In the case of the EGB theories discussed recently, $\mathcal{S}_{E}^{WZ}$ is
the only contribution derived above in $\mathcal{S}_{WZ}$ that plays a role in
those analysis. One notices the overlap betyween the regularization proposed
in [127, 129, 128] and the one discussed here and in previous works [121].
Obviously, the latter are deprived of the quantum correction contained in
$\mathcal{S}_{f}$.

#####  7.7.2 Anomaly constraints

At this point, we are in condition to derive the anomaly constraints on
$\mathcal{S}^{WZ}$ and on $\mathcal{S}_{A}$, clarifying some of the
intermediate steps in the derivations. We recall that the usual relation

$2g_{\mu\nu}\frac{\delta}{\delta g_{\mu\nu}}=\frac{\delta}{\delta\phi},$
(7.7.41)

holds only if a functional is a function of the entire metric $g_{\mu\nu}$. In
general, for a regulated functional, separate function of $\bar{g}$ and
$\phi$, say $F(\bar{g},\phi)$, this relation is not valid any longer. Given a
functional $F(\bar{g})$, we use the relation
$\bar{g}_{\mu\nu}=g_{\mu\nu}e^{-2\phi}$, to derive the expressions

$\frac{\delta\bar{g}_{\mu\nu}(x)}{\delta
g_{\alpha\beta}(y)}=\delta^{\alpha\beta}_{\mu\nu}e^{-2\phi}\delta^{d}(x-y),\qquad\frac{\delta
F}{\delta g_{\alpha\beta}}=\frac{\delta
F}{\delta\bar{g}_{\alpha\beta}}e^{-2\phi},\qquad
2\bar{g}_{\alpha\beta}\frac{\delta
F}{\delta\bar{g}_{\alpha\beta}}=2g_{\alpha\beta}\frac{\delta F}{\delta
g_{\alpha\beta}}.$ (7.7.42)

The regularization, by separating the dependence on the two components of $g$,
leads to identities between the functional variations w.r.t. $\phi$ and
$\bar{g}$ that, as we are going to show, are related by the anomaly. In EGB
theories, the link is expressed only by the Euler density, but the constrain
between the two variations is similar. This implies that the functional
expansion of $\mathcal{S}_{R}$ is far more involved, similar to (7.2.6), but
with the dilaton appearing on the external legs of the correlation functions.
This requires a separate analysis of the anomalous CWIs, that we will discuss
elsewhere. In general, $\mathcal{S}_{R}$ can be expressed either in terms of
the couple $(\bar{g},\phi)$, but also in terms of $g$ and $\phi$, where the
two variables shuould be treated as independent. We may easily derive several
relations.
For instance, from $\mathcal{S}_{E}^{WZ}$ we obtain

$2g_{\mu\nu}\frac{\delta\mathcal{S}_{E}^{WZ}}{\delta
g_{\mu\nu}}=\frac{1}{\epsilon}\left(2g_{\mu\nu}\frac{\delta V_{E}(g,d)}{\delta
g_{\mu\nu}}-2g_{\mu\nu}\frac{\delta V_{E}(\bar{g},d)}{\delta
g_{\mu\nu}}\right).$ (7.7.43)

Now using

$2g_{\mu\nu}\frac{\delta}{\delta
g_{\mu\nu}}V_{E}(\bar{g},d)=2\bar{g}_{\mu\nu}\frac{\delta}{\delta\bar{g}_{\mu\nu}}V_{E}(\bar{g},d)=\sqrt{\bar{g}}\bar{E},$
(7.7.44)

together with (7.7.20), we obtain

$2g_{\mu\nu}\frac{\delta\mathcal{S}_{E}^{WZ}}{\delta
g_{\mu\nu}}=\sqrt{g}E-\sqrt{\bar{g}}\bar{E}.$ (7.7.45)

On the other end, we have

$\frac{\delta}{\delta\phi}V_{E}(g,d)=\epsilon\sqrt{g}E,\qquad\frac{\delta}{\delta\phi}V_{E}(\bar{g},d)=0,$
(7.7.46)

giving

$\frac{\delta}{\delta\phi}\mathcal{S}_{E}^{WZ}=\sqrt{g}E.$ (7.7.47)

Therefore, in the case of a GB theory one derives a constraint between the
equation of motion of the dilaton and the trace of the stress energy tensor of
the fiducial metric

$2g_{\mu\nu}\frac{\delta\mathcal{S}_{E}^{WZ}}{\delta
g_{\mu\nu}}-\frac{\delta\mathcal{S}_{E}^{WZ}}{\delta\phi}=-\sqrt{\bar{g}}\bar{E}.$
(7.7.48)

This relation shows that the Weyl variation and the trace/metric variation are
not identical, if we perform a regularization.
A similar relation holds for $\mathcal{S}_{C^{2}}^{WZ}$

$2g_{\mu\nu}\frac{\delta\mathcal{S}_{C^{2}}^{WZ}}{\delta
g_{\mu\nu}}-\frac{\delta\mathcal{S}_{C^{2}}^{WZ}}{\delta\phi}=-\sqrt{\bar{g}}\bar{C}^{2}.$
(7.7.49)

In summary, a WZ anomaly action, following the definitions above, will satisfy
the anomaly condition

$2g_{\mu\nu}\frac{\delta\mathcal{S}^{WZ}}{\delta
g_{\mu\nu}}=b^{\prime}\sqrt{g}E+b\sqrt{g}C^{2}-\left(b^{\prime}\sqrt{\bar{g}}\bar{E}+b\sqrt{\bar{g}}\bar{C}^{2}\right),$
(7.7.50)

and

$\frac{\delta\mathcal{S}^{WZ}}{\delta\phi}=b^{\prime}\sqrt{g}E+b\sqrt{g}C^{2}$
(7.7.51)

and, more generally, the condition

$2g_{\mu\nu}\frac{\delta\mathcal{S}^{WZ}}{\delta
g_{\mu\nu}}-\frac{\delta\mathcal{S}^{WZ}}{\delta\phi}=-\left(b^{\prime}\sqrt{\bar{g}}\bar{E}+b\sqrt{\bar{g}}\bar{C}^{2}\right).$
(7.7.52)

It differs from the DR anomaly-induced action by terms

$\left(2g_{\mu\nu}\frac{\delta\mathcal{S}^{A}}{\delta
g_{\mu\nu}}-2g_{\mu\nu}\frac{\delta\mathcal{S}^{WZ}}{\delta
g_{\mu\nu}}\right)=\frac{\partial}{\partial
d}V_{E}(\bar{g},d)\mid_{d=4}+\frac{\partial}{\partial
d}V_{C^{2}}(\bar{g},d)\mid_{d=4},$ (7.7.53)

that are Weyl invariant, since they do not depend on the conformal factor
$\phi$.

#####  7.7.3 $\mathcal{S}_{B}(d)$ for $d\to 4$

$\mathcal{S}_{B}(d)$ describes the finite quantum corrections that develop a
singularity in the $d\to 4$ limit. Notice that this is a functional of the
entire metric before the limit is taken, and it is Weyl-invariant. Therefore
it satisfies the constraint

$2g_{\mu\nu}\frac{\delta\mathcal{S}_{B}}{\delta
g_{\mu\nu}}=\frac{\delta\mathcal{S}_{B}}{\delta\phi}=0.$ (7.7.54)

This relation implies that $\mathcal{S}_{B}$ is only a functional of
$\bar{g}$. Using the relations

$2g_{\mu\nu}\frac{\delta V_{E/C^{2}}(g,4)}{\delta
g_{\mu\nu}}=2\bar{g}_{\mu\nu}\frac{\delta
V_{E/C^{2}}(\bar{g},4)}{\delta\bar{g}_{\mu\nu}}=0,$ (7.7.55)

which are valid at $d=4$, and using the definition in DR

$\mathcal{S}_{f}(d)=S_{B}(g,d)-b^{\prime}\frac{1}{\epsilon}V_{E}(\bar{g},4)-b\frac{1}{\epsilon}V_{C^{2}}(\bar{g},4),$
(7.7.56)

one finds that $\mathcal{S}_{f}(d)$ is only a function of the fiducial metric,
and its stress energy tensor has a vanishing trace. This property continues to
hold in the $d\to 4$ limit and henceforth

$\mathcal{S}_{f}(4)\equiv\mathcal{S}_{f}(\bar{g}).$ (7.7.57)

This implies that $\mathcal{S}_{f}(\bar{g})$ does not contribute to the
anomaly and we have consistently

$2\bar{g}_{\mu\nu}\frac{\delta
S_{f}(4)}{\delta\bar{g}_{\mu\nu}}=0,\qquad\frac{\delta\mathcal{S}_{f}(4)}{\delta\phi}=0.$
(7.7.58)

In the WZ case, on the other end we define

$\tilde{\mathcal{S}}_{f}(d)=S_{B}(g,d)+b^{\prime}\frac{1}{\epsilon}V_{E}(\bar{g},d)+b\frac{1}{\epsilon}V_{E}(\bar{g},d),$
(7.7.59)

and in this case, the response of this functional metric and conformal
variations are similar, but now

$\frac{\delta\tilde{\mathcal{S}}_{f}}{\delta\phi}=0,$ (7.7.60)

since the subtractions $V_{E/C^{2}}(\bar{g},d)$ do not depend on the conformal
factor, as well as the bare action $\mathcal{S}_{B}$. On the other hand we
have

$2\bar{g}_{\mu\nu}\frac{\delta
V_{E}(\bar{g},d)}{\delta\bar{g}_{\mu\nu}}=\sqrt{\bar{g}}E$ (7.7.61)

and similarly for $V_{C^{2}}(\bar{g},d)$, thereby obtaining

$2\bar{g}_{\mu\nu}\frac{\delta\tilde{\mathcal{S}}_{f}(\bar{g},d)}{\delta\bar{g}_{\mu\nu}}=b^{\prime}\sqrt{\bar{g}}\bar{E}+b\sqrt{\bar{g}}\bar{C}^{2}$
(7.7.62)

#####  7.7.4 The complete quantum action

It is clear that the conformal backreaction is associated with the entire
renormalized effective action $\mathcal{S}_{R}$, rather than to the anomaly-
induced actions $\mathcal{S}_{A}$ or the WZ action $S^{WZ}$. The difference
between $\mathcal{S}_{R}$ and the previous two actions is, again, given by
Weyl invariant terms. We recall that as far as we stay in $d$ dimensions,
$S_{R}(g,d)$, defined by the sum of $\mathcal{S}_{B}(g,d)$ and of the two
counterterms $1/\epsilon V_{E}(g,d)$ and $1/\epsilon V_{C^{2}}$, under a Weyl
variation behaves as

$\frac{\delta}{\delta\phi}\mathcal{S}_{R}(g,d)=\frac{\delta\mathcal{S}_{B}(g,d)}{\delta\phi}+\frac{\delta}{\delta\phi}\Bigg{(}\frac{1}{\epsilon}\left(b^{\prime}V_{E}(g,d)+bV_{C^{2}}(g,d)\right)\Bigg{)}.$
(7.7.63)

Using the invariance of $\mathcal{S}_{B}$ (7.7.54), we obtain

$\frac{\delta\mathcal{S}_{R}(g,d)}{\delta\phi}=2g_{\mu\nu}\frac{\delta\mathcal{S}_{R}(g,d)}{\delta
g_{\mu\nu}}=b^{\prime}\sqrt{g}E+b\sqrt{g}C^{2}.$ (7.7.64)

This equation is modified by the separation into poles plus finite terms in DR
in the form

$\frac{\delta\mathcal{S}_{R}(g,4)}{\delta\phi}=\frac{\delta\mathcal{S}_{f}(g,4)}{\delta\phi}+\frac{\delta\mathcal{S}_{A}(g,4)}{\delta\phi}.$
(7.7.65)

In this case both a $\phi$\- variation and trace-metric variation coincide and
give

$\frac{\delta\mathcal{S}_{A}(4)}{\delta\phi}=b^{\prime}\sqrt{g}E+b\sqrt{g}C^{2}$
(7.7.66)

while

$2g_{\mu\nu}\frac{\delta\mathcal{S}_{A}(4)}{\delta
g_{\mu\nu}}=b^{\prime}\sqrt{g}E+b\sqrt{g}C^{2}.$ (7.7.67)

The finite part of $\mathcal{S}_{R}$, given by $\mathcal{S}_{f}$, has no
dependence on the conformal factor and its stress energy tensor has zero
trace. Therefore the anomaly, in this case, is all generated by
$\mathcal{S}_{A}$

$\frac{\delta\mathcal{S}_{R}}{\delta{\phi}}=2g_{\mu\nu}\frac{\delta\mathcal{S}_{R}}{\delta
g_{\mu\nu}}=b^{\prime}\sqrt{g}E+b\sqrt{g}C^{2}.$ (7.7.68)

If we perform the renormalization of $\mathcal{S}_{B}$ using (7.7.5) and
(7.7.28), then (7.7.68) is obviously still valid. However both terms,
$\tilde{\mathcal{S}}_{f}$ and $\mathcal{S}_{WZ}$ on the rhs of (7.7.28) will
contribute to the anomaly, with extra Weyl-invariant terms that carry opposite
signs, as shown in (7.7.50) and (7.7.62), their sum reproducing again Eq.
(7.7.68).

####  7.8 Expansions in DR

In this section we turn to a discussion of the possible ways to reduce the
$d$-dimensional integrals appearing in the counterterms, down to $d=4$, by
selecting specific metrics. We start with the topological counterterm, by
mentioning that from (7.7.15), using (7.7.16) and (7.7.17) we obtain the
relation

$\bar{\nabla}_{\mu}\phi\bar{J}^{\mu}-\bar{K}=4\bar{R}^{\mu\nu}(\bar{\nabla}_{\mu}\phi\bar{\nabla}_{\nu}\phi)-2\bar{R}\bar{\square}\phi+2(\bar{\nabla}_{\lambda}\phi\bar{\nabla}^{\lambda}\phi)^{2}+4\bar{\,}\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi\bar{\nabla}_{\lambda}\phi\bar{\nabla}^{\lambda}\phi,$ (7.8.1)

written as a four-dimensional expression, but also valid in the embedding
space. As far as we do not specify the extra dimensional metric in some way,
the Greek indices may be used, with no confusion, to describe the invariants
over the entire embedding space.

After an integration by parts, we get the final form of the counterterm
contibution, up to $O(\epsilon)$ terms, given by

$\displaystyle\frac{1}{\epsilon}V_{E}(g,d)=$
$\displaystyle\frac{1}{\epsilon}\int d^{d}x\sqrt{\bar{g}}\ \bar{E}+\int
d^{d}x\sqrt{\bar{g}}\
\Big{[}\phi\bar{E}-(4\bar{G}^{\mu\nu}(\bar{\nabla}_{\mu}\phi\bar{\nabla}_{\nu}\phi)$
$\displaystyle+2(\bar{\nabla}_{\lambda}\phi\bar{\nabla}^{\lambda}\phi)^{2}+4\bar{\,}\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi\bar{\nabla}_{\lambda}\phi\bar{\nabla}^{\lambda}\phi)\Big{]}.$
(7.8.2)

This result holds in $d$ spacetime dimensions.

If we intend to take the $d\to 4$ limit with more rigour, then we need to be
more specific about the choice of the metric. In our case, for definiteness,
we will consider a manifold of the form
$\mathcal{M}_{4}\times\mathcal{M}_{e}$, split into a 4- and
$(d-4)$-dimensional part. We will denote the $d$ dimensional indices as $M,N$,
saving the Greek indices for the 4-dimensional part. The $d$ dimensional
metric is decomposed in the Weyl gauge and split into the direct sum of the
metrics of the two submanifolds. ${}_{e}g_{mn}$ is the extra dimensional
metric used for the regularization of the integral and
${}_{4}\tilde{g}_{\mu\nu}$ is its 4-dimensional part. The extra coordinates
will be denoted as $y$. For example, we choose

${}_{(d)}g_{MN}(x,y)=e^{2\phi(x)}\begin{bmatrix}{}_{4}\tilde{g}_{\mu\nu}(x)&0\\\
0&{}_{(e)}g_{mn}(y)\end{bmatrix}=e^{2\phi(x)}{}_{(d)}\bar{g}_{MN}.$ (7.8.3)

${}_{d}\bar{g}_{\mu\nu}$ is the $d$-dimensional fiducial metric, from which we
have extracted a conformal factor $\phi$, with the indices decomposed as
$M=(\mu,m),\,(N=\nu,n)\ldots$, and so on. The original scaling relation
(7.7.15) in $d$ dimensions can be expressed, in this spacetime manifold, in
the form

$\int d^{4}xd^{d-4}y\sqrt{{}_{(d)}g}\ {}_{(d)}E=\int
d^{4}xd^{d-4}y\sqrt{{}_{(d)}\bar{g}}\
e^{(d-4)\phi}\left({}_{(d)}\bar{E}+(d-3)\bar{\nabla}_{M}{}_{(d)}\bar{J}^{M}+(d-3)(d-4){}_{(d)}\bar{K}\right),$
(7.8.4)

where barred terms, including the covariant derivatives, are relative to the
fiducial metric ${}_{(d)}\bar{g}_{MN}$. Since, with our choice, the dilaton
does not appear in the extra dimensional part of the metric, the two blocks
that make up the full $d$-dimensional metric ${}_{(d)}\bar{g}_{MN}$ are only
dependent on the coordinates of the submanifold which they belong to. The same
is true for every curvature tensor, hence we have that
${}_{4}\tilde{R}_{\mu\nu\rho\sigma}$, the Riemann tensor of the base space,
depends only on the $x$-coordinates, while ${}_{e}R_{abcd}$ depends only on
the $y$-coordinates. Moreover, the connection has no mixed terms, hence the
squared curvature tensors are decoupled

$\displaystyle{}_{(d)}\bar{R}^{ABCD}{}_{(d)}\bar{R}_{ABCD}=$
$\displaystyle{}_{4}\tilde{R}^{\mu\nu\rho\sigma}{}_{4}\tilde{R}_{\mu\nu\rho\sigma}+{}_{(e)}R^{abcd}{}_{(e)}R_{abcd},$
$\displaystyle{}_{(d)}\bar{R}^{AB}{}_{(d)}\bar{R}_{AB}=$
$\displaystyle{}_{4}\tilde{R}^{\mu\nu}{}_{4}\tilde{R}_{\mu\nu}+{}_{(e)}R^{ab}{}_{(e)}R_{ab},$
$\displaystyle{}_{(d)}\bar{R}^{2}=$
$\displaystyle{}_{4}\tilde{R}^{2}+{}_{(e)}R^{2}+2{}_{4}\tilde{R}{}_{(e)}R.$
(7.8.5)

The Gauss-Bonnet density becomes

${}_{(d)}\bar{E}={}_{4}\tilde{E}+{}_{(e)}E+2{}_{4}\tilde{R}{}_{(e)}R.$ (7.8.6)

From the definitions (7.7.16) and (7.7.17), since $\phi(x)$ is a function only
of the coordinates of the 4-dimensional subspace, we obtain

$\displaystyle{}_{(d)}\bar{J}^{\mu}=$
$\displaystyle{}_{4}\tilde{J}^{\mu}-4{}_{(e)}R\
\tilde{\nabla}^{\mu}\phi,\qquad{}_{(e)}\bar{J}^{m}=0,$
$\displaystyle{}_{(d)}\bar{K}=$ $\displaystyle{}_{4}\tilde{K}-2{}_{(e)}R\
\tilde{\nabla}^{\lambda}\phi\tilde{\nabla}_{\lambda}\phi,$ (7.8.7)

where $\tilde{\nabla}$ are covariant derivatives associated with the
4-dimensional metric ${}_{4}\tilde{g}_{\mu\nu}$.
We may rewrite (7.8.4) as

$\displaystyle\int d^{4}xd^{d-4}y\sqrt{{}_{(d)}g}\ {}_{(d)}E=$
$\displaystyle\int d^{4}xd^{d-4}y\sqrt{{}_{(d)}\bar{g}}\
e^{(d-4)\phi}\Big{(}{}_{4}\tilde{E}+(d-3)\tilde{\nabla}_{\mu}\,{}_{4}\tilde{J}^{\mu}+(d-3)(d-4){}_{4}\tilde{K}$
$\displaystyle+2\,{}_{4}\tilde{R}\
{}_{(e)}R+{}_{(e)}E-(d-3){}_{(e)}R[4\tilde{\square}\phi+2(d-4)(\tilde{\nabla}_{\lambda}\phi\tilde{\nabla}^{\lambda}\phi)]\Big{)}.$
(7.8.8)

This equation is the starting point in order to proceed with a dimensional
reduction of the fields in (7.4.2).

###### Weyl flat metric in $\mathcal{M}_{e}$ and DR

In dimensional reduction, as already pointed out, one usually assumes that all
the $d$-dimensional fields do not depend on the coordinates of the extra
dimensional manifold. The structure of the reduced theory carries symmetries
which are decomposed with respect to the original ones. A classical example is
that of $\mathcal{N}=1$ supersymmetric Yang-Mills theory in $d=10$, which
turns into an $\mathcal{N}=4$ at $d=4$.
In our case, with our metric choice, DR is implemented by assuming that the
${}_{e}g_{mn}$ metric becomes flat, in order to obtain a pure 4-dimensional
integrand. In this case all the external curvatures ${}_{e}R$, ${}_{e}R_{mn}$
and so on, will obviously vanish.
We split the integration measure into
$\sqrt{{}_{(d)}g}=\sqrt{{}_{4}g}\sqrt{{}_{(e)}g}$ and take the flat limit in
${}_{e}g$ ($\sqrt{{}_{e}{g}}\to 1$). We also reinsert the $\mu^{\epsilon}$
renormalization scale in the original defintion (7.3), to obtain

$\displaystyle\frac{1}{d-4}V_{E}(g,d)=$
$\displaystyle\frac{1}{\epsilon}\left({L}{\mu}\right)^{\epsilon}\int
d^{4}x\sqrt{g}\ {}_{4}\bar{E}+\left({L}{\mu}\right)^{\epsilon}\int
d^{4}x\sqrt{g}\
\Big{[}\phi{}_{4}\bar{E}-(4{}G^{\mu\nu}(\bar{\nabla}_{\mu}\phi\bar{\nabla}_{\nu}\phi)$
$\displaystyle+2(\nabla_{\lambda}\phi\nabla^{\lambda}\phi)^{2}+4\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi\nabla_{\lambda}\phi\nabla^{\lambda}\phi)\Big{]},$ (7.8.9)

where all the terms in the integrands are 4-dimensional and $L$ is a space
cutoff in the $d-4$ extra dimensions. $L^{\epsilon}$ is the volume of the
extra space. Taking the $\epsilon\to 0$ limit, and going back to the ordinary
4-$d$ notation, $g_{\mu\nu}=\bar{g}_{\mu\nu}e^{2\phi}$ for the fiducial
metric, we finally derive the expressions

$\displaystyle\hat{V}^{\prime}_{E}(g,\phi)$
$\displaystyle\equiv\mathcal{S}_{E}^{WZ}=\frac{1}{\epsilon}\left(V_{E}(g,d)-V_{E}(\bar{g},d)\right)$
$\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{g}\Big{[}\phi{}_{4}E-(4{}G^{\mu\nu}(\bar{\nabla}_{\mu}\phi\bar{\nabla}_{\nu}\phi)+2(\nabla_{\lambda}\phi\nabla^{\lambda}\phi)^{2}+4\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi\nabla_{\lambda}\phi\nabla^{\lambda}\phi)\Big{]},$ (7.8.10)

and

$V^{\prime}_{E}=\hat{V}^{\prime}_{E}(\bar{g},\phi,d)+\log(L\mu)\int
d^{4}x\sqrt{\bar{g}}\bar{E}.$ (7.8.11)

The dilaton appears with vertices up to order four. The result reproduces in
details the analysis in (7.7.53). In this case we identify the term

$\frac{\partial V_{E}(\bar{g},d)}{\partial d}\mid_{d=4}=\log(L\mu)\int
d^{4}x\sqrt{\bar{g}}\bar{E}$ (7.8.12)

which is the Weyl invariant mismatch between the regularization obtained via
the WZ action and the one present in $\mathcal{S}_{R}$ using standard DR. The
example treated above defines a combined DR regularization for such types of
actions.

A similar analysis can be performed for $V_{C^{2}}$. In this case, under a
Weyl rescaling

$V_{C^{2}}=\int d^{4}xd^{d-4}y\sqrt{{}_{(d)}g}\ {}_{(d)}C^{2}=\int
d^{4}xd^{d-4}y\sqrt{{}_{(d)}\bar{g}}\ e^{(d-4)\phi}{}_{(d)}\bar{C}^{2}.$
(7.8.13)

As in the previous case, in the ${}_{(d)}\bar{g}_{MN}$ metric, the squared
curvatures are of the form (7.8.5), hence the $d$-dimensional Weyl tensor
squared is expanded as

${}_{(d)}\bar{C}^{2}={}_{4}\tilde{C}^{2}+{}_{(e)}C^{2}+\frac{4}{(d-1)(d-2)}{}_{4}\tilde{R}{}_{(e)}R.$
(7.8.14)

Then the countertem reads

$\int d^{4}xd^{d-4}y\sqrt{{}_{(d)}g}\ {}_{(d)}C^{2}=\int
d^{4}xd^{d-4}y\sqrt{{}_{(d)}\bar{g}}\
e^{(d-4)\phi}\left({}_{4}\tilde{C}^{2}+{}_{(e)}C^{2}+\frac{4}{(d-1)(d-2)}{}_{4}\tilde{R}{}_{(e)}R\right).$
(7.8.15)

Similarly we obtain

$\displaystyle V_{C^{2}}=\int d^{4}xd^{d-4}y\sqrt{{}_{(d)}g}\ {}_{(d)}C^{2}=$
$\displaystyle\int
d^{4}xd^{d-4}y\sqrt{{}_{4}\tilde{g}}e^{(d-4)\phi}{}_{4}\tilde{C}^{2}$
$\displaystyle=$ $\displaystyle L^{\epsilon}\int
d^{4}x\sqrt{{}_{4}\tilde{g}}e^{(d-4)\phi}{}_{4}\tilde{C}^{2}.$ (7.8.16)

Reinserting the $\mu$ dependence, the expansion in $\epsilon$ of this term
generates two contributions

$\frac{1}{d-4}V_{C^{2}}(d)=\frac{1}{\epsilon}\left(\mu L\right)^{\epsilon}\int
d^{4}x\sqrt{{g}}{}C^{2}+O(\epsilon)+\int d^{4}x\sqrt{{g}}\phi C^{2},$ (7.8.17)

the first of them relevant for the cancellation of the singular behaviour of
$\mathcal{S}_{B}(d)$ as $d$ goes to 4. Differently from the similar
counterterm in $V_{E}$, this is necessary in order to regulate the divergences
of $\mathcal{S}_{B}(g,d)$ in the $d\to 4$ limit.
Also in this case, by defining

$\hat{V}^{\prime}_{C^{2}}\equiv
S_{C^{2}}^{WZ}=\frac{1}{\epsilon}\left(V_{C^{2}}(g,d)-V_{C^{2}}(\bar{g},d)\right),$
(7.8.18)

we have

$\hat{V}^{\prime}_{C^{2}}(\bar{g},d)=\int
d^{d}x\sqrt{{\bar{g}}}\phi\bar{C}^{2}$ (7.8.19)

and

$V^{\prime}_{C^{2}}(4)=\hat{V}^{\prime}_{C^{2}}(\bar{g},d)+\log(L\mu)\int
d^{4}x\sqrt{{\bar{g}}}\bar{C}^{2},$ (7.8.20)

giving from (7.7.53)

$\left(2g_{\mu\nu}\frac{\delta\mathcal{S}^{A}}{\delta
g_{\mu\nu}}-2g_{\mu\nu}\frac{\delta\mathcal{S}^{WZ}}{\delta
g_{\mu\nu}}\right)=\log(L\mu)\int
d^{4}x\sqrt{\bar{g}}\left(b^{\prime}\bar{E}+b\bar{C}^{2}\right).$ (7.8.21)

The conformal backreaction, identified in $\mathcal{S}_{R}$, can then be
expressed in the final form

$\displaystyle\mathcal{S}_{R}=$ $\displaystyle\mathcal{S}_{f}+b^{\prime}\int
d^{4}x\sqrt{g}\Big{[}\phi{}_{4}E-(4{}G^{\mu\nu}(\bar{\nabla}_{\mu}\phi\bar{\nabla}_{\nu}\phi)+2(\nabla_{\lambda}\phi\nabla^{\lambda}\phi)^{2}+4\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi\nabla_{\lambda}\phi\nabla^{\lambda}\phi)\Big{]}$
$\displaystyle+b\int d^{d}x\sqrt{{\bar{g}}}\phi\bar{C}^{2}+\log(L\mu)\int
d^{4}x\sqrt{\bar{g}}\left(b^{\prime}\bar{E}+b\bar{C}^{2}\right),$ (7.8.22)

valid in DR, where the only missing term is $\mathcal{S}_{f}$. Both
$\mathcal{S}_{f}$ and the log-contribution are Weyl-invariant (i.e.
$\phi$-independent) terms which are part of the regulated action. If we limit
our attention only to a GB theory, with a classical singular rescaling of the
GB coupling, as in ordinary $d=2$ gravity [117], then $\mathcal{S}_{f}$ and
the log terms are obviously absent, while at the same time we need to set
$b=0$.

#####  7.8.1 Including the Einstein-Hilbert term: $EGBW_{1}$ and variants

Extending $\mathcal{S}_{R}$ in order to derive a Einsten GB/Weyl theory is
quite straightforward, but it is not a unique procedure. At the same time,
this theory can be accompanied by Lagriangian terms of various types. If one
intends to adhere and modify the Lovelock classification at $d=4$, one can add
the EH term.
We recall that the EH term may be expressed either in terms of the fiducial
metric, as $\mathcal{S}_{EH}(\bar{g},4)$, where

$\mathcal{S}_{EH}(\bar{g},4)\equiv\int
d^{4}x\sqrt{\bar{g}}\left(M_{P}^{2}\bar{R}+2\Lambda\right),$ (7.8.23)

generating an action at $d=4$ of the form

$\displaystyle\mathcal{S}_{EGBW_{1}}\equiv\mathcal{S}_{EH}(\bar{g},4)+\mathcal{S}_{R}(\bar{g},\phi),$
(7.8.24)

with

$\mathcal{S}_{R}(\bar{g},\phi)=\mathcal{S}_{f}(\bar{g})+\mathcal{S}_{A}(\bar{g},\phi),$
(7.8.25)

and $\mathcal{S}_{A}$ given by (7.8.22), or, alternatively, by promoting the
entire EGBW theory to $d$-dimensions and performing the $d\to 4$ limit on all
of its components.
In this second case, if we perform a Weyl trasformation also on the EH action,
we derive the ordinary form of the dilaton gravity action

$\displaystyle\int d^{d}x\sqrt{g}(M_{P}^{2}R-2\Lambda)=$ $\displaystyle\int
d^{d}x\sqrt{\bar{g}}\
e^{(d-2)\phi}\left(M_{P}^{2}[_{d}\bar{R}-2(d-1)\bar{\square}\phi-(d-1)(d-2)\bar{\nabla}_{\lambda}\phi\bar{\nabla}^{\lambda}\phi]-2e^{2\phi}\Lambda\right)$
$\displaystyle=\int
d^{d}x\sqrt{\bar{g}}e^{(d-2)\phi}\left(M_{P}^{2}[\bar{R}+(d-1)(d-2)\bar{\nabla}_{\lambda}\phi\bar{\nabla}^{\lambda}\phi]-2e^{2\phi}\Lambda\right).$
(7.8.26)

Dimensional reduction of this action leads to the ordinary dilaton gravity
$\mathcal{S}_{EHd}$ in the Jordan (string) frame

$\mathcal{S}_{EHd_{1}}(\bar{g},\phi)=\int
d^{4}x\sqrt{\bar{g}}e^{2\phi}\left(M_{P}^{2}[\bar{R}+6\bar{\nabla}_{\lambda}\phi\bar{\nabla}^{\lambda}\phi]-2e^{2\phi}\Lambda\right).$
(7.8.27)

Logarithmic, scale dependent terms are absent in this action, since we can
smoothly take the $d\to 4$ limit in DR from $\mathcal{S}_{EH}$ in $d$
dimensions, due to finiteness.

We can add $\mathcal{S}_{R}$ to this action, obtaining the corresponding EGBW
action -denoted as $EGBW_{1}$ -

$\displaystyle\mathcal{S}_{EGBW_{1}}\equiv\mathcal{S}_{EH{d_{1}}}(\bar{g},\phi)+\mathcal{S}_{R}(\bar{g},\phi),$
(7.8.28)

with $\mathcal{S}_{R}(\bar{g},\phi)$ given by (7.8.22). We have also observed
that there are variants of the theory in which the logarithmic $\log(\mu L)$
scale dependent terms are absent from the counterterms. We work in the context
of this variant, which corresponds to a redefiniton of the renormalized
quantum effective action $\mathcal{S}_{R}$ in the form

$\tilde{\mathcal{S}}_{R}=\mathcal{S}_{f}(4)+\mathcal{S}^{WZ},$ (7.8.29)

giving

$\displaystyle\tilde{\mathcal{S}}_{EGBW_{1}}\equiv\mathcal{S}_{EH{d_{1}}}(\bar{g},\phi)+\tilde{\mathcal{S}}_{R}(\bar{g},\phi).$
(7.8.30)

This theory is defined according (7.8.22) by the choice $\mu=1/L$, which
removes the log terms in DR. In plane terms, in this case the subtraction is
performed using the fiducial metric $\bar{g}$ in $d$ dimensions rather than at
$d=4$. The need of performing an expansion in $\epsilon$ with dimensionless
logs, justifies the presence of a second cutoff, since $\mu L$ needs to be
dimensionless. The results is that extra Weyl invariant terms are present in
the expansion an cannot be neglected.

#####  7.8.2 A dilaton in $\mathcal{M}_{e}$

As already mentioned, one of the pressing issues in DR using higher
dimensional metrics for the regularization of a certain action, is the
ambiguity in the choice of the fiducial metric in the $d$-dimensional space.
From this point of view, selecting a metric embedding such as the one
discussed in the previous sections appears to be quite simple and appealing,
but it is not unique. One criterion might be to ensure that fluctuations
around flat spacetime, once the DR procedure is applied, do not include ghost-
like degrees of freedom. We now bring an example that illustrates some of the
problems encountered when the metric structure introduced above is modified by
the inclusion of a dilator field only in the extradimensional manifold
$\mathcal{M}_{e}$. Indeed, the expression of the effective action depends
critically on the way the dilaton is extracted from the metric. We choose the
metric of the form

${}_{(d)}g_{MN}(x,y)=\begin{bmatrix}{}_{4}g_{\mu\nu}(x)&0\\\
0&e^{2\phi(x)}{}_{(e)}g_{mn}(y)\end{bmatrix},$ (7.8.31)

Now, consider the $d$-dimensional $V_{E}$ counterterm that we will compute by
iterating the procedure (7.8). First, we factor the exponential in the $(d-4)$
metric

${}_{(d)}g_{MN}(x,y)=e^{2\phi(x)}\begin{bmatrix}e^{-2\phi(x)}{}_{4}g_{\mu\nu}(x)&0\\\
0&{}_{(e)}g_{mn}(y)\end{bmatrix}.$ (7.8.32)

In this way, the $d$-dimensional metric ${}_{(d)}g_{MN}$ can be written as an
overall exponential $e^{2\phi(x)}$ times a fiducial metric as in (7.8.3),
where now we define

${}_{4}\tilde{g}_{\mu\nu}=e^{-2\phi}{}_{4}g_{\mu\nu}.$ (7.8.33)

Such rescaling affects $V_{E}$, since it corespond just to a Weyl
transformation in $4$ dimensions, with a switch of the overall sign in front
of $\phi$. Our goal now is to extract that factor from the metric..
For full conversion, we have to act with this reverse transformation on every
single term of (7.8).
We start with the ${}_{4}\tilde{E}$ term. We thus expand
$\sqrt{{}_{d}g}=\sqrt{{}_{4}g}\sqrt{{}_{(e)}g}$ and then use (7.7.15) to get

$\displaystyle\int d^{4}xd^{d-4}y\sqrt{{}_{(d)}\bar{g}}\
e^{(d-4)\phi}{}_{(4)}\bar{E}=$ $\displaystyle\int
d^{4}xd^{d-4}y\sqrt{{}_{(4)}\tilde{g}}\sqrt{{}_{(e)}g}\
e^{(d-4)\phi}{}_{4}\tilde{E}$ $\displaystyle=$ $\displaystyle\int
d^{4}xd^{d-4}y\sqrt{{}_{4}g}\sqrt{{}_{(e)}g}\
e^{(d-4)\phi}\left({}_{4}E-(d-4)\nabla_{\mu}\phi\,\,{}_{4}J^{\mu}_{-}\right),$
(7.8.34)

where in the last equality we have integrated by parts. The $(-)$ notation
${}_{4}J^{\mu}_{-}$ is a remainder hat we have inverted the sign of $\phi$
inside the $J$ term

${}_{4}J^{\mu}_{-}=-8{}_{4}R^{\mu\nu}\nabla_{\nu}\phi+4{}_{4}R\nabla^{\mu}\phi+8(\nabla^{\mu}\phi\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi-\nabla^{\mu}(\nabla^{\nu}\phi)\nabla_{\nu}\phi-\nabla^{\mu}\phi\nabla_{\lambda}\phi\nabla^{\lambda}\phi).$
(7.8.35)

Next, we deal with the second and the third term of (7.8). We integrate by
parts the term $\tilde{\nabla}_{\mu}{}_{4}\tilde{J}^{\mu}$ and expand using
(7.7.16) and (7.7.17) to obtain

$\displaystyle\Sigma$ $\displaystyle\equiv\int
d^{4}xd^{d-4}y\sqrt{{}_{(d)}\bar{g}}\
e^{(d-4)\phi}(d-3)(d-4)[-\tilde{\nabla}_{\mu}\phi{}_{4}\tilde{J}^{\mu}+{}_{4}\tilde{K}]$
$\displaystyle=$ $\displaystyle\int
d^{4}xd^{d-4}y\sqrt{{}_{4}\tilde{g}}\sqrt{{}_{(e)}g}\
e^{(d-4)\phi}(d-3)(d-4)\biggl{[}-4\left({}_{4}\tilde{R}^{\mu\nu}-\frac{1}{2}{}_{4}\tilde{g}^{\mu\nu}{}_{4}\tilde{R}\right)\tilde{\nabla}_{\nu}\phi\tilde{\nabla}_{\mu}\phi$
$\displaystyle+4(d-2)\tilde{\nabla}^{\mu}(\tilde{\nabla}^{\nu}\phi)\tilde{\nabla}_{\nu}\phi\tilde{\nabla}_{\mu}\phi+(d-2)(d-5)(\tilde{\nabla}^{\lambda}\phi\tilde{\nabla}_{\lambda}\phi)^{2}\biggl{)}\biggl{]}.$
(7.8.36)

At this point we must expand
$\left({}_{4}\tilde{R}^{\mu\nu}-\frac{1}{2}{}_{4}\tilde{g}^{\mu\nu}{}_{4}\tilde{R}\right)$
using (7.7.7) and (7.7.8), taking care of the sign flip
$\phi\rightarrow-\phi$, therefore obtaining

$\left({}_{4}\tilde{R}^{\mu\nu}-\frac{1}{2}{}_{4}\tilde{g}^{\mu\nu}{}_{4}\tilde{R}\right)\tilde{\nabla}_{\nu}\phi\tilde{\nabla}_{\mu}\phi={}_{4}G^{\mu\nu}\nabla_{\nu}\phi\nabla_{\mu}\phi-(d-7)(\nabla_{\lambda}\phi\nabla^{\lambda}\phi)^{2}-3\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi(\nabla_{\lambda}\phi\nabla^{\lambda}\phi).$

After an integration by parts we finally get

$\displaystyle\Sigma=$ $\displaystyle\int
d^{4}xd^{d-4}y\sqrt{{}_{4}g}\sqrt{{}_{(e)}g}e^{(d-4)\phi}(d-3)(d-4)\biggl{[}-4{}_{(4)}G^{\mu\nu}\nabla_{\nu}\phi\nabla_{\mu}\phi+4(d-7)(\nabla\phi)^{4}+12\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi(\nabla\phi)^{2}$
$\displaystyle-2(d-2)(d-4)(\nabla\phi)^{4}-2(d-2)(\nabla\phi)^{2}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi+4(d-2)(\nabla\phi)^{4}+(d-2)(d-5)(\nabla\phi)^{4}\biggl{)}\biggl{]}.$

At this point, only the second line of (7.8) remains to be treated. We use
(7.7.8) to obtain

$\displaystyle\int d^{4}xd^{d-4}y\sqrt{{}_{(d)}\bar{g}}\
e^{(d-4)\phi}{}_{(e)}R\ {}_{(4)}\tilde{R}=\int
d^{4}xd^{d-4}y\sqrt{{}_{(4)}g}\sqrt{{}_{(e)}g}\ e^{(d-6)\phi}{}_{(e)}R\
\left({}_{(4)}R+6(d-7)\nabla_{\lambda}\phi\nabla^{\lambda}\phi\right),$

where we have integrated by part in the last line, remembering that
${}_{(e)}R$ does not depend on $x$. For the last term of (7.8) we easily
obtain

$\displaystyle\int d^{4}x\,d^{d-4}y\sqrt{{}_{(4)}\tilde{g}}\sqrt{{}_{(e)}g}\
e^{(d-4)\phi}{}_{(e)}R[4\tilde{\square}\phi+2(d-4)(\tilde{\nabla}_{\lambda}\phi\tilde{\nabla}^{\lambda}\phi)]$
$\displaystyle=$ $\displaystyle\int
d^{4}x\,d^{d-4}y\sqrt{{}_{(4)}g}\sqrt{{}_{(e)}g}\
e^{(d-6)\phi}{}_{(e)}R[4\square\phi-(d-2)(\nabla_{\lambda}\phi\nabla^{\lambda}\phi)+2(d-4)(\nabla_{\lambda}\phi\nabla^{\lambda}\phi)]$
$\displaystyle=$ $\displaystyle-6(d-4)\int
d^{4}x\,d^{d-4}y\sqrt{{}_{4}g}\sqrt{{}_{e}g}e^{(d-6)\phi}{}_{(e)}R(\nabla_{\lambda}\phi\bar{\nabla}^{\lambda}\phi).$
(7.8.39)

By adding all the terms, the expanded counterterm takes the form

$\displaystyle V_{E}(d)=$ $\displaystyle\int
d^{4}x\,d^{d-4}y\sqrt{{}_{(4)}g}\sqrt{{}_{(e)}g}\
e^{(d-4)\phi}\left[{}_{4}E-4(d-4)(d-5){}_{(4)}G^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi\right.$
$\displaystyle-(d-4)(d-5)(d-6)[2(\nabla\phi)^{2}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi+(d-5)(\nabla_{\lambda}\phi\nabla^{\lambda}\phi)^{2}]+{}_{(e)}E$
$\displaystyle\left.+2e^{-2\phi}{}_{(e)}R\left({}_{4}R+3(d-10)\nabla_{\lambda}\phi\nabla^{\lambda}\phi\right)\right].$
(7.8.40)

Also in this case, to derive the renormalized EGB action, we consider a flat
extra space ${}_{4}g_{mn}=\delta_{mn}$. With this choice, all the curvatures
in $\mathcal{M}_{e}$ vanish. Finally, we drop the last line of (7.8.2) since
${}_{(e)}R=0$ and obtain

$\displaystyle V_{E}(d)=$ $\displaystyle\int d^{4}x\,d^{d-4}y\sqrt{4_{g}}\
e^{(d-4)\phi}\biggl{[}{}_{4}E-4(d-4)(d-5){}_{4}G^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$
$\displaystyle-(d-4)(d-5)(d-6)[2(\nabla\phi)^{2}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi+(d-5)(\nabla\phi)^{4}]\biggl{]}.$ (7.8.41)

At this point, we need to expand (7.8.2) around $d=4$. As in the previous
section, also in this case we need to intoduce a cutoff $L^{\epsilon}$ on the
volume of $\mathcal{M}_{e}$, which will be removed in the process of
dimensional reduction, since all the integrands are only supported on
$\mathcal{M}_{4}$.

Indeed we obtain

$\displaystyle\frac{1}{\epsilon}V_{E}(d)=$
$\displaystyle\frac{1}{\epsilon}\left(\mu L\right)^{\epsilon}\int
d^{4}x\sqrt{{}_{4}g}\,{}_{4}E+\left(L\mu\right)^{\epsilon}\int
d^{4}x\sqrt{{}_{4}g}[\phi\,{}_{4}E+4{}_{4}G^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi)$
$\displaystyle-4\nabla_{\lambda}\phi\nabla^{\lambda}\phi\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi+2(\nabla_{\lambda}\phi\nabla^{\lambda}\phi)^{2}],$ (7.8.42)

from which we extract the finite contribution of $V_{E}$ to $\mathcal{S}_{R}$,
obtaining for this type of metric the following expression of the topological
contribution to $\mathcal{S}^{WZ}$

$\displaystyle\hat{V}^{\prime}_{E}=$ $\displaystyle\int
d^{4}x\sqrt{{}_{4}g}[\phi\,{}_{4}E+4\,{}_{4}G^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi)-4\nabla_{\lambda}\phi\nabla^{\lambda}\phi\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi+2(\nabla_{\lambda}\phi\nabla^{\lambda}\phi)^{2}].$ (7.8.43)

Also in this case, $\hat{V}^{\prime}_{E}$ will differ from $V^{\prime}_{E}$,
by Weyl-invariant terms, exaclty as in (7.8.11). For the computation of the
Weyl counterterm we use (7.8.15) and extract a conformal factor from the
metric (7.8.33), obtaining

$\displaystyle\int d^{4}xd^{d-4}y\sqrt{{}_{(d)}\bar{g}}\
e^{(d-4)\phi}{}_{(4)}\tilde{C}^{2}=\int d^{d}x\sqrt{{}_{4}g}\
e^{(d-4)\phi}{}_{4}C^{2}.$ (7.8.44)

For the ${}_{4}\tilde{R}$ term, we resort again to (7.7.8) obtaining

$\displaystyle\int d^{4}xd^{d-4}y\sqrt{{}_{(d)}\bar{g}}\ {}_{(e)}R{}_{(4)}\
\tilde{R}=$ $\displaystyle\int d^{4}xd^{d-4}y\sqrt{{}_{(4)}g}\sqrt{{}_{(e)}g}\
e^{(d-6)\phi}{}_{(e)}R\left({}_{(4)}R-6\square\phi-6\nabla_{\lambda}\phi\nabla^{\lambda}\phi\right)$
$\displaystyle=$ $\displaystyle\int
d^{4}xd^{d-4}y\sqrt{{}_{(4)}g}\sqrt{{}_{(e)}g}\
e^{(d-6)\phi}{}_{(e)}R\left({}_{4}R+6(d-7)\nabla_{\lambda}\phi\nabla^{\lambda}\phi\right).$
(7.8.45)

The full $C^{2}$ counterterm thus reads

$V_{C^{2}}=\int
d^{4}xd^{d-4}y\sqrt{{}_{4}g}e^{(d-4)\phi}\left[{}_{4}C^{2}+{}_{(e)}C^{2}+\frac{4}{(d-1)(d-2)}e^{-2\phi}{}_{(e)}R\left({}_{4}R+6(d-7)\nabla_{\lambda}\phi\nabla^{\lambda}\phi\right)\right].$
(7.8.46)

Choosing a flat extra space ${}_{(e)}g_{mn}=\delta_{mn}$, (7.8.46) reduces to
the expression

$V_{C^{2}}=\int d^{4}xd^{d-4}y\sqrt{{}_{4}g}\ e^{(d-4)\phi}\ {}_{4}C^{2}.$
(7.8.47)

Also for this term, after factorizing out an overall volume on
$\mathcal{M}_{e}$ we can reduce the integral to 4 dimensions. Expanding the
exponential we get

$\frac{1}{\epsilon}V_{C^{2}}=\frac{(L\mu)^{\epsilon}}{\epsilon}\int
d^{4}x\sqrt{{}_{4}g}\left({}_{4}C^{2}+\phi\,{}_{4}C^{2}\right)+L\mu)^{\epsilon}\int
d^{4}x\phi\,{}_{4}C^{2},$ (7.8.48)

and the finite contribution coming from the Weyl counterterm, is here
expressed only in terms of the fiducial metric ($g\to\bar{g}$)

$\hat{V}^{\prime}_{C^{2}}=\int d^{4}x\sqrt{\bar{g}}\phi\,{}_{4}\bar{C}^{2}.$
(7.8.49)

To obtain the complete EGB action, we Weyl gauge the Einstein-Hilbert term by
the same choice of metric. In fact, under a rescaling, as described in
(7.7.8), we obtain

$\int d^{4}xd^{d-4}y\sqrt{{}_{(d)}g}\ \left({}_{(d)}R-2\Lambda\right)=\int
d^{4}xd^{d-4}y\sqrt{{}_{(4)}g}\sqrt{{}_{(e)}g}\
e^{(d-4)\phi}\left[{}_{4}R-2e^{4\phi}\Lambda+(d-4)(d-5)(\nabla\phi)^{2}\right].$
(7.8.50)

From this expression we can factorize a volume $L^{\epsilon}$, if we take the
extra dimensional metric flat ($g_{mn}(y)=\delta_{mn}$), giving a typical
dilaton gravity action

$\mathcal{S}_{EH}\to\mathcal{S}_{EHd_{2}}=\left(L\mu\right)^{\epsilon}\int
d^{4}xe^{(d-4)\phi}\left[R-2e^{4\phi}\Lambda+(d-4)(d-5)(\nabla\phi)^{2}\right],$
(7.8.51)

which at $d=4$ becomes

$\mathcal{S}_{EH}\to\mathcal{S}_{EHd_{2}}=\int
d^{4}x\left[R-2e^{4\phi}\Lambda\right].$ (7.8.52)

There are some clear differences respect to $\mathcal{S}_{EHd_{1}}$ in
(7.8.27), since the dilaton kinetic term in (7.8.51) is ghost-like at
$O(\epsilon)$. Obviously, exactly at $d=4$, the instability is not present,
but in (7.8.52) we miss a quadratic kinetic term for $\phi$ altogether. This
clearly indicates that a consistent DR regularization should include metrics
exhibiting a correct kinetic term for the dilaton, and this is clearly not
always possible. At the same time, we need to exclude from the procedure those
metrics which are not fully consistent, once we perform the flat spacetime
limit at $d=4$.

#####  7.8.3 Equations of motion for $\widetilde{EGBW_{1}}$

To derive the equations of motion we set $\kappa^{2}\equiv 1/(16\pi G_{N})=1$.
We use the modified definition of $\tilde{\mathcal{S}}_{EGBW_{1}}$ as given by
(7.8.30), which is deprived of logarithmic terms. As already remarked, this
amounts to a finite renormalization of the effective action in field space. We
obtain

$\left(G^{\mu\nu}+A^{\mu\nu}\right)-W^{\mu\nu}=\langle
T^{\mu\nu}\rangle_{f}+\frac{2}{\sqrt{g}}\left[b^{\prime}{\hat{V}^{\prime}}_{E}^{\mu\nu}+b{\hat{V}^{\prime}}_{C^{2}}^{\mu\nu}\right]$
(7.8.53)

where $G^{\mu\nu}$ is the Einstein tensor,

$\displaystyle A_{\mu\nu}$
$\displaystyle=\frac{2}{\sqrt{g}}\frac{\delta}{\delta g^{\mu\nu}}\left[\int
d^{4}x\sqrt{g}\
e^{2\phi}\left([R+6\nabla_{\lambda}\phi\nabla^{\lambda}\phi]-2e^{2\phi}\Lambda\right)\right]$
$\displaystyle=e^{2\phi}\left(-3g_{\mu\nu}\nabla_{\lambda}\phi\nabla^{\lambda}\phi+3\nabla_{\mu}\phi\nabla^{\mu}\phi+e^{2\phi}g_{\mu\nu}\Lambda\right),$
(7.8.54)

and

$W_{\mu\nu}=\frac{2}{\sqrt{g}}\frac{\delta}{\delta g^{\mu\nu}}\left[\int
d^{4}x\sqrt{\bar{g}}\Big{(}\phi(\alpha\bar{E}+\alpha^{\prime}\bar{C}^{2})+\alpha[\bar{G}^{\mu\nu}(\bar{\nabla}_{\mu}\phi\bar{\nabla}_{\nu}\phi)+2(\bar{\nabla}\phi)^{4}+4\bar{\,}\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi\bar{\nabla}\phi^{2})]\Big{)}\right].$ (7.8.55)

Explicitly

$\displaystyle
W_{\mu\nu}=2\bar{R}(\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\phi-\bar{\nabla}_{\mu}\phi\bar{\nabla}_{\nu}\phi)+2\bar{G}_{\mu\nu}\left((\bar{\nabla}\phi)^{2}-2\bar{\,}\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi)\right)+4\bar{G}_{\nu\alpha}(\bar{\nabla}^{\alpha}\bar{\nabla}_{\mu}\phi-\bar{\nabla}^{\alpha}\phi\bar{\nabla}_{\mu}\phi)$
$\displaystyle+4\bar{G}_{\mu\alpha}\left(\bar{\nabla}^{\alpha}\bar{\nabla}_{\nu}\phi-\bar{\nabla}^{\alpha}\phi\bar{\nabla}_{\nu}\phi\right)+4\bar{R}_{\mu\alpha\nu\beta}\left(\bar{\nabla}^{\beta}\bar{\nabla}^{\alpha}\phi-\bar{\nabla}^{\alpha}\phi\bar{\nabla}^{\beta}\phi\right)+4\bar{\nabla}_{\alpha}\bar{\nabla}_{\nu}\phi\left(\bar{\nabla}^{\alpha}\phi\bar{\nabla}_{\mu}\phi-\bar{\nabla}^{\alpha}\bar{\nabla}_{\mu}\phi\right)$
$\displaystyle+4\bar{\nabla}_{\alpha}\bar{\nabla}_{\mu}\phi\bar{\nabla}^{\alpha}\phi\bar{\nabla}_{\nu}\phi-4\bar{\nabla}_{\mu}\phi\bar{\nabla}_{\nu}\phi\left((\bar{\nabla}\phi)^{2}+\bar{\,}\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi)\right)+4\bar{\,}\raise 0.5pt\hbox{$\vbox{\hrule\hbox{\vrule
height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi\bar{\nabla}_{\nu}\bar{\nabla}_{\mu}\phi-
g_{\mu\nu}\biggl{[}2R\left((\bar{\,}\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi-\bar{\nabla}\phi)^{2})\right)$
$\displaystyle+4\bar{G}^{\alpha\beta}(\bar{\nabla}_{\beta}\bar{\nabla}_{\alpha}\phi-\bar{\nabla}_{\alpha}\phi\bar{\nabla}_{\beta}\phi)+2(\bar{\,}\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi)^{2}-(\bar{\nabla}\phi)^{4}+2\bar{\nabla}_{\beta}\bar{\nabla}_{\alpha}\phi(2\bar{\nabla}^{\alpha}\phi\bar{\nabla}^{\beta}\phi-\bar{\nabla}^{\beta}\bar{\nabla}^{\alpha}\phi)\biggl{]}+$
$\displaystyle+\phi[\alpha
H_{\mu\nu}+\alpha^{\prime}B_{\mu\nu}]+\alpha^{\prime}\Big{(}-4R^{\mu\nu\alpha\beta}\nabla_{a}\nabla_{b}\phi-4R^{\lambda(\mu}\nabla_{\lambda}^{\nu)}\phi+2R^{\mu\nu}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi+g^{\mu\nu}R^{\alpha\beta}\nabla_{a}\nabla_{b}\phi$
$\displaystyle-\frac{2}{3}RR^{\mu\nu}\phi+\frac{2}{3}Rg^{\mu\nu}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi-\frac{2}{3}R\nabla^{\mu}\nabla^{\nu}\phi\Big{)},$ (7.8.56)

where

$H_{\mu\nu}=2\left[\bar{R}\bar{R}_{\mu\nu}-2\bar{R}_{\mu\alpha\nu\beta}\bar{R}^{\alpha\beta}+\bar{R}_{\mu\alpha\beta\kappa}\bar{R}_{\nu}{}^{\alpha\beta\kappa}-2\bar{R}_{\mu\alpha}\bar{R}^{\alpha}_{\nu}-\frac{1}{4}g_{\mu\nu}\bar{E}\right]$
(7.8.57)

vanishes at $d=4$. Here, the renormalized quantum average of the stress energy
tensor takes the form

$\displaystyle\langle T^{\mu\nu}\rangle_{R}=$ $\displaystyle\langle
T^{\mu\nu}\rangle_{B}+\lim_{d\rightarrow
4}\frac{2}{\sqrt{g}}\frac{1}{(d-4)}\left[b^{\prime}\hat{V}_{E}^{\mu\nu}+b\hat{V}_{C^{2}}^{\mu\nu}\right]$
$\displaystyle=$ $\displaystyle\langle
T^{\mu\nu}\rangle_{f}+\frac{2}{\sqrt{g}}\left[b^{\prime}{\hat{V}^{\prime}}_{E}^{\mu\nu}+b{\hat{V}^{\prime}}_{C^{2}}^{\mu\nu}\right]$
(7.8.58)

where ${\hat{V}^{\prime}}^{\mu\nu}\equiv\frac{\delta}{\delta
g_{\mu\nu}}\hat{V}^{\prime}$. The counterterms induce a cancellation of the
singularities present in $\langle T^{\mu\nu}\rangle_{B}$, the stress energy
tensor associated with $\mathcal{S}_{B}$, as we perform the $d\to 4$ limit.
The anomaly relation

$g_{\mu\nu}\langle
T^{\mu\nu}\rangle_{R}=\mathcal{A}(x),\qquad\mathcal{A}(x)\equiv\sqrt{g}\left(b^{\prime}E+bC^{2}\right),$
(7.8.59)

is generated by the $\hat{V}^{\prime}$ counterterms since

$g_{\mu\nu}\langle T^{\mu\nu}\rangle_{f}=0$ (7.8.60)

and henceforth

$\frac{2}{\sqrt{g}}g_{\mu\nu}\left[b^{\prime}{\hat{V}^{\prime}}_{E}^{\mu\nu}+b{\hat{V}^{\prime}}_{C^{2}}^{\mu\nu}\right]=\mathcal{A}(x).$
(7.8.61)

####  7.9 The quartic dilaton action and the conformal breaking scale

In this section we are going to address one aspect of the dilaton gravity
action that we have indicated as $\tilde{\mathcal{S}}_{EGBW_{1}}$, introduced
in the previous sections. One of the most important issues concerns the
presence of constrains between the trace equation of motion of the fiducial
metric and the equation of the conformal factor, as shown by (7.7.52). This
relation is induced by the renormalization procedure and is obviously related
to the anomaly, that breaks the residual invariance of the conformal
decomposition (7.7.1). Notice that $\phi$ does not carry any dimension and it
is clear that a correct normalization of this field requires the introduction
of a scale $f$. Therefore, the selection of a given fiducial metric $\bar{g}$
is directly linked to the emergence of $f$, which breaks the conformal
symmetry. To investigate this point, we proceed as follows.
Before expanding we send $\phi\rightarrow-\phi$, obtaining

$\displaystyle\tilde{S}_{EGBW_{1}}=\frac{1}{16\pi G}\int d^{4}x\sqrt{g}\
e^{-2\phi}\left([R+6\nabla_{\lambda}\phi\nabla^{\lambda}\phi]-2e^{-2\phi}\Lambda\right)+\mathcal{S}_{f}(4)$
$\displaystyle+\int
d^{4}x\sqrt{g}\biggl{[}-\phi(b^{\prime}E+bC^{2})-b^{\prime}\left(4G^{\mu\nu}(\nabla_{\mu}\phi\nabla_{\nu}\phi)+2(\nabla_{\lambda}\phi\nabla^{\lambda}\phi)^{2}-4\bar{\,}\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi\nabla_{\lambda}\phi\nabla^{\lambda}\phi\right)\biggl{]}.$
(7.9.1)

We omit, for simplicity, the bar symbol on the gravitational metric. It is
quite straightforward to show that such a Lagrangian describe a spontaneously
broken phase, due to the presence of a bilinear mixing between the scalar
field $\phi$ and the metric.
To show this, it is convenient to introduce the field redefinition

$e^{-2\phi}=1-\frac{\tilde{\phi}}{f}\qquad\phi=-\frac{1}{2}\log(1-\frac{\tilde{\phi}}{f})$
(7.9.2)

for which the action is rewritten as

$\displaystyle\tilde{S}_{EGBW_{1}}=\frac{M_{P}^{2}}{2}\int
d^{4}x\sqrt{g}\left(R-\frac{1}{f}\tilde{\phi}R+\frac{3}{2}\frac{1}{(1-\frac{\tilde{\phi}}{f})f^{2}}\partial_{\lambda}\tilde{\phi}\partial^{\lambda}\tilde{\phi}-2(1-\frac{\tilde{\phi}}{f})\Lambda\right)+\mathcal{S}_{f}(4)$
$\displaystyle+\int
d^{4}x\sqrt{g}\Biggl{[}\frac{1}{2}\log(1-\frac{\tilde{\phi}}{f})(b^{\prime}E+bC^{2})-b^{\prime}\left(G^{\mu\nu}\frac{1}{(1-\frac{\tilde{\phi}}{f})^{2}f^{2}}(\partial_{\mu}\tilde{\phi}\partial_{\nu}\tilde{\phi})+\frac{1}{8(1-\frac{\tilde{\phi}}{f})^{4}f^{4}}(\partial_{\lambda}\tilde{\phi}\partial^{\lambda}\tilde{\phi})^{2}\right.$
$\displaystyle\qquad\qquad\left.-\frac{1}{2(1-\frac{\tilde{\phi}}{f})^{3}f^{3}}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,_{0}\tilde{\phi}\partial_{\lambda}\tilde{\phi}\partial^{\lambda}\tilde{\phi}-\frac{1}{2(1-\frac{\tilde{\phi}}{f})^{4}f^{4}}\partial_{\mu}\tilde{\phi}\partial^{\mu}\tilde{\phi}\partial_{\nu}\tilde{\phi}\partial^{\nu}\tilde{\phi}+\frac{1}{2(1-\frac{\tilde{\phi}}{f})^{2}f^{3}}\Gamma^{\lambda}\partial_{\lambda}\tilde{\phi}\partial_{\sigma}\tilde{\phi}\partial^{\sigma}\tilde{\phi}\right)\Bigg{]},$
(7.9.3)

where $M_{P}^{2}=1/(8\pi G_{N})$ is the reduced Planck mass and
$\Gamma^{\lambda}\equiv
g^{\mu\nu}\Gamma^{\lambda}_{\mu\nu}\equiv-\frac{1}{\sqrt{g}}\partial_{\mu}(\sqrt{g}g^{\lambda\mu})$.
Notice that the coupling of the action can be organized in terms of
interactions of increasing mass-dimensions in an expansion in
$\tilde{\phi}/f$. The presence of a bilinear mixing in the EH part of the
effective action $(\sim M_{P}^{2}/f)\tilde{\phi}R$ is indicating that we are
describing a spontaneously broken phase. A solution of the equations of motion
can be obtained by setting $\phi$ constant, and taking a flat fiducial metric
$\bar{g}_{\mu\nu}=\delta_{\mu\nu}$, (i.e. a Weyl flat $g_{\mu\nu}$). In this
case

$\phi=v,\qquad R_{\mu\nu}=\frac{1}{4}Rg_{\mu\nu},\qquad R=24\lambda v^{2}.$
(7.9.4)

An alternative approach is to proceed by itroducing a different field
redefinition of the form

$e^{-\phi}=\bar{\chi}(x),$ (7.9.5)

where $\bar{\chi}(x)$ can be related to a mass dimension-1 scalar as
$\bar{\chi}(x)={\chi(x)}/({\sqrt{3}f})$ generating the coupling

$\mathcal{L}\supset\frac{M_{P}^{2}}{2f^{2}}\sqrt{g}\left(\frac{1}{2}g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi+\frac{1}{6}R\chi^{2}\ldots\right).$
(7.9.6)

It is easy to realize that the $R\chi^{2}$ term carries the wrong sign, since
for slowly varying curvature behaves essentially as a mass term with
$m^{2}\sim R$. The presence either of mixing terms or of mass terms with the
wrong sign are the signatures that the procedure of Weyl gauging generates a
Lagrangian in a broken phase.
Concerning the asymptotic structure of $\tilde{\mathcal{S}}_{EGBW_{1}}$, it is
convenient to organize the terms appearing in it as an expansion in the two
scales $1/f$ and $1/(f^{n}M_{P}^{2})$, obtaining

$\displaystyle\tilde{\mathcal{S}}_{EGBW_{1}}=$
$\displaystyle\frac{M_{P}^{2}}{2}\int
d^{4}x\sqrt{g}\left(R-\frac{\tilde{\phi}}{\bar{f}}R+\frac{1}{2\bar{f}^{2}}(\partial_{\mu}\tilde{\phi})^{2}+2\frac{\tilde{\phi}}{\bar{f}}\Lambda-2\Lambda+O(1/\bar{f}^{2})\right.$
$\displaystyle\left.\qquad\qquad\qquad-\frac{\tilde{\phi}}{\bar{f}M_{P}^{2}}(\alpha
E+\alpha^{\prime}C^{2})+O(1/(\bar{f}^{2}M_{P}^{2})\right),$ (7.9.7)

where we have redefined $\tilde{\phi}\to\tilde{\phi}/\sqrt{3}$ and
$\bar{f}=\sqrt{3}f$.

At large $\bar{f}$, with $f\ll M_{P}$, the dilaton field can be expressed in
terms of the fiducial metric using the nonlocal relation

$\tilde{\phi}\sim\frac{1}{\,\raise 0.5pt\hbox{$\vbox{\hrule\hbox{\vrule
height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,}\left(-\bar{f}({R}+{\Lambda})-\frac{\bar{f}}{M_{P}^{2}}(b^{\prime}E+bC^{2})\right),$
(7.9.8)

where the asymptotic expression of the field can be removed on-shell, via an
auxiliary nonlocal interaction containing suppressed - by $\bar{f}/M_{P}^{2}$
\- nonlocal couplings to the anomaly ($\frac{1}{\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,}(\alpha E+\alpha^{\prime}C^{2})$) and to the curvature
($\frac{1}{\,\raise 0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule} \hrule}$}\,}R$). If the conformal breaking scale $f$ grows
towards $M_{P}$, the leading behaviour of the dilaton contribution is
described by $\frac{1}{\,\raise 0.5pt\hbox{$\vbox{\hrule\hbox{\vrule
height=6.50403pt\kern 6.50403pt\vrule} \hrule}$}\,}R$. Similar couplings are
found if one looks for a solution of the anomaly equation in four spacetime
dimensions [106], although in this case, such behaviour is found simply as a
result of the .
We have seen that the structure of the effective action depends on the way we
perform the $d\to 4$ limit and on the choice of the metric that is used for
evaluating the finite contributions $V^{\prime}_{E}$ and $V^{\prime}_{C^{2}}$
or $\hat{V}^{\prime}_{E}$ and $\hat{V}^{\prime}_{C^{2}}$, resulting from the
renormalization procedure.
Eq. (7.9.8) indicates that after the spontaneous breaking of the conformal
symmetry, the coupling of the dilaton to the anomaly is suppressed, compared
to its coupling to the curvature or to the cosmological constant. One may
observe that the enhanced coupling of such field to $R$, is a result of the
Weyl gauging of the EH action, and it is not related to the inclusion of the
quantum corrections. It shows up as a purely classical effect, which is
expected to be present in any dilaton gravity model, given the generality of
the procedure.

#####  7.9.1 Comments on (7.9.8)

It is important to pause for a moment and address an issue which is always
present in these actions, concerning their local versus their nonlocal
structure. It is expected that an anomaly action is nonlocal, since there are
no local operators that allow to remove any anomaly in quantum field theory.
For this reason, the terms $V^{\prime}_{E}$ and $V^{\prime}_{C^{2}}$ or
$\hat{V}^{\prime}_{E}$ and $\hat{V}^{\prime}_{C^{2}}$ which result from the
regularization procedure are local only apparently.
If we were able to solve for $\phi$ in terms of the background fiducial
metric, then we would again derive a nonlocal action, expressed only in terms
of the fiducial metric $\bar{g}$. One can recover a nonlocal action only by
removing the dilaton $\tilde{\phi}$ from these expressions, generalizing the
procedure that we have summarised in Eq. (7.9.8). Anomaly actions are all
nonlocal, since the equation of motion of the dilaton field is sourced by the
anomaly.
Another important point concerns the role of the regularization-dependent
$\,\raise 0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule} \hrule}$}\,R$ term in the anomaly induced action, associated
with the variation of local $R^{2}$ operators. As we are going to discuss in
Section 7.11, such terms may be introduced by a local renormalization of the
topological density $\sqrt{g}E$, in order to generate a quadratic action in
$\phi$. This procedure, obviously, remains valid for the classical EGB actions
discussed in the recent literature, as illustrated in the Introduction.
It is clear that our redefinitions hide some important steps that are beyond
our control, as $f$ was not present in the original action and it has been
introduced by hand through a dimensional redefinition of $\phi$. It is also
obvious that a nonzero vev for the dilaton field can only be induced by a
breaking of the Weyl simmetry (7.7.1). In both cases these actions are missing
a dynamical mechanism which may justify the origin of $f$.
In a classical EGB analysis, which does not contemplate the inclusion of any
quantum correction, there is no obvious way to infer how such a scale should
emerge. As we have emphasized, a classical renormalization of a singular
action is equivalent to a phenomenological parameterization of an effect
which, in general, is controlled by some extra potential, but does not
correspond to its fundamental and complete description. Similar features have
been discussed in Stuckelberg models [116] in the past in anomaly actions
which emerge in the field theory limit of the Green-Schwarz mechanism of
string theory. As we have already mentioned, in chiral anomaly models the
solution of this conundrum forces us to advocate an extra potential whose
origin, in that case, is attributed to the instanton sector of nonabelian
gauge theories. Several analysis of this phenomenon have been performed in the
past about this specific point. Here, essentially, we are after an analogous,
but far more complex, gravitational one. It is well known, however, that
critical phenomena in classical gravity [170] are possible, and it is not
excluded that the breaking of the $\sigma$-symmetry may be associated with the
presence of classical unstable modes. We also remark that a parametric
dependence in classical EGB models may be generated by the inclusion of finite
boundary effects. The emergence of $f$ as a fundamental breaking scale in a
scaleless GB theory could be communicated by the Einstein action, through the
Planck scale, which is part of the EGB theory.

####  7.10 Conformal constraints

The contributions to the gravitational backreaction are due both to
$\mathcal{S}_{f}$, the Weyl-invariant part of the renormalized action
$\mathcal{S}_{R}$, and to $\mathcal{S}_{A}$, as defined by (7.5.8). The
computations of $\mathcal{S}_{B}$, and of its finite part $\mathcal{S}_{f}$,
can be performed explicitly in particular backgrounds, using specific free
field theory realizations.
Conformal Ward identities ( CWIs) constrain $\mathcal{S}_{R}$ in a significant
way. Starting from the one-point function, under a Weyl rescaling (7.3.2) and
diffeomorhisms of the metric $x^{\mu}\to x^{\mu}+\epsilon(x)$

$\delta_{\epsilon}g_{\mu\nu}=-\nabla_{\mu}\epsilon_{\nu}-\nabla_{\nu}\epsilon_{\mu},$
(7.10.1)

the responses of $\mathcal{S}_{B}$ are assocated with the stress energy tensor

$\frac{\delta\mathcal{S}_{B}}{\delta\sigma(x)}=\sqrt{g}\,g_{\mu\nu}\,\langle
T^{\mu\nu}\rangle_{B},\qquad$ (7.10.2)

and its invariances are summarised by the relations

$\delta_{\sigma}\mathcal{S}=0\qquad\delta_{\epsilon}\mathcal{S}=0,$ (7.10.3)

imposing trace and conservation conditions of the quantum averages of
$T^{\mu\nu}$

$\langle T^{\mu}_{\mu}\rangle_{B}=0\qquad\qquad\nabla_{\mu}\langle
T^{\mu\nu}\rangle_{B}=0.$ (7.10.4)

The conservation equation, in terms of the action takes the form

$\partial_{\nu}\left(\frac{\delta\mathcal{S}_{B}(g)}{\delta
g_{\mu\nu}(x)}\right)+\Gamma^{\mu}_{\nu\lambda}\left(\frac{\delta\mathcal{S}_{B}(g)}{\delta
g_{\lambda\nu}(x)}\right)=0,$ (7.10.5)

where $\Gamma^{\mu}_{\lambda\nu}$ is the Christoffel connection for the
general background metric $g_{\mu\nu}$.
This equation is not affected by the process of renormalization and holds also
for the renormalized action $S_{R}$

$\partial_{\nu}\left(\frac{\delta\mathcal{S}_{R}(g)}{\delta
g_{\mu\nu}(x)}\right)+\Gamma^{\mu}_{\nu\lambda}\left(\frac{\delta\mathcal{S}_{R}(g)}{\delta
g_{\lambda\nu}(x)}\right)=0.$ (7.10.6)

#####  7.10.1 Conformal constraints in a flat background

In a flat background both equations become ordinary conservation equation for
the quantum averaged stress energy tensor

$\partial_{\nu}\left(\frac{\delta\mathcal{S}_{R/B}(g)}{\delta
g_{\mu\nu}(x)}\right)=0.$ (7.10.7)

These are accompanied by special CWIs

$\displaystyle\left(2d\,x_{1}^{\kappa}+2x_{1}^{\kappa}\,x^{\mu}_{1}\frac{\partial}{\partial
x_{1}^{\mu}}+x_{1}^{2}\frac{\partial}{\partial x_{1\kappa}}\right)\langle
T^{\mu_{1}\nu_{1}}(x_{1})\rangle$
$\displaystyle\quad+2\bigg{(}x_{1\lambda}\,\delta^{\mu_{1}\kappa}-x_{1}^{\mu_{1}}\delta^{\kappa}_{\lambda}\bigg{)}\langle
T^{\lambda\nu_{1}}(x_{1})\rangle_{B}+2\bigg{(}x_{1\lambda}\,\delta^{\nu_{1}\kappa}-x_{1}^{\nu_{1}}\delta^{\kappa}_{\lambda}\bigg{)}\langle
T^{\mu_{1}\lambda\rangle(x_{1})}_{B}=0,$ (7.10.8)

which becomes anomalous after renormalization in which
$\mathcal{S}_{B}\to\mathcal{S}_{R}$. The two equations above, obviously, are
regularization dependent, and need to be considered with care. In dimensional
regularization, for instance, the computation of the 1-point function reduces
to a combination of masselss tadpoles and vanishes identically before that any
trace is taken. This is consistent from the point of view of the trace anomaly
relation, since the anomaly functional $\mathcal{A}(x)$, that we will define
below, vanishes in flat space. We need at least three derivatives in order for
the anomaly contribution to appear in the special CWIs and trace WIs, which
means that the first correlation function affected by a trace anomaly in their
corresponding WIs is the $TTT$. The dilatation WI, on the other end, in flat
space shows up in the $TT$. Notice that the trace anomaly of the $TT$ can be
removed by a finite renormalization of the counterterms $\sim\int
d^{d}x\sqrt{g}R^{2}$.

Trace and CWI’s can be derived from the equations above by functional
differentiations of $\mathcal{S}_{R}(g)$ with respect to the metric
background. The anomalous trace WIs can be derived by allowing for an anomaly
contribution on the rhs of the $\sigma$ variation in (7.10.3)

$\delta_{\sigma}\mathcal{S}_{R}=\int
d^{4}x\sqrt{g}\,\sigma\,\bar{\mathcal{A}}(x),\qquad\langle
T^{\mu}_{\mu}\rangle_{R}=\bar{\mathcal{A}}(x),$ (7.10.9)

which violates Weyl invariance. Here, $\sqrt{g}\,\bar{\mathcal{A}}(x)$ is the
anomaly functional with

$\mathcal{A}(x)=\sqrt{-g(x)}\bigg{[}b\,C^{2}(x)+b^{\prime}E(x)\bigg{]}.$
(7.10.10)

In a generic background $g$, following the renormalization procedure of
(7.5.5) (LABEL:ren), the 1-point function is decomposed as

$\langle
T^{\mu\nu}\rangle_{R}=\frac{2}{\sqrt{g}}\frac{\delta\mathcal{S}_{R}}{\delta
g_{\mu\nu}}=\langle
T^{\mu\nu}\rangle_{A}+\langle\overline{T}^{\mu\nu}\rangle_{f},$ (7.10.11)

with

$g^{\mu\nu}\frac{\delta\mathcal{S}_{R}}{\delta
g^{\mu\nu}}=g^{\mu\nu}\frac{\delta\mathcal{S}_{A}}{\delta
g^{\mu\nu}}\equiv\frac{\sqrt{g}}{2}g_{\mu\nu}\langle T^{\mu\nu}\rangle_{A}$
(7.10.12)

being the trace anomaly equation, and
$\langle\overline{T}^{\mu\nu}\rangle_{f}$ is the traceless stress energy
tensor derived from the Weyl-invariant functional $\mathcal{S}_{f}$.
Constraints on the entire expansions of $\mathcal{S}_{R}$ or
$\mathcal{S}_{B}$, in the form of canonical WI, can be derived from the
functional differentiation of 1-point functions in a rather direct way. For
instance, 4-point functions are related to 3-point correlators by the
hierarchical equations

$\displaystyle\partial_{\nu_{1}}\langle
T^{\mu_{1}\nu_{1}}(x_{1})T^{\mu_{2}\nu_{2}}(x_{2})T^{\mu_{3}\nu_{3}}(x_{3})T^{\mu_{4}\nu_{4}}(x_{4})\rangle_{R}=$
$\displaystyle=$
$\displaystyle-\left[2\left(\frac{\delta\Gamma^{\mu_{1}}_{\lambda\nu_{1}}(x_{1})}{\delta
g_{\mu_{2}\nu_{2}}(x_{2})}\right)_{g=\delta}\langle
T^{\lambda\nu_{1}}(x_{1})T^{\mu_{3}\nu_{3}}(x_{3})T^{\mu_{4}\nu_{4}}(x_{4})\rangle_{R}+(23)+(24)\right]$
$\displaystyle-\left[4\left(\frac{\delta^{2}\Gamma^{\mu_{1}}_{\lambda\nu_{1}}(x_{1})}{\delta
g_{\mu_{2}\nu_{2}}(x_{2})\delta
g_{\mu_{3}\nu_{3}}(x_{3})}\right)_{g=\delta}\langle
T^{\lambda\nu_{1}}(x_{1})T^{\mu_{4}\nu_{4}}(x_{4})\rangle_{R}+(24)+(34)\right],$
(7.10.13)

where the correlators can either be the bare ones or the renormalized ones,
the latter containing also the contributions coming from the functional
differentiations of $\mathcal{S}_{A}$, the anomaly part. In general, such
equations can be expanded in any background, but they simplify significantly
in a flat one, since the derivative of the connections take the form

$\displaystyle\left(\frac{\delta\Gamma^{\mu_{1}}_{\lambda\nu_{1}}(x_{1})}{\delta
g_{\mu_{i}\nu_{i}}(x_{i})}\right)_{g=\delta}$
$\displaystyle=\frac{1}{2}\left(\delta^{\mu_{1}(\mu_{i}}\delta^{\nu_{i})}_{\nu_{1}}\,\partial_{\lambda}\delta_{x_{1}x_{i}}+\delta^{\mu_{1}(\mu_{i}}\delta^{\nu_{i})}_{\lambda}\,\partial_{\nu_{1}}\delta_{x_{1}x_{i}}-\delta^{(\mu_{i}}_{\lambda}\delta^{\nu_{i})}_{\nu_{1}}\,\partial^{\mu_{1}}\delta_{x_{1}x_{i}}\right)$
(7.10.14)
$\displaystyle\left(\frac{\delta^{2}\Gamma^{\mu_{1}}_{\lambda\nu_{1}}(x_{1})}{\delta
g_{\mu_{i}\nu_{i}}(x_{i})\delta g_{\mu_{j}\nu_{j}}(x_{j})}\right)_{g=\delta}$
$\displaystyle=$
$\displaystyle=-\frac{\delta_{x_{1}x_{i}}}{2}\delta^{\mu_{1}(\mu_{i}}\delta^{\nu_{i})\epsilon}\left(\delta^{(\mu_{j}}_{\epsilon}\delta^{\nu_{j})}_{\nu_{1}}\,\partial_{\lambda}\delta_{x_{1}x_{j}}+\delta^{(\mu_{j}}_{\epsilon}\delta^{\nu_{j})}_{\lambda}\,\partial_{\nu_{1}}\delta_{x_{1}x_{j}}-\delta^{(\mu_{j}}_{\lambda}\delta^{\nu_{j})}_{\nu_{1}}\,\partial_{\epsilon}\delta_{x_{1}x_{j}}\right)+(ij),$
(7.10.15)

and in particular in momentum space, where they become polynomial in the
external momenta.
The inclusion of the backreaction in flat space is rather straightforward and
can be addressed using the formalism developed in [171]. The two contributions
indicated as $V^{\prime}_{E}$ and ${V^{\prime}}_{C^{2}}$ are free of
ambiguities in a flat backgound. By differentiation we get

$\frac{1}{\epsilon}(V_{E}(d))^{\mu_{1}\nu_{1}\ldots\mu_{n}\nu_{n}}=\frac{1}{\epsilon}(V_{E}(4))^{\mu_{1}\nu_{1}\ldots\mu_{n}\nu_{n}}+(V^{\prime}_{E}(4))^{\mu_{1}\nu_{1}\ldots\mu_{n}\nu_{n}}$
(7.10.16)

and collect the $O(\epsilon^{0})$ terms that automatically appear in the
expansion, once we perform the $g_{\mu\nu}\to\delta_{\mu\nu}$ limit. The
approach does not need an explicit evaluation of $V_{E}(d)$, on the countrary
of what required in the case of a general background metric
$\bar{g}_{\mu\nu}$. The cancellations of the evanescent $0/0$ contributions
are automatic, and can be checked in momentum space by a direct evaluation of
$(V_{E}(4))^{\mu_{1}\nu_{1}\ldots\mu_{n}\nu_{n}}$. In this respect, the
specification of a metric which separates the four-dimensional part from the
extra-dimensional one will not be essential.
Terms of order $O(\epsilon^{0})$ are generated only by the trace of the
$d$-dimensional metric, using the condition $\delta^{\mu}_{\mu}=\epsilon+4$,
in the flat limit. We can also readily move to momentum space by defining

$\displaystyle
V_{E}^{\mu_{1}\nu_{1}\dots\mu_{n}\nu_{n}}(p_{1},\dots,\bar{p}_{n})$
$\displaystyle\equiv
2^{n}\big{[}\sqrt{-g}\,E\big{]}^{\mu_{1}\nu_{1}\dots\mu_{n}\nu_{n}}(p_{1},\dots,\bar{p}_{n})$
$\displaystyle=2^{n}\int\,d^{d}x_{1}\,\dots\,d^{d}x_{n}\,d^{d}x\,\bigg{(}\mbox{\small$\displaystyle\frac{\delta^{n}(\sqrt{-g}E)(x)}{\delta
g_{\mu_{1}\nu_{1}}(x_{1})\dots\delta
g_{\mu_{n}\nu_{n}}(x_{n})}$}\bigg{)}_{g=\delta}\,e^{-i(p_{1}\,x_{1}+\dots+p_{n}x_{n})}.$
(7.10.17)

Then, the relation

$\frac{\delta}{\delta g_{\mu_{1}\nu_{1}}}\ldots\frac{\delta}{\delta
g_{\mu_{n}\nu_{n}}}\frac{\delta}{\delta\sigma}V_{E}=\epsilon(\sqrt{g}E)^{\mu_{1}\nu_{1}\ldots\mu_{n}\nu_{n}},$
(7.10.18)

in momentum space turns into the trace relation

$\displaystyle\delta_{\mu_{1}\nu_{1}}\,V_{E}^{\mu_{1}\nu_{1}\dots\mu_{n}\nu_{n}}(p_{1},\dots,p_{n})=2^{n-1}(d-4)\,\left[\sqrt{-g}E\right]^{\mu_{2}\nu_{2}\dots\mu_{n}\nu_{n}}(p_{2},\dots,p_{n})$
$\displaystyle\qquad\qquad-2\bigg{[}V_{E}^{\mu_{2}\nu_{2}\dots\mu_{n}\nu_{n}}(p_{1}+p_{2},p_{3},\dots,p_{n})+V_{E}^{\mu_{2}\nu_{2}\dots\mu_{n}\nu_{n}}(p_{2},p_{1}+p_{3},\dots,p_{n})$
$\displaystyle\qquad\qquad\qquad+\dots+V_{E}^{\mu_{2}\nu_{2}\dots\mu_{n}\nu_{n}}(p_{2},p_{3},\dots,p_{1}+p_{n})\bigg{]}.$
(7.10.19)

Of the two terms generated in the expression above, the second, proportional
to $n-1$-point functions is obtained by the action of the functional
derivative on the metric.
A simple example illustrating the result above is given by the following
identity

$\frac{\delta}{\delta
g_{\mu_{2}\nu_{2}}(x_{2})}\left(2g_{\mu_{1}\nu_{1}}(x_{1})\frac{\delta}{\delta
g_{\mu_{1}\nu_{1}}(x_{1})}V_{E}(x_{1})\right)=2\left(\sqrt{g(x_{1})}V_{E}\right)^{\mu_{2}\nu_{2}}\delta(x_{2}-x_{1})+2g_{\mu_{1}\nu_{1}}(x_{1})\left(\sqrt{g(x_{1})}V_{E}\right)^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}},$
(7.10.20)

where the last term on the rhs of this expression is further simplified by
using the trace WIs. and expressed in terms of the one-point function.

The correlation functions extracted by the renormalized action can be
expressed as the sum of a finite $(f)$ correlator and of an anomaly term
(anomaly) in the form

$\displaystyle\langle T^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}\ldots
T^{\mu_{n}\nu_{n}}\rangle_{Ren}=$ $\displaystyle\bigg{[}\langle
T^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}\ldots
T^{\mu_{n}\nu_{n}}\rangle_{bare}+\langle
T^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}\ldots
T^{\mu_{n}\nu_{n}}\rangle_{count}\bigg{]}_{d\to 4}=$ $\displaystyle=\langle
T^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}\ldots
T^{\mu_{n}\nu_{n}}\rangle^{(d=4)}_{f}+\langle
T^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}\ldots
T^{\mu_{n}\nu_{n}}\rangle^{(d=4)}_{anomaly}$ (7.10.21)

This equation describes (7.5.5) the correlators itroduced by the functional
expansion of $\mathcal{S}_{R}$, separated in terms of a Weyl-invariant
$\mathcal{S}_{f}$ and the anomaly part $\mathcal{S}_{A}$.
Approaching the flat limit from a specific parameterization of a background
metric, only requires some rearrangements of the functional expansion (7.2.6).
Indeed, if we parameterize the metric in terms of a conformal factor and of a
fiducial background $\bar{g}_{\mu\nu}$ in $d$ dimensions, the flat limit is
equivalently obtained by setting $\phi(x)\to 0$ and
$\bar{g}_{\mu\nu}\to\delta_{\mu\nu}$.
Notice that the conformal factor, in this case, multiplies all the components
of the metric, as described by Eq. (7.7.1). A dedicated comment deserves the
case in which the extra dimensional space is characterised by a dimensionful
constant. For instance, this could be the constant curvature of the extra
space, as in Einstein spaces. In this case the inclusion of a dilaton
component in the internal metric is necessary in order to preserve the scale
invariance of the effective action in $d$ dimensions. We recall that scale
invariance is preserved, in a theory which contain dimensionful constants,
only if such constants are promoted to $x$-dependent fields.

####  7.11 Moving towards the UV: The reconstruction at $d=4$ for GB

The reconstruction of the anomaly action in $d=4$ follows the standard
procedure introduced long ago by Riegert, that we will review and extend to
the GB case, in order to underline the difference between the various possible
effective actions that may follow. Therefore, the regularization of the GB
term can indeed generate regulated GB actions which can either take a local or
a nonlocal form, depending on the way the conformal factor is treated in the
regularization procedure. Both $V^{\prime}_{E}$ and $V^{\prime}_{C^{2}}$ are
local expressions of the fiducial metric $\bar{g}_{\mu\nu}$ and of the field
$\phi$. As already pointed out, their nonlocal structure will be apparent only
if we are able to remove $\phi$, by re-expressing it in terms of the original
metric $g_{\mu\nu}$, and this is not always possible. The case discussed by
Riegert is one in which the conformal scaling relation (7.7.15) is linear in
$\phi$, and the dilaton can be removed by an integration procedure. This is a
consequence of the fact that the rescaling is performed at $d=4$. Indeed, in
this case the rescaling gives

$\sqrt{g}\,\Big{(}E-\frac{2}{3}\,\,\raise 0.5pt\hbox{$\vbox{\hrule\hbox{\vrule
height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R\Big{)}\,=\,\sqrt{\bar{g}}\,\Big{(}{\bar{E}}-\frac{2}{3}\,{\bar{\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,}}{\bar{R}}+4{\bar{\Delta_{4}}}\phi\Big{)}\,,$ (7.11.1)

where $\Delta_{4}$ is the fourth order self-adjoint operator, which is
conformal invariant when it acts on a scalar function of vanishing scaling
dimensions

$\Delta_{4}=\nabla^{2}+2\,R^{\mu\nu}\nabla_{\mu}\nabla_{\nu}-\frac{2}{3}\,R{\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,}+\frac{1}{3}\,(\nabla^{\mu}R)\nabla_{\mu}\,.$ (7.11.2)

and satisfies the relation

$\sqrt{-g}\,\mathcal{D}_{4}\chi=\sqrt{-\bar{g}}\,\bar{\mathcal{D}}_{4}\chi,$
(7.11.3)

if $\chi$ is invariant (i.e. has scaling equal to zero) under a Weyl
transformation.
Eq. (7.11.1) is crucial for the elimination of $\phi$ from the effective
action.

Riegert’s approach, obviously, can be modified by turning to $d$ dimensions,
and the ambiguity in the derivation of the effective action, that may proceed
either in d=4 or for $d=4+\epsilon$, is an intrinsic part of the
regularization. There are variants of $E$ that can be introduced in order to
satisfy (7.7.18) and allow to eliminate $\phi$, quite closely to (7.11.1), as
we are going to discuss in the sections below. We can introduce for instance
the modified and extended version of $E$ in the form

$E^{ext}\equiv E+\frac{\epsilon}{2(d-1)^{2}}R^{2},$ (7.11.4)

which is useful in order to investigate the contribution of the $V_{E}$
counterterm - and of its variants - to the effective action. Notice that the
two extra terms that appear on the rhs of (7.11.4), correspond to a boundary
contribution $(\,\raise 0.5pt\hbox{$\vbox{\hrule\hbox{\vrule
height=6.50403pt\kern 6.50403pt\vrule} \hrule}$}\,R)$, and to an $O(\epsilon)$
modification $(\sim\epsilon R^{2})$ that vanish if we ensure either trivial
boundary conditions on the metric, or we perform the $d\to 4$ limit. $E^{ext}$
plays a role in the identification of a form of the effective action which is
quite close to Riegert’s action.
The scaling relation (7.11.1) is rather unusual, in the sense that its metric
variation links boundary terms in the two metrics $g_{\mu\nu}$ and
$\bar{g}_{\mu\nu}$. One can show that in $d=4$, under a metric variation
$\delta$

$\frac{1}{4}\delta(\sqrt{g}E)=\sqrt{g}\,\nabla_{\sigma}\delta
X^{\sigma},\qquad\delta
X^{\sigma}=\varepsilon^{\mu\nu\alpha\beta}\varepsilon^{\sigma\lambda\gamma\tau}\delta\Gamma^{\eta}_{\nu\lambda}g_{\mu\eta}R_{\alpha\beta\gamma\tau},\qquad\varepsilon^{\mu\nu\alpha\beta}=\frac{\epsilon^{\mu\nu\alpha\beta}}{\sqrt{g}}$
(7.11.5)

$\delta(\sqrt{g}\,\raise 0.5pt\hbox{$\vbox{\hrule\hbox{\vrule
height=6.50403pt\kern 6.50403pt\vrule} \hrule}$}\,R)=\sqrt{g}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\delta\zeta,\qquad\delta\zeta=-R^{\mu\nu}\delta
g_{\mu\nu}+\nabla^{\mu}\nabla^{\nu}\delta g_{\mu\nu}-\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,(g^{\mu\nu}\delta g_{\mu\nu}).$ (7.11.6)

These relations follow after some integration by parts, having observed that
the conformal factor varies like a scalar under the Weyl rescalings in two
different frames $x$ and $x^{\prime}$. This results from the fact that a
fiducial metric transforms as an ordinary tensor in the two frames, hence

$g_{\mu\nu}(x)=\bar{g}_{\mu\nu}(x)e^{2\phi(x)}\qquad
g^{\prime}_{\mu\nu}(x^{\prime})=\bar{g}^{\prime}_{\mu\nu}(x^{\prime})e^{2\phi^{\prime}(x^{\prime})}\qquad\phi^{\prime}(x^{\prime})=\phi(x).$
(7.11.7)

It is convenient to define

$\delta\Sigma^{\sigma}=\sqrt{g}g^{\sigma\beta}\partial_{\beta}\delta\zeta\qquad$
(7.11.8)

and vary both sides of (7.11.1) to obtain, using

$\delta_{\phi}\left(\sqrt{g}\,\raise 0.5pt\hbox{$\vbox{\hrule\hbox{\vrule
height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R\right)=\epsilon\delta\phi\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R+(d-6)\sqrt{g}\nabla^{\lambda}R\nabla_{\lambda}\delta\phi-2\sqrt{g}R\nabla^{2}\delta\phi-2(d-1)\sqrt{g}\nabla^{4}\delta\phi$
(7.11.9)

the scaling relation at $d=4$

$\displaystyle\delta_{\phi}\left(\frac{1}{4}\sqrt{-g}\left(E-\frac{2}{3}\square\,R\right)\right)=\sqrt{-g}\mathcal{D}_{4}\delta\phi,$
(7.11.10)

which simplifies in the form

$\partial_{\sigma}\left(\delta_{\phi}X^{\sigma}-\frac{1}{6}\delta_{\phi}\Sigma^{\sigma}\right)=\sqrt{-g}\mathcal{D}_{4}\delta\phi,$
(7.11.11)

if we use (7.11.5) to relate it to a boundary contribution. Here we have used
the general variation of $\delta X^{\sigma}$ specialised to changes in the
dilaton field ($\delta_{\phi}$)

$\delta_{\phi}X^{\sigma}=\varepsilon^{\mu\nu\alpha\beta}\varepsilon^{\sigma\lambda\gamma\tau}\delta_{\phi}\Gamma^{\eta}_{\nu\lambda}g_{\mu\eta}R_{\alpha\beta\gamma\tau}.$
(7.11.12)

We have defined

$\delta_{\phi}\Gamma_{\mu\nu}^{\lambda}=\delta_{\mu}^{\lambda}\nabla_{\nu}\delta\phi+\delta^{\lambda}_{\nu}\nabla_{\mu}\delta\phi-e^{-2\phi}g_{\mu\nu}\nabla^{\lambda}\delta\phi,$
(7.11.13)

derived from (7.6.1), using
$\bar{\nabla}_{\mu}\delta\phi=\nabla_{\mu}\delta\phi$ on scalars, while and
analogous variation $\delta\Sigma^{\sigma}$ in (7.11.8) is specialised in the
form

$\delta_{\phi}\Sigma^{\sigma}=\sqrt{g}g^{\sigma\beta}\partial_{\beta}\delta_{\phi}\zeta,$
(7.11.14)

where

$\delta_{\phi}\zeta=-R^{\mu\nu}\delta_{\phi}g_{\mu\nu}+\nabla^{\mu}\nabla^{\nu}\delta_{\phi}g_{\mu\nu}-\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,(g^{\mu\nu}\delta_{\phi}g_{\mu\nu}),\qquad\textrm{with}\qquad\delta_{\phi}g_{\mu\nu}=2g_{\mu\nu}\delta\phi.$
(7.11.15)

Eq. (7.11.11), integrated over spacetime, gives consistently

$\int
d^{4}x\partial_{\sigma}\left(\delta_{\phi}X^{\sigma}-\frac{1}{6}\delta_{\phi}\Sigma^{\sigma}\right)=0,$
(7.11.16)

if we assume asymptotic flatness, and therefore

$\int d^{4}x\sqrt{-g}\mathcal{D}_{4}\delta\phi=0,$ (7.11.17)

that follows from the self-adjointness of $\Delta_{4}$

$\int d^{4}x\sqrt{-g}\,\psi(\mathcal{D}_{4}\xi)=\int
d^{4}x\sqrt{-g}\,(\mathcal{D}_{4}\psi)\xi,$ (7.11.18)

where $\xi$ and $\psi$ are scalar fields of zero scaling dimensions.

The scaling relation $\eqref{119}$ is strictly valid at $d=4$ and clearly is
much simplified compared to (7.7.15), which is valid in $d$ dimensions
Clearly, Eq. (7.11.1) is not directly related to a DR procedure, but simply
takes the expression of the anomaly as a given fundamental 4-dimensional
result and integrates out the dilaton field from the scaling relation, to
derive the nonlocal form of the action.
We briefly review this point. It is convenient to redefine (7.11.1) in the
form

$J(x)=\bar{J}(x)+4\sqrt{g}\Delta_{4}\phi(x),\qquad\bar{J}(x)\equiv\sqrt{\bar{g}}\left(\bar{E}-\frac{2}{3}\bar{\,}\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\bar{R}\right),\qquad J(x)\equiv\sqrt{g}\left(E-\frac{2}{3}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R\right)$ (7.11.19)
$(\sqrt{-g}\,\mathcal{D}_{4})_{x}D_{4}(x,y)=\delta^{4}(x,y).$ (7.11.20)

We invert (7.11.1) using the properties of the operator $\Delta_{4}$ to find
the explicit form of the function $\phi(x)$, obtaining

$\phi(x)=\frac{1}{4}\int d^{4}y\,D_{4}(x,y)(J(y)-\bar{J}(y)).$ (7.11.21)

This sets $\phi$ on-shell. The derivation of $\mathcal{S}_{WZ}$ requires the
solution of the equation

$\frac{\delta\mathcal{S}_{WZ}^{(GB)}}{\delta\phi}=J,$ (7.11.22)

clearly identified in the form

$\mathcal{S}_{WZ}=\int
d^{4}x\sqrt{\bar{g}}\left(\bar{J}\phi+2\phi\Delta_{4}\phi\right).$ (7.11.23)

At this stage it is just matter of inserting the on-shell expression of $\phi$
(7.11.21) into this equation to obtain the WZ action, in the form

$\mathcal{S}_{WZ}=\mathcal{S}_{anom}(g)-\mathcal{S}_{anom}(\bar{g}),$
(7.11.24)

with

$\mathcal{S}_{anom}(g)=\frac{1}{8}\int d^{4}xd^{4}yJ(x)D_{4}(x,y)J(y),$
(7.11.25)

and a similar expression for $\mathcal{S}_{anom}(\bar{g})$. Using the explicit
expression of $\phi$, and including the contributon from the rescaled $C^{2}$
term, we finally find the nonlocal and covariant anomaly effective action as

$\mathcal{S}_{\rm
anom}(g)=\frac{1}{8}\\!\int\\!d^{4}x\sqrt{-g_{x}}\,\left(E-\frac{2}{3}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R\right)_{\\!x}\int\\!d^{4}x^{\prime}\sqrt{-g_{x^{\prime}}}\,D_{4}(x,x^{\prime})\left[\frac{b^{\prime}}{2}\,\left(E-\frac{2}{3}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R\right)+b\,C^{2}\right]_{x^{\prime}}.$ (7.11.26)

####  7.12 Modified Euler density and the nonlocal GB action

Notice that if the rescaling is performed at $d=4$, and the extra field $\phi$
is reabsorbed into the definition of $g_{\mu\nu}$, giving a nonlocal action,
then no scale of expansion is present in $\mathcal{S}_{A}(4)$. If we move away
from 4 dimensions, and this is clearly allowed in DR, then it is obvious that
extra components of the metric will be present in the expressions of
$V^{\prime}_{E}$ and $V^{\prime}_{C^{2}}$, and the computation of the
effective action will be affected by the choice of the fiducial metric over
which we integrate in $d$ dimensions.
In the context of DR and, in particular, in the analysis of the effective
actions, it is clear that variants of the topological terms are possible.
The functional differential form $V_{E}$, constructed out of $E_{4}\equiv E$
is not the only possible one. It is clearly exact since

$\delta_{\phi}(\sqrt{g}E_{4})=\epsilon\sqrt{g}E_{4}\delta\phi,$ (7.12.1)

that can be verified directly by taking the trace of $V_{E}^{\mu\nu}$. Another
way is to introduce separately the variations

$\delta_{\phi}\left(\sqrt{g}R^{2}\right)=\delta\phi\left(\epsilon\sqrt{g}R^{2}-4(d-1)\sqrt{g}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R\right),$ (7.12.2)

giving under integration

$\delta_{\phi}\int
d^{d}x\sqrt{g}R^{2}=\epsilon\sqrt{g}R^{2}-4(d-1)\sqrt{g}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R.$ (7.12.3)

Similarly

$\delta_{\phi}\left(\sqrt{g}(R_{\mu\nu\alpha\beta})^{2}\right)=\delta\phi\left(\epsilon\sqrt{g}(R_{\mu\nu\alpha\beta})^{2}-8\sqrt{g}\nabla_{\mu}\nabla_{\nu}R^{\mu\nu}\right),$
(7.12.4)

$\delta_{\phi}\left(\sqrt{g}(R_{\mu\nu})^{2}\right)=\epsilon\delta\phi\sqrt{g}(R_{\mu\nu})^{2}-2\sqrt{g}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R\delta\phi-2(d-2)\sqrt{g}\nabla_{\mu}\nabla_{\nu}R^{\mu\nu}\delta\phi,$
(7.12.5)

and combine them in the definition of $E_{4}$ using also the derivative
Bianchi identity $\nabla_{\mu}R^{\mu}_{\nu}=\frac{1}{2}\nabla_{\nu}R$ to
derive

$\delta_{\phi}\int d^{d}x\sqrt{g}E_{4}=\epsilon\sqrt{g}E_{4}.$ (7.12.6)

There are no boundary terms produced in the variation. Alternatively, from
(3.2.25) we obtain the two contributions

$\frac{\delta}{\delta\phi}\int d^{d}x\sqrt{g}E_{4}=\epsilon\sqrt{g}E_{4}+\int
d^{d}x\sqrt{\bar{g}}e^{\epsilon\phi}\left((d-3)\frac{\delta}{\delta\phi}\bar{\nabla}_{\mu}\bar{J}^{\mu}+\epsilon(d-3)\frac{\delta}{\delta\phi}\bar{K}\right),$
(7.12.7)

and only the first term on the rhs of this equation contributes at
$O(\epsilon)$. Consistency with (7.12.6) induces the identities

$(d-3)\int
d^{d}x\sqrt{\bar{g}}\left(\phi(x)\bar{\nabla}_{\mu}\frac{\delta\bar{J}^{\mu}(x)}{\delta\phi(y)}+\epsilon\frac{\delta\bar{K}(x)}{\delta\phi(y)}\right)=0,$
(7.12.8)

where, we recall, $\bar{J}=\bar{J}(\bar{g},\phi)$ and
$\bar{K}=\bar{K}(\bar{g},\phi)$. Here we have used the boundary relation

$\int
d^{d}x\sqrt{\bar{g}}\frac{\delta}{\delta\phi}\bar{\nabla}_{\mu}\bar{J}^{\mu}=0.$
(7.12.9)

Therefore (7.12.7) implies that

$(d-3)\int
d^{d}x\sqrt{g}e^{\epsilon\sigma}\left(\frac{\delta}{\delta\sigma}\bar{\nabla}_{\mu}\bar{J}^{\mu}+\epsilon\frac{\delta}{\delta\sigma}\bar{K}\right)=O(\epsilon^{2}).$
(7.12.10)

Obviously, as we move away from $d=4$, modifications of such densities are
possible.
In general, we can modify such forms either by boundary terms, which play a
role only if we include a spacetime boundary and/or by additional
diffeomorphism invariant contributions of $O(\epsilon)$.
If we consider the extended expression of $E_{4}$ given by (7.11.4), in this
case we define the counterterm

$\tilde{V}_{E}=\int
d^{d}x\sqrt{g}\left(E_{4}+\epsilon\frac{R^{2}}{2(d-1)^{2}}\right).$ (7.12.11)

Using the variations above, one obtains

$\delta_{\phi}(\sqrt{g}E_{ext})=\delta\phi\epsilon\left(\sqrt{g}E_{ext}-\frac{2}{d-1}\sqrt{g}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R\right),$ (7.12.12)

giving under integration

$\delta_{\phi}\int
d^{d}x\sqrt{g}E_{ext}=\epsilon\sqrt{g}(E_{ext}-\frac{2}{d-1}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R).$ (7.12.13)

Also in this case one needs to be careful about the $d\to 4$ limit since the
metric is still $d$-dimensional and one has to proceed with an accurate
definition of the corresponding invariants. One possibility is to perform a
dimensional reduction as in (7.4.2). As discussed in the previous sections,
this introduces a cutofl $L^{\epsilon}$ that can be consistently removed as
$\epsilon\to 0$.

#####  7.12.1 The nonlocal EGB expansion

One of the standing issues concerning the consistency of the scaling approach
introduced in (7.11.1) is that it is possible to make it consistent with DR,
promoting to $d$ dimensions from $d=4$. This point can be addressed and solved
by a redefinition of $S_{WZ}$ using $E_{ext}$ with $V_{E}\to\tilde{V}_{E}$,
obtaining

$\tilde{V}_{E}=\int d^{d}x\sqrt{g}E_{ext}\,,$ (7.12.14)

$\mathcal{S}^{(WZ)}_{GB}=\frac{\alpha}{\epsilon}\left(\tilde{V}_{E}(\bar{g}_{\mu\nu}e^{2\phi},d)-\tilde{V}_{E}(\bar{g}_{\mu\nu},d\right).$
(7.12.15)

To derive its nonlocal expression, we can use the relation

$\frac{\delta}{\delta\phi}\frac{1}{\epsilon}\tilde{V}_{E}(g_{\mu\nu},d)=\sqrt{g}\left(E-\frac{2}{3}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R+\epsilon\frac{R^{2}}{2(d-1)^{2}}\right)$ (7.12.16)

in (7.12.15), to obtain

$\displaystyle\frac{\delta\mathcal{S}^{(WZ)}_{GB}}{\delta\phi}=$
$\displaystyle\alpha\sqrt{g}\left(E-\frac{2}{3}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R\right)$ $\displaystyle=$
$\displaystyle\alpha\sqrt{\bar{g}}\left(\bar{E}-\frac{2}{3}\bar{\,}\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\bar{R}+4\bar{\Delta}_{4}\phi\right),$ (7.12.17)

and henceforth

$\mathcal{S}^{(WZ)}_{GB}=\alpha\int\,d^{4}x\,\sqrt{-\bar{g}}\,\left\\{\left(\overline{E}-{2\over
3}\bar{\,\raise 0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule}
\hrule}$}\,}\overline{R}\right)\phi+2\,\phi\bar{\Delta}_{4}\phi\right\\},\,$
(7.12.18)

As before, we can solve for $\phi$, deriving the regulated GB action

$\displaystyle\mathcal{S}^{(WZ)}_{GB}$ $\displaystyle={\alpha\over 8}\int
d^{4}x\,\sqrt{-g}\,\int
d^{4}x^{\prime}\,\sqrt{-g^{\prime}}\,\left(E_{4}-{2\over 3}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R\right)_{x}\,$ $\displaystyle\qquad\times
D_{4}(x,x^{\prime})\left(E-{2\over 3}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R\right)_{x^{\prime}},\,$ (7.12.19)

that coincides with the result provided in [121] by Mazur and Mottola.

The nonlocal EGB action can be expanded, at least around a flat spacetime, in
terms of the combination of the product of scalar curvature $R$ and the
inverse of the D’Alembertian of flat space, i.e. of the variable $R\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,^{-1}$, which is dimensionless [123, 90, 171]. This results both
from perturbative computations performed around flat space and from studies of
the hierarchical structure of the CWIs. The nonlocal action ca be derived by
the standard approach. One considers the anomaly action written in the form

$\displaystyle\mathcal{S}_{\rm
anom}(g,\phi)\equiv-\mbox{\small$\displaystyle\frac{1}{2}$}\int
d^{4}x\,\sqrt{-g}\,\Big{[}(\,\raise 0.5pt\hbox{$\vbox{\hrule\hbox{\vrule
height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi)^{2}-2\big{(}R^{\mu\nu}-\tfrac{1}{3}Rg^{\mu\nu}\big{)}(\nabla_{\mu}\phi)(\nabla_{\nu}\phi)\Big{]}$
$\displaystyle\hskip
42.67912pt+\,\mbox{\small$\displaystyle\frac{1}{2}$}\,\int d^{4}x\,\sqrt{-g}\
\Big{[}\big{(}E-\tfrac{2}{3}\,\raise 0.5pt\hbox{$\vbox{\hrule\hbox{\vrule
height=6.50403pt\kern 6.50403pt\vrule} \hrule}$}\,R\big{)}\Big{]}\,\phi,$
(7.12.20)

that can be varied with respect to $\phi$, giving

$\sqrt{-g}\,\mathcal{D}_{4}\,\phi=\sqrt{-g}\left[\mbox{\small$\displaystyle\frac{E}{2}$}-\mbox{\small$\displaystyle\frac{\\!\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R\\!}{3}$}\right].$ (7.12.21)

The metric can be expanded perturbatively in the form

$\displaystyle g_{\mu\nu}=$ $\displaystyle
g_{\mu\nu}^{(0)}+g_{\mu\nu}^{(1)}+g_{\mu\nu}^{(2)}+\dots\equiv\eta_{\mu\nu}+h_{\mu\nu}+h_{\mu\nu}^{(2)}+\dots$
(7.12.22a) $\displaystyle\phi=$
$\displaystyle\phi^{(0)}+\phi^{(1)}+\phi^{(2)}+\dots$ (7.12.22b)

The expansion above should be interpreted as a collection of terms generated
by setting

$g_{\mu\nu}=\delta_{\mu\nu}+\kappa h_{\mu\nu}$ (7.12.23)

having reinstated the coupling expansion $\kappa$, with $h$ of mass-dimension
one, and collecting all the higher order terms in the functional expansion of
(LABEL:loc) of the order $h^{2}$, $h^{3}$ and so on. A similar expansion holds
for $\phi$ if we redefine
$\phi^{(1)}=\kappa\bar{\phi}^{(1)},\phi^{(2)}=\kappa^{2}\bar{\phi}^{(2)}$ and
so on. One obtains the relations

$\displaystyle\hskip 142.26378pt\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule} \hrule}}$}\,^{2}\phi^{(0)}=0$ (7.12.24a)
$\displaystyle(\sqrt{-g}\mathcal{D}_{4})^{(1)}\phi^{(0)}+\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule}
\hrule}}$}\,^{2}\phi^{(1)}=\left[\sqrt{-g}\left(\mbox{\small$\displaystyle\frac{E}{2}$}-\mbox{\small$\displaystyle\frac{\\!\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R\\!}{3}$}\right)\right]^{(1)}=-\mbox{\small$\displaystyle\frac{\\!1\\!}{3}$}\,\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule} \hrule}}$}\,R^{(1)}$ (7.12.24b)
$\displaystyle(\sqrt{-g}\mathcal{D}_{4})^{(2)}\phi^{(0)}+(\sqrt{-g}\mathcal{D}_{4})^{(1)}\phi^{(1)}+\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule}
\hrule}}$}\,^{2}\phi^{(2)}=\left[\sqrt{-g}\left(\mbox{\small$\displaystyle\frac{E}{2}$}-\mbox{\small$\displaystyle\frac{\\!\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R\\!}{3}$}\right)\right]^{(2)}$ $\displaystyle\hskip
156.49014pt=\mbox{\small$\displaystyle\frac{1}{2}$}E^{(2)}-\mbox{\small$\displaystyle\frac{1}{3}$}\,[\sqrt{-g}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R]^{(2)},$ (7.12.24c)

where $\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule} \hrule}}$ is the d’Alembert wave operator in flat Minkowski
spacetime, and we have used the fact that $E$ and $C^{2}$ are second order in
the fluctuations while the Ricci scalar $R$ starts at first order

$\phi^{(1)}=-\mbox{\small$\displaystyle\frac{\\!1\\!}{3\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule} \hrule}}$}\,}$}\,R^{(1)}$ (7.12.25)

and the solution of (7.12.24c) is

$\phi^{(2)}=\mbox{\small$\displaystyle\frac{1}{\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule}
\hrule}}$}\,^{2}}$}\left\\{(\sqrt{-g}\mathcal{D}_{4})^{(1)}\mbox{\small$\displaystyle\frac{\\!1\\!}{3\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule}
\hrule}}$}\,}$}\,R^{(1)}+\mbox{\small$\displaystyle\frac{1}{2}$}E^{(2)}-\mbox{\small$\displaystyle\frac{1}{3}$}\,[\sqrt{-g}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R]^{(2)}\right\\}.$ (7.12.26)

In this way we obtain the quadratic term

$\mathcal{S}_{\rm anom}^{(2)}=-\mbox{\small$\displaystyle\frac{1}{2}$}\,\int
d^{4}x\,\phi^{(1)}\,\raise 0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule
height=6.50403pt\kern 6.50403pt\vrule}
\hrule}}$}\,^{2}\phi^{(1)}+\mbox{\small$\displaystyle\frac{1}{2}$}\,\int
d^{4}x\,\left(-\mbox{\small$\displaystyle\frac{2}{3}$}\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule}
\hrule}}$}\,R^{(1)}\right)\phi^{(1)}=\mbox{\small$\displaystyle\frac{1}{18}$}\,\int
d^{4}x\,\left(R^{(1)}\right)^{2},$ (7.12.27)

which is purely local, since all propagators cancel. The third order terms in
the expansion of the anomaly action are

$\displaystyle\mathcal{S}_{\rm anom}^{(3)}=$
$\displaystyle-\mbox{\small$\displaystyle\frac{1}{2}$}\int
d^{4}x\,\left\\{2\,\phi^{(1)}\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule}
\hrule}}$}\,^{2}\phi^{(2)}+\phi^{(1)}\big{(}\sqrt{-g}\mathcal{D}_{4}\big{)}^{(1)}\,\phi^{\\!(1)}\right\\}$
$\displaystyle+\mbox{\small$\displaystyle\frac{1}{2}$}\int
d^{4}x\left\\{\left(-\mbox{\small$\displaystyle\frac{2}{3}$}\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule}
\hrule}}$}\,R^{(1)}\right)\phi^{(2)}+\left(E^{(2)}-\mbox{\small$\displaystyle\frac{2}{3}$}\,\sqrt{-g}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R\right)^{\\!(2)}\phi^{(1)}\right\\}.$ (7.12.28)

The remaining terms in (7.12.28) yield

$\displaystyle\mathcal{S}_{\rm
anom}^{(3)}=-\mbox{\small$\displaystyle\frac{1}{18}$}\int
d^{4}x\,\left\\{R^{(1)}\mbox{\small$\displaystyle\frac{1}{\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule}
\hrule}}$}\,}$}\big{(}\sqrt{-g}\mathcal{D}_{4}\big{)}^{\\!(1)}\,\mbox{\small$\displaystyle\frac{1}{\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule}
\hrule}}$}\,}$}R^{(1)}\right\\}-\mbox{\small$\displaystyle\frac{b^{\prime}}{6}$}\int
d^{4}x\left(E-\mbox{\small$\displaystyle\frac{2}{3}$}\,\sqrt{-g}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R\right)^{\\!(2)}\,\mbox{\small$\displaystyle\frac{1}{\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule} \hrule}}$}\,}$}R^{(1)}.$ $\displaystyle\hskip 56.9055pt.$
(7.12.29)

In the variation of $\Delta_{4}$ it is convenient first to rewrite

$\Delta_{4}=\,\raise 0.5pt\hbox{$\vbox{\hrule\hbox{\vrule
height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,^{2}+2\nabla_{\mu}(R^{\mu\nu}\nabla^{\nu})-\frac{2}{3}\nabla_{\mu}(R\nabla^{\mu}),$
(7.12.30)

having used the Leibnitz rule and the derivative Bianchi identity
$\nabla_{\mu}R^{\mu\nu}=1/2\nabla^{\nu}R$. An expansion of this operator to
first order in $\delta g_{\mu\nu}$ gives

$\big{(}\sqrt{-g}\mathcal{D}_{4}\big{)}^{\\!(1)}=\big{(}\sqrt{-g}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,^{2}\big{)}^{\\!(1)}+2\,\partial_{\mu}\left(R^{\mu\nu}-\mbox{\small$\displaystyle\frac{1}{3}$}\eta^{\mu\nu}R\right)^{\\!(1)}\partial_{\nu}.$
(7.12.31)

An integration by parts gives

$\displaystyle\mathcal{S}_{\rm
anom}^{(3)}\\!=\\!-\mbox{\small$\displaystyle\frac{1}{18}$}\\!\int\\!d^{4}x\left\\{\\!R^{(1)}\\!\mbox{\small$\displaystyle\frac{1}{\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule} \hrule}}$}\,}$}\big{(}\sqrt{-g}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,^{2}\big{)}^{\\!(1)}\mbox{\small$\displaystyle\frac{1}{\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule}
\hrule}}$}\,}$}R^{(1)}\\!\right\\}+\mbox{\small$\displaystyle\frac{1}{9}$}\\!\int\\!d^{4}x\left\\{\\!\partial_{\mu}R^{(1)}\\!\mbox{\small$\displaystyle\frac{1}{\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule}
\hrule}}$}\,}$}\\!\left(\\!R^{(1)\mu\nu}\\!-\\!\mbox{\small$\displaystyle\frac{1}{3}$}\eta^{\mu\nu}R^{(1)}\\!\right)\\!\mbox{\small$\displaystyle\frac{1}{\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule} \hrule}}$}\,}$}\partial_{\nu}R^{(1)}\\!\right\\}$
$\displaystyle-\mbox{\small$\displaystyle\frac{1}{6}$}\\!\int\\!d^{4}xE^{\\!(2)}\mbox{\small$\displaystyle\frac{1}{\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule}
\hrule}}$}\,}$}R^{(1)}+\mbox{\small$\displaystyle\frac{1}{9}$}\\!\int\\!d^{4}x\,R^{(1)}\mbox{\small$\displaystyle\frac{1}{\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule} \hrule}}$}\,}$}\left(\sqrt{-g}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\right)^{\\!(1)}R^{(1)}+\mbox{\small$\displaystyle\frac{1}{9}$}\\!\int\\!d^{4}x\,R^{\\!(2)}R^{(1)},$
(7.12.32)

which contains only single propagator poles. At this stage, using the
covariant equation $\nabla_{\mu}\sqrt{g}=0$ on the tensor density $\sqrt{g}$
we rewrite

$\left(\sqrt{g}\,\raise 0.5pt\hbox{$\vbox{\hrule\hbox{\vrule
height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,^{2}\right)^{(1)}\equiv\delta\left(\sqrt{g}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,^{2}\right)=\left(\frac{(\sqrt{g}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,)^{2}}{\sqrt{g}}\right)^{(1)}\qquad\delta\left(\frac{1}{\sqrt{g}}\right)(\sqrt{g}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,)^{2}=-\delta(\sqrt{g})\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,^{2},$ (7.12.33)

and

$\left(\sqrt{g}\,\raise 0.5pt\hbox{$\vbox{\hrule\hbox{\vrule
height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,^{2}\right)^{(1)}=-\delta(\sqrt{g})\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,^{2}+\frac{1}{\sqrt{g}}\delta(\sqrt{g}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,)(\sqrt{g}\,\raise 0.5pt\hbox{$\vbox{\hrule\hbox{\vrule
height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,)+\frac{1}{\sqrt{g}}(\sqrt{g}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,)\left(\delta(\sqrt{g}\,\raise 0.5pt\hbox{$\vbox{\hrule\hbox{\vrule
height=6.50403pt\kern 6.50403pt\vrule} \hrule}$}\,)\right),$ (7.12.34)

that in the flat limit becomes

$\left(\sqrt{g}\,\raise 0.5pt\hbox{$\vbox{\hrule\hbox{\vrule
height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,^{2}\right)^{(1)}=-(\sqrt{g})^{(1)}\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule} \hrule}}$}\,^{2}+(\sqrt{g}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,)^{(1)}\,\raise 0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule
height=6.50403pt\kern 6.50403pt\vrule} \hrule}}$}\,+\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule} \hrule}}$}\,(\sqrt{g}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,)^{(1)},$ (7.12.35)

obtaining finally

$\displaystyle\mathcal{S}_{\rm
anom}^{(3)}=\mbox{\small$\displaystyle\frac{1}{9}$}\int\\!d^{4}x\int\\!d^{4}x^{\prime}\\!\int\\!d^{4}x^{\prime\prime}\\!\left\\{\big{(}\partial_{\mu}R^{(1)})_{x}\left(\mbox{\small$\displaystyle\frac{1}{\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule}
\hrule}}$}\,}$}\right)_{\\!xx^{\prime}}\\!\left(R^{(1)\mu\nu}\\!-\\!\mbox{\small$\displaystyle\frac{1}{3}$}\eta^{\mu\nu}R^{(1)}\right)_{x^{\prime}}\\!\left(\mbox{\small$\displaystyle\frac{1}{\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule}
\hrule}}$}\,}$}\right)_{\\!x^{\prime}x^{\prime\prime}}\\!\big{(}\partial_{\nu}R^{(1)})_{x^{\prime\prime}}\right\\}$
$\displaystyle-\mbox{\small$\displaystyle\frac{1}{6}$}\\!\int\\!d^{4}x\\!\int\\!d^{4}x^{\prime}\\!\left(\,E^{\\!(2)}\right)_{\\!x}\\!\left(\mbox{\small$\displaystyle\frac{1}{\,\raise
0.5pt\hbox{$\overline{\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule}
\hrule}}$}\,}$}\right)_{\\!xx^{\prime}}\\!R^{(1)}_{x^{\prime}}+\mbox{\small$\displaystyle\frac{1}{18}$}\\!\int\\!d^{4}x\,R^{(1)}\left(2\,R^{\\!(2)}+(\sqrt{-g})^{(1)}R^{(1)}\right),$
(7.12.36)

where the last term is purely local. This action describes graviton
interactions up to trilinear fluctuations in the graviton field. (7.12.20) can
be expanded, with some extra effort, to quartic and higher orders, providing a
consistent definition of the EGB theory, now in a completely nonlocal form and
without a dilaton. The appearance of the Green’s function of a quadratic
$\,\raise 0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern
6.50403pt\vrule} \hrule}$}\,_{-1}$ operator, once the expansion is performed
around flat space has been shown in [56, 57, 59] for 3-point functions in the
$TJJ$ case and in [90] for the 3-graviton vertex TTT.

####  7.13 Conclusions

In this chapter we have presented a discussion of the structure of the
effective action and of its renormalization in DR in some detail, illustrating
the main features of the procedure that allow to identify its explicit
expression. The conformal backreaction on a metric is characterised not just
from the part of the action which is derived by the anomaly constraints, but
also by important Weyl-invariant terms which should be explicitly computed in
specific spacetimes.
In the expansion around flat space, these are given by the set of correlation
functions that have been discussed in the past in free field theory
realizations, such as in the TT, TTT, and the 4T cases, but similar
computations can be performed in other cases, such as in Weyl-flat
backgrounds. The inclusion of these contributions would allow to obtain a
consistent description of the backreaction, investigating its impact in a
realistic cosmological context.
One of the objectives of this analysis was to establish a link between the
anomalous actions in which the presence of the dilaton is manifest and those
in which the dilaton can be removed. The latter, originally derived by Riegert
by a rescaling at $d=4$ of the metric, can be reconsidered in a complete DR
scheme, by a redefinition of the topological density, as shown by Mazur and
Mottola. We have pointed out that the former are expected to describe
effective actions valid around a conformal breaking scale, a scale whose
origin remains, at this time, not justified by the theory itself. The former
are suitable for a description of the conformal anomaly action in the UV, and
may play an important role in the analysis of the conformal backreaction in
that domain.

We have also shown that the presence of a $R\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,^{-1}$ behaviour emerges from the quartic dilaton action derived in
the ”local” WZ approach, as we push the breaking scale up towards the Planck
scale. This allows to establish a connection between the two formulations.
Such behaviour shows that the description of anomaly actions can be realized
without the inclusion of any scale, manifesting the nonlocal character of the
action, reproducing, in the case of the conformal anomaly, a pattern already
known for the chiral anomaly.
We have also shown how our results impact a class of theories, the EGB
theories, in which a classical singular limit on the Gauss-Bonnet term is
performed. These theories have recently received significant attention for
defining local actions of dilaton gravities of Horndeski type which bypass
Lovelock’s theorem. We have shown that the relation between the equation of
motion for the conformal factor and those of the metric, in theze theories,
are constrained by the topological density. They can be classified as typical
WZ actions. Also for such classical theories, a nonlocal formulations is
possible.

### Epilogue

In this thesis we have studied correlators in Conformal Field Theories. First,
we demonstrated a method for deriving n-point scalar in coordinate space in
section 1.3.2. Then we turned our attention to correlators in momentum space,
which was the main focus of this thesis. Our goal, in both cases, was to
illustrate in a systematical way the methods one needs to study such
correlators in general CFTs. In the two cases we studied in momentum space we
focused on tensorial four-point correlators. While in coordinate space, these
correlators have been studied in great detail and by using various methods
such as OPE and holography in momentum space we were among the first groups to
deal with such correlators. We have used the tensor decomposition along with
the correlator invariance to find the tensor structure of each correlator. We
derived the CWI and the trace and conservation WI for both cases and then for
each correlator we focused our analysis in different aspects. For the $TOOO$,
we focused on the structure of the CWI in two different scattering processes
and then we studied various kinematical limits. Regarding the $TTTT$
correlator, due to the presence of the conformal anomaly in $d=4$, we analyzed
the counterterms along with the WI that they have to satisfy and we continued
with the renormalization procedure from which we derived the anomaly part of
the correlator. In the final chapter, we have extended our analysis to
different theories of gravities and turned our attention to conformal
backreaction and topological terms. We have studied their implications with
the goal to provide a clear explanation about anomalous actions with the
presence of a dilaton and action in which the dilaton can be removed.
During the years of these studies there have been significant growing interest
in CFT in momentum space for reason we have already discussed. Some works have
focused on the OPE in momentum space [172] while other have investigated
conformal/Polyakov blocks [173, 36]. Also, there have been works that exploit
the formalism developed in momentum space to link CFT with cosmology [37, 38,
174, 175, 49, 50]. In our work we based our analysis in CWI in flat space to
study the correlators in interest. Moreover, there has been significant
progress in the n-point scalar correlators [22, 105]. Although, the analysis
of CFT in momentum space is not developed as the one in coordinate space,
there are many interesting directions one could take. Apart from the ones we
mentioned above one could also investigate perturbative realizations of four-
point or higher function. Another possible direction is further investigation
of such correlators in curved spaces such as DeSitter and the study of
anomalous actions in such spaces. More details for future direction are given
in the conclusion of the previous chapters.
We hope that this thesis provides a structured and detailed study for the
readers and that it will inspire them to delve deeper in this promising and
exciting research area of Theoretical Physics.

###  Appendix A Appendix for the $\langle TOOO\rangle$ correlator

####  A.1 Primary Conformal Ward Identities

#####  A.1.1 Primary Conformal Ward Identities in $\bar{p}_{1}$

Here we present the explicit expressions of the Primary Conformal Ward
Identites in the case of $\bar{p}_{1}$ dependency. The $C_{ij}$ are given by

$\displaystyle C_{11}$
$\displaystyle=\Bigg{[}K_{2}+\frac{p_{3}^{2}-p_{4}^{2}}{s\,t}\frac{\partial}{\partial
s\partial t}-\frac{p_{3}^{2}-p_{4}^{2}}{s\,u}\frac{\partial}{\partial
s\partial u}+\frac{1}{t}\frac{\partial}{\partial
t}\left(p_{2}\frac{\partial}{\partial p_{2}}+p_{3}\frac{\partial}{\partial
p_{3}}-p_{4}\frac{\partial}{\partial
p_{4}}\right)+(d-\Delta)\left(\frac{1}{t}\frac{\partial}{\partial
t}+\frac{1}{u}\frac{\partial}{\partial u}\right)$
$\displaystyle+\frac{1}{u}\frac{\partial}{\partial
u}\left(p_{2}\frac{\partial}{\partial p_{2}}-p_{3}\frac{\partial}{\partial
p_{3}}+p_{4}\frac{\partial}{\partial
p_{4}}\right)+\frac{2p_{2}^{2}+p_{3}^{2}+p_{4}^{2}-s^{2}-t^{2}-u^{2}}{t\,u}\frac{\partial}{\partial
t\partial u}\Bigg{]}\,A(p_{2},p_{3},p_{4},s,t,u)=0$ (A.1.1) $\displaystyle
C_{12}$
$\displaystyle=\Bigg{[}K_{2}+\frac{p_{3}^{2}-p_{4}^{2}}{s\,t}\frac{\partial}{\partial
s\partial t}-\frac{p_{3}^{2}-p_{4}^{2}}{s\,u}\frac{\partial}{\partial
s\partial u}+\frac{1}{t}\frac{\partial}{\partial
t}\left(p_{2}\frac{\partial}{\partial p_{2}}+p_{3}\frac{\partial}{\partial
p_{3}}-p_{4}\frac{\partial}{\partial
p_{4}}\right)+(d-\Delta)\left(\frac{1}{t}\frac{\partial}{\partial
t}+\frac{1}{u}\frac{\partial}{\partial u}\right)$
$\displaystyle+\frac{1}{u}\frac{\partial}{\partial
u}\left(p_{2}\frac{\partial}{\partial p_{2}}-p_{3}\frac{\partial}{\partial
p_{3}}+p_{4}\frac{\partial}{\partial
p_{4}}\right)+\frac{2p_{2}^{2}+p_{3}^{2}+p_{4}^{2}-s^{2}-t^{2}-u^{2}}{t\,u}\frac{\partial}{\partial
t\partial u}\Bigg{]}\,A(p_{3},p_{2},p_{4},u,t,s)$
$\displaystyle+\frac{2}{t}\frac{\partial}{\partial
t}\bigg{(}A(p_{2},p_{3},p_{4},s,t,u)+A(p_{3},p_{2},p_{4},u,t,s)\bigg{)}-\frac{2}{u}\frac{\partial}{\partial
u}\bigg{(}A(p_{2},p_{3},p_{4},s,t,u)+A(p_{3},p_{2},p_{4},u,t,s)\bigg{)}$
(A.1.2) $\displaystyle C_{13}$
$\displaystyle=\Bigg{[}K_{2}+\frac{p_{3}^{2}-p_{4}^{2}}{s\,t}\frac{\partial}{\partial
s\partial t}-\frac{p_{3}^{2}-p_{4}^{2}}{s\,u}\frac{\partial}{\partial
s\partial u}+\frac{1}{t}\frac{\partial}{\partial
t}\left(p_{2}\frac{\partial}{\partial p_{2}}+p_{3}\frac{\partial}{\partial
p_{3}}-p_{4}\frac{\partial}{\partial
p_{4}}\right)+(d-\Delta)\left(\frac{1}{t}\frac{\partial}{\partial
t}+\frac{1}{u}\frac{\partial}{\partial u}\right)$
$\displaystyle+\frac{1}{u}\frac{\partial}{\partial
u}\left(p_{2}\frac{\partial}{\partial p_{2}}-p_{3}\frac{\partial}{\partial
p_{3}}+p_{4}\frac{\partial}{\partial
p_{4}}\right)+\frac{2p_{2}^{2}+p_{3}^{2}+p_{4}^{2}-s^{2}-t^{2}-u^{2}}{t\,u}\frac{\partial}{\partial
t\partial u}\Bigg{]}\,A(p_{4},p_{3},p_{2},t,s,u)$
$\displaystyle-\frac{2}{t}\frac{\partial}{\partial
t}\bigg{(}A(p_{2},p_{3},p_{4},s,t,u)+A(p_{4},p_{3},p_{2},t,s,u)\bigg{)}+\frac{2}{u}\frac{\partial}{\partial
u}\bigg{(}A(p_{2},p_{3},p_{4},s,t,u)+A(p_{4},p_{3},p_{2},t,s,u)\bigg{)}$
(A.1.3)

and

$\displaystyle C_{21}$
$\displaystyle=\Bigg{[}K_{3}+\frac{p_{2}^{2}-p_{4}^{2}}{t\,u}\frac{\partial}{\partial
t\partial u}-\frac{p_{2}^{2}-p_{4}^{2}}{s\,u}\frac{\partial}{\partial
s\partial u}+\frac{1}{t}\frac{\partial}{\partial
t}\left(p_{2}\frac{\partial}{\partial p_{2}}+p_{3}\frac{\partial}{\partial
p_{3}}-p_{4}\frac{\partial}{\partial
p_{4}}\right)+(d-\Delta)\left(\frac{1}{t}\frac{\partial}{\partial
t}+\frac{1}{s}\frac{\partial}{\partial s}\right)$
$\displaystyle+\frac{1}{s}\frac{\partial}{\partial
s}\left(p_{3}\frac{\partial}{\partial p_{3}}+p_{4}\frac{\partial}{\partial
p_{4}}-p_{2}\frac{\partial}{\partial
p_{2}}\right)+\frac{2p_{3}^{2}+p_{2}^{2}+p_{4}^{2}-s^{2}-t^{2}-u^{2}}{s\,t}\frac{\partial}{\partial
s\partial t}\Bigg{]}\,A(p_{2},p_{3},p_{4},s,t,u)$
$\displaystyle+\frac{2}{t}\frac{\partial}{\partial
t}\bigg{(}A(p_{2},p_{3},p_{4},s,t,u)+A(p_{3},p_{2},p_{4},u,t,s)\bigg{)}-\frac{2}{s}\frac{\partial}{\partial
s}\bigg{(}A(p_{2},p_{3},p_{4},s,t,u)+A(p_{3},p_{2},p_{4},u,t,s)\bigg{)}$
(A.1.4) $\displaystyle C_{22}$
$\displaystyle=\Bigg{[}K_{3}+\frac{p_{2}^{2}-p_{4}^{2}}{t\,u}\frac{\partial}{\partial
t\partial u}-\frac{p_{2}^{2}-p_{4}^{2}}{s\,u}\frac{\partial}{\partial
s\partial u}+\frac{1}{t}\frac{\partial}{\partial
t}\left(p_{2}\frac{\partial}{\partial p_{2}}+p_{3}\frac{\partial}{\partial
p_{3}}-p_{4}\frac{\partial}{\partial
p_{4}}\right)+(d-\Delta)\left(\frac{1}{t}\frac{\partial}{\partial
t}+\frac{1}{s}\frac{\partial}{\partial s}\right)$
$\displaystyle+\frac{1}{s}\frac{\partial}{\partial
s}\left(p_{3}\frac{\partial}{\partial p_{3}}+p_{4}\frac{\partial}{\partial
p_{4}}-p_{2}\frac{\partial}{\partial
p_{2}}\right)+\frac{2p_{3}^{2}+p_{2}^{2}+p_{4}^{2}-s^{2}-t^{2}-u^{2}}{s\,t}\frac{\partial}{\partial
s\partial t}\Bigg{]}\,A(p_{3},p_{2},p_{4},u,t,s)$ (A.1.5) $\displaystyle
C_{23}$
$\displaystyle=\Bigg{[}K_{3}+\frac{p_{2}^{2}-p_{4}^{2}}{t\,u}\frac{\partial}{\partial
t\partial u}-\frac{p_{2}^{2}-p_{4}^{2}}{s\,u}\frac{\partial}{\partial
s\partial u}+\frac{1}{t}\frac{\partial}{\partial
t}\left(p_{2}\frac{\partial}{\partial p_{2}}+p_{3}\frac{\partial}{\partial
p_{3}}-p_{4}\frac{\partial}{\partial
p_{4}}\right)+(d-\Delta)\left(\frac{1}{t}\frac{\partial}{\partial
t}+\frac{1}{s}\frac{\partial}{\partial s}\right)$
$\displaystyle+\frac{1}{s}\frac{\partial}{\partial
s}\left(p_{3}\frac{\partial}{\partial p_{3}}+p_{4}\frac{\partial}{\partial
p_{4}}-p_{2}\frac{\partial}{\partial
p_{2}}\right)+\frac{2p_{3}^{2}+p_{2}^{2}+p_{4}^{2}-s^{2}-t^{2}-u^{2}}{s\,t}\frac{\partial}{\partial
s\partial t}\Bigg{]}\,A(p_{4},p_{3},p_{2},t,s,u)$
$\displaystyle+\frac{2}{s}\frac{\partial}{\partial
s}\bigg{(}A(p_{4},p_{3},p_{2},t,s,u)+A(p_{3},p_{2},p_{4},u,t,s)\bigg{)}-\frac{2}{t}\frac{\partial}{\partial
t}\bigg{(}A(p_{4},p_{3},p_{2},t,s,u)+A(p_{3},p_{2},p_{4},u,t,s)\bigg{)}$
(A.1.6)

and finally

$\displaystyle C_{31}$
$\displaystyle=\Bigg{[}K_{4}+\frac{p_{2}^{2}-p_{3}^{2}}{t\,u}\frac{\partial}{\partial
t\partial u}-\frac{p_{2}^{2}-p_{3}^{2}}{s\,t}\frac{\partial}{\partial
s\partial t}+\frac{1}{u}\frac{\partial}{\partial
u}\left(p_{2}\frac{\partial}{\partial p_{2}}-p_{3}\frac{\partial}{\partial
p_{3}}+p_{4}\frac{\partial}{\partial
p_{4}}\right)+(d-\Delta)\left(\frac{1}{u}\frac{\partial}{\partial
u}+\frac{1}{s}\frac{\partial}{\partial s}\right)$
$\displaystyle+\frac{1}{s}\frac{\partial}{\partial
s}\left(p_{3}\frac{\partial}{\partial p_{3}}+p_{4}\frac{\partial}{\partial
p_{4}}-p_{2}\frac{\partial}{\partial
p_{2}}\right)+\frac{2p_{4}^{2}+p_{2}^{2}+p_{3}^{2}-s^{2}-t^{2}-u^{2}}{s\,u}\frac{\partial}{\partial
s\partial u}\Bigg{]}\,A(p_{2},p_{3},p_{4},s,t,u)$
$\displaystyle-\frac{2}{s}\frac{\partial}{\partial
s}\bigg{(}A(p_{2},p_{3},p_{4},s,t,u)+A(p_{4},p_{3},p_{2},t,s,u)\bigg{)}+\frac{2}{u}\frac{\partial}{\partial
u}\bigg{(}A(p_{2},p_{3},p_{4},s,t,u)+A(p_{4},p_{3},p_{2},t,s,u)\bigg{)}$
(A.1.7) $\displaystyle C_{32}$
$\displaystyle=\Bigg{[}K_{4}+\frac{p_{2}^{2}-p_{3}^{2}}{t\,u}\frac{\partial}{\partial
t\partial u}-\frac{p_{2}^{2}-p_{3}^{2}}{s\,t}\frac{\partial}{\partial
s\partial t}+\frac{1}{u}\frac{\partial}{\partial
u}\left(p_{2}\frac{\partial}{\partial p_{2}}-p_{3}\frac{\partial}{\partial
p_{3}}+p_{4}\frac{\partial}{\partial
p_{4}}\right)+(d-\Delta)\left(\frac{1}{u}\frac{\partial}{\partial
u}+\frac{1}{s}\frac{\partial}{\partial s}\right)$
$\displaystyle+\frac{1}{s}\frac{\partial}{\partial
s}\left(p_{3}\frac{\partial}{\partial p_{3}}+p_{4}\frac{\partial}{\partial
p_{4}}-p_{2}\frac{\partial}{\partial
p_{2}}\right)+\frac{2p_{4}^{2}+p_{2}^{2}+p_{3}^{2}-s^{2}-t^{2}-u^{2}}{s\,u}\frac{\partial}{\partial
s\partial u}\Bigg{]}\,A(p_{3},p_{2},p_{4},u,t,s)$
$\displaystyle+\frac{2}{s}\frac{\partial}{\partial
s}\bigg{(}A(p_{3},p_{2},p_{4},u,t,s)+A(p_{4},p_{3},p_{2},t,s,u)\bigg{)}-\frac{2}{u}\frac{\partial}{\partial
u}\bigg{(}A(p_{3},p_{2},p_{4},u,t,s)+A(p_{4},p_{3},p_{2},t,s,u)\bigg{)}$
$\displaystyle C_{33}$
$\displaystyle=\Bigg{[}K_{4}+\frac{p_{2}^{2}-p_{3}^{2}}{t\,u}\frac{\partial}{\partial
t\partial u}-\frac{p_{2}^{2}-p_{3}^{2}}{s\,t}\frac{\partial}{\partial
s\partial t}+\frac{1}{u}\frac{\partial}{\partial
u}\left(p_{2}\frac{\partial}{\partial p_{2}}-p_{3}\frac{\partial}{\partial
p_{3}}+p_{4}\frac{\partial}{\partial
p_{4}}\right)+(d-\Delta)\left(\frac{1}{u}\frac{\partial}{\partial
u}+\frac{1}{s}\frac{\partial}{\partial s}\right)$
$\displaystyle+\frac{1}{s}\frac{\partial}{\partial
s}\left(p_{3}\frac{\partial}{\partial p_{3}}+p_{4}\frac{\partial}{\partial
p_{4}}-p_{2}\frac{\partial}{\partial
p_{2}}\right)+\frac{2p_{4}^{2}+p_{2}^{2}+p_{3}^{2}-s^{2}-t^{2}-u^{2}}{s\,u}\frac{\partial}{\partial
s\partial u}\Bigg{]}\,A(p_{4},p_{3},p_{2},t,s,u).$ (A.1.9)

#####  A.1.2 Primary Conformal Ward Identities in $\bar{p}_{4}$

We present the remaining Primary CWIs of Section 5.4.5. We obtain

$\displaystyle\tilde{C}_{12}=$
$\displaystyle\Bigg{[}\frac{\partial^{2}}{\partial
p_{4}^{2}}+\frac{d-2\Delta+1}{p_{4}}\frac{\partial}{\partial
p_{4}}-\frac{\partial^{2}}{\partial
p_{1}^{2}}-\frac{1-d}{p_{1}}\frac{\partial}{\partial
p_{1}}+\frac{1}{s}\frac{\partial}{\partial
s}\left(p_{4}\frac{\partial}{\partial p_{4}}+p_{3}\frac{\partial}{\partial
p_{3}}-p_{1}\frac{\partial}{\partial p_{1}}-p_{2}\frac{\partial}{\partial
p_{2}}\right)$ $\displaystyle+\frac{d-\Delta+2}{s}\frac{\partial}{\partial
s}+\frac{p_{3}^{2}-p_{2}^{2}}{st}\frac{\partial^{2}}{\partial s\partial
t}\Bigg{]}F(p_{1},p_{4},p_{3},p_{2},t,s),$ $\displaystyle\tilde{C}_{13}=$
$\displaystyle\Bigg{[}\frac{\partial^{2}}{\partial
p_{4}^{2}}+\frac{d-2\Delta+1}{p_{4}}\frac{\partial}{\partial
p_{4}}-\frac{\partial^{2}}{\partial
p_{1}^{2}}-\frac{1-d}{p_{1}}\frac{\partial}{\partial
p_{1}}+\frac{1}{s}\frac{\partial}{\partial
s}\left(p_{4}\frac{\partial}{\partial p_{4}}+p_{3}\frac{\partial}{\partial
p_{3}}-p_{1}\frac{\partial}{\partial p_{1}}-p_{2}\frac{\partial}{\partial
p_{2}}\right)$ $\displaystyle+\frac{d-\Delta-2}{s}\frac{\partial}{\partial
s}+\frac{p_{3}^{2}-p_{2}^{2}}{st}\frac{\partial^{2}}{\partial s\partial
t}\Bigg{]}F(p_{1},p_{2},p_{4},p_{3},s,\tilde{u}),$ (A.1.11)
$\displaystyle\tilde{C}_{22}=$
$\displaystyle\Bigg{[}\frac{\partial^{2}}{\partial
p_{4}^{2}}+\frac{d-2\Delta+1}{p_{4}}\frac{\partial}{\partial
p_{4}}-\frac{\partial^{2}}{\partial
p_{2}^{2}}-\frac{d-2\Delta+1}{p_{2}}\frac{\partial}{\partial
p_{2}}+\frac{1}{s}\frac{\partial}{\partial
s}\left(p_{3}\frac{\partial}{\partial p_{3}}+p_{4}\frac{\partial}{\partial
p_{4}}-p_{1}\frac{\partial}{\partial p_{1}}-p_{2}\frac{\partial}{\partial
p_{2}}\right)$ $\displaystyle+\frac{\Delta-d}{t}\frac{\partial}{\partial
t}+\frac{d-\Delta+2}{s}\frac{\partial}{\partial
s}+\frac{1}{t}\frac{\partial}{\partial t}\left(p_{1}\frac{\partial}{\partial
p_{1}}+p_{4}\frac{\partial}{\partial p_{4}}-p_{2}\frac{\partial}{\partial
p_{2}}-p_{3}\frac{\partial}{\partial p_{3}}\right)$
$\displaystyle+\frac{p_{4}^{2}-p_{2}^{2}}{st}\frac{\partial^{2}}{\partial
s\partial t}\Bigg{]}F(p_{1},p_{4},p_{3},p_{2},t,s)+\frac{2}{t}\frac{\partial
F(p_{1},p_{2},p_{4},p_{3},s,\tilde{u})}{\partial t}-\frac{2}{t}\frac{\partial
F(p_{1},p_{2},p_{3},p_{4},s,t)}{\partial t},$ $\displaystyle\tilde{C}_{23}=$
$\displaystyle\Bigg{[}\frac{\partial^{2}}{\partial
p_{4}^{2}}+\frac{d-2\Delta+1}{p_{4}}\frac{\partial}{\partial
p_{4}}-\frac{\partial^{2}}{\partial
p_{2}^{2}}-\frac{d-2\Delta+1}{p_{2}}\frac{\partial}{\partial
p_{2}}+\frac{1}{s}\frac{\partial}{\partial
s}\left(p_{3}\frac{\partial}{\partial p_{3}}+p_{4}\frac{\partial}{\partial
p_{4}}-p_{1}\frac{\partial}{\partial p_{1}}-p_{2}\frac{\partial}{\partial
p_{2}}\right)$ $\displaystyle+\frac{\Delta-d+2}{t}\frac{\partial}{\partial
t}+\frac{d-\Delta-2}{s}\frac{\partial}{\partial
s}+\frac{1}{t}\frac{\partial}{\partial t}\left(p_{1}\frac{\partial}{\partial
p_{1}}+p_{4}\frac{\partial}{\partial p_{4}}-p_{2}\frac{\partial}{\partial
p_{2}}-p_{3}\frac{\partial}{\partial p_{3}}\right)$
$\displaystyle+\frac{p_{4}^{2}-p_{2}^{2}}{st}\frac{\partial^{2}}{\partial
s\partial t}\Bigg{]}F(p_{1},p_{2},p_{4},p_{3},s,\tilde{u}),$
$\displaystyle\tilde{C}_{32}=$
$\displaystyle\Bigg{[}\frac{\partial^{2}}{\partial
p_{4}^{2}}+\frac{d-2\Delta+1}{p_{4}}\frac{\partial}{\partial
p_{4}}-\frac{\partial^{2}}{\partial
p_{3}^{2}}-\frac{d-2\Delta+1}{p_{3}}\frac{\partial}{\partial
p_{3}}+\frac{p_{1}^{2}-p_{2}^{2}}{st}\frac{\partial^{2}}{\partial s\partial
t}+\frac{\Delta-d+2}{t}\frac{\partial}{\partial t}$
$\displaystyle+\frac{1}{t}\frac{\partial}{\partial
t}\left(p_{1}\frac{\partial}{\partial p_{1}}+p_{4}\frac{\partial}{\partial
p_{4}}-p_{2}\frac{\partial}{\partial p_{2}}-p_{3}\frac{\partial}{\partial
p_{3}}\right)\Bigg{]}F(p_{1},p_{4},p_{3},p_{2},t,s),$
$\displaystyle\tilde{C}_{33}=$
$\displaystyle\Bigg{[}\frac{\partial^{2}}{\partial
p_{4}^{2}}+\frac{d-2\Delta+1}{p_{4}}\frac{\partial}{\partial
p_{4}}-\frac{\partial^{2}}{\partial
p_{3}^{2}}-\frac{d-2\Delta+1}{p_{3}}\frac{\partial}{\partial
p_{3}}+\frac{p_{1}^{2}-p_{2}^{2}}{st}\frac{\partial^{2}}{\partial s\partial
t}+\frac{\Delta-d}{t}\frac{\partial}{\partial t}$
$\displaystyle+\frac{1}{t}\frac{\partial}{\partial
t}\left(p_{1}\frac{\partial}{\partial p_{1}}+p_{4}\frac{\partial}{\partial
p_{4}}-p_{2}\frac{\partial}{\partial p_{2}}-p_{3}\frac{\partial}{\partial
p_{3}}\right)-\frac{2}{s}\frac{\partial}{\partial
s}\Bigg{]}F(p_{1},p_{2},p_{4},p_{3},t,\tilde{u})$
$\displaystyle+\frac{2}{s}\frac{\partial
F(p_{1},p_{2},p_{3},p_{4},s,t)}{\partial s}-\frac{2}{s}\frac{\partial
F(p_{1},p_{4},p_{3},p_{2},t,s)}{\partial s}$
$\displaystyle+\frac{2}{t}\frac{\partial
F(p_{1},p_{4},p_{3},p_{2},t,s)}{\partial t}-\frac{2}{t}\frac{\partial
F(p_{1},p_{2},p_{3},p_{4},s,t)}{\partial t}.$ (A.1.15)

####  A.2 The hypergeometric system from (5.5.10)

Rewriting (5.5.10), with quadratic ratios
$x=q_{1}^{2}/q_{3}^{2},y=q_{2}^{2}/q_{3}^{2}$ for the correlator
$\Phi(q_{1},q_{2},q_{3})=\langle OOO\rangle$ with scaling dimensions
$\Delta_{i}=\Delta,\,i=1,2,3$

$K_{13}\Phi=0\qquad K_{23}\Phi=0,$

one obtains the system of equations

$\displaystyle\begin{cases}\bigg{[}x(1-x)\frac{\partial^{2}}{\partial
x^{2}}-y^{2}\frac{\partial^{2}}{\partial
y^{2}}-2\,x\,y\frac{\partial^{2}}{\partial x\partial
y}+\left[\gamma-(\alpha+\beta+1)x\right]\frac{\partial}{\partial
x}-(\alpha+\beta+1)y\frac{\partial}{\partial
y}-\alpha\,\beta\bigg{]}\Phi(x,y)=0\,,\\\
\bigg{[}y(1-y)\frac{\partial^{2}}{\partial
y^{2}}-x^{2}\frac{\partial^{2}}{\partial
x^{2}}-2\,x\,y\frac{\partial^{2}}{\partial x\partial
y}+\left[\gamma^{\prime}-(\alpha+\beta+1)y\right]\frac{\partial}{\partial
y}-(\alpha+\beta+1)x\frac{\partial}{\partial
x}-\alpha\,\beta\bigg{]}\Phi(x,y)=0\,,\end{cases}$ (A.2.1)

with parameters $\alpha(a,b),\beta(a,b),\gamma(a,b),\gamma^{\prime}(a,b)$
given in (2.5.13) and (4.1.8), which are solved by ansätz of the form
$x^{a}y^{b}G(x,y)$. $G$ is an Appell function of type
$F_{4}(\alpha,\beta,\gamma,\gamma^{\prime},x,y)$, given by

$F_{4}(\alpha,\beta,\gamma,\gamma^{\prime};x,y)=\sum_{m,n=0}^{\infty}\frac{(\alpha)_{m+n}(\beta)_{m+n}}{(\gamma)_{m}(\gamma^{\prime})_{n}m!n!}x^{m}y^{n}.$
(A.2.2)

Both in the case of quadratic or quartic (5.5.5) ratios, in the variables $x$
and $y$, the structure of (A.2.1) is preserved, with appropriate values of the
parameters $\alpha(a,b),\beta(a,b),\gamma(a,b),\gamma^{\prime}(a,b)$ and
indices $a,b$.

#####  A.2.1 The Lauricella system and 4K

In the $s,t\to\infty$ limit the equations for the $K_{ij}$ operators, for
arbitrary scalings $\Delta_{i}$, can be organized in the form

$\textup{K}_{14}\phi=0,\qquad\textup{K}_{24}\phi=0,\qquad\textup{K}_{34}\phi=0$
(A.2.3)

where

$\displaystyle\textup{K}_{i}$ $\displaystyle=\frac{\partial^{2}}{\partial
p_{i}^{2}}+\frac{(d-2\Delta_{i}+1)}{p_{i}}\frac{\partial}{\partial
p_{i}},\qquad i=1,\dots,4\ ,$ (A.2.4) $\displaystyle\textup{K}_{ij}$
$\displaystyle=\textup{K}_{i}-\textup{K}_{j}\ .$ (A.2.5)

One can choose an arbitrary momentum as pivot in the ansatz for the solution
of such system, for instance $(x,y,z,p_{4}^{2})$, where

$x=\mbox{\small$\displaystyle\frac{p_{1}^{2}}{p_{4}^{2}}$},\quad
y=\mbox{\small$\displaystyle\frac{p_{2}^{2}}{p_{4}^{2}}$},\quad
z=\mbox{\small$\displaystyle\frac{p_{3}^{2}}{p_{4}^{2}}$}$ (A.2.6)

are dimensionless quadratic ratios. The ansätz for the solution can be taken
of the form

$\phi(p_{1},p_{2},p_{3},p_{4})=(p_{4}^{2})^{n_{s}}\,x^{a}\,y^{b}\,z^{c}\,F(x,y,z),$
(A.2.7)

satisfying the dilatation Ward identity with the condition

$n_{s}=\frac{\Delta_{t}}{2}-\frac{3d}{2}\qquad\Delta_{t}=\sum_{i=1}^{4}\Delta_{i}.$
(A.2.8)

With this ansätz the conformal Ward identities takes the form

$\displaystyle\textup{K}_{14}\phi=$ $\displaystyle
4p_{4}^{\Delta_{t}-3d-2}\,x^{a}\,y^{b}\,z^{c}\,\bigg{[}(1-x)x\mbox{\small$\displaystyle\frac{\partial^{2}}{\partial
x^{2}}$}-2x\,y\mbox{\small$\displaystyle\frac{\partial^{2}}{\partial x\partial
y}$}-y^{2}\mbox{\small$\displaystyle\frac{\partial^{2}}{\partial
y^{2}}$}-2x\,z\mbox{\small$\displaystyle\frac{\partial^{2}}{\partial x\partial
z}$}-z^{2}\mbox{\small$\displaystyle\frac{\partial^{2}}{\partial
z^{2}}$}-2y\,z\mbox{\small$\displaystyle\frac{\partial^{2}}{\partial y\partial
z}$}$ $\displaystyle\hskip
56.9055pt+(Ax+\gamma)\mbox{\small$\displaystyle\frac{\partial}{\partial
x}$}+Ay\mbox{\small$\displaystyle\frac{\partial}{\partial
y}$}+Az\mbox{\small$\displaystyle\frac{\partial}{\partial
z}$}+\left(E+\mbox{\small$\displaystyle\frac{G}{x}$}\right)\bigg{]}F(x,y,z)=0,$
(A.2.9)

with

$\displaystyle A$
$\displaystyle=\Delta_{1}+\Delta_{2}+\Delta_{3}-\mbox{\small$\displaystyle\frac{5}{2}$}d-2(a+b+c)-1$
(A.2.10a) $\displaystyle E$
$\displaystyle=-\mbox{\small$\displaystyle\frac{1}{4}$}\big{(}3d-\Delta_{t}+2(a+b+c)\big{)}\big{(}2d+2\Delta_{4}-\Delta_{t}+2(a+b+c)\big{)}$
(A.2.10b) $\displaystyle G$
$\displaystyle=\mbox{\small$\displaystyle\frac{a}{2}$}\,\left(d-2\Delta_{1}+2a\right)$
(A.2.10c) $\displaystyle\gamma$
$\displaystyle=\mbox{\small$\displaystyle\frac{d}{2}$}-\Delta_{1}+2a+1.$
(A.2.10d)

Similar constraints are obtained from the equations $\textup{K}_{24}\phi=0$
and $\textup{K}_{34}\phi=0$.
The reduction to the hypergeometric form requires that all the $1/x,1/y$ and
$1/z$ terms of the equations vanish. This implies that the Fuchsian points
$a,b,c$ have values

$\displaystyle a$
$\displaystyle=0,\,\Delta_{1}-\mbox{\small$\displaystyle\frac{d}{2}$}$
(A.2.11) $\displaystyle b$
$\displaystyle=0,\,\Delta_{2}-\mbox{\small$\displaystyle\frac{d}{2}$}$
(A.2.12) $\displaystyle c$
$\displaystyle=0,\,\Delta_{3}-\mbox{\small$\displaystyle\frac{d}{2}$}$
(A.2.13)

and

$\displaystyle\alpha(a,b,c)$
$\displaystyle=d+\Delta_{4}-\mbox{\small$\displaystyle\frac{\Delta_{t}}{2}$}+a+b+c$
$\displaystyle\beta(a,b,c)$
$\displaystyle=\mbox{\small$\displaystyle\frac{3d}{2}$}-\mbox{\small$\displaystyle\frac{\Delta_{t}}{2}$}+a+b+c$
(A.2.14)

$\gamma(a)=\frac{d}{2}-\Delta_{1}+2a+1\,,\qquad\gamma^{\prime}(b)=\frac{d}{2}-\Delta_{2}+2b+1\,,\qquad\gamma^{\prime\prime}(c)=\frac{d}{2}-\Delta_{3}+2c+1.$
(A.2.15)

With this redefinition of the coefficients, the equations are then expressed
in the form

$\left\\{\begin{matrix}&x_{j}(1-x_{j})\mbox{\small$\displaystyle\frac{\partial^{2}F}{\partial
x_{j}^{2}}$}+\sum\limits_{\begin{subarray}{c}\hskip 36.98866pts\neq j\
\text{for}\ r=j\end{subarray}}x_{r}\hskip 5.69046pt\sum x_{s}\hskip
2.15277pt\mbox{\small$\displaystyle\frac{\partial^{2}F}{\partial x_{r}\partial
x_{s}}$}+\left[\gamma_{j}-(\alpha+\beta+1)x_{j}\right]\mbox{\small$\displaystyle\frac{\partial
F}{\partial x_{j}}$}-(\alpha+\beta+1)\sum\limits_{k\neq
j}\,x_{k}\mbox{\small$\displaystyle\frac{\partial F}{\partial
x_{k}}$}-\alpha\,\beta\,F=0\\\\[12.91663pt] &(j=1,2,3)\end{matrix}\right.$

(A.2.16)

where we have set $\gamma_{1}=\gamma$, $\ \gamma_{2}=\gamma^{\prime}$ and
$\gamma_{3}=\gamma^{\prime\prime}$ and $x_{1}=x$, $x_{2}=y$ and $x_{3}=z$. The
system of equations admits as solutions hypergeometric functions of three
variables, the Lauricella functions, of the form

$F_{C}(\alpha,\beta,\gamma,\gamma^{\prime},\gamma^{\prime\prime},x,y,z)=\sum\limits_{m_{1},m_{2},m_{3}}^{\infty}\,\frac{(\alpha)_{m_{1}+m_{2}+m_{3}}(\beta)_{m_{1}+m_{2}+m_{3}}}{(\gamma)_{m_{1}}(\gamma^{\prime})_{m_{2}}(\gamma^{\prime\prime})_{m_{3}}m_{1}!\,m_{2}!\,m_{3}!}x^{m_{1}}y^{m_{2}}z^{m_{3}}.$
(A.2.17)

where the Pochhammer symbol $(\lambda)_{k}$ with an arbitrary $\lambda$ and
$k$ a positive integer is defined as

$(\alpha)_{k}=\frac{\Gamma(\alpha+k)}{\Gamma(\alpha)}=\alpha(\alpha+1)\dots(\alpha+k-1).$
(A.2.18)

The convergence region of this series is defined by the condition

$\left|\sqrt{x}\right|+\left|\sqrt{y}\right|+\left|\sqrt{z}\right|<1.$
(A.2.19)

The function $F_{C}$ is the generalization of the Appell $F_{4}$ for the case
of three variables. The system of equations (A.2.16) admits 8 independent
particular integrals (solutions) listed below. Finally, the solution for
$\phi$ can be written as

$\phi(p_{i}^{2})=p_{4}^{\Delta_{t}-3d}\sum_{a,b}C_{i\,(a,b)}\
x^{a}y^{b}z^{c}F_{C}(\alpha(a,b),\beta(a,b),\gamma(a,b),\gamma^{\prime}(as,b),\gamma^{\prime\prime}(a,b),x,y,z)$
(A.2.20)

where $C_{i}$ are arbitrary constants. The sum runs over all the possible
triple $(a,b,c)$ identified in (A.2.11). Introducing the 4K integral

$I_{\alpha\\{\beta_{1},\beta_{2},\beta_{3},\beta_{4}\\}}(p_{1},p_{2},p_{3},p_{4})=\int_{0}^{\infty}\,dx\,x^{\alpha}\,\prod_{i=1}^{4}(p_{i})^{\beta_{i}}\,K_{\beta_{i}}(p_{i}\,x),$
(A.2.21)

the same solution can be re-expressed in the form

$\displaystyle\phi(p_{1},p_{2},p_{3},p_{4})$
$\displaystyle=C\,I_{d-1\left\\{\Delta_{1}-\frac{d}{2},\Delta_{2}-\frac{d}{2},\Delta_{3}-\frac{d}{2},\Delta_{4}-\frac{d}{2}\right\\}}(p_{1},p_{2},p_{3},p_{4})$
$\displaystyle=\int_{0}^{\infty}\,dx\,x^{d-1}\,\prod_{i=1}^{4}(p_{i})^{\Delta_{i}-\frac{d}{2}}\,K_{\Delta_{i}-\frac{d}{2}}(p_{i}\,x),$
(A.2.22)

where $C$ is a undetermined constant.

###  Appendix B Appendix for the $\langle TTTTT\rangle$ correlator

In this appendix, we provide useful details that complement the 6 chapter.

####  B.1 Contact terms

Contact terms generated by 4th variation of a fundamental action are generated
in the form

$\displaystyle\left\langle\frac{\delta^{4}S}{\delta g(d)\delta g(c)\delta
g(b)\delta
g(a)}\right\rangle=\frac{1}{2}\sqrt{\text{dg}(a)}\left\langle\frac{\delta^{3}T(a)}{\delta
g(b)\delta
g(c)\text{$\delta$g}(d)}\right\rangle+\frac{1}{8}g(b)\sqrt{\text{dg}(c)}g(c)\delta(b,a)\delta(c,b)\left\langle\frac{\text{$\delta$T}(a)}{\delta
g(d)}\right\rangle$
$\displaystyle+\frac{1}{8}g(b)\sqrt{\text{dg}(d)}g(d)\delta(b,a)\delta(d,b)\left\langle\frac{\text{$\delta$T}(a)}{\delta
g(c)}\right\rangle+\frac{1}{8}g(c)\sqrt{\text{dg}(d)}g(d)\delta(c,a)\delta(d,c)\left\langle\frac{\text{$\delta$T}(a)}{\text{$\delta$g}(b)}\right\rangle$
$\displaystyle+\frac{1}{8}\sqrt{\text{dg}(d)}g(d)\delta(b,a)\delta(c,b)s(c,b)\delta(d,b)\langle
T(a)\rangle+\frac{1}{8}\sqrt{\text{dg}(c)}g(c)\delta(b,a)\delta(c,b)\delta(d,b)s(d,b)\langle
T(a)\rangle$
$\displaystyle+\frac{1}{8}g(b)\sqrt{\text{dg}(c)}\delta(b,a)\delta(c,b)\delta(d,c)s(d,c)\langle
T(a)\rangle+\frac{1}{4}\sqrt{\text{dg}(b)}g(b)\delta(b,a)\left\langle\frac{\delta^{2}T(a)}{\text{$\delta$g}(c)\delta
g(d)}\right\rangle$
$\displaystyle+\frac{1}{4}\sqrt{\text{dg}(c)}g(c)\delta(c,a)\left\langle\frac{\delta^{2}T(a)}{\delta
g(b)\text{$\delta$g}(d)}\right\rangle+\frac{1}{4}\sqrt{\text{dg}(d)}g(d)\delta(d,a)\left\langle\frac{\delta^{2}T(a)}{\delta
g(b)\delta g(c)}\right\rangle$
$\displaystyle+\frac{1}{16}g(b)g(c)\sqrt{\text{dg}(d)}g(d)\delta(b,a)\delta(c,b)\delta(d,c)\langle
T(a)\rangle+\frac{1}{4}\sqrt{\text{dg}(b)}\delta(b,a)\delta(c,b)s(c,b)\left\langle\frac{\text{$\delta$T}(a)}{\delta
g(d)}\right\rangle$
$\displaystyle+\frac{1}{4}\sqrt{\text{dg}(b)}\delta(b,a)\delta(d,b)s(d,b)\left\langle\frac{\text{$\delta$T}(a)}{\delta
g(c)}\right\rangle+\frac{1}{4}\sqrt{\text{dg}(c)}\delta(c,a)\delta(d,c)s(d,c)\left\langle\frac{\text{$\delta$T}(a)}{\text{$\delta$g}(b)}\right\rangle$
$\displaystyle+\frac{1}{4}\sqrt{\text{dg}(b)}\delta(b,a)\delta(c,b)\delta(d,b)\delta(d,c)\langle
T(a)\rangle s(d,c,b).$ (B.1.1)

In the flat limit we obtain

$\displaystyle\langle\frac{\delta^{4}S}{\delta g(a)\delta g(b)\delta
g(c)\delta g(d)}\rangle=$
$\displaystyle+\frac{1}{8}\delta^{b}\delta^{c}\delta(b,a)\delta(c,b)\left\langle\frac{\text{$\delta$T}(a)}{\delta
g(d)}\right\rangle+\frac{1}{8}\delta^{b}\delta^{d}\delta(b,a)\delta(d,b)\left\langle\frac{\text{$\delta$T}(a)}{\delta
g(c)}\right\rangle$
$\displaystyle+\frac{1}{8}\delta^{d}\delta^{c}\delta(c,a)\delta(d,c)\left\langle\frac{\text{$\delta$T}(a)}{\delta
g(b)}\right\rangle+\frac{1}{4}\delta(b,a)\delta(c,b)s(c,b)\left\langle\frac{\text{$\delta$T}(a)}{\delta
g(d)}\right\rangle$
$\displaystyle+\frac{1}{4}\delta(b,a)\delta(d,b)s(d,b)\left\langle\frac{\text{$\delta$T}(a)}{\text{$\delta$g}(c)}\right\rangle+\frac{1}{4}\delta(c,a)\delta(d,c)s(d,c)\left\langle\frac{\text{$\delta$T}(a)}{\delta
g(b)}\right\rangle$
$\displaystyle+\frac{1}{4}\delta^{d}\delta(d,a)\left\langle\frac{\delta^{2}T(a)}{\delta
g(b)\text{$\delta$g}(c)}\right\rangle+\frac{1}{4}\delta^{c}\delta(c,a)\left\langle\frac{\delta^{2}T(a)}{\delta
g(b)\delta g(d)}\right\rangle$
$\displaystyle+\frac{1}{4}\delta^{b}\delta(b,a)\left\langle\frac{\delta^{2}T(a)}{\delta