomorphism on the scalar field (without
derivatives). This is why in the form of writing the operator $\Delta_{4}$ we
need to use only two quadratic in curvature invariants in $d\neq 4$ (and we
have chosen them as $R^{2}$ and $R_{\mu\nu}R^{\mu\nu})$. Notice that in
critical dimensions the terms with no derivatives acting on the scalar $\phi$
(so with two gravitational curvatures) are never needed for the construction
of the minimal $\Delta_{4}$ operator in such a critical dimension $d=4$, but
the additional term $C^{2}\phi$ can be added here for free and with arbitrary
chosen value of the real front coefficient. We call the operator $\Delta_{4}$
without such addition $C^{2}\phi$ as minimal one, contrary to the non-minimal
case. This addition can be realized in any dimension $d\geqslant 4$.

Note that the general Paneitz operator in (19) is a four-derivative kinetic
operator (acting on the scalar field) non-minimally coupled to various
gravitational curvatures of the background geometry and without the mass since
a constant mass term again would generally break here the conformal symmetry
of the model. (Actually here one can treat effectively the curvature terms,
like $R^{2}$, $R_{\mu\nu}R^{\mu\nu}$ and $\square R$ as a possibly spacetime-
dependent mass terms for the scalar field $\phi$). One sees that the non-
minimal coupling to the geometry is realized by more than one term (one term
with Ricci scalar curvature was the case for the $\Delta_{2}$ operator).
Moreover, terms with lower number of derivatives than 4 still acting non-
endomorphically on the scalar field are present in (19). They are generally
possible and contain two or one background covariant derivative still acting
on the scalar field $\phi$ on the right. The terms with three derivatives are
not possible to construct here due to the requirement of general covariance
symmetry of the form for the operator in (19) since we cannot construct local
GR-covariant terms not being derivatives with precisely one unit of energy
dimension. The same type of argument excluded terms with one covariant
derivative still acting on the scalar field in the case of the Ørsted
conformal operator $\Delta_{2}$ \- there we cannot have any additional
differential operators constructed with covariant derivatives besides the
leading one, that is a kinetic term $\partial^{2}$ with the highest number of
derivatives. The form of the operator in (19) constitutes the case of a
generalized conformal coupling of a scalar field with four-derivative action
and four-derivative kinetic term (on flat spacetime) to background
gravitational field. The conformally invariant action of the theory is given
by (14) with the operator $\Delta=\Delta_{4}$ for this case.

Such a four-derivative operator in (19) can be defined in any spacetime
dimension, except the case of $d=2$. It can be easily found here that
$w_{\Delta_{2}}=-4$, so in the result with (20), we find that
$w^{\prime}=-\frac{d+4}{2}$, which is the weight of the scalar
$\Delta_{2}\phi$. The action in (14) for the case of Paneitz operator
$\Delta=\Delta_{4}$ is conformally invariant (has effectively $w=0$) because
of the compensating total conformal weight of the metric density $\sqrt{|g|}$
as in (10). The action based on the kinetic operator as in (19) describes just
a free scalar field with four-derivative kinetic term and without self-
interactions and only coupled in a conformal way to the background geometry.
This action consists of the four-derivative kinetic term for scalars, which is
the leading term in the number of derivatives and also some other subleading
terms in the number of derivatives which therefore must be accompanied in
their construction by some curvature terms. Moreover, only the leading in
derivatives four-derivative term $\square^{2}$ survives in a limiting case of
flat spacetime backgrounds giving rise to a scalar flat space propagator with
four derivatives in momentum space.

In $d=4$, the action of the Paneitz operator $\Delta_{4}$ from (18) still
contains various terms expressing non-trivial and non-minimal couplings to the
background geometry. These couplings still give rise to a renormalizable
quantum theory of scalar matter coupled to quantum gravity, provided that the
scalar field is taken as dimensionless, so then it is not canonically
normalized scalar field. The coupling of this type is naturally generated when
one puts massless dimensionless scalar matter (with a four-derivative kinetic
operator) from flat spacetime background on a non-trivial gravitational
configuration. These are some additional terms of interactions between scalars
and gravitation that have to be added by the token of conformal version of the
DeWitt-Utiyama arguments for higher derivatives [86]. So these coupling terms
are still renormalizable, but non-minimal and required by the consistency of
the coupled quantum matter and gravitational theory. Moreover, the couplings
of the forms as in (18) are the most general ones, that one can add to the
four-derivative scalar theory, but still being consistent on the quantum level
with renormalizability of the coupled theory.

One notices that for the critical dimension of $d=4$ all the endomorphic
terms, so not containing derivatives when acting on the scalar field $\phi$,
so terms proportional to the Ricci scalar curvature squared $R^{2}$, Ricci
tensor squared $R_{\mu\nu}R^{\mu\nu}$ and the GR-covariant box operator acting
on a scalar curvature $\square R$ in (19) drop down.

Another interesting thing is that for the action of the Paneitz operator in
(19) on a scalar field $\phi$, the conformal weight $w$ of the scalar field in
interesting and physically sensible dimensions, namely for $d\geqslant 4$, is
fixed and completely determined by the dimension of spacetime and it is also
always non-positive, reaching zero only in the special case of $d=4$. As we
will show later this conformal weight is closely related to the energy
dimensionality of the scalar field with the same kinetic term on the flat
spacetime, so in our case this is just the $(\partial^{2})^{2}$ operator and
this is canonically normalized massless scalar field on flat spacetime with
four-derivative action. Another known speciality of the $d=4$ case (both for
Euclidean and Minkowskian signature of the metric) is that there such a scalar
field is dimensionless. From this we derive that its conformal weight vanishes
too, so this is a case for a conformally invariant scalar field $\phi$.

##  III Hamada operator in $d=6$

The operator with six-derivatives $\Delta_{6}$ was first found by Hamada in
the critical number of dimensions, so for the dimensionless scalar field in
$d=6$ with the vanishing conformal weight $w=0$. As explained above, terms
with energy dimensions $E^{6}$ in background quantities are not needed in its
construction. Still, the structure of the operator contains 20 terms (in the
basis chosen below). Its form is

$\Delta_{6}=\square^{3}+4R_{\mu\nu}\nabla^{\mu}\nabla^{\nu}\square-R\square^{2}+4\nabla_{\rho}R_{\mu\nu}\nabla^{\rho}\nabla^{\mu}\nabla^{\nu}+4\square
R_{\mu\nu}\nabla^{\mu}\nabla^{\nu}-\frac{3}{5}\square R\square+$
$+\zeta_{1}R_{\mu\rho\sigma\tau}R_{\nu}{}^{\rho\sigma\tau}\nabla_{\mu}\nabla_{\nu}+\zeta_{2}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\square+\zeta_{1}R_{\rho\sigma}R_{\mu}{}^{\rho\sigma}{}_{\nu}\nabla^{\mu}\nabla^{\nu}+\left(6-\frac{3}{4}\zeta_{1}\right)R_{\mu\rho}R_{\nu}{}^{\rho}\nabla^{\mu}\nabla^{\nu}+$
$+\left(-1+\frac{1}{8}\zeta_{1}-\zeta_{2}\right)R_{\mu\nu}R^{\mu\nu}\square+\left(-2+\frac{1}{4}\zeta_{1}\right)RR_{\mu\nu}\nabla^{\mu}\nabla^{\nu}+\left(\frac{9}{25}-\frac{1}{40}\zeta_{1}+\frac{1}{10}\zeta_{2}\right)R^{2}\square+$
$+\frac{2}{5}\nabla_{\mu}\square
R\nabla^{\mu}+\left(\zeta_{1}+4\zeta_{2}\right)R_{\mu\nu\rho\sigma}\nabla^{\sigma}R^{\mu\nu\rho}{}_{\tau}\nabla^{\tau}-\zeta_{1}R_{\mu\nu\rho\sigma}\nabla^{\mu}R^{\nu\rho}\nabla^{\sigma}+\left(6+\frac{1}{4}\zeta_{1}\right)R_{\mu\nu}\nabla^{\mu}R^{\nu}{}_{\rho}\nabla^{\rho}+$
$+\left(-2-\frac{3}{4}\zeta_{1}-2\zeta_{2}\right)R_{\mu\nu}\nabla_{\rho}R^{\mu\nu}\nabla^{\rho}+\left(1-\frac{1}{8}\zeta_{1}\right)R_{\mu\nu}\nabla^{\mu}R\nabla^{\nu}+\left(-\frac{7}{25}+\frac{3}{40}\zeta_{1}+\frac{1}{5}\zeta_{2}\right)R\nabla_{\mu}R\nabla^{\mu}.$
(21)

One observes that the operator $\Delta_{6}$ is not unique in $d=6$ since it
depends on two arbitrary real parameters $\zeta_{1}$ and $\zeta_{2}$ which can
take values, which may simplify in some special cases the form of the
operator. This two-parameter family of conformal operators is a common feature
since in higher dimensions we have more ways to construct gravitational
conformal invariants (we have more ways to contract various Weyl
tensors)555For example, in $d=4$, there is only one way of contracting two
Weyl tensors $C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$. Using the identity
$C_{\mu\nu\rho\sigma}C^{\mu\rho\nu\sigma}=\frac{1}{2}C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$
due to cyclicity properties of the Weyl tensor, the second possible
contraction $C_{\mu\nu\rho\sigma}C^{\mu\rho\nu\sigma}$ is eliminated. In
dimensions $d=6$, we have three independent ways to construct gravitational
conformal invariants [100]. In the sector of non-differentiated Weyl tensors
only, they reduce to two different contractions of the cube of the Weyl
tensor, namely
$C_{\mu\nu\rho\sigma}C^{\rho\sigma}{}_{\kappa\lambda}C^{\kappa\lambda\mu\nu}$
and the other
$C_{\mu\rho\sigma\nu}C^{\rho\kappa\lambda\sigma}C_{\kappa\mu\nu\lambda}$ (the
other possible contractions again reduce to the two above by exploiting the
cyclicity). Only in lower dimensions, like in $d=4$, there exists an identity
relating
$C_{\mu\rho\sigma\nu}C^{\rho\kappa\lambda\sigma}C_{\kappa\mu\nu\lambda}$ and
$C_{\mu\nu\rho\sigma}C^{\rho\sigma}{}_{\kappa\lambda}C^{\kappa\lambda\mu\nu}$.
Surely the number of free parameters in the construction of the $\Delta_{6}$
operator is related to the number of invariants that can be constructed by
contractions of three Weyl tensors.

###  III.1 Terms in the generalization of Hamada operator (away from $d=6$)

The generalization of the Hamada operator to arbitrary dimensions $d$ was not
known before. First, one has to complete the basis of invariants as given
above in the expansion of the $\Delta_{6}$ operator in the critical dimension
$d=6$. Two terms are additionally possible which contain free derivatives
acting on the scalar field (which is always understood to be on the right of
the operator), namely

$\nabla_{\mu}R\nabla^{\mu}\square\quad{\rm
and}\quad\nabla_{\mu}\nabla_{\nu}R\nabla^{\mu}\nabla^{\nu}.$ (22)

The basis of all together 22 terms containing derivatives is now complete. The
other possible terms that one would think should be added are reduced by the
following identities

$R_{\nu\rho\sigma\tau}\nabla_{\mu}R^{\nu\sigma\rho\tau}\nabla^{\mu}\phi=\frac{1}{2}R_{\nu\rho\sigma\tau}\nabla_{\mu}R^{\nu\rho\sigma\tau}\nabla^{\mu}\phi,$
(23)
$R_{\rho\sigma\tau\mu}\nabla^{\mu}R^{\rho\tau\sigma}{}_{\nu}\nabla^{\nu}\phi=\frac{1}{2}R_{\rho\sigma\tau\mu}\nabla^{\mu}R^{\rho\sigma\tau}{}_{\nu}\nabla^{\nu}\phi,$
(24)
$R_{\nu\rho\sigma\tau}\nabla_{\mu}R^{\nu\sigma\rho\tau}\nabla^{\mu}\phi=\frac{1}{2}R_{\nu\rho\sigma\tau}\nabla_{\mu}R^{\nu\rho\sigma\tau}\nabla^{\mu}\phi$
(25)

arising from the usage of cyclicity of Riemann tensor
($R_{\mu\nu\rho\sigma}+R_{\mu\rho\sigma\nu}+R_{\mu\sigma\nu\rho}=0$) and

$R_{\nu\rho\sigma\tau}\nabla_{\mu}R^{\nu\rho\sigma\tau}\nabla^{\mu}\phi=2R_{\rho\sigma\tau\mu}\nabla^{\mu}R^{\rho\sigma\tau}{}_{\nu}\nabla^{\nu}\phi,$
(26)
$R_{\nu\rho\sigma\tau}\nabla^{\rho}R^{\mu\nu\sigma\tau}=-\frac{1}{2}R_{\nu\rho\sigma\tau}\nabla^{\mu}R^{\nu\rho\sigma\tau}$
(27)

arising from use of Bianchi identity
($\nabla_{\mu}R_{\nu\rho\sigma\tau}+\nabla_{\nu}R_{\rho\mu\sigma\tau}+\nabla_{\rho}R_{\mu\nu\sigma\tau}=0$)
satisfied by Riemann tensor.

Therefore, following Hamada, the basis we choose to write the full expansion
of the derivative part of the $\Delta_{6}$ operator reads

$\Delta_{6,{\rm\,der}}=\square^{3}+z_{1}R_{\mu\nu}\nabla^{\mu}\nabla^{\nu}\square+z_{2}R\square^{2}+z_{3}\nabla_{\rho}R_{\mu\nu}\nabla^{\rho}\nabla^{\mu}\nabla^{\nu}+z_{4}\nabla_{\mu}R\nabla^{\mu}\square+z_{5}\nabla_{\mu}\nabla_{\nu}R\nabla^{\mu}\nabla^{\nu}+$
$+z_{6}\square R_{\mu\nu}\nabla^{\mu}\nabla^{\nu}+z_{7}\square
R\square+z_{8}R_{\mu\rho\sigma\tau}R_{\nu}{}^{\rho\sigma\tau}\nabla_{\mu}\nabla_{\nu}+z_{9}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\square+$
$+z_{10}R_{\rho\sigma}R_{\mu}{}^{\rho\sigma}{}_{\nu}\nabla^{\mu}\nabla^{\nu}+z_{11}R_{\mu\rho}R_{\nu}{}^{\rho}\nabla^{\mu}\nabla^{\nu}+z_{12}R_{\mu\nu}R^{\mu\nu}\square+z_{13}RR_{\mu\nu}\nabla^{\mu}\nabla^{\nu}+z_{14}R^{2}\square+$
$+z_{15}\nabla_{\mu}\square
R\nabla^{\mu}+z_{16}R_{\mu\nu\rho\sigma}\nabla^{\sigma}R^{\mu\nu\rho}{}_{\tau}\nabla^{\tau}+z_{17}R_{\mu\nu\rho\sigma}\nabla^{\mu}R^{\nu\rho}\nabla^{\sigma}+z_{18}R_{\mu\nu}\nabla^{\mu}R^{\nu}{}_{\rho}\nabla^{\rho}+$
$+z_{19}R_{\mu\nu}\nabla_{\rho}R^{\mu\nu}\nabla^{\rho}+z_{20}R_{\mu\nu}\nabla^{\mu}R\nabla^{\nu}+z_{21}R\nabla_{\mu}R\nabla^{\mu},$
(28)

where we already set the coefficient of the term leading in the number of
derivatives (so of the $\square^{3}$ term) to unity. One sees that the choice
of the basis made by Hamada with two arbitrary parameters $\zeta_{1}$ and
$\zeta_{2}$ corresponds to allowing for completely arbitrary coefficients in
front of the terms

$R_{\mu\rho\sigma\tau}R_{\nu}{}^{\rho\sigma\tau}\nabla_{\mu}\nabla_{\nu}\quad{\rm
and}\quad R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\square$ (29)

and these two terms can be traded to two ones containing only contractions of
the Weyl tensors and no ordinary covariant derivatives acting on Weyl tensors
(though they still will act on the scalar field $\phi$ on the right).
Actually, the change of a basis to the one dominated by Weyl tensors is
possible here as well. However, for further computation we have to stick to
one basis. Of course, the results for the existence of the operator, etc. do
not depend on the choice of such a basis. And we here adopt the same choice as
of Hamada. One sees a posteriori that the two new elements (22) of the basis
(28) are generated in general dimension $d\neq 6$.

However, in general dimension $d$ one also expects terms in $\Delta_{6}$ which
are without any derivative acting on the scalar (so they are endomorphisms).
The construction of the basis for these terms is another tedious task, which
was solved in Gilkey. Before we list all the terms present in this additional
non-derivative part of the operator $\Delta_{6}$, we discuss the issue related
to the general construction of the action for scalars, which supposed to be
conformally invariant. The action we consider is quadratic in the scalar
fields $\phi$ and therefore has the general form

$S=\\!\int\\!d^{d}x\sqrt{|g|}\phi\Delta_{6}\phi$ (30)

in general dimension $d$. We can decompose the operator $\Delta_{6}$ as
follows

$\Delta_{6}=\Delta_{6,\,{\rm der}}+\Delta^{\prime}_{6}.$ (31)

Of course, all terms in $\Delta_{6,\,{\rm der}}$ contain derivatives (from one
to six) acting on the right scalar field. For such terms we cannot perform
integration by parts of covariant derivatives, if we want to preserve the
structure of the general action (30), that is that the left scalar field is
not differentiated at all. This fixes the possible form of the
$\Delta_{6,\,{\rm der}}$ operator. However, in $\Delta^{\prime}_{6}$ there are
no derivatives and one may think that a complete basis of terms consists of
all six-dimensional gravitational invariants, which can give rise to
conformally invariant gravitational action in $d=6$ dimensions. Or to globally
scale-invariant gravitational actions in $d=6$ dimensions. It is known that
there are precisely 10 such independent terms. However, the basis for
$\Delta^{\prime}_{6}$ operator has to be extended by inclusion of 7 new terms,
which on the level of gravitational action in $d=6$ would be total derivatives
(or related to them by subtracting terms from the original basis with 10
elements). On the level of the scalar action (conformally coupled to
gravitation) in (30) these terms are not total derivatives because of the
presence of two scalar fields (both left and right). Again if we require the
conservation of the structure as in (30), we cannot allow for doing of
integration by parts and this 10-element basis must be extended to the general
17-element one. This last basis we can call as Gilkey basis.

Having discussed this issue, we give below the explicit form of terms in this
basis,

$\Delta^{\prime}_{6}=x_{1}R\square R+x_{2}R_{\mu\nu}\square
R^{\mu\nu}+x_{3}R^{3}+x_{4}RR_{\mu\nu}R^{\mu\nu}+x_{5}RR_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+x_{6}R_{\mu\nu}R^{\mu}{}_{\rho}R^{\nu\rho}+$
$+x_{7}R_{\mu\nu}R_{\rho\sigma}R^{\mu\rho\nu\sigma}+x_{8}R_{\mu\nu}R^{\mu}{}_{\rho\sigma\tau}R^{\nu\rho\sigma\tau}+x_{9}R_{\mu\nu\rho\sigma}R^{\rho\sigma}{}_{\tau\omega}R^{\tau\omega\mu\nu}+x_{10}R_{\mu\nu\rho\sigma}R^{\mu\tau\rho\omega}R^{\nu}{}_{\tau}{}^{\sigma}{}_{\omega}+$
$+x_{11}\nabla_{\mu}\nabla_{\nu}RR^{\mu\nu}+x_{12}\nabla_{\mu}R\nabla^{\mu}R+x_{13}\nabla_{\mu}R_{\nu\rho}\nabla^{\mu}R^{\nu\rho}+x_{14}\square^{2}R+x_{15}\nabla_{\mu}R_{\nu\rho}\nabla^{\nu}R^{\mu\rho}+$
$+x_{16}\nabla_{\mu}\nabla_{\nu}R_{\rho\sigma}R^{\mu\rho\nu\sigma}+x_{17}\nabla_{\mu}R_{\nu\rho\sigma\tau}\nabla^{\mu}R^{\nu\rho\sigma\tau}.$
(32)

The other possible terms one could think about like

$R_{\mu\nu\rho\sigma}\square
R^{\mu\nu\rho\sigma},\,\nabla_{\mu}R_{\nu\rho\sigma\tau}\nabla^{\mu}R^{\nu\sigma\rho\tau},\,\nabla_{\mu}R_{\nu\rho\sigma\tau}\nabla^{\nu}R^{\mu\rho\sigma\tau},\,\nabla_{\mu}R_{\nu\rho\sigma\tau}\nabla^{\nu}R^{\mu\sigma\rho\tau},\,R_{\mu\nu}R^{\mu}{}_{\rho\sigma\tau}R^{\nu\sigma\rho\tau}$
(33)

are reduced by performing commutation of covariant derivatives, using
cyclicity or Bianchi identities (sometimes also in the contracted forms) of
the Riemann tensor. Explicitly, we use the following formulas which are valid
at any point of the spacetime, where we do not have to permit integration by
parts:

$R_{\mu\nu\rho\sigma}\square R^{\mu\nu\rho\sigma}=$
$=2R_{\mu\nu}R^{\mu}{}_{\rho\sigma\tau}R^{\nu\rho\sigma\tau}+4\nabla_{\mu}\nabla_{\nu}R_{\rho\sigma}R^{\mu\rho\nu\sigma}-R_{\mu\nu\rho\sigma}R^{\rho\sigma}{}_{\tau\omega}R^{\tau\omega\mu\nu}-4R_{\mu\nu\rho\sigma}R^{\mu\tau\rho\omega}R^{\nu}{}_{\tau}{}^{\sigma}{}_{\omega},$
(34)
$\nabla_{\mu}R_{\nu\rho\sigma\tau}\nabla^{\mu}R^{\nu\sigma\rho\tau}=\frac{1}{2}\nabla_{\mu}R_{\nu\rho\sigma\tau}\nabla^{\mu}R^{\nu\rho\sigma\tau},$
(35)
$\nabla_{\mu}R_{\nu\rho\sigma\tau}\nabla^{\nu}R^{\mu\rho\sigma\tau}=\frac{1}{2}\nabla_{\mu}R_{\nu\rho\sigma\tau}\nabla^{\mu}R^{\nu\rho\sigma\tau},$
(36)
$\nabla_{\mu}R_{\nu\rho\sigma\tau}\nabla^{\nu}R^{\mu\sigma\rho\tau}=\frac{1}{2}\nabla_{\mu}R_{\nu\rho\sigma\tau}\nabla^{\nu}R^{\mu\rho\sigma\tau}=\frac{1}{4}\nabla_{\mu}R_{\nu\rho\sigma\tau}\nabla^{\mu}R^{\nu\rho\sigma\tau},$
(37) ${\rm and}\quad
R_{\mu\nu}R^{\mu}{}_{\rho\sigma\tau}R^{\nu\sigma\rho\tau}=\frac{1}{2}R_{\mu\nu}R^{\mu}{}_{\rho\sigma\tau}R^{\nu\rho\sigma\tau}.$
(38)

##  IV Derivation of $\Delta_{6}$ in arbitrary $d$

###  IV.1 Method of derivation of $\Delta_{6}$ in general $d$

Since now we have selected and described the basis of terms (both in
$\Delta_{6,\,{\rm der}}$ and in $\Delta^{\prime}_{6}$) which will be present
in the generalized form of the Hamada operator in an arbitrary dimension $d$,
we can describe shortly our method of derivation of the conformal scalar
operator $\Delta_{6}$. The operator will be determined (up to possible
ambiguities like in two parameters $\zeta_{1}$ and $\zeta_{2}$ in (21))
completely by specifying the values of all coefficients $z_{1},\ldots,z_{21}$
and $x_{1},\ldots,x_{17}$ where the normalization is kept by fixing the
coefficient in front of $\square^{3}$ to unity in the expansion of the
$\Delta_{6}$ operator. We will solve the system of linear equations needed to
determine the $21+17=38$ coefficients $z_{1},\ldots,z_{21}$ and
$x_{1},\ldots,x_{17}$ or relations between them, if there will be some free
parameters here.

The condition of conformal invariance of the action (30) implies that

$\delta_{c}S=\frac{\delta S}{\delta
g_{\mu\nu}}\delta_{c}g_{\mu\nu}+\frac{\delta S}{\delta\phi}\delta_{c}\phi=0.$
(39)

This last condition for conformally invariant theories is also known as
Noether identity for local conformal symmetry. Remembering that under
infinitesimal local conformal transformations we have that

$\delta_{c}g_{\mu\nu}=2\Omega g_{\mu\nu}$ (40)

and

$\delta_{c}\phi=w\Omega\phi,$ (41)

this condition is rewritten as

$0=\delta_{c}S=2\frac{\delta S}{\delta g_{\mu\nu}}\Omega
g_{\mu\nu}+\frac{\delta S}{\delta\phi}w\Omega\phi$ (42)

or defining the densitized trace of the energy-momentum tensor of the system
and the scalar equation of motion (EOM)

$\sqrt{|g|}T=\sqrt{|g|}g_{\mu\nu}T^{\mu\nu}=2\frac{\delta S}{\delta
g_{\mu\nu}}$ (43)

and

$E=\frac{\delta S}{\delta\phi},$ (44)

this condition is written as

$0=\delta_{c}S=\sqrt{|g|}T\Omega+Ew\Omega\phi=\Omega\left(\sqrt{|g|}T+Ew\phi\right).$
(45)

One sees explicitly that the trace of the EMT of the matter action defined in
(30) is not vanishing, but is balanced by the term proportional to the scalar
classical equation of motion. When $w=0$, so we consider the situation in
critical spacetime dimensions $d=n$ (as we also explain in the next section),
then the trace $T$ of EMT of the scalar matter system must vanish identically.
If we are not in a critical dimension, then the trace $T$ of the EMT in matter
models conformally coupled to gravitation, vanishes but only on-shell (that is
using EOM from the matter sector). In general off-shell situation one finds
the following expression for the densitized EMT in conformally coupled matter
models

$\sqrt{|g|}T=-wE\phi.$ (46)

However, in our derivation for the algebraic system of equations, we decide
not to use integration by parts after performing the conformal transformations
on terms and this implies that the equation in (39) has only a formal meaning.
The conformal variations of metric and of the scalar fields will be still
under the derivatives in our variation of the action $\delta_{c}S$. Instead,
we remark that such operations of integration by parts under spacetime volume
integral of the action of the model are typically performed to obtain the
expressions for the EMT and for the matter EOM as considered above.

###  IV.2 Consequences of global scale-invariance

First, it is instructive to analyze the situation for global scale invariance,
so the case in which $\Omega$ being the parameter of infinitesimal conformal
transformations is constant ($\Omega={\rm const}$) and hence all derivatives
acting on it vanish identically. Then conformal invariance agrees with the
results of dimensional analysis and this is only an algebraic constraint on
the possible value of the conformal weight $w$ of the scalar field. Here, we
can analyze the situation for the first leading term $\square^{\frac{n}{2}}$
(in number of derivatives) of the $\Delta_{n}$ operator in $d$ dimensions.
This part of the action now reads

$\int d^{d}x\sqrt{|g|}\phi\square^{\frac{n}{2}}\phi.$ (47)

Requiring for this to be dimensionless quantity, we find the energy dimension
of the scalar field

$[\phi]=E^{\frac{d-n}{2}}$ (48)

or in terms of the conformal weight

$w=-\frac{d-n}{2}=\frac{n-d}{2}.$ (49)

This counting of energy dimensions agrees precisely with the counting of
coefficients in front of $\Omega$ for the infinitesimal transformations. And
similarly for conformal invariance we want to have that
$\delta_{c}S\propto\Omega$ with the proportionality coefficient equal to zero.
For this one has to recall that for global infinitesimal conformal
transformation

$\delta_{c}\left(d^{d}x\sqrt{|g|}\right)=d\Omega d^{d}x\sqrt{|g|},$ (50)
$\delta_{c}\left(\square\right)=\delta_{c}\left(g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\right)=-2\Omega
g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}=-2\Omega\square,$ (51) ${\rm
and}\quad\delta_{c}\phi=w\Omega\phi$ (52)

in precise opposite analogy (the minus sign is only different) to the energy
dimension assignments of the same elements of the action, namely

$\left[d^{d}x\sqrt{|g|}\right]=E^{-d},$ (53)
$\left[\square\right]=\left[g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\right]=\left[\nabla\right]^{2}=E^{2},$
(54) ${\rm and}\quad[\phi]=E^{\frac{d-n}{2}}.$ (55)

The conformal weight $w=\frac{n-d}{2}$ of the scalar field $\phi$ derived this
way must be uniformly valid for all terms in the operator $\Delta_{n}$, hence
this is already a first result for the issue of determination of $\Delta_{6}$
in general dimensions $d$. This is of course in agreement with the general
methodology of our construction of the scalar conformal operators $\Delta_{n}$
in general spacetime of dimensionality $d$. First, we need to assign the
conformal weight $w$ to the scalar field $\phi$ and only after this we can
search for coefficients of all terms in the expansion of the operator
$\Delta_{n}.$ As emphasized above the first task is easily solved by
exploiting global scale-invariance (or dimensional analysis).

###  IV.3 Linearized infinitesimal conformal transformations

For other terms in the conformal variation of the action $\delta_{c}S$, where
the derivatives act on spacetime-dependent parameter of local conformal
transformation $\Omega=\Omega(x)$, it is enough to resort to considering
invariance of the action under infinitesimal transformations, so to the linear
level in $\Omega$, where we neglect terms with higher powers of it. This
corresponds to the analysis of the algebra of conformal transformations.
Finite conformal transformations are obtained by exponentiation of
infinitesimal ones and we expect no problem with invariance with respect to
them, if the first is established provided that there are no any topological
obstacles to finite conformal transformations. The detail computation shows
that the scalar actions with $\Delta_{6}$ as we find them below are without
any problems invariant also under finite conformal rescalings.

After performing the conformal transformation, we need to order terms
resulting from the variation and keep only these linear in $\Omega$. For this
we can only keep terms linear in the derivatives of the scalar $\Omega$. The
global part was analyzed above with the results for $w$ only. Next, we need to
use Bianchi identities, cyclicity of Riemann tensor and commutation of
derivatives to reduce the terms and find only linearly independent combination
of them in the final result for $\delta_{c}S$ written in an irreducible basis.
This step is very important to get a correct solution for the unknown
coefficients $z_{1},\ldots,z_{21}$ and $x_{1},\ldots,x_{17}$. We remark that
the number of equations we get is typically bigger than $21+17=38$, because
the number of independent terms in the results for $\delta_{c}S$ is bigger
than the total number of invariants in (28) and in (32). For example, in
$\delta_{c}S$ we have scalars with derivatives on $\Omega$, which are neither
present in the basis in (28), nor in (32). Hence the basis of all possible
terms in $\delta_{c}S$ is a completely different and unrelated to the ones in
(28), (32).

Only for the global case, when $\Omega={\rm const}$ we find in $\delta_{c}S$
the same terms as in $\Delta_{6}$ just multiplied by $\Omega$. Therefore, we
must have precisely $22+17=39$ terms in the results for $\delta_{c}S$ with
$\Omega={\rm const}$666One more term without associated a numerical $z$
coefficient in the basis (28) is simply $\square^{3}$.. All of the conditions
for vanishing of these terms hold, when $w=\frac{n-d}{2}$ is plugged in.

For terms with derivatives of $\Omega$ we find in $\delta_{c}S$ precisely
$37+22=59$ independent terms (this decomposition will be explained below).
Hence, at the end we get a system of $59$ equations for $38$ unknown
coefficients. The check that this is not a contradictory system is a very
important verification step for our method of computation (there are $21$
excess equations). At the end, we find that this system still has some free
parameters (like $\zeta_{1}$ and $\zeta_{2}$ in the case of the original
Hamada operator). We will not write explicitly the independent terms in the
basis of these $59$ elements of expansion of $\delta_{c}S$, since this would
be quite lengthy and moreover this basis will not be used by us further. We
will need it just to specify the system of linear equations for $38$
coefficients.

Actually, here one can easily determine and distinguish the form of terms in
$\delta_{c}S$, which come from conformal variation of the part
$\Delta^{\prime}_{6}$ of the $\Delta_{6}$ operator. These terms will of course
carry with them the linear dependence on the $x_{1},\ldots,x_{17}$
coefficients. It is clear from their character that they will result
exclusively in terms, which will be proportional to $\phi^{2}$, so no
derivatives will act on the scalar field, while they may act on $\Omega$ or on
other gravitational curvatures. We can single out these basis terms, which are
proportional to $\phi^{2}$. We find precisely $22$ of them. The other $39$
terms are built with derivatives on one of the scalar field (right field) and
on $\Omega$. We find the ordering of terms there in number of derivatives
acting on scalars quite useful. To deal with these terms and with the big
systems of linear equations we find help from special Mathematica packages.

As a matter of fact one sees that the system of coefficients
$x_{1},\ldots,x_{17}$ can be solved (or relations between them found) based on
mentioned above $22$ equations arising from $22$ independent terms in
$\delta_{c}S$ proportional to $\phi^{2}$. On the other hand, for the
coefficients $z_{1},\ldots,z_{21}$, we need to solve or reduce the bigger
system of $37$ equations. At the end, two systems of equations are coupled, so
the solutions for $x_{1},\ldots,x_{17}$ depend also on solutions found for
$z_{1},\ldots,z_{21}$.

###  IV.4 Solutions for coefficients of $\Delta_{6}$

The results for the solutions of these systems are as follows. The system of
$z_{1},\ldots,z_{21}$ coefficients has the form

$z_{1}=\frac{16}{d-2}$
$z_{2}=-\frac{3d^{2}-12d+44}{4(d-1)(d-2)}=-\frac{3(d-6)^{2}+24(d-6)+80}{4(d-1)(d-2)}$
$z_{3}=\frac{16}{d-2}$
$z_{4}=-\frac{(d-6)(3d-10)}{2(d-1)(d-2)}=-\frac{(d-6)[3(d-6)+8]}{2(d-1)(d-2)}$
$z_{5}=-\frac{(d-6)(d-8)}{(d-1)(d-4)}=-\frac{(d-6)[(d-6)-2]}{(d-1)(d-4)}$
$z_{6}=\frac{8}{d-4}$
$z_{7}=-\frac{3d^{2}-26d+72}{4(d-1)(d-4)}=-\frac{3(d-6)^{2}+10(d-6)+24}{4(d-1)(d-4)}$
$z_{10}=\frac{4\left[\left(z_{8}+4\right)d-4\left(z_{8}+6\right)\right]}{(d-2)(d-4)}=\frac{4\left[\left(z_{8}+4\right)(d-6)+2z_{8}\right]}{(d-2)(d-4)}$
$z_{11}=-\frac{2\left(z_{8}-8\right)d}{(d-2)^{2}}=-\frac{2\left(z_{8}-8\right)[(d-6)+6]}{(d-2)^{2}}$
$z_{12}=-\frac{2\left[2\left(z_{9}+1\right)d^{2}-\left(z_{8}+12z_{9}+16\right)d+4\left(z_{8}+4z_{9}+10\right)\right]}{(d-2)^{2}(d-4)}=$
$=-\frac{2\left[2\left(z_{9}+1\right)(d-6)^{2}+\left(-z_{8}+12z_{9}+8\right)(d-6)-2\left(z_{8}-8z_{9}-8\right)\right]}{(d-2)^{2}(d-4)}$
$z_{13}=-\frac{4\left[d^{3}-\left(z_{8}+8\right)d^{2}+\left(5z_{8}+36\right)d-4\left(z_{8}+16\right)\right]}{(d-1)(d-2)^{2}(d-4)}=$
$=-\frac{4\left[(d-6)^{3}-\left(z_{8}-10\right)(d-6)^{2}+\left(-7z_{8}+48\right)(d-6)-10\left(z_{8}-8\right)\right]}{(d-1)(d-2)^{2}(d-4)}$
$z_{14}=\frac{1}{16(d-1)^{2}(d-2)^{2}(d-4)}\left\\{3d^{5}-36d^{4}+8\left(4z_{9}+29\right)d^{3}-32\left(z_{8}+7z_{9}+30\right)d^{2}+\right.$
$\left.+16\left(10z_{8}+28z_{9}+155\right)d-64\left(2z_{8}+4z_{9}+39\right)\right\\}=$
$=\frac{1}{16(d-1)^{2}(d-2)^{2}(d-4)}\left\\{3(d-6)^{5}+54(d-6)^{4}+32\left(z_{9}+14\right)(d-6)^{3}-\right.$
$\left.-32\left(z_{8}-11z_{9}-60\right)(d-6)^{2}-32\left(7z_{8}-38z_{9}-136\right)(d-6)-64\left(5z_{8}-20z_{9}-72\right)\right\\}$
$z_{15}=-\frac{d-8}{d-1}=-\frac{(d-6)-2}{d-1}$ $z_{16}=z_{8}+4z_{9}$
$z_{17}=-\frac{2(d-4)z_{8}}{d-2}=-\frac{2[(d-6)+2]z_{8}}{d-2}$
$z_{18}=\frac{2\left[\left(z_{8}+8\right)d-4z_{8}\right]}{(d-2)^{2}}=\frac{2\left[\left(z_{8}+8\right)(d-6)+2\left(z_{8}+24\right)\right]}{(d-2)^{2}}$
$z_{19}=-\frac{4\left[\left(z_{8}+2z_{9}+4\right)d-3z_{8}-4z_{9}-16\right]}{(d-2)^{2}}=-\frac{4\left[\left(z_{8}+2z_{9}+4\right)(d-6)+3z_{8}+8z_{9}+8\right]}{(d-2)^{2}}$
$z_{20}=-\frac{d^{3}+\left(z_{8}-10\right)d^{2}+\left(-5z_{8}+4\right)d+4\left(z_{8}+10\right)}{(d-1)(d-2)^{2}}=$
$=-\frac{(d-6)^{3}+\left(z_{8}+8\right)(d-6)^{2}+\left(7z_{8}-8\right)(d-6)+10\left(z_{8}-8\right)}{(d-1)(d-2)^{2}}$
$z_{21}=\frac{3d^{4}-40d^{3}+8\left(2z_{8}+4z_{9}+27\right)d^{2}-32\left(2z_{8}+3z_{9}+25\right)d+16\left(3z_{8}+4z_{9}+55\right)}{8(d-1)^{2}(d-2)^{2}}=$
$=\frac{3(d-6)^{4}+32(d-6)^{3}+16\left(z_{8}+2z_{9}+9\right)(d-6)^{2}+32\left(4z_{8}+9z_{9}+2\right)(d-6)+16\left(15z_{8}+40z_{9}-56\right)}{8(d-1)^{2}(d-2)^{2}},$
(56)

while the system of solutions for $x_{1},\ldots,x_{17}$ is

$x_{1}=\frac{1}{16(d-1)^{2}(d-2)^{2}(d-4)}\left\\{3d^{5}-44d^{4}+8\left(z_{8}+4z_{9}+37\right)d^{3}-8\left(11z_{8}+44z_{9}-32x_{17}+152\right)d^{2}+\right.$
$\left.+16\left(17z_{8}+68z_{9}-80x_{17}+163\right)d-64\left(3z_{8}+12z_{9}-16x_{17}+33\right)\right\\}=$
$=\frac{1}{16(d-1)^{2}(d-2)^{2}(d-4)}\left\\{3(d-6)^{5}+46(d-6)^{4}+8\left(z_{8}+4z_{9}+40\right)(d-6)^{3}+\right.$
$\left.+8\left(7z_{8}+28z_{9}+32x_{17}+136\right)(d-6)^{2}+16\left(5z_{8}+20z_{9}+112z_{17}+88\right)(d-6)+2560x_{17}\right\\}$
$x_{2}=\frac{-\left(z_{8}+4z_{9}+4\right)d^{2}+2\left(5z_{8}+20z_{9}-16x_{17}+16\right)d-8\left(3z_{8}+12z_{9}-16x_{17}+6\right)}{(d-2)^{2}(d-4)}=$
$=-\frac{\left(z_{8}+4z_{9}+4\right)(d-6)^{2}+2\left(z_{8}+4z_{9}+16x_{17}+6\right)(d-6)+64x_{17}}{(d-2)^{2}(d-4)}$
$x_{3}=-\frac{1}{64(d-1)^{3}(d-2)^{3}(d-4)}\left\\{d^{7}-16d^{6}+4\left(8z_{9}+35\right)d^{5}-16\left(6z_{9}-4x_{10}+53\right)d^{4}-\right.$
$-16\left(132z_{9}+64x_{9}+16x_{10}-192x_{17}-231\right)d^{3}+64\left(190z_{9}+104x_{9}-5x_{10}-304x_{17}-186\right)d^{2}-$
$\left.-64\left(300z_{9}+184x_{9}-24x_{10}-512x_{17}-309\right)d+256\left(36z_{9}+24x_{9}-4x_{10}-64x_{17}-45\right)\right\\}=$
$=-\frac{1}{64(d-1)^{3}(d-2)^{3}(d-4)}\left\\{(d-6)^{7}+26(d-6)^{6}+32\left(z_{9}+10\right)(d-6)^{5}+\right.$
$+32\left(27z_{9}+2x_{10}+71\right)(d-6)^{4}+64\left(111z_{9}-16x_{9}+20x_{10}+48x_{17}+156\right)(d-6)^{3}+$
$+64\left(352z_{9}-184x_{9}+139x_{10}+560x_{17}+408\right)(d-6)^{2}+256\left(90z_{9}-166x_{9}+99x_{10}+512x_{17}+120\right)(d-6)-$
$\left.-1280\left(36x_{9}-19x_{10}-112x_{17}\right)\right\\}$
$x_{4}=\frac{1}{(d-1)(d-2)^{3}(d-4)}\left\\{\left(z_{9}+1\right)d^{4}+\left(2z_{9}+3x_{10}-12\right)d^{3}+\right.$
$+\left(-112z_{9}-48x_{9}-9x_{10}+128x_{17}+72\right)d^{2}+8\left(56z_{9}+33x_{9}-3x_{10}-84x_{17}-34\right)d+$
$\left.+16\left(-24z_{9}-18x_{9}+3x_{10}+40x_{17}+21\right)\right\\}=$
$=\frac{1}{(d-1)(d-2)^{3}(d-4)}\left\\{\left(z_{9}+1\right)(d-6)^{4}+\left(26z_{9}+3x_{10}+12\right)(d-6)^{3}+\right.$
$+\left(140z_{9}-48x_{9}+45x_{10}+128x_{17}+72\right)(d-6)^{2}+8\left(23z_{9}-39x_{9}+24x_{10}+108x_{17}+20\right)(d-6)+$
$\left.+4\left(-108x_{9}+57x_{10}+304x_{17}\right)\right\\}$
$x_{5}=-\frac{z_{9}d(d-4)-12z_{9}-24x_{9}+6x_{10}+32x_{17}}{4(d-1)(d-2)}=-\frac{z_{9}(d-6)^{2}+8z_{9}(d-6)+2\left(-12x_{9}+3x_{10}+16x_{17}\right)}{4(d-1)(d-2)}$
$x_{6}=\frac{z_{8}d^{2}+2\left(-4z_{8}-4z_{9}+8x_{9}-3x_{10}\right)d+4\left(3z_{8}+12z_{9}-4x_{9}+2x_{10}-16x_{17}\right)}{(d-2)^{3}}=$
$=\frac{z_{8}(d-6)^{2}+2\left(2z_{8}-4z_{9}+8x_{9}-3x_{10}\right)(d-6)+4\left(20x_{9}-7x_{10}-16x_{17}\right)}{(d-2)^{3}}$
$x_{7}=-\frac{1}{(d-2)^{2}(d-4)}\left\\{\left(z_{8}+12z_{9}+3x_{10}\right)d^{2}-2\left(5z_{8}+60z_{9}+12x_{9}+6x_{10}-48x_{17}-8\right)d+\right.$
$\left.+24\left(z_{8}+12z_{9}+4x_{9}-16x_{17}-4\right)\right\\}=$
$=-\frac{1}{(d-2)^{2}(d-4)}\left\\{\left(z_{8}+12z_{9}+3x_{10}\right)(d-6)^{2}-2\left(-z_{8}-12z_{9}+12x_{9}-12x_{10}-48x_{17}-8\right)(d-6)-\right.$
$\left.-12\left(4x_{9}-3x_{10}-16x_{17}\right)\right\\}$
$x_{8}=\frac{2z_{9}(d-6)-12x_{9}+3x_{10}+16x_{17}}{d-2}=\frac{2z_{9}(d-6)-12x_{9}+3x_{10}+16x_{17}}{d-2}$
$x_{11}=-\frac{2(d-6)^{2}}{(d-1)(d-2)(d-4)}$
$x_{12}=\frac{(d-6)\left(d^{3}-\left(z_{8}+10\right)d^{2}+\left(z_{8}-16z_{9}+32x_{17}+52\right)d+8\left(2z_{9}-4x_{17}-7\right)\right)}{8(d-1)^{2}(d-2)^{2}}=$
$=\frac{(d-6)}{8(d-1)^{2}(d-2)^{2}}\left\\{(d-6)^{3}+\left(-z_{8}+8\right)(d-6)^{2}+\right.$
$\left.+\left(-11z_{8}-16z_{9}+32x_{17}+40\right)(d-6)+2\left(-15z_{8}-40z_{9}+80x_{17}+56\right)\right\\}$
$x_{13}=\frac{z_{8}d^{2}-2\left(3z_{8}-8z_{9}+12x_{17}+4\right)d-16\left(6z_{9}-11x_{17}-3\right)}{2(d-2)^{2}}=$
$=\frac{z_{8}(d-6)^{2}+2\left(3z_{8}+8z_{9}-4-12x_{17}\right)(d-6)+32x_{17}}{2(d-2)^{2}}$
$x_{14}=-\frac{d-6}{4(d-1)}$
$x_{15}=-\frac{z_{8}d^{2}-2\left(3z_{8}-8z_{9}+8x_{17}\right)d-32\left(3z_{9}-5x_{17}\right)}{2(d-2)^{2}}=$
$=-\frac{z_{8}(d-6)^{2}+2\left(3z_{8}+8z_{9}-8x_{17}\right)(d-6)+64x_{17}}{2(d-2)^{2}}$
$x_{16}=\frac{\left(z_{8}+4z_{9}\right)d-2\left(3z_{8}-12z_{9}+16x_{17}\right)}{d-2}=\frac{\left(z_{8}+4z_{9}\right)(d-6)+32x_{17}}{d-2}.$
(57)

One has to make several observations pertaining to the new results obtained
above. First, in solving for the system of $z_{1},\ldots,z_{21}$ we find again
two-parameter freedom. We have chosen $z_{9}$ and $z_{10}$ as free parameters
following the choice made by Hamada motivated firstly in $d=6$ dimensions. All
solutions for $z$ coefficients: from $z_{1}$ to $z_{8}$ and from $z_{11}$ to
$z_{21}$ depend only on these two parameters and the dimension of spacetime
$d$. The system is solved for $19$ coefficients in terms of $2$ parameters
$z_{9}$, $z_{10}$ and the dimension $d$.

For the system of $17$ a priori unknown coefficients $x_{1},\ldots,x_{17}$ we
find that we were able to determine $14$ of them in terms of $3$ free
parameters from the set $x_{1},\ldots,x_{17}$, with addition of $z_{9}$ and
$z_{10}$ from the previous set and the dimension of spacetime $d$. As free
parameters here we decide to choose $x_{9}$, $x_{10}$ and $x_{17}$. The reason
for such a choice will be explained later. Therefore we solved for $x$
coefficients from $x_{1}$ to $x_{8}$ and from $x_{11}$ to $x_{16}$.

Now, we comment on the dependence on dimension in the results above. First,
one notices that some coefficients explode in dimensions $d=4$ and $d=2$ and
$d=1$ due to denominators, where we find only decomposable polynomials of the
dimension $d$. The first two problematic dimensions are even dimensions lower
than the critical dimension $d=6$ for the $\Delta_{6}$ conformal operator. One
sees that generally one is unable to define the conformal operator in
dimensions $d=2$ and $d=4$ by any limiting procedure. Below we can prove as a
theorem that for dimensions smaller by two than the critical one for a given
operator this is impossible and one cannot have the operator $\Delta_{n}$ in
dimensions $d=n-2$ obtained using this method. In numerators of the
expressions above we find polynomials of $d$ and of parameters $z_{9}$ and
$z_{10}$ (and also of $x_{9}$, $x_{10}$ and $x_{17}$ for the solved $x$
coefficients). The degree in $d$ variable of these polynomials is up to fifth
in solved $z$ coefficients (and up to seventh in $x$ coefficients), while in
other variables $z_{9}$ and $z_{10}$ (and also $x_{9}$, $x_{10}$ and $x_{17}$)
the polynomials are linear.

##  V Singularities in construction of conformal operators

###  V.1 Hamada operator in $d=6$ reconsidered

Next, we consider the limiting situation in critical dimension $d=6,$ which is
special for the operator $\Delta_{6}$. To facilitate this we wrote above when
this was possible the expansion of the polynomials in the numerator in the
shifted variable $(d-6)$. One notices that the constant term of this last
expansion is what really matters for the situation in $d=6$. We therefore
reproduce exactly the coefficients as found by Hamada. In the basis chosen by
him they take the form

$z_{1}=4,\quad z_{2}=-1,\quad z_{3}=4,\quad z_{4}=0,\quad z_{5}=0,\quad
z_{6}=4,\quad z_{7}=-\frac{3}{5},$ $z_{8}=\zeta_{1},\quad
z_{9}=\zeta_{2},\quad z_{10}=z_{8},\quad
z_{11}=-\frac{3}{4}\left(z_{8}-8\right),\quad
z_{12}=\frac{1}{8}\left(z_{8}-8z_{9}-8\right),\quad
z_{13}=\frac{1}{4}\left(z_{8}-8\right),$
$z_{14}=-\frac{1}{200}\left(5z_{8}-20z_{9}-72\right),\quad
z_{15}=\frac{2}{5},\quad z_{16}=z_{8}+4z_{9},\quad z_{17}=-z_{8},\quad
z_{18}=\frac{1}{4}\left(z_{8}+24\right),$
$z_{19}=-\frac{1}{4}\left(3z_{8}+8z_{9}+8\right),\quad
z_{20}=-\frac{1}{8}\left(z_{8}-8\right),\quad
z_{21}=\frac{1}{200}\left(15z_{8}+40z_{9}-56\right).$ (58)

Notice that the coefficients $z_{4}$ and $z_{5}$ vanish in $d=6$ since their
general dimension expressions are both proportional to the $(d-6)$ factor.
Also almost all coefficients after $z_{10}$ (except $z_{15}$) carry some
dependence on $z_{8}$ and $z_{9}$ coefficients, so when we set them both to
zero a few of coefficients after $z_{10}$ get vanish. This may have some
explanation below. Finally, one also notices that the particular combination
$\left(z_{8}-8\right)$ appears above quite frequently, so again for the
special choice $z_{8}=8$ more coefficients vanish.

When one analyzes the situation with $x_{1},\ldots,x_{17}$ coefficients again
in the limit $d=6$, one finds that a few terms there drop out (namely
$x_{11}$, $x_{12}$ and $x_{14}$), namely we have

$x_{1}=\frac{x_{17}}{5},\quad x_{2}=-2x_{17},\quad
x_{3}=\frac{36x_{9}-19x_{10}-112x_{17}}{800},\quad
x_{4}=\frac{-108x_{9}+57x_{10}+304x_{17}}{160},$
$x_{5}=\frac{12x_{9}-3x_{10}-16x_{17}}{40},\quad
x_{6}=\frac{20x_{9}-7x_{10}-16x_{17}}{16},\quad
x_{7}=\frac{3\left(4x_{9}-3x_{10}-16x_{17}\right)}{8},$
$x_{8}=\frac{-12x_{9}+3x_{10}+16x_{17}}{4},\quad x_{11}=0,\quad x_{12}=0,\quad
x_{13}=x_{17},\quad x_{14}=0,\quad x_{15}=-2x_{17},\quad x_{16}=8x_{17},$ (59)

while the other are linear functions of only the coefficients from the $x$
set, that is of $x_{9}$, $x_{10}$ and $x_{17}$. The constant terms in the
numerator (independent of the above three variables) do not appear and there
is no any dependence on the two free parameters $z_{8}$ and $z_{9}$. This
proves that it is consistent to set all three $x_{9}$, $x_{10}$ and $x_{17}$
to zero and end up with Hamada operator as a conformal scalar operator with
six derivatives in $d=6$ without a need to include terms in
$\Delta^{\prime}_{6}$ there. Here, instead we provide the generalization of
the Hamada operator still in $d=6$, when we include additional three-parameter
family of endomorphism terms acting on scalar $\phi$ described by these
$x_{9}$, $x_{10}$ and $x_{17}$ free coefficients. We will show below that when
we change the basis this extension of the Hamada operator has a natural
interpretation. In a sense the size of this freedom has to do with the number
of Weyl co-cycles that one can create with cubes of Weyl tensors [119, 117].

Although the system of equations as analyzed before was derived from the
invariance under infinitesimal conformal transformations, we also checked that
generalized Hamada operator is invariant under finite transformations too.
Therefore this proves that there is not any topological obstacle in the group
space of the full conformal group and we can easily extend the results
obtained in the vicinity of identity (or origin in the conformal algebra) to
the full group.

###  V.2 Theorem about the nonexistence of $\Delta_{n}$ in $d=n-2$

We already have seen that this theorem works for $n=4$ and $d=2$ (singular
case of the generalized Paneitz operator) and for $n=6$ and $d=4=n-2$ and
$d=2=n-4$ from the generalized Hamada operator.

To prove this theorem in a bigger generality, one has to recall a small number
of facts about the construction of the operator $\Delta_{n}$ in general
dimension $d$. The leading (in derivatives) term of this operator must
necessarily have the form $\square^{\frac{n}{2}}$. Moreover, following the
listing of terms as written by Hamada we know that the following term with
exactly two derivatives acting on the scalar field $\phi$ can be present

$z_{*}\square^{\frac{n-4}{2}}R_{\mu\nu}\nabla^{\mu}\nabla^{\nu}\phi,$ (60)

of course, among many others. But for the sake of the proof it is necessary to
concentrate on these two terms in $\Delta_{n}$. Next, one recalls the
infinitesimal conformal transformation laws for the scalar field $\phi$ (with
for the moment arbitrary conformal weight $w$) and for the covariant metric
tensor of spacetime, which are

$\delta_{c}\phi=w\Omega\phi,$ (61) $\delta_{c}g_{\mu\nu}=2\Omega g_{\mu\nu}.$
(62)

The main idea of the proof concentrates on the following term

$X=\square^{\frac{n-4}{2}}R_{\mu\nu}\nabla^{\mu}\Omega\nabla^{\nu}\phi$ (63)

in the long expression for the infinitesimal local conformal variation of the
action of the operator on the scalar field, so on the quantity
$\delta_{c}\left(\Delta_{n}\phi\right)$. We will not write other terms in
$\delta_{c}\left(\Delta_{n}\phi\right)$ since they are irrelevant and, of
course, for increasing $n$ the number of these terms grows very fast.

The first pertinent observation is that only two terms in the general
expansion of the operator $\Delta_{n}$ contribute anything to the $X$ term in
$\delta_{c}\left(\Delta_{n}\phi\right)$. They are precisely: the leading term
(with fixed overall coefficient)

$\square^{\frac{n}{2}}\phi$ (64)

and the one we have singled out already

$z_{*}\square^{\frac{n-4}{2}}R_{\mu\nu}\nabla^{\mu}\nabla^{\nu}\phi.$ (65)

That all other possible terms in $\Delta_{n}\phi$ do not contribute to $X$ one
can easily convince oneself by considering the structure of the background of
the $X$ term. The choice $\square^{\frac{n-4}{2}}R_{\mu\nu}$ as the background
quantity in $X$ is very peculiar since the number of derivatives acting on
Ricci tensor is here maximal (if in the $X$ term we have to have derivatives
of conformal parameter $\Omega$) and hence it is very difficult to obtain it
as a result of conformal variation from other terms in $\Delta_{6}$.

In order to arrive to such results of the expansion of the conformal variation
of the action of the operator on the scalar field $\Delta_{n}\phi$, one must
pick up some convention how to order derivatives in various terms. We decide
to pick up one which seems to us a very natural (and it is also motivated by
the work of Barvinsky, Vilkovisky and others). First, derivative operators
should act on a single quantity (not on a product of them), like on a scalar
field $\phi$, parameter $\Omega$ or on some gravitational curvatures. To
achieve this one has to sufficiently many times perform expansion of the
action of derivatives using the Leibniz rules. When derivatives acting on the
same quantity have indices contracted, these two derivatives should be
collected in powers of the GR-covariant box operator
($\square=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$). When derivatives and box
differential operators act on the same quantity, then the box operators should
act with priority. To achieve this order of covariant derivatives and boxes
acting before covariant derivatives, one has to sufficiently many times use
commutation of covariant derivatives, producing of course terms with
gravitational curvatures. This is a convention of writing all terms in
$\delta_{c}\left(\Delta_{n}\phi\right)$. This convention still leaves some
ambiguity (of how to order free uncontracted derivatives), however for the
term $X$ this is completely unambiguous (since there are no more than one free
covariant derivative acting on scalars $\phi$ and $\Omega$ there). Moreover,
to continue with the proof some convention has to be chosen, but of course,
the final result of nonexistence does not depend at all on the choice of
convention of how the terms are written in
$\delta_{c}\left(\Delta_{n}\phi\right)$, because different ways of writing all
of them are perfectly equivalent and they related by identities from
differential geometry.

Having discussed the convention of writing terms in
$\delta_{c}\left(\Delta_{n}\phi\right)$, we can now give final results. We
find that the conformal variation of the leading term
$\square^{\frac{n}{2}}\phi$ contains

$\delta_{c}\left(\square^{\frac{n}{2}}\phi\right)=\delta_{c}\left(\square^{\frac{n}{2}-2}\square^{2}\phi\right)\supset\square^{\frac{n}{2}-2}\left(\delta_{c}\left(\square^{2}\phi\right)\right)\supset
2(2w+d-2)\square^{\frac{n}{2}-2}R_{\mu\nu}\nabla^{\mu}\Omega\nabla^{\nu}\phi,$
(66)

where the inclusion symbol $\supset$ denotes that of course there are also
other terms in the conformal variation, but they do not interest us here. To
obtain this formula we use first the general formula for the conformal
variation of the scalar box acting on some scalar field $Y$

$\delta_{c}\left(\square Y\right)=\square\delta_{c}Y-2\Omega\square
Y+(d-2)\nabla_{a}\Omega\nabla_{a}Y,$ (67)

where $Y$ is a scalar from the point of view of GR, but it does not even have
to have assigned a conformal weight777It is only important everything which is
encoded in the transformation $\delta_{c}Y$.. Of course, when $Y=\phi$, we
know that conformal weight can be assigned, and then $w(Y)=w(\phi)$, but for
example such a weight cannot be defined when $Y=\square\phi$, since the last
does not transform co-covariantly. Here we treat the operator
$\square^{\frac{n}{2}-2}$ as a spectator acting on the variation of
$\square^{2}\phi$ and that is why nothing in the coefficient of the result
depend on $n$. We order terms with derivatives in this conformal variation and
we use the commutation rules

$\left[\square,\nabla_{\mu}\right]\phi=R_{\mu\nu}\nabla^{\nu}\phi$ (68)

and

$\left[\square,\nabla_{\mu}\right]\Omega=R_{\mu\nu}\nabla^{\nu}\Omega.$ (69)

Similarly, we find that

$\delta_{c}\left(\square^{\frac{n-4}{2}}R_{\mu\nu}\nabla^{\mu}\nabla^{\nu}\phi\right)\supset\square^{\frac{n-4}{2}}R_{\mu\nu}\delta_{c}\left(\nabla^{\mu}\nabla^{\nu}\phi\right)\supset
2(w-1)\square^{\frac{n}{2}-2}R_{\mu\nu}\nabla^{\mu}\Omega\nabla^{\nu}\phi.$
(70)

It is important for the proof to notice the exact coefficients in front of the
$X$ term in the conformal variation $\delta_{c}\left(\Delta_{n}\phi\right)$.
In the second expression we see that the final result does not completely
depend on $n$ since $\square^{\frac{n}{2}-2}R_{\mu\nu}$ is treated here like a
spectator for this derivation. Similarly, there is no dependence on the
dimension $d$ of the spacetime, since the dimension is not needed in the
linearized conformal variation

$\delta_{c}\left(\nabla_{\mu}\nabla_{\nu}\phi\right)=\nabla_{\mu}\nabla_{\nu}\delta_{c}\phi-2\nabla_{(\mu}\Omega\nabla{}_{\nu)}\phi+g_{\mu\nu}\nabla_{\rho}\Omega\nabla^{\rho}\phi.$
(71)

Now, the key point for the proof is that the conformal weight of the scalar
field $\phi$ is in $d=n-2$ equal precisely to $w=1$. Then this implies that
the contribution of the leading term is non-vanishing in
$\delta_{c}\left(\Delta_{n}\phi\right)$

$\delta_{c}\left(\square^{\frac{n}{2}}\phi\right)\supset
2d\square^{\frac{n}{2}-2}R_{\mu\nu}\nabla^{\mu}\Omega\nabla^{\nu}\phi,$ (72)

while from the other term we find

$\delta_{c}\left(\square^{\frac{n-4}{2}}R_{\mu\nu}\nabla^{\mu}\nabla^{\nu}\phi\right)\supset
2\left.(w-1)\right|_{w=1}\square^{\frac{n}{2}-2}R_{\mu\nu}\nabla^{\mu}\Omega\nabla^{\nu}\phi=0.$
(73)

So no contribution from the second term! Hence it is impossible by playing
with the front coefficient $z_{*}$ (of the second term) to cancel the
contribution

$2d\square^{\frac{n}{2}-2}R_{\mu\nu}\nabla^{\mu}\Omega\nabla^{\nu}\phi$ (74)

of the leading term to the conformal variation
$\delta_{c}\left(\Delta\phi\right)$ on a general non-Ricci-flat background.
The other possible terms in $\Delta_{n}$ cannot help with the cancellation of
this contribution either. Therefore by any choice of the arbitrary
coefficients (like $z_{*}$) in the expansion of the operator $\Delta_{n}$ we
are unable to make the whole local conformal variation vanish. In conclusion,
such a conformal operator $\Delta_{n}$ acting on scalar $\phi$ does not exist
in $d=n-2$ (so two below the critical dimension $d=n$ where the scalar field
is dimensionless). This completes the proof of the absence of $\Delta_{n}\phi$
conformal kinetic operator for scalars in spacetime of dimensionality $d=n-2$.
Probably the operator also does not exist for any other even dimension
$d<n-2$. We therefore may conjecture that for even dimensionalities $d<n$
smaller than critical $d_{{\rm crit}}=n$, the operator $\Delta_{n}$ is not
constructable in this way.

The particular examples of application of this theorem with $n=6$ (in $d=4$)
and with $n=4$ (in $d=2$) we have seen on explicit forms of the generalized
Paneitz and Hamada operators respectively.

##  VI Wunsch conformal covariant derivatives

In this section we discuss various other ways how the general Hamada operator
can be presented. In the sense we will employ different bases than this one
used in section IV and V.

###  VI.1 Review of the construction of conformal covariant derivative due to
Wunsch

Firstly, we comment on the results obtained by Wunsch [105, 106]. The main
element of his construction is the usage of the specially constructed
conformal covariant derivative, denoted by $\overset{c}{\nabla}_{\mu}$ and
differential operators built with it. In order to introduce the mathematical
description of this new covariant derivative, we need a few additional
technical details. We define the Schouten tensor $P_{\mu\nu}$ by

$P_{\mu\nu}=\frac{1}{d-2}\left(R_{\mu\nu}-\frac{1}{2(d-1)}g_{\mu\nu}R\right)$
(75)

in any dimension $d\neq 1,2$, where $R_{\mu\nu}$ and $R$ denote above standard
Ricci tensor and Ricci (curvature) scalar. The derived property of the
Schouten tensor is its trace, which equals to
$P=g^{\mu\nu}P_{\mu\nu}=\frac{R}{2(d-1)}$ in any dimension $d\neq 1$. In
particular, the last formula, holds also for the case $d=2$, despite the
formula in (75) being apparently singular in $d=2$. Knowing that in $d=2$ the
Ricci tensor is not independent since it is completely expressed via the Ricci
scalar with the formula $R_{\mu\nu}=\frac{1}{2}g_{\mu\nu}R$ and the same for
other curvature tensors, symmetric and with two indices (so also
$P_{\mu\nu}=\frac{1}{2}g_{\mu\nu}P$), one can reexpress the Schouten tensor
there. We have that in $d=2$, $P=\frac{1}{2}R$, hence
$P_{\mu\nu}=\frac{R}{4}g_{\mu\nu}$ and this proves that the Schouten tensor is
not singular in $d=2$ and its expression is finite and does not vanish since
for 2-dimensional manifolds Ricci scalar $R$ can take arbitrary values.
However, we have to be very careful with the expressions for Schouten tensor
and its trace in $d=2,$ as we also show later (with the discussion of the
impossibility of the construction of the Paneitz operator in $d=2$).

An important property of the Schouten tensor is that under infinitesimal
conformal transformation its variation takes quite simple form, namely
$\delta_{c}P_{\mu\nu}=-\nabla_{\mu}\nabla_{\nu}\Omega$, where $\Omega$ is the
parameter of transformation. Below we will work with differential operators
acting only on scalar fields $\phi$, which have already good transformation
law under conformal symmetry. That is we assume, as earlier, that

$\delta_{c}\phi=w_{0}\Omega\phi.$ (76)

The idea of the construction with conformal covariant derivatives acting in
sequence and eventually on a scalar field $\phi$ is the following. We treat a
sequence of conformal covariant derivatives with various Lorentz indices as
one differential operator, which always acts on a scalar $\phi$ and in this
way we do not have to consider conformal derivatives acting on different more
complicated representations of the Poincare group. For example, the expression
$\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi$ (to be defined
shortly) we treat as one differential operator acting on $\phi$ and not as a
Wunsch conformal covariant derivative $\overset{c}{\nabla}_{\nu}$ acting on a
GR vector $v_{\mu}=\overset{c}{\nabla}_{\mu}\phi$.

We initiate the explicit construction of Wunsch conformal covariant
derivatives by the simple case of one derivative. Then we have the definition
that

$\overset{c}{\nabla}_{\mu}\phi=\nabla_{\mu}\phi.$ (77)

Below we summarize the important conformal transformation properties of the
first conformal covariant derivative. We have first that

$\delta_{c}\left(\overset{c}{\nabla}_{\mu}\phi\right)=w_{0}\nabla_{\mu}\Omega\phi+w_{0}\Omega\nabla_{\mu}\phi.$
(78)

The derivative on the scalar field $\phi$ can be for free promoted to the
conformal one here. We can decompose this expression of the infinitesimal
conformal transformation (always linear in the parameter $\Omega$) for the
part without and with only first derivatives of the parameter $\Omega$
according to

$\delta_{c}\left(\overset{c}{\nabla}_{\mu}\phi\right)=\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\mu}\phi\right)\right)\right|_{\Omega}+\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\mu}\phi\right)\right)\right|_{\nabla\Omega},$
(79)

where we also find now the identifications that

$\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\mu}\phi\right)\right)\right|_{\Omega}=w_{0}\Omega\nabla_{\mu}\phi=w_{0}\Omega\overset{c}{\nabla}_{\mu}\phi$
(80)

(the last equality is written to be consistent with the formalism of all
conformal derivatives) and

$\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\mu}\phi\right)\right)\right|_{\nabla\Omega}=w_{0}\nabla_{\mu}\Omega\phi=\nabla_{\gamma}\Omega
X^{\gamma}{}_{\mu}\phi,$ (81)

where in the last equality we also have defined the new tensor $X$, here with
two indices in the mixed position $X^{\gamma}{}_{\mu}$, so the first GR-
covariant derivative of the parameter $\Omega$ can be with arbitrary index
$\gamma$ and there is a linear matrix multiplication involved to get
$\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\mu}\phi\right)\right)\right|_{\nabla\Omega}$
from the first derivatives of $\Omega$. Explicitly, we can define formally the
tensor $X$ as the result of the following linear operation of double
variational derivative:

$X^{\gamma}{}_{\mu}=\frac{\delta^{2}\left(\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\mu}\phi\right)\right)\right|_{\nabla\Omega}\right)}{\delta\left(\nabla_{\gamma}\Omega\right)\delta\phi}$
(82)

and its expression reads $X^{\gamma}{}_{\mu}=w_{0}\delta_{\mu}^{\gamma}$ in
this simple case.

Next, we proceed recursively, definining a sequence of two conformal
derivative as

$\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi=\nabla_{\nu}\overset{c}{\nabla}_{\mu}\phi+P_{\nu\gamma}X^{\gamma}{}_{\mu}\phi=\nabla_{\nu}\nabla_{\mu}\phi+P_{\nu\gamma}X^{\gamma}{}_{\mu}\phi,$
(83)

where the tensor $X^{\gamma}{}_{\mu}=w_{0}\delta_{\mu}^{\gamma}$ we use from
the previous definitions for the first conformal derivative. One can pause
here for a moment to contemplate the properties of the just defined new
differential operators with respect to conformal transformations. We have them
explicitly. The infinitesimal conformal variation of
$\overset{c}{\nabla}_{\mu}\phi$ contains only terms with no derivatives on the
parameter $\Omega$ or with precisely first derivative on $\Omega$. Regarding
the transformation of $\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi$
one can convince oneself that the same is true, mainly because of the
properties of the Schouten tensor and the fact that the $X^{\gamma}{}_{\mu}$
tensor is not touched by conformal transformation since it is built with
Kronecker delta(s). For the last case, the speciality is that there are no
terms like $\nabla_{\alpha}\nabla_{\beta}\Omega$ generated in the expression
for
$\delta_{c}\left(\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi\right)$.
This is the defining property and feature of Wunsch construction. His
conformal covariant derivative operators $\overset{c}{\nabla}_{\mu}$ transform
infinitesimally only up to the first derivative of the parameter $\Omega$ and
higher derivatives are not allowed. (For the specification and counting of
derivatives on $\Omega$ we can just use standard covariant derivatives from
GR, that is $\nabla_{\mu}$.)

One can see here that this notion of “conformal” covariant derivative is
generalized since in standard gauge theory (like in QED, QCD) the respective
covariant derivatives transform without terms with derivatives of the
parameters of gauge transformations; they are constructed in such a way to
transform precisely as the matter field they are acting upon. This is there
obtained by introducing special gauge connections which balance the first and
higher derivatives of the parameter of gauge transformations in the law for
gauge transformations of total gauge covariant derivatives acting on some
matter field charged with respect to a gauge group. Here, with conformal
symmetry, we are with a different set of tricks. Firstly, we do not employ
additional conformal connection, special for the conformal transformations,
although this is possible and it would lead to the conformal gauge theory.
Secondly, since we accept that the first (in a sequence, that is the most
internal or the most closed to the scalar field $\phi)$ conformal covariant
derivative is identical to the standard GR-covariant derivative (and also
identical here to the partial derivative since on a scalar we get
$\nabla_{\mu}\phi=\partial_{\mu}\phi$), then we must also accept that this
derivative has the conformal transformation law with precisely first
derivative on $\Omega$. And there is no a theoretical way to cancel these
first derivatives keeping general covariance of the results (this is why here
we cannot use for compensation the Christoffel symbols of the first kind, that
is $\Gamma^{\mu}=g^{\nu\rho}\Gamma^{\mu}{}_{\nu\rho}$ or
$\Gamma_{\rho}=\Gamma^{\mu}{}_{\mu\rho}$). Had we also accepted that the
sequence of two derivatives is identical with two normal GR-derivatives, then
we would have to accept inevitability of two derivatives on $\Omega$ in the
conformal transformation law and so on for higher terms. We decided, following
Wunsch, not to use a special conformal connection, which would be a GR vector
field $A_{\mu}$ quite similar to the electromagnetic potential in QED, which
is alien to the GR framework; however, we could still use the geometric
elements: tensors and connections as all used in framework of differential
geometry. As a matter of fact, we cannot compensate and modify the first
conformal covariant derivative $\overset{c}{\nabla}_{\mu}$ and we must accept
the fact of the presence of derivatives of $\Omega$ in the transformation
laws. However, for the definition of the sequence of two conformal derivatives
we may employ some addition of a special tensor, as in (83), chosen in such a
way to precisely cancel the terms with two derivatives of the conformal
parameter $\Omega$. In a sense, the Schouten tensor plays here the “role” of
conformal compensator for two derivatives (so it acts like a conformal
“connection” but it is built entirely out of the elements available in GR,
does not require new additional connection, and it behaves as a tensor with
respect to the diffeomorphism transformations of GR, differently than for
example the Levi-Civita connection in GR). The Schouten tensor is the first
object that we could use here from GR, which is built with tensors and has a
useful infinitesimal conformal transformation law. The fact that it is a
tensor helps because then also the new differential operator
$\overset{c}{\nabla}_{\mu}$ behaves like a good tensorial (or precisely
vectorial) operatorial expression from the point of view of GR.

Consistently with the previous discussion, here we also list the properties
under infinitesimal conformal transformations of the just defined sequence of
two Wunsch conformal derivatives. We have first that

$\delta_{c}\left(\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi\right)=w_{0}\Omega\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi+2\left(w_{0}-1\right)\nabla_{(\nu}\Omega\overset{c}{\nabla}_{\mu)}\phi+g_{\nu\mu}\nabla_{\gamma}\Omega\overset{c}{\nabla}{}^{\gamma}\phi.$
(84)

The derivatives on the scalar field $\phi$ are all written as conformal ones
here. We can decompose this expression of the infinitesimal conformal
transformation (always linear in the parameter $\Omega$) for the part without
and with only first derivatives of the parameter $\Omega$ according to

$\delta_{c}\left(\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi\right)=\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi\right)\right)\right|_{\Omega}+\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi\right)\right)\right|_{\nabla\Omega},$
(85)

where we also find now the following identifications that

$\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi\right)\right)\right|_{\Omega}=w_{0}\Omega\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi$
(86)

and

$\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi\right)\right)\right|_{\nabla\Omega}=\left(w_{0}-1\right)\nabla_{\nu}\Omega\overset{c}{\nabla}_{\mu}\phi+\left(w_{0}-1\right)\nabla_{\mu}\Omega\overset{c}{\nabla}_{\nu}\phi+g_{\nu\mu}\nabla_{\gamma}\Omega\overset{c}{\nabla}{}^{\gamma}\phi$
(87)

(the last equality is written to be consistent with the formalism of all
conformal derivatives on $\phi$). The last equation can be also written in the
formal way as

$\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi\right)\right)\right|_{\nabla\Omega}=\nabla_{\gamma}\Omega
X^{\gamma}{}_{\nu\mu}{}^{\alpha}\overset{c}{\nabla}_{\alpha}\phi,$ (88)

where in the last equality we also have defined the new tensor $X$, here with
four indices in the mixed position $X^{\gamma}{}_{\nu\mu}{}^{\alpha}$, here
two indices are covariant and two others are contravariant, but these are all
not at all equivalent indices and their positions matter. Therefore, in the
expression (88) the first GR-covariant derivative of the parameter $\Omega$
can be with arbitrary index $\gamma$ and there is a linear matrix
multiplication involved to get
$\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi\right)\right)\right|_{\nabla\Omega}$
from the first derivatives of $\Omega$. Moreover, we have also a linear matrix
multiplication to produce
$\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi\right)\right)\right|_{\nabla\Omega}$
from the first conformal covariant derivatives of the scalar field $\phi$
here, i.e. $\overset{c}{\nabla}_{\alpha}\phi$ with an arbitrary index
$\alpha$. This form of the equation as in (88) containing two first
derivatives of $\Omega$ and $\phi$ respectively is also required based on
dimensional arguments. Explicitly, here we can define formally the tensor $X$
as the result of the following linear operation of double variational
derivative:

$X^{\gamma}{}_{\nu\mu}{}^{\alpha}=\frac{\delta^{2}\left(\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi\right)\right)\right|_{\nabla\Omega}\right)}{\delta\left(\nabla_{\gamma}\Omega\right)\delta\left(\overset{c}{\nabla}_{\alpha}\phi\right)}$
(89)

and its expression based on (87) reads

$X^{\gamma}{}_{\nu\mu}{}^{\alpha}=\left(w_{0}-1\right)\delta_{\nu}^{\gamma}\delta_{\mu}^{\alpha}+\left(w_{0}-1\right)\delta_{\mu}^{\gamma}\delta_{\nu}^{\alpha}+g_{\nu\mu}g^{\gamma\alpha}$
(90)

in this case. By inspection of the above formula, we see that this tensor is
accidentally symmetric in $\mu$ and $\nu$ pair of covariant indices, together
with explicit symmetry in contravariant indices $\gamma$ and $\alpha$.
However, the role of the two contravariant indices $\gamma$ and $\alpha$ is
different, hence we view this last symmetry as without deeper meaning which is
also proved by considerations for higher number of derivatives, where it is
absent.

Now, we discuss the next step in the recurrence procedure of defining the
sequence of conformal covariant derivatives. For three consecutive
derivatives, we postulate that

$\overset{c}{\nabla}_{\rho}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi=\nabla_{\rho}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi+P_{\rho\gamma}X^{\gamma}{}_{\nu\mu}{}^{\alpha}\overset{c}{\nabla}_{\alpha}\phi,$
(91)

where this time more complicated version of the $X$ tensor with four indices
we used from the formula (90). We see that in the definition (91) we invoke
the expression with two conformally covariant derivatives (83), when the most
external (most distant from the scalar field $\phi$) derivative, i.e. here
$\overset{c}{\nabla}_{\rho}$, is opened. At the same time, in (91), the form
of the $X$ tensor is more complicated because of addition of two new indices
and finally due to dimensional arguments, since the $X$ tensor is always
constructed as dimensionless, at the end the expression must be proportional
to the first conformal covariant derivative on the scalar field $\phi$, i.e.
$\overset{c}{\nabla}_{\alpha}\phi$. Here this last in the expression, the
first covariant derivative comes with a Lorentz index $\alpha$ that is a dummy
index and contracted with the last contravariant index $\alpha$ on the $X$
tensor. The more complicated form of the $X$ tensor is required because of new
indices, however its form is unique to establish the fact that the sequence of
three Wunsch conformal covariant derivatives on the scalar field
$\overset{c}{\nabla}_{\rho}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi$
under conformal transformation changes maximally only up to the first
derivatives of the parameter $\Omega$. One notices that the $X$ tensor here is
covariantly conserved that is
$\nabla_{\beta}X^{\gamma}{}_{\nu\mu}{}^{\alpha}=0$ (because it is constructed
out of Kronecker deltas and metric tensors) and that it does not at all
transform under conformal transformations, i.e.
$\delta_{c}X^{\gamma}{}_{\nu\mu}{}^{\alpha}=0$ (because it has the same number
of covariant and contravariant indices – all on deltas or on metrics).

We also have the following properties of three conformal covariant derivatives
under the transformations of conformal symmetry. We have first the known
decomposition

$\delta_{c}\left(\overset{c}{\nabla}_{\rho}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi\right)=\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\rho}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi\right)\right)\right|_{\Omega}+\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\rho}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi\right)\right)\right|_{\nabla\Omega},$
(92)

where we obviously and naturally find that

$\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\rho}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi\right)\right)\right|_{\Omega}=w_{0}\Omega\overset{c}{\nabla}_{\rho}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi,$
(93)

since this is the part without derivatives on $\Omega$, so effectively when
the parameter of conformal transformation $\Omega$ is constant, so we consider
here only global conformal transformations. Obviously $w_{0}$ is the conformal
weight of the scalar $\phi$ and of any sequence of conformal (or GR-covariant,
or partial) derivatives acting on it, because for the definition of the weight
only global transformations matter. For the part with precisely one derivative
on $\Omega$, the last equation can be also written in the formal way as

$\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\rho}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi\right)\right)\right|_{\nabla\Omega}=\nabla_{\gamma}\Omega
X^{\gamma}{}_{\rho\nu\mu}{}^{\alpha\beta}\overset{c}{\nabla}_{\alpha}\overset{c}{\nabla}_{\beta}\phi,$
(94)

where in the last equality we also have defined the new tensor $X$, here with
six indices (twice the number of conformal derivatives in (94)) in the mixed
position $X^{\gamma}{}_{\rho\nu\mu}{}^{\alpha\beta}$, here three indices are
covariant (coming from the indices on conformal derivatives on the LHS and we
preserved their order) and three others are contravariant, but these are all
not at all equivalent indices and their positions matter. Therefore, in the
expression (94) the first GR-covariant derivative of the parameter $\Omega$
can be with arbitrary index $\gamma$ and there is a linear matrix
multiplication involved to get
$\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\rho}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi\right)\right)\right|_{\nabla\Omega}$
from the first derivatives of $\Omega$. Moreover, we have also a linear matrix
multiplication to produce
$\left.\left(\delta_{c}\left(\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\nu}\overset{c}{\nabla}_{\mu}\phi\right)\right)\right|_{\nabla\Omega}$
from the two conformal covariant derivatives of the scalar field $\phi$ here,
they are in a
sequence$\overset{c}{\nabla}_{\alpha}\overset{c}{\nabla}_{\beta}\phi$ with
arbitrary indices names $\alpha$ and $\beta$. This form of the equation as in
(94) containing exactly one first derivative of $\Omega$ and one second order
conformal covariant derivative of $\phi$ respectively is also required based