# Notes on conformal anomaly, nonlocal effective action and the metamorphosis
of the running scale

A. O. Barvinsky<EMAIL_ADDRESS>Theory Department, Lebedev Physics
Institute, Leninsky Prospect 53, Moscow 119991, Russia  Institute for
Theoretical and Mathematical Physics, Moscow State University, Leninskie Gory,
GSP-1, Moscow, 119991, Russia  W. Wachowski<EMAIL_ADDRESS>Theory
Department, Lebedev Physics Institute, Leninsky Prospect 53, Moscow 119991,
Russia

###### Abstract

We discuss the structure of nonlocal effective action generating the conformal
anomaly in classically Weyl invariant theories in curved spacetime. By the
procedure of conformal gauge fixing, selecting the metric representative on a
conformal group orbit, we split the renormalized effective action into
anomalous and Weyl invariant parts. A wide family of thus obtained anomalous
actions is shown to include two special cases of Riegert–Fradkin–Tseytlin and
Fradkin–Vilkovisky actions. Both actions are shown to be contained in the
first three orders of the curvature expansion for a generic one-loop effective
action obtained by covariant perturbation theory. The complementary Weyl
invariant part of the action is given by the “conformization” of the full
effective action—restricting its argument to the conformally invariant
representative of the orbit of the conformal group. This is likely to resolve
a long-standing debate between the proponents of the Riegert action and
adherents of the perturbation expansion for the effective action with typical
nonlocal logarithmic form factors. We derive the relation between quantum
stress tensors on conformally related metric backgrounds, which generalizes
the known Brown-Cassidy equation to the case of nonzero Weyl tensor, and
discuss applications of this relation in the cosmological model driven by
conformal field theory. We also discuss the issue of renormalization group
running for the cosmological and gravitational coupling constants and show
that it exhibits a kind of a metamorphosis to the nonlocal form factors of the
so-called partners of the cosmological and Einstein terms—nonlocal curvature
squared terms of the effective action.

###### Contents

  1. 1 Introduction
  2. 2 Conformal gauge fixing
    1. 2.1 Riegert–Fradkin–Tseytlin gauge
    2. 2.2 Fradkin-Vilkovisky gauge
  3. 3 Conformal anomaly and covariant curvature expansion
    1. 3.1 Cubic order
    2. 3.2 Conformal resummation: Fradkin–Vilkovisky anomaly action
    3. 3.3 The problem of double poles and global conformal transformations
  4. 4 Stress tensor in conformally related spacetimes
    1. 4.1 Conformal anomaly from the divergent part of the effective action
    2. 4.2 Minimal form of Wess-Zumino action and a-theorem
    3. 4.3 Renormalized stress tensors
  5. 5 Conformally flat spacetime
    1. 5.1 Anomaly driven cosmology
  6. 6 Renormalization group and the metamorphosis of the running scale
  7. 7 Conclusions

To the memory of Stanley Deser

##  1 Introduction

The status of local Weyl anomalies is widely considered to be fully settled in
current literature. However, the issue of their relevance to concrete physical
effects, as opposed to a mere criterion of consistency at the quantum level of
the classically Weyl invariant theories, often remains a subject of the
debate. The manifestation of the conformal anomaly in physical applications
usually occurs within the effective action formalism, and there is extending
over years debate on the structure of this action, taking place between the
pioneers of the conformal anomaly and adherents of perturbation theory. The
nature of this debate consists in a seemingly contradictory difference between
the known expression for the anomaly action and the form of the nonlocal
effective action obtained by Feynman diagrammatic technique.

As is well known, the one-loop conformal anomaly for classically Weyl
invariant 4-dimensional theory having in Euclidean curved spacetime the
covariantly renormalized effective action $\varGamma[\,g_{\mu\nu}]$ reads as
[1, 2, 3, 4, 5, 6, 7]

$\displaystyle\langle\,T^{\mu}_{\mu}\,\rangle\equiv\frac{2\,g_{\mu\nu}}{\sqrt{g}}\frac{\delta\varGamma}{\delta
g_{\mu\nu}}=\frac{1}{16\pi^{2}}\big{(}\alpha C^{2}+\beta E+\gamma\Box
R\big{)},$ (1.1)
$\displaystyle\;E=R_{\mu\nu\alpha\gamma}R^{\mu\nu\alpha\gamma}-4R_{\mu\nu}R^{\mu\nu}+R^{2},$
(1.2)

where $\sqrt{g}E$ denotes the Gauss–Bonnet density, $C_{\mu\nu\alpha\beta}$ is
the Weyl tensor, $C^{2}=C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}$, and
$\alpha$, $\beta$ and $\gamma$ are the numerical coefficients depending on the
spin of the quantum field.111We work in Euclidean signature spacetime, and our
notations are
$R^{\alpha}{}_{\beta\mu\nu}=\partial_{\mu}\varGamma^{\alpha}_{\nu\beta}-\cdots$,
$R_{\mu\nu}=R^{\alpha}{}_{\mu\alpha\nu}$,
$\Box=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$. For simplicity we do not include in
the anomaly the contribution $F_{\mu\nu}^{2}$ of the vector gauge field and
$\varphi^{4}$-contribution of the self-interacting conformal scalar field. The
anomalous action $\varGamma_{A}[\,g_{\mu\nu}]$ generating this anomaly was
first derived in the nonlocal form by Riegert [8] and by Fradkin and Tseytlin
[9] in the local form of the conformal Wess-Zumino action involving an
auxiliary scalar field—the dilaton responsible for intetwining two conformally
related metrics. The nonlocal form of the Riegert–Fradkin–Tseytlin (RFT)
action reads as

$\displaystyle\varGamma_{A}[\,g\,]$ $\displaystyle=\frac{1}{64\pi^{2}}\int
d^{4}x\,\sqrt{g}\,\Big{(}\alpha\,C^{2}+\frac{\beta}{2}\mathcal{E}_{4}\Big{)}\frac{1}{\Delta}_{4}\mathcal{E}_{4}$
$\displaystyle-\frac{1}{32\pi^{2}}\Big{(}\frac{\gamma}{6}+\frac{\beta}{9}\Big{)}\,\int
d^{4}x\,\sqrt{g}R^{2},$ (1.3)

where

${\cal E}_{4}\equiv E-\tfrac{2}{3}\Box R,$ (1.4)

$\Delta_{4}$ denotes the so-called Paneitz operator [10]

$\Delta_{4}=\Box^{2}+2R^{\mu\nu}\nabla_{\mu}\nabla_{\nu}-\frac{2}{3}R\,\Box+\frac{1}{3}(\nabla^{\mu}R)\,\nabla_{\mu}$
(1.5)

and $1/\Delta_{4}$ implies its inverse—the notation for the operation of
acting by its Green’s function ${\cal G}(x,y)$ on a generic test function
$\psi(y)$, $\Delta_{4}{\cal G}(x,y)=\delta(x,y)$,
$\tfrac{1}{\Delta}_{4}\psi(x)=\int d^{4}y\,{\cal G}(x,y)\,\psi(y)$.

Some time after the invention of the RFT action the attention to it was drawn
by Antoniadis, Mazur and Mottola due to several applications in gravity theory
[11, 12], but this caused a serious criticism [13] of the expression (1.3) in
view of its drastic structural difference from the renormalized effective
action built within perturbation theory in powers of spacetime curvature. This
expansion begins with [3]

$\displaystyle\varGamma_{\rm ren}$ $\displaystyle=\frac{1}{32\pi^{2}}\int
dx\,\sqrt{g}\Big{[}-\alpha\,C_{\mu\nu\alpha\beta}\ln\Big{(}\\!-\frac{\Box}{\mu^{2}}\Big{)}C^{\mu\nu\alpha\beta}$
$\displaystyle-\frac{\gamma}{6}R^{2}\,\Big{]}+O(\Re^{3}),$ (1.6)

$\Re$ collectively denoting here the Riemann, Ricci and scalar curvature, and
does not at all resemble the form of (1.3). This criticism was maintained by
objections against short distance behavior of stress tensor correlation
functions generated by the RFT action, which were shown to contradict the
conformal Ward identities for these correlator [14]. Another criticism was
associated with the objections against the double pole structure of the
Green’s function of the operator (1.5), $\sim 1/\Box^{2}$ [15]. Although these
objections were disclaimed in [16] by explicit calculations of $\langle
TTT\rangle$-correlators, the question might still be hovering unsettled in the
literature [17].

The goal of this paper will be to discuss the status of the effective action
responsible for the generation of the Weyl anomaly. To begin with we will
focus on a wide variety of nonlocal anomalous actions by including the RFT
action in their functional family. The idea of this construction is similar to
gauge fixing applied to the ambiguity of the conformal split of the metric
argument of the action functional, which was suggested rather long ago in
[18]. The resulting class of anomaly actions will be parameterized by the
conformal gauge selecting the representative on the orbit of the local
conformal group. We will explicitly demonstrate that the difference between
the members of this class is a Weyl invariant functional—a point of departure
between various suggestions for the anomalous action. Two particular gauges
will be considered, one of them exactly corresponding to the RFT action (1.3)
and another associated with the Weyl invariant nonlocal rescaling of the
metric field suggested by Fradkin and Vilkovisky. This rescaling, which is
directly applicable in asymptotically flat spacetimes, was designed as a
remedy against the trace anomaly [19]—the analogue of the Yamabe problem of a
local Weyl transformation to the metric with a vanishing scalar curvature.

Then we show how the Fradkin–Vilkovisky version of the anomaly action arises
in the first three orders of the covariant curvature expansion for a generic
one-loop effective action. We discuss the associated mechanism of partial
summation of scalar curvature terms of this expansion [20] along with the
double pole problem for the Green function of the Paneitz operator (1.5).

Lack of uniqueness of the anomaly action defined only up to a Weyl invariant
functional raises, of course, the question of its incompleteness in concrete
applications. This also poses the question of whether the RFT action or its
modifications within the above class provides an optimal description of the
physical problem in question. For example, it is well known that in two
dimensions the stress tensor trace anomaly and the associated nonlocal
Polyakov action are fully responsible for the Hawking radiation of the two
dimensional black holes [21]. On the contrary, in higher dimensions the
anomaly action is insufficient to describe this phenomenon. Still there is a
strong belief [11, 12, 16] that at distances of the horizon scale gravity
theory is essentially modified due to large infrared effects of the conformal
mode described by the action (1.3). These effects might dominate macroscopic
physics at such scales, like for instance the near black hole horizon behavior
of quantum stress tensor [22], the contribution to the scalar sector of
gravitational waves [23] or dynamical vacuum energy in effective theory of
gravity [24]. Though it is not entirely clear how complete is the setup in
these problems, there are physical situations when the conformal mode really
runs the whole show, and we consider as a direct application of (1.3) two
examples of such a situation. These are the calculation of the metric stress
tensor in a generic conformally flat spacetime [25] and the Friedmann metric
cosmology driven by the trace anomaly of conformal invariant fields [26], the
latter playing important role in the model of initial conditions for
inflationary cosmology [27, 28].

A related issue in the problem of nonlocal effective action is the question of
renormalization group (RG) running of the cosmological and gravitational
constants. Though the issue of running scale and its relation to the
cosmological constant problem has already become a byword in current
literature, it becomes increasingly clearer that this running should not be
interpreted in the usual sense of RG theory [29, 30]. The notion of “scale” is
so ambiguous in physics that its running nature actually looses universality
when addressing various physical setups, like for example associating
cosmological inflation with RG running [31]. Serious arguments against running
nature of the cosmological and gravitational couplings in [29, 30] have led to
the notion of cosmological constant partners [32] interpreted in [33] in terms
of separation of scales or decoupling of heavy modes [34, 35]. Still, it is
customary to have nontrivial solutions of RG equations in renormalizable
gravity models [36, 37] with running scale dependent $\varLambda$ and $G$.
Therefore a natural question arises how these solutions have to be interpreted
when the tadpole structure of the covariant cosmological and Einstein terms
preclude them from their actual dependence on the momentum [30].

So one of the goals of this paper is an attempt to clarify this issue within a
special version of the notion of the “scale”. Looking forward to the final
conclusion, we might formulate the suggestion for the notions of running
$\varLambda$ and $G$ couplings as their conversion or metamorphosis into their
nonlocal partners similar to those introduced by J. Donoghue in [32]. Within
perturbation scheme the cosmological and Einstein terms start manifesting
themselves as nonlocal curvature squared terms very different from their
original form.

The paper is organized as follows. In Sect. 2 we decompose the quantum
effective action into anomalous and Weyl invariant parts by imposing the
conformal gauge for the choice of the representative on the orbit of the
conformal group. This allows one to build the whole class of nonlocal
anomalous actions, functionally parameterized by the choice of this gauge and
including the RFT action (1.3) and the Fradkin–Vilkovisky action suggested in
[20]. Sect. 3 contains the discussion of the covariant curvature expansion of
[38, 39, 40] and the way how it contains the anomalous action in the lowest
orders of this expansion. In particular, it is shown that the
Fradkin–Vilkovisky version of this action performs a resummation of the
covariant curvature series in powers of the Ricci scalar [20]. In Sect. 4 we
give a direct and, apparently, not very well known derivation from the RFT
action of the vacuum stress-tensor behavior at the orbit of the conformal
group—a good example of direct applicability of (1.3). Here we also comment on
the application of the anomalous conformal Wess-Zumino action to the a-theorem
[41, 42] and present the generalization of the Brown–Cassidy formula [25] for
the stress tensor to the case of a nonzero Weyl tensor, see Eq.(4.34).
Applications of the anomaly action in conformally flat spacetime are presented
in Sect. 5. It is shown how this action underlies the construction of the
inflation scenario starting from the cosmological initial state in the form of
the mircocanonical density matrix [27, 28, 43, 44, 45], recently reviewed in
[46]. Important feature of this application is the value of the Casimir vacuum
energy which is also determined by the coefficients of the anomalous trace
(1.1) [47, 48, 49, 50].

In Sect.6 we discuss the problem of scale dependence of the gravitational and
cosmological constants related to the ideas of [29, 30, 32] and [51, 52]. Here
we show that in the UV regime the RG analysis of the cosmological and Einstein
terms strongly points out to the conversion of their scale dependence into the
nonlocal form factors of their UV partners represented by curvature squared
terms with dimensionless nonlocal coefficients. We call this phenomenon a
metamorphosis of the running scale, which we derive by using a special scaling
operator. In IR domain the same analysis leads to the low energy partners
depending on mass scale of the theory. These nonlocal partners were suggested
in [32] by J. Donoghue for the cosmological constant term and blueprinted for
the Einstein term in [51, 52] in the form of the long distance modification of
Einstein gravity.

In the concluding section we briefly recapitulate the above observations and
dwell on related potential problems and applications. We start by discussing
the role of Weyl anomaly in the problem of cosmological initial conditions for
the inflation scenario driven by a conformal field theory [46]. This scenario
motivates introduction of numerous conformal higher spin (CHS) fields whose
Weyl anomaly is generated only in the one-loop approximation and, thus,
acquires a kind of nonperturbative status. Then we discuss the uniqueness for
the nonlocal scaling operator used for the derivation of the above
metamorphosis phenomenon. In particular, we show that in the curvature squared
terms of the action it is nearly uniquely determined due to general covariance
of the theory, though in Lorentz symmetry violating models like Hořava gravity
[53] it may be rather ambiguous.

##  2 Conformal gauge fixing

The splitting of the renormalized effective action of a classically
conformally invariant theory into the anomaly part $\varGamma_{A}$ generating
the trace anomaly (1.1) and the Weyl invariant part $\varGamma^{\rm conf}$,
$g_{\mu\nu}\delta\varGamma^{\rm conf}/\delta g_{\mu\nu}=0$,

$\varGamma_{\rm ren}=\varGamma_{A}+\varGamma^{\rm conf},$ (2.1)

is obviously not unique and admits the freedom

$\varGamma_{A}\to\varGamma_{A}+W^{\rm conf},\qquad\varGamma^{\rm
conf}\to\varGamma^{\rm conf}-W^{\rm conf},$ (2.2)

with an arbitrary conformally invariant functional $W^{\rm conf}$,

$g_{\mu\nu}\frac{\delta W^{\rm conf}}{\delta g_{\mu\nu}}=0.$ (2.3)

The freedom in the choice of $W^{\rm conf}[\,g_{\mu\nu}]$ arises as a
functional integration constant for the first order variational equation that
can be written down for $\varGamma_{A}[\,g_{\mu\nu}]$ or for the renormalized
effective action $\varGamma[\,g_{\mu\nu}]\equiv\varGamma_{\rm
ren}[\,g_{\mu\nu}]$. At the orbit of the conformal group passing through the
metric $g_{\mu\nu}$—the argument of the effective action—and parameterized by
the local conformal parameter $\sigma=\sigma(x)$,

$g_{\mu\nu}=e^{2\sigma}\bar{g}_{\mu\nu},$ (2.4)

the renormalized action $\varGamma_{\rm ren}[\,e^{\sigma}\bar{g}\,]$ satisfies
the equation

$\frac{\delta\varGamma_{\rm
ren}[e^{2\sigma}\bar{g}]}{\delta\sigma}=\frac{\sqrt{g}}{16\pi^{2}}\big{(}\alpha
C^{2}+\beta E+\gamma\Box
R\big{)}\Big{|}_{g_{\mu\nu}=e^{\sigma}\bar{g}_{\mu\nu}},$ (2.5)

which can be integrated to give conformal Wess-Zumino action [9]

$\displaystyle\Delta\varGamma[\,\bar{g},\sigma\,]\equiv\varGamma_{\rm
ren}[\,g\,]-\varGamma_{\rm ren}[\,\bar{g}\,]$
$\displaystyle\qquad=\frac{1}{16\pi^{2}}\int
d^{4}x\,\sqrt{\bar{g}}\left\\{\Big{[}\alpha\bar{C}^{2}+\beta\bar{\mathcal{E}}_{4}\Big{]}\sigma+2\beta\sigma\bar{\Delta}_{4}\sigma\right\\}$
$\displaystyle\qquad-\frac{1}{32\pi^{2}}\Big{(}\frac{\gamma}{6}+\frac{\beta}{9}\Big{)}\,\int
d^{4}x\,\Big{(}\sqrt{g}R^{2}-\sqrt{\bar{g}}\bar{R}^{2}\Big{)},$ (2.6)

where the two metrics $g_{\mu\nu}$ and $\bar{g}_{\mu\nu}$ are related by the
equation (2.4), all barred quantities are built in terms of $\bar{g}_{\mu\nu}$
and $\bar{\Delta}_{4}$ is the barred version of the fourth-order Paneitz
operator (1.5). This expression $\varGamma_{\rm ren}-\bar{\varGamma}_{\rm
ren}=\varGamma_{A}-\bar{\varGamma}_{A}$ can also be rewritten in the other
form

$\displaystyle\varGamma_{A}[\,g\,]-\varGamma_{A}[\,\bar{g}\,]$
$\displaystyle\qquad=\frac{1}{16\pi^{2}}\int
d^{4}x\sqrt{g}\,\left\\{\,\Big{[}\,\alpha\,C^{2}+\beta\mathcal{E}_{4}\Big{]}\,\sigma-2\beta\,\sigma\Delta_{4}\sigma\,\right\\}$
$\displaystyle\qquad-\frac{1}{32\pi^{2}}\Big{(}\frac{\gamma}{6}+\frac{\beta}{9}\Big{)}\,\int
d^{4}x\,\Big{(}\sqrt{g}R^{2}-\sqrt{\bar{g}}\bar{R}^{2}\Big{)},$ (2.7)

if one takes into account two important properties of the Paneitz
operator—Weyl invariance of its densitized form,

$\sqrt{\bar{g}}\,\bar{\Delta}_{4}=\sqrt{g}\,\Delta_{4},$ (2.8)

and the finite conformal transformation of ${\cal E}_{4}$—the Gauss-Bonnet
density modified by $\sqrt{g}\,\Box R$ term (1.4),

$\sqrt{g}\,\mathcal{E}_{4}=\sqrt{\bar{g}}\,\bar{\mathcal{E}}_{4}+4\sqrt{\bar{g}}\,\bar{\Delta}_{4}\sigma.$
(2.9)

These two properties are consistent with each other because the last equation
should obviously remain valid under the interchange of $g_{\mu\nu}$ and
$\bar{g}_{\mu\nu}$ accompanied by flipping the sign of $\sigma$.

There is also the third form of the Wess-Zumino action, which will be given
below in Eq.(4.24). It exists for a special renormalization converting to zero
the coefficient $\gamma$ of the $\Box R$ term in (1.1), and underlies the
proof of the so-called $a$-theorem for the monotonic RG flow of the
coefficient $a=\beta/16\pi^{2}$ of the topological term in the trace anomaly
[41, 42].

Modulo a nonvanishing conformal anomaly all points on the orbit of the
conformal group (2.4) are physically equivalent, and this typical situation of
a broken local gauge invariance can be managed by introducing the gauge
condition which uniquely selects $\bar{g}_{\mu\nu}$ as the representative of
the equivalence class of metrics (2.4). If we denote this gauge condition as
$\chi[\bar{g}]=0$ then this representative should be uniquely selected by the
solution of the equation for the conformal parameter $\sigma$,

$\chi[\,\bar{g}\,]=\chi[\,ge^{-2\sigma}]=0,$ (2.10)

this solution being a functional of the metric
$\varSigma_{\chi}[\,\bar{g}\,]$, labelled by the gauge symbol $\chi$,

$\sigma=\varSigma_{\chi}[\,\bar{g}\,].$ (2.11)

The representative of the conformal orbit $\bar{g}_{\mu\nu}[g]$ as a
functional of a given metric $g_{\mu\nu}$ (through which the orbit is passing)
becomes Weyl invariant,

$\bar{g}_{\mu\nu}[\,g\,]\equiv
g_{\mu\nu}\,e^{-2\varSigma_{\chi}[\,\bar{g}\,]},\quad
g_{\alpha\beta}\frac{\delta\bar{g}_{\mu\nu}[\,\bar{g}\,]}{\delta
g_{\alpha\beta}}=0,$ (2.12)

because under any local Weyl rescaling $g_{\mu\nu}\to e^{2\sigma}g_{\mu\nu}$
the conformal parameter transforms as
$\varSigma_{\chi}[g]\to\varSigma_{\chi}[g]+\sigma$ in view of the identity
$\chi\big{[}ge^{-\varSigma_{\chi}[g]}\big{]}\equiv 0$, so that

$\delta_{\sigma}\varSigma_{\chi}[\,\bar{g}\,]=\sigma,$ (2.13)

where $\delta_{\sigma}$ is the operator of the conformal variation

$\delta_{\sigma}\equiv 2\int
d^{4}x\;\sigma(x)\,g_{\mu\nu}(x)\frac{\delta}{\delta g_{\mu\nu}(x)}.$ (2.14)

For the uniqueness of such conformal gauge fixing procedure (in spacetime and
at least in some finite domain of the space of metrics) the Faddeev–Popov
operator $Q_{\chi}=Q_{\chi}(x,y)$, corresponding to the gauge $\chi[g]$,
$\delta_{\omega}\chi(x)=\int d^{4}y\,Q_{\chi}(x,y)\,\omega(y)$, should be
nondegenerate.

Thus, the terms of (2.7)

$\displaystyle W^{\rm
conf}[\,g\,]=\varGamma_{A}[\,\bar{g}\,]+\frac{1}{32\pi^{2}}\Big{(}\frac{\gamma}{6}+\frac{\beta}{9}\Big{)}\\!\int
d^{4}x\sqrt{\bar{g}}\,\bar{R}^{2}$ (2.15)

taken at $\bar{g}_{\mu\nu}[\,g\,]$ can be considered as an irrelevant Weyl
invariant integration “constant”, while the rest of the terms can be
identified with the anomaly action after the substitution of
$\sigma=\varSigma_{\chi}[\,g\,]$. This set of anomaly actions
$\varGamma_{A}[\,g\,]\equiv\varGamma_{\chi}[\,g\,]$ parameterized and labelled
by conformal gauge conditions $\chi$ reads as

$\displaystyle\varGamma_{\chi}[\,g\,]=\frac{1}{16\pi^{2}}\int
d^{4}x\,\sqrt{g}\,\Big{\\{}\big{(}\alpha
C^{2}+\beta\mathcal{E}_{4}\big{)}\varSigma_{\chi}$
$\displaystyle\;\;\;\;-2\beta\varSigma_{\chi}\Delta_{4}\varSigma_{\chi}\Big{\\}}-\frac{1}{32\pi^{2}}\Big{(}\frac{\gamma}{6}+\frac{\beta}{9}\Big{)}\\!\int
d^{4}x\,\sqrt{g}R^{2}.$ (2.16)

The difference between various members of this set is, of course, a Weyl
invariant functional. For two arbitrary conformal gauges one has

$\displaystyle\varGamma_{\chi_{1}}-\varGamma_{\chi_{2}}$
$\displaystyle=\frac{1}{16\pi^{2}}\int
d^{4}x\,\sqrt{g}\,\big{(}\varSigma_{\chi_{1}}-\varSigma_{\chi_{2}}\big{)}$
$\displaystyle\times\Big{[}\alpha\,C^{2}+\beta\mathcal{E}_{4}-2\beta\Delta_{4}\big{(}\varSigma_{\chi_{1}}+\varSigma_{\chi_{2}}\big{)}\Big{]}.$
(2.17)

Conformal variation of this expression is vanishing, because of the
transformation law (2.13) for $\varSigma_{1,2}$, Weyl invariance of the
density $\sqrt{g}\,C^{2}$ and the relation (2.9) which in the infinitesimal
form reads as

$\delta_{\sigma}\Big{[}\sqrt{g}\,\mathcal{E}_{4}\Big{]}=4\sqrt{g}\,\Delta_{4}\sigma,$
(2.18)

so that using all the above properties
$\delta_{\sigma}\big{(}\varGamma_{\chi_{1}}-\varGamma_{\chi_{2}}\big{)}=0$.

Note that with our definition of the anomaly action (2.16) the way it enters
the full quantum action can be represented as

$\varGamma[\,g\,]=\varGamma_{\chi}[\,g\,]+\varGamma[\,\bar{g}\,]+\frac{1}{32\pi^{2}}\Big{(}\frac{\gamma}{6}+\frac{\beta}{9}\Big{)}\int
d^{4}x\sqrt{\bar{g}}\,\bar{R}^{2},$ (2.19)

where $\bar{g}_{\mu\nu}[\,g\,]=e^{-2\varSigma_{\chi}[\,g\,]}g_{\mu\nu}$

###  2.1 Riegert–Fradkin–Tseytlin gauge

An obvious choice of the conformal gauge associated with the Gauss–Bonnet
density and the Branson curvature is the Riegert–Fradkin–Tseytlin gauge

$\chi_{{\vphantom{L}}_{\rm RFT}}[\,\bar{g}\,]\equiv\bar{\mathcal{E}}_{4}=0.$
(2.20)

It can be imposed for topologically simple spacetime manifolds with a
vanishing bulk part of the Euler characteristics (see Eq.(2.38) and footnote 3
below), to which in particular belongs asymptotically flat spacetime to be
mainly considered throughout the paper. The advantage of this gauge is that it
is exactly solvable due to the transformation law for the Branson curvature
(2.9). Applying this gauge and using Eq. (2.8) we obtain a linear equation on
$\varSigma_{{\vphantom{L}}_{\rm RFT}}$ which has a solution in terms of the
inverse Paneitz operator

$\varSigma_{{\vphantom{L}}_{\rm
RFT}}=\frac{1}{4}\,\frac{1}{\Delta}_{4}\mathcal{E}_{4}.$ (2.21)

Formally substituting this expression to (2.16) we obtain exactly the RFT
action (1.3).

This RFT action and the inverse Paneitz operator are well defined and exist in
asymptotically flat spacetime under Dirichlet boundary conditions at infinity
when treated within perturbation theory in powers of the curvatures whose
collection is denoted below as $\Re$. Indeed, in this case

$\frac{1}{\Delta}_{4}=\frac{1}{\Box^{2}}+O(\Re),$ (2.22)

and this operator works well when it is applied to the functions of the
Branson curvature type $\sim\mathcal{E}_{4}$. Because of the double-pole
nature of the operator $1/\Box^{2}$ its action on generic functions may be
badly defined due to infrared divergences, but when the function is
represented by the total derivative structure it generates, when acted upon by
$1/\Box^{2}$, well defined multipole expansion valid in four dimensions at
spacetime infinity [39].222 As discussed in [39], the operator $1/\Box^{n}$ in
$D$-dimensional space with $D<2n$ is ill defined unless the functions it acts
upon are of the form $\partial_{\alpha_{1}}...\partial_{\alpha_{m}}j(x)$,
$m=2n-D+1$ with the function $j(x)$ having an asymptotic behavior
$j(x)=O(1/|x|^{D})$, $|x|\to\infty$. This property can be explained by the
fact that in the multipole expansion of
$\tfrac{1}{\Box}\partial_{\alpha_{1}}...\partial_{\alpha_{m}}j(x)$ the first
few multipoles vanish, which improves the fall-off properties of the result at
infinity and makes possible a repeated action by $1/\Box$. But the
Gauss–Bonnet density and $\sqrt{g}\,\Box R$ are both locally a total
derivative which makes $1/\Delta_{4}$ well defined in the expression (2.21)
for $\varSigma_{{\vphantom{L}}_{\rm RFT}}$. This in fact implies the
invertibility of the Faddeev–Popov operator in this gauge, which up to
coefficient coincides with the Paneitz operator, $Q_{{\vphantom{L}}_{\rm
RFT}}=4\Delta_{4}$, and thus guarantees local uniqueness of conformal gauge
fixing procedure.

Moreover, the above observation serves as a repudiation of the harmful role of
double poles in the RFT action that was claimed in [15]. Absence of infrared
dangerous double poles is explicit in the lowest order of the curvature
expansion for $\varSigma_{{\vphantom{L}}_{\rm RFT}}$ which reads

$\varSigma_{{\vphantom{L}}_{\rm RFT}}=-\frac{1}{6\,\Box}R+O(\Re^{2}),$ (2.23)

in view of the fact that the Gauss–Bonnet density is quadratic in the
curvature $\sqrt{g}E=O(\Re^{2})$. Higher orders of this expansion are also
safe because of the total derivative nature of $\sqrt{g}\,E$. Regarding the
lowest order quadratic in curvature part, with the above approximation for
$\varSigma_{{\vphantom{L}}_{\rm RFT}}$ it equals

$\varGamma_{{\vphantom{L}}_{\rm RFT}}[\,g\,]=-\frac{\gamma}{192\pi^{2}}\int
d^{4}x\sqrt{g}\,R^{2}+O(\Re^{3}),$ (2.24)

because all the terms depending on the parameter $\beta$ completely cancel
out, and what remains coincides with the last quadratic term of (1.6). This
coincidence fully matches with the linear in curvature part of the trace
anomaly (1.1) (its $\gamma$-term) generated by the quadratic action (1.6).
Indeed, the conformal transformation of its nonlocal Weyl term contributes
only to $O(\Re^{2})$-part of the anomaly due to the fact that only its form
factor $\ln(-\Box/\mu^{2})$ is not Weyl invariant, and the whole $\gamma$-term
of the anomaly entirely comes from the $R^{2}$-part of (1.6).

###  2.2 Fradkin-Vilkovisky gauge

Another conformal gauge arises in context of conformal off-shell extension of
Einstein gravity suggested in [19] and corresponds to the 4-dimensional
version of the Yamabe problem. The representative of the conformal group orbit
is chosen to be the metric with a vanishing scalar curvature

$\chi_{{\vphantom{L}}_{\rm FV}}[\,\bar{g}\,]=\bar{R},$ (2.25)

which implies a nonlinear but still explicitly solvable equation for
$\varSigma_{{\vphantom{L}}_{\rm FV}}$ ,

$R[e^{-2\varSigma_{{\vphantom{L}}_{\rm
FV}}}g_{\mu\nu}]=e^{3\varSigma_{{\vphantom{L}}_{\rm
FV}}}\big{(}R-6\,\Box\big{)}\,e^{-\varSigma_{{\vphantom{L}}_{\rm FV}}}=0.$
(2.26)

This solution reads

$\displaystyle\varSigma_{{\vphantom{L}}_{\rm
FV}}=-\ln\Big{(}1+\frac{1}{6}\frac{1}{\Box-R/6}R\Big{)},$ (2.27)
$\displaystyle\lim\limits_{|x|\to\infty}e^{-\varSigma_{{\vphantom{L}}_{\rm
FV}}}=1$ (2.28)

in terms of the inverse of the conformal second order operator
$\Box-\tfrac{1}{6}R$ subject to zero boundary conditions at infinity. This
inverse operator also admits covariant curvature expansion and in the lowest
order yields the function $\varSigma_{{\vphantom{L}}_{\rm FV}}$ coinciding
with that of the RFT gauge (2.23),

$\varSigma_{{\vphantom{L}}_{\rm FV}}=\varSigma_{{\vphantom{L}}_{\rm
RFT}}+O(\Re^{2}),$ (2.29)

and, therefore, generates in the quadratic order the same expression for the
anomaly action

$\varGamma_{{\vphantom{L}}_{\rm FV}}=\varGamma_{{\vphantom{L}}_{\rm
RFT}}+O(\Re^{3}).$ (2.30)

Using Eqs. (2.17) and (2.21) it is easy to see that the difference between RFT
and FV actions is given by the exact expression

$\displaystyle\varGamma_{{\vphantom{L}}_{\rm
RFT}}-\varGamma_{{\vphantom{L}}_{\rm FV}}$
$\displaystyle=\frac{1}{16\pi^{2}}\int
d^{4}x\,\sqrt{g}\big{(}\varSigma_{{\vphantom{L}}_{\rm
RFT}}-\varSigma_{{\vphantom{L}}_{\rm FV}}\big{)}$
$\displaystyle\times\Big{[}\,\alpha\,C^{2}+2\beta\,\Delta_{4}\big{(}\varSigma_{{\vphantom{L}}_{\rm
RFT}}-\varSigma_{{\vphantom{L}}_{\rm FV}}\big{)}\Big{]},$ (2.31)

bilinear in the local Weyl squared term and conformally invariant nonlocal
functional

$\displaystyle\varSigma_{{\vphantom{L}}_{\rm
RFT}}-\varSigma_{{\vphantom{L}}_{\rm FV}}$
$\displaystyle=\frac{1}{4}\,\frac{1}{\Delta}_{4}\mathcal{E}_{4}$
$\displaystyle+\ln\Big{(}1+\frac{1}{6}\frac{1}{\Box-R/6}R\Big{)}=O(\Re^{2}).$
(2.32)

Therefore within perturbation theory these two actions remain coinciding even
in the cubic order and become different only starting from the fourth order in
the curvature.

Perturbatively both terms of (2.32) produce similar nonlocal structures of
tree-like nature, that is the terms characteristic of the tree-level
approximation in field theory. Such terms are composed of the powers of
inverse d’Alembertians acting on the curvature tensor structures or on the
products of similar nonlocal tensor structures built according to the same
pattern. However, taken separately as exact entities they have essentially
different types of nonlocality. RFT action formalism involves the Green’s
function of the fourth order Paneitz operator, whereas the FV version of the
action is based on the Green’s function of the second order operator
$\Box-\tfrac{1}{6}R$. Both operators are conformally covariant, but the Weyl
transformation of $\Box-\tfrac{1}{6}R$ is different from (2.8)

$\Box-\tfrac{1}{6}R=e^{-3\sigma}\,\big{(}\bar{\Box}-\tfrac{1}{6}\bar{R}\big{)}\,e^{\sigma},\qquad
g_{\mu\nu}=e^{2\sigma}\bar{g}_{\mu\nu}.$ (2.33)

Moreover, FV action formalism involves a special logarithmic nonlinearity
absent in RFT gauge fixing. The action of the Paneitz operator derivatives in
(2.16) can destroy this logarithmic structure, but the
$\varSigma_{{\vphantom{L}}_{\rm FV}}C^{2}$-term in
$\varGamma_{{\vphantom{L}}_{\rm FV}}$ still contains it intact.

A further comparison of the RFT and FV actions can be done along the lines of
their “naturalness”. RFT gauge (2.20) is based on structures organically
belonging to the conformal anomaly formalism in the sense that it involves the
same fundamental objects—the Branson curvature $\mathcal{E}_{4}$ and the
relevant Paneitz operator $\Delta_{4}$ which are immanently present in the
flow of the anomalous action along the conformal group orbit (2.6). One could
even interpret this gauge as the one providing the extremum of $\beta$-terms
in this expression with respect to the variation of the orbit parameter
$\sigma$. This interpretation is, however, erroneous because $g_{\mu\nu}$,
$\bar{g}_{\mu\nu}$ and $\sigma$ cannot be treated as independent variables in
Eq. (2.6).

On the contrary, FV gauge (2.25) uses a somewhat extraneous entity—the scalar
curvature—which is singled out only by the fact that it turns out to be the
bearer of the metric conformal mode. As the result the advantage of FV gauge
is that it does not involve higher than second order derivatives and does not
produce double pole nonlocalities. Another advantage is that the equation
(2.19) disentangling the FV anomaly action from the full effective action
becomes in view of $\bar{R}=0$ much simpler

$\varGamma[\,g\,]=\varGamma_{{\vphantom{L}}_{\rm
FV}}[\,g\,]+\varGamma[\,\bar{g}\,]\,\big{|}_{\bar{g}_{\mu\nu}[\,g\,]},$ (2.34)

where $\bar{g}_{\mu\nu}[\,g\,]=e^{-2\varSigma_{{\vphantom{L}}_{\rm
FV}}[\,g\,]}g_{\mu\nu}$, which is obviously consistent with the fact that
$\varGamma_{{\vphantom{L}}_{\rm FV}}[\,\bar{g}\,]=0$ because
$\varSigma_{{\vphantom{L}}_{\rm FV}}[\,\bar{g}\,]\equiv 0$.

As compared to the FV version, among technical disadvantages of the RFT gauge
and the action is the presence of fourth order derivatives of the Paneitz
operator. Due to this the RFT version turns out to be vulnerable from the
viewpoint of possible generalizations. For example, a modification of the
gauge (2.20) by the additional Weyl squared term, $\chi_{{\vphantom{L}}_{\rm
RFT}}\to\chi_{{\vphantom{L}}_{\rm RFT}}+aC^{2}$ would not work, because the
relevant modification $\varSigma_{{\vphantom{L}}_{\rm
RFT}}\to\varSigma_{{\vphantom{L}}_{\rm RFT}}+a\,(2\Delta_{4})^{-1}C^{2}$ is
badly defined for the reasons described above in the footnote 2—the additional
term should have a total derivative structure.

The generalization to spacetimes with nontrivial topology is also not
straightforward, because the condition (2.20) should not contradict
nonvanishing Euler number of the manifold, which for compact manifolds without
a boundary reads $e_{E}=\tfrac{1}{32\pi^{2}}\int d^{4}x\sqrt{g}\,E(x)$. Say,
for a compact manifold of a finite volume $V=\int d^{4}x\,\sqrt{g}$ the gauge
(2.20) can be chosen to be

$\chi(\bar{g})=\sqrt{\bar{g}}\,\Big{(}\bar{E}-\frac{2}{3}\bar{\Box}\bar{R}-32\pi^{2}\frac{e_{E}}{\bar{V}}\Big{)},$
(2.35)

but this leads to a nonlinear integro-differential equation for the relevant
$\varSigma$

$\displaystyle 4\sqrt{g}\Delta_{4}\varSigma$
$\displaystyle=\sqrt{g}\bigg{(}E-\frac{2}{3}\Box
R-32\pi^{2}\frac{e^{-4\varSigma}}{\langle\,e^{-4\varSigma}\,\rangle}\frac{e_{E}}{V}\bigg{)},$
(2.36) $\displaystyle\langle\,e^{-4\varSigma}\,\rangle$
$\displaystyle\equiv\frac{1}{V}\int d^{4}x\,\sqrt{g}e^{-4\varSigma},$ (2.37)

which apparently can be solved analytically only by perturbations in
$e_{E}/V$.

Unless stated otherwise, below we consider asymptotically flat spacetime with
a trivial topology, whose Euler characteristics should be modified by the
boundary term. For generic 4-dimensional manifolds with a smooth boundary it
reads

$e_{E}=\tfrac{1}{32\pi^{2}}\Big{(}\int_{\cal
M}d^{4}x\sqrt{g}\,E(x)+\int_{\cal\partial
M}d^{3}x\sqrt{\gamma}\varOmega(x)\Big{)},$ (2.38)

where $\gamma=\det\gamma_{ab}$ and $\gamma_{ab}$ is the induced metric on
$\partial{\cal M}$. For asymptotically flat case due to the contribution of
$\partial{\cal M}$ at infinity $|\,x\,|\to\infty$ it equals $1$, so that
everywhere in what follows the bulk part of the Euler characteristics is
$\tfrac{1}{32\pi^{2}}\int d^{4}x\sqrt{g}\,E(x)\equiv
e^{\prime}_{E}=e_{E}-1=0$.333I am grateful for this observation to M.Duff.
Explicit and simple expression for the boundary term of the Euler
characteristics in the 4-dimensional case can be found in [36],
$\varOmega=\tfrac{1}{4}R_{a\perp b\perp}K^{ab}+16\det K^{a}_{b}$, where
$K_{ab}=\nabla_{a}n_{b}$ is the extrinsic curvature of the boundary, and
$\perp$ denotes the projection on the outward pointing normal vector
$n^{\mu}$. The last term in $\varOmega$ exactly reproduces the value of the
Euler number $e_{E}=1$ for flat and asymptotically flat spaces [54].

##  3 Conformal anomaly and covariant curvature expansion

Despite the diversity of nonlocal structures of RFT and FV versions of anomaly
action, neither of them seem to appear in conventional perturbation theory for
quantum effective action. The covariant form of this perturbation theory in
curved spacetime (1.6) was pioneered in [3], but its logarithmic nonlocal
formfactor did not resemble the nonlocal operators of the RFT action (1.3).
Here we show how in spite of these discrepancies the anomaly action originates
from covariant perturbation theory of [38, 39, 40].

This perturbation theory arose as a concrete implementation of the ideas of
[3] as an expansion in powers of covariant tensors of spacetime and fibre
bundle curvatures and other covariant background field objects. This expansion
is completely equivalent to standard Feynman diagrammatic technique and
represents its resummation converting the original perturbation series in
noncovariant odjects, like matter and metric field perturbations on top of
flat and empty spacetime background, into the series in powers of covariant
fields strengths denoted collectively below by $\Re$ and including spacetime
and fibre bundle curvature.

To be more specific, consider the theory with the inverse propagator on top of
the nontrivial field background $\hat{F}(\nabla)=F^{A}_{B}(\nabla)$, hat
denoting the matrix structure of the operator acting in the space of fields
$\varphi=\varphi^{A}(x)$ with a generic spin-tensor index $A$ and
$\nabla=\nabla_{\mu}$ denoting the covariant derivative with respect to the
corresponding fibre bundle connection,

$\hat{F}(\nabla)=\Box+\hat{P}-\frac{\hat{1}}{6}R,\qquad\Box=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}.$
(3.1)

This operator is characterized by the “curvatures”—metric Riemann tensor with
its Ricci contractions, fibre bundle curvature $\hat{\cal R}_{\mu\nu}$
determining the commutator of covariant derivatives,
$[\nabla_{\mu},\nabla_{\nu}]\,\varphi=\hat{\cal R}_{\mu\nu}\,\varphi$, and the
potential term $\hat{P}$ (the term $-\tfrac{\hat{1}}{6}R$ is disentangled from
the operator potential for reasons of convenience),

$\Re=(R^{\mu}{}_{\nu\alpha\beta},R_{\mu\nu},R,\hat{\cal R}_{\mu\nu},\hat{P}).$
(3.2)

In covariant perturbation theory the one-loop effective action gets expanded
in powers of these curvatures

$\varGamma=\frac{1}{2}{\rm Tr}\ln
F(\nabla)=\\!\\!\\!\\!\stackrel{{\scriptstyle\text{local power
div}}}{{\overbrace{\varGamma_{0}+\varGamma_{1}}}}\\!\\!\\!\\!\\!\\!+\,\varGamma_{2}+\varGamma_{3}+O(\Re^{4}),$
(3.3)

where $\varGamma_{n}\sim\Re^{n}$. Within dimensional regularization of
$2\omega$-dimensional spacetime, $\omega\to 2$, the zeroth and first order
terms of the expansion represent pure power divergences (note that we consider
the case of a massless theory, or the theory where the mass matrix is included
in the potential term $\hat{P}$ and treated by perturbations), so that these
two terms are annihilated by the regularization, while the second order term
is given by the expression [38]

$\displaystyle\varGamma^{(2)}_{\rm dim\;reg}$
$\displaystyle=-\frac{\Gamma(2-\omega)\Gamma(\omega+1)\Gamma(\omega-1)}{2(4\pi)^{\omega}\Gamma(2\omega+2)}\,\mu^{4-2\omega}$
$\displaystyle\times\int dx\,\sqrt{g}\,{\rm
tr}\,\Big{\\{}R_{\mu\nu}(-\Box)^{\omega-2}R^{\mu\nu}\hat{1}$
$\displaystyle-\frac{1}{18}(4-\omega)(\omega+1)R(-\Box)^{\omega-2}R\hat{1}$
$\displaystyle-\frac{2}{3}(2-\omega)(2\omega+1)\,\hat{P}(-\Box)^{\omega-2}R$
$\displaystyle+2(4\omega^{2}-1)\,\hat{P}(-\Box)^{\omega-2}\hat{P}$
$\displaystyle+(2\omega+1)\hat{\cal R}_{\mu\nu}(-\Box)^{\omega-2}\hat{\cal
R}^{\mu\nu}\Big{\\}},$ (3.4)

where $\omega=\tfrac{d}{2}\to 2$. Here ${\rm tr}$ denotes the matrix trace and
the concrete coefficients implement the originally conjectured structure of
dimensionally regularized effective action Lagrangian,
$\Re(-\Box)^{\omega-2}\Re$, that was blueprinted in [3]. What is important and
should be especially emphasized is that
$\Box=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$ means here the full covariant
d’Alembertian acting on a respective scalar $R$, tensor $R_{\mu\nu}$ or
spintensor $\hat{\cal R}_{\mu\nu}$ and $\hat{P}$ objects.

For brevity we will consider the case of a single conformal scalar field with
$\hat{1}=1$, $\hat{P}=0$, $\hat{\cal R}_{\mu\nu}=0$ and the following values
of the trace anomaly coefficients444The coefficients have the opposite sign to
those of $b=-\alpha/16\pi^{2}$ and $b^{\prime}=-\beta/16\pi^{2}$ in [4],
because in our case the stress tensor is defined with respect to the Euclidean
effective action $\varGamma=-i\varGamma_{L}$ in contrast to the definition of
$T^{\mu\nu}=2g^{-1/2}\delta\varGamma_{L}/\delta g_{\mu\nu}$ in the Lorentzian
signature spacetime of [4]. Comparison with [5] should also take into account
another sign of the stress tensor defined by the variation with respect to the
contravariant metric.

$\alpha=-\frac{1}{120},\quad\beta=\frac{1}{360},\quad\gamma=-\frac{1}{180},$
(3.5)

for which the action (3.4) takes the form—a particular case of (1.6),

$\displaystyle\varGamma^{(2)}_{\rm ren}$
$\displaystyle=\frac{1}{32\pi^{2}}\int
dx\,\sqrt{g}\,\Big{\\{}\frac{1}{60}\Big{[}R_{\mu\nu}\gamma(-\Box)R^{\mu\nu}$
$\displaystyle-\frac{1}{3}R\gamma(-\Box)R\Big{]}+\frac{R^{2}}{1080}\Big{\\}}$
$\displaystyle=\frac{1}{32\pi^{2}}\int
dx\,\sqrt{g}\,\Big{\\{}\frac{1}{120}C_{\mu\nu\alpha\beta}\gamma(-\Box)C^{\mu\nu\alpha\beta}$
$\displaystyle+\frac{R^{2}}{1080}\Big{\\}}+O(\Re^{3}).$ (3.6)

Here $\gamma(-\Box)$ is the nonlocal formfactor (in minimal subtraction scheme
with $\ln(4\pi)$ and Euler constants absorbed in $\mu$)

$\gamma(-\Box)=\ln\Big{(}\\!-\frac{\Box}{\mu^{2}}\Big{)}-\frac{16}{15},$ (3.7)

and the transition to the last line is valid up to the higher order terms in
curvature and based on the nonlocal generalization of the identity

$\int d^{4}x\,\sqrt{g}\,C^{2}=2\int
d^{4}x\,\sqrt{g}\,(R_{\mu\nu}R^{\mu\nu}-\tfrac{1}{3}R^{2})$ (3.8)

derived in [39, 40] by integration by parts and use of the nonlocal
representation of the Riemann tensor in terms of the Ricci one (see footnote 5
below).

The first term of this action is obviously conformal invariant in quadratic
order, so that the linear in curvature part of the anomaly originates from the
last term which is the RFT (or FV) action (2.24) in the quadratic
approximation with $\gamma=-1/180$. Thus, the RFT or FV action is fully
recovered in this approximation from perturbation theory and, as expected,
turns out to be local.

###  3.1 Cubic order

Quadratic order of the covariant curvature expansion is, in fact, a trivial
generalization of the flat space expressions for self-energy operators of
Feynman diagrammatic technique, because $\ln(-\Box/\mu^{2})$ is just a
straightforward replacement of the typical momentum space formfactor
$\ln(p^{2}/\mu^{2})$ by its position space version. At higher orders the
situation becomes much more complicated and usually represented in terms of
correlators of stress-tensor and other observables, written down in momentum
space representation, see [55, 56, 57] for the treatment of generic conformal
field theories. These correlators are, of course, contained in the effective
action expanded in curvatures which, for reasons of general covariance, we
prefer to consider in coordinate representation.

In this representation the effective action becomes for each order $N$ in the
curvature a sum of nonlocal monomials

$\int d^{4}x_{1}\cdots
d^{4}x_{N}\,F(x_{1},\ldots,x_{N})\nabla...\nabla\Re(x_{1})...\Re(x_{N})$ (3.9)

with nonlocal multiple-point coefficients and covariant derivatives somehow
acting on the product of curvatures at their various points. The absence of
convenient and generally covariant momentum space representation makes us to
work in coordinate representation and invent a special language which would
simplify the formalism and make it manageable [38, 39, 40]. This language is
based on the operator representation of nonlocal formfactors,

$\displaystyle F(x_{1},\ldots,x_{N})$
$\displaystyle=\varGamma(\nabla_{1},\ldots,\nabla_{N})$
$\displaystyle\times\delta(x_{1},x_{2})\,\delta(x_{1},x_{2})\cdots\delta(x_{1},x_{N}),$
(3.10)

where $\varGamma(\nabla_{1},\ldots,\nabla_{N})$ is the operator valued
function of $N$ independent covariant derivatives such that each $\nabla_{i}$
is acting on its own $x_{i}$. This allows one to write the orders of
perturbation theory as

$\displaystyle\varGamma^{(N)}$
$\displaystyle=\frac{1}{2(4\pi)^{2}}\int\\!d^{4}x\,\sqrt{g}\sum\limits_{M}\varGamma_{M}(\nabla_{1},\ldots,\nabla_{N})$
$\displaystyle\times I_{M}(x_{1},\ldots,x_{N})\,\Big{|}_{\\{x\\}=x},$ (3.11)

where summation runs over all invariant monomials in curvatures of a given
$n$-th order

$I_{M}(x_{1},\ldots,x_{N})\sim\nabla\cdots\nabla\Re(x_{1})\cdots\Re(x_{N})$
(3.12)

and after the action of all independent derivatives on their arguments all
these arguments $\\{x\\}=(x_{1},\ldots x_{N})$ have to be identified.

In the cubic order for the full set of curvatures (3.2) there are 29 such
invariant structures built of these curvatures and their covariant derivatives
with all indices fully contracted with each other. Moreover, in view of the
scalar (no free indices) nature of the formfactors and the formal identity
$\nabla_{1}+\nabla_{2}+\nabla_{3}=0$ (reflecting the possibility of
integration by parts without surface terms, which is a counterpart to the
momentum conservation in Feynman diagrams) the formfactors of
$\varGamma^{(3)}$ can be written down as functions of three d’Alembertians
$\Box_{1}$, $\Box_{2}$ and $\Box_{3}$ independently acting on three arguments
of $I_{M}(x_{1},x_{2},x_{3})$. Thus, cubic order reads as

$\displaystyle\varGamma^{(3)}$
$\displaystyle=\frac{1}{2(4\pi)^{2}}\int\\!dx\,\sqrt{g}\sum\limits^{29}_{M=1}\varGamma_{M}(\Box_{1},\Box_{2},\Box_{3})$
$\displaystyle\times I_{M}(x_{1},x_{2},x_{3})\,\Big{|}_{\\{x\\}=x}.$ (3.13)

The list of cubic invariants and their formfactors is presented in [40, 18,
20]. It is very long and, as its details are not necessary for our purposes,
we will not fully present it here. We only give the general structure of the
nonlocal formfactors of these invariants. It reads as a sum of three different
groups of terms

$\displaystyle\varGamma_{M}(\Box_{1},\Box_{2},\Box_{3})$
$\displaystyle=A_{M}\,\varGamma(\Box_{1},\Box_{2},\Box_{3})$
$\displaystyle+\sum_{1\leq
i<k}^{3}\frac{D^{ik}_{M}}{(\Box_{i}-\Box_{k})}\ln\frac{\Box_{i}}{\Box_{k}}+B_{M}.$
(3.14)

Here $\varGamma(\Box_{1},\Box_{2},\Box_{3})$ is the fundamental cubic
formfactor corresponding to the triangular Feynman graph of massless theory
with unit vertices [58],

$\displaystyle\varGamma(\Box_{1},\Box_{2},\Box_{3})$
$\displaystyle=\int\limits_{\alpha\geq
0}\frac{d^{3}\alpha\,\delta(1-\alpha_{1}-\alpha_{2}-\alpha_{3})}{\alpha_{1}\alpha_{2}(-\Box_{3})+\alpha_{1}\alpha_{3}(-\Box_{2})+\alpha_{2}\alpha_{3}(-\Box_{1})},$
(3.15)

which cannot be reduced to an elementary function. The operator-valued
coefficients $A_{M}$, $B_{M}$ and $D_{M}^{ik}$ are rational functions of three
$\Box$-arguments with a polynomial numerator $P(\Box_{1},\Box_{2},\Box_{3})$
and the denominator containing together with the product
$\Box_{1}\Box_{2}\Box_{3}$ also the powers of a special quadratic form of
these arguments $D$,

$\displaystyle
A_{M},\;D^{ik}_{M},\;B_{M}\sim\frac{P(\Box_{1},\Box_{2},\Box_{3})}{\Box_{1}\Box_{2}\Box_{3}\,D^{L}},\;L\leq
6,$ (3.16) $\displaystyle
D={\Box_{1}}^{2}\\!+{\Box_{2}}^{2}\\!+{\Box_{3}}^{2}\\!-\\!2\Box_{1}\Box_{2}\\!-\\!2\Box_{1}\Box_{3}\\!-\\!2\Box_{2}\Box_{3}.$
(3.17)

In this cubic order of the curvature expansion the conformal anomaly (1.1),
which is quadratic in curvatures, was explicitly derived by the direct
variation of the metric in [59]. Though this derivation has demonstrated
nontrivial localization of the nonlocal terms under straightforward tracing
the metric variational derivative, it still remained rather technical and not
very illuminating because it has not revealed the anomalous part of the
action. It turns out, however, that the transition to another basis of
curvature invariants, suggested in [20, 18], explicitly disentangles this
part.

###  3.2 Conformal resummation: Fradkin–Vilkovisky anomaly action

The recovery of the anomaly part of the action and its conformal invariant
part is based on a simple idea that the latter should consist of the series of
Weyl invariant structures. The construction of Weyl invariants can be done by
the gauge fixing procedure of the above type—choosing the representative
metric on the group orbit by imposing the conformal gauge. Obviously the set
of invariants surviving after imposing this gauge will be minimal if the gauge
would explicitly annihilate the maximum number of invariants in their original
full set. For this reason the FV gauge (2.25) is much easier to use for the
separation of the total set of invariants into the Weyl type ones and those
which vanish when the gauge is enforced. As $R$ is one of the curvatures in
the set of $\Re$ the FV gauge is more useful for the purpose of such a
separation than the RFT gauge (2.20) which nonlinearly intertwines all the
curvatures. Intuitively it is also clear because $R$, in contrast to
$C^{\alpha}{}_{\beta\mu\nu}$, is a bearer of the conformal mode.

In the purely metric sector such a separation is attained by the transition to
the new curvature basis [20],

$\Re=\big{(}\,R^{\mu}_{\;\nu\alpha\beta},R_{\mu\nu},R\,\big{)}\to\tilde{\Re}=\big{(}\,C^{\alpha}_{\;\;\beta\mu\nu},R\,\big{)},$
(3.18)

via expressing Ricci tensor in terms of the Weyl tensor and the Ricci
scalar555In fact, the original basis and the curvature expansion of [38, 39,
40] consisted of $R_{\mu\nu}$ and $R$ because in asymptotically flat Euclidean
spacetime Riemann tensor can be expressed as nonlocal power series in the
Ricci tensor,
$R_{\alpha\beta\mu\nu}=\frac{1}{\Box}\big{(}\nabla_{\mu}\nabla_{\alpha}R_{\nu\beta}-\nabla_{\nu}\nabla_{\alpha}R_{\mu\beta}\big{)}-(\alpha\leftrightarrow\beta)+O(\Re^{2})$,—the
corollary of contracted Bianchi identity.. This expression follows from the
contracted Bianchi identity which for the Weyl tensor reads as

$\nabla^{\beta}\nabla^{\alpha}C_{\alpha\mu\beta\nu}=\frac{1}{2}\Box
R_{\mu\nu}-\frac{1}{6}\nabla_{\mu}\nabla_{\nu}R-\frac{g_{\mu\nu}}{12}\Box
R+O(\Re^{2}).$ (3.19)

This equation can be solved by iterations for Ricci tensor in terms of
nonlocal series in powers of two objects—Ricci scalar $R$ and the new
traceless (and up to quadratic order transverse) tensor $C_{\mu\nu}$ which is
itself a nonlocal derivative of Weyl,

$C_{\mu\nu}=\frac{2}{\Box}\nabla^{\beta}\nabla_{\alpha}C^{\alpha}_{\;\;\mu\beta\nu}.$
(3.20)

The resulting series begins with

$R_{\mu\nu}=C_{\mu\nu}+\frac{1}{3}\nabla_{\mu}\nabla_{\nu}\frac{1}{\Box}R+\frac{1}{6}g_{\mu\nu}R+O(\tilde{\Re}^{2}).$
(3.21)

Effective action reexpansion imples the transition from
$I_{M}(x_{1},\ldots,x_{n})$ to a new basis of invariants

$\tilde{I}_{M}(x_{1},...x_{n})\sim\nabla...\nabla\tilde{\Re}(x_{1})...\tilde{\Re}(x_{n}),$
(3.22)

which can be separated in the set of monomials $I_{C}(x_{1},\ldots,x_{n})$
involving only $C_{\mu\nu}$ and the set of monomials
$I_{R}(x_{1},\ldots,x_{n})$ containing at least one scalar curvature factor,

$\displaystyle I_{C}(x_{1},...x_{n})\sim$ $\displaystyle\nabla...\nabla
C(x_{1})...C(x_{n}),$ (3.23) $\displaystyle I_{R}(x_{1},...x_{n})\sim$
$\displaystyle\nabla...\nabla R(x_{1})C(x_{2})...C(x_{n}),$
$\displaystyle\nabla...\nabla R(x_{1})R(x_{2})C(x_{3})...C(x_{n}),\;...$
(3.24)

Expansion in the new basis of invariants implies, of course, the transition to
a new set of their relevant formfactors

$\varGamma_{M}(\nabla_{1},...\nabla_{n})\to\varGamma_{C}(\nabla_{1},...\nabla_{n}),\varGamma_{R}(\nabla_{1},...\nabla_{n}),$
(3.25)

and the new expansion takes the form

$\varGamma=W+\varGamma_{R},$ (3.26)

where $W$ is the Weyl and $\varGamma_{R}$ is the mixed Weyl–Ricci scalar parts
of the whole expansion, which we write in abbreviated form (omitting multiple
spacetime arguments and the operation of equating them)

$\displaystyle W$
$\displaystyle=\frac{1}{32\pi^{2}}\int\\!d^{4}x\,\sqrt{g}\sum\limits_{n,C}\varGamma_{C}^{(n)}I_{C}^{(n)},$
(3.27) $\displaystyle\varGamma_{R}$
$\displaystyle=\frac{1}{32\pi^{2}}\int\\!d^{4}x\,\sqrt{g}\sum\limits_{n,R}\varGamma_{R}^{(n)}I_{R}^{(n)}.$
(3.28)

Note that $W$ and its Weyl basis invariants are not Weyl invariant, because
apart from Weyl tensors they contain covariant derivatives and nontrivial
formfactors which do not possess conformal invariance properties.

The main statement on the conformal decomposition of the effective action of
[20] is that

$\varGamma[\,g\,]=\varGamma_{{\vphantom{L}}_{\rm
FV}}[\,g\,]+W[\,\bar{g}\,]\,\Big{|}_{\,\bar{g}_{\mu\nu}=e^{-2\varSigma_{{\vphantom{L}}_{\rm
FV}}[\,g\,]}g_{\mu\nu}},$ (3.29)

where $\varGamma_{{\vphantom{L}}_{\rm FV}}[\,g\,]$ is exactly the FV anomaly
action introduced above666One can check that the last four lines of Eq. (24)
in [20] form exact expression for $\varGamma_{{\vphantom{L}}_{\rm FV}}[\,g\,]$
by taking into account that the function $Z$ in this equation coincides with
$-\varSigma_{{\vphantom{L}}_{\rm FV}}$ and satisfies the equation $\Box
Z+\tfrac{1}{2}(\nabla Z)^{2}=\tfrac{1}{3}R$.. Conformally invariant part is
obtained by the “conformization” of $W$, while the rest of the effective
action is exhausted by the Fradkin-Vilkovisky anomaly action. Invariant
meaning of this representation is that the Ricci part of the full action is
not independent, but fully determined by the anomaly and Weyl parts of the
action. This representation looks as the realization of Eq. (2.34) within
perturbation theory in curvatures. This result is likely to resolve a long-
standing debate between the proponents of the Riegert action and adherents of
the flat space perturbation expansion for the effective action with typical
nonlocal logarithmic form factors of the form (3.7). Note that these form
factors do not contribute to the anomaly even though their coefficients are
directly related to its expression (1.1). Rather they become Weyl invariant
under the substitution of $\bar{g}_{\mu\nu}$ as their functional argument.

Validity of the representation (3.29) was checked in the cubic order
approximation for the effective action in [20]. The transition to the new
basis of invariants in the second order leads to (see the second line of Eq.
(3.6)),

$\displaystyle W^{(2)}[\,g\,]$ $\displaystyle=\frac{1}{32\pi^{2}}\int
dx\,\sqrt{g}\frac{1}{120}C_{\mu\nu\alpha\beta}\,\gamma(-\Box)C^{\mu\nu\alpha\beta},$
(3.30) $\displaystyle\varGamma^{(2)}_{R}[\,g\,]$
$\displaystyle=\frac{1}{32\pi^{2}}\int dx\,\sqrt{g}\frac{1}{1080}R^{2},$
(3.31)

whereas in the third order it results in a great simplification of the “Ricci
scalar” formfactors $\varGamma_{R}^{(3)}$ as compared to the original
ones—they become much simpler and, moreover, in their expressions of the form
(3.14) the coefficients $A,D^{ik}_{M},B_{M}$ of (3.16) completely loose powers
of the function $D$ in the denominator. Thus, modulo the contributions of
$\ln(\Box_{i}/\Box_{k})/(\Box_{i}-\Box_{k})$ the formfactors
$\varGamma_{R}^{(3)}$ acquire the tree-level structure. The terms with these
factors get, however, completely absorbed with accuracy $O(\Re^{4})$ by the
replacement $W^{(2)}[\,g_{\mu\nu}\,]\to W^{(2)}[\,\bar{g}_{\mu\nu}\,]$ in view
of the following relation [40]

$\displaystyle W^{(2)}[\,g\,]-W^{(2)}[\,\bar{g}\,]$ $\displaystyle\sim\int
dx\,\sqrt{g}\,C_{\mu\nu\alpha\beta}\Big{[}\ln(-\Box)-\ln(-\bar{\Box})\Big{]}C^{\mu\nu\alpha\beta}$
$\displaystyle=\int
dx\,\sqrt{g}\frac{\ln(\Box_{1}/\Box_{2})}{\Box_{1}-\Box_{2}}[\Box_{2}-\bar{\Box}_{2}]C_{1\,\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}_{2}+O(\Re^{4}),$
$\displaystyle\Box_{2}-\bar{\Box}_{2}\sim\Re_{3}+O(\Re^{2}),$ (3.32)

where the right hand side is the set of relevant cubic order terms with the
above factor acting on two Weyl tensors out of three curvatures in $RCC$-type
invariants. What remains in the sector of cubic $I^{(3)}_{R}$-invariants is
the set of tree-like nonlocal form factors which comprise the curvature
expansion of FV action up to $\tilde{\Re}^{3}$ order inclusive. This
observation done in [20] can be formalized as the following sequence of
identical transformations

$\displaystyle\varGamma[\,g\,]=W^{(2+3)}[\,g\,]+\varGamma^{(2+3)}_{R}[\,g\,]+O(\Re^{4})=W^{(2+3)}[\,\bar{g}\,]$
$\displaystyle\;\;\;\;+\mathop{\underbrace{\varGamma^{(2+3)}_{R}[\,g\,]+\\!\big{(}W^{(2)}[\,g\,]-W^{(2)}[\,\bar{g}\,]\big{)}}}_{{\varGamma^{(2+3)}_{{\vphantom{L}}_{\rm
FV}}+O(\Re^{4})}}+O(\Re^{4}),$ (3.33)

where the group of the last three terms forms Fradkin–Vilkovisky anomaly
action expanded with $\tilde{\Re}^{3}$-accuracy. Explicitly the cubic part of
$\varGamma_{{\vphantom{L}}_{\rm FV}}$ for the model of a single conformal
scalar field with (3.5) reads [20]

$\displaystyle\varGamma^{(3)}_{{\vphantom{L}}_{\rm FV}}$
$\displaystyle=-\frac{1}{32\pi^{2}}\int
dx\,\sqrt{g}\bigg{\\{}\frac{1}{19440}\Big{(}\frac{2}{\Box}_{3}-\frac{\Box_{1}}{\Box_{2}\Box_{3}}\Big{)}R_{1}R_{2}R_{3}$
$\displaystyle+\frac{1}{1620\,\Box_{2}\Box_{3}}C_{1}^{\alpha\beta}\nabla_{\alpha}R_{2}\nabla_{\beta}R_{3}$
$\displaystyle+\frac{1}{540}\Big{(}\frac{4}{\Box}_{2}-\frac{1}{\Box}_{3}-\frac{2\,\Box_{1}}{\Box_{2}\Box_{3}}-\frac{\Box_{3}}{\Box_{1}\Box_{2}}\Big{)}C_{1}^{\mu\nu}C_{2\,\mu\nu}R_{3}$
$\displaystyle+\frac{1}{135}\Big{(}\frac{1}{\Box_{1}\Box_{2}}-\frac{2}{\Box_{2}\Box_{3}}\Big{)}\nabla^{\mu}C_{1}^{\nu\alpha}\nabla_{\nu}C_{2\,\mu\alpha}R_{3}$
$\displaystyle-\frac{1}{135\,\Box_{1}\,\Box_{2}\Box_{3}}\nabla_{\alpha}\nabla_{\beta}C_{1}^{\mu\nu}\nabla_{\mu}\nabla_{\nu}C_{2}^{\alpha\beta}R_{3}\bigg{\\}}\bigg{|}_{\,\\{x\\}=x},$
(3.34)

where $C_{\mu\nu}$ is the “Weyl” part (3.20) of Ricci tensor (3.21)

###  3.3 The problem of double poles and global conformal transformations

The expression (3.34) shows that in the cubic order the anomalous effective
action is free from double pole nonlocal terms. For the FV action this is
obviously true to all orders of the curvature expansion, since all its tree
type nonlocalities originate from the Green’s function of the conformal scalar
operator $\Box-\tfrac{1}{6}R$. However, for the RFT action double poles
formally appear starting from the fourth order in the curvature because the
metric variation of $\varSigma_{\chi}=\varSigma_{\rm RTF}$ in (2.16) leads to
the action of the inverse Paneitz operator upon the square of the Weyl tensor
$C^{2}=C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}$ due to a formal variational
rule

$\int d^{4}x\,\sqrt{g}\,C^{2}\delta\varSigma_{\rm RFT}=\int
d^{4}x\,\sqrt{g}\,(\Delta_{4}^{-1}C^{2})\delta(\ldots).$ (3.35)

This operation is not well defined, because $C^{2}$ is not a total derivative
and the repeated action of $1/\Box$ upon generic test functions in four
dimensions leads to IR divergent integrals—see footnote 2. In the cubic order
of $\varGamma_{\rm RFT}$ this problem does not arise because of the extra
$\Box$ factor in $\Box R$, as it was checked in [16] by explicit calculations
of $\langle\,TTT\,\rangle$ correlators, but one is not granted to be free from
this difficulty for higher order correlators.

In fact this is a typical situation of IR divergences in two dimensions, where
the kernel of $1/\Box$ has a logarithmic dependence at infinity, and the
correlators of undifferentiated conformal fields $\phi$ are UV divergent,
while the correlators $\langle\partial\phi(x)\partial\phi(y)\cdots\rangle$
stay well defined. Apparently, the same property in four dimensions also
underlies absence of unitarity in dipole theories with $1/\Box^{2}$-type
propagators recently discussed in [60]. The mechanism of transition from
operators to their derivatives in shift symmetric theories actually helps to
justify the RFT action as a source of well defined stress tensor correlators
and extend the validity of results in [16] to all higher orders.

This follows from the observation that the Paneitz operator reads

$\sqrt{g}\Delta_{4}=\partial_{\mu}\Big{[}\sqrt{g}\big{(}\nabla^{\mu}\nabla_{\nu}+2R^{\mu\nu}-\frac{2}{3}Rg^{\mu\nu}\big{)}\Big{]}\partial_{\nu}$
(3.36)

and, therefore, perturbatively on the flat space background can be represented
as

$\sqrt{g}\Delta_{4}=\tilde{\Box}^{2}+V,\;\tilde{\Box}=\delta^{\mu\nu}\partial_{\mu}\partial_{\nu},\;V=\overrightarrow{\partial_{\mu}V^{\mu\nu}\partial_{\nu}},$
(3.37)

where the perturbation $V=O(\Re)$ has a special form—another differential
operator $V^{\mu\nu}$ sandwiched between two derivatives with all derivatives
acting to the right (which is indicated by the arrow). Within perturbation
theory in powers of $V$ the action of the inverse operator on a generic test
function $\psi$—scalar density—could have been understood as the expansion

$\displaystyle\phi=\frac{1}{\sqrt{g}\Delta_{4}}\psi$
$\displaystyle=\sum\limits_{n=0}^{\infty}\frac{(-1)^{n}}{\tilde{\Box}^{2}}\left(V\frac{1}{\tilde{\Box}^{2}}\right)^{n}\psi$
$\displaystyle=\sum\limits_{n=0}^{\infty}\frac{(-1)^{n}}{\tilde{\Box}^{2}}\left(\overrightarrow{\partial_{\mu}V^{\mu\nu}}\frac{1}{\tilde{\Box}^{2}}\partial_{\nu}\right)^{n}\psi,$
(3.38)

where we deliberately permuted the factors of $\partial_{\nu}$ and
$1/\tilde{\Box}^{2}$ using their formal commutativity in order to provide the
action of $1/\tilde{\Box}^{2}$ on the total derivative function. Thus all
terms of this expansion except the first one become infrared finite. The first
term $(1/\tilde{\Box}^{2})\psi$, however, makes this function $\phi$ ill
defined. On the contrary, its derivative $\partial_{\alpha}\phi$ becomes
consistent if one understands the first term of the expansion as
$(1/\tilde{\Box}^{2})\partial_{\alpha}\psi$, so that the prescription for the
operation of $\partial_{\alpha}(1/\sqrt{g}\Delta_{4})$ on a generic non-
derivative type test function reads as

$\partial_{\alpha}\frac{1}{\sqrt{g}\Delta_{4}}\psi=\sum\limits_{n=0}^{\infty}\frac{(-1)^{n}}{\tilde{\Box}^{2}}\partial_{\alpha}\left(\overrightarrow{\partial_{\mu}V^{\mu\nu}}\frac{1}{\tilde{\Box}^{2}}\partial_{\nu}\right)^{n}\psi.$
(3.39)

With this prescription the term $C^{2}\varSigma_{\rm RFT}$ in the RFT action
becomes perturbatively well defined to all orders of expansion. Indeed, this
term with $\varSigma_{\rm RFT}$ given by (2.21) and on account of total
derivative structure $\sqrt{g}(E-\tfrac{2}{3}\,\Box
R\big{)}=\partial_{\alpha}E^{\alpha}$ can be rewritten by integration by parts
as

$4\\!\int d^{4}x\sqrt{g}\,C^{2}\varSigma_{\rm RFT}=-\int
d^{4}x\,\sqrt{g}E^{\alpha}\,\partial_{\alpha}\frac{1}{\sqrt{g}\Delta_{4}}\big{(}\sqrt{g}\,C^{2}\big{)}$
(3.40)

with the above prescription (3.39). This confirms a well defined nature of all
multiple point correlators of stress tensor generated by RFT action.

Finally, it is worth discussing the effective action behavior under global
conformal transformations with $\sigma_{0}={\rm const}$. Higher order
curvature terms of the effective action scale as negative powers of
$e^{\sigma_{0}}$ and therefore are irrelevant in the IR limit. In [61] this
was a main argument in favor of a dominant role of the Wess–Zumino action
(2.6) in this limit because $\Delta\varGamma[g,\sigma]$ behaves linearly in
$\sigma_{0}$ (or logarithmically in the distance). Indeed,

$\displaystyle\Delta\varGamma[g,\sigma+\sigma_{0}]$
$\displaystyle=\Delta\varGamma[g,\sigma]$
$\displaystyle+\sigma_{0}\Big{(}\frac{\gamma}{32\pi^{2}}\int
d^{4}x\,\sqrt{g}\,C^{2}+\beta\,e^{\prime}_{E}\Big{)},$ (3.41)

where $e^{\prime}_{E}$ is the Euler characteristics of the manifold modulo its
boundary contribution (see footnote 3). Note, however, that this behavior
cannot be captured within the nonlocal RTF form of the anomaly action (1.3)
because it is valid only under Dirichlet boundary conditions for the Green’s
function of $\Delta_{4}$ (which would be violated by the $\sigma_{0}$-shift).
In other words, the expression (1.3) lacks the contribution of the zero mode
of the Paneitz operator, which on the contrary is explicitly featuring in
(3.41). For compact manifolds with possibly nontrivial topology global Weyl
transformations would not contradict boundary conditions, and these
transformations will obviously show up in the generalized RFT gauge (2.35) as
an ambiguity of the solution for Eq. (2.37),
$\varSigma\to\varSigma+\sigma_{0}$.

##  4 Stress tensor in conformally related spacetimes

Equations (2.19) and (2.34) show that the anomalous action makes sense as an
object specifying the difference of effective actions on conformally related
metrics and other fields. Outside of this context this action, being a subject
of shifting by an arbitrary conformal invariant functional $W^{\rm
conf}[\,g\,]$, as in Eq. (2.2), is not very instructive because such a shift
can include essential physical information on conformally invariant degrees of
freedom. Anomaly action $\varGamma_{\chi}$, or it would be better to say, the
Wess–Zumino type action (2.6)—the generating functional of
$\varGamma_{\chi}$—is really useful in situations when the physics of a
conformally related spacetime with the metric $\bar{g}_{\mu\nu}$ is fully
known. Then the effective action at $g_{\mu\nu}$ can be completely recovered
from the knowledge of the Weyl anomaly.

The simplest situation belongs to the class of conformally flat spacetimes
when $\bar{g}_{\mu\nu}$ can be associated with flat metric for which all the
metric field invariants are vanishing and $\varGamma[\,\bar{g}\,]$ is either
exactly zero or calculable for quantum matter fields in flat spacetime. In
particular, the fundamental observable which can then be obtained is the UV
renormalized expectation value of the stress tensor of classically conformally
invariant fields,

$\sqrt{g}\,\big{\langle}\,T^{\alpha\beta}\big{\rangle}=2\frac{\delta\varGamma_{\rm
ren}}{\delta g_{\alpha\beta}}$ (4.1)

provided $\langle\,\bar{T}^{\alpha\beta}\rangle=0$ or known from flat space
physics. Here we derive from (2.6) the expression for the difference of
(densitized) stress tensors
$\sqrt{g}\,\langle\,T^{\alpha}_{\beta}\rangle-\sqrt{\bar{g}}\,\langle\,\bar{T}^{\alpha}_{\beta}\rangle$,
which for a conformally flat spacetime coincides with a well-known
Brown–Cassidy expression [25] and generalizes it to the case of a nonvanishing
Weyl tensor.

###  4.1 Conformal anomaly from the divergent part of the effective action

To derive the behavior of the renormalized stress tensor on the conformal
group orbit we, first, have to trace the origin of conformal anomaly as the
result of subtracting UV divergences from covariantly regularized effective
action, $\varGamma_{\rm ren}=\varGamma_{\rm reg}-\varGamma_{\infty}$. In
dimensional regularization, $\varGamma_{\rm reg}={}^{(d)}\\!\varGamma$, these
divergences are given by

$\displaystyle\varGamma_{\infty}$
$\displaystyle=-\frac{1}{16\pi^{2}\epsilon}\int d^{d}x\,\sqrt{g}\,a_{2}$
$\displaystyle=\frac{1}{16\pi^{2}\epsilon}\int
d^{d}x\,\sqrt{g}\,\big{(}\alpha\,{}^{(4)}\\!C^{2}+\beta\,{}^{(4)}\\!E\,\big{)},$
(4.2)

where $\epsilon=4-d$, ${}^{(4)}\\!C^{2}$ and ${}^{(4)}\\!E$ are the four-
dimensional invariants formally continued to $d$-dimensions and $a_{2}$ is the
relevant second Schwinger–DeWitt coefficient of the corresponding heat kernel
expansion for the inverse propagator of the theory [62, 63, 64],

$\displaystyle\;a_{2}=-\big{(}\alpha\,{}^{(4)}\\!C^{2}+\beta\,{}^{(4)}E+\gamma\,\Box
R\big{)},$ (4.3)
$\displaystyle{}^{(4)}C^{2}=R_{\mu\nu\alpha\beta}^{2}-2R_{\mu\nu}^{2}+\frac{1}{3}R^{2},$
(4.4)
$\displaystyle{}^{(4)}\\!E=R_{\mu\nu\alpha\beta}^{2}-4R_{\mu\nu}^{2}+R^{2}.$
(4.5)

This structure of $a_{2}$ follows from the local conformal invariance of the
pole residue of $\varGamma_{\infty}$ at $d=4$ and associated with the
integrability (or conformal Wess–Zumino) condition for a conformal anomaly. It
includes the topological Gauss–Bonnet density $\sqrt{g}E$, Weyl tensor squared
and the total derivative $\Box R$ terms.

Conformal anomaly arises as a contribution of the conformal transformation of
the one-loop counterterm (4.2) subtracted from the regularized effective
action

$\sqrt{g}\,\big{\langle}\,T^{\alpha}_{\alpha}\big{\rangle}=-2g_{\alpha\beta}\frac{\delta\varGamma_{\infty}}{\delta
g_{\alpha\beta}},$ (4.6)

because the regularized (but not yet renormalized by counterterm subtracting)
action $\varGamma_{\rm reg}$ is assumed to be conformally invariant777Or the
Weyl invariance violation of dimensionally regularized $\varGamma_{\rm reg}$
is proportional to $(d-4)^{2}$ as it happens for spin one case [4], so that it
does not contribute to the residue of the simple pole in dimensionality.. The
$\Box R$ term does not contribute to the divergences but it appears in the
conformal anomaly in view of the conformal transformation of the Weyl squared
term continued to $d$ dimensions. Moreover, within the above subtraction
scheme its coefficient $\gamma$ in the anomaly turns out to be determined by
the coefficient $\alpha$ of the Weyl term [4].

Indeed, introduce conformally covariant Weyl tensor in $d$ dimensions

$\displaystyle{}^{(d)}\\!C_{\mu\nu\alpha\beta}$
$\displaystyle=R_{\mu\nu\alpha\beta}+2P_{\beta[\mu}g_{\nu]\alpha}-2P_{\alpha[\mu}g_{\nu]\beta},$
(4.7) $\displaystyle{}^{(d)}C^{\mu}{}_{\nu\alpha\beta}$
$\displaystyle={}^{(d)}\\!\bar{C}^{\mu}{}_{\nu\alpha\beta},$ (4.8)

which is written down in terms of the Schouten tensor

$P_{\mu\nu}\equiv\frac{1}{d-2}\Big{(}R_{\mu\nu}-\frac{Rg_{\mu\nu}}{2(d-1)}\Big{)}.$
(4.9)

In view of the relation between the square of Weyl tensors
${}^{(d)}C^{2}\equiv{}^{(d)}C_{\mu\nu\alpha\beta}^{2}$ and
$C^{2}\equiv{}^{(4)}\\!C^{2}_{\mu\nu\alpha\beta}$ (both formally continued to
$d$ dimensions) [25]

${}^{(4)}\\!C^{2}={}^{(d)}\\!C^{2}-\frac{\epsilon}{2}\big{(}E-C^{2}-\tfrac{1}{9}R^{2}\big{)}+O(\epsilon^{2})$
(4.10)

one has

$\displaystyle\frac{\delta}{\delta g_{\mu\nu}}\int
d^{d}x\,\sqrt{g}\,C^{2}=\frac{\delta}{\delta g_{\mu\nu}}\int
d^{d}x\,\sqrt{g}\,{}^{(d)}\\!C^{2}$
$\displaystyle\;\;\;\;+\frac{\epsilon}{2}\frac{\delta}{\delta g_{\mu\nu}}\int
d^{4}x\,\sqrt{g}\,\big{(}C^{2}+\tfrac{1}{9}R^{2}\big{)}+O(\epsilon^{2}).$
(4.11)

Then, since the tensor ${}^{(d)}C_{\mu\nu\alpha\beta}$ is conformally
covariant in any dimension, $g_{\mu\nu}(\delta/\delta g_{\mu\nu})\int
d^{d}x\,\sqrt{g}\,^{(d)}C^{2}=-\tfrac{\epsilon}{2}\sqrt{g}\,{}^{(d)}C^{2}$, we
have

$\frac{1}{\epsilon}g_{\mu\nu}\frac{\delta}{\delta g_{\mu\nu}}\int
d^{d}x\,\sqrt{g}C^{2}=-\frac{1}{2}\sqrt{g}\Big{(}C^{2}+\frac{2}{3}\Box
R\Big{)}+O(\epsilon).$ (4.12)

Using this in (4.6) one recovers the $C^{2}$ and the $\Box R$ terms in the
expression for the anomaly

$\sqrt{g}\,\big{\langle}T^{\alpha}_{\alpha}\big{\rangle}=-\frac{1}{16\pi^{2}}\sqrt{g}\,a_{2},$
(4.13)

with the parameter $\gamma$ related to the coefficient $\alpha$ of the Weyl
squared term [4]

$\gamma=\frac{2}{3}\alpha.$ (4.14)

This simple expression for the trace anomaly in terms of the second
Schwinger–DeWitt coefficient also follows from the zeta-function
regularization [65].

The Gauss–Bonnet part of the anomaly follows from the conformal variation of
the ${}^{(4)}\\!E$-term in the divergent part of the action. Just like $\Box
R$, as the residue of the pole in $\varGamma_{\infty}$ the integral of
$\sqrt{g}{}^{(4)}\\!E$ at least naively does not contribute to the stress
tensor, because in four dimensions this integral is a constant Euler
characteristics of the manifold. But in a covariant renormalization procedure
the coefficient of $1/\epsilon$ in $\varGamma_{\infty}$ cannot be treated
other than as a $d$-dimensional object, so that $\int
d^{d}x\sqrt{g}{}^{(4)}\\!E$ is no longer a topological invariant, and its
metric variation is nontrivial. Therefore, rewriting, similarly to (4.10), the
dimensionally continued Gauss–Bonnet density in terms of ${}^{(d)}\\!C^{2}$,

$\displaystyle{}^{(4)}\\!E$
$\displaystyle=R_{\mu\nu\alpha\beta}^{2}-4R_{\mu\nu}^{2}+R^{2}$
$\displaystyle={}^{(d)}\\!C^{2}-(2-3\epsilon)\big{(}R_{\mu\nu}^{2}-\tfrac{1}{3}R^{2}\big{)}+O(\epsilon^{2}),$
(4.15)

one has

$\displaystyle\frac{1}{\epsilon}\frac{\delta}{\delta g_{\alpha\beta}}\int
d^{d}x\,\sqrt{g}\,{}^{(4)}\\!E=$
$\displaystyle-\sqrt{g}\big{(}\,\tfrac{1}{2}W^{\alpha\beta}+^{(3)}\\!\\!H^{\alpha\beta}$
$\displaystyle+2R_{\mu\nu}C^{\mu\alpha\nu\beta}\big{)}+O(\epsilon),$ (4.16)

where the two new tensors arise

$\displaystyle\\!\\!\\!{}^{(3)}\\!H^{\alpha\beta}=R^{\alpha\mu}R^{\beta}_{\mu}-\frac{2}{3}RR^{\alpha\beta}-\frac{1}{2}g^{\alpha\beta}R_{\mu\nu}^{2}+\frac{1}{4}g^{\alpha\beta}R^{2},$
(4.17) $\displaystyle W^{\alpha\beta}=\lim\limits_{\epsilon\to
0}\frac{1}{\epsilon}\left(4\,{}^{(d)}\\!C^{\alpha}{}_{\mu\nu\lambda}{}^{(d)}\\!C^{\beta\mu\nu\lambda}-g^{\alpha\beta}\,{}^{(d)}\\!C^{2}\right).$
(4.18)

The limit to $d=4$ for the tensor $W^{\alpha\beta}$ is regular here because at
$d=4$ there is the important identity

$4\,{}^{(4)}\\!C^{\alpha}{}_{\mu\nu\lambda}{}^{(4)}\\!C^{\beta\mu\nu\lambda}=g^{\alpha\beta}{}^{(4)}\\!C^{2}$
(4.19)

—it can be proven by antisymmetrization over five indices in the four-
dimensional spacetime [66]. Tensors ${}^{(3)}H^{\alpha\beta}$ and
$W^{\alpha\beta}$ have the following traces

${}^{(3)}\\!H^{\alpha}_{\alpha}=\tfrac{1}{3}R^{2}-R_{\mu\nu}^{2}=\tfrac{1}{2}(E-C^{2}),\quad
W_{\alpha}^{\alpha}=C^{2}.$ (4.20)

Thus from (4.1) and (4.20) we have the relation

$\frac{2}{\epsilon}g_{\alpha\beta}\frac{\delta}{\delta g_{\alpha\beta}}\int
d^{d}x\,\sqrt{g}\,{}^{(4)}\\!E=-\sqrt{g}{}^{(4)}\\!E+O(\epsilon),$ (4.21)

which recovers the contribution of $E$-term in the conformal anomaly (4.13)
with the expression (4.3) for $a_{2}$.

###  4.2 Minimal form of Wess-Zumino action and a-theorem

Of course there is a big ambiguity in the above analytic continuation of the
coefficients relating 4-dimensional objects to their $d$-dimensional
counterparts. This ambiguity reduces to the renormalization by finite
4-dimensional counterterms $\int d^{4}x\sqrt{g}\,R_{\mu\nu\alpha\beta}^{2}$,
$\int d^{4}x\sqrt{g}\,R_{\mu\nu}^{2}$ and $\int d^{4}x\sqrt{g}\,R^{2}$ among
which in view of the total-derivative nature of the Gauss-Bonnet density only
one counterterm can additionally break Weyl invariance and change the
coefficient $\gamma$ of the $\Box R$ term in the conformal anomaly. This is
because the combination $\int d^{4}x\,\sqrt{g}(C^{2}-E)=2\int
d^{4}x\,\sqrt{g}(R_{\mu\nu}^{2}-\tfrac{1}{3}R^{2})$ is Weyl invariant, and
such a counterterm can be chosen as the square of the curvature scalar,
satisfying

$g_{\mu\nu}\frac{\delta}{\delta g_{\mu\nu}}\int
d^{4}x\sqrt{g}\,R^{2}=-6\sqrt{g}\,\Box R.$ (4.22)

Therefore this finite local counterterm can be used to alter the coefficient
$\gamma$ and, in particular, put it to zero by a special finite
renormalization which we will denote by a subscript $\rm Ren$,

$\varGamma_{\rm ren}[\,g\,]\to\varGamma_{\rm Ren}[\,g\,]\equiv\varGamma_{\rm
ren}[\,g\,]+\frac{\gamma}{192\pi^{2}}\int d^{4}x\sqrt{g}\,R^{2}.$ (4.23)

Regularization and subtraction scheme dependence of $\gamma$-coefficient
manifests itself in the violation of the relation (4.14) for the dimensionally
regularized electromagnetic vector field [5], but ultimately does not change
the physics of the theory because of the locality of the covariant counterterm
$\int d^{4}x\sqrt{g}\,R^{2}$, whose subtraction point should be determined
from the comparison with the observable value of its coupling constant. In the
cosmological example considered below the above renormalization (4.23)
corresponds to fixing the coupling constant in the Starobinsky $R^{2}$-model
[43].

The renormalization (4.23) has an important consequence – with $\gamma=0$ the
terms with quartic derivatives of $\sigma$, contained in the combination
$\tfrac{\beta}{16\pi^{2}}\int
d^{4}x\,\big{(}4\sqrt{\bar{g}}\,\sigma\bar{\Delta}_{4}\sigma-\frac{1}{9}\sqrt{g}\,R^{2}\big{)}$
of (2.6), completely cancel out, and the resulting minimal Wess-Zumino action
does not acquire extra hihger-derivative degrees of freedom,

$\displaystyle\varGamma_{\rm Ren}[\,g\,]-\varGamma_{\rm
Ren}[\,\bar{g}\,]=\frac{\alpha}{16\pi^{2}}\int
d^{4}x\,\sqrt{\bar{g}}\,\bar{C}_{\mu\nu\alpha\beta}^{2}\sigma$
$\displaystyle\quad+\frac{\beta}{16\pi^{2}}\int
d^{4}x\,\sqrt{\bar{g}}\,\Big{\\{}\bar{E}\,\sigma-4\,\big{(}\bar{R}^{\mu\nu}-\tfrac{1}{2}\bar{g}^{\mu\nu}\bar{R}\,\big{)}\,\partial_{\mu}\sigma\,\partial_{\nu}\sigma$
$\displaystyle\quad-4\,\bar{\Box}\sigma\,(\bar{\nabla}^{\mu}\sigma\,\bar{\nabla}_{\mu}\sigma)-2\,(\bar{\nabla}^{\mu}\sigma\,\bar{\nabla}_{\mu}\sigma)^{2}\Big{\\}}.$
(4.24)

This minimal version of the action for the dilaton field $\sigma$ was
discussed in [41] and used in the derivation of the $a$-theorem in [42, 67] –
monotonically decreasing coefficient $a=\beta/16\pi^{2}$ in the RG flow of the
theory from UV to IR domains. This theorem is based on the sign of the last
quartic interaction term for this field, related to the cross section of the
forward $2\to 2$ dilaton scattering which should be positive in unitary
theory, its unitarity being related to the absence of higher-derivative ghosts
in (4.24).

###  4.3 Renormalized stress tensors

The behavior of the stress tensor on the orbit of the conformal group can be
obtained by using the commutativity of the following functional variations

$\left[\,g_{\mu\nu}(y)\frac{\delta}{\delta
g_{\mu\nu}(y)},g_{\beta\gamma}(x)\frac{\delta}{\delta
g_{\alpha\gamma}(x)}\right]=0,$ (4.25)

which allows one to write

$\displaystyle\frac{\delta}{\delta\sigma(y)}\sqrt{g}\left\langle
T^{\alpha}_{\beta}(x)\right\rangle=2g_{\beta\gamma}(x)\frac{\delta}{\delta
g_{\alpha\gamma}(x)}\frac{\delta\varGamma_{\rm
ren}}{\delta\sigma(y)}\bigg{|}_{g_{\mu\nu}=e^{2\sigma}\bar{g}_{\mu\nu}}$
$\displaystyle\qquad=g_{\beta\gamma}(x)\frac{\delta}{\delta
g_{\alpha\gamma}(x)}\sqrt{g}(y)\left\langle
T^{\mu}_{\mu}(y)\right\rangle\Big{|}_{g_{\mu\nu}=e^{2\sigma}\bar{g}_{\mu\nu}}.$
(4.26)

Bearing in mind that $g_{\beta\gamma}\delta/\delta
g_{\alpha\gamma}=\bar{g}_{\beta\gamma}\delta/\delta\bar{g}_{\alpha\gamma}$ at
fixed $\sigma$ and functionally integrating this relation over $\sigma$ one
has

$\sqrt{g}\,\big{\langle}T^{\alpha}_{\beta}\big{\rangle}-\sqrt{\bar{g}}\,\big{\langle}\bar{T}^{\alpha}_{\beta}\big{\rangle}=2\bar{g}_{\beta\gamma}\frac{\delta}{\delta\bar{g}_{\alpha\gamma}}\Delta\varGamma[\,\bar{g},\sigma\,],$
(4.27)

where $\Delta\varGamma[\,\bar{g},\sigma\,]=\varGamma_{\rm
ren}-\bar{\varGamma}_{\rm ren}$ is given by (2.6).

Before calculating this difference by the metric variation of
$\Delta\varGamma[\bar{g},\sigma]$ it is instructive to obtain it directly from
the divergent part of the action as it was done in [25]. Note that
$\varGamma_{\rm ren}-\bar{\varGamma}_{\rm
ren}=-(\varGamma_{\infty}-\bar{\varGamma}_{\infty})$ because $\varGamma_{\rm
reg}$ does not contribute to the anomaly (see footnote 7). Therefore,

$\sqrt{g}\,\big{\langle}\,T^{\alpha}_{\beta}\big{\rangle}\,\Big{|}_{\,\bar{g}}^{\,g}=-2\,g_{\beta\gamma}\frac{\delta\varGamma_{\infty}}{\delta
g_{\alpha\gamma}}\,\Big{|}_{\,\bar{g}}^{\,g}$ (4.28)

To calculate the contribution of the ${}^{(4)}C^{2}$-term in
$\varGamma_{\infty}$ we rewrite it in terms of ${}^{(d)}C^{2}$ and use Eq.
(4.11). This leads to the contribution of the first term of this equation

$\displaystyle\frac{\delta}{\delta g_{\mu\nu}}\int
d^{d}x\,\sqrt{g}\,{}^{(d)}C^{2}=-\frac{\epsilon}{2}\sqrt{g}\,W^{\mu\nu}-4\sqrt{g}\,{}^{(d)}B^{\mu\nu},$
(4.29)
$\displaystyle{}^{(d)}B^{\mu\nu}=\Big{(}\frac{1}{d-2}R_{\alpha\beta}+\nabla_{(\alpha}\nabla_{\beta)}\Big{)}C^{\mu\alpha\nu\beta},$
(4.30)

where the tensor $W^{\mu\nu}$ is defined by Eq. (4.18) and
${}^{(d)}\\!B^{\mu\nu}$ is the $d$-dimensional Bach tensor. Assembling this
with the second term of Eq. (4.11) we get on the orbit of the conformal group

$\frac{1}{\epsilon}g_{\beta\gamma}\frac{\delta}{\delta g_{\alpha\gamma}}\int
d^{d}x\,\sqrt{g}\,^{(4)}\\!C^{2}\,\Big{|}_{\,\bar{g}}^{\,g}\\\
=-\sqrt{g}\Big{[}\,\frac{4}{\epsilon}{}^{(d)}\\!B^{\alpha}_{\beta}+\frac{1}{18}{}^{(1)}\\!H^{\alpha}_{\beta}\,\Big{]}_{\,\bar{g}}^{\,g}+O(\epsilon),$
(4.31)

where the tensor ${}^{(1)}\\!H^{\alpha}_{\beta}$ is given by the equation

$\displaystyle{}^{(1)}\\!H^{\alpha}_{\beta}=\frac{1}{\sqrt{g}}g^{\alpha\gamma}\frac{\delta}{\delta
g^{\beta\gamma}}\int d^{4}x\,\sqrt{g}R^{2}$
$\displaystyle=-\frac{1}{2}\delta^{\alpha}_{\beta}R^{2}+2RR^{\alpha}_{\beta}+2\delta^{\alpha}_{\beta}\Box
R-2\nabla^{\alpha}\nabla_{\beta}R,$ (4.32)

and we took into account that the both tensor densities
$\sqrt{g}\,W^{\alpha}_{\beta}$ and $\sqrt{g}\,B^{\alpha}_{\beta}$ in four
dimensions are invariant on the conformal orbit. Outside of four dimensions
the Bach tensor density transforms on this orbit as (here as above
$g_{\mu\nu}=e^{2\sigma}\bar{g}_{\mu\nu}$)

$\sqrt{g}\,{}^{(d)}\\!B^{\alpha}_{\beta}\,\Big{|}_{\bar{g}}^{g}=-\frac{\epsilon}{2}\sqrt{\bar{g}}\big{(}\bar{R}^{\mu\nu}+2\bar{\nabla}^{(\mu}\bar{\nabla}^{\nu)}\big{)}\big{(}\sigma\bar{C}^{\alpha}{}_{\mu\beta\nu}\big{)}+O(\epsilon^{2}),$
(4.33)

which makes the first term on the right hand side of (4.31) well defined at
$d\to 4$. Note that the expression
$\sqrt{\bar{g}}(\bar{R}^{\mu\nu}+2\bar{\nabla}^{(\mu}\bar{\nabla}^{\nu)}\big{)}\big{(}\sigma\bar{C}^{\alpha}{}_{\mu\beta\nu})$
treated as a functional of independent $\bar{g}_{\mu\nu}$ and $\sigma$ is Weyl
invariant under local conformal transformations of the barred metric. This can
be easily inferred from the invariance of Eq.(4.33) under the interchange
$g_{\mu\nu}\leftrightarrow\bar{g}_{\mu\nu}$ and $\sigma\to-\sigma$ or directly
checking the conformal transformation of $\bar{g}_{\mu\nu}$ (with a fixed
scalar $\sigma$).

The contribution of Gauss–Bonnet term to the stress tensor behavior on the
conformal orbit is obtained from using (4.1)–(4.17). Collecting this
contribution with the contribution (4.31) of the Weyl tensor squared part we
finally have

$\displaystyle\sqrt{g}\,\big{\langle}T^{\alpha}_{\beta}\big{\rangle}\Big{|}_{\,\bar{g}}^{\,g}=-\frac{\alpha}{4\pi^{2}}\sqrt{\bar{g}}\,\big{(}\bar{R}^{\mu\nu}+2\bar{\nabla}^{(\mu}\bar{\nabla}^{\nu)}\big{)}\big{(}\sigma\bar{C}^{\alpha}{}_{\mu\beta\nu}\big{)}+\frac{1}{8\pi^{2}}\sqrt{g}\,\Big{[}\,\beta\,^{(3)}\\!H^{\alpha}_{\beta}+\frac{\alpha}{18}\,^{(1)}\\!H^{\alpha}_{\beta}+2\beta
R^{\mu\nu}C^{\alpha}{}_{\mu\beta\nu}\,\Big{]}_{\,\bar{g}}^{\,g}.$ (4.34)

This is a generalization of the Brown–Cassidy formula to the case of a nonzero
Weyl tensor. The first term of this expression is Weyl invariant in view of
the above remark and can be represented by its unbarred version.

The check of consistency of this formula with the original expression for the
conformal anomaly is trivial in view of
${}^{(3)}\\!H^{\alpha}_{\alpha}=(E-C^{2})/2$,
${}^{(1)}\\!H^{\alpha}_{\alpha}=6\Box R$ and tracelessness of the Weyl tensor,

$\sqrt{g}\,\big{\langle}T^{\alpha}_{\alpha}\big{\rangle}\Big{|}_{\,\bar{g}}^{\,g}=\frac{\sqrt{g}}{16\pi^{2}}\\!\Big{[}\beta
E-\beta C^{2}+\tfrac{2\alpha}{3}\Box
R\Big{]}_{\bar{g}}^{g}=-\frac{\sqrt{g}\,a_{2}}{16\pi^{2}}\Big{|}_{\,\bar{g}}^{\,g},$
(4.35)

where the last equality follows from the conformal invariance of the density
$\sqrt{g}\,C^{2}$ and from the relation (4.14) between the coefficients
$\gamma$ and $\alpha$, $\alpha=\tfrac{3}{2}\gamma$.

The recovery of (4.34) from the direct variation of the Wess–Zumino action
(4.27) goes as follows. We use metric variational formulae

$\displaystyle\frac{\delta}{\delta g_{\alpha\beta}}\\!\int
d^{4}x\,\sqrt{g}\,C^{2}\sigma=-2\sqrt{g}\big{(}R_{\mu\nu}\\!+2\nabla_{(\mu}\\!\nabla_{\nu)}\big{)}\big{(}\sigma
C^{\alpha\mu\beta\nu}\big{)},$ (4.36) $\displaystyle\frac{\delta}{\delta
g_{\alpha\beta}}\int
d^{4}x\,\sqrt{g}\,\mathcal{E}_{4}\sigma=\sqrt{g}\,\Delta^{\alpha\beta}\sigma,$
(4.37) $\displaystyle\frac{\delta}{\delta g_{\alpha\beta}}\int
d^{4}x\,\sqrt{g}\,\varphi\Delta_{4}\sigma=-\frac{\sqrt{g}}{2}D^{\alpha\beta}[\varphi,\sigma],$
(4.38)

which hold for generic scalar test functions $\sigma$ and $\varphi$ with the
differential operator $\Delta^{\alpha\beta}$ acting on $\sigma$,

$\displaystyle\Delta_{\alpha\beta}$
$\displaystyle=\frac{1}{3}(g_{\alpha\beta}\Box-\nabla_{\alpha}\nabla_{\beta})\,\Box$
$\displaystyle+\left[\,2(g_{\alpha\beta}P_{\mu\nu}-g_{\alpha\mu}P_{\beta\nu}-g_{\alpha\nu}P_{\beta\mu})+\frac{8}{3}g_{\mu\nu}P_{\alpha\beta}\right.$
$\displaystyle\left.+2Pg_{\alpha\mu}g_{\beta\nu}-\frac{5}{3}Pg_{\alpha\beta}g_{\mu\nu}-2W_{\alpha\mu\beta\nu}\right]\nabla^{\mu}\nabla^{\nu}$
$\displaystyle+\big{(}\,g_{\alpha\beta}g_{\mu\nu}-g_{\alpha\mu}g_{\beta\nu}-g_{\alpha\nu}g_{\beta\mu}\big{)}(\nabla^{\mu}P)\nabla^{\nu},$
(4.39)

and the bilinear form $D^{\alpha\beta}(\varphi,\sigma)$,

$\displaystyle D_{\alpha\beta}[\varphi,\sigma]$
$\displaystyle=-\frac{1}{2}g_{\alpha\beta}\Box\varphi\,\Box\sigma-2\sigma_{\alpha\beta}\Box\varphi$
$\displaystyle+2\sigma_{\alpha}\Box\varphi_{\beta}-\frac{1}{3}g_{\alpha\beta}\sigma_{\mu}\Box\varphi^{\mu}-\frac{2}{3}\varphi_{\mu(\alpha\beta)}\sigma^{\mu}$
$\displaystyle+\left[\,2W_{\alpha\mu\beta\nu}+\frac{1}{3}(g_{\mu\nu}R_{\alpha\beta}-g_{\alpha\mu}g_{\beta\nu}R)\right]\varphi^{(\mu}\sigma^{\nu)}$
$\displaystyle+\frac{1}{3}\big{(}\,4\varphi_{\alpha\mu}\sigma^{\mu}_{\beta}-g_{\alpha\beta}\varphi_{\mu\nu}\sigma^{\mu\nu}\big{)}+\big{(}\,\varphi\Leftrightarrow\sigma\,\big{)},$
(4.40)

where $\varphi_{\alpha}\equiv\nabla_{\alpha}\varphi$,
$\sigma_{\alpha\beta}\equiv\nabla_{\beta}\nabla_{\alpha}\sigma$,
$\varphi_{\alpha\beta\gamma}\equiv\nabla_{\gamma}\nabla_{\beta}\nabla_{\alpha}\varphi$,
etc. Note that the trace of $\Delta^{\alpha\beta}$ coincides with the Paneitz
operator, $g_{\alpha\beta}\Delta^{\alpha\beta}=\Delta_{4}$, which matches with
the conformal variation (2.9), and the bilinear form
$D^{\alpha\beta}(\varphi,\sigma)$ is traceless in view of conformal invariance
of $\sqrt{g}\Delta_{4}$.

Using these relations we get from (4.27) and (2.6)

$\displaystyle\sqrt{g}\,\big{\langle}T^{\alpha}_{\beta}\big{\rangle}\,\Big{|}_{\,\bar{g}}^{\,g}=-\frac{\alpha}{4\pi^{2}}\sqrt{g}\big{(}R^{\mu\nu}+2\nabla^{(\mu}\nabla^{\nu)}\big{)}\big{(}\sigma
C^{\alpha}{}_{\mu\beta\nu}\big{)}$
$\displaystyle+\frac{\sqrt{g}}{8\pi^{2}}\\!\Big{(}2\beta\Delta^{\alpha}_{\beta}\sigma+\beta
D^{\alpha}_{\beta}[\sigma,\sigma]\Big{)}+\sqrt{g}\left(\tfrac{\gamma}{12}+\tfrac{\beta}{18}\right){}^{(1)}\\!H^{\alpha}_{\beta}\,\Big{|}_{\,\bar{g}}^{\,g}.$
(4.41)

The term in the first line here coincides with its barred version in
(4.34)—this easily follows from the relation (4.36) where the integrand can be
identically replaced by the barred one. The
$\frac{\gamma}{12}{}^{(1)}\\!H^{\alpha}_{\beta}$ term here matches with the
$\frac{\alpha}{18}{}^{(1)}\\!H^{\alpha}_{\beta}$ term of (4.34) in view of the
relation $\alpha=\tfrac{3}{2}\gamma$. And finally, the identity holds

$\displaystyle\sqrt{g}\,\Big{[}{}^{(3)}\\!H^{\alpha}_{\beta}+\tfrac{1}{18}{}^{(1)}\\!H^{\alpha}_{\beta}$
$\displaystyle+2R^{\mu\nu}C^{\alpha}{}_{\mu\beta\nu}\Big{]}_{\,\bar{g}}^{\,g}$
$\displaystyle=\sqrt{g}\Big{(}2\Delta^{\alpha}_{\beta}\sigma+D^{\alpha}_{\beta}[\sigma,\sigma]\Big{)},$
(4.42)

which completely reconciles the two expressions (4.34) and (4.41) for the
stress tensor behavior on the orbit of the conformal group.

##  5 Conformally flat spacetime

The generalization (4.34) of Brown-Cassidy formula to the case of a
nonvanishing Weyl tensor might be not very useful, because in the general case
not much can be said about $\langle\,T^{\alpha}_{\beta}\rangle\,|_{\bar{g}}$.
Therefore we will restrict ourselves with the case of the conformally flat
spacetime for which the conformal transformation of the metric can lead to the
metric $\bar{g}_{\mu\nu}$ of flat spacetime, where
$\langle\,\bar{T}^{\alpha}_{\beta}\rangle$ is either zero or can be obtained
from flat space physics. Interestingly, in this case the parameter of the
conformal transformation $\sigma$ making this transition satisfies the
equation

$\Delta_{4}\,\sigma=\frac{1}{4}\mathcal{E}_{4}$ (5.1)

and in asymptotically flat case with Dirichle boundary conditions has a unique
solution (2.21), $\sigma=\varSigma_{\rm RFT}$. This, apparently not very well
known fact, can be proven by using the equation for the conformal
transformation of the four-dimensional Schouten tensor (4.8)
($g_{\mu\nu}=e^{2\sigma}\bar{g}_{\mu\nu}$)

$P_{\mu\nu}-\bar{P}_{\mu\nu}=-\sigma_{\mu\nu}-\sigma_{\mu}\sigma_{\nu}+\frac{1}{2}\sigma_{\alpha}\sigma^{\alpha}g_{\mu\nu},$
(5.2)

where $\sigma_{\mu}\equiv\nabla_{\mu}\sigma$ and
$\sigma_{\mu\nu}\equiv\nabla_{\nu}\nabla_{\mu}\sigma$. Assuming that
$\bar{g}_{\mu\nu}$ is flat space metric with $\bar{P}_{\mu\nu}=0$,
differentiating twice and again using this relation to express $P_{\mu\nu}$ in
terms of the derivatives of $\sigma$ one has

$\nabla^{\mu}\nabla^{\nu}\left(P_{\mu\nu}+\sigma_{\mu\nu}+\sigma_{\mu}\sigma_{\nu}-\frac{1}{2}\sigma_{\alpha}\sigma^{\alpha}g_{\mu\nu}\right)\\\
=\Delta_{4}\sigma-\frac{1}{4}\mathcal{E}_{4}=0,$ (5.3)

whence it follows that the conformal invariant metric (2.12) in the RFT gauge
(2.20) is actually the flat space one when the Weyl tensor is zero

$\bar{R}^{\alpha}_{\;\;\beta\mu\nu}=0,\quad\bar{g}_{\mu\nu}=e^{-2\varSigma_{\rm
RFT}[\,g\,]}g_{\mu\nu}\big{|}_{\,C_{\alpha\beta\mu\nu}=0}.$ (5.4)

Note that $\bar{g}_{\mu\nu}$ here is not automatically diagonal unit matrix
$\delta_{\mu\nu}$, because this is the invariant statement which is valid in
any coordinate system.

###  5.1 Anomaly driven cosmology

Applications of the conformal anomaly in the cosmological context have a long
history, see for example [26, 68, 9, 69, 70, 71, 72]. In particular, cosmology
with the Friedman–Robertson–Walker (FRW) metric represents the situation when
the anomalous action $\Delta\varGamma[\bar{g},\sigma]$ entirely determines the
physics of the field model and via effective equations of motion produces a
nontrivial back reaction of quantum matter on the dynamical metric background.
The most interesting example is, perhaps, the case when
$\varGamma[\,\bar{g}\,]$ in (2.6) nontrivially contributes to this back
reaction effect rather than just serves as an inert flat space background.

This is the spatially closed cosmology driven by a conformal field theory
(CFT) from the initial state in the form of a special microcanonical density
matrix, which was orginally suggested in [27] and recently reviewed in [46].
With the density matrix defined as the projector on the space of solutions of
the Wheeler–DeWitt equations [28, 73] the statistical sum in this model has a
representation of the Euclidean quantum gravity (EQG) path integral

$Z=\int D[\,g_{\mu\nu},\phi\,]\;e^{-S[\,g_{\mu\nu},\phi\,]},$ (5.5)

where integration runs over the metric $g_{\mu\nu}$ and matter fields $\phi$
which are periodic on the Euclidean spacetime of topology $S^{1}\times S^{3}$
with the time $\tau$ compactified to a circle $S^{1}$.

When the classical action $S[\,g_{\mu\nu},\phi\,]$ is dominated by numerous
CFT fields $\varPhi$ with their action
$S_{C\\!F\\!T}[\,g_{\mu\nu},\varPhi\,]$, the statistical sum can be
approximated by the contribution of the saddle point of this integral. This is
the extremum of the total action including the tree-level gravitational
Einstein–Hilbert action $S_{EH}[\,g_{\mu\nu}]$ and the effective action
$\varGamma[\,g_{\mu\nu}]$ of these CFT fields888Disregarding the graviton
loops can be justified by the domination of conformal fields outnumbering the
metric, and retaining the Einstein–Hilbert term obviously follows from the
fact that this term with renormalized gravitational and cosmological constants
is anyway induced from the quantum conformal sector.,

$\displaystyle\varGamma_{\rm tot}[\,g_{\mu\nu}]$
$\displaystyle=S_{EH}[\,g_{\mu\nu}]+\varGamma[\,g_{\mu\nu}],$ (5.6)
$\displaystyle e^{-\varGamma[\,g_{\mu\nu}]}$ $\displaystyle=\int
D\varPhi\,e^{-S_{C\\!F\\!T}[\,g_{\mu\nu},\varPhi\,]}.$ (5.7)

Choosing as $g_{\mu\nu}$ the FRW metric with the scale factor $a(\tau)$ and
the lapse function $N$ ($\varOmega^{2}_{(3)}$ is the metric of the
3-dimensional sphere of a unit radius),

$ds^{2}=N^{2}d\tau^{2}+a^{2}d\varOmega^{2}_{(3)}=a^{2}(\tau)\big{(}d\eta^{2}+d\varOmega^{2}_{(3)}\big{)},$
(5.8)

one immediately finds that in terms of the conformal time variable $\eta$,
related to the Euclidean time $\tau$ by the relation $d\eta=d\tau/a(\tau)$,
this metric is conformally equivalent to the metric $\bar{g}_{\mu\nu}\equiv
g_{\mu\nu}^{EU}$ of the Einstein static universe with spatial sections—the
3-dimensional spheres of some constant radius $a_{0}$,

$\displaystyle
d\bar{s}^{2}=a_{0}^{2}\big{(}d\eta^{2}+d\varOmega^{2}_{(3)}\big{)}\equiv
g_{\mu\nu}^{EU}dx^{\mu}dx^{\nu},$ (5.9) $\displaystyle
ds^{2}=e^{2\sigma}d\bar{s}^{2},\quad
g_{\mu\nu}=e^{2\sigma}g_{\mu\nu}^{EU},\quad\sigma=\ln\frac{a}{a_{0}}.$ (5.10)

Therefore the CFT effective action expresses in terms of the same action on a
static Einstein universe $\varGamma[\,g_{\mu\nu}^{EU}]\equiv\varGamma_{EU}$
and Wess–Zumino action (2.6) with the above conformal parameter $\sigma$

$\varGamma[\,g_{\mu\nu}]=\Delta\varGamma[\,g_{\mu\nu}^{EU},\sigma\,]+\varGamma_{EU}.$
(5.11)

The calculation of $\varGamma_{EU}$ is strongly facilitated by the static
nature of the background, but it still yields a nontrivial result in view of
compactification of time on $S^{1}$. To begin with, note that although
$g_{\mu\nu}^{EU}$ explicitly depends on the size $a_{0}$ of $S^{3}$, the value
of $\varGamma_{EU}$ is $a_{0}$-independent for a fixed period of the conformal
time $\eta=\oint d\eta$. This follows from the invariance of the effective
action under global conformal transformations (3.41) for conformally flat
spacetimes with zero bulk part of the Euler characteristics (which is the case
of $S^{1}\times S^{3}$). This also can be confirmed by using scaling
properties of the conformal fields. Indeed, the energies of conformal quanta
on a static spacetime scale as $1/a_{0}$ and their Hamiltonian reads,

$\hat{H}=\sum_{\omega}\frac{\omega}{a_{0}}\Big{(}\hat{a}^{\dagger}_{\omega}\hat{a}_{\omega}\pm\frac{1}{2}\Big{)},$
(5.12)

where summation runs over all quantum numbers (and spins) of the energies
$\omega/a_{0}$ of all field oscillator modes on a static 3-dimensional sphere
of the radius $a_{0}$ and $\hat{a}^{\dagger}_{\omega}$ and $\hat{a}_{\omega}$
are the relevant creation-annihilation operators ($\pm$ signs correspond to
bosons or fermions). The path integral over (anti)periodic conformal (fermion)
boson fields with a period ${\cal T}=\oint d\tau N$ on a static metric
background is exactly calculable and equals the equilibrium statistical sum at
the temperature $1/{\cal T}$ which expresses as a function of the conformal
time period $\eta={\cal T}/a_{0}$

$e^{-\varGamma_{EU}}=\int
D\varPhi\,e^{-S_{C\\!F\\!T}[\,g_{\mu\nu}^{EU},\varPhi\,]}\\\ ={\rm
Tr}\,e^{-{\cal T}\hat{H}}=\exp\big{(}-\eta E_{\rm vac}-F(\eta)\big{)}.$ (5.13)

Here $F(\eta)$ is the free energy of the gas of conformal particles and
$E_{\rm vac}$ is a UV divergent Casimir energy which should be covariantly
renormalized

$\displaystyle F(\eta)$ $\displaystyle=\sum_{\omega}\Big{[}\pm\,\ln\big{(}1\mp
e^{-\omega\eta}\big{)}\,\Big{]},$ (5.14) $\displaystyle E_{\rm vac}$
$\displaystyle=\Big{(}\sum_{\omega}\frac{\pm\,\omega}{2}\Big{)}_{\rm ren}.$
(5.15)

Thus, the dependence on $a_{0}$ is absorbed into the dependence on $\eta$
which should be fixed under the rescaling of $a_{0}$. Note that it is $\eta$
that should be kept fixed under the global conformal transformation which
simultaneously rescales the lapse function $N$ and $a_{0}$ in the definition
of the conformally invariant $\eta=\oint d\tau\,N/a_{0}$.

Remarkably, the covariant renormalization of the vacuum Casimir energy $E_{\rm
vac}$ also follows from the behavior of the effective action on the orbit of
the conformal group. The Einstein universe extending from $-\infty$ to
$+\infty$ in $\eta$ is mapped to flat space by the transition to the radial
coordinate $\rho$

$\eta\mapsto\rho=a_{0}e^{\eta},\quad-\infty<\eta<+\infty,\quad
0\leq\rho<\infty,$ (5.16)

with the conformal relation between the two metrics

$\displaystyle ds^{2}_{EU}=e^{2\sigma}ds_{\rm
flat}^{2},\quad\sigma=-\eta=\ln\frac{a_{0}}{\rho},$ (5.17) $\displaystyle
ds_{\rm flat}^{2}=d\rho^{2}+\rho^{2}d\varOmega^{2}_{(3)}.$ (5.18)

For the vacuum state (the limit $\eta\to\infty$ and $F(\eta)\to 0$ in Eq.
(5.13)) $\varGamma_{EU}\to E_{\rm vac}\eta$. On the other hand, from Eq. (2.6)
with the above expression for $\sigma$

$\displaystyle\Delta\varGamma[\,g_{\rm
flat},\sigma\,]=\frac{\beta}{8\pi^{2}}\int d^{4}x\sqrt{g_{\rm
flat}}\big{(}\Box_{\rm flat}\sigma\big{)}^{2}$
$\displaystyle\qquad\qquad\quad-\frac{1}{32\pi^{2}}\Big{(}\frac{\gamma}{6}+\frac{\beta}{9}\Big{)}\,\int
d^{4}x\sqrt{g_{EU}}R^{2}_{EU}.$ (5.19)

Bearing in mind that $\Box_{\rm flat}\sigma=-2/\rho^{2}$, $\int
d^{4}x\sqrt{g_{\rm flat}}\\!\mapsto 2\pi^{2}\int d\rho\,\rho^{3}$,
$R_{EU}=6/a_{0}^{2}$ and $\int d^{4}x\sqrt{g_{EU}}\mapsto
2\pi^{2}a_{0}^{4}\int d\eta$, one has

$\displaystyle\varGamma_{EU}-\varGamma_{\rm flat}=\Delta\varGamma[\,g_{\rm
flat},\sigma\,]$
$\displaystyle=\beta\\!\int\frac{d\rho}{\rho}-\Big{(}\frac{3}{8}\gamma+\frac{\beta}{4}\Big{)}\\!\int
d\eta$ $\displaystyle=\frac{3}{4}\,\Big{(}\beta-\frac{\gamma}{2}\Big{)}\int
d\eta.$ (5.20)

Therefore, under an obvious assumption that $\varGamma_{\rm flat}=0$ one has

$E_{\rm vac}=\frac{3}{4}\,\Big{(}\beta-\frac{\gamma}{2}\Big{)}.$ (5.21)

In other words, after covariant renormalization by covariant counterterms the
Casimir energy gets the value compatible with the behavior of the renormalized
effective action on the conformal group orbit (or with the Brown–Cassidy
formula for the vacuum stress tensor). This compatibility was indeed checked
by direct renormalization of the UV divergent sum over field modes in (5.15)
[47, 48, 49, 50].

Let us now turn to the contribution of the conformal transformation from the
generic FRW metric to that of the static Einstein universe in (5.11). To begin
with we use the freedom of finite renormalization (4.23) which reduces the
theory to the case of anomaly (1.1) with $\gamma=0$ and, in particular,
renders $E_{\rm vac}=\tfrac{3}{4}\beta$. In the cosmological context this
freedom corresponds to the adjustment of the coupling constant of the
Starobinsky $R^{2}$-action [68] which plays an important role in inflation
theory and the dark energy model. Then, with $\gamma=0$ and $\sigma$ given by
(5.17) the Wess–Zumino term in (5.11) takes the form [27]

$\varGamma_{\rm Ren}[\,g\,]-\varGamma_{\rm
Ren}[\,g_{EU}]=\frac{3\beta}{2}\oint d\tau\,N\left(\frac{a^{\prime
2}}{a}-\frac{a^{\prime 4}}{6\,a}\right),$ (5.22)

when written down in terms of the original FRW coordinates with the notation
for the invariant time derivative $a^{\prime}=da/Nd\tau$. Note that the result
is again independent of the constant $a_{0}$ because it contains only
differentiated $\sigma$ and, moreover, it does not involve higher order
derivatives of $a(\tau)$. The last property is entirely due to the fact of
$\gamma$ being renormalized to zero and due to the cancellation of higher
derivative terms in the minimal form of Wess-Zumino action (4.24).

Now we assemble together the Einstein-Hilbert action (with the reduced Planck
mass $M_{\rm P}=1/\sqrt{8\pi G}$ and the cosmological constant $\varLambda$),
the action on the Einstein universe space (5.13) and (5.22). This leads to the
total effective action on the generic Euclidean FRW background periodic in
Euclidean time with the period $\eta$ measured in units of the conformal time

$\displaystyle\varGamma_{\rm tot}[\,a,N\,]=6\pi^{2}M_{P}^{2}\oint
d\tau\,N\bigg{\\{}-aa^{\prime 2}-a+\frac{\varLambda}{3}a^{3}$
$\displaystyle+\frac{\beta}{4\pi^{2}M_{P}^{2}}\left(\frac{a^{\prime
2}}{a}-\frac{a^{\prime 4}}{6a}+\frac{1}{2a}\right)\bigg{\\}}+F(\eta),$ (5.23)
$\displaystyle\eta=\oint\frac{d\tau N}{a}.$ (5.24)

Here the contribution of the conformal anomaly and Casimir energy (5.21) (with
$\gamma=0$) are both weighted by the parameter $\beta$ of the topological term
in the conformal anomaly. The free energy of the gas of conformal particles
$F(\eta)$ is a function of the effective (“comoving”) temperature of this gas
– the inverse of the circumference $\eta$ of the cosmological instanton
(5.24). Despite essentially non-stationary metric background this gas stays in
equilibrium state because of scaling properties of its particles and produces
back reaction on the Friedmann metric background.

Applications of the action (5.23) have been considered in the number of papers
[27, 43, 44] and recently reviewed in [46]. Physics of the CFT driven
cosmology is entirely determined by this effective action and the effective
(Euclidean) Friedmann equation. The latter follows from the action by varying
the lapse $N(\tau)$ and expressing the Hubble factor in terms of the energy
density. In cosmic type gauge $N=1$, $\dot{a}=da/d\tau$, it reads

$\displaystyle\frac{1}{a^{2}}-\frac{\dot{a}^{2}}{a^{2}}=\frac{\varepsilon}{3M_{\pm}^{2}\big{(}\varepsilon\big{)}},$
(5.25)
$\displaystyle\varepsilon=M_{P}^{2}\varLambda+\frac{1}{2\pi^{2}a^{4}}\sum_{\omega}\frac{\omega}{e^{\eta\omega}-1},$
(5.26) $\displaystyle
M_{\pm}^{2}(\varepsilon)=\frac{M_{P}^{2}}{2}\Big{(}1\pm\sqrt{1-\beta\varepsilon/6\pi^{2}M_{P}^{4}}\,\Big{)},$
(5.27)

where the total energy density $\varepsilon$ includes the cosmological
constant contribution and the radiation density of conformal field modes
distributed over Planckian spectrum with the comoving temperature $1/\eta$.
The nonlinear effect of the Weyl anomaly manifests itself in the effective
Planck mass squared explicitly depending on $\varepsilon$ which takes two
possible values $M_{\pm}^{2}(\varepsilon)$.999To avoid mixup of the signs in
$M_{\pm}^{2}$ and sign factors associated with the statistics of conformal
$\omega$-modes we present here the radiation spectrum only for bosonic case.
These equations should be amended by the expression for the conformal time
period that interpolates between the turning points of the solution with
$\dot{a}(\tau)=0$. Note that the right hand side of the Friedmann equation
does not contain Casimir energy density – it turns out to be fully screened
due to the dynamical effect of the Weyl anomaly. This is the result of the
finite renormalization (4.23) leading to a particular value of the anomaly
coefficient of $\Box R$, $\gamma=0$.

For the choice of $+$ sign in $M_{\pm}^{2}$ the solutions of this quantum
Friedmann equation turn out to be the so-called garlands – the cosmological
instantons of $S^{1}\times S^{3}$ topology, which have the periodic scale
factor $a(\tau)$ oscillating on $S^{1}$ between maximal and minimal values
$a_{\pm}$ [27]. These instantons serve as initial conditions for the
cosmological evolution in the physical Lorentzian spacetime. This evolution
follows from $a(\tau)$ by the analytic continuation $a_{L}(t)=a(\tau_{+}+it)$,
$(da_{L}/dt)^{2}=-\dot{a}^{2}$, to the complex plane of the Euclidean time at
the turning point with the maximal scale factor $a_{+}=a(\tau_{+})$. It can
incorporate a finite inflationary stage if the model is generalized to the
case when a primordial cosmological constant is replaced by the potential of
the inflaton field $\phi$, $\varLambda\to V(\phi)/M_{P}^{2}$, staying in the
slow-roll regime during the inflationary stage101010Alternatively, the role of
inflaton can be played by Ricci curvature in the Starobinsky $R^{2}$-model,
the coupling of the $R^{2}$ term being subject to the renormalization
respecting the zero value of $\alpha$ in the total Weyl anomaly [43]. and
decaying in the end of inflation by a usual exit scenario [43, 44]. The energy
scale of inflation – its Hubble parameter $H\sim\sqrt{\varLambda/3}$ turns out
to be bounded from above by $\sqrt{2}\pi M_{P}/\sqrt{\beta}$, so that to solve
the problem of hierarchy between the Planck and inflation scales one needs
$\beta\gg 1$ which matches with the previously adopted assumption that
numerous conformal fields drastically outnumber all other fields and dominate
over their loop corrections.

For the negative sign in $M_{\pm}^{2}$ the solutions represent vacuum
$S^{4}$-instantons of the no-boundary type with the vanishing minimal value of
the scale factor $a_{-}=0$. They correspond to the diverging
$\eta\sim\int_{0}^{a_{+}}da/a\dot{a}\to\infty$ or zero temperature. These
solutions, however, do not contribute to the statistical sum because of their
infinitely positive action $\varGamma_{\rm tot}\to+\infty$ — the quantum
effect of the trace anomaly which flips the sign of the negative tree-level
action of the Hartle-Hawking instantons [74] and sends it to $+\infty$ [27].
Thus the CFT cosmology scenario is free from the infrared catastrophe of the
no-boundary quantum state which would imply that the origin of an infinitely
big Universe is infinitely more probable than that of a finite one.

##  6 Renormalization group and the metamorphosis of the running scale

This section has essentially discussion nature and is associated with the
covariant perturbation theory of the above type. One of the motivations for
this discussion is that, in spite of a widespread concept of running
cosmological and gravitational constants, which is especially popular within
the asymptotic safety approach, there is a very profound and persuading
criticism of this concept [30]. It is based on numerous arguments of the
tadpole structure of the cosmological and Einstein terms, on concrete results
for graviton scattering amplitudes [29] which cannot be interpreted in terms
of a universal scaling of $\varLambda$ and $G$, etc.

At the same time in renormalizable gravity models with multiple couplings the
solution of the full set of RG equations includes running cosmological and
gravitational constants [36]. So the question arises how to interpret their
running scale. Here is the attempt to do this in terms of the covariant
curvature expansion developed in [38, 39, 40].

We start with the classical action which is the sum of local curvature
invariants of growing dimensionality $(4+m)$ in units of the mass

$S[\,g_{\mu\nu}]=\sum\limits_{m,N}\varLambda^{(m)}_{N}\int
d^{4}x\,\sqrt{g}\,\Re^{(4+m)}_{N}(x).$ (6.1)

They are monomials of $N$-th order in curvature tensors which are acted upon
by covariant derivatives

$\displaystyle\Re^{(m)}_{N}(x)=\mathop{\underbrace{\nabla...\nabla}_{m-2N}}\mathop{\overbrace{\Re(x)...\Re(x)}^{N}},$
(6.2) $\displaystyle{\rm
dim}\;\Re^{(m)}_{N}(x)\equiv\big{[}\,\Re^{(m)}_{N}(x)\,\big{]}=m.$ (6.3)

The curvature monomials enter the action with coupling constants
$\varLambda^{(m)}_{N}$ of the decreasing (with growing $m$) dimensionality

$[\,\varLambda^{(m)}_{N}\,]=d-m,\quad m=0,1,\ldots.$ (6.4)

Summation in (6.1) can run over finite set of terms providing the
renormalizability of the theory, or formally extended to the infinite set in
the framework of generalized RG theory with infinite set of couplings
$\\{\varLambda\\}=\varLambda^{(m)}_{N}$.

Within covariant perturbation theory the full metric is decomposed as a sum of
the flat spacetime metric $\tilde{g}_{\mu\nu}$ and the perturbation
$h_{\mu\nu}$

$g_{\mu\nu}=\tilde{g}_{\mu\nu}+h_{\mu\nu},$ (6.5)

so that each curvature invariant becomes expanded as an infinite series in
powers of $h_{\mu\nu}$ forming a new set of $h$-monomials on the flat space
background

$\displaystyle\int
d^{4}x\,\sqrt{g}\,\Re^{(m)}_{N}=\sum\limits_{M=N}^{\infty}\int
d^{4}x\,\sqrt{\tilde{g}}\,I_{M}^{(m)}(h),$ $\displaystyle
I_{M}^{(m)}(h)\propto\mathop{\underbrace{\tilde{\nabla}...\tilde{\nabla}}_{m}}\mathop{\overbrace{h(x)...h(x)}^{M}}.$
(6.6)

Then in the notations of the covariant perturbation theory the calculation of
the renormalized effective action leads to the same sequence of monomials
acted upon by the operator form factors
$\varGamma_{n}^{(i)}\big{(}\\{\varLambda\\},\tilde{\nabla}_{1},...\tilde{\nabla}_{1}\big{)}$
which make them nonlocal, $\\{\varLambda\\}$ denoting the full set of
couplings (6.4). Within dimensional regularization these renormalized coupling
constants get rescaled by the normalization parameter $\mu$ and expressed in
terms of their dimensionless analogues $\lambda^{(m)}_{N}(\mu)$

$\varLambda^{(m)}_{N}=\mu^{d-m}\lambda^{(m)}_{N}(\mu),$ (6.7)

and the perturbation theory form factors also express as the functions of
dimensionless arguments

$\varGamma_{M}^{(m)}\big{(}\\{\varLambda\\},\tilde{\nabla}_{1},...\tilde{\nabla}_{M}\big{)}=\mu^{d-m}\gamma_{M}^{(m)}\Big{(}\\{\lambda(\mu)\\},\tfrac{\tilde{\nabla}_{1}}{\mu},...\tfrac{\tilde{\nabla}_{M}}{\mu}\Big{)}$
(6.8)

Correspondingly the effective action becomes

$\displaystyle\varGamma[\,g_{\mu\nu}]=\sum\limits_{(m)}\mu^{d-m}\sum\limits_{M=0}^{\infty}\int
d^{d}x\,\sqrt{\tilde{g}}$
$\displaystyle\times\gamma_{M}^{(m)}\big{(}\\{\lambda(\mu)\\},\tfrac{\tilde{\nabla}_{1}}{\mu},...\tfrac{\tilde{\nabla}_{M}}{\mu}\big{)}I_{M}^{(m)}(h_{1},h_{2},...h_{M})\,\big{|}_{\,\\{x\\}=x},$
(6.9)

where $I_{M}^{(m)}(h_{1},h_{2},...h_{M})$ is the analogue of the invariant
(6.6) with split spacetime arguments. A typical assumption of the RG theory
that the renormalized action is independent of the running scale then leads to
the set of equations for $\lambda^{(m)}_{N}(\mu)$ with the beta functions
following from the residues of spacetime dimension poles in the formfactors
$\varGamma_{M}^{(m)}\big{(}\\{\lambda(\mu)\\},\\{\tilde{\nabla}/\mu\\}\big{)}$,

$\mu\frac{d}{d\mu}\varGamma[\,g_{\mu\nu}]=0\to\mu\frac{d}{d\mu}\lambda^{(m)}_{N}(\mu)=\beta^{(m)}_{N}(\mu)\big{(}\\{\lambda(\mu)\\}\big{)}.$
(6.10)

A critical step now consists in the choice of the running scale which could
probe the high energy limit of the theory and embrace a simultaneous scaling
of all formfactors and invariant monomials of (6.9). Then the replacement of
the parameter $\mu$ by this scale will identically bring the effective action
to the form explicitly revealing its UV limit. The choice of this scaling
object can be very different depending on the concrete physical setup. If the
theory has a dimensional scalar field $\phi$ with a nonvanishing and slowly
varying mean value it would be natural to identify RG normalization $\mu$ with
$\phi$. This would lead to the nontrivially “running” in $\phi$ of the
cosmological and Einstein terms, $\varLambda\to\varLambda(\phi)$ and $G\to
G(\phi)$, (amended of course by a gradient expansion series in derivatives of
$\phi$), but of course these terms acquire the interpretation of the Coleman-
Weinberg type potential and nonminimal coupling of $\phi$ to the scalar
curvature.

We, however, are interested in the UV scaling of all derivatives
$\tilde{\nabla}\to\infty$, which in momentum space representation of
scattering amplitudes is conventionally represented by the high energy
Mandelstam invariants or some other combinations of external momenta. In the
coordinate representation of the covariant perturbation theory of [38, 39, 40]
the role of this scale should be played by some operator. So we suggest as a
candidate for this object the following nonlocal operator $\tilde{D}$ which
also formally tends to infinity in the limit of $\tilde{\nabla}\to\infty$ and
in fact embraces a simultaneous scaling of all invariant monomials in (6.9),

$\tilde{D}\equiv\Big{(}-\sum_{N=1}^{\infty}\tilde{\Box}_{N}\Big{)}^{1/2},\quad\tilde{\Box}_{N}\equiv\tilde{g}^{\mu\nu}\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}.$
(6.11)

Though being very formal, this operator is well defined in each $N$-th
monomial order because it becomes truncated to the finite sum when acting on
the monomial of $N$ perturbations $h_{1},...h_{N}$, and for $N=0$ it is just
zero because of its action on an independent of $x$ constant,

$\tilde{D}_{N}\equiv\bigg{(}-\sum_{M=1}^{N}\tilde{\Box}_{M}\bigg{)}^{1/2},\quad\tilde{D}_{0}=0.$
(6.12)

In the UV domain $\tilde{\nabla}_{n}\to\infty$, when
$\tilde{\nabla}_{n}/\tilde{D}_{N}=O(1)$, $n\leq N$, the formfactors in each
$N$-th order become after the replacement $\mu\to\tilde{D}$ the functions of a
single operator variable $\tilde{D}_{N}$,

$\displaystyle\mu^{4-m}\gamma_{N}^{(m)}\big{(}\lambda(\mu)\,\big{|}\,\tfrac{\tilde{\nabla}_{1}}{\mu},...\tfrac{\tilde{\nabla}_{N}}{\mu}\big{)}\,\Big{|}_{\,\mu\to\tilde{D}_{N}}\;\to$
$\displaystyle\;(\tilde{D}_{N})^{4-m}\gamma_{N}^{(m)}\big{(}\lambda(\tilde{D}_{N})\,\big{|}\,O(1)\big{)}\equiv(\tilde{D}_{N})^{4-m}\lambda_{N}^{(m)}\big{(}\tilde{D}_{N}\big{)},$
(6.13)

and the expansion of the formally independent of $\mu$ action takes the form

$\displaystyle\varGamma[\,g_{\mu\nu}]\,\Big{|}_{\,\mu\to\tilde{D}}\to\,\sum\limits_{m}\sum\limits_{N=0}^{\infty}\int
d^{4}x\,\sqrt{\tilde{g}}$
$\displaystyle\times\,(\tilde{D}_{N})^{4-m}\lambda_{N}^{(m)}(\tilde{D}_{N})\,I_{N}^{(m)}(h_{1},h_{2},...h_{N})\,\Big{|}_{\,\\{x\\}=x}.$
(6.14)

The next step consists in the recovery of the covariant form of the expansion
in terms of the original spacetime curvature. Curiously, despite the fact that
the covariant perturbation theory of [38, 39, 40] is rather often being
referred to in literature, subtle details of this step are usually disregarded
which leads to confusing statements on the ambiguity of this procedure,
dependence on the gauge by which the metric perturbation $h_{\mu\nu}$ is
related to the curvature [33], etc. At the same time, this procedure is
unique, provided that one does not treat $\tilde{g}_{\mu\nu}$ and
$\tilde{\nabla}_{\mu}$ as Cartesian $\delta_{\mu\nu}$ and $\partial_{\mu}$,
but rather proceeds in generic coordinate system and uses the only invariant
statements that the curvature of the tilded metric is vanishing
$\tilde{R}^{\alpha}_{\;\;\beta\mu\nu}=0$. This is the covariant equation for
$\tilde{g}_{\mu\nu}$ in terms of the curved metric $g_{\mu\nu}$ and its
curvature $R^{\alpha}_{\;\;\beta\mu\nu}$, whose solution exists as
perturbation expansion in $R^{\alpha}_{\;\;\beta\mu\nu}$ and also requires
imposing the gauge [38, 39]. But the result of substituting this solution back
into manifestly noncovariant (double field) series (6.6) is gauge independent
because of the implicit invariance of the left hand side of (6.6).

In the convenient DeWitt type gauge
$\tilde{\nabla}^{\nu}h_{\mu\nu}-\tfrac{1}{2}\nabla_{\mu}h=O[\,h^{2}]$,
$h\equiv\tilde{g}^{\alpha\beta}h_{\alpha\beta}$, the solution for $h_{\mu\nu}$
and $\tilde{\nabla}_{\mu}$ in terms of $g_{\mu\nu}$ and $\nabla_{\mu}$ reads
in the lowest order as [38, 39]

$h_{\mu\nu}=-\frac{2}{\Box}R_{\mu\nu}+O[\,\Re^{2}],\quad\tilde{\nabla}_{\mu}=\nabla_{\mu}+O[\,\Re\,].$
(6.15)

Using this in (6.14) we get the replacement of $h$-monomials by the covariant
curvature monomials along with the replacement of $\tilde{D}_{N}$ by $D_{N}$,

$\displaystyle I_{N}^{(m)}(h_{1},h_{2},...h_{N})\to$
$\displaystyle\qquad\quad\frac{1}{\Box_{1}...\Box_{N}}\Re^{(m+2N)}_{N}(x_{1},...x_{N})+O[\,\Re^{N+1}],$
(6.16) $\displaystyle\tilde{D}_{N}\to D_{N}+O[\,\Re\,],$ (6.17)

where $D_{N}$ is obviously defined by (6.12) in terms of full-fledged
covariant d’Alembertians $\Box=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$, and we
reabsorb the coefficient $(-2)^{n}$ into the symbolic definition of the $N$-th
order covariant monomial – the analogue of the local $\Re^{(m)}_{N}(x)$, see
Eq. (6.2), with split $N$ spacetime arguments

$\Re^{(m)}_{N}(x_{1},...x_{N})=\mathop{\underbrace{\nabla...\nabla}_{m-2N}}\Re(x_{1})...\Re(x_{N}),\,N\geq
1.$ (6.18)

For $N=0$ this monomial can be defined as an irrelevant constant bringing no
contribution in the UV limit.

Thus the UV limit of the effective action takes the form

$\displaystyle\varGamma[\,g_{\mu\nu}]\to\int
d^{4}x\,\sqrt{g}\sum\limits_{m,N\geq
0}^{\infty}\\!\frac{\lambda_{N}^{(m)}(D_{N})(D_{N})^{4-m}}{\Box_{1}...\Box_{N}}$
$\displaystyle\qquad\qquad\qquad\times\,\Re^{(m+2N)}_{N}(x_{1},\cdots
x_{N})\,\Big{|}_{\,\\{x\\}=x},$ (6.19)

where we remind that the dimensionless formfactors $\lambda_{N}^{(m)}(D_{N})$
follow from the running RG couplings of the theory $\lambda_{N}^{(m)}(\mu)$ by
the replacement of $\mu$ with the operator $D_{N}$.

Let us consider application of this result to the cosmological constant sector
involving the metric invariants of dimensionality $m=0$ and
$\varLambda^{(4)}_{0}=\varLambda/16\pi G$. This classical cosmological term
gives rise to the infinite set of zero dimension invariants

$\displaystyle\int d^{4}x\,\sqrt{g}=\sum\limits_{n=0}^{\infty}\int
d^{4}x\,\sqrt{\tilde{g}}\,I_{n}^{(0)}(\tilde{g},h),$ (6.20)
$\displaystyle\;\;I_{0}^{(0)}(\tilde{g},h)=1,\;\;I_{1}^{(0)}(\tilde{g},h)=-\tfrac{1}{2}h,$
$\displaystyle\;\;I_{2}^{(0)}(\tilde{g},h)=\tfrac{1}{4}h^{2}-\tfrac{1}{2}h_{\mu\nu}^{2},\ldots$
(6.21)

(indices are contracted by the flat metric and
$h=\tilde{g}^{\mu\nu}h_{\mu\nu}$), whereas at the quantum level they generate
the sequence of high energy $m=0$ structures of (6.19)

$\int\\!d^{4}x\sqrt{g}\sum\limits_{N=2}^{\infty}\lambda_{N}^{(0)}(D_{N})\frac{(D_{N})^{4}}{\Box_{1}...\Box_{N}}\,\Re^{(2N)}_{N}(x_{1},...\,x_{N})\Big{|}_{\\{x\\}=x},$
(6.22)

where the zeroth order term is zero in view of $D_{0}=0$ (see Eq.(6.11) and
the first order term is also absent due to its tadpole (total derivative)
nature – remember that $D_{1}=(-\Box_{1})^{1/2}$ and
$D_{1}^{4}/\Box_{1}=\Box_{1}$ is acting on
$\Re^{(2)}_{1}(x_{1})$.111111Important caveat is necessary here concerning the
annihilation of the total derivative terms. The surface terms at infinity
should be vanishing, which is equivalent to a good IR behavior of the nonlocal
form factor $\lambda_{1}^{(0)}(D_{1})$ at $\Box\to 0$. We will assume this
property basing on the maximum logarithmic singularity of
$\lambda_{1}^{(0)}(D_{1})$ which is a function of $\log(-\Box)$ solving the RG
equation. The same also applies to integrations by parts considered in what
follows. Otherwise, the procedure of subtracting the boundary terms, like the
Gibbons-Hawking surface action at asymptotically flat infinity, will be
needed, which we briefly discuss below.

The expansion starts at $N=2$ with the term which has the following structure

$\displaystyle 4\sum\int
d^{4}x\,\sqrt{g}\,\Re^{(2)}(x)\,\lambda_{2}^{(0)}\big{(}\sqrt{-2\Box}\big{)}\,\Re^{(2)}(x)$
$\displaystyle\;\;=\int
d^{4}x\sqrt{g}\Big{(}R_{\mu\nu}F_{1}(\Box)R^{\mu\nu}+RF_{2}(\Box)R\Big{)}\\!+\\!O[\,\Re^{3}].$
(6.23)

Here we took into account that the set of invariants
$\Re^{(4)}_{2}(x_{1},x_{2})$ can be represented as a sum of terms factored out
into the products of Ricci tensors and Ricci scalars with some
coefficients121212Bilinear in Riemann curvature terms under the integration
sign also reduce to bilinear combinations of $R_{\mu\nu}$ and $R$ by using the
expression for Riemann tensor in terms of the Ricci one [38, 39], see footnote
5. $a$ and $b$,

$\Re^{(4)}_{2}(x_{1},x_{2})=aR_{\mu\nu}(x_{1})\,R^{\mu\nu}(x_{2})+bR(x_{1})R(x_{2}),$
(6.24)

and also used an obvious corollary of integration by parts

$\displaystyle\int
d^{4}x\sqrt{g}\,F(\Box_{1},\Box_{2})\Re(x_{1})\Re(x_{2})\,\Big{|}_{\,\\{x\\}=x}$
$\displaystyle\qquad\qquad=\int d^{4}x\sqrt{g}\,\Re(x)F(\Box,\Box)\Re(x).$
(6.25)

Remarkable feature of the expression (6.23) is that the power-law operator
factors in $(D_{N})^{4}/\Box_{1}...\Box_{N}$ at $N=2$ completely cancelled out
to give the dimensionless formfactors $F_{1}(\Box)$ and $F_{2}(\Box)$ which
originate as linear combinations of relevant running
$\lambda_{2}^{(0)}\big{(}\sqrt{-2\Box}\big{)}$ obtained by solving the RG
equation. Even more remarkable is the fact that this is a nonlocal term which
is quadratic in the curvature even though it has originated from the sector of
cosmological term expanded in the series of zero dimension invariants. This is
what can be called as metamorphosis to high-energy partners of the
cosmological constant suggested by J.Donoghue in [32]. Their structure is a
direct corollary of the dimensionality arguments within RG approach. The
arising form factors of the curvature squared terms are the descendants of RG
running couplings of the zero dimension invariants which participate in the
decomposition of the cosmological constant term.

In fact, the same structure (6.23) gets reproduced for the contribution of any
dimension $m$ in the expansion (6.19). For even dimensionality131313For the
set of 2-dimensional curvatures $\Re$ only even dimensions $m$ enter the
expansion (6.19), but this can always be generalized to the case of odd-
dimensional “curvatures”, like for example the extrinsic curvature in Hořava
gravity models., $m\to 2m$, this can be easily demonstrated by decomposing any
$(2m+4)$-dimensional quadratic invariant as this was done above

$\Re^{(2m+4)}_{2}(x_{1},x_{2})=\\!\\!\\!\\!\sum_{m_{1}+m_{2}=2m}\\!\\!\\!\\!\Re^{(m_{1}+2)}_{1}(x_{1})\Re^{(m_{2}+2)}_{1}(x_{2}).$
(6.26)

Using this in (6.19) one has complete cancellation of the dimensional factor
$(D_{2})^{4-2m}/\Box^{2}\sim\Box^{-m}$ in the expression

$\displaystyle\int
d^{4}x\,\sqrt{g}\\!\sum\limits_{m_{1}+m_{2}=2m}\Re^{(m_{1}+2)}_{1}(x)$
$\displaystyle\qquad\times\frac{\lambda_{2}^{(2m)}(D_{2})(D_{2})^{4-m_{1}-m_{2}}}{\Box^{2}}\,\Re^{(m_{2}+2)}_{1}(x)$
$\displaystyle=\\!\int
d^{4}x\sqrt{g}\Big{(}R_{\mu\nu}F_{1}(\Box)R^{\mu\nu}+RF_{2}(\Box)R\Big{)}\\!+O[\,\Re^{3}].$
(6.27)

Noting that with
$\Re^{(m+2)}_{1}=\mathop{\overbrace{\nabla\cdots\nabla}^{m}}\Re^{(2)}_{1}$
this follows from integration by parts and the use of various corollaries of
contracted Bianchi identity
($\nabla^{\nu}R_{\mu\nu}=\tfrac{1}{2}\nabla_{\mu}R$, etc.),

$\displaystyle\int
d^{4}x\,\sqrt{g}\\!\sum\limits_{m_{1}+m_{2}=2m}\mathop{\underbrace{\nabla\cdots\nabla}_{m_{1}}}\Re(x)F(\Box)\mathop{\underbrace{\nabla\cdots\nabla}_{m_{2}}}\Re(x)$
$\displaystyle=\int
d^{4}x\,\sqrt{g}\Big{(}R_{\mu\nu}\,\Box^{m}F_{1}(\Box)R^{\mu\nu}+R\,\Box^{m}F_{2}(\Box)R\Big{)}$
$\displaystyle\qquad+O[\,\Re^{3}].$ (6.28)

Here the operators $F_{1}(\Box)$ and $F_{2}(\Box)$ have the same dimension as
$F(\Box)$ and originate from $F(\Box)$ by the algebra of contracting the
indices of covariant derivatives. Using this relation in the left hand side of
(6.27) one gets the right hand side with completely cancelled powers of
$\Box$.

Thus, Eq.(6.27) with $m=2$ implies the conversion of the gravitational
coupling constant into the dimensionless formfactors of the Einstein term
partners. These partners have the same structure as the cosmological term
partners quadratic in curvatures. This is again the metamorphosis of RG
running of the form $1/16\pi G(\mu)=\mu^{2}\lambda^{(2)}_{2}(\mu)\to
F_{1,2}(\Box)$.

Note that all this takes place in the UV limit where all curvatures in their
monomials are rapidly varying in spacetime with their derivatives
$\nabla\to\infty$. At intermediate energies, when the mass scale $M$ surfaces
up, the scaling (6.13) ceases to make sense and roughly should be replaced
with $D\sim M$, and instead of (6.23) one gets exactly the cosmological
constant partners of Donoghue [32] which have the structure of

$M^{4}\\!\int d^{4}x\,\sqrt{g}\Big{(}R_{\mu\nu}\frac{F_{1}^{\rm
part}(\Box)}{\Box^{2}}R^{\mu\nu}+R\,\frac{F_{2}^{\rm
part}(\Box)}{\Box^{2}}R\Big{)}.$ (6.29)

The dimensionless form factors $F^{\rm part}_{1,2}(\Box)$ here are
accumulating loop corrections with nonlocal logarithmic structures of the form

$F^{\rm part}(\Box)\sim\ln\frac{M^{2}-\Box}{M^{2}}.$ (6.30)

Note that these partners are still in high-energy domain $-\Box\geq M^{2}$,
but they are subdominant as compared to the leading contribution (6.23) with
dimensionless form factors which incorporate the logarithmically running
solutions of RG equations. This is because the partners (6.29) are suppressed
by power law factors $M^{4}/\Box^{2}$. Exact form of these formfactors at
intermediate scales was derived at one-loop order in [33] for rather generic
theory of massive fields by using the heat kernel technique of [38, 39]. In IR
domain $-\Box\ll M^{2}$ they are of course expandable in local gradient series
reflecting the decoupling phenomenon [34, 35, 33].

Similarly, the gravitational constant partner in IR reads as

$M^{2}\int
d^{4}x\,\sqrt{g}\Big{(}R_{\mu\nu}\frac{F_{1}(\Box)}{\Box}R^{\mu\nu}+R\,\frac{F_{2}(\Box)}{\Box}R\Big{)},$
(6.31)

which reminds the construction of the nonlocal action for long-distance
modifications of gravity theory in [51, 52]. This differs from the
cosmological constant partner by another powers of $M$ and the power of $\Box$
in the denominator.

One should be more careful at this point – while the case of (6.31) is well
defined in asymptotically flat spacetime, the cosmological constant partner
(6.29) is IR divergent for the reasons discussed above. The action of
$\tfrac{1}{\Box^{2}}$ is not well defined in four dimensions (or,
equivalently, $\int d^{4}x\sqrt{g}(\tfrac{1}{\Box}\Re)^{2}$ is IR divergent),
so that the perturbation expansion in the dimension zero sector should be
critically reconsidered. To trace the origin of this difficulty note that the
first three terms of the cosmological term expansion (6.20) are divergent,
whereas a similar expansion for the Einstein term becomes well defined only
after the subtraction of the Gibbons-Hawking surface term
$\int_{\infty}d^{3}\sigma^{\mu}\big{(}\partial_{\mu}h-\partial^{\nu}h_{\mu\nu}\big{)}$
at the infinity of asymptotically flat spacetime. Due to this subtraction we
can write for the integral of the invariant $\Re^{(2)}_{1}(x)=-R(x)$, weighted
in the Einstein action by $\varLambda^{(2)}_{1}=1/16\pi G$, a legitimate
expansion (6.6) starting with the quadratic order in $h_{\mu\nu}$,

$\displaystyle\int
d^{4}x\sqrt{g}(-R)-\int_{\infty}d^{3}\sigma^{\mu}\big{(}\partial_{\mu}h-\partial^{\nu}h_{\mu\nu}\big{)}$
$\displaystyle\qquad\qquad\quad\quad\;=\sum\limits_{M=2}^{\infty}\int
d^{4}x\sqrt{\tilde{g}}\,I_{M}^{(2)}(\tilde{g},h),$ (6.32) $\displaystyle
I_{2}^{(2)}(\tilde{g},h)=\\!-\tfrac{1}{4}h_{\mu\nu}\tilde{\Box}h^{\mu\nu}\\!+\tfrac{1}{8}h\tilde{\Box}h-\tfrac{1}{2}\big{(}\tilde{\nabla}^{\nu}h_{\mu\nu}\\!-\tfrac{1}{2}\tilde{\nabla}_{\mu}\tilde{h}\big{)}^{2}.$
(6.33)

Then the above calculational strategy leads to the effective action (6.31)
whose tree level IR limit should match low energy physics with the Planck mass
cutoff $M^{2}$ and the form factors $F_{1}(0)=1$ and $F_{2}(0)=1/2$. This tree
level answer up to $\Re^{3}$-corrections directly corresponds to the above
expression for $I_{2}^{(2)}(\tilde{g},h)$ with $h_{\mu\nu}$ given by Eq.
(6.15) in terms of the curved space metric $g_{\mu\nu}$ [51, 52].

To the best of our knowledge, no such subtraction is known for cosmological
term expansion (6.20), so that its rigorous treatment is still to be done. It
is interesting if new structures can be generated by the regularization of
this IR behavior. Apparently, this should be based on the analogue of the
Graham-Fefferman construction for asymptotically AdS spaces [75, 76] and
deserves further studies.

In any case, the UV behavior of both cosmological and gravitational constant
partners, which should be not sensitive to IR problems, is determined by
curvature squared terms (6.23) with running dimensionless “couplings”. Their