related to $F_{4}$. The only exact statement that can be made concerning the
structure of such systems of equations, as we are going to show, will be that
Lauricella functions - i.e. hypergeometric functions of three variables - are
exact solutions of all these systems of equations and can be interpreted as
homogeneous (i.e. particular) solutions of such CWIs for arbitrary scaling
dimensions of the scalar operators.
We start our discussion by recalling that for the $OOOO$, the two CWIs take
the form (general scalar CWIs) [26]

$\displaystyle S_{1}=$ $\displaystyle=\bigg{\\{}\frac{\partial^{2}}{\partial
p_{2}^{2}}+\frac{(d-2\Delta_{2}+1)}{p_{2}}\frac{\partial}{\partial
p_{2}}-\frac{\partial^{2}}{\partial
p_{4}^{2}}-\frac{(d-2\Delta_{4}+1)}{p_{4}}\frac{\partial}{\partial p_{4}}$
$\displaystyle\qquad+\frac{1}{s}\frac{\partial}{\partial
s}\left(p_{1}\frac{\partial}{\partial p_{1}}+p_{2}\frac{\partial}{\partial
p_{2}}-p_{3}\frac{\partial}{\partial p_{3}}-p_{4}\frac{\partial}{\partial
p_{4}}\right)+\frac{\Delta_{3412}}{s}\frac{\partial}{\partial s}$
$\displaystyle\qquad+\frac{1}{t}\frac{\partial}{\partial
t}\left(p_{2}\frac{\partial}{\partial p_{2}}+p_{3}\frac{\partial}{\partial
p_{3}}-p_{1}\frac{\partial}{\partial p_{1}}-p_{4}\frac{\partial}{\partial
p_{4}}\right)+\frac{\Delta_{1423}}{t}\frac{\partial}{\partial t}$
$\displaystyle\qquad+\frac{(p_{2}^{2}-p_{4}^{2})}{st}\frac{\partial^{2}}{\partial
s\partial t}\bigg{\\}}\,\Phi(p_{1},p_{2},p_{3},p_{4},s,t)=0$ (5.5.1)

$\displaystyle S_{2}$ $\displaystyle=\bigg{\\{}\frac{\partial^{2}}{\partial
p_{1}^{2}}+\frac{(d-2\Delta_{1}+1)}{p_{1}}\frac{\partial}{\partial
p_{1}}-\frac{\partial^{2}}{\partial
p_{3}^{2}}-\frac{(d-2\Delta_{3}+1)}{p_{3}}\frac{\partial}{\partial p_{3}}$
$\displaystyle\qquad+\frac{1}{s}\frac{\partial}{\partial
s}\left(p_{1}\frac{\partial}{\partial p_{1}}+p_{2}\frac{\partial}{\partial
p_{2}}-p_{3}\frac{\partial}{\partial p_{3}}-p_{4}\frac{\partial}{\partial
p_{4}}\right)+\frac{\Delta_{3412}}{s}\frac{\partial}{\partial s}$
$\displaystyle\qquad+\frac{1}{t}\frac{\partial}{\partial
t}\left(p_{1}\frac{\partial}{\partial p_{1}}+p_{4}\frac{\partial}{\partial
p_{4}}-p_{2}\frac{\partial}{\partial p_{2}}-p_{3}\frac{\partial}{\partial
p_{3}}\right)+\frac{\Delta_{1423}}{t}\frac{\partial}{\partial t}$
$\displaystyle\qquad+\frac{(p_{1}^{2}-p_{3}^{2})}{st}\frac{\partial^{2}}{\partial
s\partial t}\bigg{\\}}\,\Phi(p_{1},p_{2},p_{3},p_{4},s,t)=0.$ (5.5.2)

$\displaystyle S_{3}$ $\displaystyle=\bigg{\\{}\frac{\partial^{2}}{\partial
p_{1}^{2}}+\frac{(d-2\Delta_{1}+1)}{p_{1}}\frac{\partial}{\partial
p_{1}}-\frac{\partial^{2}}{\partial
p_{4}^{2}}-\frac{(d-2\Delta_{4}+1)}{p_{4}}\frac{\partial}{\partial p_{4}}$
$\displaystyle\qquad+\frac{1}{s}\frac{\partial}{\partial
s}\left(p_{1}\frac{\partial}{\partial p_{1}}+p_{2}\frac{\partial}{\partial
p_{2}}-p_{3}\frac{\partial}{\partial p_{3}}-p_{4}\frac{\partial}{\partial
p_{4}}\right)+\frac{\Delta_{3412}}{s}\frac{\partial}{\partial s}$
$\displaystyle\qquad+\frac{(p_{2}^{2}-p_{3}^{2})}{st}\frac{\partial^{2}}{\partial
s\partial t}\bigg{\\}}\,\Phi(p_{1},p_{2},p_{3},p_{4},s,t)=0$ (5.5.3)

where

$\Delta_{ijkl}=\Delta_{i}+\Delta_{j}-\Delta_{k}-\Delta_{l}$ (5.5.4)

is a specific combination of the scaling parameters of the primary scalar
operators $(O)$, which plays a special role in the derivation of the dcc
solutions. In [26] the discussion dealt with two possible cases for the $OOOO$
in which the scaling combinations in (5.5.4) vanish: 1) the equal scaling case
with $\Delta_{i}=\Delta$ (i=1,2,3,4) and 2) the case in which two operators
are pairwise of equal scalings. In both cases, the solutions satisfy the
condition of being conformal and dual conformal invariant.
The vanishing of (5.5.4) is necessary in order to remove the
$\partial/\partial s$ and $\partial/\partial t$ terms and reduce the three
$S_{i}$’s to a hypergeometric system of equations (A.2.1). Notice that
differently from the case of 3-point functions, where a similar system has
been identified [20], as shown in Eq. (5.5.10), the variables are quartic -
rather than quadratic - ratios of the invariants.
In order to derive such a system, which is extracted from the $S_{i}$’s, we
need a product ansätz based on a quartic pivot $(s^{2}t^{2})$ with variables
[26][21]

$x=\frac{p_{1}^{2}p_{3}^{2}}{s^{2}t^{2}}\qquad
y=\frac{p_{2}^{2}p_{4}^{2}}{s^{2}t^{2}},$ (5.5.5)

and observe that this choice sets automatically to zero the mixed derivative
terms in $p_{i}$ and $s$ and $t$ in Eqs. (5.5.1),(5.5.2) and (5.5.3). The
ansätz for the solution is based on the product of a function G(x,y) and of
powers of x and y - given by (5.5.5) - of the form

$\Phi\sim x^{a}y^{b}G(x,y),$ (5.5.6)

for suitable $a$ and $b$, quite similarly to the case of a scalar 3-point
function. On any function $G(x,y)$, terms of the form

$\left(p_{1}\frac{\partial}{\partial p_{1}}+p_{2}\frac{\partial}{\partial
p_{2}}-p_{3}\frac{\partial}{\partial p_{3}}-p_{4}\frac{\partial}{\partial
p_{4}}\right)G(x,y)=0$ (5.5.7)

vanish, if we choose $x$ and $y$ as the quartic ratios (5.5.5). If we use the
definition of the $K_{ij}$ operators (2.4.6) and the ansätz based on $G(x,y)$
as defined above, the three equations take the form (intermediate scalar CWIs)

$\left\\{\begin{split}&\bigg{(}K_{24}+\frac{(p_{2}^{2}-p_{4}^{2})}{st}\frac{\partial^{2}}{\partial
s\partial
t}\bigg{)}\Phi(p_{1},p_{2},p_{3},p_{4},s,t)=-\bigg{(}\frac{\Delta_{3412}}{s}\frac{\partial}{\partial
s}+\frac{\Delta_{1423}}{t}\frac{\partial}{\partial
t}\bigg{)}\,\Phi(p_{1},p_{2},p_{3},p_{4},s,t)\\\
&\bigg{(}K_{13}+\frac{(p_{1}^{2}-p_{3}^{2})}{st}\frac{\partial^{2}}{\partial
s\partial
t}\bigg{)}\Phi(p_{1},p_{2},p_{3},p_{4},s,t)=-\bigg{(}\frac{\Delta_{3412}}{s}\frac{\partial}{\partial
s}+\frac{\Delta_{1423}}{t}\frac{\partial}{\partial
t}\bigg{)}\,\Phi(p_{1},p_{2},p_{3},p_{4},s,t)\\\
&\bigg{(}K_{14}+\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)=-\frac{\Delta_{3412}}{s}\frac{\partial}{\partial
s}\,\Phi(p_{1},p_{2},p_{3},p_{4},s,t)\\\ \end{split}\right.\\\ $ (5.5.8)

where we have removed all the mixed derivative terms in
$(s,p_{i}^{2}),(t,p_{i}^{2})$, thanks to (5.5.7). Explicit dcc solutions of
this system of equations are obtained if $\Delta_{ijkl}=0$, and the operators
$K_{24}$ and $K_{13}$ depend separately on a single scaling variable, that is
if $\Delta_{2}=\Delta_{4}$ and $\Delta_{1}=\Delta_{3}$. Notice that this
condition is compatible with the vanishing of $\Delta_{3412}$ and
$\Delta_{1423}$ and takes to a hypergeometric system of equations, which are
again solved in terms of hypergeometrics of the variables $x$ and $y$ given in
(5.5.5). In this case we could rewrite the system in the form (reduced scalar
CWIs)

$\left\\{\begin{split}&\bigg{(}K_{24}(\Delta_{2})+\frac{(p_{2}^{2}-p_{4}^{2})}{st}\frac{\partial^{2}}{\partial
s\partial t}\bigg{)}\Phi(p_{1},p_{2},p_{3},p_{4},s,t)=0\\\
&\bigg{(}K_{13}(\Delta_{1})+\frac{(p_{1}^{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\\\
&\bigg{(}K_{14}+\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\end{split}\right.\\\ $
(5.5.9)

where the $K_{ij}(\Delta_{i})$ indicates that such operators depend on a
single scaling constant.
It is important to observe that the system (5.5.9) admits explicit dcc
solutions which are expressed as hypergeometric functions, or, equivalently,
as 3K integrals, but the entire set of dcc solutions is not just composed of
these functions..
Dual conformal symmetry constrains a certain ansätz (the dual conformal
ansätz) to be expressed only in terms of the two quartic ratios $x$ and $y$,
via a function $G(x,y)$. Functions $G$ of such ratios will then necessarily
satisfy the condition (5.5.7), and henceforth the reduced system (LABEL:CC4).
The solutions of the three constraints in (LABEL:CC4) of the form $G(x,y)$,
will then characterize the most general set of dcc solutions for scalar
primary operators, of which special cases are those found in [26] and reported
below in Eq. (5.5.13). The additional reduction of the system (LABEL:CC4) to
(5.5.9) obviously, allows us to work with explicit expressions which are all
related by analytic continuations and therefore describe a unique solution, as
shown in [26]. Therefore, they are optimal for the study of several
kinematical limits of the scalar correlator, that we are now going to
investigate.

#####  5.5.1 Limits for equal scalings and $\Delta_{ijkl}=0$

As we have mentioned, the choice $\Delta_{ijkl}=0$ is what renders the system
(5.5.9) a variant of the ordinary hypergeometric system, which in general
takes the form

$K_{13}\Phi=0\qquad K_{23}\Phi=0,\qquad\textrm{where}\qquad
K_{ij}=K_{i}-K_{j}.$ (5.5.10)

and it is solved by quadratic - rather than quartic - ratios of invariants.
Once this gets reduced to (5.5.9), as already mentioned, the complete ansätz
for the general solution of such system is constructed by multiplying the
function $G(x,y)$ by the pivot, raised to a power $n_{s}$, fixed by the
dilatation WI

$\displaystyle\bigg{[}(\Delta_{t}-3d)-\sum_{i=1}^{4}p_{i}\frac{\partial}{\partial
p_{i}}-s\frac{\partial}{\partial s}-t\frac{\partial}{\partial
t}\bigg{]}\Phi(p_{1},p_{2},p_{3},p_{4},s,t)=0,$ (5.5.11)

with $\Delta_{t}$ denoting the total scaling. If we choose as a pivot
$s^{2}t^{2}$, the solution indeed will take the form

$\Phi(p_{i},s,t)=(s^{2}t^{2})^{n_{s}}G(x,y)\qquad
n_{s}=\frac{\Delta_{t}-3d}{4}.$ (5.5.12)

Few additional comments are in order concerning the homogeneous case
$(\Delta_{ijkl}=0)$ and the system (5.5.9). We remark that the third equation
of such system is identically satisfied if the first and the second equations
are, which is the case if an ansätz of type (5.5.12) is chosen. This is
clearly consistent with the fact the four functionally independent solutions
of an Appell system of equations (for $F_{4}$) is based only on two
independent equations (A.2.1).

The solution of the homogeneous system (5.5.9), as already mentioned, can be
written in terms of 4 Appell functions $F_{4}$ of the $x$ and $y$ ratios given
in (5.5.5) [26]

$\displaystyle\langle{O(p_{1})O(p_{2})O(p_{3})O(p_{4})}\rangle$
$\displaystyle=2^{\frac{d}{2}-4}\ \
C\,\sum_{\lambda,\mu=0,\Delta-\frac{d}{2}}\xi(\lambda,\mu)\bigg{[}\big{(}s^{2}\,t^{2}\big{)}^{\Delta-\frac{3}{4}d}\left(\frac{p_{1}^{2}p_{3}^{2}}{s^{2}t^{2}}\right)^{\lambda}\left(\frac{p_{2}^{2}p_{4}^{2}}{s^{2}t^{2}}\right)^{\mu}$
$\displaystyle\times\,F_{4}\left(\frac{3}{4}d-\Delta+\lambda+\mu,\frac{3}{4}d-\Delta+\lambda+\mu,1-\Delta+\frac{d}{2}+\lambda,1-\Delta+\frac{d}{2}+\mu,\frac{p_{1}^{2}p_{3}^{2}}{s^{2}t^{2}},\frac{p_{2}^{2}p_{4}^{2}}{s^{2}t^{2}}\right)$
$\displaystyle+\big{(}s^{2}\,u^{2}\big{)}^{\Delta-\frac{3}{4}d}\left(\frac{p_{2}^{2}p_{3}^{2}}{s^{2}u^{2}}\right)^{\lambda}\left(\frac{p_{1}^{2}p_{4}^{2}}{s^{2}u^{2}}\right)^{\mu}$
$\displaystyle\times\,F_{4}\left(\frac{3}{4}d-\Delta+\lambda+\mu,\frac{3}{4}d-\Delta+\lambda+\mu,1-\Delta+\frac{d}{2}+\lambda,1-\Delta+\frac{d}{2}+\mu,\frac{p_{2}^{2}p_{3}^{2}}{s^{2}u^{2}},\frac{p_{1}^{2}p_{4}^{2}}{s^{2}u^{2}}\right)$
$\displaystyle+\big{(}t^{2}\,u^{2}\big{)}^{\Delta-\frac{3}{4}d}\left(\frac{p_{1}^{2}p_{2}^{2}}{t^{2}u^{2}}\right)^{\lambda}\left(\frac{p_{3}^{2}p_{4}^{2}}{t^{2}u^{2}}\right)^{\mu}$
$\displaystyle\times\,F_{4}\left(\frac{3}{4}d-\Delta+\lambda+\mu,\frac{3}{4}d-\Delta+\lambda+\mu,1-\Delta+\frac{d}{2}+\lambda,1-\Delta+\frac{d}{2}+\mu,\frac{p_{1}^{2}p_{2}^{2}}{t^{2}u^{2}},\frac{p_{3}^{2}p_{4}^{2}}{t^{2}u^{2}}\right)\bigg{]},$
(5.5.13)

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

$\begin{split}\xi\left(0,0\right)&=\left[\Gamma\left(\frac{3}{4}d-\Delta\right)\right]^{2}\left[\Gamma\left(\Delta-\frac{d}{2}\right)\right]^{2}\\\
\xi\left(0,\Delta-\frac{d}{2}\right)&=\xi\left(\Delta-\frac{d}{2},0\right)=\left[\Gamma\left(\frac{d}{4}\right)\right]^{2}\Gamma\left(\Delta-\frac{d}{2}\right)\Gamma\left(\frac{d}{2}-\Delta\right)\\\
\xi\left(\Delta-\frac{d}{2},\Delta-\frac{d}{2}\right)&=\left[\Gamma\left(\Delta-\frac{d}{4}\right)\right]^{2}\left[\Gamma\left(\frac{d}{2}-\Delta\right)\right]^{2},\end{split}$
(5.5.14)

which is explicitly symmetric under all the possible permutations of the
momenta and it is fixed up to one undetermined constant $C$.
As shown in [26], (5.5.13) can be re-expressed in the form

$\displaystyle
I_{\frac{d}{2}-1\\{\Delta-\frac{d}{2},\Delta-\frac{d}{2},0\\}}(p_{1}p_{3},p_{2}p_{4},st)=$
$\displaystyle\qquad=\,(p_{1}p_{3})^{\Delta-\frac{d}{2}}(p_{2}p_{4})^{\Delta-\frac{d}{2}}\int_{0}^{\infty}\,dx\,x^{\frac{d}{2}-1}\,K_{\Delta-\frac{d}{2}}(p_{1}p_{3}\,x)\,K_{\Delta-\frac{d}{2}}(p_{2}p_{4}\,x)\,K_{0}(st\,x).$
(5.5.15)

i.e. as a 3K integrals of quadratic $(p_{1}p_{3},st,p_{2}p_{4})$ variables,
which are solutions of a system of the form (A.2.1) with quartic ratios $x,y$.

#####  5.5.2 Comparison between the general, the intermediate and the reduced
systems

To address the asymptotic behaviour of this solution of the general system of
Eqs. (5.5.1)(5.5.2)(5.5.3) (the $S_{i}$ constraints) and compare it with the
intermediate (LABEL:CC4) and the reduced (5.5.9) ones, we clearly need to
perform a special asymptotic limit. We can reasonably assume that at large $s$
and $t$ the general solution of the $S_{i}^{\prime}s$ equations decays as
$\sim 1/(st)^{\alpha}$, with $\alpha>0$.
Both for the $S_{i}$ and for the intermediate system (LABEL:CC4), the action
of the derivative operators $(1/s)\partial/\partial s$ and
$(1/t)\partial/\partial t$ is suppressed by two additional powers of the
kinematic invariant $s$ and $t$ and can reasonably be set to zero
asymptotically.
If we neglect such contributions, the equations in (LABEL:CC4) turn again into
a homogeneous system (5.5.13) which, however, is not hypergeometric any
longer, nor the third equation is dependent from the previous two, as found in
the $\Delta_{ijkl}=0$ case for the reduced system (5.5.9). Although the three
systems, general intermediate and reduced, look pretty similar in such limit,
we can only safely state that their solutions have to share the same
asymptotic behaviour. This is fixed by the scaling power
$n_{s}=\Delta_{t}-3/4d$, extracted from the dilatation WI in the form

$\Phi(p_{1},p_{2},p_{3},p_{4})\sim\frac{1}{(s^{2}t^{2})^{-n_{s}}}+O(1/(s^{2}t^{2}))$
(5.5.16)

which requires that $n_{s}$ be negative.
In the two sections below we will try to characterize the behaviour of the dcc
solution of (5.5.9) in various limits before coming back again to the three
systems of equations, discussing some approximate factorised solutions of such
equations.

#####  5.5.3 IR and equal mass limits of the dcc solutions

The analysis of the infrared or soft limits at small $s$ and $t$ of the dual
conformal solution, with $\Delta_{ijkl}=0$, for $\Delta_{i}=\Delta,i=1,2,3,4$
can be discussed using a second version of the solution given by (5.5.13), but
completely equivalent to it, obtained by a sequence of analytic continuations
[26]

$\begin{aligned}
\Phi&=C_{1}\bigg{\\{}\left(p_{1}^{2}\,p_{3}^{2}\right)^{\Delta-\frac{3}{4}d}\bigg{[}F_{4}\left(\frac{d}{4}\,,\,\frac{3}{4}d-\Delta\,,\,1\,,\,\frac{d}{2}-\Delta+1\,;\frac{s^{2}t^{2}}{p_{1}^{2}p_{3}^{2}}\,,\,\frac{p_{2}^{2}p_{4}^{2}}{p_{1}^{2}p_{3}^{2}}\right)\\\\[5.16663pt]
&+\tau_{1}\left(\frac{p_{2}^{2}p_{4}^{2}}{p_{1}^{2}p_{3}^{2}}\right)^{\Delta-\frac{d}{2}}F_{4}\left(\Delta-\frac{d}{4}\,,\,\frac{d}{4}\,,\,1\,,\,1-\frac{d}{2}+\Delta\,;\frac{s^{2}t^{2}}{p_{1}^{2}p_{3}^{2}}\,,\,\frac{p_{2}^{2}p_{4}^{2}}{p_{1}^{2}p_{3}^{2}}\right)\bigg{]}\\\\[5.16663pt]
&+\left(p_{2}^{2}\,p_{3}^{2}\right)^{\Delta-\frac{3}{4}d}\bigg{[}F_{4}\left(\frac{d}{4}\,,\,\frac{3}{4}d-\Delta\,,\,1\,,\,\frac{d}{2}-\Delta+1\,;\frac{s^{2}u^{2}}{p_{2}^{2}p_{3}^{2}}\,,\,\frac{p_{1}^{2}p_{4}^{2}}{p_{2}^{2}p_{3}^{2}}\right)\\\\[5.16663pt]
&+\tau_{1}\left(\frac{p_{1}^{2}p_{4}^{2}}{p_{2}^{2}p_{3}^{2}}\right)^{\Delta-\frac{d}{2}}F_{4}\left(\Delta-\frac{d}{4}\,,\,\frac{d}{4}\,,\,1\,,\,1-\frac{d}{2}+\Delta\,;\frac{s^{2}u^{2}}{p_{2}^{2}p_{3}^{2}}\,,\,\frac{p_{1}^{2}p_{4}^{2}}{p_{2}^{2}p_{3}^{2}}\right)\bigg{]}\\\\[5.16663pt]
&+\left(p_{1}^{2}\,p_{2}^{2}\right)^{\Delta-\frac{3}{4}d}\bigg{[}F_{4}\left(\frac{d}{4}\,,\,\frac{3}{4}d-\Delta\,,\,1\,,\,\frac{d}{2}-\Delta+1\,;\frac{u^{2}t^{2}}{p_{1}^{2}p_{2}^{2}}\,,\,\frac{p_{3}^{2}p_{4}^{2}}{p_{1}^{2}p_{2}^{2}}\right)\\\\[5.16663pt]
&+\tau_{1}\left(\frac{p_{3}^{2}p_{4}^{2}}{p_{1}^{2}p_{2}^{2}}\right)^{\Delta-\frac{d}{2}}F_{4}\left(\Delta-\frac{d}{4}\,,\,\frac{d}{4}\,,\,1\,,\,1-\frac{d}{2}+\Delta\,;\frac{u^{2}t^{2}}{p_{1}^{2}p_{2}^{2}}\,,\,\frac{p_{3}^{2}p_{4}^{2}}{p_{1}^{2}p_{2}^{2}}\right)\bigg{]}\bigg{\\}}.\end{aligned}$

(5.5.17)

$\tau_{1}=\frac{\Gamma\left(\Delta-\frac{d}{4}\right)\Gamma\left(1+\Delta-\frac{3}{4}d\right)\Gamma\left(1-\Delta+\frac{d}{2}\right)}{\Gamma\left(\Delta-\frac{3}{4}d\right)\Gamma\left(1-\Delta+\frac{3}{4}d\right)\Gamma\left(1+\Delta-\frac{d}{2}\right)}$
(5.5.18)

From this expression, we can keep $t^{2}$ fixed and of order $O(p_{i}^{2})$
and send $s^{2}\to 0$ to derive the soft behaviour

$\Phi\sim(p_{1}^{2}p_{3}^{2})^{\Delta-\frac{3}{4}d}+O(s^{2}/p_{i}^{2})$
(5.5.19)

if the external mass invariants $p_{i}^{2}$ are kept larger than the invariant
$s^{2}$.

#####  5.5.4 Equal mass limit with $p_{i}^{2}=M^{2}>\,s^{2},t^{2}$

The equal mass limit is obtained by taking $p_{i}^{2}=M^{2}$ for all the
external invariants. In this case, using the relation between $F_{4}$ and the
Gauss hypergeometric $F_{21}(a,b,c,x)$

$F_{4}(\alpha,\beta,\gamma,\gamma^{\prime},x,y)=\sum_{m=0}^{\infty}\frac{(\alpha)_{m}(\beta)_{m}}{(\gamma)_{m}m!}F_{21}(\alpha+m,\beta+m,\gamma^{\prime},y)x^{m}$
(5.5.20)

and

$F_{21}(a,b,c,1)=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)},$
(5.5.21)

from (5.5.17) we then obtain the simplified expression

$\begin{split}&\Phi=M^{4\Delta-3d}\sum_{m=0}^{\infty}\frac{1}{m!}\Bigg{(}\frac{\Gamma\left(\frac{d}{2}-\Delta+1\right)\Gamma\left(-\frac{d}{2}-2m+1\right)\Gamma\left(\frac{d}{4}+m\right)\Gamma\left(\frac{3d}{4}-\Delta+m\right)}{\Gamma\left(\frac{d}{4}\right)\Gamma(m+1)\Gamma\left(\frac{3d}{4}-\Delta\right)\Gamma\left(-\frac{d}{4}-m+1\right)\Gamma\left(\frac{d}{4}-\Delta-m+1\right)}\\\
&\frac{\Gamma\left(\frac{d}{2}-\Delta+1\right)\Gamma\left(-\frac{3d}{4}+\Delta+1\right)\Gamma\left(-\frac{d}{2}-2m+1\right)\Gamma\left(\frac{d}{4}+m\right)\Gamma\left(-\frac{d}{4}+\Delta+m\right)}{\Gamma\left(\frac{d}{4}\right)\Gamma(m+1)\Gamma\left(\frac{3d}{4}-\Delta+1\right)\Gamma\left(\delta-\frac{3d}{4}\right)\Gamma\left(-\frac{d}{4}-m+1\right)\Gamma\left(-\frac{3d}{4}+\Delta-m+1\right)}\Bigg{)}\times\\\
\\\
&\times\Bigg{(}\left(\frac{s^{2}t^{2}}{M^{4}}\right)^{m}+\left(\frac{s^{2}u^{2}}{M^{4}}\right)^{m}+\left(\frac{u^{2}t^{2}}{M^{4}}\right)^{m}\Bigg{)}\\\
\end{split}$

which in $d=4$ becomes

$\begin{split}&\Phi=M^{4\Delta-3d}\sum_{m=0}^{\infty}\frac{1}{m!}\left(\frac{\Gamma(-2m-1)\Gamma(-\Delta+m+3)}{\Gamma(-m)\Gamma(-\Delta-m+2)}+\frac{\Gamma(3-\Delta)\Gamma(\Delta-2)\Gamma(-2m-1)\Gamma(\Delta+m-1)}{\Gamma(4-\Delta)\Gamma(\Delta-3)\Gamma(-m)\Gamma(\Delta-m-2)}\right)\times\\\
&\qquad\qquad\times\Bigg{(}\left(\frac{s^{2}t^{2}}{M^{4}}\right)^{m}+\left(\frac{s^{2}u^{2}}{M^{4}}\right)^{m}+\left(\frac{u^{2}t^{2}}{M^{4}}\right)^{m}\Bigg{)}\\\
\end{split}$

and is convergent as far as $M^{2}\gg s^{2},t^{2}$. Explicitly (in $d=4$)

$\Phi=M^{4\Delta-3d}\left[d_{0}+d_{1}(\frac{s^{2}t^{2}}{M^{4}}+\frac{s^{2}u^{2}}{M^{4}}+\frac{u^{2}t^{2}}{M^{4}})+\ldots\right],$
(5.5.22)

where

$d_{0}=\frac{1}{2}\left(\frac{\Gamma(\Delta-1)}{\Gamma(\Delta-2)}-\frac{\Gamma(3-\Delta)}{\Gamma(2-\Delta)}\right)$
(5.5.23)

and

$\begin{split}&d_{1}=\frac{1}{12}\left(\frac{\Gamma(4-\Delta)}{\Gamma(1-\Delta)}-\frac{\Gamma(\Delta)}{\Gamma(\Delta-3)}\right).\end{split}$
(5.5.24)

#####  5.5.5 The equal mass limit with $s^{2},t^{2}>M^{2}$

A similar limit can be performed starting from (5.5.13). We can take
$s^{2},t^{2},u^{2}>M^{2}$, which in (5.5.13) takes to a univariate expression
of $F_{4}$, $F_{4}(a,b,c,c^{\prime};x,x)$. It can be expressed as a single
series in $x$ using the relation

$F_{4}(a,b,c,c^{\prime};x,x)={}_{4}F_{3}\left(a,b,\frac{c+c^{\prime}}{2},\frac{c+c^{\prime}-1}{2}|\,c,c^{\prime},c+c^{\prime}-1;4x\right)$
(5.5.25)

due to Burchnall, as reported in [99].
Setting $x_{1}=M^{4}/(s^{2}t^{2})\sim M^{4}/(s^{2}u^{2})\sim
M^{4}/(u^{2}t^{2})$ and choosing, for instance, a scaling dimension of a
scalar operator $\phi^{2}$ (with $\Delta=d-2$), in $d=4$ one obtains a simple
expression for $\Phi$

$\Phi=C\left(\frac{\log^{2}\left(\frac{M^{4}}{s^{2}t^{2}}\right)}{s^{2}t^{2}}+\frac{\log^{2}\left(\frac{M^{4}}{s^{2}u^{2}}\right)}{s^{2}u^{2}}+\frac{\log^{2}\left(\frac{M^{4}}{t^{2}u^{2}}\right)}{t^{2}u^{2}}+\frac{\pi^{2}}{3s^{2}t^{2}}+\frac{\pi^{2}}{3s^{2}u^{2}}+\frac{\pi^{2}}{3t^{2}u^{2}}\right)+O(x_{1}^{2})$
(5.5.26)

and in $d=3$

$\begin{split}&\Phi=C\left(\frac{\pi\Gamma\left(\frac{1}{4}\right)^{2}}{M^{4}\sqrt{s}\sqrt{t}}+\frac{\pi\Gamma\left(\frac{1}{4}\right)^{2}}{M^{4}\sqrt{s}\sqrt{u}}+\frac{\pi\Gamma\left(\frac{1}{4}\right)^{2}}{M^{4}\sqrt{t}\sqrt{u}}-\frac{4\pi\Gamma\left(\frac{3}{4}\right)^{2}}{M^{2}s^{3/2}t^{3/2}}-\frac{4\pi\Gamma\left(\frac{3}{4}\right)^{2}}{M^{2}s^{3/2}u^{3/2}}\right.\\\
\\\
&\qquad\qquad\left.-\frac{4\pi\Gamma\left(\frac{3}{4}\right)^{2}}{M^{2}t^{3/2}u^{3/2}}+\frac{\pi\Gamma\left(\frac{1}{4}\right)^{2}}{4s^{5/2}t^{5/2}}+\frac{\pi\Gamma\left(\frac{1}{4}\right)^{2}}{4s^{5/2}u^{5/2}}+\frac{\pi\Gamma\left(\frac{1}{4}\right)^{2}}{4t^{5/2}u^{5/2}}\right)+O(x_{1}^{2})\end{split}$

####  5.6 Large $s$ and $t$ limits and the Lauricella system

We have already mentioned that the system of Eqs. (5.5.1), (5.5.2) (5.5.3)
reduces to (LABEL:CC4) if we choose a combination of invariants given by
(5.5.5). The system (LABEL:CC4) turns into hypergeometric if
$\Delta_{ijkl}=0$, with only two independent equations, as pointed out above.
However, for a generic $\Delta_{ijkl}$ it is possible to uncover an
approximate hypergeometric structure in the equations only in the large $s$
and $t$ limit, if we neglect the coupling between the $s,t$ and $p_{i}^{2}$
invariants. At the same time we could assume that $\Delta_{ijkl}\ll 1$, which
allows to drop the $1/s\partial/\partial_{s}$ and $1/t\partial/\partial_{t}$
terms in the differential operator. This approximate factorization has been
discussed in [26], where it has been shown to take to a hypergeometric system
of Lauricella type in three variables (see section (A.1.2)). This asymptotic
analysis is based on the ansätz

$\Phi(p_{1}^{2},p_{2}^{2},p_{3}^{2},p_{4}^{2},s,t)\sim\phi(p_{1}^{2},p_{2}^{2},p_{3}^{2},p_{4}^{2})\chi(s,t)$
(5.6.1)

and invokes the separability of the asymptotic system (LABEL:CC4)

$\begin{split}&K_{24}\phi=0\\\ &K_{13}\phi=0\\\ &K_{14}\phi=0\\\
&\frac{1}{st}\frac{\partial^{2}}{\partial s\partial t}\,\chi(s,t)=0.\\\
\end{split}$ (5.6.2)

The Lauricella system corresponds to the first three equations of (5.6.2).
Lauricella systems have recently appeared also in CFT in coordinate space
[30]. One can easily realize that they characterize a homogenous solution in
the variables $p_{i}^{2}$ of the entire (complete) system (5.5.1),
(5.5.2)(5.5.3) as well as of (LABEL:CC4). They are exact solutions of such
systems, before an asymptotic limit. Notice that logarithmic terms of the form
$f(1/(s^{2}t^{2}))\log^{k}(s^{2}/t^{2})$ (k¿0) are also compatible with the
asymptotic structure of such systems, which are generically expected for
scattering at fixed angle in perturbation theory (see [100] for an example).

#####  5.6.1 The general primary CWIs in the $2\to 2$ and $1\to 3$
formulations for the $TOOO$ and asymptotics

In order to get further insight into the structure of the CWIs for the $TOOO$,
we proceed with a rearrangement of their expressions in order to reduce them
to homogeneous equations, following the same approach of section 5.5, adopted
in the scalar case. We will proceed by generalizing the CWIs for different
scalar coefficients from the equal scaling case presented in section 5.4.5.
For simplicity we set

$F\equiv F(p_{1},p_{2},p_{3},p_{4},s,t),\,\,\,F(p_{2}\leftrightarrow
p_{4})\equiv F(p_{1},p_{2},p_{4},p_{3},t,s),\,\,\,F(p_{3}\leftrightarrow
p_{4})\equiv F(p_{1},p_{4},p_{2},p_{3},s,\tilde{u}).$ (5.6.3)

If we allow for different scaling $\Delta_{i}$, with $\Delta_{1}=d$ for the
stress-energy tensor, then the equations given in (5.4.5)-(5.4.5) can be
generalized as follows

$\begin{split}\tilde{C}_{11}-\tilde{C}_{21}\to
B_{1}=&\Bigg{(}K_{21}+\frac{\Delta_{1423}+2}{t}\frac{\partial}{\partial
t}-\frac{1}{t}\frac{\partial}{\partial t}\left(p_{1}\frac{\partial}{\partial
p_{1}}+p_{4}\frac{\partial}{\partial p_{4}}-p_{2}\frac{\partial}{\partial
p_{2}}-p_{3}\frac{\partial}{\partial p_{3}}\right)\\\
&+\frac{p_{3}^{2}-p_{4}^{2}}{st}\frac{\partial^{2}}{\partial s\partial
t}\Bigg{)}F(1,2,3,4)=0.\end{split}$ (5.6.4)

The other homogeneous equations for $F(p_{1},p_{2},p_{3},p_{4},s,t)$ are
similarly derived in the form

$\displaystyle\tilde{C}_{11}-\tilde{C}_{31}\to B_{2}=$
$\displaystyle\Bigg{(}K_{31}+\frac{\Delta_{1423}+2}{t}\frac{\partial}{\partial
t}+\frac{\Delta_{1234}-2}{s}\frac{\partial}{\partial
s}-\frac{1}{t}\frac{\partial}{\partial t}\left(p_{1}\frac{\partial}{\partial
p_{1}}+p_{4}\frac{\partial}{\partial p_{4}}-p_{2}\frac{\partial}{\partial
p_{2}}-p_{3}\frac{\partial}{\partial p_{3}}\right)$
$\displaystyle+\frac{1}{s}\frac{\partial}{\partial
s}\left(p_{3}\frac{\partial}{\partial p_{3}}+p_{4}\frac{\partial}{\partial
p_{4}}-p_{1}\frac{\partial}{\partial p_{1}}-p_{2}\frac{\partial}{\partial
p_{2}}\right)+\frac{p_{3}^{2}-p_{1}^{2}}{st}\frac{\partial^{2}}{\partial
s\partial t}\Bigg{)}F(1,2,3,4)=0,$ (5.6.5)

and

$\displaystyle\tilde{C}_{21}-\tilde{C}_{31}\to B_{3}=$
$\displaystyle\Bigg{(}K_{32}+\frac{\Delta_{1234}-2}{s}\frac{\partial}{\partial
s}+\frac{1}{s}\frac{\partial}{\partial s}\left(p_{3}\frac{\partial}{\partial
p_{3}}+p_{4}\frac{\partial}{\partial p_{4}}-p_{1}\frac{\partial}{\partial
p_{1}}-p_{2}\frac{\partial}{\partial p_{2}}\right)$ (5.6.6)
$\displaystyle+\frac{p_{4}^{2}-p_{1}^{2}}{st}\frac{\partial^{2}}{\partial
s\partial t}\Bigg{)}F(1,2,3,4)=0.$ (5.6.7)

One can show that $B_{1},B_{2},B_{3}$ are not independent, in fact

$B_{1}+B_{3}=B_{2},$ (5.6.8)

indicating that there are only two independent homogeneous equations involving
the $F$ form factor.
Finally, one has to consider the system of three differential equations,
composed of ($B_{1},B_{2})$ together with the analogous of $\tilde{C}_{21}$,
given in (5.4.5), now for different $\Delta_{i}$’s, which can be written as

$\left\\{\begin{split}&\Bigg{(}K_{31}-\frac{1}{t}\frac{\partial}{\partial
t}\left(p_{1}\frac{\partial}{\partial p_{1}}+p_{4}\frac{\partial}{\partial
p_{4}}-p_{2}\frac{\partial}{\partial p_{2}}-p_{3}\frac{\partial}{\partial
p_{3}}\right)+\frac{1}{s}\frac{\partial}{\partial
s}\left(p_{3}\frac{\partial}{\partial p_{3}}+p_{4}\frac{\partial}{\partial
p_{4}}-p_{1}\frac{\partial}{\partial p_{1}}-p_{2}\frac{\partial}{\partial
p_{2}}\right)\\\ &+\frac{p_{3}^{2}-p_{1}^{2}}{st}\frac{\partial^{2}}{\partial
s\partial t}\Bigg{)}F=-\left(\frac{\Delta_{1423}+2}{t}\frac{\partial}{\partial
t}+\frac{\Delta_{1234}-2}{s}\frac{\partial}{\partial s}\right)F\\\\[8.61108pt]
&\Bigg{[}K_{42}+\frac{1}{s}\frac{\partial}{\partial
s}\left(p_{3}\frac{\partial}{\partial p_{3}}+p_{4}\frac{\partial}{\partial
p_{4}}-p_{1}\frac{\partial}{\partial p_{1}}-p_{2}\frac{\partial}{\partial
p_{2}}\right)+\frac{1}{t}\frac{\partial}{\partial
t}\left(p_{1}\frac{\partial}{\partial p_{1}}+p_{4}\frac{\partial}{\partial
p_{4}}-p_{2}\frac{\partial}{\partial p_{2}}-p_{3}\frac{\partial}{\partial
p_{3}}\right)\\\ &+\frac{p_{4}^{2}-p_{2}^{2}}{st}\frac{\partial^{2}}{\partial
s\partial t}\Bigg{]}F=\left(\frac{\Delta_{1423}+2}{t}\frac{\partial}{\partial
t}-\frac{\Delta_{1234}}{s}\frac{\partial}{\partial
s}\right)F-\frac{2}{s}\frac{\partial}{\partial s}\big{(}F(p_{2}\leftrightarrow
p_{4})-F(p_{3}\leftrightarrow p_{4})\big{)}\\\\[8.61108pt]
&\left(K_{21}-\frac{1}{t}\frac{\partial}{\partial
t}\left(p_{1}\frac{\partial}{\partial p_{1}}+p_{4}\frac{\partial}{\partial
p_{4}}-p_{2}\frac{\partial}{\partial p_{2}}-p_{3}\frac{\partial}{\partial
p_{3}}\right)+\frac{p_{3}^{2}-p_{4}^{2}}{st}\frac{\partial^{2}}{\partial
s\partial t}\right)F=-\frac{\Delta_{1423}+2}{t}\frac{\partial}{\partial
t}F\end{split}\right.$

We will try to extract some information about the structure of such equations
by discussing some possible limits.
In the characterization of the nature of the system we begin by considering
the case in which all the scalings are different and work our way starting
from the left hand side of (5.6.1). We have different options. For instance,
if we are looking for factorised solutions such as those discussed in the
scalar case, of the form (5.6.1), then we could consider the asymptotic limit
$s,t\to\infty$ and identify the Lauricella component of such solutions, since
the equations above turn homogenous, and the left hand side, exactly as in
(5.6.2), reduces to a Lauricella system of hypergeometrics (A.1.2). This holds
independently of the values of the scalings $\Delta_{i}$. On the other hand,
it is possible to identify, at least asymptotically, some hypergeometric
solutions, different from the Lauricella’s, but we need to constrain the
scaling dimensions in such a way that the operators $K_{31}$ and $K_{42}$ are
each characterised by a single conformal scaling ($\Delta_{4}=\Delta_{2}$ and
$\Delta_{3}=\Delta_{1}$). As discussed in the previous sections, we can choose
as variable the scale invariant ratios (5.5.5) for $x$ and $y$ in the ansätz
for the solution, reobtaining the same left hand side of (LABEL:CC4). This
approximate solutions would again be quite similar to those discussed in the
scalar case. However, in the general case, as we have already mentioned, even
for large $s$ and $t$, when we keep the scalings generic, one can show that
the left hand side of such system of equations is not of hypergeometric form,
and the explicit form of such solutions is unknown.

Figure 5.5: The $TOOO$ in a kinematical region in which can be described as a
$1\to 3$ process.

#####  5.6.2 The $1\to 3$ case

It is possible to perform other limits on the same form factor of the $TOOO$
in order to simplify the primary CWIs presented in the previous sections. We
are going to focus our discussion on the $1\to 3$ formulation, which is
symmetric in the momenta of the three scalar operators and provides a clear
separation of the parametric dependence of the correlator in terms of a
function of the external invariants $p_{i}^{2}$ times a function of $s,t$ and
$u$, in analogy with the discussion presented in the $2\to 2$ case.
This kinematic choice is illustrated in Fig. 1.
In order to proceed with the investigation of this limit, it is convenient to
perform an analytic continuation of the CWIs to the Minkowski region from
their Euclidean definition, and take all the invariants $t^{2}$ and $u^{2}$
and $s^{2}$ to be positive. The kinematical region of interest, in this case,
is delimited by the conditions

$\displaystyle(p_{2}+p_{3})^{2}\leq t^{2}\leq\left(p_{1}-p_{4}\right)^{2}$
$\displaystyle(p_{3}+p_{4})^{2}\leq s^{2}\leq\left(p_{1}-p_{2}\right)^{2}$
$\displaystyle(p_{2}+p_{4})^{2}\leq u^{2}\leq\left(p_{1}-p_{3}\right)^{2},$
(5.6.9)

with the usual relation

$\displaystyle s^{2}+t^{2}+u^{2}=p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+p_{4}^{2}.$
(5.6.10)

We will be performing the large $p_{1}$ limit, where the invariant mass of the
virtual graviton line gets asymptotically large, and assume that the
invariants $s^{2}\sim t^{2}\sim u^{2}\sim p_{1}^{2}$ grow large with
$p_{1}^{2}$. In this limit the primary CWIs simplify, and the equations become
approximately separable in their dependence on the external
$p_{i}^{2}(i=2,3,4)$ and the remaining $(s,t,u)$ invariants. For this reason
we choose asymptotic solutions of the form

$A(p_{2},p_{3},p_{4},s,t,u)\sim\Phi(p_{2},p_{3},p_{4})\chi(s,t,u).$ (5.6.11)

We study now the form of the $\chi(s,t,u)$. The corresponding equations for
the $(s,t,u)$ invariants, from the primary conformal WI’s, take the form

$\frac{\partial^{2}\chi}{\partial s\partial
t}=0,\qquad\frac{\partial^{2}\chi}{\partial s\partial
u}=0,\qquad\frac{\partial^{2}\chi}{\partial t\partial u}=0,$ (5.6.12)

with the additional constraint imposed by the dilatation Ward identity. In
particular, in this limit, the dilatation WI for the $(s,t,u)$ invariants
takes the form

$\displaystyle\left[s\,\frac{\partial}{\partial s}+t\,\frac{\partial}{\partial
t}+u\,\frac{\partial}{\partial u}\right]\chi(s,t,u)=0.$ (5.6.13)

Notice that the remaining contribution to the dilatation WI is satisfied
separately by the scale invariant condition on $\Phi(p_{2},p_{3},p_{4})$

$\left[p_{2}\,\frac{\partial}{\partial p_{2}}+p_{3}\,\frac{\partial}{\partial
p_{3}}+p_{4}\,\frac{\partial}{\partial
p_{4}}\right]\Phi(p_{2},p_{3},p_{4})=(\Delta_{t}-3d-2)\Phi(p_{2},p_{3},p_{4}),$
(5.6.14)

which takes to generalized hypergeometric $F_{4}$ solutions, functions of the
ratios $p_{2}^{2}/p_{4}^{2}$ and $p_{3}^{2}/p_{4}^{2}$, as given in (2.5.16).
The choice of the pivot ($p_{4}$ in this case) is arbitrary.
By differentiating (5.6.13) with respect $s$ and using (5.6.12), one finds
another constraint. Similar constraints are obtained by repeating the
procedure with respect to $t$ and $u$. The resulting three equations obtained
in this manner can be written in the form

$\displaystyle\left[s\frac{\partial^{2}}{\partial\,s^{2}}+\,\frac{\partial}{\partial
s}\right]\chi(s,t,u)$ $\displaystyle=0$
$\displaystyle\left[\,t\,\frac{\partial^{2}}{\partial\,t^{2}}+\,\frac{\partial}{\partial
t}\right]\chi(s,t,u)$ $\displaystyle=0$
$\displaystyle\left[u\frac{\partial^{2}}{\partial
u^{2}}+\frac{\partial}{\partial u}\right]\chi(s,t,u)$ $\displaystyle=0,$
(5.6.15)

giving the solution for $\chi$ of the form

$\displaystyle\chi(s,t,u)=c_{1}\log(s)+c_{2}\log(t)+c_{3}\log(u)+c_{4},$
(5.6.16)

where $c_{1},\,c_{2},\,c_{3},\,c_{4}$ are undetermined constants. Imposing the
dilatation WI on this solution we find some relations between the undetermined
coefficients, with the solution rewritten in the form

$\chi(s,t,u)=c_{1}\log\left(\frac{s}{t}\right)+c_{2}\log\left(\frac{u}{t}\right)+c_{4}.$
(5.6.17)

Finally, we also impose the symmetry constraint on the form factor $A$ of the
$TOOO$

$A(p_{2},p_{4},p_{3},s,u,t)=A(p_{2},p_{3},p_{4},s,t,u)$ (5.6.18)

which implies that

$\displaystyle\phi(p_{2},p_{3},p_{4})\chi(s,t,u)=\phi(p_{2},p_{4},p_{3})\chi(s,u,t),$
(5.6.19)

and recalling that the $\phi(p_{2},p_{3},p_{4})$ is symmetric under the
permutation of $\\{p_{2},p_{3},p_{4}\\}$, we obtain the condition
$\chi(s,t,u)=\chi(s,u,t)$ or

$\displaystyle\big{(}2c_{2}+c_{1}\big{)}\log\left(\frac{u}{t}\right)=0.$
(5.6.20)

Therefore the $\chi(s,t,u)$ function acquires the final form

$\chi(s,t,u)=c_{1}\log\left(\frac{u\,t}{s^{2}}\right)+c_{4}.$ (5.6.21)

As we have seen from the last and the previous cursory analysis of such
systems, it is possible to identify an approximate behaviour of such
solutions, in one specific asymptotic limit in which the invariant $s$ and $t$
get large and of the same size.
In this approximate analysis the only exact statement that one can make is
that Lauricella functions are indeed special solutions of such equations, and
correspond to particular solutions of such inhomogenous systems.
We have been careful to rewrite all the CWIs for generic scalings
$\Delta_{i}$, in such as way that the left hand sides of thse systems carry a
close resemblance to those of 3-point functions, except for an extra term
proportional to a double derivative in $s$ and $t$, $\sim
1/(st)\partial^{2}/(\partial s\partial t)$, which is new for 4-point functions
and absent in 3-point functions.
As we have stressed in the previous sections in the case of dcc solutions,
this term does preserve the hypergeometric structure of the corresponding
equations, although such solutions have little in common with those derived
for genuine 3-point functions, for being quartic -rather than quadratic -
ratios of momenta.
The discovery of such solutions may not be accidental in the context of CFT’s,
since in ordinary perturbation theory similar dependences have been uncovered
in the analysis of ladder diagrams [101]. However, one can easily check,
following the discussion in [26], that box-like master integrals with
propagators raised to generic powers, cannot be special cases of such dcc
solutions, except for the ordinary box diagram. On general grounds, one
expects that the simplified CWIs, which are found in the scalar case for the
dcc solutions, are related to an underlying Yangian symmetry [102], which is
manifesting here in a bosonic, non supersymmetric, context. In the $TOOO$ such
a symmetry, differently from the scalar case, is violated by the presence of a
single stress-energy tensor. It could be restored in tensor correlators
characterised by a single primary operator, such as the $JJJJ$ or the $TTTT$.

####  5.7 Conclusions

The investigation of the CWIs of four point functions of a generic CFT in
momentum space in $d>2$ is a new challenging domain of research, with the
possibility of establishing a direct connection with the analysis of
scattering amplitudes in Lagrangian field theories. As in the case of lower
point functions, one could envision several areas where such studies could
find direct physical applications, from cosmology to condensed matter theory
[103, 6], due to the interplay, in the latter case, of quantum anomalies in
transport phenomena. These studies need to be accompanied by investigations of
the operator product expansion in the same variables, in order to develop a
bootstrap program, as in coordinate space.
Obviously, while in coordinate space the operatorial expansion is well-behaved
at separate spacetime points, in momentum space we gather information on such
operators from all the spacetime regions, including those in which the
external coordinates of a correlator coalesce. This makes the analysis in
momentum space more demanding, and we have to worry about anomalies and
address the issue of how to regulate a given theory.
It is then natural to advance our knowledge in this area starting from the
analysis of simpler correlation functions, the scalar and the tensor/scalar
cases being the first on the list.
For this reason we have derived the CWIs for the $TOOO$ and discussed their
relation to those obtained in the case of 4 scalars. In both cases we have
discussed their expressions in various limits, showing the hypergeometric
character of the asymptotic solutions, if certain constraints on the scaling
dimensions are respected.
While, obviously, we do not expect that a given correlator can be uniquely
identified by these equations, neverthless they constrain quite significantly
the structure of the possible solutions.
As mentioned, in our analysis we have concentrated on the structure of the
equations in several kinematical limits, in order to gather some information
about the behaviour of the corresponding solutions. In such limits, the
differential operators take a simplified but a still nontrivial form.
The comparison between the $TOOO$ and scalar cases, allows to uncover some
common features of the systems of equations that they need to satisfy. In this
context, of particular significance are those solutions which are dual
conformal and conformal at the same time (or dcc solutions), which take a
unique expression. Several different ansätze take to the same hypergeometric
form of such solutions, which are related by analytic continuations, and, as
we have shown, turn useful for their study in specific kinematical limits. For
such a reason they play a strategic role, since they can be used to
investigate the behaviour of scalar 4-point functions in a rather direct way
and allow to underscore some similarities between the CWIs both in the tensor
and in the scalar contexts.
Specific features of such dcc solutions, extracted in several asymptotic
limits, are expected to provide some indication on the behaviour of the more
general (and unknown) solutions of the equations satisfied by scalar operators
- the $OOOO$ for instance - for generic scaling dimensions of the primaries
$O$. Both correlators are characterised by a single form factors, allowing
particular solutions of Lauricella type. This suggests the presence of a more
general underlying hypergeometric structure in such systems of equations. It
could be of interest to investigate from a purely mathematical point of view
the structure such equations in order to classify the structure of such
solutions.
Our investigations can be extended in several directions, for instance to the
study of the renormalization of the corresponding form factors, which requires
a separate investigation, as in the case of 3-point functions [71]. There are
also other and quite direct implications of our results and equations for the
analysis of the decomposition of such correlators in terms of CP-symmetric
(Polyakov) blocks. Indeed the CWIs that we have derived can be applied to
constrain the block decomposition [28].

###  Chapter 6 The $\langle TTTT\rangle$ correlator: Renormalization and the
Conformal Anomaly Action to Fourth Order in $d=4$

####  6.1 Introduction

In this chapter, we will demonstrate an analysis of the $\langle TTTT\rangle$
correlator in momentum space. It is based on the work [53]. As we move from 3-
to 4-point functions, the solutions of the CWIs are affected by the appearance
of arbitrary functions, which is a general feature of CFT, and this approach
is not of immediate help, in the sense that even if such bilinear mixings
would be found in the perturbative description of a certain correlator, they
would not find an immediate counterpart in the general solution of the
conformal constraints.
Obviously, we would like to provide a proof of such behaviour with no
reference to free field theory. We recall that the CWIs impose, at any $n$,
hierarchical relations connecting $n$ to $n-1$ point functions, which we are
going to investigate in great detail in momentum space. While these relations
are not sufficient, for a generic CFT, to reconstruct the expression of
$\mathcal{S}(g)$ beyond $O(\delta g^{3})$ (i.e. $n=3$), they nevertheless
constrain significantly its structure after renormalization.
As we are going to show, the CWIs predict a well-defined structure for the
anomaly contribution to $\mathcal{S}_{A}$ at the level of the 4T, once we
assume that their renormalization proceeds via the counterterm action
(3.2.24). As already mentioned, this analysis does not require a complete
classification of all the form factors which appear in the correlator, but
just a careful study of the structure of the CWIs satisfied by it and by the
3T.

#####  6.1.1 Bilinear mixings

Bilinear mixings are, even in ordinary field theory, the signature that the
functional expansion of the effective action is taking place in a nontrivial
vacuum, as for the Higgs mechanism.
In an ordinary gauge theory such mixings are removed by a suitable gauge
choice, such as the ‘t Hooft or the unitary gauge. In the conformal case, a
massless pole represents a virtual (nonlocal) interaction, which is directly
coupled to the anomaly and defines a skeleton expansion which can be taken as
a possible definition the anomaly action.
As we are going to elaborate in a following section, such terms correspond to
an expansion of the same action - in coordinate space and in the flat limit-
in the dimensionless variable $R\,\square^{-1}$. Here, $R$ is the scalar
curvature, which has very often appeared in the analysis of nonlocal
cosmologies, such as in $f(R\,\square^{-1})$ models [104]. In this case, the
goal has been that of explaining the late-time dark energy dominance of our
universe.
These analysis have been limited in the past to 3-point functions [105, 62,
26]. Four-point functions have received attention only more recently [62, 86].
For 3-point functions, the analysis of conformal and non-conformal correlators
in realistic theories such as QED and QCD, at one-loop, has covered the $TJJ$
and the $TTT$ (3T)[56, 24, 25, 57].
In this chapter we are going to show that this feature is generic. No complete
solution of the CWIs at quartic level ($O(h^{4})$ in flat space, with $h$ the
metric fluctuation) is needed in order to extract such behaviour, at least in
flat space, once the contribution of the anomaly is correctly taken into
account.
Notice that the advantage of solving explicitly the CWIs, as in the case of
conformal 3-point functions, beside its indisputable value, is that it gives
the opportunity to apply and verify the renormalization procedure on the
solution, using the known gravitational counterterms, as given in Eq. (3.2.24)
below. However, for the rest, the method becomes increasingly prohibitive if
the goal is to infer the all-order behaviour of the anomaly action.
The purpose of the chpater is to show that a certain amount of information is
hidden in the structure of the WIs, if we assume that they can be renormalized
using the standard (known) counterterms typical of the anomaly functional. We
work our way from this assumption backward, in order to characterize the
structure of the CWIs and identify the implications for the structure of the
anomaly action to a given order in $h^{\mu\nu}$ $(h)$, the fluctuation of the
metric around its flat spacetime limit.
Our approach is, in a way, quite direct, and can be extended to n-point
functions and to any spacetime dimension, following the same strategy. Since
we intend to characterize the method in its simplest formulation, we will
focus in $d=4$ spacetime dimensions, and we will rely on an expansion of the
anomaly functional around a flat background in Dimensional Regularization
(DR).

#####  6.1.2 Nonlocal interactions in the 3T

Figure 6.1: Expansion of the anomaly contributions to the renormalized vertex
for the 3T.

In the case of the 3T all the contributions to the trace anomaly of the
renormalized vertex can be summarised by the three diagrams shown in Fig. 6.1
and will be discussed below.
In this figure, the dashed lines indicate the inclusion of an operator

$\hat{\pi}^{\mu\nu}=g^{\mu\nu}p^{2}-p^{\mu}p^{\nu}$ (6.1.1)

on the external lines, which projects on the subspace transverse to a given
momentum $p$. The projector is accompanied by a pole. Together, $\hat{\pi}$
and the pole define the ordinary transverse projector

$\pi^{\mu\nu}=\hat{\pi}^{\mu\nu}\frac{1}{p^{2}}$ (6.1.2)

inserted on all the external lines in one, two and three copies. Poles in
separate variables $1/p_{i}^{2}$, $1/(p_{i}^{2}p_{j}^{2})(i\neq j)$ and
$1/(p_{1}^{2}p_{2}^{2}p_{3}^{2})$, are connected to separate external graviton
lines, and each momentum invariant appears as a single pole.
This result had been obtained in [24] by performing a complete perturbative
analysis of the same vertex using free field theory realizations. In this
case, the inclusion of 3 sectors, a scalar, a fermion and a spin 1, allows to
obtain the most general expression of this correlator, and its
renormalization, performed by the addition of the general counterterm
(3.2.24), has been verified.
We will show that the Weyl-variant contribution to $\mathcal{S}_{A}$ can be
identified directly from the CWIs, under the assumption that
$\mathcal{S}_{ct}$, as defined in (3.2.24), is all that is needed in order to
proceed with the renormalization of a 3- or a 4-point function of correlators
of stress energy tensors in $d=4$, in the flat spacetime limit.
If we move to coordinate space from momentum space and attach the metric
fluctuations $h^{\mu\nu}$ on the external lines, it is easy to show that the
inclusion of a projector of the form (6.1.2) induces a bilinear mixing between
the graviton and a virtual scalar propagating in the intermediate virtual
(dashed) line. The interaction takes the form

$\frac{1}{p^{2}}\hat{\pi}^{\mu\nu}\leftrightarrow R^{(1)}\frac{1}{\square}$
(6.1.3)

In the 3T case, as one can immediately figure out from Fig. 6.1, such
expressions can be easily traced back to nonlocal terms in the anomaly action
$\mathcal{S}_{A}$ of the form

$\mathcal{S}_{A}\sim\int
d^{4}x\,d^{4}yR^{(1)}(x)\left(\frac{1}{\square}\right)(x,y)\left(b^{\prime}\,E_{4}^{(2)}(y)+b\,(C^{2})^{(2)}(y)\right),$
(6.1.4)

where the labels $(1)$, $(2)$ refer to the first and second variations of the
invariants $R$, the scalar curvature, $C^{2}$ and $E_{4}$.

Figure 6.2: Expansion of the anomaly contributions to the renormalized vertex
for the TJJ. Left: The collinear region in momentum space responsible for the
origin of the pole. Right: Its interpretation as a scalar exchange.

Analogously, in the case of the TJJ correlator in QED, the anomaly
contribution, extracted by a complete perturbative analysis, is shown in Fig.
6.2 and takes the form

$\mathcal{S}_{A}\sim\beta(e)\int
d^{4}x\,d^{4}y\,R^{(1)}(x)\left(\frac{1}{\square}\right)(x,y)F^{\mu\nu}F_{\mu\nu}(y),$
(6.1.5)

with $F^{\mu\nu}$ being the QED field strength and $\beta(e)$ the
corresponding beta-function of the gauge coupling. Notice that in both
examples the intermediate propagating virtual state is directly coupled to the
anomaly.
This characterization of the anomaly action $\mathcal{S}_{A}$ at cubic order,
in each of the two cases, has been obtained by a complete analysis of the
conformal constraints, using the explicit expression of their solutions, which
for 3 point functions depends only on three constant for the 3T and on two for
the TJJ. In both cases, the solutions, which are given by hypergeometric
functions $F_{4}$, can be mapped to general free field theory realizations,
characterized by the inclusion at 1-loop level of an arbitrary number of
scalars, fermions and spin-1 fields in the Feynman expansion. The map between
the two approaches is an exact one in $d=3,5$.
In [90] it has been shown that such nonzero trace contributions are
automatically generated by a nonlocal conformal anomaly action of Riegert type
[106] expanded to first order in $\delta g$ and to second order in the
external gauge field $A_{\mu}$, respectively.
As we move from 3- to 4-point functions, the solutions of the CWIs are
affected by the appearance of arbitrary functions, which is a general feature
of CFT, and this approach is not of immediate help, in the sense that even if
such bilinear mixings would be found in the perturbative description of a
certain correlator, they would not find an immediate counterpart in the
general solution of the conformal constraints.
We recall that the CWIs impose, at any $n$, hierarchical relations connecting
$n$ to $n-1$ point functions, which we are going to investigate in great
detail in momentum space. While these relations are not sufficient, for a
generic CFT, to reconstruct the expression of $\mathcal{S}(g)$ beyond
$O(\delta g^{3})$ (i.e. $n=3$), they nevertheless constrain significantly its
structure after renormalization.
The CWIs predict a well-defined structure for the anomaly contribution to
$\mathcal{S}_{A}$ at the level of the 4T, once we assume that their
renormalization proceeds via the counterterm action (3.2.24) as we will
demonstrate. This analysis does not require a complete classification of all
the form factors which appear in the correlator, but just a careful study of
the structure of the CWIs satisfied by it and by the 3T.
For this reason, we are going to illustrate how this simplified procedure
works first in the case of the 3T, before moving to the 4T in this new
approach, showing how the structures of $\mathcal{S}_{A}$ is constrained by
the renormalized CWIs to assume a form very similar to the 3T case.
As in previous studies in momentum space, we will rely on decompositions of
the 3T and 4T into a transverse traceless, a longitudinal, and a trace part,
extending the approach formulated in [21] for 3-point functions to 4-point
functions.

#####  6.1.3 Contents of the chapter

This chapter is organised as follows. We will define our conventions and
characterise the main features of the anomaly action highlighting all the
simplifications that are present when we consider the flat spacetime limit in
the definition of the correlation functions. A complementary and more detailed
analysis of 6.2 can be found in chapter 3.There the discussion, is carried out
in full generality in regard to the choice of the background metric, and
provides a wider perspective on the structure of the effective action
$(\mathcal{S})$ for arbitrary metrics, and on the increased complexity of the
CWIs around such backgrounds.
It is well known that in more general backgrounds, for instance for Weyl
flat/conformally flat metrics, the structure of the quantum average of the
energy momentum tensor (its one-point function), is affected by tadpole
contributions which are fixed by the anomaly, and need to be included in the
analysis, and call for a generalization of our method.
Then we continue with a review of the structure of the CWIs in the
longitudinal/transverse separation introduced in [21] and developed further in
[23, 71, 97], followed by the analysis of the anomalous CWIs. In Section 6.6 a
new derivation, performed directly in $d=4$, of the special CWIs is presented
for the $4T$. The approach extends the one developed in [90] for the 3T to the
new case.
The renormalization of the 4T and its longitudinal/transverse decomposition is
worked out afterwards, where, just for completeness, we also classify the
singular form factors which are affected by renormalization.
The structure of the anomaly action is slowly built starting from the 2-point
function, that we review. The vanishing of the anomaly contribution to the
anomaly action $\mathcal{S}$ at $O(h^{2})$ is illustrated in great detail.
Then we derive the structure of the local (i.e. $t_{loc}$ or longitudinal
stress energy tensor) contributions to the CWI in the renormalization
procedure, and separate the anomaly contributions from the unknown but finite,
renormalized parts. This is the content of Section 6.17.
Even if the finite local terms are not explicitly given, we show that they are
not necessary in order to identify the structure of the anomaly action at the
level of the 4T.
The result is obtained by working directly with the longitudinal components
$t_{loc}$ of the counterterms to the same 4T correlator, combined with their
trace WIs. We demonstrate that a structure with bilinear mixings appears quite
naturally from the decomposition, together with an extra, trace-free
contribution. This extra term, absent in the 3-point function, appears for the
first time at the level of the 4T.

####  6.2 Renormalization in momentum space

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 renormalized effective action $\mathcal{S}_{R}$ is then defined by the sum
of the two terms

$\mathcal{S}_{R}(g)=\mathcal{S}(g)+\mathcal{S}_{ct}(g)$ (6.2.1)
$\displaystyle\mathcal{S}_{ct}=\raisebox{-3.44444pt}{\includegraphics[width=65.04034pt]{Figures/ct2p}}+\raisebox{-21.52771pt}{\includegraphics[width=65.04034pt]{Figures/ct3p}}+\raisebox{-25.83325pt}{\includegraphics[width=65.04034pt]{Figures/ct4p}}+...$
(6.2.2)

with $\mathcal{S}_{ct}$ shown in (6.2.2). Both terms of $\mathcal{S}_{R}(g)$
are expanded in the metric fluctuations as in (6.2.2). If we resort to a path
integral definition of a certain CFT, it is clear that any renormalized
correlation function appearing in the expansion of $\mathcal{S}_{R}(g)$ would
be expressed in terms of a bare contribution accompanied by a counterterm
vertex.
The correlation functions extracted by the renormalized action can be
expressed as the sum of a finite $(f)$ correlator and of an anomaly term
($anomaly$) in the form

$\displaystyle\langle{T^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}\ldots
T^{\mu_{n}\nu_{n}}}\rangle_{Ren}=$
$\displaystyle\bigg{[}\langle{T^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}\ldots
T^{\mu_{n}\nu_{n}}}\rangle_{bare}+\langle{T^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}\ldots
T^{\mu_{n}\nu_{n}}}\rangle_{count}\bigg{]}_{d\to 4}=$
$\displaystyle=\langle{T^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}\ldots
T^{\mu_{n}\nu_{n}}}\rangle^{(d=4)}_{f}+\langle{T^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}\ldots
T^{\mu_{n}\nu_{n}}}\rangle^{(d=4)}_{anomaly}$ (6.2.3)

The renormalized correlator shown above satisfies anomalous CWIs.
To characterize the anomaly contribution to each correlation function, we
start from the 1-point function. In a generic background $g$, the 1-point
function is decomposed as

$\langle
T^{\mu\nu}\rangle_{Ren}=\frac{2}{\sqrt{g}}\frac{\delta\mathcal{S}_{Ren}}{\delta
g_{\mu\nu}}=\langle
T^{\mu\nu}\rangle_{A}+\langle\overline{T}^{\mu\nu}\rangle_{f}$ (6.2.4)

with

$g^{\mu\nu}\frac{\delta\mathcal{S}}{\delta
g^{\mu\nu}}=g^{\mu\nu}\frac{\delta\mathcal{S}_{A}}{\delta
g^{\mu\nu}}\equiv\frac{\sqrt{g}}{2}g_{\mu\nu}\langle T^{\mu\nu}\rangle_{A}$
(6.2.5)

being the trace anomaly equation, and
$\langle\overline{T}^{\mu\nu}\rangle_{f}$ is the Weyl-invariant (traceless)
term.
Following the discussion in (3.2.3), these scaling violations may be written
for the 1-point function in the form

$\langle{T^{\mu}_{\ \ \mu}(x)}\rangle=\mathcal{A}(x)$ (6.2.6)

\- 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

$\mathcal{A}(x)=\sqrt{-g(x)}\bigg{[}b\,C^{2}(x)+b^{\prime}E(x)\bigg{]}$
(6.2.7)

being the anomaly functional. We will be needing several differentiation of
this functional, evaluated in the flat limit. This procedure generates
expressions which are polynomial in the momenta, that can be found in the
Appendix B.4. In general, one also finds additional dimension-4 local
invariants $\mathcal{L}_{i}$, if there are couplings to other background
fields, as for instance in the QED and QCD cases, with coefficients related to
the $\beta$ functions of the corresponding gauge couplings.
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, it is possible to circumvent this limiting procedure by working
out the equations directly in $d=4$, with the inclusion of the contributions
coming from the anomaly functional, as we are going to show below. This second
approach has been formulated in [90] and will be extended to the 4-point
function in Section 6.5.
We recall that the counterterm vertex for the nT correlator, in DR, in
momentum space takes the form

$\langle{T^{\mu_{1}\nu_{1}}(p_{1})\dots
T^{\mu_{n}\nu_{n}}(\bar{p}_{n})}\rangle_{count}=-\frac{\mu^{-\varepsilon}}{\varepsilon}\bigg{(}b\,V_{C^{2}}^{\mu_{1}\nu_{1}\dots\mu_{n}\nu_{n}}(p_{1},\dots,\bar{p}_{n})+b^{\prime}\,V_{E}^{\mu_{1}\nu_{1}\dots\mu_{n}\nu_{n}}(p_{1},\dots,\bar{p}_{n})\bigg{)},$
(6.2.8)

where

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

and

$\displaystyle
V_{E}^{\mu_{1}\nu_{1}\dots\mu_{n}\nu_{n}}(p_{1},\dots,\bar{p}_{n})$
$\displaystyle\equiv
2^{n}\big{[}\sqrt{-g}\,E\big{]}^{\mu_{1}\nu_{1}\dots\mu_{n}\nu_{n}}(p_{1},\dots,\bar{p}_{n})$
$\displaystyle=2^{n}\int\,d^{d}x_{1}\,\dots\,d^{d}x_{n}\,d^{d}x\,\bigg{(}\mbox{\small$\displaystyle\frac{\delta^{n}(\sqrt{-g}E)(x)}{\delta
g_{\mu_{1}\nu_{1}}(x_{1})\dots\delta
g_{\mu_{n}\nu_{n}}(x_{n})}$}\bigg{)}_{g=\delta}\,e^{-i(p_{1}\,x_{1}+\dots+p_{n}x_{n})}.$
(6.2.10)

are the expressions of the two contributions present in (6.2.8) in momentum
space. One can also verify the following trace relations

$\displaystyle\delta_{\mu_{1}\nu_{1}}\,V_{C^{2}}^{\mu_{1}\nu_{1}\dots\mu_{n}\nu_{n}}(p_{1},\dots,p_{n})=2^{n-1}(d-4)\,\left[\sqrt{-g}C^{2}\right]^{\mu_{2}\nu_{2}\dots\mu_{n}\nu_{n}}(p_{2},\dots,p_{n})$
$\displaystyle-2\bigg{[}V_{C^{2}}^{\mu_{2}\nu_{2}\dots\mu_{n}\nu_{n}}(p_{1}+p_{2},p_{3},\dots,p_{n})+V_{C^{2}}^{\mu_{2}\nu_{2}\dots\mu_{n}\nu_{n}}(p_{2},p_{1}+p_{3},\dots,p_{n})+\dots+V_{C^{2}}^{\mu_{2}\nu_{2}\dots\mu_{n}\nu_{n}}(p_{2},p_{3},\dots,p_{1}+p_{n})\bigg{]}$
(6.2.11)
$\displaystyle\delta_{\mu_{1}\nu_{1}}\,V_{E}^{\mu_{1}\nu_{1}\dots\mu_{n}\nu_{n}}(p_{1},\dots,p_{n})=2^{n-1}(d-4)\,\left[\sqrt{-g}E\right]^{\mu_{2}\nu_{2}\dots\mu_{n}\nu_{n}}(p_{2},\dots,p_{n})$
$\displaystyle-2\bigg{[}V_{E}^{\mu_{2}\nu_{2}\dots\mu_{n}\nu_{n}}(p_{1}+p_{2},p_{3},\dots,p_{n})+V_{E}^{\mu_{2}\nu_{2}\dots\mu_{n}\nu_{n}}(p_{2},p_{1}+p_{3},\dots,p_{n})+\dots+V_{E}^{\mu_{2}\nu_{2}\dots\mu_{n}\nu_{n}}(p_{2},p_{3},\dots,p_{1}+p_{n})\bigg{]}$
(6.2.12)

that hold in general $d$ dimensions.
Obviously, the effective action that results from the renormalization can be
clearly separated in terms of two contributions, as evident from (6.2.4),

$\mathcal{S}_{R}(g)=\mathcal{S}_{A}(g)+{\mathcal{S}}_{f}(g),$ (6.2.13)

corresponding to an anomaly part $\mathcal{S}_{A}[g]$ and to a finite, Weyl-
invariant term, which can be expanded in terms of fluctuations over a
background $\bar{g}$ as for the entire effective action $\mathcal{S}(g)$

$\mathcal{S}_{f}(g)=\mathcal{S}(\bar{g})+\sum_{n=1}^{\infty}\frac{1}{2^{n}n!}\int
d^{d}x_{1}\ldots d^{d}x_{n}\sqrt{g_{1}}\ldots\sqrt{g_{n}}\,\langle
T^{\mu_{1}\nu_{1}}\ldots\,T^{\mu_{n}\nu_{n}}\rangle_{f}\delta
g_{\mu_{1}\nu_{1}}(x_{1})\ldots\delta g_{\mu_{n}\nu_{n}}(x_{n}).$ (6.2.14)

This functional collects finite correlators (7.10.1) in $d=4$. A similar
expansion holds also for $\mathcal{S}_{A}$, the anomaly part.
The anomaly effective action $\mathcal{S}_{A}$ that results from this analysis
in momentum space is a rational function of the external momenta,
characterised by well-defined tensor structures, and it is free of logarithmic
terms, as shown in direct perturbative studies of the TJJ and 3T [56, 57,
107].

####  6.3 Conservation Ward identities

The anomaly action $\mathcal{S}_{A}(g)$ is constrained by a hierarchical set
of equations which can be derived by the symmetries of the general effective
action $\mathcal{S}$. We proceed assuming that the correlation functions can
be derived by varying the path integral definition of $\mathcal{S}(g)$ (7.2.2)
(3.1.3) as in (7.2.7).
Starting from the covariant definition of the stress-energy tensor, expressed
in terms of a fundamental action as in (7.2), but for the rest, generic

$\langle{T^{\mu\nu}(x)}\rangle_{g}=\frac{2}{\sqrt{-g(x)}}\frac{\delta
S(g)}{\delta g_{\mu\nu}(x)},$ (6.3.1)

this 1-point function satisfies the fundamental Ward identity of covariant
conservation in an arbitrary background $g$

$^{(g)}\nabla_{\mu}\langle T^{\mu\nu}(x)\rangle_{g}=0,\qquad
i.e.\qquad\delta_{\epsilon}\mathcal{S}(g)=0$ (6.3.2)

as a consequence of the invariance of $\mathcal{S}(g)$ under diffeomorphisms.
Here ${}^{(g)}\nabla_{\mu}$ denotes the covariant derivative in the general
background metric $g_{\mu\nu}(x)$. It can be expressed in the form

$\partial_{\nu}\left(\frac{\delta S(g)}{\delta
g_{\mu\nu}(x)}\right)+\Gamma^{\mu}_{\nu\lambda}\left(\frac{\delta S(g)}{\delta
g_{\lambda\nu}(x)}\right)=0,$ (6.3.3)

where $\Gamma^{\mu}_{\lambda\nu}$ is the Christoffel connection for the
general background metric $g_{\mu\nu(x)}$.
Our definitions and conventions are summarised in an Appendix B.3. In order to
derive the conservation WIs for higher point correlation functions, one has to
consider additional variations with respect to the metric of (6.3.2) and then
move to flat space, obtaining

$\displaystyle\partial_{\nu_{1}}\langle
T^{\mu_{1}\nu_{1}}(x_{1})T^{\mu_{2}\nu_{2}}(x_{2})T^{\mu_{3}\nu_{3}}(x_{3})T^{\mu_{4}\nu_{4}}(x_{4})\rangle=$
$\displaystyle=$
$\displaystyle-\left[2\left(\frac{\delta\Gamma^{\mu_{1}}_{\lambda\nu_{1}}(x_{1})}{\delta
g_{\mu_{2}\nu_{2}}(x_{2})}\right)_{g=\delta}\langle
T^{\lambda\nu_{1}}(x_{1})T^{\mu_{3}\nu_{3}}(x_{3})T^{\mu_{4}\nu_{4}}(x_{4})\rangle+(23)+(24)\right]$
$\displaystyle-\left[4\left(\frac{\delta^{2}\Gamma^{\mu_{1}}_{\lambda\nu_{1}}(x_{1})}{\delta
g_{\mu_{2}\nu_{2}}(x_{2})\delta
g_{\mu_{3}\nu_{3}}(x_{3})}\right)_{g=\delta}\langle{T^{\lambda\nu_{1}}(x_{1})T^{\mu_{4}\nu_{4}}(x_{4})}\rangle+(24)+(34)\right],$
(6.3.4)

where

$\displaystyle\left(\frac{\delta\Gamma^{\mu_{1}}_{\lambda\nu_{1}}(x_{1})}{\delta
g_{\mu_{i}\nu_{i}}(x_{i})}\right)_{g=\delta}$
$\displaystyle=\frac{1}{2}\left(\delta^{\mu_{1}(\mu_{i}}\delta^{\nu_{i})}_{\nu_{1}}\,\partial_{\lambda}\delta_{x_{1}x_{i}}+\delta^{\mu_{1}(\mu_{i}}\delta^{\nu_{i})}_{\lambda}\,\partial_{\nu_{1}}\delta_{x_{1}x_{i}}-\delta^{(\mu_{i}}_{\lambda}\delta^{\nu_{i})}_{\nu_{1}}\,\partial^{\mu_{1}}\delta_{x_{1}x_{i}}\right)$
(6.3.5)
$\displaystyle\left(\frac{\delta^{2}\Gamma^{\mu_{1}}_{\lambda\nu_{1}}(x_{1})}{\delta
g_{\mu_{i}\nu_{i}}(x_{i})\delta g_{\mu_{j}\nu_{j}}(x_{j})}\right)_{g=\delta}$
$\displaystyle=$
$\displaystyle=-\frac{\delta_{x_{1}x_{i}}}{2}\delta^{\mu_{1}(\mu_{i}}\delta^{\nu_{i})\epsilon}\left(\delta^{(\mu_{j}}_{\epsilon}\delta^{\nu_{j})}_{\nu_{1}}\,\partial_{\lambda}\delta_{x_{1}x_{j}}+\delta^{(\mu_{j}}_{\epsilon}\delta^{\nu_{j})}_{\lambda}\,\partial_{\nu_{1}}\delta_{x_{1}x_{j}}-\delta^{(\mu_{j}}_{\lambda}\delta^{\nu_{j})}_{\nu_{1}}\,\partial_{\epsilon}\delta_{x_{1}x_{j}}\right)+(ij),$
(6.3.6)

are the first and second functional derivatives of the connection, in the flat
limit. We have explicitly indicated the symmetrization with respect to the
relevant indices using the permutation $(ij)\equiv(i\leftrightarrow j)$. We
have defined $\delta^{(\mu_{i}}_{\lambda}\delta^{\nu_{i})}_{\nu_{1}}\equiv
1/2(\delta^{\mu_{i}}_{\lambda}\delta^{\nu_{i}}_{\nu_{1}}+\delta^{\nu_{i}}_{\lambda}\delta^{\mu_{i}}_{\nu_{1}})$,
and introduced a simplified notation for the Dirac delta
$\delta_{x_{i}x_{j}}\equiv\delta(x_{i}-x_{j})$. All the derivative (e.g.
$\partial_{\lambda}$) are taken with respect to the coordinate $x_{1}$
$(e.g.\partial/\partial x_{1}^{\lambda})$.
We Fourier transform to momentum space with the convention

$\displaystyle\langle{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\overline{p}_{4})}\rangle$
$\displaystyle=\int d^{4}x_{1}d^{4}x_{2}d^{4}x_{3}e^{-i(p_{1}\cdot
x_{1}+p_{2}\cdot x_{2}+p_{3}\cdot x_{3})}$
$\displaystyle\times\langle{T^{\mu_{1}\nu_{1}}(x_{1})T^{\mu_{2}\nu_{2}}(x_{2})T^{\mu_{3}\nu_{3}}(x_{3})T^{\mu_{4}\nu_{4}}(0)}\rangle.$
(6.3.7)

Here we have used the translational invariance of the correlator in flat
space, which allows to use momentum conservation to express one of the momenta
(in our convention $p_{4}$) as combination of the remaining ones
$\bar{p}_{4}=-p_{1}-p_{2}-p_{3}$. Details on the elimination of one of the
momenta in the derivation of the CWIs and on the modification of the Leibnitz
rule in the differentiation of such correlators in momentum space can be found
in [25].
The conservation Ward Identity (7.10.13) in flat spacetime may be Fourier
transformed, giving the CWIs in momentum space

$\displaystyle
p_{1\nu_{1}}\langle{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle=$
$\displaystyle=\Big{[}4\,\mathcal{B}^{\mu_{1}\hskip
11.38092pt\mu_{2}\nu_{2}\mu_{3}\nu_{3}}_{\hskip
8.5359pt\lambda\nu_{1}}(p_{2},p_{3})\langle{T^{\lambda\nu_{1}}(p_{1}+p_{2}+p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle+(34)+(24)\Big{]}$
$\displaystyle\hskip 14.22636pt+\Big{[}2\,\mathcal{C}^{\mu_{1}\hskip
11.38092pt\mu_{2}\nu_{2}}_{\hskip
8.5359pt\lambda\nu_{1}}(p_{2})\langle{T^{\lambda\nu_{1}}(p_{1}+p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle+(23)+(24)\Big{]},$
(6.3.8)

where we have defined

$\displaystyle\mathcal{B}^{\mu_{1}\hskip
11.38092pt\mu_{2}\nu_{2}\mu_{3}\nu_{3}}_{\hskip
8.5359pt\lambda\nu_{1}}(p_{2},p_{3})$
$\displaystyle\equiv-\frac{1}{2}\delta^{\mu_{1}(\mu_{2}}{\delta^{\nu_{2})\epsilon}}\left(\delta_{\epsilon}^{(\mu_{3}}\delta^{\nu_{3})}_{\nu_{1}}\,p_{3\,\lambda}+\delta_{\epsilon}^{(\mu_{3}}\delta^{\nu_{3})}_{\lambda}\,p_{3\,\nu_{1}}-\delta_{\lambda}^{(\mu_{3}}\delta^{\nu_{3})}_{\nu_{1}}\,p_{3\,\epsilon}\right)+(23)$
(6.3.9) $\displaystyle\mathcal{C}^{\mu_{1}\hskip
11.38092pt\mu_{2}\nu_{2}}_{\hskip 8.5359pt\lambda\nu_{1}}(p_{2})$
$\displaystyle\equiv\frac{1}{2}\left(\delta^{\mu_{1}(\mu_{2}}\delta^{\nu_{2})}_{\nu_{1}}\,p_{2\,\lambda}+\delta^{\mu_{1}(\mu_{2}}\delta^{\nu_{2})}_{\lambda}p_{2\,\nu_{1}}-\delta^{(\mu_{2}}_{\lambda}\delta^{\nu_{2})}_{\nu_{1}}p_{2}^{\mu_{1}}\right),$
(6.3.10)

related to the second and first functional derivatives of the Christoffel,
connection respectively.

#####  6.3.1 Conservation WI’s for the counterterms

To illustrate the conservation WI in detail, we turn to the expression of the
counterterm action (3.2.24), which generates counterterm vertices of the form

$\displaystyle\langle{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}_{count}=$
$\displaystyle\qquad=-\frac{\mu^{-\varepsilon}}{\varepsilon}\bigg{(}b\,V_{C^{2}}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{1},p_{2},p_{3},\bar{p}_{4})+b^{\prime}\,V_{E}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{1},p_{2},p_{3},\bar{p}_{4})\bigg{)},$
(6.3.11)

where on the rhs of the expression above we have introduced the counterterm
vertices (with $P=p_{1}+\ldots p_{4}$)

$\displaystyle
V_{C^{2}}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{1},p_{2},p_{3},\bar{p}_{4})\,\delta^{4}(P)\equiv
16\,\delta^{4}(P)\,\big{[}\sqrt{-g}\,C^{2}\big{]}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{1},p_{2},p_{3},\bar{p}_{4})$
$\displaystyle=$ $\displaystyle
16\int\,d^{d}x_{1}\,\,d^{d}x_{2}\,d^{d}x_{3}\,d^{d}x_{4}\,d^{d}x\,\bigg{(}\mbox{\small$\displaystyle\frac{\delta^{4}(\sqrt{-g}C^{2})(x)}{\delta
g_{\mu_{1}\nu_{1}}(x_{1})\delta g_{\mu_{2}\nu_{2}}(x_{2})\delta
g_{\mu_{3}\nu_{3}}(x_{3})\delta
g_{\mu_{4}\nu_{4}}(x_{4})}$}\bigg{)}_{g=\delta}\,e^{-i(p_{1}\,x_{1}+p_{2}\,x_{2}+p_{3}\,x_{3}+p_{4}x_{4})},$
$\displaystyle
V_{E}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{1},p_{2},p_{3},\bar{p}_{4})\delta^{4}(P)\equiv
16\,\,\delta^{4}(P)\big{[}\sqrt{-g}\,E\big{]}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{1},p_{2},p_{3},\bar{p}_{4})$
$\displaystyle=$ $\displaystyle
16\int\,d^{d}x_{1}\,\,d^{d}x_{2}\,d^{d}x_{3}\,d^{d}x_{4}\,d^{d}x\,\bigg{(}\mbox{\small$\displaystyle\frac{\delta^{4}(\sqrt{-g}E)(x)}{\delta
g_{\mu_{1}\nu_{1}}(x_{1})\delta g_{\mu_{2}\nu_{2}}(x_{2})\delta
g_{\mu_{3}\nu_{3}}(x_{3})\delta
g_{\mu_{4}\nu_{4}}(x_{4})}$}\bigg{)}_{g=\delta}\,e^{-i(p_{1}\,x_{1}+p_{2}\,x_{2}+p_{3}\,x_{3}+p_{4}x_{4})},$
(6.3.12)

evaluated in the flat spacetime limit. These vertices share some properties
when contracted with flat metric tensors and the external momenta as we have
already seen. In particular, from (6.2) and (7.10.1), when $n=4$ and in $d$
dimensions we have

$\displaystyle\delta_{\mu_{1}\nu_{1}}\,V_{C^{2}}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{1},p_{2},p_{3},\bar{p}_{4})$
$\displaystyle=8(d-4)\,\left[\sqrt{-g}C^{2}\right]^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{2},p_{3},\bar{p}_{4})$
$\displaystyle-2V_{C^{2}}^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{1}+p_{2},p_{3},\bar{p}_{4})-2V_{C^{2}}^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{2},p_{1}+p_{3},\bar{p}_{4})-2V_{C^{2}}^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{2},p_{3},p_{1}+\bar{p}_{4}),$
(6.3.13)
$\displaystyle\delta_{\mu_{1}\nu_{1}}\,V_{E}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{1},p_{2},p_{3},\bar{p}_{4})$
$\displaystyle=8(d-4)\,\left[\sqrt{-g}E\right]^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{2},p_{3},\bar{p}_{4})$
$\displaystyle-2V_{E}^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{1}+p_{2},p_{3},\bar{p}_{4})-2V_{E}^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{2},p_{1}+p_{3},\bar{p}_{4})-2V_{E}^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{2},p_{3},p_{1}+\bar{p}_{4}),$
(6.3.14)

which play a key role in the renormalization procedure. Furthermore, the
contraction of these vertices with the external momenta generates conservation
WIs in $d$ dimensions, similar to (6.3.8),

$\displaystyle
p_{1\,\nu_{1}}\,V_{C^{2}}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{1},p_{2},p_{3},\bar{p}_{4})=$
$\displaystyle=\Big{[}4\,\mathcal{B}^{\mu_{1}\hskip
11.38092pt\mu_{2}\nu_{2}\mu_{3}\nu_{3}}_{\hskip
8.5359pt\lambda\nu_{1}}(p_{2},p_{3})V_{C^{2}}^{\lambda\nu_{1}\mu_{4}\nu_{4}}(p_{1}+p_{2}+p_{3},\bar{p}_{4})+(34)+(24)\Big{]}$
$\displaystyle\hskip 14.22636pt+\Big{[}2\,\mathcal{C}^{\mu_{1}\hskip
11.38092pt\mu_{2}\nu_{2}}_{\hskip
8.5359pt\lambda\nu_{1}}(p_{2})V_{C^{2}}^{\lambda\nu_{1}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{1}+p_{2},p_{3},\bar{p}_{4})+(23)+(24)\Big{]}$
(6.3.15) $\displaystyle
p_{1\,\nu_{1}}\,V_{E}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{1},p_{2},p_{3},\bar{p}_{4})=$
$\displaystyle=\Big{[}4\,\mathcal{B}^{\mu_{1}\hskip
11.38092pt\mu_{2}\nu_{2}\mu_{3}\nu_{3}}_{\hskip
8.5359pt\lambda\nu_{1}}(p_{2},p_{3})V_{E}^{\lambda\nu_{1}\mu_{4}\nu_{4}}(p_{1}+p_{2}+p_{3},\bar{p}_{4})+(34)+(24)\Big{]}$
$\displaystyle\hskip 14.22636pt+\Big{[}2\,\mathcal{C}^{\mu_{1}\hskip
11.38092pt\mu_{2}\nu_{2}}_{\hskip
8.5359pt\lambda\nu_{1}}(p_{2})V_{E}^{\lambda\nu_{1}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{1}+p_{2},p_{3},\bar{p}_{4})+(23)+(24)\Big{]},$
(6.3.16)

where $\mathcal{C}$ and $\mathcal{B}$ are given in (6.3.9) and (6.3.10). These
equations can be generalized to the case of $n$-point functions.

#####  6.3.2 Conformal Ward Identities

Turning to the ordinary (i.e. non anomalous) trace and conformal WIs, these
can be obtained directly in flat space using the expression of operators of
the dilatation and special conformal transformations. The dilatation WI’s for
the 4T can be easily constructed from the condition of Weyl invariance of the
effective action $\mathcal{S}$, or equivalently, in the ordinary operatorial
approach (see [25])

$\left[4d+\sum_{j=1}^{4}\,x_{j}^{\alpha}\mbox{\small$\displaystyle\frac{\partial}{\partial
x_{j}^{\alpha}}$}\right]\langle{T^{\mu_{1}\nu_{1}}(x_{1})T^{\mu_{2}\nu_{2}}(x_{2})T^{\mu_{3}\nu_{3}}(x_{3})T^{\mu_{4}\nu_{4}}(x_{4})}\rangle=0,$
(6.3.17)

where we have used the explicit expression of the scaling dimension of the
stress energy tensor $\Delta_{T}=d$. Analogously, the special conformal WIs,
corresponding to special conformal transformations, can be derived in the
operatorial approach, applied to an ordinary CFT in flat space, relying on the
change of $T^{\mu\nu}$ under a special conformal transformation, with a
generic parameter $b_{\mu}$, and $\sigma=-2b\cdot x$

$\delta T^{\mu\nu}(x)=-(b^{\alpha}x^{2}-2x^{\alpha}b\cdot
x)\,\partial_{\alpha}T^{\mu\nu}(x)-\Delta_{T}\sigma
T^{\mu\nu}(x)+2(b_{\mu}x_{\alpha}-b_{\alpha}x_{\mu})T^{\alpha\nu}+2(b_{\nu}x_{\alpha}-b_{\alpha}x_{\nu})\,T^{\mu\alpha}(x).$
(6.3.18)

The action of the special conformal operator $\mathcal{K}^{\kappa}$ on $T$ in
its finite form is obtained differentiating respect to the parameter
$b_{\kappa}$

$\displaystyle\mathcal{K}^{\kappa}T^{\mu\nu}(x)$ $\displaystyle\equiv$
$\displaystyle\delta_{\kappa}T^{\mu\nu}(x)=\frac{\partial}{\partial
b^{\kappa}}(\delta T^{\mu\nu})$
$\displaystyle=-(x^{2}\partial_{\kappa}-2x_{\kappa}x\cdot\partial)T^{\mu\nu}(x)+2\Delta_{T}x_{\kappa}T^{\mu\nu}(x)+2(\delta_{\mu\kappa}x_{\alpha}-\delta_{\alpha\kappa}x_{\mu})T^{\alpha\nu}(x)$
$\displaystyle+2(\delta_{\kappa\nu}x_{\alpha}-\delta_{\alpha\kappa}x_{\nu})T^{\mu\alpha}.$
(6.3.19)

By using the Leibniz rule for the variation on correlation functions of
multiple T’s, it can be distributed over the entire correlator as

$\mathcal{K}^{\kappa}\langle{T^{\mu_{1}\nu_{1}}(x_{1})T^{\mu_{2}\nu_{2}}(x_{2})T^{\mu_{3}\nu_{3}}(x_{3})T^{\mu_{4}\nu_{4}}(x_{4})}\rangle=\sum_{i=1}^{4}\langle
T(x_{1})\ldots\delta_{\kappa}T(x_{i})\ldots T(x_{4})\rangle=0$ (6.3.20)

which takes the form

$\displaystyle 0=$
$\displaystyle\sum_{j=1}^{n}\left(2d\,x_{j}^{k}+2x_{j}^{\kappa}\,x_{j}^{\alpha}\mbox{\small$\displaystyle\frac{\partial}{\partial
x_{j}\alpha}$}-x_{j}^{2}\mbox{\small$\displaystyle\frac{\partial}{\partial
x_{jk}}$}\right)\,\langle{T^{\mu_{1}\nu_{1}}(x_{1})T^{\mu_{2}\nu_{2}}(x_{2})T^{\mu_{3}\nu_{3}}(x_{3})T^{\mu_{4}\nu_{4}}(x_{4})}\rangle$
$\displaystyle+2\sum_{j=1}^{4}\left[(x_{j})_{\alpha_{j}}\delta^{\kappa\mu_{j}}-x_{j}^{\mu_{j}}\delta^{\kappa}_{\alpha_{j}}\right]\,\langle{T^{\mu_{1}\nu_{1}}(x_{1})\dots
T^{\alpha_{j}\nu_{j}}(x_{j})\dots T^{\mu_{4}\nu_{4}}(x_{4})}\rangle$
$\displaystyle+2\sum_{j=1}^{4}\left[(x_{j})_{\alpha_{j}}\delta^{\kappa\nu_{j}}-x_{j}^{\nu_{j}}\delta^{\kappa}_{\alpha_{j}}\right]\,\langle{T^{\mu_{1}\nu_{1}}(x_{1})\dots
T^{\mu_{j}\alpha_{j}}(x_{j})\dots T^{\mu_{4}\nu_{4}}(x_{4})}\rangle,$ (6.3.21)

where $\kappa$ is now a free Lorentz index. Notice that in order to be allowed
to use an operatorial approach, one needs to rely on correlators defined via
direct insertions of T’s. Such correlation functions, are, in general,
different from the definition given above in (7.2.7) due to possible contact
terms and the presence of nonvanishing tadpoles, not contemplated in (6.3.20).
For this reason the CWIs derived by this operatorial method and by the
functional method that we will present below, are naive expressions which are
perfectly well-defined and equivalent, only in the presence of a suitable
regularization scheme and of a flat background. In DR, which is well-defined
in a flat spacetime, the vanishing of the 1-point function and the inclusion
of vertex counterterms shows that we don’t need to worry about such issues.
Notice that these constraints are directly written in momentum space as

$\displaystyle 0=$ $\displaystyle
D\,\langle{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle=$
$\displaystyle\left(d-\sum_{j=1}^{3}\,p_{j}^{\alpha}\mbox{\small$\displaystyle\frac{\partial}{\partial
p_{j}^{\alpha}}$}\right)\langle{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}({p}_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle$
(6.3.22)

and

$\displaystyle 0=$
$\displaystyle\mathcal{K}^{\kappa}\langle{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{3}\left((p_{j})^{\kappa}\frac{\partial}{\partial
p_{j}^{\alpha}}\frac{\partial}{\partial
p_{j\,\alpha}}-2p_{j}^{\alpha}\frac{\partial}{\partial
p_{j}^{\alpha}}\frac{\partial}{\partial
p_{j\,\kappa}}\right)\langle{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle\hskip
28.45274pt+4\left(\delta^{\kappa(\mu_{1}}\frac{\partial}{\partial
p_{1}^{\alpha_{1}}}-\delta^{\kappa}_{\alpha_{1}}\Delta_{\lambda}^{(\mu_{1}}\frac{\partial}{\partial
p_{1\,\lambda}}\right)\langle{T^{\nu_{1})\alpha_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle\hskip
28.45274pt+4\left(\delta^{\kappa(\mu_{2}}\frac{\partial}{\partial
p_{2}^{\alpha_{2}}}-\delta^{\kappa}_{\alpha_{2}}\delta^{(\mu_{2}}_{\lambda}\frac{\partial}{\partial
p_{2\,\lambda}}\right)\langle{T^{\nu_{2})\alpha_{2}}(p_{2})T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle\hskip
28.45274pt+4\left(\delta^{\kappa(\mu_{3}}\frac{\partial}{\partial
p_{3}^{\alpha_{3}}}-\delta^{\kappa}_{\alpha_{3}}\delta^{(\mu_{3}}_{\lambda}\frac{\partial}{\partial
p_{3\,\lambda}}\right)\langle{T^{\nu_{3})\alpha_{3}}(p_{3})T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle,$
(6.3.23)

in terms of a dilatation operator $D$ and a special conformal transformation
operator $K^{\kappa}$. The action of the differential operators on the momenta
is implicit on the 4th momentum, as discussed in the case of 3-point functions
in previous works [20, 21, 25], with a modification of the Leibnitz rule.

####  6.4 Trace and conformal anomalous Ward identities

The CWIs become anomalous as we move from $d$ spacetime dimensions to 4. In
$d$ dimensions the conformal symmetry of the correlator 4T is preserved and
this property is reflected in the trace identity

$\langle{T^{\mu}_{\ \
\mu}(x)}\rangle_{g}=g_{\mu\nu}\langle{T^{\mu\nu}(x)}\rangle_{g}=0,$ (6.4.1)

which generates, as we have already mentioned, a hierarchy of equations by
functional differentiation of this result respect to the background metric
$g$. Equivalently, the same equations can be derived from the condition of
Weyl invariance of the effective action.
Following the same procedure as for the conservation WIs, we may derive the
trace Ward identities for the four-point function 4T, in general $d$
dimensions, as

$\displaystyle\delta_{\mu_{1}\nu_{1}}\,\langle{T^{\mu_{1}\nu_{1}}(x_{1})T^{\mu_{2}\nu_{2}}(x_{2})T^{\mu_{3}\nu_{3}}(x_{3})T^{\mu_{4}\nu_{4}}(x_{4})}\rangle$
$\displaystyle=$
$\displaystyle=-2\Big{[}\delta_{x_{1}x_{2}}\langle{T^{\mu_{2}\nu_{2}}(x_{1})T^{\mu_{3}\nu_{3}}(x_{3})T^{\mu_{4}\nu_{4}}(x_{4})}\rangle+(23)+(24)\Big{]},$
(6.4.2)

that may be written in momentum space, after a Fourier transform, as

$\displaystyle\delta_{\mu_{1}\nu_{1}}\langle{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle=-2\langle{T^{\mu_{2}\nu_{2}}(p_{1}+p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle\hskip
56.9055pt-2\langle{T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{1}+p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle-2\langle{T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(p_{1}+\bar{p}_{4})}\rangle.$
(6.4.3)

We have omitted an overall $\delta$ function, having replaced $p_{4}$ with
$\bar{p}_{4}$.
In $d=4$ the equations need to be renormalized, by adding local covariant
counterterms which will be generated from the action (3.2.24).
If general covariance is respected by this procedure, the conservation WIs
remain valid for the renormalized effective action and for its variations.
This is reflected on the hierarchical structure of the equations, which remain
identical to the bare (naive) case.
Trace identities of the correlation functions involving at least three stress
energy tensor operators are instead affected by the anomaly, due to the
scaling violations induced by the regularization/renormalization procedure. In
$d=4$ the corresponding anomalous Ward identities for the trace can be
obtained by a functional variation of the equation (6.2.6) with respect to the
background metric.
In this case (6.4.2) is characterised by new contributions on its rhs, coming
from the anomaly $\mathcal{A}(x)$

$\displaystyle\delta_{\mu_{1}\nu_{1}}\langle{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle=-2\langle{T^{\mu_{2}\nu_{2}}(p_{1}+p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle\hskip
56.9055pt-2\langle{T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{1}+p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle-2\langle{T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(p_{1}+\bar{p}_{4})}\rangle$
$\displaystyle\hskip
56.9055pt+8\mathcal{A}^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{2},p_{3},\bar{p}_{4}),$
(6.4.4)

where the trace anomaly functional $\mathcal{A}$ is given in Eq. (7.10.10),
and the Fourier transform of its variation in the flat spacetime limit takes
the form

$\displaystyle\mathcal{A}^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{2},p_{3},\bar{p}_{4})\delta^{4}(P)=\int
d^{4}x_{1}d^{4}x_{2}d^{4}x_{3}d^{4}x_{4}\,\frac{\delta^{3}\mathcal{A}(x_{1})}{\delta
g_{\mu_{2}\nu_{2}}(x_{2})\delta g_{\mu_{3}\nu_{3}}(x_{3})\delta
g_{\mu_{4}\nu_{4}}(x_{4})}\Bigg{|}_{flat,d=4}e^{i(p_{1}x_{1}+\dots+p_{4}x_{4})}.$
(6.4.5)

In the following we will also be using the simpler general definition

$\mathcal{A}^{\mu_{1}\nu_{1}\dots\mu_{n}\nu_{n}}(p_{1},\dots,p_{n})\equiv\bigg{[}\sqrt{-g}\bigg{(}b\,C^{2}+b^{\prime}\,E\bigg{)}\bigg{]}^{\mu_{1}\nu_{1}\dots\mu_{n}\nu_{n}}(p_{1},\dots,p_{n})$
(6.4.6)

to denote the anomaly contributions to 3- and 4-point functions. Notice that
these are genuine 4-dimensional terms, left over by the procedure of
renormalization.

####  6.5 The anomalous CWIs using conformal Killing vectors

The expressions of the anomalous conformal WIs can be derived in an
alternative way following the formulation of [90], that here we are going to
extend to the 4-point function case.
The derivation of such identities relies uniquely on the effective action and
can be obtained as follows. We illustrate it first in the TT case, and then
move to the 4T.
We start from the conservation of the conformal current as derived in (3.1.23)

$\int
d^{d}x\sqrt{g}\,\nabla^{\alpha}\left(\epsilon_{\alpha}\frac{2}{\sqrt{g}}\frac{\delta\mathcal{S}}{\delta
g_{\mu\alpha}}\right)=\int
d^{4}x\sqrt{g}\,\nabla_{\mu}\langle{\epsilon_{\alpha}T^{\alpha\nu}}\rangle=0.$
(6.5.1)

In the TT case the derivation of the special CWIs is simplified, since there
is no trace anomaly if the counterterm action is defined as in (3.2.24), a
point that we will address in Section 6.13. We rely on the fact that the
conservation of the conformal current $J^{\mu}_{(K)}$ implies the conservation
equation

$\displaystyle
0=\int\,d^{d}x\,\sqrt{-g}\,\,\nabla_{\mu}\,\langle{J^{\mu}_{(K)}(x)\,T^{\mu_{1}\nu_{1}}(x_{1})}\rangle.$
(6.5.2)

By making explicit the expression $J^{\mu}(x)=K_{\nu}(x)\,T^{\mu\nu}(x)$, with
$\epsilon\to K$ in the flat limit, the previous relation takes the form

$\displaystyle
0=\int\,d^{d}x\,\bigg{(}\partial_{\mu}K_{\nu}\,\langle{T^{\mu\nu}(x)\,T^{\mu_{1}\nu_{1}}(x_{1})}\rangle+K_{\nu}\,\partial_{\mu}\,\langle{T^{\mu\nu}(x)\,T^{\mu_{1}\nu_{1}}(x_{1})}\rangle\bigg{)}.$
(6.5.3)

We recall that $K_{\nu}$ satisfies the conformal Killing equation in flat
space

$\displaystyle\partial_{\mu}K_{\nu}+\partial_{\nu}K_{\mu}=\frac{2}{d}\delta_{\mu\nu}\,\left(\partial\cdot
K\right),$ (6.5.4)

and by using this equation (7.7.52) can be re-written in the form

$\displaystyle
0=\int\,d^{d}x\,\bigg{(}K_{\nu}\partial_{\mu}\,\langle{T^{\mu\nu}(x)\,T^{\mu_{1}\nu_{1}}(x_{1})}\rangle+\frac{1}{d}\big{(}\partial\cdot
K\big{)}\,\langle{T(x)\,T^{\mu_{1}\nu_{1}}(x_{1})}\rangle\bigg{)}.$ (6.5.5)

We can use in this previous expression the conservation and trace Ward
identities for the two-point function $\langle{TT}\rangle$, that in the flat
spacetime limit are explicitly given by

$\displaystyle\partial_{\mu}\langle{T^{\mu\nu}(x)T^{\mu_{1}\nu_{1}}(x_{1})}\rangle$
$\displaystyle=\bigg{(}\delta^{(\mu_{1}}_{\mu}\delta^{\nu_{1})}_{\lambda}\partial^{\nu}\delta(x-x_{1})-2\delta^{\nu(\mu_{1}}\delta^{\nu_{1})}_{\mu}\partial_{\lambda}\delta(x-x_{1})\bigg{)}\langle{T^{\lambda\mu}(x)}\rangle,$
(6.5.6)
$\displaystyle\delta_{\mu\nu}\langle{T^{\mu\nu}(x)T^{\mu_{1}\nu_{1}}(x_{1})}\rangle$
$\displaystyle\equiv\langle{T(x)T^{\mu_{1}\nu_{1}}(x_{1})}\rangle=-2\delta(x-x_{1})\langle{T^{\mu_{1}\nu_{1}}(x)}\rangle$
(6.5.7)

and the explicit expression of the Killing vector $K^{(C)}_{\nu}$ for the
special conformal transformations

$\begin{split}K^{(C)\,\kappa}_{\mu}&=2x^{\kappa}\,x_{\mu}-x^{2}\delta^{\kappa}_{\mu}\\\
\partial\cdot K^{(C)\,\kappa}&=2d\,x^{\kappa}\end{split}$ (6.5.8)

where $\kappa=1,\dots,d$. By using (6.5.8) in the integral (6.5.5), we can
rewrite that expression as

$\displaystyle
0=\int\,d^{d}x\,\bigg{[}\big{(}2x^{\kappa}\,x_{\nu}-x^{2}\delta^{\kappa}_{\nu}\big{)}\partial_{\mu}\,\langle{T^{\mu\nu}(x)\,T^{\mu_{1}\nu_{1}}(x_{1})}\rangle+2\,x^{\kappa}\,\langle{T(x)\,T^{\mu_{1}\nu_{1}}(x_{1})}\rangle\bigg{]},$
(6.5.9)

A final integrating by parts finally gives the relations

$\displaystyle\left(2d\,x_{1}^{\kappa}+2x_{1}^{\kappa}\,x^{\mu}_{1}\frac{\partial}{\partial
x_{1}^{\mu}}+x_{1}^{2}\frac{\partial}{\partial
x_{1\kappa}}\right)\langle{T^{\mu_{1}\nu_{1}}(x_{1})}\rangle$
$\displaystyle\quad+2\bigg{(}x_{1\lambda}\,\delta^{\mu_{1}\kappa}-x_{1}^{\mu_{1}}\delta^{\kappa}_{\lambda}\bigg{)}\langle{T^{\lambda\nu_{1}}(x_{1})}\rangle+2\bigg{(}x_{1\lambda}\,\delta^{\nu_{1}\kappa}-x_{1}^{\nu_{1}}\delta^{\kappa}_{\lambda}\bigg{)}\langle{T^{\mu_{1}\lambda}(x_{1})}\rangle=0$
(6.5.10)

that are the special CWIs for the 1-point function
$\langle{T^{\mu_{1}\nu_{1}}(x_{1})}\rangle$.

####  6.6  4-point functions

The derivation above can be extended to n-point functions, starting from the
identity

$\int d^{d}x\sqrt{g}\nabla_{\alpha}(x)\langle
J^{\alpha}_{c}(x)T^{\mu_{1}\nu_{1}}(x_{1})\ldots
T^{\mu_{n}\nu_{n}}(x_{n})\rangle=0.$ (6.6.1)

We have used the conservation of the conformal current in d dimensions under
variations of the metric, induced by the conformal Killing vectors.
In absence of an anomaly, the conservation of the current $J^{\mu}_{c}$
follows from the conservation of the stress energy tensor plus the zero trace
condition. As in the example illustrated above, we consider (6.6.1) in the
flat limit

$\displaystyle\int
dx^{d}\,\partial_{\nu}\bigg{[}K_{\mu}(x)\langle{T^{\mu\nu}(x)T^{\mu_{1}\nu_{1}}(x_{1})\dots
T^{\mu_{4}\nu_{4}}(x_{4})}\rangle\bigg{]}=0,$ (6.6.2)

where we are assuming that the surface terms vanish, due to the fast fall-off
behaviour of the correlation function at infinity. Expanding (6.6.2) we obtain
an expression similar to (6.5.5)

$\displaystyle 0=\int
d^{d}x\left\\{K_{\mu}(x)\partial_{\nu}\langle{T^{\mu\nu}(x)T^{\mu_{1}\nu_{1}}(x_{1})\dots
T^{\mu_{4}\nu_{4}}(x_{4})}\rangle+\frac{1}{d}\big{(}\partial\cdot
K\big{)}\delta_{\mu\nu}\langle{T^{\mu\nu}(x)T^{\mu_{1}\nu_{1}}(x_{1})\dots
T^{\mu_{4}\nu_{4}}(x_{4})}\rangle\right\\}.$ (6.6.3)

Starting from this expression, the dilatation CWI is obtained by the choice of
the CKV characterising the dilatations

$K^{(D)}_{\mu}(x)=x_{\mu},\qquad\partial\cdot K^{(D)}=d$ (6.6.4)

and (6.6.3) becomes

$\displaystyle 0=\int
d^{d}x\bigg{\\{}x_{\mu}\,\partial_{\nu}\langle{T^{\mu\nu}(x)T^{\mu_{1}\nu_{1}}(x_{1})\dots
T^{\mu_{4}\nu_{4}}(x_{4})}\rangle+\delta_{\mu\nu}\langle{T^{\mu\nu}(x)T^{\mu_{1}\nu_{1}}(x_{1})\dots
T^{\mu_{4}\nu_{4}}(x_{4})}\rangle\bigg{\\}}.$ (6.6.5)

At this stage, we use the conservation and trace Ward identities in $d=4$ for
the $4$-point function written as

$\displaystyle\partial_{\nu}\langle{T^{\mu\nu}(x)T^{\mu_{1}\nu_{1}}(x_{1})\dots
T^{\mu_{4}\nu_{4}}(x_{4})}\rangle=$ $\displaystyle=$
$\displaystyle-8\bigg{\\{}\left[\Gamma^{\mu}_{\nu\lambda}(x)\right]^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(x_{1},x_{2},x_{3})\langle{T^{\lambda\nu}(x)T^{\mu_{4}\nu_{4}}(x_{4})}\rangle+(14)+(24)+(34)\bigg{\\}}$
$\displaystyle-4\bigg{\\{}\left[\Gamma^{\mu}_{\nu\lambda}(x)\right]^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(x_{1},x_{2})\langle{T^{\lambda\nu}(x)T^{\mu_{3}\nu_{3}}(x_{3})T^{\mu_{4}\nu_{4}}(x_{4})}\rangle+(13)+(23)+(14)+(24)+(34)\bigg{\\}}$
$\displaystyle-2\bigg{\\{}\left[\Gamma^{\mu}_{\nu\lambda}(x)\right]^{\mu_{1}\nu_{1}}(x_{1})\langle{T^{\lambda\nu}(x)T^{\mu_{2}\nu_{2}}(x_{2})T^{\mu_{3}\nu_{3}}(x_{3})T^{\mu_{4}\nu_{4}}(x_{4})}\rangle+(12)+(13)+(14)\bigg{\\}}$
(6.6.6)

and

$\displaystyle\delta_{\mu\nu}\langle{T^{\mu\nu}(x)T^{\mu_{1}\nu_{1}}(x_{1})\dots
T^{\mu_{4}\nu_{4}}(x_{4})}\rangle$
$\displaystyle=-2\bigg{\\{}\delta_{xx_{1}}\langle{T^{\mu_{1}\nu_{1}}(x)T^{\mu_{2}\nu_{2}}(x_{2})\dots
T^{\mu_{4}\nu_{4}}(x_{4})}\rangle+(12)+(13)+(14)\bigg{\\}}$
$\displaystyle\hskip
56.9055pt+2^{4}\big{[}\mathcal{A}(x)\big{]}^{\mu_{1}\nu_{1}\dots\mu_{4}\nu_{4}}(x_{1},\dots,x_{4}).$
(6.6.7)

to finally derive the dilatation WI from (6.6.5) in the form

$\displaystyle\left(4d+\sum_{j=1}^{4}\,x_{j}^{\alpha}\frac{\partial}{\partial
x_{j}^{\alpha}}\right)\langle{T^{\mu_{1}\nu_{1}}(x_{1})\dots
T^{\mu_{4}\nu_{4}}(x_{4})}\rangle=2^{4}\int
dx\big{[}\mathcal{A}(x)\big{]}^{\mu_{1}\nu_{1}\dots\mu_{4}\nu_{4}}(x_{1},\dots,x_{4})$
(6.6.8)

where $d=4$. It is worth mentioning that (6.6.8) is valid in any even
spacetime dimension if we take into account the particular structure of the
trace anomaly in that particular dimension.

The special CWIs correspond to the $d$ special conformal Killing vectors in
flat space given in (6.5.8), as in the $TT$ case. Also in this case we derive
the identity

$\displaystyle 0=$ $\displaystyle\int
d^{d}x\bigg{\\{}\big{(}2x^{\kappa}\,x_{\mu}-x^{2}\delta^{\kappa}_{\mu}\big{)}\,\partial_{\nu}\langle{T^{\mu\nu}(x)T^{\mu_{1}\nu_{1}}(x_{1})\dots
T^{\mu_{4}\nu_{4}}(x_{4})}\rangle$ $\displaystyle\hskip
113.81102pt+2x^{\kappa}\delta_{\mu\nu}\langle{T^{\mu\nu}(x)T^{\mu_{1}\nu_{1}}(x_{1})\dots
T^{\mu_{4}\nu_{4}}(x_{4})}\rangle\bigg{\\}}.$ (6.6.9)

By using the relations (6.6.6) and (6.6.7) and performing the integration over
$x$ explicitly in the equation above, the anomalous special CWIs for the
$4$-point function take the form

$\displaystyle\sum_{j=1}^{4}\left[2x_{j}^{\kappa}\left(d+x_{j}^{\alpha}\frac{\partial}{\partial
x_{j}^{\alpha}}\right)-x_{j}^{2}\,\delta^{\kappa\alpha}\frac{\partial}{\partial
x_{j}^{\alpha}}\right]\langle{T^{\mu_{1}\nu_{1}}(x_{1})\dots
T^{\mu_{4}\nu_{4}}(x_{4})}\rangle$
$\displaystyle+2\sum_{j=1}^{4}\left(\delta^{\kappa\mu_{j}}x_{j\,\alpha}-\delta^{\kappa}_{\alpha}x_{j}^{\mu_{j}}\right)\langle{T^{\mu_{1}\nu_{1}}(x_{1})\dots
T^{\nu_{j}\alpha}(x_{j})\dots T^{\mu_{4}\nu_{4}}(x_{4})}\rangle$
$\displaystyle+2\sum_{j=1}^{4}\left(\delta^{\kappa\nu_{j}}x_{j\,\alpha}-\delta^{\kappa}_{\alpha}x_{j}^{\nu_{j}}\right)\langle{T^{\mu_{1}\nu_{1}}(x_{1})\dots
T^{\mu_{j}\alpha}(x_{j})\dots T^{\mu_{4}\nu_{4}}(x_{4})}\rangle=$
$\displaystyle=2^{5}\,\int
dx\,x^{\kappa}\big{[}\mathcal{A}(x)\big{]}^{\mu_{1}\nu_{1}\dots\mu_{4}\nu_{4}}(x_{1},\dots,x_{4}),$
(6.6.10)

where the presence of the anomaly term comes from the inclusion of the trace
WI, exactly as in the TT case.
At this stage, these equations can be transformed to momentum space, giving
the final expressions of the CWIs in the form

$\displaystyle\left(d-\sum_{j=1}^{3}\,p_{j}^{\alpha}\frac{\partial}{\partial
p_{j}^{\alpha}}\right)\langle{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle=2^{4}\,\mathcal{A}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{1},p_{2},p_{3},\bar{p}_{4})$
(6.6.11)

for the dilatation, and

$\displaystyle\sum_{j=1}^{3}\left(p_{j}^{\kappa}\frac{\partial^{2}}{\partial
p_{j}^{\alpha}\partial p_{j\alpha}}-2p_{j}^{\alpha}\frac{\partial}{\partial
p_{j}^{\alpha}\partial
p_{j\kappa}}\right)\langle{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle+2\sum_{j=1}^{3}\left(\delta^{\kappa\mu_{j}}\frac{\partial}{\partial
p_{j\,\alpha}}-\delta^{\kappa}_{\alpha}\frac{\partial}{\partial
p_{j}^{\mu_{j}}}\right)\langle{T^{\mu_{1}\nu_{1}}(p_{1})\dots
T^{\nu_{j}\alpha}(p_{j})\dots T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle+2\sum_{j=1}^{3}\left(\delta^{\kappa\nu_{j}}\frac{\partial}{\partial
p_{j\,\alpha}}-\delta^{\kappa}_{\alpha}\frac{\partial}{\partial
p_{j}^{\nu_{j}}}\right)\langle{T^{\mu_{1}\nu_{1}}(p_{1})\dots
T^{\mu_{j}\alpha}(p_{j})\dots T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle=-2^{5}\,\left[\frac{\partial}{\partial
p_{4\kappa}}\,\mathcal{A}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}(p_{1},p_{2},p_{3},p_{4})\right]_{\bar{p_{4}}=-p_{1}-p_{2}-p_{3}},$
(6.6.12)

for the special CWIs, having used the definition (6.4.6).
When this procedure is applied to the $n$-point function, one finds that the
anomalous CWIs are written as

$\displaystyle\left(d-\sum_{j=1}^{n-1}\,p_{j}^{\alpha}\frac{\partial}{\partial
p_{j}^{\alpha}}\right)\langle{T^{\mu_{1}\nu_{1}}(p_{1})\dots
T^{\mu_{n}\nu_{n}}(\bar{p}_{n})}\rangle=2^{n}\,\mathcal{A}^{\mu_{1}\nu_{1}\dots\mu_{n}\nu_{n}}(p_{1},\dots,\bar{p}_{n})$
(6.6.13)

for the dilatation and

$\displaystyle\sum_{j=1}^{n-1}\left(p_{j}^{\kappa}\frac{\partial^{2}}{\partial
p_{j}^{\alpha}\partial p_{j\alpha}}-2p_{j}^{\alpha}\frac{\partial}{\partial
p_{j}^{\alpha}\partial
p_{j\kappa}}\right)\langle{T^{\mu_{1}\nu_{1}}(p_{1})\dots
T^{\mu_{n}\nu_{n}}(\bar{p}_{n})}\rangle$
$\displaystyle+2\sum_{j=1}^{n-1}\left(\delta^{\kappa\mu_{j}}\frac{\partial}{\partial
p_{j\,\alpha}}-\delta^{\kappa}_{\alpha}\frac{\partial}{\partial
p_{j}^{\mu_{j}}}\right)\langle{T^{\mu_{1}\nu_{1}}(p_{1})\dots
T^{\nu_{j}\alpha}(p_{j})\dots T^{\mu_{n}\nu_{n}}(\bar{p}_{n})}\rangle$
$\displaystyle+2\sum_{j=1}^{n-1}\left(\delta^{\kappa\nu_{j}}\frac{\partial}{\partial
p_{j\,\alpha}}-\delta^{\kappa}_{\alpha}\frac{\partial}{\partial
p_{j}^{\nu_{j}}}\right)\ \langle T^{\mu_{1}\nu_{1}}(p_{1})\dots
T^{\mu_{j}\alpha}(p_{j})\dots T^{\mu_{n}\nu_{n}}(\bar{p}_{n})\rangle$
$\displaystyle=-2^{n+1}\left[\frac{\partial}{\partial
p_{n\kappa}}\,\mathcal{A}^{\mu_{1}\nu_{1}\dots\mu_{n}\nu_{n}}(p_{1},\dots,p_{n})\right]_{p_{n}=\bar{p}_{n}},$
(6.6.14)

for the special conformal Ward identities, where
$\bar{p}_{n}=-\sum_{i=1}^{n-1}p_{i}$ and we have used the definition (6.4.6).

####  6.7 Decomposition of the 4T

The analysis in momentum space allows to identify the contributions generated
by the breaking of the conformal symmetry, after renormalization, in a direct
manner. For this purpose we will be using the longitudinal transverse L/T
decomposition of the correlator presented in [21] for 3-point functions,
extending it to the 4T. This procedure has been investigated in detail for
3-point functions in [24] in the context of a perturbative approach [59]. The
perturbative analysis in free field theory shows how renormalization acts on
the two L/T subspaces, forcing the emergence of a trace in the longitudinal
sector.
Due to the constraint imposed by conformal symmetry (i.e. their CWIs), the
correlation functions can be decomposed into a transverse-traceless and a
semilocal part. The term semilocal refers to contributions which are obtained
from the conservation and trace Ward identities. Of an external off- shell
graviton only its spin-2 component will couple to transverse-traceless part.
In general, by decomposing the gravitational fluctuations into their
transverse-traceless and spin-1 and spin-0 components one finds an interesting
separation of the anomaly effective action which can be useful also in a
phenomenological context. We will address this point in a forthcoming paper.
The split of the energy momentum operator in terms of a transverse traceless
($tt$) part and of a longitudinal (local) part [21] is defined in the form

$T^{\mu_{i}\nu_{i}}(p_{i})\equiv
t^{\mu_{i}\nu_{i}}(p_{i})+t_{loc}^{\mu_{i}\nu_{i}}(p_{i})$ (6.7.1)

with

$\displaystyle t^{\mu_{i}\nu_{i}}(p_{i})$
$\displaystyle=\Pi^{\mu_{i}\nu_{i}}_{\alpha_{i}\beta_{i}}(p)\,T^{\alpha_{i}\beta_{i}}(p_{i})$
(6.7.2) $\displaystyle t_{loc}^{\mu_{i}\nu_{i}}(p_{i})$
$\displaystyle=\Sigma^{\mu_{i}\nu_{i}}_{\alpha_{i}\beta_{i}}(p)\,T^{\alpha_{i}\beta_{i}}(p_{i}).$
(6.7.3)

We have introduced the transverse-traceless ($\Pi$), transverse-trace $(\tau)$
and longitudinal ($\mathcal{I}$) projectors, given respectively by

$\displaystyle\pi^{\mu}_{\alpha}$
$\displaystyle=\delta^{\mu}_{\alpha}-\frac{p^{\mu}p_{\alpha}}{p^{2}},\qquad\tilde{\pi}^{\mu}_{\alpha}=\frac{1}{d-1}\pi^{\mu}_{\alpha}$
(6.7.4) $\displaystyle\ \Pi^{\mu\nu}_{\alpha\beta}$
$\displaystyle=\frac{1}{2}\left(\pi^{\mu}_{\alpha}\pi^{\nu}_{\beta}+\pi^{\mu}_{\beta}\pi^{\nu}_{\alpha}\right)-\frac{1}{d-1}\pi^{\mu\nu}\pi_{\alpha\beta},$
(6.7.5) $\displaystyle\mathcal{J}^{\mu\nu}_{\alpha\beta}$
$\displaystyle=\frac{1}{p^{2}}p_{\beta}\left(p^{\mu}\delta^{\nu}_{\alpha}+p^{\nu}\delta^{\mu}_{\alpha}-\frac{p_{\alpha}}{d-1}(\delta^{\mu\nu}+(d-2)\frac{p^{\mu}p^{\nu}}{p^{2}})\right)$
(6.7.6) $\displaystyle\mathcal{I}^{\mu\nu}_{\alpha\beta}$
$\displaystyle=\frac{1}{2}\left(\mathcal{J}^{\mu\nu}_{\alpha\beta}+\mathcal{J}^{\mu\nu}_{\beta\alpha}\right)\qquad\tau^{\mu\nu}_{\alpha\beta}=\tilde{\pi}^{\mu\nu}\delta_{\alpha\beta}$
(6.7.7) $\displaystyle\mathcal{I}^{\mu\nu}_{\alpha}$
$\displaystyle=\frac{1}{p^{2}}\left(p^{\mu}\delta^{\nu}_{\alpha}+p^{\nu}\delta^{\mu}_{\alpha}-\frac{p_{\alpha}}{d-1}(\delta^{\mu\nu}+(d-2)\frac{p^{\mu}p^{\nu}}{p^{2}}\right)$
(6.7.8) $\displaystyle\mathcal{I}^{\mu\nu}_{\alpha\beta}$
$\displaystyle=\frac{1}{2}\left(p_{\beta}\mathcal{I}^{\mu\nu}_{\alpha}+p_{\alpha}\mathcal{I}^{\mu\nu}_{\beta}\right)$
(6.7.9)

with

$\displaystyle\delta^{(\mu}_{\alpha}\delta^{\nu)}_{\beta}$
$\displaystyle=\Pi^{\mu\nu}_{\alpha\beta}+\Sigma^{\mu\nu}_{\alpha\beta}$
(6.7.10) $\displaystyle\Sigma^{\mu_{i}\nu_{i}}_{\alpha_{i}\beta_{i}}$
$\displaystyle\equiv\mathcal{I}^{\mu_{i}\nu_{i}}_{\alpha_{i}\beta_{i}}+\tau^{\mu_{i}\nu_{i}}_{\alpha_{i}\beta_{i}}$
$\displaystyle=\frac{1}{p_{i}^{2}}\Big{[}2\delta^{(\nu_{i}}_{(\alpha_{i}}p_{i}^{\mu_{i})}p_{i\,\beta_{i})}-\frac{p_{i\alpha_{i}}p_{i\beta_{i}}}{(d-1)}\left(\delta^{\mu_{i}\nu_{i}}+(d-2)\frac{p_{i}^{\mu_{i}}p_{i}^{\nu_{i}}}{p_{i}^{2}}\right)\Big{]}+\frac{1}{(d-1)}\pi^{\mu_{i}\nu_{i}}(p_{i})\delta_{\alpha_{i}\beta_{i}}.$
(6.7.11)

Notice that we have combined together the operators $\mathcal{I}$ and $\tau$
into a projector $\Sigma$ which defines the local components of a given tensor
$T$, according to (6.7.2), which are proportional both to a given momentum $p$
(the longitudinal contribution) and to the trace parts. Both $\Pi$ and $\tau$
are transverse by construction, while $\mathcal{I}$ is longitudinal and of
zero trace.

The projectors induce a decomposition respect to a specific momentum $p_{i}$.
By using (6.7.1), the entire correlator is written as

$\displaystyle\langle{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle=$
$\displaystyle=\langle{t^{\mu_{1}\nu_{1}}(p_{1})t^{\mu_{2}\nu_{2}}(p_{2})t^{\mu_{3}\nu_{3}}(p_{3})t^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle+\langle{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle_{loc}$
(6.7.12)

where the first contribution is the transverse-traceless part which satisfies
by construction the conditions

$\begin{split}p_{i\,\mu_{i}}\langle{t^{\mu_{1}\nu_{1}}(p_{1})t^{\mu_{2}\nu_{2}}(p_{2})t^{\mu_{3}\nu_{3}}(p_{3})t^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle&=0,\qquad
i=1,2,3,4\,,\\\
\delta_{\mu_{i}\nu_{i}}\langle{t^{\mu_{1}\nu_{1}}(p_{1})t^{\mu_{2}\nu_{2}}(p_{2})t^{\mu_{3}\nu_{3}}(p_{3})t^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle&=0,\qquad
i=1,2,3,4.\end{split}$ (6.7.13)

It is clear now that only the second term in (6.7.12) contributes entirely to
the conservation WIs. Thus, the proper new information on the form factors of
the 4-point function is entirely encoded in its transverse-traceless ($tt$)
part, since the remaining longitudinal + trace contributions, corresponding to
the local term, are related to lower point functions.

####  6.8 Projecting the Conformal Ward Identities

The action of $D$ and $\mathcal{K}$ on the 4T in (6.3.22) and (6.3.23), after
the projection on the transverse traceless component, simplifies. We start by
considering at the dilatation operator $D$ which has the property of leaving
unchanged the two subspaces identified by the $\Pi$ and $\Sigma$ projectors.
These properties can be summarized by the relations

$\begin{split}&\Sigma^{\mu_{i}\nu_{i}}_{\rho_{i}\sigma_{i}}(p_{i})\,D\,\langle{t^{\mu_{1}\nu_{1}}(p_{1})t^{\mu_{2}\nu_{2}}(p_{2})t^{\mu_{3}\nu_{3}}(p_{3})t^{\mu_{4}\nu_{4}}(p_{4})}\rangle=0,\\\
&\Pi^{\mu_{i}\nu_{i}}_{\rho_{i}\sigma_{i}}(p_{i})\,D\,\langle{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(p_{4})}\rangle_{loc}=0,\end{split}\qquad\qquad
i=1,\dots,4.$ (6.8.1)

having used the properties of orthogonality and idempotence of the projectors.
For this reason, if one wants to project the dilatation equation (6.3.22), by
using the transverse-traceless and longitudinal projectors, there are only two
ways of doing it. These are the cases where we have either four $\Pi$’s or
four $\Sigma$’s, due to the relations in (6.8.1). Therefore, the only relevant
projected dilatation WIs are

$\displaystyle\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(p_{1})\Pi^{\mu_{2}\nu_{2}}_{\alpha_{2}\beta_{2}}(p_{2})\Pi^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}}(p_{3})\Pi^{\mu_{4}\nu_{4}}_{\alpha_{4}\beta_{4}}(\bar{p}_{4})\,D\,\langle{T^{\alpha_{1}\beta_{1}}(p_{1})T^{\alpha_{2}\beta_{2}}(p_{2})T^{\alpha_{3}\beta_{3}}(p_{3})T^{\alpha_{4}\beta_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle=0$ (6.8.2)
$\displaystyle\Sigma^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(p_{1})\Sigma^{\mu_{2}\nu_{2}}_{\alpha_{2}\beta_{2}}(p_{2})\Sigma^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}}(p_{3})\Sigma^{\mu_{4}\nu_{4}}_{\alpha_{4}\beta_{4}}(\bar{p}_{4})\,D\,\langle{T^{\alpha_{1}\beta_{1}}(p_{1})T^{\alpha_{2}\beta_{2}}(p_{2})T^{\alpha_{3}\beta_{3}}(p_{3})T^{\alpha_{4}\beta_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle=0$ (6.8.3)

that can be simplified as

$\displaystyle\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(p_{1})\Pi^{\mu_{2}\nu_{2}}_{\alpha_{2}\beta_{2}}(p_{2})\Pi^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}}(p_{3})\Pi^{\mu_{4}\nu_{4}}_{\alpha_{4}\beta_{4}}(\bar{p}_{4})\,D\,\langle{t^{\alpha_{1}\beta_{1}}(p_{1})t^{\alpha_{2}\beta_{2}}(p_{2})t^{\alpha_{3}\beta_{3}}(p_{3})t^{\alpha_{4}\beta_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle=0,$ (6.8.4)
$\displaystyle\Sigma^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(p_{1})\Sigma^{\mu_{2}\nu_{2}}_{\alpha_{2}\beta_{2}}(p_{2})\Sigma^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}}(p_{3})\Sigma^{\mu_{4}\nu_{4}}_{\alpha_{4}\beta_{4}}(\bar{p}_{4})\,D\,\langle{T^{\alpha_{1}\beta_{1}}(p_{1})T^{\alpha_{2}\beta_{2}}(p_{2})T^{\alpha_{3}\beta_{3}}(p_{3})T^{\alpha_{4}\beta_{4}}(\bar{p}_{4})}\rangle_{loc}$
$\displaystyle=0,$ (6.8.5)

once we insert the decomposition of the 4T and use Eqs. (6.8.1). It is worth
mentioning that (6.8.5) does not impose any additional constraints on the
$4$-point function. This because the longitudinal part of the correlator is
explicitly given in terms of lower point functions, and (6.8.5) is related to
the dilatation WIs of the $3$\- and $2$-point functions. The constraints on
the 4-point function will be derived from (6.8.4), which is related to the
transverse traceless part of the correlator.

Turning our attention towards the special CWIs, we observe that the action of
the $\mathcal{K}^{\kappa}$ operator on the transverse-traceless part gives a
result that it is still transverse and traceless

$\begin{split}&p_{i\mu_{i}}\,\mathcal{K}^{\kappa}\langle{t^{\mu_{1}\nu_{1}}(p_{1})t^{\mu_{2}\nu_{2}}(p_{2})t^{\mu_{3}\nu_{3}}(p_{3})t^{\mu_{4}\nu_{4}}(p_{4})}\rangle=0,\\\
&\delta_{\mu_{i}\nu_{i}}\,\mathcal{K}^{\kappa}\langle{t^{\mu_{1}\nu_{1}}(p_{1})t^{\mu_{2}\nu_{2}}(p_{2})t^{\mu_{3}\nu_{3}}(p_{3})t^{\mu_{4}\nu_{4}}(p_{4})}\rangle=0\end{split}\qquad
i=1,\dots,4$ (6.8.6)

This can be shown exactly as in the case of the 3T discussed in previous
works. This property of $\mathcal{K}^{\kappa}$ allows us to identify the
relevant subspace where the special CWIs act.
As already pointed out, the transverse-traceless part of the 4T is the only
part of the correlator which is really unknown. Indeed, the longitudinal
components can be expressed in terms of 2- and 3-point functions, by using the
conservation and trace WIs. Therefore, the special CWIs will be constraining
that unknown part, which can be parametrized in terms of a certain number of
independent for factors, as in the 3T case.
By using the properties of the projectors $\Sigma$ and $\Pi$, from (6.8.6) one
derives the relation

$\Sigma^{\rho_{i}\sigma_{i}}_{\mu_{i}\nu_{i}}(p_{i})K^{\kappa}\langle{t^{\mu_{1}\nu_{1}}(p_{1})t^{\mu_{2}\nu_{2}}(p_{2})t^{\mu_{3}\nu_{3}}(p_{3})t^{\mu_{4}\nu_{4}}(p_{4})}\rangle=0,\qquad
i=1,\dots,4,$ (6.8.7)

since the action of $K^{\kappa}$, as just mentioned, is endomorphic on the
transverse traceless sector. With this result in mind, using the projectors
$\Pi$ and $\Sigma$ we project (6.3.23) into all the possible subspaces and
observe that when at least one $\Sigma$ is present, the equations reduce to
the form

$\displaystyle 0$
$\displaystyle=\Sigma^{\rho_{i}\sigma_{i}}_{\mu_{i}\nu_{i}}(p_{i})\mathcal{K}^{\kappa}\langle{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(p_{4})}\rangle$
$\displaystyle=\Sigma^{\rho_{i}\sigma_{i}}_{\mu_{i}\nu_{i}}(p_{i})\mathcal{K}^{\kappa}\bigg{[}\langle{t^{\mu_{1}\nu_{1}}(p_{1})t^{\mu_{2}\nu_{2}}(p_{2})t^{\mu_{3}\nu_{3}}(p_{3})t^{\mu_{4}\nu_{4}}(p_{4})}\rangle+\langle{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(p_{4})}\rangle_{loc}\bigg{]}$
$\displaystyle=\Sigma^{\rho_{i}\sigma_{i}}_{\mu_{i}\nu_{i}}(p_{i})\mathcal{K}^{\kappa}\bigg{(}\langle{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(p_{4})}\rangle_{loc}\bigg{)},\qquad
i=1,\dots,4.$ (6.8.8)

The equations that result

$\Sigma^{\rho_{i}\sigma_{i}}_{\mu_{i}\nu_{i}}(p_{i})\mathcal{K}^{\kappa}\bigg{(}\langle{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(p_{4})}\rangle_{loc}\bigg{)}=0,\qquad
i=1,\dots,4$ (6.8.9)

will involve only 3- and 2-point functions, via the canonical Ward identities.
For this reason, being the structure of the equations hierarchical, if we have
already solved for the correlators of lower orders, no new constraint will be
induced by (6.8.9).
The only significant constraint will be derived when acting with 4 $\Pi$
projectors. For this reason we are interested in studying the equation

$\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(p_{1})\Pi^{\mu_{2}\nu_{2}}_{\alpha_{2}\beta_{2}}(p_{2})\Pi^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}}(p_{3})\Pi^{\mu_{4}\nu_{4}}_{\alpha_{4}\beta_{4}}(\bar{p}_{4})\,\mathcal{K}^{\kappa}\,\langle{T^{\alpha_{1}\beta_{1}}(p_{1})T^{\alpha_{2}\beta_{2}}(p_{2})T^{\alpha_{3}\beta_{3}}(p_{3})T^{\alpha_{4}\beta_{4}}(\bar{p}_{4})}\rangle=0.$
(6.8.10)

If we insert the decomposition (6.7.12) into the equation above, one can prove
that terms containing two or more $t_{loc}$ operators will vanish when
projected on the transverse traceless component, for instance

$\displaystyle\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(p_{1})\Pi^{\mu_{2}\nu_{2}}_{\alpha_{2}\beta_{2}}(p_{2})\Pi^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}}(p_{3})\Pi^{\mu_{4}\nu_{4}}_{\alpha_{4}\beta_{4}}(\bar{p}_{4})\,\mathcal{K}^{\kappa}\,\langle{t^{\alpha_{1}\beta_{1}}_{loc}(p_{1})t^{\alpha_{2}\beta_{2}}_{loc}(p_{2})T^{\alpha_{3}\beta_{3}}(p_{3})T^{\alpha_{4}\beta_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle=0,$ (6.8.11)

where the definitions (6.7.5) and (6.7.11) have been used.

Proceeding with the reconstruction program, we need the action of the special
conformal transformations on the correlators with a single $t_{loc}$, after
projecting on the transverse traceless sector. After a lengthy but
straightforward calculations we obtain

$\displaystyle\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(p_{1})\Pi^{\mu_{2}\nu_{2}}_{\alpha_{2}\beta_{2}}(p_{2})\Pi^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}}(p_{3})\Pi^{\mu_{4}\nu_{4}}_{\alpha_{4}\beta_{4}}(\bar{p}_{4})\,\,\mathcal{K}^{\kappa}\langle{T^{\alpha_{1}\beta_{1}}(p_{1})T^{\alpha_{2}\beta_{2}}(p_{2})T^{\alpha_{3}\beta_{3}}(p_{3})t_{loc}^{\alpha_{4}\beta_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle=4\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(p_{1})\Pi^{\mu_{2}\nu_{2}}_{\alpha_{2}\beta_{2}}(p_{2})\Pi^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}}(p_{3})\Pi^{\mu_{4}\nu_{4}}_{\alpha_{4}\beta_{4}}(\bar{p}_{4})$
$\displaystyle\bigg{\\{}\bigg{[}(d-1)\delta^{\kappa\alpha_{4}}+L_{1}^{\kappa\alpha_{4}}+L_{2}^{\kappa\alpha_{4}}+L_{3}^{\kappa\alpha_{4}}\bigg{]}\left(\frac{p_{4\rho_{4}}}{p_{4}^{2}}\langle{T^{\alpha_{1}\beta_{1}}(p_{1})T^{\alpha_{2}\beta_{2}}(p_{2})T^{\alpha_{3}\beta_{3}}(p_{3})T^{\rho_{4}\beta_{4}}(\bar{p}_{4})}\rangle\right)$
$\displaystyle\hskip
28.45274pt+\left(S^{\kappa\alpha_{4}}\right)^{\alpha_{1}}_{\hskip
6.02777pt\rho_{1}}\left(\frac{p_{4\rho_{4}}}{p_{4}^{2}}\langle{T^{\rho_{1}\beta_{1}}(p_{1})T^{\alpha_{2}\beta_{2}}(p_{2})T^{\alpha_{3}\beta_{3}}(p_{3})T^{\rho_{4}\beta_{4}}(\bar{p}_{4})}\rangle\right)$
$\displaystyle\hskip
28.45274pt+\left(S^{\kappa\alpha_{4}}\right)^{\alpha_{2}}_{\hskip
6.02777pt\rho_{2}}\left(\frac{p_{4\rho_{4}}}{p_{4}^{2}}\langle{T^{\alpha_{1}\beta_{1}}(p_{1})T^{\rho_{2}\beta_{2}}(p_{2})T^{\alpha_{3}\beta_{3}}(p_{3})T^{\rho_{4}\beta_{4}}(\bar{p}_{4})}\rangle\right)$
$\displaystyle\hskip
28.45274pt+\left(S^{\kappa\alpha_{4}}\right)^{\alpha_{3}}_{\hskip
6.02777pt\rho_{3}}\left(\frac{p_{4\rho_{4}}}{p_{4}^{2}}\langle{T^{\alpha_{1}\beta_{1}}(p_{1})T^{\alpha_{2}\beta_{2}}(p_{2})T^{\rho_{3}\beta_{3}}(p_{3})T^{\rho_{4}\beta_{4}}(\bar{p}_{4})}\rangle\right)\bigg{\\}},$
(6.8.12)

where

$\displaystyle L_{i}^{\kappa\alpha_{4}}\equiv
p_{i}^{\alpha_{4}}\frac{\partial}{\partial
p_{i\,\kappa}}-p_{i}^{\kappa}\frac{\partial}{\partial
p_{i\,\alpha_{4}}},\qquad i=1,2,3$ (6.8.13)

are the $SO(4)$ (Lorentz) generators and $S_{\mu\nu}$ is the spin part, for
which

$\displaystyle\big{(}S^{\mu\nu}\big{)}^{\rho\sigma}=\delta^{\mu\rho}\delta^{\nu\sigma}-\delta^{\mu\sigma}\delta^{\nu\rho}.$
(6.8.14)

By using the Lorentz Ward identities we obtain the expression

$\displaystyle\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(p_{1})\Pi^{\mu_{2}\nu_{2}}_{\alpha_{2}\beta_{2}}(p_{2})\Pi^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}}(p_{3})\Pi^{\mu_{4}\nu_{4}}_{\alpha_{4}\beta_{4}}(\bar{p}_{4})\,\,\mathcal{K}^{\kappa}\langle{T^{\alpha_{1}\beta_{1}}(p_{1})T^{\alpha_{2}\beta_{2}}(p_{2})T^{\alpha_{3}\beta_{3}}(p_{3})t_{loc}^{\alpha_{4}\beta_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle=\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(p_{1})\Pi^{\mu_{2}\nu_{2}}_{\alpha_{2}\beta_{2}}(p_{2})\Pi^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}}(p_{3})\Pi^{\mu_{4}\nu_{4}}_{\alpha_{4}\beta_{4}}(\bar{p}_{4})\left(\frac{4\,d}{p_{4}^{2}}\,\delta^{\alpha_{4}\kappa}\,p_{4\rho_{4}}\langle{T^{\alpha_{1}\beta_{1}}(p_{1})T^{\alpha_{2}\beta_{2}}(p_{2})T^{\alpha_{3}\beta_{3}}(p_{3})T^{\rho_{4}\beta_{4}}(\bar{p}_{4})}\rangle\right).$
(6.8.15)

Analogous results are obtained for the other terms involving one $t_{loc}$
operator.
In summary, when we project (6.3.23) on the transverse traceless components we
find

$\displaystyle 0=$
$\displaystyle\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(p_{1})\Pi^{\mu_{2}\nu_{2}}_{\alpha_{2}\beta_{2}}(p_{2})\Pi^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}}(p_{3})\Pi^{\mu_{4}\nu_{4}}_{\alpha_{4}\beta_{4}}(\bar{p}_{4})\,\,\mathcal{K}^{\kappa}\langle{T^{\alpha_{1}\beta_{1}}(p_{1})T^{\alpha_{2}\beta_{2}}(p_{2})T^{\alpha_{3}\beta_{3}}(p_{3})T^{\alpha_{4}\beta_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle=\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(p_{1})\Pi^{\mu_{2}\nu_{2}}_{\alpha_{2}\beta_{2}}(p_{2})\Pi^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}}(p_{3})\Pi^{\mu_{4}\nu_{4}}_{\alpha_{4}\beta_{4}}(\bar{p}_{4})\Bigg{[}\mathcal{K}^{\kappa}\langle{t^{\alpha_{1}\beta_{1}}(p_{1})t^{\alpha_{2}\beta_{2}}(p_{2})t^{\alpha_{3}\beta_{3}}(p_{3})t^{\alpha_{4}\beta_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle\hskip
56.9055pt+\frac{4\,d}{p_{1}^{2}}\,\delta^{\alpha_{1}\kappa}\,p_{1\rho_{1}}\langle{T^{\rho_{1}\beta_{1}}(p_{1})T^{\alpha_{2}\beta_{2}}(p_{2})T^{\alpha_{3}\beta_{3}}(p_{3})T^{\alpha_{4}\beta_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle\hskip
56.9055pt+\frac{4\,d}{p_{2}^{2}}\,\delta^{\alpha_{2}\kappa}\,p_{2\rho_{2}}\langle{T^{\alpha_{1}\beta_{1}}(p_{1})T^{\rho_{2}\beta_{2}}(p_{2})T^{\alpha_{3}\beta_{3}}(p_{3})T^{\alpha_{4}\beta_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle\hskip
56.9055pt+\frac{4\,d}{p_{3}^{2}}\,\delta^{\alpha_{3}\kappa}\,p_{3\rho_{3}}\langle{T^{\alpha_{1}\beta_{1}}(p_{1})T^{\alpha_{2}\beta_{2}}(p_{2})T^{\rho_{3}\beta_{3}}(p_{3})T^{\alpha_{4}\beta_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle\hskip
56.9055pt+\frac{4\,d}{p_{4}^{2}}\,\delta^{\alpha_{4}\kappa}\,p_{4\rho_{4}}\langle
T^{\alpha_{1}\beta_{1}}(p_{1})T^{\alpha_{2}\beta_{2}}(p_{2})T^{\alpha_{3}\beta_{3}}(p_{3})T^{\rho_{4}\beta_{4}}(\bar{p}_{4})\rangle\Bigg{]}.$
(6.8.16)

It is worth mentioning that the last four terms in the previous equations are
completely expressible in terms of lower point functions via the longitudinal
WIs. Then, (6.8.16) imposes some constraints on the transverse traceless part
of the 4T and connects this part of the correlator with the lower point
functions 3T and $TT$.

####  6.9 Identifying the divergent form factors and their reduction in $d=4$

The general expression of the $tt$-contributions can be identified by imposing
the transversality and trace-free (6.7.13) conditions on all the possible
tensor structures which are allowed by the symmetries of the correlation
function. In this section we are going first to proceed with the
classification of the divergent ones which are present in the decomposition
and are affected by the renormalization.
We introduce a general decomposition of the counterterm for the 4T in general
$d$ dimensions, in order to identify such form factors. Their number gets
reduced once we move to $d=4$, due to the possibility of expressing the
Kronecker $\delta^{\mu\nu}$ in terms of 3 of the 4 momenta of the correlator,
and of a linearly independent 4-vector $n^{\mu}$. The latter is defined via a
generalization of the external product by the $\epsilon^{\mu\nu\rho\sigma}$,
as we are going to illustrate below (see [21] for the case $d=3$)

The decomposition can be generically written in the form

$\displaystyle\langle{t^{\mu_{1}\nu_{1}}(p_{1})t^{\mu_{2}\nu_{2}}(p_{2})t^{\mu_{3}\nu_{3}}(p_{3})t^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle\equiv\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(p_{1})\Pi^{\mu_{2}\nu_{2}}_{\alpha_{2}\beta_{2}}(p_{2})\Pi^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}}(p_{3})\Pi^{\mu_{4}\nu_{4}}_{\alpha_{4}\beta_{4}}(\bar{p}_{4})$
$\displaystyle\times\Big{\\{}\sum_{a,b\in\\{2,3\\}}\,\sum_{c,d\in\\{3,4\\}}\,\sum_{e,f\in\\{4,1\\}}\,\sum_{g,h\in\\{1,2\\}}A^{(8p)}_{a\,b\,c\,d\,e\,f\,g\,h}\
\
p_{a}^{\alpha_{1}}\,p_{b}^{\beta_{1}}\,p_{c}^{\alpha_{2}}\,p_{d}^{\beta_{2}}\,p_{e}^{\alpha_{3}}\,p_{f}^{\beta_{3}}\,p_{g}^{\alpha_{4}}\,p_{h}^{\beta_{4}}$
$\displaystyle\qquad+\Bigg{[}\delta^{\alpha_{1}\alpha_{2}}\sum_{a\in\\{2,3\\}}\sum_{b\in\\{3,4\\}}\sum_{c,d\in\\{4,1\\}}\sum_{e,f\in\\{1,2\\}}A^{(6p)}_{a\,b\,c\,d\,e\,f}\,p_{a}^{\beta_{1}}\,p_{b}^{\beta_{2}}\,p_{c}^{\alpha_{3}}\,p_{d}^{\beta_{3}}\,p_{e}^{\alpha_{4}}\,p_{f}^{\beta_{4}}+\text{(permutations)}\Bigg{]}$
$\displaystyle\qquad+\Bigg{[}\delta^{\alpha_{1}\alpha_{2}}\delta^{\beta_{1}\beta_{2}}\sum_{a,b\in\\{4,1\\}}\sum_{c,d\in\\{1,2\\}}A^{(I,4p)}_{a\,b\,c\,d}\,p_{a}^{\alpha_{3}}\,p_{b}^{\beta_{3}}\,p_{c}^{\alpha_{4}}\,p_{d}^{\beta_{4}}+\text{(permutations)}\Bigg{]}$
$\displaystyle\qquad+\Bigg{[}\delta^{\alpha_{1}\alpha_{2}}\delta^{\beta_{2}\beta_{3}}\sum_{a\in\\{2,3\\}}\sum_{b\in\\{4,1\\}}\sum_{c,d\in\\{1,2\\}}A^{(II,4p)}_{a\,b\,c\,d}\,p_{a}^{\beta_{1}}\,p_{b}^{\alpha_{3}}\,p_{c}^{\alpha_{4}}\,p_{d}^{\beta_{4}}+\text{(permutations)}\Bigg{]}$
$\displaystyle\qquad+\Bigg{[}\delta^{\alpha_{1}\alpha_{2}}\delta^{\alpha_{3}\alpha_{4}}\sum_{a\in\\{2,3\\}}\sum_{b\in\\{3,4\\}}\sum_{c\in\\{4,1\\}}\sum_{d\in\\{1,2\\}}A^{(III,4p)}_{a\,b\,c\,d}\,p_{a}^{\beta_{1}}\,p_{b}^{\beta_{2}}\,p_{c}^{\beta_{3}}\,p_{d}^{\beta_{4}}+\text{(permutations)}\Bigg{]}$
$\displaystyle\qquad+\Bigg{[}\delta^{\alpha_{1}\beta_{2}}\delta^{\alpha_{2}\beta_{3}}\delta^{\alpha_{3}\beta_{1}}\sum_{a,b\in\\{1,2\\}}A^{(I,2p)}_{a\,b}\,p_{a}^{\alpha_{4}}\,p_{b}^{\beta_{4}}+\text{(permutations)}\Bigg{]}$
$\displaystyle\qquad+\Bigg{[}\delta^{\alpha_{1}\beta_{2}}\delta^{\alpha_{2}\beta_{1}}\delta^{\alpha_{3}\alpha_{4}}\sum_{a\in\\{4,1\\}}\sum_{b\in\\{1,2\\}}A^{(II,2p)}_{a\,b}\,p_{a}^{\beta_{3}}\,p_{b}^{\beta_{4}}+\text{(permutations)}\Bigg{]}$
$\displaystyle\qquad+\Bigg{[}\delta^{\alpha_{1}\beta_{2}}\delta^{\alpha_{2}\beta_{4}}\delta^{\alpha_{4}\beta_{3}}\sum_{a\in\\{2,3\\}}\sum_{b\in\\{4,1\\}}A^{(III,2p)}_{a\,b}\,p_{a}^{\beta_{1}}\,p_{b}^{\alpha_{3}}+\text{(permutations)}\Bigg{]}$
$\displaystyle\qquad+\Bigg{[}\delta^{\alpha_{1}\beta_{2}}\delta^{\alpha_{2}\beta_{3}}\delta^{\alpha_{3}\beta_{4}}\delta^{\alpha_{4}\beta_{1}}A^{(I,0p)}+\text{(permutations)}\Bigg{]}+\Bigg{[}\delta^{\alpha_{1}\beta_{2}}\delta^{\alpha_{2}\beta_{1}}\delta^{\alpha_{3}\beta_{4}}\delta^{\alpha_{4}\beta_{3}}A^{(II,0p)}+\text{(permutations)}\Bigg{]}\Bigg{\\}}$
(6.9.1)

in terms of form factors $A$. We have explicitly labeled the form factors with
an index counting the number of momenta that each of them multiplies in the
decomposition. This notation will be useful in our discussion below. The
choice of the independent momenta of the expansion, similarly to the case of
3-point functions, can be different for each set of uncontracted tensor
indices. We will choose

$\displaystyle(\alpha_{1},\beta_{1})\leftrightarrow
p_{2},p_{3},\quad(\alpha_{2},\beta_{2})\leftrightarrow p_{3},p_{4}$
$\displaystyle(\alpha_{3},\beta_{3})\leftrightarrow
p_{4},p_{1},\quad(\alpha_{4},\beta_{4})\leftrightarrow p_{1},p_{2}.$ (6.9.2)

as basis of the expansion for each pair of indices shown above. The linear
dependence of $p_{4}$, which we will impose at a later stage, is not in
contradiction with this choice, which allows to reduce the number of form
factors, due to the presence of a single $tt$ projector for each external
momentum. This strategy has been introduced in [21] for 3-point functions and
it allows to reduce the number of form factors. These, in eq. (6.9.1) are
functions of the six kinematic invariants

$\displaystyle p_{i}=\sqrt{p_{i}^{2}},\quad\,i=1,2,3,4,\qquad
s=\sqrt{(p_{1}+p_{2})^{2}},\ t=\sqrt{(p_{2}+p_{3})^{2}}$ (6.9.3)

or equivalently, in a completely symmetric formulation, they are functions of
the six invariants

$\displaystyle s_{ij}=\sqrt{p_{i}\cdot p_{j}},\quad\,i\neq j\in\\{1,2,3,4\\}.$
(6.9.4)

As already mentioned, the local part of the 4T can be expressed entirely in
terms of three- and two-point functions due to the transverse and trace WIs.
The explicit form of the local contribution is indeed given by the expression

$\displaystyle\langle{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle_{loc}=$
$\displaystyle=\Bigg{[}\langle{t_{loc}^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle+(1\,2)+(1\,3)+(1\,4)\Bigg{]}$
$\displaystyle-\Bigg{[}\langle{t_{loc}^{\mu_{1}\nu_{1}}(p_{1})t_{loc}^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle+(1\,3)+(1\,4)+(2\,3)+(2\,4)+(1\,3)(2\,4)\Bigg{]}$
$\displaystyle+\Bigg{[}\langle{t_{loc}^{\mu_{1}\nu_{1}}(p_{1})t_{loc}^{\mu_{2}\nu_{2}}(p_{2})t_{loc}^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle+(1\,4)+(2\,4)+(3\,4)\Bigg{]}$
$\displaystyle-\langle{t_{loc}^{\mu_{1}\nu_{1}}(p_{1})t_{loc}^{\mu_{2}\nu_{2}}(p_{2})t_{loc}^{\mu_{3}\nu_{3}}(p_{3})t_{loc}^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle,$
(6.9.5)

where the insertion of $t_{loc}$ gives

$\displaystyle\langle{t_{loc}^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle=\frac{1}{p_{1}^{2}}\Big{[}2\delta^{(\nu_{1}}_{(\alpha_{1}}p_{1}^{\mu_{1})}p_{1\,\beta_{1})}-\frac{p_{1\alpha_{1}}p_{1\beta_{1}}}{(d-1)}\left(\delta^{\mu_{1}\nu_{1}}+(d-2)\frac{p_{1}^{\mu_{1}}p_{1}^{\nu_{1}}}{p_{1}^{2}}\right)\Big{]}\langle{T^{\alpha_{1}\beta_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle$
$\displaystyle\qquad+\frac{1}{(d-1)}\pi^{\mu_{1}\nu_{1}}(p_{1})\delta_{\alpha_{1}\beta_{1}}\langle{T^{\alpha_{1}\beta_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})T^{\mu_{4}\nu_{4}}(\bar{p}_{4})}\rangle.$
(6.9.6)

Notice that the right-hand-side of (6.9.6) is entirely expressed in terms of
lower-point correlation functions, due to the WI’s (6.3.8) and (6.4.3).
Similar relations hold for all the other contributions contained in (6.9.5).

####  6.10 Divergences and Renormalization

In order to investigate the implications of the CWIs on the anomaly
contributions of the 4T, we turn to 4 spacetime dimensions and discuss the
anomaly form of such equations. We start from the dilatation WIs.
The scale invariance of the correlator is expressed through the Dilatation
Ward Identity (6.3.22), which in terms of the corresponding form factors takes
to scalar equations of the form

$\displaystyle\left(d-n_{p}-\sum_{i=1}^{3}\,p_{i}^{\mu}\frac{\partial}{\partial
p_{i}^{\mu}}\right)A^{(np)}_{a\dots}=0,$ (6.10.1)

where $n_{p}$ is the number of momenta multiplying the form factors in the
decomposition (6.9.1). Eqs. (6.10.1) characterize the scaling behaviour of the
form factors, and allow to identify quite easily those among all which will be
manifestly divergent in the UV. For instance, the form factor corresponding to
eight momenta in (6.9.1) has degree $d-8$ and is finite in $d=4$. This simple
dimensional counting can be done for all the form factors allowed by the
symmetry of the correlator.
We have summarised the UV behaviour in the table below

Form Factor | $A^{(8p)}$ | $A^{(6p)}$ | $A^{(4p)}$ | $A^{(2p)}$ | $A^{(0p)}$
---|---|---|---|---|---
Degree | $d-8$ | $d-6$ | $d-4$ | $d-2$ | $d$
UV divergent in $d=4$ | ✗ | ✗ | ✓ | ✓ | ✓

The expected form factors that will manifest divergences in $d=4$ are those of
the form $A^{(4p)}$, $A^{(2p)}$ and $A^{(0p)}$ in (6.9.1), which will show up
as single poles in the regulator $\varepsilon$. The procedure of
renormalization, obtained by the inclusion of the counterterm (3.2.24), will
remove these divergences and will generate an anomaly. An explicit check of
this cancellation is contained in [86], in the case of a conformally coupled
free scalar theory.

####  6.11 Explicit form of the divergences

Being the anomaly generated by the renormalization procedure, it is possible
to derive the structure of the anomaly contributions and the form of the
anomalous CWIs’ by applying the reconstruction procedure to the counterterms.
One can also work out the explicit structure of the counterterms for each of
the divergent form factors $A^{(np)}$ identified above. Their renormalization
is obviously guaranteed by the general counterterm Lagrangian (3.2.24). For
example, considering the decomposition identified in (6.9.1), the
corresponding counterterms to the $A^{(np)}$ form factors, expanded in series
of power in $\varepsilon=d+4$, can be determined in the form

$\displaystyle A_{4411,\,count}^{(I,4p)}=$
$\displaystyle\frac{2}{\varepsilon}\,b^{\prime}+O(\varepsilon^{0})$ (6.11.1)
$\displaystyle A_{2411,\,count}^{(II,4p)}=$
$\displaystyle\frac{4}{\varepsilon}\,b^{\prime}+O(\varepsilon^{0})$ (6.11.2)
$\displaystyle A_{2341,\,count}^{(III,4p)}=$
$\displaystyle-\frac{5}{8\,\varepsilon}\,\left(b-6\,b^{\prime}\right)+O(\varepsilon^{0})$
(6.11.3) $\displaystyle A_{11,\,count}^{(I,2p)}=$
$\displaystyle-\frac{7}{16\,\varepsilon}\bigg{[}b\left(p_{1}^{2}-4p_{2}^{2}+p_{3}^{2}+3s^{2}+3t^{2}\right)-b^{\prime}\left(p_{1}^{2}+4p_{2}^{2}+p_{3}^{2}+4p_{4}^{2}-s^{2}-t^{2}\right)\bigg{]}+O(\varepsilon^{0})$
(6.11.4) $\displaystyle A_{41\,,count}^{(II,2p)}=$