integrate the action of a generic CFT over a field $(\phi)$ in a background
metric $g_{\mu\nu}$,

$Z(g)=N\int D\phi e^{-S_{0}(g,\phi)},\qquad Z(\eta)=1.$ (3.1.3)

Its logarithm, $\mathcal{S}(g)$, is our definition of the effective action,
while $S_{0}(g,\phi)$ is the classical action. As usual $Z(g)$ will contain
both connected and disconnected graphs, while $\mathcal{S}(g)$ collects only
connected graphs. One can verify that this collection corresponds also to 1PI
(1 particle irreducible) graphs only in the case of free field theories
embedded in external (classical) gravity.

The emergence of bilinear mixing on the external graviton lines, as we are
going to realize at the end of our analysis, should then be interpreted as a
dynamical response of the theory, induced by the process of renormalization,
with the generation of an intermediate dynamical degree of freedom propagating
with the $1/\square$ operator. For this reason, the presence of such terms
does not invalidate the 1PI nature of this functional.

#####  3.1.1 The scalar case

We may assume that $S(\phi,g)$ describes a free scalar field $\phi$ in a
generic background. The action, in this case, is given by

$S_{0}=\frac{1}{2}\int\,d^{d}x\,\sqrt{-g}\left[g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi-\chi\,R\,\phi^{2}\right],$
(3.1.4)

where $\chi(d)=\frac{1}{4}\frac{(d-2)}{(d-1)}$ is the conformal coupling. This
particular choice of $\chi(d)$ guarantees the conformal invariance of this
action in $d$ dimensions and generates a term of improvement for the stress-
energy tensor in the flat limit, which becomes symmetric and traceless in this
limit. For a more detailed perturbative analysis of this term when the
Standard Model is coupled to gravity, we refer to [88, 89]. One can verify
explicitly that the ordinary counterterms which renormalize the Lagrangian of
the Standard Model, renormalize also insertions of the stress energy tensor of
the theory, only if the Higgs sector is conformally coupled.
The stress energy tensor, takes the form

$T^{\mu\nu}_{scalar}\equiv\frac{2}{\sqrt{g}}\frac{\delta S_{0}}{\delta
g_{\mu\nu}}\\\
=\nabla^{\mu}\phi\,\nabla^{\nu}\phi-\frac{1}{2}\,g^{\mu\nu}\,g^{\alpha\beta}\,\nabla_{\alpha}\phi\,\nabla_{\beta}\phi+\chi\bigg{[}g^{\mu\nu}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,-\nabla^{\mu}\,\nabla^{\nu}+\frac{1}{2}\,g^{\mu\nu}\,R-R^{\mu\nu}\bigg{]}\,\phi^{2}.$

A conformal free field theory realization stays conformal at quantum level,
except for the appearance of an anomaly in even dimensions. For an interacting
theory, on the other hand, anomalous dimensions appear as soon as we switch-on
an interaction in (7.2.3).
In the thesis, $n$-point correlation functions are defined as

$\langle T^{\mu_{1}\nu_{1}}(x_{1})\ldots
T^{\mu_{n}\nu_{n}}(x_{n})\rangle\equiv\frac{2}{\sqrt{g_{1}}}\ldots\frac{2}{\sqrt{g_{n}}}\frac{\delta^{n}\mathcal{S}[g]}{\delta
g_{\mu_{1}\nu_{1}}(x_{1})\delta g_{\mu_{2}\nu_{2}}(x_{2})\ldots\delta
g_{\mu_{n}\nu_{n}}(x_{n})},$ (3.1.5)

with $\sqrt{g_{1}}\equiv\sqrt{|\textrm{det}\,g_{{\mu_{1}\nu_{1}}}(x_{1})}$ and
so on.
The action $\mathcal{S}$ collects all the connected contributions of the
correlation functions in the expansion with respect to the metric
fluctuations, and may as well be expressed in a covariant expansion as

$\mathcal{S}(g)=\mathcal{S}(\bar{g})+\sum_{n=1}^{\infty}\frac{1}{2^{n}n!}\int
d^{d}x_{1}\ldots d^{d}x_{n}\sqrt{g_{1}}\ldots\sqrt{g_{n}}\,\langle
T^{\mu_{1}\nu_{1}}\ldots\,T^{\mu_{n}\nu_{n}}\rangle_{\bar{g}}\delta
g_{\mu_{1}\nu_{1}}(x_{1})\ldots\delta g_{\mu_{n}\nu_{n}}(x_{n}).$ (3.1.6)

For a scalar theory in a flat background, in terms of Feynman diagrams

$\displaystyle\mathcal{S}(g)=$
$\displaystyle\sum_{n}\quad\raisebox{-29.7082pt}{{\includegraphics[width=86.72267pt]{Figures/np}}}\,,$
(3.1.7)

and defines the bare action in $d$ spacetime dimensions. As $d\to 4$ it will
develop singularities in the form of single poles in $1/\epsilon$ that will be
removed by the action of the counterterms. The external weavy lines represent
gravitational fluctuations, in terms of contributions that are classified as
tadpoles, two-point , three-point and $n$-point correlation functions of
stress energy tensors. Tadpoles are removed in dimensional regularization (DR)
in flat space, and the sum in (3.1.7) starts from two-point functions. The
anomaly contributions start from three-point functions.
Indeed, in the case of the scalar free field theory presented above, the
topological contributions coming from the 4-$T$ are summarized by the
following vertices

$\displaystyle\mathcal{S}_{4}(g)=\quad\raisebox{-21.09712pt}{{\includegraphics[width=86.72267pt]{Figures/mt}}}+\quad\raisebox{-19.37494pt}{{\includegraphics[width=65.04034pt]{Figures/kt}}}+\quad\raisebox{-15.0694pt}{{\includegraphics[width=65.04034pt]{Figures/it}}}+\quad\raisebox{-25.83325pt}{{\includegraphics[width=86.72267pt]{Figures/et}}}$

which can be directly computed in perturbation theory [52]. A counterterm
action will be included for the process of renormalization. For example, in
the action (7.2.3) the counterterm action will be determined by two
gravitational counterterms, which will be defined below (3.2.24), and the
renormalized partition functional will be defined as

$Z[g]=\mathcal{N}\int D\phi
e^{i(S_{0}(g,\phi)+S_{ct}(g))}=\mathcal{N}e^{iS_{ct}(g)}\int D\phi
e^{i(S_{0}(g,\phi)},$ (3.1.9)

where $\mathcal{N}$ is a normalization constant. The role of the two
counterterms is to remove the $1/\epsilon$ singularities present in the bare
effective action $\mathcal{S}_{B}$

$\mathcal{S}_{B}(g,d)=-\log\left(\int D\Phi
e^{-S(\Phi,g)}\right)+\log\mathcal{N},$ (3.1.10)

and allow to define the regularized effective action in the form

$\mathcal{S}_{R}(g,d)=\mathcal{S}_{B}(g,d)+b^{\prime}\frac{1}{\epsilon}V_{E}(g,d)+b\frac{1}{\epsilon}V_{C^{2}}(g,d).$
(3.1.11)

$V_{E}$ and $V_{C^{2}}$ are related to the Euler density $E$ and to the Weyl
tensor squared $C^{2}$, respectively, defined by the expressions

$\displaystyle V_{C^{2}}(g,d)\equiv$
$\displaystyle\mu^{\varepsilon}\int\,d^{d}x\,\sqrt{-g}\,C^{2},$ $\displaystyle
V_{E}(g,d)\equiv$ $\displaystyle\mu^{\varepsilon}\int\,d^{d}x\,\sqrt{-g}\,E,$
(3.1.12)

where $\mu$ is a renormalization scale while $\varepsilon=d-4$. The
counterterm vertices will be simply obtained by multiple differentiations of
the two expressions above. We will start our analysis of the terms from the
next section.

For the case of the scalar theory, this implies that the loop contributions
are renormalized just by using the vertices obtained by the functional
differentiation of $S_{ct}$ with respect to the metric. Specific choices of
different backgrounds allows connect with cosmological scenarios. For
instance, if $\bar{g}$ is chosen to be Weyl-flat, giving the opportunity for
studying the conformal back reaction in De Sitter cosmology.

#####  3.1.2 The effective action and space time symmetries

The analysis until this section is mainly focused in flat space time.
Especially the derivation of the CWI can be done if we start from the
effective action in a curved background and take the flat limit. These results
will be of course equivalent to the corresponding results that one can obtain
from a conformal field theory. This is rather obvious since a CFT doesnt
require a particular curved background to be formulated. The infinitesimal
Weyl variation of the metric is

$\delta_{\sigma}g_{\mu\nu}=2\sigma g_{\mu\nu},$ (3.1.13)

and its variation under diffeomorphisms ($x^{\mu}\to x^{\mu}+\epsilon(x)$) is

$\delta_{\epsilon}g_{\mu\nu}=-\nabla_{\mu}\epsilon_{\nu}-\nabla_{\nu}\epsilon_{\mu}.$
(3.1.14)

The variation of the action with respect to $\sigma$ is given by

$\frac{\delta\mathcal{S}}{\delta\sigma(x)}=\sqrt{g}\,g_{\mu\nu}\,\langle
T^{\mu\nu}\rangle,\qquad\textrm{where}\qquad\langle
T^{\mu\nu}\rangle=\frac{2}{\sqrt{g}}\frac{\delta\mathcal{S}}{\delta
g_{\mu\nu}},$ (3.1.15)

Thus Weyl and diffeomorphism invariance of $\mathcal{S}$ mean

$\delta_{\sigma}\mathcal{S}=0,\qquad\delta_{\epsilon}\mathcal{S}=0,$ (3.1.16)

are then summarised by the relations

$\langle T^{\mu}_{\mu}\rangle=0,\qquad\textrm{and}\qquad\nabla_{\mu}\langle
T^{\mu\nu}\rangle=0.$ (3.1.17)

By taking the functional derivative of $\mathcal{S}[g]$ with respect to the
metric background, we can derive the trace and the conformal WI. In the
presence of an anomaly, the equation take the form

$\delta_{\sigma}\mathcal{S}=\int
d^{4}x\sqrt{g}\bar{\mathcal{A}}(x),\qquad\langle
T^{\mu}_{\mu}\rangle=\bar{\mathcal{A}}(x),$ (3.1.18)

which violates Weyl invariance. $\sqrt{g}\bar{\mathcal{A}}(x)$ is the anomaly
functional,that will be discussed later. Now, taking functional derivatives of
the anomaly equation will produce the anomalous WI that play an important role
in this thesis.

It is certainly interesting to examine the implications of spacetime
symmetries which that the effective action invariant. Two example are provided
by metrics which allow Killing (KV) or conformal Killing vectors (CKV). The
requirement that the action $\mathcal{S}$ is invariant under such symmetries
allows to define some conserved currents. These are constructed by contracting
the stress energy tensor derived from $\mathcal{S}$ with such 4-vector fields.
The relations that one derives are valid at quantum level, since the stress
energy tensor derived by varying $\mathcal{S}$ is a quantum average. In
particular, the use of conformal Killing vectors allows to define a conformal
current $J_{c}$ which plays a key role in the derivation of the CWIs and of
the corresponding CWIs in the flat spacetime limit.
In Minkowski space, scale invariance of a certain theory implies that the
corresponding stress energy tensor has zero trace, and one can then define the
conserved current

$J^{\mu}=x_{\nu}T^{\mu\nu},\qquad\partial\cdot J=T^{\mu}_{\mu}=0.$ (3.1.19)

At quantum level, in a curved background, we can generalize this approach by
defining similar currents

$\langle J^{\mu}\rangle=\epsilon_{\nu}^{(K)}\langle T^{\mu\nu}\rangle,$
(3.1.20)

where $\epsilon_{\mu}^{(K)}(x)$ is a Killing or a conformal Killing vector
field of the metric $g$. The proof of their conservation at quantum level
follows closely the derivation at classical level. For instance, if we assume
that $g$ allows vector isometries $\epsilon_{\mu}^{(K)}(x)$, which for changes
of coordinates $x\to x+\epsilon^{(K)}_{\mu}$ leave the metric invariant

$\delta_{\sigma}g_{\mu\nu}=-(\nabla_{\mu}\epsilon^{(K)}_{\nu}+\nabla_{\nu}\epsilon^{(K)}_{\mu})=0,$
(3.1.21)

then the requirement of diffeomorphism invariance of the effective action
$\mathcal{S}[g]$ in that metric implies that the quantum average of
$T^{\mu\nu}$ is conserved

$\nabla_{\mu}\langle T^{\mu\nu}\rangle=0.$ (3.1.22)

Combining this condition with the requirement that $\epsilon^{(K)}_{\mu}$ are
Killing vectors, then $J$ in (3.1.20) is conserved at quantum level

$\nabla\cdot\langle J\rangle=0.$ (3.1.23)

A conformal current $J_{c}$ can be defined analogously to (3.1.20) by assuming
that the background metric $g$ allows conformal Killing vectors. In this more
general case, we recall that the CKVs are solution of the equation

$(ds^{\prime})^{2}=e^{2\sigma(x)}(ds)^{2}\qquad\leftrightarrow\qquad\nabla_{\mu}\epsilon_{\nu}^{(K)}+\nabla_{\nu}\epsilon_{\mu}^{(K)}=2\sigma\delta_{\mu\nu}\qquad\sigma=\frac{1}{4}\nabla\cdot\epsilon^{(K)}.$
(3.1.24)

We remark that if we introduce a conformal current, now using the CKVs of the
background metric as in (3.1.20) in order to define $J_{c}$, if conditions
(3.1.17) are respected by $\mathcal{S}$, then $J_{c}$ is conserved as in the
isometric case (3.1.23).
On the other hand, anomalous CWIs are generated if we allow a Weyl-variant
term in $\mathcal{S}$, which takes place in $d=4$, after renormalization, as
in the case of an anomaly action.
In this case the anomaly induces a non-zero trace, and modifies the semi-
classical condition (3.1.23) into the new form

$\displaystyle\nabla\cdot\langle
J_{c}\rangle=\frac{1}{4}\nabla\cdot\epsilon^{(K)}\langle
T^{\mu}_{\mu}\rangle+\epsilon^{(K)}_{\nu}\nabla_{\mu}\langle
T^{\mu\nu}\rangle=\frac{1}{4}\nabla\cdot\epsilon^{(K)}\langle
T^{\mu}_{\mu}\rangle.$ (3.1.25)

This relation can still be used for the derivation of the special CWIs of
$n$-point functions, as shown in [90, 53]. Notice that $\sigma(x)$ is, at the
beginning, a generic scalar function, which in a Taylor expansion around a
given point $x^{\mu}$ is characterized by an infinite and arbitrary number of
constants. Their number gets drastically reduced if we require that the
spacetime manifold with metric $g$ allows a tangent space at each of its
points, endowed with a flat conformal symmetry.

In an equivalent, more general definition, one introduces an infinite
dimensional abelian Weyl group $\mathcal{G}$ of transformations, with
generators $J_{x}$ acting on the space of fields

$J_{x}=-2g^{\mu\nu}\frac{\delta}{\delta g^{\mu\nu}(x)}-\Delta_{f}\Phi_{f}(x),$
(3.1.26)

summed over the fields $\Phi_{f}$, where $\Delta_{f}$ denote their scaling
dimensions. The finite action of an element of $\mathcal{G}$ can be expressed
as

$e^{\sigma(x)\cdot J_{x}},\qquad\textrm{where}\qquad{\sigma(x)\cdot
J_{x}}\equiv\int d^{4}x\sigma(x)J_{x},$ (3.1.27)

with Weyl invariance of a generic action action expressed in the form

$J_{x}\mathcal{S}_{0}[g,\Phi_{f}]=0.$ (3.1.28)

In the case of the anomaly action $\Gamma[g]$, the expression (3.1.26)
includes only the variation of the metric, since all the remaining fields
$\Phi_{f}$ have been integrated out,

$J_{x}\mathcal{S}[g]=0.$ (3.1.29)

In flat space, the conformal Killing equation identify CKVs $\epsilon^{\mu}$
which are at most quadratic in $x$, are expressed in terms of the fifteen
parameters $(a^{\mu},\omega^{\mu\nu},\lambda_{s},b^{\mu})$ of the conformal
group, indicated as $K^{\mu}(x)$

$\epsilon^{\mu}(x)\vline_{flat}\to
K^{\mu}(x)=a^{\alpha}+\omega^{\mu\nu}x^{\nu}+\lambda_{s}x^{\mu}+b^{\mu}x^{2}-2x^{\mu}b\cdot
x.$ (3.1.30)

Using such CKVs, the derivation of the special CWIs, following the approach of
[90], can be performed directly in $d=4$, and takes to anomalous special CWIs.

####  3.2 A first look on the counterterms

Around a flat metric background, if we use a mass-independent regularization
scheme, the structure of the counterterms is polynomial in momentum space.
As we have seen, the breaking of Weyl invariance due to the presence of an
anomaly produces

$\delta_{\sigma}\Gamma=\frac{1}{(4\pi)^{2}}\int
d^{4}x\sqrt{g}\left(c_{1}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+c_{2}R_{\mu\nu}R^{\mu\nu}+c_{3}R^{2}\right).$
(3.2.1)

The exact form is constrained by the Wess-Zumino consistency condition

$\left[\delta_{\sigma_{1}},\delta_{\sigma_{2}}\right]\Gamma=0.$ (3.2.2)

Applying this condition we arrive at

$\delta_{\sigma}\Gamma=\frac{1}{(4\pi)^{2}}\int
d^{4}x\sqrt{g}\sigma\left(b_{1}C^{(4)}_{\mu\nu\rho\sigma}C^{(4)\mu\nu\rho\sigma}+b_{2}E_{4}+b_{3}\square
R\right),$ (3.2.3)

where

$\displaystyle E_{4}$ $\displaystyle\equiv
R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}-4R_{\mu\nu}R^{\mu\nu}+R^{2},$
(3.2.4) $\displaystyle(C^{(4)})^{2}$ $\displaystyle\equiv
R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}-2R_{\mu\nu}R^{\mu\nu}+\frac{1}{3}R^{2},$
(3.2.5)

which are the Euler-Gauss-Bonnet (GB) invariant and the square of the Weyl
conformal tensor, respectively, in $d=4$. In dimensional regularization, we
need an expansion of $d$ around 4 for a proper definition of $C^{2}$. The
dependence of $C^{2}$ on $d$ allows to introduce both a $(C^{(d)})^{2}$, and a
$(C^{(4)})^{2}$ operator, both defined in $d$ dimensions. Consequently the
contraction of the indices in both cases are all performed in $d$ spacetime
dimensions while the coefficients in their definition are parametrically
dependent on $d$

$C^{(d)\alpha\beta\gamma\delta}C^{(d)}_{\alpha\beta\gamma\delta}=R^{\alpha\beta\gamma\delta}R_{\alpha\beta\gamma\delta}-\frac{4}{d-2}R^{\alpha\beta}R_{\alpha\beta}+\frac{2}{(d-2)(d-1)}R^{2},$
(3.2.6)

with

$C^{(d)}_{\alpha\beta\gamma\delta}=R_{\alpha\beta\gamma\delta}-\frac{2}{d-2}(g_{\alpha\gamma}\,R_{\delta\beta}+g_{\alpha\delta}\,R_{\gamma\beta}-g_{\beta\gamma}\,R_{\delta\alpha}-g_{\beta\delta}\,R_{\gamma\alpha})+\frac{2}{(d-1)(d-2)}\,(g_{\alpha\gamma}\,g_{\delta\beta}+g_{\alpha\delta}\,g_{\gamma\beta})R\,.$
(3.2.7)

We prefer to separate the parametric dependence from the range of variability
of the tensor indices of this operator, since different choices of their range
(and of the parametric dependence) induce finite renormalization of the
effective action and change the anomaly by local terms. Here the term “local”
refers to terms which are obtained from the Weyl variation of a local action.
The remaining contributions, usually termed “non-local”, refer to operators
appearing in the anomaly functional which can be derived by varying a non-
local action. An example of such action is the Riegert action. In general one
could define

$\displaystyle C^{(d)}_{\lambda\mu\nu\rho}\qquad\lambda,\mu,\nu,\rho=0,1\ldots
d-1,$ $\displaystyle
C^{(4)}_{\lambda\mu\nu\rho}\qquad\lambda,\mu,\nu,\rho=0,1\ldots d-1,$
$\displaystyle C^{(4)r}_{\lambda\mu\nu\rho},\qquad
C^{(d)r}_{\lambda\mu\nu\rho}\qquad\lambda,\mu,\nu,\rho=0,1,2,3\qquad\textrm{restricted
forms},$ (3.2.8)

as possible operators appearing in the counterterm action that renormalizes
the effective action. After differentiation of these expression with respect
to the metric, only a contraction of the result with the metric itself will
set a difference between these different definitions. If the indices of the
differentiation are left open, there will be no extra dependence on $d$ which
is generated in the various cases.
The operators above have a parametric dependence on the dimension and an index
variability, and both can be 4 or $d$.

The Weyl-scaling of the metric

$g_{\mu\nu}=e^{2\phi}\bar{g}_{\mu\nu}.$ (3.2.9)

gives

$C{}_{\lambda\mu\nu\rho}=e^{2\sigma}\bar{C}_{\lambda\mu\nu\rho},\qquad
C^{\lambda\mu\nu\rho}=e^{-6\sigma}\bar{C}^{\lambda\mu\nu\rho},$ (3.2.10)

and corresponds to Weyl covariance. By contraction, one gets

$\displaystyle C^{2}\equiv
g^{\alpha\mu}g^{\beta\nu}C^{\lambda}_{\gamma\alpha\beta}C^{\gamma}_{\lambda\mu\nu}=$
$\displaystyle
e^{-4\sigma(x)}\bar{C}^{\lambda}_{\gamma\alpha\beta}\bar{C}^{\gamma}_{\lambda\mu\nu}=e^{-4\sigma(x)}\bar{C}^{2}.$
(3.2.11)

Moreover, this tensor carries the symmetries of the Riemann tensor along with
the corresponding Bianchi identity.

$C_{\mu\nu\rho\sigma}=C_{\rho\sigma\mu\nu},$ (3.2.12)
$C_{\mu\nu\rho\sigma}=-C_{\nu\mu\rho\sigma}=-C_{\mu\nu\sigma\rho},$ (3.2.13)
$C_{\mu\nu\rho\sigma}+C_{\mu\sigma\nu\rho}+C_{\mu\rho\sigma\nu}=0.$ (3.2.14)

The choice of the Weyl counterterm is affected by the prescription dependence.
One could choose a counterterm of the form

$\frac{1}{\epsilon}V_{C^{2}}=\frac{\mu^{-\epsilon}}{\epsilon}\int
d^{d}x\sqrt{g}(C^{(d)})^{2},$ (3.2.15)

or

$\frac{1}{\epsilon}\tilde{V}_{C^{2}}=\frac{\mu^{-\epsilon}}{\epsilon}\int
d^{d}x\sqrt{g}(C^{(4)})^{2}.$ (3.2.16)

Let’s examine the differences between the two choices. Setting $d=4-\epsilon$
in (3.2.15), we get

$C^{(d)\alpha\beta\gamma\delta}C_{(d)\alpha\beta\gamma\delta}=R^{\alpha\beta\gamma\delta}R_{\alpha\beta\gamma\delta}-2R^{\alpha\beta}R_{\alpha\beta}+\frac{1}{3}R^{2}+\epsilon\left(-R^{\alpha\beta}R_{\alpha\beta}+\frac{5}{18}R^{2}\right),$
(3.2.17)

which can be written as

$(C^{(4-\epsilon)})^{2}=(C^{(4)})^{2}+\epsilon\left(-(R_{\mu\nu})^{2}+\frac{5}{18}R^{2}\right).$
(3.2.18)

Using the scaling property of the integration measure

$g=\textrm{det}g_{\alpha\beta}=\epsilon^{\mu_{1}\ldots\mu_{d}}g_{0\mu_{1}}\ldots
g_{0\mu_{d}}\to e^{2d\sigma}g,$ (3.2.19)

one obtains

$\frac{\delta}{\delta\sigma(x)}\int
d^{d}x\sqrt{-g}(C^{4)})^{2}(x)=\epsilon\sqrt{g}(C^{(4)})^{2},$ (3.2.20)

as well as

$\frac{\delta}{\delta\sigma(x)}\int
d^{d}x\sqrt{-g}(C^{d)})^{2}(x)=\epsilon\sqrt{g}(C^{(d)})^{2}.$ (3.2.21)

The use of (3.2.18) before the Weyl variation in $\sigma$, produces a
different result, since the operation of expanding in $d$ and the variation do
not commute. Looking separately to the variation of the second term of
(3.2.18) we have

$\frac{\delta}{\delta\sigma(x)}\int
d^{d}x\sqrt{-g}\left(-R_{\mu\nu}^{2}+\frac{5}{18}R^{2}\right)=-\frac{2}{3}\epsilon\sqrt{g}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R.$ (3.2.22)

Thus we obtain

$\frac{1}{d-4}\frac{\delta}{\delta\sigma(y)}\int
d^{d}x\sqrt{g}(C^{(4-\epsilon)})^{2}=\sqrt{g}\left((C^{(4)})^{2}-\frac{2}{3}\,\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,R\right).$ (3.2.23)

The renormalization of the entire functional is then obtained by the addition
of the counterterm action

$\displaystyle\mathcal{S}_{ct}$
$\displaystyle=-\frac{\mu^{-\varepsilon}}{\varepsilon}\,\int\,d^{d}x\,\sqrt{-g}\left(b\,(C^{(d)})^{2}+b^{\prime}\,E\right),$
(3.2.24)

corresponding to the Weyl tensor squared and the Euler density in $d=4$ as in
(3.2.4) and (7.3.6). Here, $\mu$ is a renormalization scale.While the Gauss-
Bonnet term does not carry any parametric dependence in $d$ its tensorial
structure is expanded in $d$ dimensions, according to the regularization. The
scaling of $E$ is more involved

$\sqrt{g}E=\sqrt{\bar{g}}e^{(d-4)\phi}\biggl{\\{}\bar{E}+(d-3)\bar{\nabla}_{\mu}\bar{J}_{1}^{\mu}+(d-3)(d-4)\bar{K}\biggl{\\}},$
(3.2.25)

where

$\bar{J}_{1}^{\mu}=8\bar{R}^{\mu\nu}\bar{\nabla}_{\nu}\phi-4\bar{R}\bar{\nabla}^{\mu}\phi+4(d-2)(\bar{\nabla}^{\mu}\phi\bar{\,}\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi-\bar{\nabla}^{\mu}\bar{\nabla}^{\nu}\phi\bar{\nabla}_{\nu}\phi+\bar{\nabla}^{\mu}\phi\bar{\nabla}_{\lambda}\phi\bar{\nabla}^{\lambda}\phi),$
(3.2.26)

$\bar{K}=4\bar{R}^{\mu\nu}\bar{\nabla}_{\mu}\phi\bar{\nabla}_{\nu}\phi-2\bar{R}\bar{\nabla}_{\lambda}\phi\bar{\nabla}^{\lambda}\phi+4(d-2)\bar{\,}\raise
0.5pt\hbox{$\vbox{\hrule\hbox{\vrule height=6.50403pt\kern 6.50403pt\vrule}
\hrule}$}\,\phi\bar{\nabla}_{\lambda}\phi\bar{\nabla}^{\lambda}\phi+(d-1)(d-2)(\bar{\nabla}_{\lambda}\phi\bar{\nabla}^{\lambda}\phi)^{2}.$
(3.2.27)

and all the barred operators are computed respect to the fiducial metric
$\bar{g}_{\mu\nu}$.
The Weyl tensor can be used to build a conformally invariant action at $d=4$
in the form

$S_{G}=\alpha\int d^{4}x\sqrt{g}(C^{(4)})^{2},$ (3.2.28)

which is the action of Weyl gravity. In the context of anomaly actions, the
derivative of $V_{C^{2}}$ (i.e. $V^{\prime}_{C^{2}}$) with respect to the
dimension $d$ will be part of the renormalized action and the approach can be
modified by the inclusion of a finite renormalization. For the remaining
section we will choose $(C^{(d)})^{2}$ as the operator appearing in the
counterterm action, and we will be using the variation

$\frac{\delta}{(d-4)\delta\sigma(x)}\int
d^{d}x\sqrt{-g}(C^{(d)})^{2}=\sqrt{-g}(C^{(d)})^{2},$ (3.2.29)

which differs from

$\frac{\delta}{(d-4)\delta\sigma(x)}\int
d^{d}x\sqrt{-g}(C^{(4)})^{2}=\sqrt{-g}\Bigg{(}(C^{(4)})^{2}-\frac{2}{3}\square
R\Bigg{)},$ (3.2.30)

introduced in some of the literature on the conformal anomaly. We remark that
the number of Weyl invariants operators depends on the dimension. The case of
$d=6$ is discussed in several works [91, 92, 93].
One can derive an anomaly action of the Wess-Zumino form starting from those
invariants, using the Weyl gauging approach [94, 95]. An example, in $d=6$,
can be found in [76].
The anomaly action that one introduces via the regularization process, is
derived using counterterms which are defined for any even dimension ($2k$),
and analytically continued around the integer value $k$.

In generic spacetime dimensions, the structure of the counterterm Lagrangian
is modified accordingly, with the Euler/ GB density formula

$\displaystyle
E_{d}=\frac{1}{2^{d/2}}\delta_{\mu_{1}\cdots\mu_{d}}^{\nu_{1}\cdots\nu_{d}}{R^{\mu_{1}\mu_{2}}}_{\nu_{1}\nu_{2}}\cdots{R^{\mu_{d-1}\mu_{d}}}_{\nu_{d-1}\nu_{d}}\
,$ (3.2.31)

which, for $d=4$, is quadratic in the curvatures. The topological nature of
$V_{E}$ is evident only if $d=2k$, but the regularization procedure will
modify the effective action with the inclusion of a finite contribution
($V^{\prime}_{E}$) which is metric-dependent, produced by the integration of
the Euler density in $d=2k+\epsilon$ dimensions.

#####  3.2.1  Gauss-Bonnet term and finite renormalization

The following section is based on [53]. We will address the nature of the
finite renormalization induced by the Gauss-Bonnet counterterm and examine
this counterterm in more detail.

###### The finite renormalization induced by $V_{E}$

It is quite obvious that the inclusion of $V_{E}$ induces a finite
renormalization of the effective action in $d=4$. This can be simply shown by
noticing that both $V_{E}$ and $V_{C^{2}}$ manifest an explicit dependence on
$\varepsilon$, i.e.

$V_{E/C^{2}}\equiv V_{E/C^{2}}(d),$ (3.2.32)

and the counterterm contributions can be expanded around $d=4$. For this
purpose, given a scalar functional $f(d)$, it will be convenient to denote its
Taylor expansion around $d=4$ in the form

$f(d)=\left[f\right]+\varepsilon\left[f^{\prime}\right]\qquad\left[f\right]\equiv
f(4),\left[f^{\prime}\right]\equiv f^{\prime}(4)$ (3.2.33)

with coefficients which are square bracketed once they are computed at $d=4$.
The expansion of $V_{E/C^{2}}(d)$, using these notations, takes the form

$\frac{1}{\varepsilon}V_{E/C^{2}}(d)=\frac{\mu^{-\varepsilon}}{\varepsilon}\left(\left[V_{E/C^{2}}\right]+\varepsilon\left[V_{E/C^{2}}^{\prime}\right]+O(\varepsilon^{2})\right),$
(3.2.34)

where the first correction to the residue at the pole in $\varepsilon$ comes
from the derivative respect to the dimension $d$. Notice that
$\left[V_{E}\right]$ is a topological term and it is therefore metric-
independent. Its value is related to the global topology of the spacetime and
it is therefore a pure number. Then, it is clear that the $1/\varepsilon$ term
will not contribute to the renormalization of the bare 4-T vertex, since each
counterterm vertex is generated by functional differentiation of (3.2.34) with
respect to the background metric.
The only contribution of the $V_{E}$ as $\varepsilon\to 0$ is related to
$\left[V^{\prime}_{E}\right]$, and it is indeed finite, as a 0/0 contribution
in $\varepsilon$. Therefore, the inclusion of $V_{E}$, will induce only a
finite renormalization of the bare vertex, and henceforth of the entire
effective action, since this result remains valid to all orders.
Finally, we can relate $\left[V^{\prime}_{E}\right]$ to the anomaly by the
equation

$\sqrt{g}E(x)=2g_{\alpha\beta}\left[V^{\prime}_{E}\right]^{\alpha\beta}$
(3.2.35)

where the indices of $\left[V^{\prime}_{E}\right]^{\alpha\beta}$ run in four
dimensions.

###### Open indices in $V_{E}$

We can investigate this point in more detail.
The extraction of a counterterm vertex, as mentioned above, requires a
functional differentiation of (3.2.34). It will appear in the $\varepsilon\to
0$ limit, in the form

$\displaystyle\left[\frac{\delta}{\delta
g_{\mu_{1}\nu_{1}}(x_{1})}\frac{\delta}{\delta
g_{\mu_{2}\nu_{2}}(x_{2})}\ldots\frac{\delta}{\delta
g_{\mu_{n}\nu_{n}}(x_{n})}\frac{\mu^{-\varepsilon}}{\varepsilon}V_{E}(d)\right]=$
$\displaystyle\frac{\delta}{\delta
g_{\mu_{2}\nu_{2}}(x_{2})}\ldots\frac{\delta}{\delta
g_{\mu_{n}\nu_{n}}(x_{n})}\left[V_{E}^{\prime}\right]$
$\displaystyle\equiv\left[V_{E}^{\prime}\right]^{\mu_{1}\nu_{1}\ldots\mu_{n}\nu_{n}}(x_{1},\ldots
x_{n}).$

The rhs of the this expression is clearly regular as $\varepsilon\to 0$, and
defines a finite renormalization of the corresponding $n$-graviton vertex.
Obviously, the same is not true for the $V_{C^{2}}$ counterterm since
$\left[V_{C^{2}}\right]$ will be metric-dependent. In this second case the
analogous of (3.2.1) is

$\displaystyle\left[\frac{\delta}{\delta
g_{\mu_{1}\nu_{1}}(x_{1})}\frac{\delta}{\delta
g_{\mu_{2}\nu_{2}}(x_{2})}\ldots\frac{\delta}{\delta
g_{\mu_{n}\nu_{n}}(x_{n})}\frac{\mu^{-\varepsilon}}{\varepsilon}V_{C^{2}}(d)\right]=$
$\displaystyle\frac{1}{\varepsilon}\left[V_{C^{2}}\right]^{\mu_{1}\nu_{1}\ldots\mu_{n}\nu_{n}}(x_{1},\ldots
x_{n})+\left[V_{C^{2}}^{\prime}\right]^{\mu_{1}\nu_{1}\ldots\mu_{n}\nu_{n}}(x_{1},\ldots
x_{n}).$

The second, finite term of this expression ($V^{\prime}_{C^{2}}$), will appear
in all the renormalized anomalous WI that we will present below, when we will
perform the $\varepsilon\to 0$ limit of those eqs. in a flat background.
One important comment concerns possible ambiguities arising from the
differentiation of terms such as $\left[V^{\prime}_{E}\right]$ with respect to
the metric $g_{\mu\nu}$. We mention that such ambiguities arise only in the
presence of contractions with a metric, and not otherwise. We will be dealing
with this second case, in the subsection below.
As in any practical application of DR, once the renormalization of a vertex is
completed, a tensor structure with open indices, which is generated by the
procedure, is automatically dimensionally reduced to the $d=4$ subspace,
giving the final, finite expression of such vertex.
Indeed, expanding around $d=4$, (3.2.34) can be rewritten in the form

$\displaystyle\frac{\delta}{\delta
g_{\mu_{1}\nu_{1}}(x_{1})}\frac{\delta}{\delta
g_{\mu_{2}\nu_{2}}(x_{2})}\ldots\frac{\delta}{\delta
g_{\mu_{n}\nu_{n}}(x_{n})}V_{E}(d)=$
$\displaystyle\left[V_{E}\right]^{\mu_{1}\nu_{1}\ldots\mu_{n}\nu_{n}}(x_{1},\ldots
x_{n})+\varepsilon\left[V_{E}^{\prime}\right]^{\mu_{1}\nu_{1}\ldots\mu_{n}\nu_{n}}(x_{1},\ldots
x_{n}),$

which in the $d\to 4$ limit vanishes, since the two terms on the rhs vanish
separately. Notice that the first term
$\left[V_{E}\right]^{\mu_{1}\nu_{1}\ldots\mu_{n}\nu_{n}}$ is topological and
is simply obtained by the differentiation of $V_{E}$ in $d=4$. As far as we
leave all the indices open, then the following relation holds

$\displaystyle\lim_{d\to 4}\left(\frac{\delta}{\delta
g_{\mu_{1}\nu_{1}}(x_{1})}\frac{\delta}{\delta
g_{\mu_{2}\nu_{2}}(x_{2})}\ldots\frac{\delta}{\delta
g_{\mu_{n}\nu_{n}}(x_{n})}V_{E}(d)\right)=$ $\displaystyle\frac{\delta}{\delta
g_{\mu_{1}\nu_{1}}(x_{1})}\frac{\delta}{\delta
g_{\mu_{2}\nu_{2}}(x_{2})}\ldots\frac{\delta}{\delta
g_{\mu_{n}\nu_{n}}(x_{n})}\lim_{d\to 4}V_{E}(d)$ $\displaystyle=$
$\displaystyle 0$ (3.2.39)

In other words, the operation of functional differentiation with open indices
and limit to $d=4$ commute. The topological nature of this part of the 3-point
vertex counterterm was originally noted in [54] (see also [96]). The explicit
check of this topological relation requires the dimensional reduction of the
$\delta^{d}_{\mu\nu}$ to $\delta^{(4)}_{\mu\nu}$, using the $n$-$p$ basis
(discussed in Sec. 6.12) to take into account momentum degeneracies in
specific dimensions, and indeed it has been verified for $n=3$ and $n=4$ [71,
86], i.e. for three- and four-point functions. We stress once again that
(3.2.1) holds as far as we do not perform any contraction of the tensor
indices with the external metric.
In the presence of metric contractions, these results can be re-addressed
using a Weyl variation. We are going to illustrate this point in some detail
below, defining a straightforward procedure in order to handle this second
case correctly.
Contributions proportional to $V^{\prime}_{E}$ will be present once we
contract the WIs with the $\Sigma$ projector defined below in (6.7.11), using
the longitudinal-trace/ transverse-traceless decomposition of the correlators.
One finds, by a direct computation, the emergence of virtual scalar exchanges
(or mixing terms) in the decomposed WI, which signal the breaking of the
conformal symmetry induced by the conformal anomaly. Such terms are easy to
derive simply due to the presence of a combined
$\pi^{\mu\nu}\delta^{\alpha\beta}$ projector in $\Sigma$, in each of the
external legs of the gravitational vertex. This defines a coupling of a scalar
pole to the anomaly functional.

###### Closed indices

The analysis of relations involving the topological counterterm $V_{E}$ in the
presence of contraction with the metric cannot be performed as above, but, as
we have mentioned, can be addressed correctly by relating the contraction to a
Weyl variation. A crucial relation is

$\frac{\delta}{(d-4)\delta\sigma(x)}\int d^{d}x\sqrt{-g}E_{4}=\sqrt{-g}E_{4}.$
(3.2.40)

For this purpose consider the Weyl scaling relations

$\sqrt{g}=e^{d\sigma.}\sqrt{\bar{g}}\qquad E=e^{-4\sigma}\bar{E},$ (3.2.41)

from which we derive the constraints

$\frac{\delta}{\delta\sigma(x_{3})}\frac{\delta}{\delta\sigma(x_{2})}\frac{\delta}{\delta\sigma(x_{1})}V_{E}=\varepsilon^{3}\sqrt{g}E\delta^{d}(x_{1}-x_{2})\delta^{d}(x_{2}-x_{3})$
(3.2.42)

which can be generalized to any multiple derivative

$\frac{\delta}{\delta\sigma(x_{n})}\ldots\frac{\delta}{\delta\sigma(x_{2})}\frac{\delta}{\delta\sigma(x_{1})}V_{E}=\varepsilon^{n}\sqrt{g}E\delta^{d}(x_{1}-x_{2})\delta^{d}(x_{2}-x_{3})\ldots\delta^{d}(x_{n-1}-x_{n}).$
(3.2.43)

If we use

$2g_{\alpha\beta}\frac{\delta}{\delta
g_{\alpha\beta}}=\frac{\delta}{\delta\sigma(x)}$ (3.2.44)

and

$\frac{\delta}{\delta\sigma(x_{2})}g^{\mu\nu}(x_{1})=-2g^{\mu\nu}\delta^{4}(x_{2}-x_{1})$
(3.2.45)

we derive several relations. For instance, expanding the left-hand side of
(3.2.43) for $n=2$ we obtain

$\displaystyle\frac{\delta}{\delta\sigma(x_{2})}\frac{\delta}{\delta\sigma(x_{1})}V_{E}=$
$\displaystyle
4g_{\mu\nu}V_{E}^{\mu\nu}(x_{1})\delta^{d}(x_{1}-x_{2})+4g_{\alpha\beta}(x_{2})g_{\mu\nu}(x_{1})V_{E}^{\alpha\beta\mu\nu}(x_{2},x_{1})$
$\displaystyle=$
$\displaystyle\varepsilon^{2}{\sqrt{g(x_{1})}}E(x_{1})\delta^{d}(x_{1}-x_{2}).$
(3.2.46)

The expression above can be evaluated in the flat limit both in 4 and $d$
dimensions. Consider, for instance, the flat limit with $d$ generic. We use
the fact that $E$ and $V_{E}^{\mu\nu}$ vanish in the flat limit to obtain the
constraint

$\displaystyle\delta^{(d)}_{\alpha\beta}\delta^{(d)}_{\mu\nu}V_{E}^{\alpha\beta\mu\nu}=0,$
(3.2.47)

which remains valid for $d=4$ since the equation is analytic in $d$, and in
particular in $d=4$. Similarly, one can derive relations for the traces of
higher point functions, which are discussed in the Appendix B.2.

###### Open and closed indices

We can generalize the method and perform combined variations respect to
$\sigma$ and to the metric in order to derive some additional relations. For
this purpose we consider the expression

$J^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}\equiv\frac{\delta}{\delta\sigma(x_{1})}\frac{\delta}{\delta
g_{\mu_{2}\nu_{2}}(x_{2})}\frac{\delta}{\delta
g_{\mu_{3}\nu_{3}}(x_{3})}\frac{\delta}{\delta
g_{\mu_{4}\nu_{4}}(x_{4})}V_{E}.$ (3.2.48)

We can use the commutation relation

$\left[\frac{\delta}{\delta\sigma(x_{1})},\frac{\delta}{\delta
g_{\mu_{2}\nu_{2}}(x_{2})}\right]V_{E}=-2V_{E}^{\mu_{2}\nu_{2}}(x_{2})\delta^{d}(x_{1}-x_{2})$
(3.2.49)

to rearrange the differentiations in the form

$\displaystyle J^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}=$
$\displaystyle\frac{\delta}{\delta
g_{\mu_{2}\nu_{2}}(x_{2})}\frac{\delta}{\delta
g_{\mu_{3}\nu_{3}}(x_{3})}\frac{\delta}{\delta
g_{\mu_{4}\nu_{4}}(x_{4})}\frac{\delta}{\delta\sigma(x_{1})}V_{E}$
$\displaystyle-$ $\displaystyle
2V_{E}^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(x_{2},x_{3},x_{4})\left(\delta^{d}(x_{1}-x_{2})+\delta^{d}(x_{1}-x_{3})+\delta^{d}(x_{1}-x_{4})\right)$

and use (3.2.40) to derive the constraint

$\displaystyle J^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}=$
$\displaystyle(d-4)\left(\sqrt{g}E\right)^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(x_{2},x_{3},x_{4})$
$\displaystyle-$ $\displaystyle
2V_{E}^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(x_{2},x_{3},x_{4})\left(\delta^{d}(x_{1}-x_{2})+\delta^{d}(x_{1}-x_{3})+\delta^{d}(x_{1}-x_{4})\right),$

which is (7.10.1) in coordinate space. The second term in the expression
above, related to the pinched contributions $\delta(x_{1}-x_{i})\,(i=2,3,4)$
is also d-dimensional and needs to be expanded around $d=4$. The expansion is
performed as in (3.2.34)

$V_{E}^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}=\left[V_{E}^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}\right]+\varepsilon\left[{V^{\prime}_{E}}^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}\right].$
(3.2.52)

Notice that the first contribution vanishes for being purely topological,
while the second, transformed to momentum space, will appear in all the
renormalized anomalous CWIs that we will discuss in the sections below

$(d-4)\left[{V^{\prime}_{E}}^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}\right]\to(d-4)\left[{V^{\prime}}_{E}^{\mu_{2}\nu_{2}\dots\mu_{4}\nu_{4}}(p_{1}+p_{2},p_{3},p_{4})\right].$
(3.2.53)

It accounts for a finite $0/0$ contribution in the expansion of the
$1/\varepsilon\,V_{E}(d)$ counterterm in the variable $\varepsilon$.

For a generic $nT$ correlator, the only counterterm needed for its
renormalization, is obtained by the inclusion of a classical gravitational
vertex generated by the differentiation of (3.2.24) $n$ times. The
coefficients $b$ and $b^{\prime}$ will be fixed by the field content, number
of scalars, fermions and spin-$1$ field, of the quantum corrections.

#####  3.2.2 The inclusion of the anomaly

To study the anomaly contribution to each correlation function, we start from
the 1-point function. In a generic background $g$, the renormalized 1-point
function is decomposed as

$\langle
T^{\mu\nu}\rangle_{R}=\frac{2}{\sqrt{g}}\frac{\delta\mathcal{S}_{R}}{\delta
g_{\mu\nu}}=\langle
T^{\mu\nu}\rangle_{A}+\langle\overline{T}^{\mu\nu}\rangle_{f},$ (3.2.54)

with

$g^{\mu\nu}\frac{\delta\Gamma}{\delta
g^{\mu\nu}}=g^{\mu\nu}\frac{\delta\mathcal{S}_{A}}{\delta
g^{\mu\nu}}\equiv\frac{\sqrt{g}}{2}g_{\mu\nu}\langle T^{\mu\nu}\rangle_{A},$
(3.2.55)

being the trace anomaly equation, and
$\langle\overline{T}^{\mu\nu}\rangle_{f}$ is the Weyl-invariant (traceless)
term. We will omit below the subscript $R$ from the quantum average $\langle
T\rangle_{R}$.

According to (3.2.3), these scaling violations can be written in the form

$\langle{T^{\mu}_{\ \ \mu}(x)}\rangle=\mathcal{A}(x),$ (3.2.56)

\- having dropped the suffix Ren from the renormalized stress energy tensor -
where the finite terms on the right hand side of this equation denote the
anomaly contribution with

$\mathcal{A}(x)=\sqrt{-g(x)}\bigg{[}b\,C^{2}(x)+b^{\prime}E(x)\bigg{]},$
(3.2.57)

being the anomaly functional. For $n$-point functions the trace anomaly, as
well as all the other CWIs, are far more involved, and take a hierarchical
structure.

For all the other WIs, in DR the structure of the equations can be analyzed in
two different frameworks.
In one of them, we are allowed to investigate the correlators directly in $d$
spacetime dimensions, deriving ordinary (anomaly-free) CWIs, which are then
modified by the inclusion of the 4-dimensional counterterm as $d\to 4$. In
this limit, the conformal constraints become anomalous and the hierarchical
equations are modified by the presence of extra terms which are anomaly-
related.
Alternatively, one can proceed by working out the equations directly in d=4,
with the inclusion of the contributions coming from the anomaly functional.

##  Part II Four-point functions in momentum space

###  Chapter 4 A brief review of the scalar four-point correlators

In this section which is heavily based [61, 26] we are going to present some
properties of the four-point scalar function. It is well known that four-point
and higher functions are not completely constrained by the the conformal
symmetry. As we have already shown in 1, for the case of the four-point
correlator one can identify the invariant cross sections

$u(x_{i})=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}},\qquad
v(x_{i})=\frac{x_{23}^{2}x_{41}^{2}}{x_{13}^{2}x_{24}^{2}},$ (4.0.1)

The general solution can be written in the form

$\langle\mathcal{O}_{1}(x_{1})\mathcal{O}_{2}(x_{2})\mathcal{O}_{3}(x_{3})\mathcal{O}_{4}(x_{4})\rangle=\frac{h(u(x_{i}),v(x_{i}))}{\left(x_{12}^{2}\right)^{\frac{\Delta_{1}+\Delta_{2}}{2}}\left(x_{34}^{2}\right)^{\frac{\Delta_{3}+\Delta_{4}}{2}}},$
(4.0.2)

where $h(u(x_{i}),v(x_{i}))$ remains unspecified. As we have seen, in momentum
space the correlator depends on six invariants that we will normalize as
$p_{i}=|\sqrt{p_{i}\,{}^{2}}|$, $i=1,\dots,4$, representing the magnitudes of
the momenta, and $s=|\sqrt{(p_{1}+p_{2})^{2}}|$,
$t=|\sqrt{(p_{2}+p_{3})^{2}}|$ the two Mandelstam invariants redefined by a
square root, for which

$\langle
O(p_{1})\,O(p_{2})\,O(p_{3})\,O(\bar{p}_{4})\rangle=\Phi(p_{1},p_{2},p_{3},p_{4},s,t).$
(4.0.3)

This correlation function has to verify the dilatation Ward identity

$\left[\sum_{i=1}^{4}\Delta_{i}-3d-\sum_{i=1}^{3}p_{i}^{\mu}\frac{\partial}{\partial
p_{i}^{\mu}}\right]\Phi(p_{1},p_{2},p_{3},p_{4},s,t)=0,$ (4.0.4)

and the special conformal Ward identities

$\sum_{i=1}^{3}\left[2(\Delta_{i}-d)\frac{\partial}{\partial
p_{i\,\kappa}}-2p_{i}^{\alpha}\frac{\partial^{2}}{\partial
p_{i}^{\alpha}\partial
p_{i}^{\kappa}}+p_{i}^{\kappa}\frac{\partial^{2}}{\partial
p_{i}^{\alpha}\partial
p_{i\,\alpha}}\right]\Phi(p_{1},p_{2},p_{3},p_{4},s,t)=0.$ (4.0.5)

One can split these equations in terms of the invariants of the four-point
function written in (4.0.3), by using the chain rules

$\displaystyle\frac{\partial}{\partial p_{1\,\mu}}$
$\displaystyle=\frac{p_{1}^{\mu}}{p_{1}}\frac{\partial}{\partial
p_{1}}-\frac{\bar{p}_{4}^{\mu}}{p_{4}}\frac{\partial}{\partial
p_{4}}+\frac{p_{1}^{\mu}+p_{2}^{\mu}}{s}\frac{\partial}{\partial s},$ (4.0.6)
$\displaystyle\frac{\partial}{\partial p_{2\,\mu}}$
$\displaystyle=\frac{p_{2}^{\mu}}{p_{2}}\frac{\partial}{\partial
p_{2}}-\frac{\bar{p}_{4}^{\mu}}{p_{4}}\frac{\partial}{\partial
p_{4}}+\frac{p_{1}^{\mu}+p_{2}^{\mu}}{s}\frac{\partial}{\partial
s}+\frac{p_{2}^{\mu}+p_{3}^{\mu}}{t}\frac{\partial}{\partial t},$ (4.0.7)
$\displaystyle\frac{\partial}{\partial p_{3\,\mu}}$
$\displaystyle=\frac{p_{3}^{\mu}}{p_{3}}\frac{\partial}{\partial
p_{3}}-\frac{\bar{p}_{4}^{\mu}}{p_{4}}\frac{\partial}{\partial
p_{4}}+\frac{p_{2}^{\mu}+p_{3}^{\mu}}{t}\frac{\partial}{\partial t},$ (4.0.8)

where $\bar{p}_{4}^{\mu}=-p_{1}^{\mu}-p_{2}^{\mu}-p_{3}^{\mu}$. From this
prescription the dilatation WI becomes

$\displaystyle\bigg{[}(\Delta_{t}-3d)-\sum_{i=1}^{4}p_{i}\frac{\partial}{\partial
p_{i}}-s\frac{\partial}{\partial s}-t\frac{\partial}{\partial
t}\bigg{]}\Phi(p_{1},p_{2},p_{3},p_{4},s,t)=0,$ (4.0.9)

with $\Delta_{t}=\sum_{i=1}^{4}\Delta_{i}$ is the total scaling, and the
special CWIs can be written as

$\sum_{i=1}^{3}\ p_{i}^{\kappa}\,C_{i}=0,$ (4.0.10)

where the coefficients $C_{i}$ are differential equations of the second order
with respect to the six invariants previously defined. Being $p_{1}^{\kappa},\
p_{2}^{\kappa},\ p_{3}^{\kappa}$, in (4.0.10) independent variables, we derive
three scalar second order equations for each of the three $C_{i}$, which must
vanish independently.
At this stage the procedure in order to simplify the corresponding equations
is similar to the one described in [24, 25]. Thus equation are rewritten in
the form

$\displaystyle C_{1}$ $\displaystyle=\bigg{\\{}\frac{\partial^{2}}{\partial
p_{1}^{2}}+\frac{(d-2\Delta_{1}+1)}{p_{1}}\frac{\partial}{\partial
p_{1}}-\frac{\partial^{2}}{\partial
p_{4}^{2}}-\frac{(d-2\Delta_{4}+1)}{p_{4}}\frac{\partial}{\partial p_{4}}$
$\displaystyle\qquad+\frac{1}{s}\frac{\partial}{\partial
s}\left(p_{1}\frac{\partial}{\partial p_{1}}+p_{2}\frac{\partial}{\partial
p_{2}}-p_{3}\frac{\partial}{\partial p_{3}}-p_{4}\frac{\partial}{\partial
p_{4}}\right)+\frac{(\Delta_{3}+\Delta_{4}-\Delta_{1}-\Delta_{2})}{s}\frac{\partial}{\partial
s}$
$\displaystyle\qquad+\frac{(p_{2}^{2}-p_{3}^{2})}{st}\frac{\partial^{2}}{\partial
s\partial t}\bigg{\\}}\,\Phi(p_{1},p_{2},p_{3},p_{4},s,t)=0,$ (4.0.11)

for $C_{1}$ and similarly for the other coefficients.
It has been shown that these equations involve a differential operator which
is of hypergeometric type combined with other terms.

####  4.1 The dcc solutions

We consider possible solutions of the conformal constraints (4.0.11) which are
built around specific dual conformal ansätze [26]. It is noteworthy that the
property of an object to be conformally and dual conformally invariant is also
known as conformal Yangian (CY) invariance.

The number of independent equations in (4.0.10), by using the ansatz that we
are going to present below, will then reduce from three down to two. We choose
the ansatz

$\Phi(p_{i},s,t)=\big{(}s^{2}t^{2}\big{)}^{n_{s}}\,F(x,y),$ (4.1.1)

where $n_{s}$ is a coefficient (scaling factor of the ansatz) that we will fix
below by the dilatation WI, and the variables $x$ and $y$ are defined by the
quartic ratios

$x=\frac{p_{1}^{2}\,p_{3}^{2}}{s^{2}\,t^{2}},\qquad
y=\frac{p_{2}^{2}\,p_{4}^{2}}{s^{2}\,t^{2}}.$ (4.1.2)

By inserting the ansatz (4.1.1) the dilatation Ward identities, and turning to
the new variables $x$ and $y$, we obtain from (4.0.9) the condition

$\bigg{[}(\Delta_{t}-3d)-\sum_{i=1}^{4}p_{i}\frac{\partial}{\partial
p_{i}}-s\frac{\partial}{\partial s}-t\frac{\partial}{\partial
t}\bigg{]}\big{(}s^{2}t^{2}\big{)}^{c}\,F(x,y)=\big{(}s^{2}t^{2}\big{)}^{n_{s}}\big{[}(\Delta_{t}-3d)-4n_{s}\big{]}\,F(x,y)=0,$
(4.1.3)

which determines $n_{s}=(\Delta_{t}-3d)/4$, giving

$\Phi(p_{i},s,t)=\big{(}s^{2}t^{2}\big{)}^{(\Delta_{t}-3d)/4}\,F(x,y).$
(4.1.4)

The functional form of $F(x,y)$ will then be furtherly constrained.

#####  4.1.1 Equal scaling solutions

We start investigating the solutions of these equations by assuming that the
scaling dimensions of all the scalar operators are equal $\Delta_{i}=\Delta$,
$i=1,\dots,4$. The special conformal Ward identities, re-expressed in terms of
$x$ and $y$, are given in the form

$\left\\{\begin{aligned}
&\bigg{[}y(1-y)\partial_{yy}-2x\,y\,\partial_{xy}-x^{2}\partial_{xx}-(1-2n_{s})x\,\partial_{x}+\left(1-\Delta+\frac{d}{2}-y(1-2n_{s})\right)\,\partial_{y}-n_{s}^{2}\bigg{]}F(x,y)=0\\\\[8.61108pt]
&\bigg{[}x(1-x)\partial_{xx}-2x\,y\,\partial_{xy}-y^{2}\partial_{yy}-(1-2n_{s})y\,\partial_{y}+\left(1-\Delta+\frac{d}{2}-x(1-2n_{s})\right)\,\partial_{x}-n_{s}^{2}\bigg{]}F(x,y)=0,\end{aligned}\right.,$
(4.1.5)

where we recall that $n_{s}$ is the scaling of the correlation function under
dilatations, now given by

$n_{s}=\Delta-\frac{3d}{4},$ (4.1.6)

since $\Delta_{t}=4\,\Delta$. Eqs. (4.1.5) correspond to a hypergeometric
system and its solutions can be expressed as linear combinations of four
Appell functions $F_{4}$ of two variables $x$ and $y$, as in the case of
three-point functions. The general solution of such system is expressed as

$\displaystyle\Phi(p_{i},s,t)$
$\displaystyle=\big{(}s^{2}t^{2}\big{)}^{(\Delta_{t}-3d)/4}\,\sum_{a,b}c(a,b,\Delta)x^{a}y^{b}F_{4}\left(\alpha(a,b),\beta(a,b),\gamma(a),\gamma^{\prime}(b);x,y\right).$
(4.1.7)

Notice that the solution is similar to that of the three-point functions given
by (2.5.16).
The general solution (4.1.7) has been written as a linear superposition of
these, with independent constants $c(a,b)$ labeled by the exponents $a,b$

$\displaystyle a=0,\,\Delta-\frac{d}{2},$ $\displaystyle
b=0,\,\Delta-\frac{d}{2},$ (4.1.8)

which fix the dependence of the $F_{4}$

$\displaystyle\alpha(a,b)=\frac{3}{4}d-\Delta+a+b,$
$\displaystyle\beta(a,b)=\frac{3}{4}d-\Delta+a+b,$
$\displaystyle\Gamma(a)=\frac{d}{2}-\Delta+1+2a,$
$\displaystyle\gamma^{\prime}(b)=\frac{d}{2}-\Delta+1+2b.$ (4.1.9)

#####  4.1.2 More general conditions

The solution that we have identified in the equal scaling case can be extended
by relaxing the conditions on the scaling dimensions, in the form

$\Delta_{1}=\Delta_{3}=\Delta_{x},\qquad\Delta_{2}=\Delta_{4}=\Delta_{y}.$
(4.1.10)

In this case the CWIs give the system of equations

$\left\\{\begin{aligned}
&\bigg{[}y(1-y)\partial_{yy}-2x\,y\,\partial_{xy}-x^{2}\partial_{xx}-(1-2n_{s})x\,\partial_{x}+\left(1-\Delta_{y}+\frac{d}{2}-y(1-2n_{s})\right)\,\partial_{y}-n_{s}^{2}\bigg{]}F(x,y)=0\\\\[8.61108pt]
&\bigg{[}x(1-x)\partial_{xx}-2x\,y\,\partial_{xy}-y^{2}\partial_{yy}-(1-2n_{s})y\,\partial_{y}+\left(1-\Delta_{x}+\frac{d}{2}-x(1-2n_{s})\right)\,\partial_{x}-n_{s}^{2}\bigg{]}F(x,y)=0\end{aligned}\right.,$
(4.1.11)

where now $n_{s}$ is defined as

$n_{s}=\frac{\Delta_{x}}{2}+\frac{\Delta_{y}}{2}-\frac{3}{4}d,$ (4.1.12)

whose solutions are expressed as

$\Phi(p_{i},s,t)=\big{(}s^{2}t^{2}\big{)}^{(\Delta_{t}-3d)/4}\,\sum_{a,b}c(a,b,\vec{\Delta}_{t})x^{a}y^{b}F_{4}\left(\alpha(a,b),\beta(a,b),\gamma(a),\gamma^{\prime}(b);x,y\right),$
(4.1.13)

with $\vec{\Delta}_{t}=(\Delta_{x},\Delta_{y},\Delta_{x},\Delta_{y})$,
$\Delta_{t}=2\Delta_{x}+2\Delta_{y}$ and the Fuchsian points are fixed by the
conditions

$\displaystyle a=0,\,\Delta_{x}-\frac{d}{2},$ $\displaystyle
b=0,\,\Delta_{y}-\frac{d}{2},$
$\displaystyle\alpha(a,b)=\frac{3}{4}d-\frac{\Delta_{x}}{2}-\frac{\Delta_{y}}{2}+a+b,$
$\displaystyle\beta(a,b)=\frac{3}{4}d-\frac{\Delta_{x}}{2}-\frac{\Delta_{y}}{2}+a+b,$
$\displaystyle\gamma(a)=\frac{d}{2}-\Delta_{x}+1+2a,$
$\displaystyle\gamma^{\prime}(b)=\frac{d}{2}-\Delta_{y}+1+2b.$ (4.1.14)

The domain of convergence of such solutions for $F_{4}$ is bounded by the
relation

$\sqrt{x}+\sqrt{y}<1,$ (4.1.15)

which is satisfied in a significant kinematic region, and in particular at
large energy and momentum transfers. Notice that the analytic continuation of
(4.1.13) in the physical region can be simply obtained by sending
$t^{2}\to-t^{2}$ (with $t^{2}<0$) and leaving all the other invariants
untouched. In this case we get

$\sqrt{p_{1}^{2}p_{3}^{2}}+\sqrt{p_{2}^{2}p_{4}^{2}}<\sqrt{-s^{2}t^{2}}.$
(4.1.16)

At large energy and momentum transfers the correlator exhibits a power-like
behavior of the form

$\Phi(p_{i},s,t)\sim\frac{1}{(-s^{2}t^{2})^{(3d-\Delta_{t})/4}}.$ (4.1.17)

Given the connection between the function $F_{4}$ and the triple-K integrals,
we will reformulate this solution in terms of such integrals. They play a key
role in the solution of the CWIs for tensor correlators, as discussed in [21],
for three-point functions.

The link between $3$\- and four-point functions outlined in the previous
section allows to re-express the solutions in terms of a class of parametric
integrals of 3 Bessel functions, as done in the case of the scalar and tensor
correlators [21], with the due modifications. We consider the case of the
solutions characterized by
$\Delta_{1}=\Delta_{2}=\Delta_{3}=\Delta_{4}=\Delta$ or
$\Delta_{1}=\Delta_{3}=\Delta_{x}$ and $\Delta_{2}=\Delta_{4}=\Delta_{y}$. We
will show that the solution can be written in terms of triple-K integrals
which are connected to the Appell function $F_{4}$ by the relation

$\displaystyle\int_{0}^{\infty}dx\>x^{\alpha-1}K_{\lambda}(ax)K_{\mu}(bx)K_{\nu}(cx)=\frac{2^{\alpha-4}}{c^{\alpha}}\bigg{[}B(\lambda,\mu)+B(\lambda,-\mu)+B(-\lambda,\mu)+B(-\lambda,-\mu)\bigg{]},$
(4.1.18)

where

$\displaystyle B(\lambda,\mu)$
$\displaystyle=\left(\frac{a}{c}\right)^{\lambda}\left(\frac{b}{c}\right)^{\mu}\Gamma\left(\frac{\alpha+\lambda+\mu-\nu}{2}\right)\Gamma\left(\frac{\alpha+\lambda+\mu+\nu}{2}\right)\Gamma(-\lambda)\Gamma(-\mu)\times$
$\displaystyle\qquad\times
F_{4}\left(\frac{\alpha+\lambda+\mu-\nu}{2},\frac{\alpha+\lambda+\mu+\nu}{2};\lambda+1,\mu+1;\frac{a^{2}}{c^{2}},\frac{b^{2}}{c^{2}}\right),$
(4.1.19)

valid for

$\textrm{Re}\,\alpha>|\textrm{Re}\,\lambda|+|\textrm{Re}\,\mu|+|\textrm{Re}\,\nu|,\qquad\textrm{Re}\,(a+b+c)>0,\\\
$

with the Bessel functions $K_{\nu}$ satisfying the equations

$\begin{split}\frac{\partial}{\partial
p}\big{[}p^{\beta}\,K_{\beta}(p\,x)\big{]}&=-x\,p^{\beta}\,K_{\beta-1}(px),\\\
K_{\beta+1}(x)&=K_{\beta-1}(x)+\frac{2\beta}{x}K_{\beta}(x).\end{split}$
(4.1.20)

In particular the solution can be written as

$I_{\alpha\\{\beta_{1},\beta_{2},\beta_{3}\\}}(p_{1}\,p_{3};p_{2}\,p_{4};s\,t)=\int_{0}^{\infty}\,dx\,x^{\alpha}\,(p_{1}\,p_{3})^{\beta_{1}}\,(p_{2}\,p_{4})^{\beta_{2}}\,(s\,t)^{\beta_{3}}\,K_{\beta_{1}}(p_{1}\,p_{3}\,x)\,K_{\beta_{2}}(p_{2}\,p_{4}\,x)\,K_{\beta_{3}}(s\,t\,x).$
(4.1.21)

Using various relations, the dilatation WI’s (4.0.9) become

$(\Delta_{t}-3d)I_{\alpha\\{\beta_{1},\beta_{2},\beta_{3}\\}}+2p_{1}^{2}p_{3}^{2}\
I_{\alpha+1\\{\beta_{1}-1,\beta_{2},\beta_{3}\\}}+2p_{2}^{2}p_{4}^{2}\
I_{\alpha+1\\{\beta_{1},\beta_{2}-1,\beta_{3}\\}}+2s^{2}t^{2}\
I_{\alpha+1\\{\beta_{1},\beta_{2},\beta_{3}-1\\}}=0,$ (4.1.22)

where the arguments of the $I_{\alpha\\{\beta_{1}\beta_{2}\beta_{3}\\}}$
function, written explicitly in (4.1.21), have been omitted for simplicity.
The $I$ integrals satisfy the differential equations

$\displaystyle\frac{1}{s}\frac{\partial}{\partial
s}\left(p_{1}\frac{\partial}{\partial p_{1}}+p_{2}\frac{\partial}{\partial
p_{2}}-p_{3}\frac{\partial}{\partial p_{3}}-p_{4}\frac{\partial}{\partial
p_{4}}\right)I_{\alpha\\{\beta_{1},\beta_{2},\beta_{3}\\}}$ $\displaystyle=0,$
(4.1.23) $\displaystyle\frac{1}{t}\frac{\partial}{\partial
t}\left(p_{1}\frac{\partial}{\partial p_{1}}+p_{4}\frac{\partial}{\partial
p_{4}}-p_{2}\frac{\partial}{\partial p_{2}}-p_{3}\frac{\partial}{\partial
p_{3}}\right)I_{\alpha\\{\beta_{1},\beta_{2},\beta_{3}\\}}$ $\displaystyle=0,$
(4.1.24)

which can be checked using the relations given in the same appendix, and we
finally obtain

$(\Delta_{t}-3d+2\alpha+2-2\beta_{t})I_{\alpha\\{\beta_{1},\beta_{2},\beta_{3}\\}}=0,$
(4.1.25)

where $\beta_{t}=\beta_{1}+\beta_{2}+\beta_{3}$. In order to satisfy this
equation the $\alpha$ parameter has to be equal to a particular value given by

$\tilde{\alpha}\equiv\frac{3}{2}d+\beta_{t}-1-\frac{\Delta_{t}}{2}.$ (4.1.26)

In the particular case of $\Delta_{i}=\Delta$, the special conformal Ward
identities are given by

$\left\\{\begin{aligned} &\\\\[-5.16663pt]
&\bigg{[}\frac{\partial^{2}}{\partial
p_{1}^{2}}+\frac{(d-2\mathcal{D}+1)}{p_{1}}\frac{\partial}{\partial
p_{1}}-\frac{\partial^{2}}{\partial
p_{3}^{2}}-\frac{(d-2\mathcal{D}+1)}{p_{3}}\frac{\partial}{\partial
p_{3}}+\frac{(p_{1}^{2}-p_{3}^{2})}{st}\frac{\partial^{2}}{\partial s\partial
t}\bigg{]}\,I_{\tilde{\alpha}\\{\beta_{1},\beta_{2},\beta_{3}\\}}=0\\\\[4.30554pt]
&\bigg{[}\frac{\partial^{2}}{\partial
p_{2}^{2}}+\frac{(d-2\mathcal{D}+1)}{p_{2}}\frac{\partial}{\partial
p_{2}}-\frac{\partial^{2}}{\partial
p_{4}^{2}}-\frac{(d-2\mathcal{D}+1)}{p_{4}}\frac{\partial}{\partial
p_{4}}+\frac{(p_{2}^{2}-p_{4}^{2})}{st}\frac{\partial^{2}}{\partial s\partial
t}\bigg{]}\,I_{\tilde{\alpha}\\{\beta_{1},\beta_{2},\beta_{3}\\}}=0\\\\[4.30554pt]
&\bigg{[}\frac{\partial^{2}}{\partial
p_{3}^{2}}+\frac{(d-2\mathcal{D}+1)}{p_{3}}\frac{\partial}{\partial
p_{3}}-\frac{\partial^{2}}{\partial
p_{4}^{2}}-\frac{(d-2\mathcal{D}+1)}{p_{4}}\frac{\partial}{\partial
p_{4}}+\frac{(p_{2}^{2}-p_{1}^{2})}{st}\frac{\partial^{2}}{\partial s\partial
t}\bigg{]}\,I_{\tilde{\alpha}\\{\beta_{1},\beta_{2},\beta_{3}\\}}=0\\\\[-3.01385pt]
\end{aligned}\right.$ (4.1.27)

and using the properties of Bessel functions they can be rewritten in a
simpler form. The first equation, for instance, can be written as

$(p_{1}^{2}-p_{3}^{2})\bigg{(}(d-2\mathcal{D}+2\beta_{1})\,I_{\tilde{\alpha}+1\\{\beta_{1}-1,\beta_{2},\beta_{3}\\}}-2\beta_{3}\,I_{\tilde{\alpha}+1\\{\beta_{1},\beta_{2},\beta_{3}-1\\}}\bigg{)}=0,$
(4.1.28)

which is identically satisfied if the conditions

$\beta_{1}=\Delta-\frac{d}{2},\qquad\beta_{3}=0$ (4.1.29)

hold. In the same way we find that the second equation takes the form

$(p_{2}^{2}-p_{4}^{2})\bigg{(}(d-2\mathcal{D}+2\beta_{2})\,I_{\tilde{\alpha}+1\\{\beta_{1},\beta_{2}-1,\beta_{3}\\}}-2\beta_{3}\,I_{\tilde{\alpha}+1\\{\beta_{1},\beta_{2},\beta_{3}-1\\}}\bigg{)}=0,$
(4.1.30)

and it is satisfied if

$\beta_{2}=\Delta-\frac{d}{2},\qquad\beta_{3}=0.$ (4.1.31)

One can check that the third equation

$p_{2}^{2}(d-2\mathcal{D}+2\beta_{2})\,I_{\tilde{\alpha}+1\\{\beta_{1},\beta_{2}-1,\beta_{3}\\}}-p_{1}^{2}(d-2\mathcal{D}+2\beta_{1})\,I_{\tilde{\alpha}+1\\{\beta_{1}-1,\beta_{2},\beta_{3}\\}}-2(p_{2}^{2}-p_{1}^{2})\beta_{3}\,I_{\tilde{\alpha}+1\\{\beta_{1},\beta_{2},\beta_{3}-1\\}}=0,$
(4.1.32)

generates the same conditions given by (4.1.28) and (4.1.30). The solution for
the four-point function, in this particular case, can be written as

$\langle{O(p_{1})\,O(p_{2})\,O(p_{3})\,O(\bar{p}_{4})}\rangle=\,\bar{\alpha}\,I_{\frac{d}{2}-1\left\\{\Delta-\frac{d}{2},\Delta-\frac{d}{2},0\right\\}}(p_{1}\,p_{3};p_{2}\,p_{4};s\,t),$
(4.1.33)

where $\bar{\alpha}$ is an undetermined constant.

In the case $\Delta_{1}=\Delta_{3}=\Delta_{x}$ and
$\Delta_{2}=\Delta_{4}=\Delta_{y}$, the special CWIs can be written as

$\left\\{\begin{aligned} &\\\\[-8.1805pt]
&\bigg{[}\frac{\partial^{2}}{\partial
p_{1}^{2}}+\frac{(d-2\Delta_{x}+1)}{p_{1}}\frac{\partial}{\partial
p_{1}}-\frac{\partial^{2}}{\partial
p_{3}^{2}}-\frac{(d-2\Delta_{x}+1)}{p_{3}}\frac{\partial}{\partial
p_{3}}+\frac{(p_{1}^{2}-p_{3}^{2})}{st}\frac{\partial^{2}}{\partial s\partial
t}\bigg{]}\,I_{\tilde{\alpha}\\{\beta_{1},\beta_{2},\beta_{3}\\}}=0\\\\[4.30554pt]
&\bigg{[}\frac{\partial^{2}}{\partial
p_{2}^{2}}+\frac{(d-2\Delta_{y}+1)}{p_{2}}\frac{\partial}{\partial
p_{2}}-\frac{\partial^{2}}{\partial
p_{4}^{2}}-\frac{(d-2\Delta_{y}+1)}{p_{4}}\frac{\partial}{\partial
p_{4}}+\frac{(p_{2}^{2}-p_{4}^{2})}{st}\frac{\partial^{2}}{\partial s\partial
t}\bigg{]}\,I_{\tilde{\alpha}\\{\beta_{1},\beta_{2},\beta_{3}\\}}=0\\\\[4.30554pt]
&\bigg{[}\frac{\partial^{2}}{\partial
p_{3}^{2}}+\frac{(d-2\Delta_{x}+1)}{p_{3}}\frac{\partial}{\partial
p_{3}}-\frac{\partial^{2}}{\partial
p_{4}^{2}}-\frac{(d-2\Delta_{y}+1)}{p_{4}}\frac{\partial}{\partial
p_{4}}+\frac{(p_{2}^{2}-p_{1}^{2})}{st}\frac{\partial^{2}}{\partial s\partial
t}\bigg{]}\,I_{\tilde{\alpha}\\{\beta_{1},\beta_{2},\beta_{3}\\}}=0\\\\[5.59721pt]
\end{aligned}\right.$ (4.1.34)

whose solution is

$\displaystyle\langle{O(p_{1})\,O(p_{2})\,O(p_{3})\,O(\bar{p}_{4})}\rangle$
$\displaystyle=\,\bar{\bar{\alpha}}\,I_{\frac{d}{2}-1\left\\{\Delta_{x}-\frac{d}{2},\Delta_{y}-\frac{d}{2},0\right\\}}(p_{1}\,p_{3};p_{2}\,p_{4};s\,t),$
(4.1.35)

which takes a form which is typical of the three-point function.

####  4.2 Symmetric solutions as $F_{4}$ hypergeometrics or triple-K
integrals. The equal scalings case

The derivation of symmetric expressions of such correlators requires and can
be obtained by resorting to the formalism of the triple-K integrals [97].
A solution which is symmetric with respect to all the permutation of the
momenta $p_{i}$, expressed in terms of 3 of the four constants $c(a,b)$ can be
expressed in the form

$\displaystyle\langle{O(p_{1})O(p_{2})O(p_{3})O(p_{4})}\rangle$
$\displaystyle=$
$\displaystyle=\sum_{a,b}c(a,b)\Bigg{[}\,(s^{2}\,t^{2})^{\Delta-\frac{3}{4}d}\left(\frac{p_{1}^{2}p_{3}^{2}}{s^{2}t^{2}}\right)^{a}\left(\frac{p_{2}^{2}p_{4}^{2}}{s^{2}t^{2}}\right)^{b}F_{4}\left(\alpha(a,b),\beta(a,b),\gamma(a),\gamma^{\prime}(b),\frac{p_{1}^{2}p_{3}^{2}}{s^{2}t^{2}},\frac{p_{2}^{2}p_{4}^{2}}{s^{2}t^{2}}\right)$
$\displaystyle+\,(s^{2}\,u^{2})^{\Delta-\frac{3}{4}d}\,\left(\frac{p_{2}^{2}p_{3}^{2}}{s^{2}u^{2}}\right)^{a}\left(\frac{p_{1}^{2}p_{4}^{2}}{s^{2}u^{2}}\right)^{b}F_{4}\left(\alpha(a,b),\beta(a,b),\gamma(a),\gamma^{\prime}(b),\frac{p_{2}^{2}p_{3}^{2}}{s^{2}u^{2}},\frac{p_{1}^{2}p_{4}^{2}}{s^{2}u^{2}}\right)$
$\displaystyle+\,(t^{2}\,u^{2})^{\Delta-\frac{3}{4}d}\,\left(\frac{p_{1}^{2}p_{2}^{2}}{t^{2}u^{2}}\right)^{a}\,\left(\frac{p_{3}^{2}p_{4}^{2}}{t^{2}u^{2}}\right)^{b}\,F_{4}\left(\alpha(a,b),\beta(a,b),\gamma(a),\gamma^{\prime}(b),\frac{p_{1}^{2}p_{2}^{2}}{t^{2}u^{2}},\frac{p_{3}^{2}p_{4}^{2}}{t^{2}u^{2}}\right)\Bigg{]},$
(4.2.1)

where the four coefficients $c(a,b)$’s given in (4.2.1) are reduced to three
by the constraint

$\displaystyle
c\left(0,\Delta-\frac{d}{2}\right)=c\left(\Delta-\frac{d}{2},0\right).$
(4.2.2)

The final symmetric result is given by

$\displaystyle\langle{O(p_{1})O(p_{2})O(p_{3})O(p_{4})}\rangle$
$\displaystyle=C\bigg{[}\,I_{\frac{d}{2}-1\\{\Delta-\frac{d}{2},\Delta-\frac{d}{2},0\\}}(p_{1}\,p_{3},p_{2}\,p_{4},s\,t)$
$\displaystyle+\,\,I_{\frac{d}{2}-1\\{\Delta-\frac{d}{2},\Delta-\frac{d}{2},0\\}}(p_{2}\,p_{3},p_{1}\,p_{4},s\,u)+\,\,I_{\frac{d}{2}-1\\{\Delta-\frac{d}{2},\Delta-\frac{d}{2},0\\}}(p_{1}\,p_{2},p_{3}\,p_{4},t\,u)\bigg{]},$
(4.2.3)

written in terms of only one arbitrary constant overall, $C$. We can use the
relation between the triple-K integrals and the $F_{4}$ written in (4.1.18)
and (4.1.19), to re-express the final symmetric solution, originally given in
(4.2.1), in terms of a single constant in the form

$\displaystyle\langle{O(p_{1})O(p_{2})O(p_{3})O(p_{4})}\rangle$
$\displaystyle=2^{\frac{d}{2}-4}\ \
C\,\sum_{\lambda,\mu=0,\Delta-\frac{d}{2}}\xi(\lambda,\mu)\bigg{[}\big{(}s^{2}\,t^{2}\big{)}^{\Delta-\frac{3}{4}d}\left(\frac{p_{1}^{2}p_{3}^{2}}{s^{2}t^{2}}\right)^{\lambda}\left(\frac{p_{2}^{2}p_{4}^{2}}{s^{2}t^{2}}\right)^{\mu}$
$\displaystyle\times\,F_{4}\left(\frac{3}{4}d-\Delta+\lambda+\mu,\frac{3}{4}d-\Delta+\lambda+\mu,1-\Delta+\frac{d}{2}+\lambda,1-\Delta+\frac{d}{2}+\mu,\frac{p_{1}^{2}p_{3}^{2}}{s^{2}t^{2}},\frac{p_{2}^{2}p_{4}^{2}}{s^{2}t^{2}}\right)$
$\displaystyle+\big{(}s^{2}\,u^{2}\big{)}^{\Delta-\frac{3}{4}d}\left(\frac{p_{2}^{2}p_{3}^{2}}{s^{2}u^{2}}\right)^{\lambda}\left(\frac{p_{1}^{2}p_{4}^{2}}{s^{2}u^{2}}\right)^{\mu}$
$\displaystyle\times\,F_{4}\left(\frac{3}{4}d-\Delta+\lambda+\mu,\frac{3}{4}d-\Delta+\lambda+\mu,1-\Delta+\frac{d}{2}+\lambda,1-\Delta+\frac{d}{2}+\mu,\frac{p_{2}^{2}p_{3}^{2}}{s^{2}u^{2}},\frac{p_{1}^{2}p_{4}^{2}}{s^{2}u^{2}}\right)$
$\displaystyle+\big{(}t^{2}\,u^{2}\big{)}^{\Delta-\frac{3}{4}d}\left(\frac{p_{1}^{2}p_{2}^{2}}{t^{2}u^{2}}\right)^{\lambda}\left(\frac{p_{3}^{2}p_{4}^{2}}{t^{2}u^{2}}\right)^{\mu}$
$\displaystyle\times\,F_{4}\left(\frac{3}{4}d-\Delta+\lambda+\mu,\frac{3}{4}d-\Delta+\lambda+\mu,1-\Delta+\frac{d}{2}+\lambda,1-\Delta+\frac{d}{2}+\mu,\frac{p_{1}^{2}p_{2}^{2}}{t^{2}u^{2}},\frac{p_{3}^{2}p_{4}^{2}}{t^{2}u^{2}}\right)\bigg{]}.$
(4.2.4)

where the coefficients $\xi(\lambda,\mu)$ are explicitly given by

$\begin{split}\xi\left(0,0\right)&=\left[\Gamma\left(\frac{3}{4}d-\Delta\right)\right]^{2}\left[\Gamma\left(\Delta-\frac{d}{2}\right)\right]^{2},\\\
\xi\left(0,\Delta-\frac{d}{2}\right)&=\xi\left(\Delta-\frac{d}{2},0\right)=\left[\Gamma\left(\frac{d}{4}\right)\right]^{2}\Gamma\left(\Delta-\frac{d}{2}\right)\Gamma\left(\frac{d}{2}-\Delta\right),\\\
\xi\left(\Delta-\frac{d}{2},\Delta-\frac{d}{2}\right)&=\left[\Gamma\left(\Delta-\frac{d}{4}\right)\right]^{2}\left[\Gamma\left(\frac{d}{2}-\Delta\right)\right]^{2}.\end{split}$
(4.2.5)

The solution found in (4.2.4) is explicitly symmetric under all the possible
permutations of the momenta and it is fixed up to one undetermined constant
$C$. Equation (4.2.4) gives the final expression of the solution obtained from
the dual conformal ansatz (4.1.4).

###  Chapter 5 The $TOOO$ correlator: CWIs in the scalar/tensor case

The investigation of the CWIs in momentum space that we are going to present
is based on the reconstruction method of a tensor correlator starting from its
transverse/traceless (tt) sector, formulated for 3-point functions [21, 23,
97], that here we are going to extend to 4-point functions.
In particular, in [21] a complete approach for the analysis of 3-point
functions, up to the $TTT$ case, with three stress-energy tensors, valid for
tensor correlators, has been formulated. The reconstruction of the entire
correlator from its $tt$ projection involves the identification of a minimal
set of form factors in this sector, and it is accompanied by a set of
technical steps for re-assembling it in a systematic way.
This approach allows to identify primary and secondary CWIs of a tensorial
3-point function, with the former corresponding to second order partial
differential equations (pde’s) which can be solved independently in terms of a
set of arbitrary constants.
Primary CWIs are equations involving only form factors of the tt sector and
generate, for tensor correlators, inhomogeneous systems of pde’s of
hypergeometric type. Secondary CWIs, on the other end, connect the same form
factors to 3-point functions via the corresponding canonical WI’s, which
impose extra constraints on the constants appearing in the solution of the
primary equations.
The results that we present extend a previous analysis devoted to the scalar
case, involving the $OOOO$ correlator [26]. We will re-investigate the scalar
case, by taking a closer look at the structure of the equations and at their
asymptotic behaviour. We will remark few additional properties of such
correlators and highlight some properties of the asymptotic solutions of such
equations, which have not been addressed before. This will allow us to gain a
more general perspective both on the scalar and the tensor cases, especially
in view of possible future extensions of our work to correlators of higher
rank.
Scalar correlators are characterised only by primary CWIs and are therefore
simpler to handle, differently from the $TOOO$ case where both primary and
secondary equations are present.
In the scalar case the analysis of the conformal constraints will be performed
by focusing on a special class of solutions of such equations which are
conformal and dual conformal at the same time, derived in [26]. These are
obtained by imposing a specific condition on the scaling dimensions of the
scalar operators, which allow to reduce the CWIs to a hypergeometric system,
as in the case of 3-point and 4-point functions. More details can be found in
2.5 and 4.

As we move from 3- to 4-point functions, all the equations, primary and
secondary, are expressed in terms of 6 invariants, which are the external
invariant masses $p_{i}^{2}$ and the two Mandelstam invariants $s$ and $t$. As
far as we keep the external lines off-shell, and stay away from kinematical
points where an invariant is exactly zero, the equations are well-defined and
it is possible to investigate their structure. As we are going to show, the
selection of a set of specific invariants, compared to others, is particularly
beneficial if we intend to uncover the symmetries of the equations and their
redundancies under the permutations of the external momenta.
A crucial goal of our study is the identification of the asymptotic behaviour
of the solutions of such constraints in specific kinematical limits. This may
allow, in the near future, to relate results from ordinary perturbation theory
- in ordinary Lagrangian realizations, at one loop level - to those derived
from CFT’s in the same limits. For instance, in [25, 24] it is shown how to
match the general solutions of the CWIs for the $TTT$ and $TJJ$ correlators,
to free field theories with a specific content of fermions, scalars and spin 1
fields. The matching allows to re-express the solutions of such equations in
terms of simple one-loop master integrals in full generality, for any CFT.

#####  5.0.1 The search for asymptotic solutions

For this reason, the search for asymptotic solutions of the CWIs, which
acquire a simpler form in such limits, is particularly interesting. It may
allow to establish a link with the classical factorization theorems proven in
gauge theory amplitudes [98], especially if such CFT methods can be extended
to multi-point functions.
We will investigate the structure of the equations in two specific limits. The
first case that we will address will be the $1\to 3$, where the graviton line
of the T is assumed to acquire a large invariant mass $(p_{1}^{2})$ and decays
into three scalar lines with small invariants
$(p_{2}^{2},p_{3}^{2},p_{4}^{2})$, while the remaining invariants $s$ and $t$
are large. We are going to derive some approximate asymptotic solutions of the
equations which are separable in the $(p_{2}^{2},p_{3}^{2},p_{4}^{2})$ and
$(s,t,u)$ dependence. A similar analysis will be presented in the $2\to 2$
process, where one of the lines of the scalar operators is selected in the
initial state together with the graviton line and the remaining scalar lines
are in the final state.
Our work is organized as follows.
After a brief discussion of the conformal and canonical WI’s in momentum
space, we investigate the structure of the tt sector of the $TOOO$,
identifying the symmetry constraints under the permutation of the momenta of
the single form factor appearing in this correlator.
We then turn to a derivation of the primary and secondary CWIs of this
correlator, written in a form which will be useful for the derivation of their
asymptotic limits. We describe the orbits of such equations under the symmetry
permutations, which allows to identify a subset of independent equations.
The analysis is repeated from scratch in the $2\to 2$ case and it is followed
by a discussion of the asymptotic limits of such equations, after a brief
overview of the approach in the scalar case.
We start from the scalar case, discuss the system of scalar equations and
discuss its reduction to the dcc case, which can be solved exactly. The
asymptotic behaviour of the dcc solutions provides an example and a guidance
for a more general analysis first of the scalar case, and then of the tensor
case, the $TOOO$.

####  5.1 Ward identities for the $TOOO$ in coordinate and in momentum space

In this subsection, we use the formalism developed in chapter 2 to formulate
the CWI our correlator

$\Gamma^{\mu\nu}(x_{1},x_{2},x_{3},x_{4})\equiv\langle
T^{\mu\nu}(x_{1})O(x_{2})O(x_{3})O(x_{4})\rangle.$ (5.1.1)

Using (2.4.5) we have

$\displaystyle{\cal K}^{\kappa}\Gamma^{\mu\nu}(x_{1},x_{2},x_{3},x_{4})$
$\displaystyle={\cal
K}^{\kappa}_{scalar}(x_{i})\Gamma^{\mu\nu}(x_{1},x_{2},x_{3},x_{4})+2\left(\delta^{\mu\kappa}x_{1\rho}-\delta_{\rho}^{\kappa}x_{1}^{\mu}\right)\Gamma^{\rho\nu}(x_{1},x_{2},x_{3},x_{4})$
$\displaystyle+2\left(\delta^{\nu\kappa}x_{1\rho}-\delta_{\rho}^{\kappa}x_{1}^{\nu}\right)\Gamma^{\mu\rho}(x_{1},x_{2},x_{3},x_{4})=0,$

where the first contribution denotes the scalar part and the last two
contributions the spin part, which are trivially absent in the case of a
scalar correlator.
The action of $\mathcal{K}^{\kappa}$ can be summarized by the expression

$\displaystyle\sum_{j=1}^{3}\left[2(\Delta_{j}-d)\mbox{\small$\displaystyle\frac{\partial}{\partial
p_{j}^{\kappa}}$}-2p_{j}^{\alpha}\mbox{\small$\displaystyle\frac{\partial}{\partial
p_{j}^{\alpha}}$}\mbox{\small$\displaystyle\frac{\partial}{\partial
p_{j}^{\kappa}}$}+(p_{j})_{\kappa}\mbox{\small$\displaystyle\frac{\partial}{\partial
p_{j}^{\alpha}}$}\mbox{\small$\displaystyle\frac{\partial}{\partial
p_{j\alpha}}$}\right]\langle{{T^{\mu_{1}\nu_{1}}(p_{1})\,O(p_{2})\,O(p_{3})}O(\bar{p_{4}}}\rangle$
$\displaystyle\qquad+{\cal
K}^{\kappa}_{spin}\langle{T^{\mu_{1}\nu_{1}}(p_{1})\,O(p_{2})\,O(p_{3})}O(\bar{p_{4}}\rangle=0,$

where we have defined the spin part of ${\cal K}$ in momentum space as

$\displaystyle{\cal
K}^{\kappa}_{spin}\langle{{T^{\mu_{1}\nu_{1}}(p_{1})\,O(p_{2})\,O(p_{3})}O(\bar{p_{4}}}\rangle$
$\displaystyle\equiv
4\left(\delta^{\kappa(\mu_{1}}\mbox{\small$\displaystyle\frac{\partial}{\partial
p_{1}^{\alpha_{1}}}$}-\delta^{\kappa}_{\alpha_{1}}\delta^{\lambda(\mu_{1}}\mbox{\small$\displaystyle\frac{\partial}{\partial
p_{1}^{\lambda}}$}\right)\langle{{T^{\nu_{1})\alpha_{1}}(p_{1})\,O(p_{2})\,Op_{3})}O(\bar{p}_{4})}\rangle$

(symmetrization is normalized with an overall factor $1/2$).
In the previous expression we have taken $p_{4}$ as a dependent momentum
($p_{4}\to\overline{p}_{4}$), which requires an implicit differentiation if we
take $p_{1},p_{2}$ and $p_{3}$ as independent momenta. The equations can be
projected onto the three independent momenta, giving scalar equations which
can be re-expressed in terms of all the scalar invariants parameterizing the
form factors. The hypergeometric character of the 3-point functions, as well
as for 4-point functions (for the dcc solutions), emerges after such reduction
of the equations to a scalar form [20, 21, 24, 25].

#####  5.1.1 Conservation and Trace Ward Identities

The next objective is to derive the the trace and conservation Ward
Identities. The starting point is

$Z[\phi_{0},g^{\mu\nu}]=\int\mathcal{D}\Phi\mathrm{exp}\big{(}-S_{CFT}[\phi,g^{\mu\nu}]-\sum_{i}\int
d^{d}x\sqrt{g}\phi_{0}^{j}O_{j}\big{)},$ (5.1.5)

dependent on the background metric $g_{\mu\nu}$ and the classical source
$\phi_{0}(x)$ coupled to the scalar operator $O(x)$, with the 1-point
functions given by

$\displaystyle\langle T^{\mu\nu}(x)\rangle=\frac{2}{\sqrt{g(x)}}\frac{\delta
Z}{\delta g_{\mu\nu}},$ (5.1.6) $\displaystyle\langle
O_{j}(x)\rangle=-\frac{1}{\sqrt{g(x)}}\frac{\delta Z}{\delta\phi_{0}^{j}(x)}.$
(5.1.7)

In our case, in order to avoid some bulky notation, we consider only one type
of scalar operator, with a unique scaling dimension $\Delta$. We will present
the derivations of all the conformal and canonical WI’s in this specific case.
In section 5.6.1 we will then provide the expression of the same equations for
general distinct $\Delta_{i}$’s, which can be obtained by a very similar
procedure, as in the equal scaling case.
To get the transverse and trace Ward Identities, we require that the
generating functional Z is invariant under diffeomorphisms and Weyl
transformations respectively, which gives

$\displaystyle\nabla_{\nu}\langle
T^{\mu\nu}(x_{1})\rangle+\partial^{\mu}\phi_{0}\cdot\langle
O(x_{1})\rangle=0,$ (5.1.8) $\displaystyle g_{\mu\nu}\langle
T^{\mu\nu}(x_{1})\rangle+(d-\Delta)\phi_{0}\langle O(x_{1})\rangle=0.$ (5.1.9)

The WI’s for the $\langle
T^{\mu_{1}\nu_{1}}(\textbf{p}_{1})O(\textbf{p}_{2})O(\textbf{p}_{3})O(\textbf{p}_{4})\rangle$
can be derived by taking three variations of the above identities with respect
to the source $\phi_{0}$ of the scalar operator. At the end, by imposing the
flat limit $g_{\mu\nu}=\delta_{\mu\nu},\nabla_{\nu}=\partial_{\nu}$, turning
off the sources ($\phi_{0}=0$) and using the definitions

$\displaystyle\langle
T^{\mu\nu}(x_{1})O(x_{2})O(x_{3})O(x_{4})\rangle=\frac{-2}{\sqrt{g(x_{1})}\dots\sqrt{g(x_{4})}}\frac{\delta^{4}Z}{\delta
g_{\mu\nu}(x_{1})\delta\phi_{0}(x_{2})\delta\phi_{0}(x_{3})\delta\phi_{0}(x_{4})},$
(5.1.10) $\displaystyle\langle
O(x_{1})O(x_{2})O(x_{3})\rangle=\frac{-1}{\sqrt{g(x_{1})}\sqrt{g(x_{2})}\sqrt{g(x_{3})}}\frac{\delta^{3}Z}{\delta\phi_{0}(x_{3})\delta\phi_{0}(x_{2})\delta\phi_{0}(x_{1})},$
(5.1.11)

the conservation WI gives the constraint

$\begin{split}\partial_{\nu}\langle
T^{\mu\nu}(x_{1})O(x_{2})O(x_{3})O(x_{4})\rangle&=\partial^{\mu}\delta^{(d)}(x_{1}-x_{2})\langle
O(x_{1})O(x_{3})O(x_{4})\rangle\\\
&+\partial^{\mu}\delta^{(d)}(x_{1}-x_{3})\langle
O(x_{1})O(x_{2})O(x_{4})\rangle+\partial^{\mu}\delta^{(d)}(x_{1}-x_{4})\langle
O(x_{1})O(x_{2})O(x_{3})\rangle,\end{split}$ (5.1.12)

while the trace WI gives

$\displaystyle\delta_{\mu\nu}\langle
T^{\mu\nu}(x_{1})O(x_{2})O(x_{3}))(x_{4})\rangle$
$\displaystyle=(d-\Delta)\delta^{(d)}(x_{1}-x_{2})\langle
O(x_{1})O(x_{3})O(x_{4})\rangle$
$\displaystyle+(d-\Delta)\delta^{(d)}(x_{1}-x_{3})\langle
O(x_{1})O(x_{2})O(x_{4})\rangle+(d-\Delta)\delta^{(d)}(x_{1}-x_{4})\langle
O(x_{1})O(x_{2})O(x_{3})\rangle.$ (5.1.13)

The expressions of (5.1.12) and (5.1.1) in momentum space can be obtained by a
Fourier transform and are explicitly given by

$\begin{split}\delta^{(d)}\left(\sum_{i=1}^{4}\textbf{p}_{i}\right)p_{1\nu}\langle
T^{\mu\nu}(\textbf{p}_{1})O(\textbf{p}_{2})O(\textbf{p}_{3})O(\textbf{p}_{4})\rangle=&-\Bigg{(}p_{2}^{\mu}\langle
O(\textbf{p}_{1}+\textbf{p}_{2})O(\textbf{p}_{3})O(\textbf{p}_{4})\rangle\\\
&+p_{3}^{\mu}\langle
O(\textbf{p}_{1}+\textbf{p}_{3})O(\textbf{p}_{2})O(\textbf{p}_{4})\rangle+p_{4}^{\mu}\langle
O(\textbf{p}_{1}+\textbf{p}_{4})O(\textbf{p}_{2})O(\textbf{p}_{3})\rangle\Bigg{)}\delta^{(d)}\left(\sum_{i=1}^{4}\textbf{p}_{i}\right),\end{split}$
(5.1.14)

and

$\begin{split}\delta^{(d)}\left(\sum_{i=1}^{4}\textbf{p}_{i}\right)\delta_{\mu\nu}\langle
T^{\mu\nu}(\textbf{p}_{1})O(\textbf{p}_{2})O(\textbf{p}_{3})O(\textbf{p}_{4})\rangle=&(d-\Delta)\Bigg{(}\langle
O(\textbf{p}_{1}+\textbf{p}_{2})O(\textbf{p}_{3})O(\textbf{p}_{4})\rangle\\\
&+\langle
O(\textbf{p}_{1}+\textbf{p}_{3})O(\textbf{p}_{2})O(\textbf{p}_{4})\rangle+\langle
O(\textbf{p}_{1}+\textbf{p}_{4})O(\textbf{p}_{2})O(\textbf{p}_{3})\rangle\Bigg{)}\delta^{(d)}\left(\sum_{i=1}^{4}\textbf{p}_{i}\right),\end{split}$
(5.1.15)

where on the right hand side of the equations appear only scalar 3-point
functions. We will insert a bar over a momentum variable to indicate that it
is treated as a dependent one. In the following we are going to make two
separate choices of dependent momenta, respectively $\bar{p}_{1}$ and
$\bar{p}_{4}$. If we choose $\bar{p}_{1}$ as the dependent momentum, the WI’s
take the form

$\displaystyle\bar{p}_{\mu_{1}}\,\langle{T^{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})\,O(\mathbf{p}_{2})\,O(\mathbf{p}_{3})\,O(\mathbf{p}_{4})}\rangle$
$\displaystyle=-p_{2}^{\nu_{1}}\langle{O(\mathbf{p}_{3}+\mathbf{p}_{4})\,O(\mathbf{p}_{3})\,O(\mathbf{p}_{4})}\rangle$
$\displaystyle-
p_{3}^{\nu_{1}}\langle{O(\mathbf{p}_{2}+\mathbf{p}_{4})\,O(\mathbf{p}_{2})\,O(\mathbf{p}_{4})}\rangle-
p_{4}^{\nu_{1}}\langle{O(\mathbf{p}_{2}+\mathbf{p}_{3})\,O(\mathbf{p}_{2})\,O(\mathbf{p}_{3})}\rangle,$
(5.1.16a)
$\displaystyle\delta_{\mu_{1}\nu_{1}}\,\langle{T^{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})\,O(\mathbf{p}_{2})\,O(\mathbf{p}_{3})\,O(\mathbf{p}_{4})}\rangle$
$\displaystyle=(d-\Delta)\Big{[}\langle{O(\mathbf{p}_{3}+\mathbf{p}_{4})\,O(\mathbf{p}_{3})\,O(\mathbf{p}_{4})}\rangle$
$\displaystyle+\langle{O(\mathbf{p}_{2}+\mathbf{p}_{4})\,O(\mathbf{p}_{2})\,O(\mathbf{p}_{4})}\rangle+\langle{O(\mathbf{p}_{2}+\mathbf{p}_{3})\,O(\mathbf{p}_{2})\,O(\mathbf{p}_{3})}\rangle\Big{]},$
(5.1.16b)

and for $\bar{p}_{4}$

$\displaystyle p_{1,\mu_{1}}\langle
T^{\mu_{1}\nu_{1}}(\textbf{p}_{1})O(\textbf{p}_{2})O(\textbf{p}_{3})O(\mathbf{\bar{p}_{4}})\rangle=$
$\displaystyle+p_{1}^{\nu_{1}}\langle{O(\textbf{p}_{2})O(\textbf{p}_{3})O(\mathbf{p_{1}+\bar{p}_{4}})}\rangle$
$\displaystyle-p_{2}^{\nu_{1}}\Big{(}\langle
O(\textbf{p}_{1}+\textbf{p}_{2})O(\textbf{p}_{3})O(\mathbf{\bar{p}_{4}})\rangle-\langle
O(\textbf{p}_{2})O(\textbf{p}_{3})O(\mathbf{p_{1}+\bar{p}_{4}})\rangle\Big{)}$
$\displaystyle-p_{3}^{\nu_{1}}\Big{(}\langle
O(\textbf{p}_{2})O(\mathbf{p_{1}+p_{3}})O(\mathbf{\bar{p}_{4}})\rangle-\langle
O(\textbf{p}_{2})O(\textbf{p}_{3})O(\mathbf{p_{1}+\bar{p}_{4}})\rangle\Big{)},$
(5.1.17a) $\displaystyle\delta_{\mu_{1}\nu_{1}}\langle
T^{\mu_{1}\nu_{1}}(\textbf{p}_{1})O(\textbf{p}_{2})O(\textbf{p}_{3})O(\mathbf{\bar{p}_{4}})\rangle=$
$\displaystyle(d-\Delta)\Big{[}\langle
O(\textbf{p}_{1}+\textbf{p}_{2})O(\textbf{p}_{3})O(\mathbf{\bar{p}_{4}})\rangle+\langle
O(\textbf{p}_{1}+\textbf{p}_{3})O(\textbf{p}_{2})O(\mathbf{\bar{p}_{4}})\rangle$
$\displaystyle+\langle
O(\textbf{p}_{1}+\mathbf{\bar{p}_{4}})O(\textbf{p}_{2})O(\textbf{p}_{3})\rangle\Big{]}.$
(5.1.17b)

The left hand sides of these equations will be related to the form factor
identified from the $tt$ sector.

####  5.2 The reconstruction method from 3- to 4-point functions and the
$TOOO$

Following [21], we consider the four point correlation function formed by a
stress-energy tensor $T^{\mu\nu}$ and three scalar operators $O(p_{i})$ of the
same kind and with the same scaling dimensions. We define

$\displaystyle p_{i}=\sqrt{\mathbf{p}_{i}^{2}},\quad
s=\sqrt{(\mathbf{\bar{p}}_{1}+\mathbf{p}_{2})^{2}}=\sqrt{(\mathbf{p}_{3}+\mathbf{p}_{4})^{2}},\quad
t=\sqrt{(\mathbf{p}_{2}+\mathbf{p}_{3})^{2}},\quad
u=\sqrt{(\mathbf{p}_{2}+\mathbf{p}_{4})^{2}},$ (5.2.1)

and introduce the $tt$ $(\Pi)$ and local $(\Sigma)$ projectors

$\displaystyle\Pi^{\mu\nu}_{\alpha\beta}(\mathbf{p})$
$\displaystyle=\frac{1}{2}\left(\pi^{\mu}_{\alpha}(\mathbf{p})\pi^{\nu}_{\beta}(\mathbf{p})+\pi^{\mu}_{\beta}(\mathbf{p})\pi^{\nu}_{\alpha}(\mathbf{p})\right)-\frac{1}{d-1}\pi^{\mu\nu}(\mathbf{p})\pi_{\alpha\beta}(\mathbf{p})$
(5.2.2) $\displaystyle\Sigma^{\mu\nu}_{\alpha\beta}(\mathbf{p})$
$\displaystyle=\delta^{(\mu}_{\alpha}\delta^{\nu)}_{\beta}-\Pi^{\mu\nu}_{\alpha\beta}(\mathbf{p})=\frac{1}{p^{2}}\left[2\,p_{\scriptsize{\raisebox{-2.71246pt}{$(\beta$}}}\,\delta^{(\mu}_{\scriptsize{\raisebox{-2.1097pt}{$\alpha)$}}}p^{\scriptsize{\raisebox{1.50694pt}{$\nu)$}}}-\frac{p_{\alpha}p_{\beta}}{(d-1)}\left(\delta^{\mu\nu}+(d-2)\frac{p^{\mu}p^{\nu}}{p^{2}}\right)\right]+\frac{1}{d-1}\pi^{\mu\nu}(\mathbf{p})\delta_{\alpha\beta}.$
(5.2.3)

The stress-energy tensor is decomposed in its transverse traceless $(tt)$ and
local parts in the form

$T^{\mu\nu}=t^{\mu\nu}+t_{loc}^{\mu\nu}$ (5.2.4)

with

$\displaystyle t_{loc}^{\mu\nu}(p)$
$\displaystyle=\frac{p^{\mu}}{p^{2}}Q^{\nu}+\frac{p^{\nu}}{p^{2}}Q^{\mu}-\frac{p^{\mu}p^{\nu}}{p^{4}}Q+\frac{\pi^{\mu\nu}}{d-1}(T-\frac{Q}{p^{2}})$
$\displaystyle=\Sigma^{\mu\nu}_{\alpha\beta}T^{\alpha\beta}$ (5.2.5)

and

$Q^{\mu}=p_{\nu}T^{\mu\nu},\qquad T=\delta_{\mu\nu}T^{\mu\nu},\qquad
Q=p_{\nu}p_{\mu}T^{\mu\nu}.$ (5.2.6)

One can consider the decomposition of the $\langle{TOOO}\rangle$ correlation
function as

$\displaystyle\langle{T^{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})O(\mathbf{p}_{2})O(\mathbf{p}_{3})O(\mathbf{p}_{4})}\rangle$
$\displaystyle=\langle{t^{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})O(\mathbf{p}_{2})O(\mathbf{p}_{3})O(\mathbf{p}_{4})}\rangle+\langle{t_{loc}^{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})O(\mathbf{p}_{2})O(\mathbf{p}_{3})O(\mathbf{p}_{4})}\rangle$
(5.2.7)

where in bold we refer to vectors in the Euclidean $\mathbb{R}^{d}$ space, and
we are considering $\bar{p}_{1}^{\mu}=-p_{2}^{\mu}-p_{3}^{\mu}-p_{4}^{\mu}$
from momentum conservation.
The first term on the right hand side of (5.2.7) is the $tt$ part of the
correlation function, and the second represents the local ($loc$) part. The
method consists in expanding the $tt$ sector into a minimal number of form
factors, fixed by the symmetry of the correlator [21]. In our case the $tt$
and local parts take the form

$\displaystyle\langle{t^{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})O(\mathbf{p}_{2})O(\mathbf{p}_{3})O(\mathbf{p}_{4})}\rangle$
$\displaystyle=\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(\mathbf{\bar{p}}_{1})\left[A\,p_{2}^{\alpha_{1}}p_{2}^{\beta_{1}}+A(p_{2}\leftrightarrow
p_{3})\,p_{3}^{\alpha_{1}}p_{3}^{\beta_{1}}+A(p_{2}\leftrightarrow
p_{4})\,p_{4}^{\alpha_{1}}p_{4}^{\beta_{1}}\right]$ (5.2.8)
$\displaystyle\langle{t_{loc}^{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})O(\mathbf{p}_{2})O(\mathbf{p}_{3})O(\mathbf{p}_{4})}\rangle$
$\displaystyle=\Sigma^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(\mathbf{\bar{p}}_{1})\langle{T^{\alpha_{1}\beta_{1}}(\mathbf{\bar{p}}_{1})O(\mathbf{p}_{2})O(\mathbf{p}_{3})O(\mathbf{p}_{4})}\rangle.$
(5.2.9)

From these expressions, one can observe that the local term is constrained by
the conservation WI’s (5.1.16), which project on 3-point functions of the form
$OOO$, as we have discussed in the previous section. Using (5.2.3) and
(5.1.16) in (5.2.9), one can explicitly write the local term in the form

$\displaystyle\langle{t_{loc}^{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})O(\mathbf{p}_{2})O(\mathbf{p}_{3})O(\mathbf{p}_{4})}\rangle$
$\displaystyle=\frac{2}{\bar{p}_{1}^{2}}\Big{[}-\bar{p}_{1}^{(\mu_{1}}p_{2}^{\nu_{1})}\langle
O(\mathbf{p}_{3}+\mathbf{p}_{4})O(\mathbf{p}_{3})O(\mathbf{p}_{4})\rangle$
$\displaystyle-\bar{p}_{1}^{(\mu_{1}}p_{3}^{\nu_{1})}\langle{O(\mathbf{p}_{2}+\mathbf{p}_{4})O(\mathbf{p}_{2})O(\mathbf{p}_{4})}\rangle-\bar{p}_{1}^{(\mu_{1}}p_{4}^{\nu_{1})}\langle{O(\mathbf{p}_{2}+\mathbf{p}_{3})O(\mathbf{p}_{2})O(\mathbf{p}_{3})}\rangle\Big{]}+\frac{(d-\Delta)}{d-1}\pi^{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})$
$\displaystyle\times\Big{[}\langle{O(\mathbf{p}_{2}+\mathbf{p}_{4})O(\mathbf{p}_{2})O(\mathbf{p}_{4})}\rangle+\langle{O(\mathbf{p}_{2}+\mathbf{p}_{3})O(\mathbf{p}_{2})O(\mathbf{p}_{3})}\rangle+\langle{O(\mathbf{p}_{3}+\mathbf{p}_{4})O(\mathbf{p}_{3})O(\mathbf{p}_{4})}\rangle\Big{]}$
$\displaystyle-\frac{1}{(d-1)}\left(\delta^{\mu_{1}\nu_{1}}+(d-2)\frac{\bar{p}_{1}^{\mu_{1}}\bar{p}_{1}^{\nu_{1}}}{\bar{p}_{1}^{2}}\right)\left[-\frac{\mathbf{\bar{p}}_{1}\cdot\mathbf{p}_{2}}{\bar{p}_{1}^{2}}\langle{O(\mathbf{p}_{3}+\mathbf{p}_{4})O(\mathbf{p}_{3})O(\mathbf{p}_{4})}\rangle\right.$
$\displaystyle\left.-\frac{\mathbf{\bar{p}}_{1}\cdot\mathbf{p}_{3}}{\bar{p}_{1}^{2}}\langle{O(\mathbf{p}_{2}+\mathbf{p}_{4})O(\mathbf{p}_{2})O(\mathbf{p}_{4})}\rangle-\frac{\mathbf{\bar{p}}_{1}\cdot\mathbf{p}_{4}}{\bar{p}_{1}^{2}}\langle{O(\mathbf{p}_{2}+\mathbf{p}_{3})O(\mathbf{p}_{2})O(\mathbf{p}_{3})}\rangle\right].$
(5.2.10)

The scalar 3-point function appearing on the right hand side is exactly known.
In this way, the task of finding the structure of the entire
$\langle{TOOO}\rangle$ has been reduced to the identification of only its $tt$
part. In particular, as we are going to show, all the WI’s will constrain a
single form factor.
The parameterization of this form factor (A), eventually, can be chosen
according to the type of amplitude that one intends to consider, in order to
facilitate the analysis.
For instance, in the case in which one in interested in a comparison between
the conformal prediction and a free field theory realization - for example in
a 1 (graviton) $\to$ three (scalars) process - then it is convenient to adopt
the parameterization $A\equiv A(p_{2},p_{3},p_{4},s,t,u)$ and derive the
equations using such variables. This choice is the one which respects the
symmetries of the process, since the three scalars can be treated equally, and
it allows to discuss more easily its asymptotic behaviour. Notice that in this
case, momentum $p_{1}$ is treated as a dependent one $(\overline{p}_{1})$ and
needs to be differentiated implicitly in the corresponding equations.

####  5.3 Conformal Ward Identities in the $1\to 3$ formulation

Using the $1\to 3$ symmetric formulation and the parameterization presented in
(5.2.9), the $A$ form factor exhibits the following symmetries

$\displaystyle A(p_{3}\leftrightarrow p_{4})$
$\displaystyle=A(p_{2},p_{3},p_{4},s,t,u)$ (5.3.1) $\displaystyle A(p_{2}\to
p_{4}\to p_{3}\to p_{2})$ $\displaystyle=A(p_{2}\leftrightarrow p_{4})$
(5.3.2) $\displaystyle A(p_{2}\to p_{3}\to p_{4}\to p_{2})$
$\displaystyle=A(p_{2}\leftrightarrow p_{3}),$ (5.3.3)

which can be written in the form

$\displaystyle A(p_{2},p_{4},p_{3},s,u,t)$
$\displaystyle=A(p_{2},p_{3},p_{4},s,t,u)$ (5.3.4) $\displaystyle
A(p_{4},p_{2},p_{3},t,u,s)$ $\displaystyle=A(p_{4},p_{3},p_{2},t,s,u)$ (5.3.5)
$\displaystyle A(p_{3},p_{4},p_{2},u,s,t)$
$\displaystyle=A(p_{3},p_{2},p_{4},u,t,s).$ (5.3.6)

In order to extract some information on A$(p_{2},p_{3},p_{4},s,t,u)$, we turn
to the dilatation and the special conformal WI’s which it has to satisfy. In
the case of scalars of equal scaling $\Delta$ these take the simplified forms

$\displaystyle 0$
$\displaystyle=\left[(3\Delta-2d)-\sum_{j=2}^{4}\,p_{j}^{\mu}\,\frac{\partial}{\partial
p_{j}^{\mu}}\right]\,\langle{T^{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})O(\mathbf{p}_{2})O(\mathbf{p}_{3})O(\mathbf{p}_{4})}\rangle,$
(5.3.7) $\displaystyle 0$
$\displaystyle=\sum_{j=2}^{4}\left[2(\Delta-d)\frac{\partial}{\partial
p_{j\,\kappa}}-2p_{j}^{\alpha}\,\frac{\partial^{2}}{\partial
p_{j}^{\alpha}\,\partial\,p_{j\,\kappa}}+p_{j}^{\kappa}\,\frac{\partial^{2}}{\partial\,p_{j}^{\alpha}\,\partial\,p_{j\,\alpha}}\right]\,\langle{T^{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})O(\mathbf{p}_{2})O(\mathbf{p}_{3})O(\mathbf{p}_{4})}\rangle,$
(5.3.8)

where, as already mentioned, the momentum $p_{1}^{\mu}$ is taken as the
dependent one.
As discussed in [25], one of the external coordinates of the correlator can be
set to vanish by translational symmetry, and its corresponding momentum, after
Fourier transform, has to be taken as dependent on the other. For instance, in
this case, for convenience, we have chosen the coordinate of the stress-energy
tensor to vanish ($x_{1}=0$), and taken its momentum as the dependent one
($p_{1}\to\overline{p}_{1}$). This implies that the spin part of the special
conformal transformation will not act on the stress-energy tensor, and the
action of this generator is reduced to a pure scalar.
The differentiation is performed only respect to the independent momenta,
using the chain rule while differentiating $\overline{p}_{1}$. This choice is
optimal if we intend to derive symmetric equations for the $TOOO$, in which we
treat the three scalar operators equally, as is the case if we intend to
investigate this correlator in a $1\to 3$ kinematical configuration. In
section 5.4 we will reverse this choice, by taking one of the scalar momenta
($p_{4}$) as the dependent one, which is equivalent to choosing $x_{4}=0$ in
coordinate space. In this second case the special conformal generator will act
with its spin part on the indices of the stress-energy tensor as well, being
the momentum $p_{1}$ one of the independent momenta.
The procedure that we will apply in this case follows quite closely the
approach implemented for 3-point functions, developed in [21]. Both equations
are projected onto the transverse traceless sector using the $\Pi$ projector,
whose action is endomorphic on this sector [21]. A more detailed discussion of
this point can be found in [25].
Henceforth, by applying
$\Pi^{\rho_{1}\sigma_{1}}_{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})$ on the left
of the dilatation and special conformal generators, we find

$\Pi^{\rho_{1}\sigma_{1}}_{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})\,\hat{D}\,\langle{t^{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})O(\mathbf{p}_{2})O(\mathbf{p}_{3})O(\mathbf{p}_{4})}\rangle=0$
(5.3.9)

for the dilatation WI, and

$\displaystyle 0$
$\displaystyle=\Pi^{\rho_{1}\sigma_{1}}_{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})\,\mathcal{K}^{\kappa}\,\left[\langle{t^{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})O(\mathbf{p}_{2})O(\mathbf{p}_{3})O(\mathbf{p}_{4})}\rangle+\langle{t_{loc}^{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})O(\mathbf{p}_{2})O(\mathbf{p}_{3})O(\mathbf{p}_{4})}\rangle\right]$
$\displaystyle=\Pi^{\rho_{1}\sigma_{1}}_{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})\,\mathcal{K}^{\kappa}\,\langle{t^{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})O(\mathbf{p}_{2})O(\mathbf{p}_{3})O(\mathbf{p}_{4})}\rangle+\Pi^{\rho_{1}\sigma_{1}}_{\mu_{1}\nu_{1}}\left[\frac{4d}{\bar{p}_{1}^{2}}\bar{p}_{1\,\beta}\delta^{\mu_{1}\kappa}\langle{T^{\nu_{1}\beta}(\mathbf{\bar{p}}_{1})O(\mathbf{p}_{2})O(\mathbf{p}_{3})O(\mathbf{p}_{4})}\rangle\right]$
(5.3.10)

for the conformal WI, where we have used the relation

$\Pi^{\rho_{1}\sigma_{1}}_{\mu_{1}\nu_{1}}\,{\cal
K}^{\kappa}\,\langle{t_{loc}^{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})O(\mathbf{p}_{2})O(\mathbf{p}_{3})O(\mathbf{p}_{4})}\rangle=\Pi^{\rho_{1}\sigma_{1}}_{\mu_{1}\nu_{1}}\left[\frac{4d}{\bar{p}_{1}^{2}}\,\delta^{\mu_{1}\kappa}\,\bar{p}_{1\,\beta}\,\langle{T^{\nu_{1}\beta}(\mathbf{\bar{p}}_{1})O(\mathbf{p}_{2})O(\mathbf{p}_{3})O(\mathbf{p}_{4})}\rangle\right].$
(5.3.11)

The first term in (5.3.10) can be explicitly written as

$\displaystyle\Pi^{\rho_{1}\sigma_{1}}_{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})\,\mathcal{K}^{\kappa}\,\langle{t^{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})O(\mathbf{p}_{2})O(\mathbf{p}_{3})O(\mathbf{p}_{4})}\rangle$
$\displaystyle=$
$\displaystyle=\Pi^{\rho_{1}\sigma_{1}}_{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})\Bigg{\\{}p_{2}^{\kappa}\Bigg{[}C_{11}\,p_{2}^{\mu_{1}}p_{2}^{\nu_{1}}+C_{12}\,p_{3}^{\mu_{1}}p_{3}^{\nu_{1}}+C_{13}\,p_{4}^{\mu_{1}}p_{4}^{\nu_{1}}\Bigg{]}+p_{3}^{\kappa}\Bigg{[}C_{21}\,p_{2}^{\mu_{1}}p_{2}^{\nu_{1}}+C_{22}\,p_{3}^{\mu_{1}}p_{3}^{\nu_{1}}+C_{23}\,p_{4}^{\mu_{1}}p_{4}^{\nu_{1}}\Bigg{]}$
$\displaystyle+p_{4}^{\kappa}\Bigg{[}C_{31}\,p_{2}^{\mu_{1}}p_{2}^{\nu_{1}}+C_{32}\,p_{3}^{\mu_{1}}p_{3}^{\nu_{1}}+C_{33}\,p_{4}^{\mu_{1}}p_{4}^{\nu_{1}}\Bigg{]}+\delta^{\mu_{1}\kappa}\Bigg{[}C_{41}\,p_{2}^{\nu_{1}}+C_{42}\,p_{3}^{\nu_{1}}+C_{43}\,p_{4}^{\nu_{1}}\Bigg{]}\Bigg{\\}}$
(5.3.12)

where we have used the chain rules

$\displaystyle\frac{\partial}{\partial p_{2}^{\mu}}$
$\displaystyle=\frac{p_{2}^{\mu}}{p_{2}}\frac{\partial}{\partial
p_{2}}+\frac{p_{2}^{\mu}+p_{3}^{\mu}}{t}\frac{\partial}{\partial
t}+\frac{p_{2}^{\mu}+p_{4}^{\mu}}{u}\frac{\partial}{\partial u}$ (5.3.13)
$\displaystyle\frac{\partial}{\partial p_{3}^{\mu}}$
$\displaystyle=\frac{p_{3}^{\mu}}{p_{3}}\frac{\partial}{\partial
p_{3}}+\frac{p_{3}^{\mu}+p_{4}^{\mu}}{s}\frac{\partial}{\partial
s}+\frac{p_{2}^{\mu}+p_{3}^{\mu}}{t}\frac{\partial}{\partial t}$ (5.3.14)
$\displaystyle\frac{\partial}{\partial p_{4}^{\mu}}$
$\displaystyle=\frac{p_{4}^{\mu}}{p_{4}}\frac{\partial}{\partial
p_{4}}+\frac{p_{2}^{\mu}+p_{4}^{\mu}}{s}\frac{\partial}{\partial
s}+\frac{p_{2}^{\mu}+p_{4}^{\mu}}{u}\frac{\partial}{\partial u}$ (5.3.15)

in order to write the covariant derivatives in terms of scalar derivatives
involving the invariants parameterizing $A$. The coefficients $C_{ij}$ in
(5.3.12) are linear combinations of differential operators acting on $A$. The
dilatation WI (5.3.9) can be written in scalar form as

$\displaystyle 0$
$\displaystyle=\Pi^{\rho_{1}\sigma_{1}}_{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})\,\hat{D}\,\langle{t^{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})O(\mathbf{p}_{2})O(\mathbf{p}_{3})O(\mathbf{p}_{4})}\rangle$
$\displaystyle=\Pi^{\rho_{1}\sigma_{1}}_{\mu_{1}\nu_{1}}(\mathbf{\bar{p}}_{1})\Bigg{\\{}D_{1}\,p_{2}^{\mu_{1}}p_{2}^{\nu_{1}}+D_{2}\,p_{3}^{\mu_{1}}p_{3}^{\nu_{1}}+D_{3}\,p_{4}^{\mu_{1}}p_{4}^{\nu_{1}}\Bigg{]}\Bigg{\\}}$
(5.3.16)

where $D_{i}$ are terms involving scalar derivatives acting on the form factor
$A(p_{2},p_{3},p_{4},s,t,u)$. The previous equation is satisfied if all the
$D_{i}$ vanish independently, giving a dilatation constraint on
$A(p_{2},p_{3},p_{4},s,t,u)$ of the form

$\displaystyle
D_{1}=0\,\implies\left[\sum_{j=2}^{4}\,p_{j}\frac{\partial}{\partial
p_{j}}+s\frac{\partial}{\partial s}+t\frac{\partial}{\partial
t}+u\frac{\partial}{\partial
u}\right]A(p_{i},s,t,u)=(\Delta_{t}-3d-2)\,A(p_{i},s,t,u),$ (5.3.17)

where $\Delta_{t}=\sum_{j=1}^{4}\Delta_{j}=d+3\Delta$, since $\Delta_{1}=d$
for the stress-energy tensor, and we have set
$\Delta_{2}=\Delta_{3}=\Delta_{4}=\Delta$. From the other conditions
$D_{i}=0$, with $i=2,3$, we generate the same constraint as from $D_{1}$,
modulo some permutations involving $(p_{2}\leftrightarrow p_{3})$ and
$(p_{2}\leftrightarrow p_{4})$ respectively.

#####  5.3.1 Primary Conformal Ward Identities

From the expressions of (5.3.10) and (5.3.12), after some lengthy algebraic
manipulations, we derive the primary constraints as

$\displaystyle C_{11}=0,$ $\displaystyle C_{12}=0,$ $\displaystyle C_{13}=0,$
$\displaystyle C_{21}=0,$ $\displaystyle C_{22}=0,$ $\displaystyle C_{23}=0,$
$\displaystyle C_{31}=0,$ $\displaystyle C_{32}=0,$ $\displaystyle C_{33}=0,$
(5.3.18)

which are explicitly given in A.1.

One can easily reorganize these equations by introducing the operators

$\displaystyle\bar{K}(p_{2},p_{3},p_{4},s,t,u)$ $\displaystyle\equiv
K_{2}+\frac{p_{3}^{2}-p_{4}^{2}}{s\,t}\frac{\partial}{\partial s\partial
t}-\frac{p_{3}^{2}-p_{4}^{2}}{s\,u}\frac{\partial}{\partial s\partial
u}+\frac{1}{t}\frac{\partial}{\partial t}\left(p_{2}\frac{\partial}{\partial
p_{2}}+p_{3}\frac{\partial}{\partial p_{3}}-p_{4}\frac{\partial}{\partial
p_{4}}\right)$
$\displaystyle+(d-\Delta)\left(\frac{1}{t}\frac{\partial}{\partial
t}+\frac{1}{u}\frac{\partial}{\partial
u}\right)+\frac{1}{u}\frac{\partial}{\partial
u}\left(p_{2}\frac{\partial}{\partial p_{2}}-p_{3}\frac{\partial}{\partial
p_{3}}+p_{4}\frac{\partial}{\partial
p_{4}}\right)+\frac{2p_{2}^{2}+p_{3}^{2}+p_{4}^{2}-s^{2}-t^{2}-u^{2}}{t\,u}\frac{\partial}{\partial
t\partial u}$ (5.3.19)

and

$L(s,t)\equiv\frac{2}{s}\frac{\partial}{\partial
s}-\frac{2}{t}\frac{\partial}{\partial t}$ (5.3.20)

with

$\displaystyle L(s,t)=-L(t,s),$ (5.3.21)

obtaining

$\displaystyle C_{11}$
$\displaystyle=\bar{K}(p_{2},p_{3},p_{4},s,t,u)\,A(p_{2},p_{3},p_{4},s,t,u)$
$\displaystyle C_{12}$
$\displaystyle=\bar{K}(p_{2},p_{3},p_{4},s,t,u)\,A(p_{3},p_{2},p_{4},u,t,s)+L(t,u)\bigg{(}A(p_{2},p_{3},p_{4},s,t,u)+A(p_{3},p_{2},p_{4},u,t,s)\bigg{)}$
$\displaystyle C_{13}$
$\displaystyle=\bar{K}(p_{2},p_{3},p_{4},s,t,u)\,A(p_{4},p_{3},p_{2},t,s,u)-L(t,u)\bigg{(}A(p_{2},p_{3},p_{4},s,t,u)+A(p_{4},p_{3},p_{2},t,s,u)\bigg{)}$
$\displaystyle C_{21}$
$\displaystyle=\bar{K}(p_{3},p_{2},p_{4},u,t,s)\,A(p_{2},p_{3},p_{4},s,t,u)-L(s,t)\bigg{(}A(p_{2},p_{3},p_{4},s,t,u)+A(p_{3},p_{2},p_{4},u,t,s)\bigg{)}$
$\displaystyle C_{22}$
$\displaystyle=\bar{K}(p_{3},p_{2},p_{4},u,t,s)\,A(p_{3},p_{2},p_{4},u,t,s)$
$\displaystyle C_{23}$
$\displaystyle=\bar{K}(p_{3},p_{2},p_{4},u,t,s)\,A(p_{4},p_{3},p_{2},t,s,u)+L(s,t)\bigg{(}A(p_{4},p_{3},p_{2},t,s,u)+A(p_{3},p_{2},p_{4},u,t,s)\bigg{)}$
$\displaystyle C_{31}$
$\displaystyle=\bar{K}(p_{4},p_{3},p_{2},t,s,u)\,A(p_{2},p_{3},p_{4},s,t,u)-L(s,u)\bigg{(}A(p_{2},p_{3},p_{4},s,t,u)+A(p_{4},p_{3},p_{2},t,s,u)\bigg{)}$
$\displaystyle C_{32}$
$\displaystyle=\bar{K}(p_{4},p_{3},p_{2},u,t,s)\,A(p_{3},p_{2},p_{4},u,t,s)+L(s,u)\bigg{(}A(p_{3},p_{2},p_{4},u,t,s)+A(p_{4},p_{3},p_{2},t,s,u)\bigg{)}$
$\displaystyle C_{33}$
$\displaystyle=\bar{K}(p_{4},p_{3},p_{2},u,t,s)\,A(p_{4},p_{3},p_{2},t,s,u).$

We illustrate in (5.1) pictorially the action of the permutation operators
$P_{ij}$, acting on the two momenta $p_{i}^{\mu}$ and $p_{j}^{\mu}$, on the
various $C_{ij}$ presented above. The functional dependence of the form factor
$A(p_{2},p_{3},p_{4},s,t,u)$ will vary accordingly.
The orbits connect the various coefficients $C_{ij}$ which can be reached by
the action of the various permutations. We start with $P_{23}$, $P_{24}$ and
their product $P_{234}$. The orbits describe equivalent equations and we are
allowed to choose any of the equations labeled by coefficients $C_{ij}$
belonging to separate orbits. Since there are three independent orbits under
this subgroup, we start by selecting only three primary conformal WI’s which
are not related by the action of such permutations

Figure 5.1: Orbits of the primary CWIs of the $TOOO$ under $P_{23}$ and
$P_{24}$

Figure 5.2: Equivalent pictorial representation of the orbits as in Fig. 5.1.

The equations that we are going to choose are

$\displaystyle C_{11}$
$\displaystyle=\bar{K}(p_{2},p_{3},p_{4},s,t,u)\,A(p_{2},p_{3},p_{4},s,t,u)$
$\displaystyle C_{12}$
$\displaystyle=\bar{K}(p_{2},p_{3},p_{4},s,t,u)\,A(p_{3},p_{2},p_{4},u,t,s)+L(t,u)\bigg{(}A(p_{2},p_{3},p_{4},s,t,u)+A(p_{3},p_{2},p_{4},u,t,s)\bigg{)}$
$\displaystyle C_{13}$
$\displaystyle=\bar{K}(p_{2},p_{3},p_{4},s,t,u)\,A(p_{4},p_{3},p_{2},t,s,u)-L(t,u)\bigg{(}A(p_{2},p_{3},p_{4},s,t,u)+A(p_{4},p_{3},p_{2},t,s,u)\bigg{)}.$
(5.3.22)

At this stage we include $P_{34}$, under whose action $C_{11}$ is mapped to
itself, while $C_{12}\leftrightarrow C_{13}$. The mapping is illustrated below
showing that the independent equations are only two.

Figure 5.3: Orbits of the CWIs for the $TOOO$ under $P_{34}$.

We take $C_{11}$ and $C_{12}$ as the independent ones, and all the other
equations are obtained by acting on these two with a generic permutation of
$(p_{2},p_{3},p_{4})$. Therefore we have to solve only the two equations

$\left\\{\begin{split}0=&\bar{K}(p_{2},p_{3},p_{4},s,t,u)\,A(p_{2},p_{3},p_{4},s,t,u)\\\
0=&\bar{K}(p_{2},p_{3},p_{4},s,t,u)\,A(p_{3},p_{2},p_{4},u,t,s)+L(t,u)\bigg{(}A(p_{2},p_{3},p_{4},s,t,u)+A(p_{3},p_{2},p_{4},u,t,s)\bigg{)}\end{split}\right.$
(5.3.23)

as representatives of the set of the CWI, after taking into account all the
symmetry properties of $A$. They can equivalently be set into the form

$\left\\{\begin{split}0&=\bar{K}(p_{2},p_{3},p_{4},s,t,u)\,A(p_{2},p_{3},p_{4},s,t,u)\\\
0&=\bar{K}(p_{3},p_{2},p_{4},u,t,s)\,A(p_{2},p_{3},p_{4},s,t,u)-L(s,t)\bigg{(}A(p_{2},p_{3},p_{4},s,t,u)+A(p_{3},p_{2},p_{4},u,t,s)\bigg{)}\end{split}\right.$
(5.3.24)

using the symmetry of the correlator.

#####  5.3.2 Secondary Conformal Ward Identities

The secondary CWIs for the correlator are first order differential equations
derived from the coefficients $C_{4i}$, $i=1,\,2,\,3$ in (5.3.12) together
with Eq. (5.3.10). Such coefficients take the forms

$\displaystyle C_{41}$
$\displaystyle=\left[\frac{2(p_{3}^{2}-p_{2}^{2}-t^{2})}{t}\frac{\partial}{\partial
t}+\frac{2(p_{4}^{2}-p_{2}^{2}-u^{2})}{u}\frac{\partial}{\partial
u}-4\,p_{2}\,\frac{\partial}{\partial
p_{2}}+\frac{2}{\bar{p}_{1}^{2}}\bigg{(}\frac{d(d-2)(p_{2}^{2}-s^{2})-2s^{2}}{(d-1)}\bigg{)}\right.$
$\displaystyle\hskip
28.45274pt\left.+\frac{4\Delta(d-1)-2d^{2}-3(d-2)}{(d-1)}-\frac{(d-2)}{(d-1)}\frac{\big{(}p_{2}^{2}-s^{2}\big{)}^{2}}{\bar{p}_{1}^{4}}\right]\,A(p_{2},p_{3},p_{4},s,t,u)$
$\displaystyle-\left[\frac{2}{\bar{p}_{1}^{2}}\bigg{(}\frac{d(p_{3}^{2}-u^{2})+2u^{2}}{(d-1)}\bigg{)}+\frac{(d-2)}{(d-1)}+\frac{(d-2)}{(d-1)}\frac{(p_{3}^{2}-u^{2})^{2}}{\bar{p}_{1}^{4}}\right]\,A(p_{3},p_{2},p_{4},u,t,s)$
$\displaystyle-\left[\frac{2}{\bar{p}_{1}^{2}}\bigg{(}\frac{d(p_{4}^{2}-t^{2})+2t^{2}}{(d-1)}\bigg{)}+\frac{(d-2)}{(d-1)}+\frac{(d-2)}{(d-1)}\frac{(p_{4}^{2}-t^{2})^{2}}{\bar{p}_{1}^{4}}\right]\,A(p_{4},p_{3},p_{2},t,s,u)\,,$
(5.3.25)

$\displaystyle C_{42}$
$\displaystyle=\left[\frac{2(p_{2}^{2}-p_{3}^{2}-t^{2})}{t}\frac{\partial}{\partial
t}+\frac{2(p_{4}^{2}-p_{3}^{2}-s^{2})}{s}\frac{\partial}{\partial
s}-4\,p_{3}\,\frac{\partial}{\partial
p_{3}}+\frac{2}{\bar{p}_{1}^{2}}\bigg{(}\frac{d(d-2)(p_{3}^{2}-u^{2})-2u^{2}}{(d-1)}\bigg{)}\right.$
$\displaystyle\hskip
28.45274pt\left.+\frac{4\Delta(d-1)-2d^{2}-3(d-2)}{(d-1)}-\frac{(d-2)}{(d-1)}\frac{\big{(}p_{3}^{2}-u^{2}\big{)}^{2}}{\bar{p}_{1}^{4}}\right]\,A(p_{3},p_{2},p_{4},u,t,s)$
$\displaystyle-\left[\frac{2}{\bar{p}_{1}^{2}}\bigg{(}\frac{d(p_{2}^{2}-s^{2})+2s^{2}}{(d-1)}\bigg{)}+\frac{(d-2)}{(d-1)}+\frac{(d-2)}{(d-1)}\frac{(p_{2}^{2}-s^{2})^{2}}{\bar{p}_{1}^{4}}\right]\,A(p_{2},p_{3},p_{4},s,t,u)$
$\displaystyle-\left[\frac{2}{\bar{p}_{1}^{2}}\bigg{(}\frac{d(p_{4}^{2}-t^{2})+2t^{2}}{(d-1)}\bigg{)}+\frac{(d-2)}{(d-1)}+\frac{(d-2)}{(d-1)}\frac{(p_{4}^{2}-t^{2})^{2}}{\bar{p}_{1}^{4}}\right]\,A(p_{4},p_{3},p_{2},t,s,u)$

and

$\displaystyle C_{43}$
$\displaystyle=\left[\frac{2(p_{3}^{2}-p_{4}^{2}-s^{2})}{s}\frac{\partial}{\partial
s}+\frac{2(p_{2}^{2}-p_{4}^{2}-u^{2})}{u}\frac{\partial}{\partial
u}-4\,p_{4}\,\frac{\partial}{\partial
p_{4}}+\frac{2}{\bar{p}_{1}^{2}}\bigg{(}\frac{d(d-2)(p_{4}^{2}-t^{2})-2t^{2}}{(d-1)}\bigg{)}\right.$
$\displaystyle\hskip
28.45274pt\left.+\frac{4\Delta(d-1)-2d^{2}-3(d-2)}{(d-1)}-\frac{(d-2)}{(d-1)}\frac{\big{(}p_{4}^{2}-t^{2}\big{)}^{2}}{\bar{p}_{1}^{4}}\right]\,A(p_{4},p_{3},p_{2},t,s,u)$
$\displaystyle-\left[\frac{2}{\bar{p}_{1}^{2}}\bigg{(}\frac{d(p_{3}^{2}-u^{2})+2u^{2}}{(d-1)}\bigg{)}+\frac{(d-2)}{(d-1)}+\frac{(d-2)}{(d-1)}\frac{(p_{3}^{2}-u^{2})^{2}}{\bar{p}_{1}^{4}}\right]\,A(p_{3},p_{2},p_{4},u,t,s)$
$\displaystyle-\left[\frac{2}{\bar{p}_{1}^{2}}\bigg{(}\frac{d(p_{2}^{2}-s^{2})+2s^{2}}{(d-1)}\bigg{)}+\frac{(d-2)}{(d-1)}+\frac{(d-2)}{(d-1)}\frac{(p_{2}^{2}-s^{2})^{2}}{\bar{p}_{1}^{4}}\right]\,A(p_{2},p_{3},p_{4},s,t,u)\,,$
(5.3.27)

where here $p_{1}^{2}$ is treated as a dependent variable, that is:
$\bar{p}_{1}^{2}=s^{2}+t^{2}+u^{2}-p_{2}^{2}-p_{3}^{2}-p_{4}^{2}$. The actions
of the operators enforcing the momentum permutations and the orbits of the
$C_{ij}$ are illustrated in Fig. (5.4), where a given equation is connected by
a link if there is a permutation of the momenta which relates it to a
different one.
It is clear from the figure that each single vertex of the triangle is mapped
into itself under a permutation acting on the opposite edge, showing that
there is only one independent secondary CWI. In particular, we choose as the
independent one $C_{41}$, which can be re-expressed in the form

Figure 5.4: Orbits of the secondary CWIs under permutations.

$\displaystyle L_{1}\,A(p_{2},p_{3},p_{4},s,t,u)$ $\displaystyle\qquad-
L_{2}\,A(p_{3},p_{2},p_{4},u,t,s)-L_{2}(p_{3}\leftrightarrow
p_{4})\,A(p_{4},p_{3},p_{2},t,s,u)=-\frac{4d}{\bar{p}_{1}^{2}}\langle{O(\mathbf{p}_{3}+\mathbf{p}_{4})O(\mathbf{p}_{3})O(\mathbf{p}_{4})}\rangle,$
(5.3.28)

where

$\displaystyle L_{1}$
$\displaystyle=\left[\frac{2(p_{3}^{2}-p_{2}^{2}-t^{2})}{t}\frac{\partial}{\partial
t}+\frac{2(p_{4}^{2}-p_{2}^{2}-u^{2})}{u}\frac{\partial}{\partial
u}-4\,p_{2}\,\frac{\partial}{\partial
p_{2}}+\frac{2}{\bar{p}_{1}^{2}}\bigg{(}\frac{d(d-2)(p_{2}^{2}-s^{2})-2s^{2}}{(d-1)}\bigg{)}\right.$
$\displaystyle\hskip
28.45274pt\left.+\frac{4\Delta(d-1)-2d^{2}-3(d-2)}{(d-1)}-\frac{(d-2)}{(d-1)}\frac{\big{(}p_{2}^{2}-s^{2}\big{)}^{2}}{\bar{p}_{1}^{4}}\right]$
(5.3.29) $\displaystyle L_{2}$
$\displaystyle=\left[\frac{2}{\bar{p}_{1}^{2}}\bigg{(}\frac{d(p_{3}^{2}-u^{2})+2u^{2}}{(d-1)}\bigg{)}+\frac{(d-2)}{(d-1)}+\frac{(d-2)}{(d-1)}\frac{(p_{3}^{2}-u^{2})^{2}}{\bar{p}_{1}^{4}}\right].$
(5.3.30)

From (5.3.28), one can check that the symmetry $p_{3}\leftrightarrow p_{4}$ is
explicitly manifest.
In general, the role of the secondary WI’s is to reduce the parameters of the
solutions of the primary ones. For instance, in the case of 3-point functions,
such solutions are determined from the primary equations modulo few constants,
which are then fixed by the secondary ones. In that case, the right hand side
of the secondary equations will be proportional to 2-point functions.
The constraints on the primary solutions are obtained by taking special limits
on the left hand side of the equations, in order to send two external
coordinates into coalescence. This is obtained, for 3-point functions, by
taking two of the external invariant masses large and of unit ratio -
$p_{3}^{2}/p_{2}^{2}\to 1$, for instance - which reduces the correlator to a
2-point function.
For 4-point functions this limit is far more involved, and we will be able to
say little about it, the crucial point being that the primary solutions should
contain arbitrary function(s), in this case a single function, as expected
from the analysis in coordinate space, which are not identified in our
formulation.
For this reason, we will try to discuss the asymptotic limit only of the
primary solutions, where it is possible to underscore some specific behaviors
of such solutions just by examining the structure of the equations.

####  5.4 The decomposition of the $TOOO$ in the $2\rightarrow 2$ formulation

In this section we will reconsider the $TOOO$ correlator $\langle
T^{\mu_{1}\nu_{1}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle$
but with a different choice of the dependent momentum compared to the $1\to 3$
case, which is suitable for the study of a $2\to 2$ process.
We choose $p_{4}^{\mu}$ as the dependent momentum,
$\bar{p_{4}}^{\mu}=-p_{1}^{\mu}-p_{2}^{\mu}-p_{3}^{\mu}$. Moreover, also the
Mandelstam invariant $u^{2}$, will be taken as dependent variable
$\tilde{u}^{2}=-s^{2}-t^{2}+\sum p_{i}^{2}$.
We rewrite the decomposition (5.2.7) in the forms

$\langle
T^{\mu_{1}\nu_{1}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle=\langle
t^{\mu_{1}\nu_{1}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle+\langle
t^{\mu_{1}\nu_{1}}_{\textrm{loc}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle,$
(5.4.1)

which is symmetric in $p_{2}^{\mu},p_{3}^{\mu},p_{4}^{\mu}$. Furthermore, we
require the parameterization in the $tt$ sector to be symmetric under the
exchange of the indices of the stress-energy tensor
$\mu_{1}\leftrightarrow\nu_{1}$. The $tt$ component of the $TOOO$ can then be
parameterized as

$\langle
t^{\mu_{1}\nu_{1}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle=\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(\mathbf{p_{1}})X^{\alpha_{1}\beta_{1}},$
(5.4.2)

where $X^{\alpha_{1}\beta_{1}}$ is a general rank-2 tensor built out momenta
and Kronecker’s delta’s. There are two equivalent decompositions of such $tt$
term, that we will present below, but only one of them allows to obtain
simplified expressions of the primary and secondary CWIs, which will turn very
useful for our analysis.

#####  5.4.1 First decomposition

First, we are going to derive the decomposition of the correlator by choosing
as independent momenta $p_{2}^{\mu},p_{3}^{\mu}$. Terms that include
$p_{1}^{\mu}$ will be eliminated by the transverse-traceless projector
$\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(\mathbf{p_{1}})$ and therefore
will be omitted. We obtain the parameterization

$X^{\alpha_{1}\beta_{1}}=C^{\prime}(p_{1},p_{2},p_{3},p_{4},s,t)p_{2}^{\alpha_{1}}p_{2}^{\beta_{1}}+C^{\prime\prime}(p_{1},p_{2},p_{3},p_{4},s,t)p_{3}^{\alpha_{1}}p_{3}^{\beta_{1}}+C(p_{1},p_{2},p_{3},p_{4},s,t)p_{2}^{\alpha_{1}}p_{3}^{\beta_{1}},$
(5.4.3)

expressed in temrs of form factors $C,C^{\prime},C^{\prime\prime}$. Now, by
imposing all the possible permutations (6 in total) of the momenta
$p_{2}^{\mu},p_{3}^{\mu},p_{4}^{\mu}$, we derive the constraints

$\displaystyle
C^{\prime}(p_{1},p_{2},p_{3},p_{4},s,t)=\frac{1}{2}\Big{(}C(p_{1},p_{2},p_{3},p_{4},s,t)+C(p_{1},p_{4},p_{3},p_{2},t,s)\Big{)},$
$\displaystyle
C^{\prime\prime}(p_{1},p_{2},p_{4},p_{3},s,\tilde{u})=\frac{1}{2}\Big{(}C(p_{1},p_{2},p_{3},p_{4},s,t)+C(p_{1},p_{2},p_{4},p_{3},s,\tilde{u})\Big{)}.$

Finally, such $tt$ component takes the form

$\begin{split}\langle
t^{\mu_{1}\nu_{1}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle&=\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}\Bigg{[}\frac{1}{2}\Big{(}C(p_{1},p_{2},p_{3},p_{4},s,t)+C(p_{1},p_{4},p_{3},p_{2},t,s)\Big{)}p_{2}^{\alpha_{1}}p_{2}^{\beta_{1}}\\\
&+\frac{1}{2}\Big{(}C(p_{1},p_{2},p_{3},p_{4},s,t)+C(p_{1},p_{2},p_{4},p_{3},s,\tilde{u})\Big{)}p_{3}^{\alpha_{1}}p_{3}^{\beta_{1}}+C(p_{1},p_{2},p_{3},p_{4},s,t)p_{2}^{\alpha_{1}}p_{3}^{\beta_{1}}\Bigg{]},\end{split}$
(5.4.4)

expressed in terms of a single form factor which exhibits the following
symmetries

$\begin{split}&C(p_{1},p_{2},p_{3},p_{4},s,t)=C(p_{1},p_{3},p_{2},p_{4},\tilde{u},t),\\\
&C(p_{1},p_{2},p_{4},p_{3},s,\tilde{u})=C(p_{1},p_{4},p_{2},p_{3},\tilde{u},s)\\\
&C(p_{1},p_{4},p_{3},p_{2},t,s)=C(p_{1},p_{3},p_{4},p_{2},\tilde{u},s).\end{split}$
(5.4.5)

#####  5.4.2 Second decomposition

The second decomposition is obtained by using all the available momenta. It
takes the form

$\langle
t^{\mu_{1}\nu_{1}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle=\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(\mathbf{p_{1}})\tilde{X}^{\alpha_{1}\beta_{1}},$
(5.4.6)

where

$\tilde{X}^{\alpha_{1}\beta_{1}}=F(p_{1},p_{2},p_{3},p_{4},s,t)p_{2}^{\alpha_{1}}p_{3}^{\beta_{1}}+F^{\prime}(p_{1},p_{2},p_{3},p_{4},s,t)p_{2}^{\alpha_{1}}p_{4}^{\beta_{1}}+F^{\prime\prime}(p_{1},p_{2},p_{3},p_{4},s,t)p_{3}^{\alpha_{1}}p_{4}^{\beta_{1}}.$
(5.4.7)

Taking into account all the possible permutations, we end up with the
expression

$\tilde{X}^{\alpha_{1}\beta_{1}}=F(p_{1},p_{4},p_{3},p_{2},t,s)p_{3}^{\alpha_{1}}p_{4}^{\beta_{1}}+F(p_{1},p_{2},p_{4},p_{3},s,\tilde{u})p_{2}^{\alpha_{1}}p_{4}^{\beta_{1}}+F(p_{1},p_{2},p_{3},p_{4},s,t)p_{2}^{\alpha_{1}}p_{3}^{\beta_{1}}.$
(5.4.8)

Our form factor obeys the following symmetries:

$\begin{split}&F(p_{1},p_{3},p_{4},p_{2},\tilde{u},s)=F(p_{1},p_{4},p_{3},p_{2},t,s),\\\
&F(p_{1},p_{4},p_{2},p_{3},\tilde{u},s)=F(p_{1},p_{2},p_{4},p_{3},s,\tilde{u}),\\\
&F(p_{1},p_{3},p_{2},p_{4},\tilde{u},t)=F(p_{1},p_{2},p_{3},p_{4},s,t).\end{split}$
(5.4.9)

Now, we can impose momentum conservation on the first two terms of (5.4.4).
Then comparing with (5.4.8) and using the symmetry properties of the previous
form factor $C$ in (LABEL:cc1), we obtain

$\displaystyle
F(p_{1},p_{2},p_{3},p_{4},s,t)=-\frac{1}{2}\Big{(}C(p_{1},p_{2},p_{3},p_{4},s,t)+C(p_{1},p_{2},p_{4},p_{3},s,\tilde{u})\Big{)},$
$\displaystyle
F(p_{1},p_{2},p_{4},p_{3},s,\tilde{u})=-\frac{1}{2}\Big{(}C(p_{1},p_{2},p_{3},p_{4},s,t)+C(p_{1},p_{4},p_{3},p_{2},t,s)\Big{)},$
$\displaystyle
F(p_{1},p_{4},p_{3},p_{2},t,s)=-\frac{1}{2}\Big{(}C(p_{1},p_{2},p_{3},p_{4},s,t)+C(p_{1},p_{2},p_{4},p_{3},s,\tilde{u})\Big{)}.$
(5.4.10)

The form factors $F$ and $C$ are related proving the equivalence between the
two parameterizations. However $F$ is the one which generates CWIs of a
simpler structure.

#####  5.4.3 Dilatation Ward Identity in the $2\rightarrow 2$ formulation

In this section we will proceed with the study of the dilatation WI. Using the
form factor $F$, the full correlator is given by (5.4.1) and the exact
parameterization of its $tt$ sector takes the form

$\begin{split}\langle
t^{\mu_{1}\nu_{1}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle=&\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(\mathbf{p_{1}})\Big{(}F(p_{1},p_{4},p_{3},p_{2},t,s)p_{3}^{\alpha_{1}}p_{4}^{\beta_{1}}+F(p_{1},p_{2},p_{4},p_{3},s,\tilde{u})p_{2}^{\alpha_{1}}p_{4}^{\beta_{1}}\\\
&+F(p_{1},p_{2},p_{3},p_{4},s,t)p_{2}^{\alpha_{1}}p_{3}^{\beta_{1}}\Big{)},\end{split}$
(5.4.11)

while the longitudinal sector is extracted by a contraction with the
longitudinal projector

$\langle
t^{\mu_{1}\nu_{1}}_{\textrm{loc}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle=\Sigma^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(\mathbf{p_{1}})\langle
T^{\alpha_{1}\beta_{1}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle$
(5.4.12)

as in our previous analysis of the equations for the $1\to 3$. We can express
the CWI’ s in terms of 6 invariants of the four-point function
($\sqrt{p_{i}^{2}}=p_{i}$,
$s=\sqrt{(\mathbf{p_{1}+p_{2}})^{2}},t=\sqrt{(\mathbf{p_{2}+p_{3}})^{2}}$) by
using the chain rules

$\displaystyle\frac{\partial}{\partial
p_{1\mu}}=\frac{p_{1}^{\mu}}{p_{1}}\frac{\partial}{\partial
p_{1}}-\frac{\bar{p_{4}}^{\mu}}{p_{4}}\frac{\partial}{\partial
p_{4}}+\frac{p_{1}^{\mu}+p_{2}^{\mu}}{s}\frac{\partial}{\partial s},$ (5.4.13)
$\displaystyle\frac{\partial}{\partial
p_{2\mu}}=\frac{p_{2}^{\mu}}{p_{2}}\frac{\partial}{\partial
p_{1}}-\frac{\bar{p_{4}}^{\mu}}{p_{4}}\frac{\partial}{\partial
p_{4}}+\frac{p_{1}^{\mu}+p_{2}^{\mu}}{s}\frac{\partial}{\partial
s}+\frac{p_{2}^{\mu}+p_{3}^{\mu}}{t}\frac{\partial}{\partial t},$ (5.4.14)
$\displaystyle\frac{\partial}{\partial
p_{3\mu}}=\frac{p_{3}^{\mu}}{p_{3}}\frac{\partial}{\partial
p_{3}}-\frac{\bar{p_{4}}^{\mu}}{p_{4}}\frac{\partial}{\partial
p_{4}}+\frac{p_{2}^{\mu}+p_{3}^{\mu}}{t}\frac{\partial}{\partial t}.$ (5.4.15)

Applying the dilatation WI to (5.4.1) we obtain

$\big{[}(\Delta_{\textrm{t}}-3d)-\sum_{i=1}^{3}p_{i}^{\lambda}\frac{\partial}{\partial
p_{i}^{\lambda}}\big{]}\langle
T^{\mu_{1}\nu_{1}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle=0,$
(5.4.16)

which can be projected using the $tt$ projector
$\Pi^{\rho\sigma}_{\mu_{1}\nu_{1}}(\mathbf{p_{1}})$ obtaining

$\Pi^{\rho\sigma}_{\mu_{1}\nu_{1}}(\mathbf{p_{1}})\hat{D}\langle
t^{\mu_{1}\nu_{1}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle=0.$
(5.4.17)

Using (5.4.11) and differentiating by the chain rule (5.4.15), we finally
obtain the equation

$\left[(2+3d-\Delta_{\textrm{t}})+\sum_{i=1}^{4}p_{i}\frac{\partial}{\partial
p_{i}}+s\frac{\partial}{\partial s}+t\frac{\partial}{\partial
t}\right]F(p_{1},p_{2},p_{3},p_{4},s,t)=0.$ (5.4.18)

The same equation holds also for $F(p_{1},p_{2},p_{4},p_{3},s,\tilde{u}),$ and
$F(p_{1},p_{4},p_{3},p_{2},t,s)$. The 2 in the first term of the sum
$(2+3d\ldots$), defines the tensorial dimension of the form factor, and counts
the number of momenta with which it appears in the parameterization.

#####  5.4.4 Special CWI for the $\langle
T^{\mu_{1}\nu_{1}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle$

In this section we repeat the analysis of the $1\to 3$ case, with the new
parameterization of the correlator that we have just derived, by selecting
$p_{4}$ as the dependent momentum. The action of the special conformal
generator, as before, will take the form

$\begin{split}0&=\mathcal{K}^{\kappa}\langle{T^{\mu_{1}\nu_{1}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})}\rangle\\\\[5.16663pt]
&=\mathcal{K}^{\kappa}\langle
t^{\mu_{1}\nu_{1}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\bar{\mathbf{p_{4}}})\rangle+\mathcal{K}^{\kappa}\langle{t^{\mu_{1}\nu_{1}}_{\textrm{loc}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\bar{\mathbf{p_{4}}})}\rangle,\end{split}$
(5.4.19)

We will focus now on the local part related to $t_{loc}$.
Using (5.4.12) we now have to compute

$\Pi^{\rho\sigma}_{\mu_{1}\nu_{1}}(p_{1})\mathcal{K}^{\kappa}\Sigma^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}\langle
T^{\alpha_{1}\beta_{1}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle,$
(5.4.20)

projected on the $tt$ sector. We split our results into the scalar and the
spin part of $\mathcal{K}^{\kappa}$. Acting with the projection $\Pi$ we
obtain

$\begin{split}&\Pi^{\rho\sigma}_{\mu_{1}\nu_{1}}(p_{1})\mathcal{K}^{\kappa}_{\mathrm{scalar}}\langle
t^{\mu_{1}\nu_{1}}_{\textrm{loc}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle=\\\
&=\Pi^{\rho\sigma}_{\mu_{1}\nu_{1}}(p_{1})\Bigg{[}4(\Delta_{1}-d+1)\frac{\delta^{\kappa\mu_{1}}\delta^{\nu_{1}}_{\alpha_{1}}p_{1,\beta_{1}}}{p_{1}^{2}}+4\frac{p_{1}^{\kappa}}{p_{1}^{2}}\delta^{\mu_{1}}_{\beta_{1}}\delta^{\nu_{1}}_{\alpha_{1}}\Bigg{]}\langle
T^{\alpha_{1}\beta_{1}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle,\end{split}$
(5.4.21)

and

$\begin{split}&\Pi^{\rho\sigma}_{\mu_{1}\nu_{1}}(p_{1})\mathcal{K}^{\kappa}_{\alpha,\mathrm{spin}}\langle
t^{\mu_{1}\nu_{1}}_{\textrm{loc}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle=\\\
&=\Pi^{\rho\sigma}_{\mu_{1}\nu_{1}}(\textbf{p}_{1})\Bigg{[}4(d-1)\frac{\delta^{\kappa\mu_{1}}\delta^{\nu_{1}}_{\alpha_{1}}p_{1,\beta_{1}}}{p_{1}^{2}}-4\frac{p_{1}^{\kappa}}{p_{1}^{2}}\delta^{\mu_{1}}_{\beta_{1}}\delta^{\nu_{1}}_{\alpha_{1}}\Bigg{]}\langle
T^{\alpha_{1}\beta_{1}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle.\end{split}$
(5.4.22)

In our results, we have ignored terms that include
$p_{1}^{\mu_{1}},p_{1}^{\nu_{1}},\delta^{\mu_{1}\nu_{1}}$, because we have the
freedom to apply a transverse-traceless projector of the form
$\Pi^{\rho\sigma}_{\mu_{1}\nu_{1}}(\mathbf{p}_{1})$ to (5.4.19), so these
terms will vanish. Adding the scalar and the spin contributions (and using
$\Delta_{1}=d$), we get

$\Pi^{\rho\sigma}_{\mu_{1}\nu_{1}}(p_{1})\mathcal{K}^{\kappa}\langle
t^{\mu_{1}\nu_{1}}_{\textrm{loc}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle=\Pi^{\rho\sigma}_{\mu_{1}\nu_{1}}(p_{1})\left(\frac{4d}{p_{1}^{2}}\delta^{\kappa\mu_{1}}p_{1\,\alpha}\langle
T^{\alpha\nu_{1}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle\right).$
(5.4.23)

Now, we will apply the $\mathcal{K}^{\kappa}$ operator on the $tt$ part,
followed by contraction with the $\Pi$ projector. We obtain the tensor
equation

$\begin{split}&\Pi^{\rho\sigma}_{\mu_{1}\nu_{1}}(p_{1})\mathcal{K}^{\kappa}\langle
t^{\mu_{1}\nu_{1}}(\mathbf{p_{1}})O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle=\\\
&\Pi^{\rho\sigma}_{\mu_{1}\nu_{1}}(p_{1})\Big{[}p_{1}^{\kappa}\big{(}\tilde{C}_{11}p_{2}^{\mu_{1}}p_{3}^{\nu_{1}}+\tilde{C}_{12}p_{2}^{\mu_{1}}p_{4}^{\nu_{1}}+\tilde{C}_{13}p_{3}^{\mu_{1}}p_{4}^{\mu_{1}}\big{)}+p_{2}^{\kappa}\big{(}\tilde{C}_{21}p_{2}^{\mu_{1}}p_{3}^{\nu_{1}}+\tilde{C}_{22}p_{2}^{\mu_{1}}p_{4}^{\nu_{1}}+\tilde{C}_{23}p_{3}^{\mu_{1}}p_{4}^{\mu_{1}}\big{)}\\\
&+p_{3}^{\kappa}\big{(}\tilde{C}_{31}p_{2}^{\mu_{1}}p_{3}^{\nu_{1}}+\tilde{C}_{32}p_{2}^{\mu_{1}}p_{4}^{\nu_{1}}+\tilde{C}_{33}p_{3}^{\mu_{1}}p_{4}^{\mu_{1}}\big{)}+\delta^{\mu_{1}\kappa}\big{(}\tilde{C}_{41}p_{2}^{\nu_{1}}+\tilde{C}_{42}p_{3}^{\nu_{1}}\big{)}+\delta^{\nu_{1}\kappa}\big{(}\tilde{C}_{51}p_{2}^{\mu_{1}}+\tilde{C}_{52}p_{3}^{\mu_{1}}\big{)}\Big{]}\end{split}$
(5.4.24)

which will allow us to extract the independent conformal constraints.

#####  5.4.5 Primary Conformal Ward Identities

The factors $\tilde{C}_{1j},\tilde{C}_{2j},\tilde{C}_{3j}$ are second-order
differential equations involving the form factor $F$ and its various
permutations. We see from (5.4.19) and (5.4.24) that the coefficients of the
four-momenta $p_{1}^{\kappa},p_{2}^{\kappa},p_{3}^{\kappa}$ are zero. This
translates into the equations

$\displaystyle\tilde{C}_{11}=0,$ $\displaystyle\tilde{C}_{12}=0,$
$\displaystyle\tilde{C}_{13}=0,$ $\displaystyle\tilde{C}_{21}=0,$
$\displaystyle\tilde{C}_{22}=0,$ $\displaystyle\tilde{C}_{23}=0,$
$\displaystyle\tilde{C}_{31}=0,$ $\displaystyle\tilde{C}_{32}=0,$
$\displaystyle\tilde{C}_{33}=0.$ (5.4.25)

These are the primary CWIs that we have mentioned before. Below we present the
explicit expressions involving the $F(p_{1},p_{2},p_{3},p_{4},s,t)$ form
factor. The remaining ones, which are obtained just by permutations of the
momenta, can be found in A.1. We obtain

$\displaystyle\tilde{C}_{11}=$
$\displaystyle\Bigg{[}\frac{\partial^{2}}{\partial
p_{4}^{2}}+\frac{d-2\Delta+1}{p_{4}}\frac{\partial}{\partial
p_{4}}-\frac{\partial^{2}}{\partial
p_{1}^{2}}-\frac{1-d}{p_{1}}\frac{\partial}{\partial
p_{1}}+\frac{1}{s}\frac{\partial}{\partial
s}\left(p_{4}\frac{\partial}{\partial p_{4}}+p_{3}\frac{\partial}{\partial
p_{3}}-p_{1}\frac{\partial}{\partial p_{1}}-p_{2}\frac{\partial}{\partial
p_{2}}\right)$ $\displaystyle+\frac{d-\Delta}{s}\frac{\partial}{\partial
s}+\frac{p_{3}^{2}-p_{2}^{2}}{st}\frac{\partial^{2}}{\partial s\partial
t}\Bigg{]}F(p_{1},p_{2},p_{3},p_{4},s,t)+\frac{2}{s}\frac{\partial
F(p_{1},p_{4},p_{3},p_{2},t,s)}{\partial s}-\frac{2}{s}\frac{\partial
F(p_{1},p_{2},p_{4},p_{3},s,\tilde{u})}{\partial s}$ (5.4.26)
$\displaystyle\tilde{C}_{21}=$
$\displaystyle\Bigg{[}\frac{\partial^{2}}{\partial
p_{4}^{2}}+\frac{d-2\Delta+1}{p_{4}}\frac{\partial}{\partial
p_{4}}-\frac{\partial^{2}}{\partial
p_{2}^{2}}-\frac{d-2\Delta+1}{p_{2}}\frac{\partial}{\partial
p_{2}}+\frac{1}{s}\frac{\partial}{\partial
s}\left(p_{3}\frac{\partial}{\partial p_{3}}+p_{4}\frac{\partial}{\partial
p_{4}}-p_{1}\frac{\partial}{\partial p_{1}}-p_{2}\frac{\partial}{\partial
p_{2}}\right)$ $\displaystyle+\frac{\Delta-d-2}{t}\frac{\partial}{\partial
t}+\frac{d-\Delta}{s}\frac{\partial}{\partial
s}+\frac{1}{t}\frac{\partial}{\partial t}\left(p_{1}\frac{\partial}{\partial
p_{1}}+p_{4}\frac{\partial}{\partial p_{4}}-p_{2}\frac{\partial}{\partial
p_{2}}-p_{3}\frac{\partial}{\partial p_{3}}\right)$
$\displaystyle+\frac{p_{4}^{2}-p_{2}^{2}}{st}\frac{\partial^{2}}{\partial
s\partial t}\Bigg{]}F(p_{1},p_{2},p_{3},p_{4},s,t)+\frac{2}{s}\frac{\partial
F(p_{1},p_{4},p_{3},p_{2},t,s)}{\partial s}-\frac{2}{s}\frac{\partial
F(p_{1},p_{2},p_{4},p_{3},s,\tilde{u})}{\partial s},$

and finally

$\displaystyle\tilde{C}_{31}=$
$\displaystyle\Bigg{[}\frac{\partial^{2}}{\partial
p_{4}^{2}}+\frac{d-2\Delta+1}{p_{4}}\frac{\partial}{\partial
p_{4}}-\frac{\partial^{2}}{\partial
p_{3}^{2}}-\frac{d-2\Delta+1}{p_{3}}\frac{\partial}{\partial
p_{3}}+\frac{2}{s}\frac{\partial}{\partial
s}+\frac{p_{1}^{2}-p_{2}^{2}}{st}\frac{\partial^{2}}{\partial s\partial
t}+\frac{\Delta-d-2}{t}\frac{\partial}{\partial t}$
$\displaystyle+\frac{1}{t}\frac{\partial}{\partial
t}\left(p_{1}\frac{\partial}{\partial p_{1}}+p_{4}\frac{\partial}{\partial
p_{4}}-p_{2}\frac{\partial}{\partial p_{2}}-p_{3}\frac{\partial}{\partial
p_{3}}\right)\Bigg{]}F(p_{1},p_{2},p_{3},p_{4},s,t)+\frac{2}{s}\frac{\partial
F(p_{1},p_{4},p_{3},p_{2},t,s)}{\partial s}$
$\displaystyle-\frac{2}{s}\frac{\partial
F(p_{1},p_{2},p_{4},p_{3},s,\tilde{u})}{\partial s}.$ (5.4.28)

#####  5.4.6 Secondary Conformal Ward Identities

Since our 4-point function is symmetric in $\mu_{1}\leftrightarrow\nu_{1}$,
the terms proportional to $\delta^{\mu_{1}\kappa}$ and
$\delta^{\nu_{1}\kappa}$ given by the coefficients $\tilde{C}_{41}$ and
$\tilde{C}_{51}$ identify a single constraint, as well as $\tilde{C}_{42}$ and
$\tilde{C}_{52}$, and are explicitly given by the factors $\tilde{C}_{4j}$.
They take the form

$\begin{split}&\tilde{C}_{41}=\hat{G}\Big{(}F(p_{1},p_{2},p_{3},p_{4},s,t)-F(p_{1},p_{4},p_{3},p_{2},t,s)\Big{)}+\hat{A}F(p_{1},p_{2},p_{4},p_{3},s,\tilde{u}),\end{split}$
(5.4.29)

where

$\begin{split}\hat{G}&=\frac{d(s^{2}+t^{2}-p_{4}^{2}-p_{2}^{2}-2p_{1}^{2})+2\Delta
p_{1}^{2}}{p_{1}^{2}}-\frac{t^{2}+p_{3}^{2}-p_{2}^{2}}{t}\frac{\partial}{\partial
t}+\frac{p_{2}^{2}+p_{4}^{2}-s^{2}-t^{2}}{p_{1}}\frac{\partial}{\partial
p_{1}}\\\ &-\frac{s^{2}+p_{3}^{2}-p_{4}^{2}}{s}\frac{\partial}{\partial
s}-2p_{3}\frac{\partial}{\partial p_{3}},\end{split}$ (5.4.30)

and

$\begin{split}\hat{A}&=\Bigg{(}\frac{d(s^{2}+p_{4}^{2}-p_{2}^{2}-t^{2})}{p_{1}^{2}}\Bigg{)}+\frac{t^{2}+p_{2}^{2}-p_{3}^{2}}{t}\frac{\partial}{\partial
t}+\frac{p_{2}^{2}+t^{2}-p_{4}^{2}-s^{2}}{p_{1}}\frac{\partial}{\partial
p_{1}}\\\ &+\frac{p_{3}^{2}-p_{4}^{2}-s^{2}}{s}\frac{\partial}{\partial
s}+2p_{2}\frac{\partial}{\partial p_{2}}-2p_{4}\frac{\partial}{\partial
p_{4}}.\end{split}$ (5.4.31)

Moreover, we obtain

$\begin{split}&\tilde{C}_{42}=\hat{M}\Big{(}F(p_{1},p_{2},p_{3},p_{4},s,t)-F(p_{1},p_{2},p_{4},p_{3},s,\tilde{u})\Big{)}+\hat{N}F(p_{1},p_{4},p_{3},p_{2},t,s),\end{split}$
(5.4.32)

where

$\begin{split}\hat{M}=&\left(\frac{2\Delta
p_{1}^{2}-d(p_{1}^{2}-p_{2}^{2}+s^{2})}{p_{1}^{2}}\right)-\frac{p_{1}^{2}+p_{2}^{2}-s^{2}}{p_{1}}\frac{\partial}{\partial
p_{1}}+\frac{p_{3}^{2}-p_{2}^{2}-t^{2}}{t}\frac{\partial}{\partial
t}-2p_{2}\frac{\partial}{p_{2}}\end{split}$ (5.4.33)

and

$\begin{split}\hat{N}=&\left(\frac{d\left(p_{1}^{2}+p_{2}^{2}+2p_{4}^{2}-s^{2}-2t^{2}\right)}{p_{1}^{2}}\right)+\frac{t^{2}+p_{3}^{2}-p_{2}^{2}}{t}\frac{\partial}{\partial
t}-\frac{p_{1}^{2}+p_{2}^{2}+2p_{4}^{2}-s^{2}-2t^{2}}{p_{1}}\frac{\partial}{\partial
p_{1}}\\\ &+\frac{2(p_{3}^{2}-p_{4}^{2})}{s}\frac{\partial}{\partial
s}-2p_{4}\frac{\partial}{\partial p_{4}}+2p_{3}\frac{\partial}{\partial
p_{3}}.\end{split}$ (5.4.34)

Combining (5.1.17) along with (5.4.23) and (5.4.24) we obtain the equations

$\begin{split}&\hat{G}\Big{(}F(p_{1},p_{2},p_{3},p_{4},s,t)-F(p_{1},p_{4},p_{3},p_{2},t,s)\Big{)}+\hat{A}F(p_{1},p_{2},p_{4},p_{3},s,\tilde{u})=\\\
&\hskip 56.9055pt=\frac{4d}{p_{1}^{2}}\Big{(}\langle
O(\mathbf{p_{1}+p_{2}})O(\mathbf{p_{3}})O(\mathbf{\bar{p}_{4}})\rangle-\langle
O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{p_{2}+p_{3}})\rangle\Big{)},\end{split}$
(5.4.35)

and

$\begin{split}&\hat{M}\Big{(}F(p_{1},p_{2},p_{3},p_{4},s,t)-F(p_{1},p_{2},p_{4},p_{3},s,\tilde{u})\Big{)}+\hat{N}F(p_{1},p_{4},p_{3},p_{2},t,s)=\\\
&\hskip 56.9055pt=\frac{4d}{p_{1}^{2}}\big{(}\langle
O(\mathbf{p_{2}})O(\mathbf{p_{1}+p_{3}})O(\mathbf{\bar{p}_{4}})\rangle-\langle
O(\mathbf{p_{2}})O(\mathbf{p_{3}})O(\mathbf{p_{2}+p_{3}})\rangle\big{)}.\end{split}$
(5.4.36)

These are the secondary WI’s for the $TOOO$. The 3-point function on the right
hand side of this equation is uniquely given by a combination of
hypergeometric functions and will be discussed below.

####  5.5 Asymptotics for scalar and dual conformal/conformal 4-point
functions

Our goal, from this section on, will be to identify some of the properties of
these primary and secondary equations for the $TOOO$, and for this reason it
will be compelling to consider first the $(OOOO)$ correlator, which is
slightly simpler compared to the former. Both cases show some similarities,
starting from the fact that they are both characterised by a single form
factor. The structure of the equations is expected to be similar, and indeed
in both cases we will be able to identify also a similar behaviour in the
corresponding form factors, in some kinematical limits.
The $OOOO$, as shown recently [26], allows a specific class of solutions which
are uniquely identified by enlarging the original conformal symmetry to
include a dual conformal symmetry as well. Indeed, these special solutions are
very useful for studying the hypergeometric structure of the CWIs in some
asymptotic limits. As we are going to see, hypergeometric solutions of 4-point
functions are very special, as one expects on generic grounds, and the general
CWIs, even in the scalar case, are not described by hypergeometric systems