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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right]\ ;$ (5.21)

the numerators of the other diagrams can be obtained by using Jacobi
identities. In this BCJ representation, the snowflake numerators will differ
from eq. (5.17) by some additional commutators
$[\mathcal{D}_{ij},\mathcal{D}_{jk}]$. These commutators are nonzero as
operators but annihilate the $D$-functions. Thus, they do not change the
boundary correlator. This phenomenon has already been observed for NLSM
Diwakar:2021juk . Finally, we note that the differential form of the six-point
fundamental BCJ relation

$\displaystyle 0$
$\displaystyle=\mathcal{D}_{12}A(1,2,3,4,5,6)+(\mathcal{D}_{12}+\mathcal{D}_{13})A(1,3,2,4,5,6)$
$\displaystyle\quad-(\mathcal{D}_{25}+\mathcal{D}_{26})A(1,3,4,2,5,6)-\mathcal{D}_{26}A(1,3,4,5,2,6)\
,$ (5.22)

holds as a consequence of the color-kinematics duality.

####  5.2.2 Graviton-mediated four-point correlator

We proceed to discuss the four-point AdS boundary correlator due to graviton
exchange following from the action (5.4). While we could directly apply the
results of section 4, we will proceed as in the case of gluon-mediated four-
point AdS boundary correlator and describe the various intermediate steps of
the calculation.

The s-channel graviton exchange is given by Costa:2014kfa ,

$3$$2$$1$$4$
$\displaystyle=\frac{1}{4}\int_{\textrm{AdS}}dX_{1}dX_{2}T_{AB}(P_{1},P_{2},X_{1})\mathcal{G}_{h}^{AB,CD}(X_{1},X_{2})T_{CD}(P_{3},P_{4},X_{2})\,,$
(5.23)

where the vertex function is the scalar stress tensor

$\displaystyle T_{AB}(P_{i},P_{j},X)$
$\displaystyle=\nabla_{A}E_{d}(X,P_{i})\nabla_{B}E_{d}(X,P_{j})+\nabla_{B}E_{d}(X,P_{i})\nabla_{A}E_{d}(X,P_{j})$
$\displaystyle\quad-G_{AB}\nabla_{C}E_{d}(X,P_{i})\nabla^{C}E_{d}(X,P_{j})\,,$
(5.24)

and $\mathcal{G}_{h}^{AB,CD}(X_{1},X_{2})$ is the graviton bulk-bulk
propagator. As for gluon exchange, we formally replace
$\mathcal{G}_{h}^{AB,CD}$ by the inverse of the free equation of motion acting
on a delta function. When acting on a vertex function, $\nabla^{2}$ has a
linear relation with the operator $\mathcal{D}_{ij}$ associated with the
external particles sourcing the vertex. Using this relation to trade the
former for the latter, we are left with a contact interaction in the bulk,
which can be integrated straightforwardly.

Of course, compared with gluons, gravitons in the bulk are more subtle: they
contain both a spin-2 and spin-0 component. We relegate the detailed
derivation to appendix A and give here the remarkably simple result,

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{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{15.08904pt}{-15.08904pt}\pgfsys@lineto{8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-15.08904pt}{15.08904pt}\pgfsys@lineto{-8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-15.08904pt}{-15.08904pt}\pgfsys@lineto{-8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,=\frac{1}{4}\frac{1}{\mathcal{D}_{12}+2d}\int_{\textrm{AdS}}dXT_{AB}(P_{1},P_{2},X)\Pi^{AB,CD}_{h}(X)T_{CD}(P_{3},P_{4},X)\,,$
(5.25)

where

$\displaystyle\Pi^{AB,CD}_{h}(X)=\frac{1}{2}\left(G^{AC}G^{BD}+G^{AD}G^{BC}-\frac{2}{d-1}G^{AB}G^{CD}\right)\,.$
(5.26)

We then carry out the bulk integral and write the answer in terms of the
differential operators acting on the contact diagram,

$\displaystyle\int_{\textrm{AdS}}dXT_{AB}(P_{1},P_{2},X)\Pi^{AB,CD}_{h}(X)T_{CD}(P_{3},P_{4},X)$
$\displaystyle=\frac{1}{2}\left(\mathcal{D}_{14}\mathcal{D}_{23}+\mathcal{D}_{13}\mathcal{D}_{24}-\mathcal{D}_{12}\mathcal{D}_{34}\right)D_{d,d,d,d}=\frac{1}{2}(\mathcal{D}_{13}^{2}+\mathcal{D}_{23}^{2}-\mathcal{D}_{12}^{2})D_{d,d,d,d}\,.$
(5.27)

Again, this result displays a very clear analogy with the corresponding flat
space amplitude — the numerators are related by the replacement
$s_{ij}\rightarrow\mathcal{D}_{ij}$ and the support by $\delta(p)\rightarrow
D_{d,d,d,d}$,

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{}{}\pgfsys@beginscope\pgfsys@invoke{
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{-15.08904pt}{-15.08904pt}\pgfsys@lineto{-8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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(5.28)

This time we need an additional shift in the propagator,
$\frac{1}{s_{12}}\rightarrow\frac{1}{\mathcal{D}_{12}+2d}$. It originates from
the coupling between the graviton and background curvature that appears in the
equation of motion (see appendix A). We note that $2d$ is exactly the
eigenvalue of the massless graviton’s quadratic Casimir operator. Thus, the
equation above confirms the more general prescription (4.22),
$(p_{I}^{2}+m^{2})^{-1}\rightarrow(\mathcal{D}_{I}+\mathsf{M}^{2})^{-1}$,
connecting propagators of flat space amplitudes and nonlocal products of
conformal generators. That is, while changing the momentum $p_{I}^{\mu}$ to
the conformal generator $D_{I}^{AB}$, it is necessary to generalize the mass
in the propagator to the eigenvalue of the quadratic Casimir operator. In the
next section, we will see that $\mathsf{M}^{2}_{\rm graviton}=2d$ plays a
crucial role when we convert the differential form of the correlator into a
sum of $D$-functions with even boundary dimension — it ensures that the sum
truncates to a finite number of terms.

If the scalars $\varphi_{1}$ and $\varphi_{2}$ are of one flavor and
$\tilde{\varphi}_{3}$ and $\tilde{\varphi}_{4}$ are of another, the graviton
can only mediate the s-channel interaction. Thus eq. (5.28) is the complete
four-scalar boundary correlator for the Lagrangian (5.4). We write the result
in a form that resembles a double copy (5.3b),

$\displaystyle\mathcal{M}(\varphi_{1}\varphi_{2}\tilde{\varphi}_{3}\tilde{\varphi}_{4})$
$\displaystyle=\frac{1}{8}\frac{1}{\mathcal{D}_{12}+2d}(\mathcal{D}_{13}^{2}+\mathcal{D}_{23}^{2}-\mathcal{D}_{12}^{2})D_{d,d,d,d}$
$\displaystyle=\frac{1}{16}\frac{1}{\mathcal{D}_{12}+2d}\Big{[}(\mathcal{D}_{23}-\mathcal{D}_{13})^{2}+2d\mathcal{D}_{12}\Big{]}D_{d,d,d,d}$
$\displaystyle\quad+\frac{1}{16}\frac{1}{\mathcal{D}_{23}+2d}(\mathcal{D}_{23}^{2}+2d\mathcal{D}_{23})D_{d,d,d,d}$
$\displaystyle\quad+\frac{1}{16}\frac{1}{\mathcal{D}_{13}+2d}(\mathcal{D}_{13}^{2}+2d\mathcal{D}_{13})D_{d,d,d,d}\,.$
(5.29)

If all four scalars have the same flavor, the complete correlator is a sum of
the s-, t- and u-channels,

$\displaystyle\mathcal{M}(\varphi_{1}\varphi_{2}\varphi_{3}\varphi_{4})$
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{15.08904pt}{-15.08904pt}\pgfsys@lineto{8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{-15.08904pt}{15.08904pt}\pgfsys@lineto{-8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{-15.08904pt}{-15.08904pt}\pgfsys@lineto{-8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,$ (5.30)
$\displaystyle=\frac{1}{16}\frac{1}{\mathcal{D}_{12}+2d}\Big{[}(\mathcal{D}_{23}-\mathcal{D}_{13})^{2}+2d\mathcal{D}_{12}\Big{]}D_{d,d,d,d}+(\textrm{t
and u channel})\,.$

This bears remarkable similarity with the flat space double copy relation
(5.3a) for same-flavor scalars.

Interestingly, both eq. (5.2.2) and (5.30) have the general form

$\displaystyle\mathcal{M}$
$\displaystyle=\frac{1}{16}\left[\frac{1}{\mathcal{D}_{12}+2d}(\hat{N}_{\rm
s}^{2}+2d\mathcal{D}_{12})+\frac{1}{\mathcal{D}_{23}+2d}(\hat{N}_{\rm
t}^{2}+2d\mathcal{D}_{23})\right.$
$\displaystyle\quad\quad\quad\left.+\frac{1}{\mathcal{D}_{13}+2d}(\hat{N}_{\rm
u}^{2}+2d\mathcal{D}_{13})\right]D_{d,d,d,d}\,,$ (5.31)

where $\hat{N}_{\rm s,t,u}$ are the gauge theory BCJ numerators (with suitable
replacements of Mandelstam invariants with differential operators). This
equation suggests a generic double copy prescription for scalar correlators on
AdS space, in which the straightforward extension of the flat space double
copy is further corrected by numerator terms proportional to the quadratic
Casimir of intermediate states and lower-dimension differential operators.
Since the evidence for this conjectured extension is only eqs. (5.2.2) and
(5.30), it may need to be further modified to accommodate data from higher-
point AdS scalar boundary correlators and other theories.

We note that, since the action of conformal generators on $D_{d,d,d,d}$ yields
a factor of $d$ (as can be seen from e.g. the integral representation of the
contact integral), the second term in the gravitational numerator, which
causes a departure from the naive double copy and is proportional to the
quadratic Casimir in the graviton propagator, is subleading in the large-$d$
limit. In this limit, the usual double copy relation is restored.

###  5.3 Local factor of scalar boundary correlators and remarks on color-
kinematics duality

We showed in the explicit examples discussed above, and in general in section
4, that the nonlocal part of AdS boundary correlators, which is in one-to-one
correspondence with bulk-bulk propagators, can be extracted as an overall
factor despite the superficial noncommutativity of differential operators. The
examples discussed above also show that the remaining local part can be
written as a sequence of point-wise conformal generators acting on a single
contact integral, in accordance with the expected differential form of the
correlators in section 3. Moreover, we showed that certain AdS boundary
correlators in the differential representation are formally very similar to
the corresponding flat space amplitudes. In this section, we comment on the
general form of the local differential operator factor for scalar correlators
and explore the depth of its similarity with flat space amplitudes. We will
see that the terms with the highest number of conformal generators can always
be obtained by a formal replacement of Mandelstam invariants in flat space
amplitudes.

For a generic scalar Witten diagram, after we trade the bulk-bulk propagators
for the nonlocal operators $({\mathcal{D}_{I}+\mathsf{M}^{2}})^{-1}$, the
contact interaction is schematically a linear combination of terms of the form

$\displaystyle\int_{\textrm{AdS}}dX\nabla^{m_{1}}E_{\Delta_{1}}\nabla^{m_{2}}E_{\Delta_{2}}\ldots\nabla^{m_{n}}E_{\Delta_{n}}\,.$
(5.32)

where $E_{\Delta}\equiv E_{\Delta}(X,P)=\frac{1}{(-2P\cdot X)^{\Delta}}$ is
the shorthand notation for a scalar bulk-boundary propagator. Indices are
contracted with $G_{AB}$, which are inherited from vertex functions and bulk-
bulk propagators. Because of our Berends-Giele-type construction, the equation
above implicitly contains $G_{AB}$ which are acted upon by derivatives. Before
explicitly evaluating any of the derivatives, a cubic $n$-point gluon-exchange
Witten diagram will contain $\sum_{i=1}^{n}m_{i}=n-2$ derivatives, and a
graviton exchange diagram will have $\sum_{i=1}^{n}m_{i}=2n-4$.

To obtain the differential representation, we need to translate eq. (5.32)
into boundary conformal generators $D_{i}^{AB}$ acting on a single
$D$-function. Starting from the identity

$\displaystyle\nabla_{A}E_{\Delta}=G_{A}{}^{B}(2\Delta
P_{B}E_{\Delta+1})=2\Delta P_{A}E_{\Delta+1}+\frac{\Delta
X_{A}}{X^{2}}E_{\Delta}$ (5.33)

with $\nabla_{A}$ defined in eq. (2.11), it is easy to see that

$\displaystyle\nabla_{A}E_{\Delta_{1}}\nabla^{A}E_{\Delta_{2}}=\frac{1}{X^{2}}(D_{1}\cdot
D_{2})E_{\Delta_{1}}E_{\Delta_{2}}\,.$ (5.34)

Since $D_{1}\cdot D_{2}$ acts only on the boundary data, we can factor it out
of the bulk integral. This identity is of the exact form as the flat space
relation

$\displaystyle(i\partial_{\mu})e^{ip_{1}x}(i\partial^{\mu})e^{ip_{2}x}=(p_{1}\cdot
p_{2})e^{i(p_{1}+p_{2})x}\,.$ (5.35)

Effectively, we have traded two contracted $\nabla$’s acting on
$E_{\Delta_{1}}$ and $E_{\Delta_{2}}$ for $D_{1}\cdot D_{2}$, which makes the
contact interaction one step closer to the $D$-function.

The analogy with flat space still holds for certain cases that have more
complicated contractions among $\nabla$’s, for example,

$\displaystyle\nabla_{A}\nabla_{B}E_{\Delta_{1}}\nabla^{A}E_{\Delta_{2}}\nabla^{B}E_{\Delta_{3}}=\frac{1}{X^{4}}(D_{1}\cdot
D_{2})(D_{1}\cdot D_{3})E_{\Delta_{1}}E_{\Delta_{2}}E_{\Delta_{3}}\,.$ (5.36)

There is no ordering ambiguity between $D_{1}\cdot D_{2}$ and $D_{1}\cdot
D_{3}$ because their commutator annihilates the product of bulk-boundary
propagators it acts on. We note that for gluon-exchange correlators up to six
points and for graviton exchange correlators up to four points, the contact
interaction only involves terms like eq. (5.34) and (5.36). Hence, in these
cases the numerator of the boundary correlator can be obtained directly from
replacing $p_{i}\cdot p_{j}$ by $D_{i}\cdot D_{j}$ in the flat space
amplitude, as we found in the explicit examples discussed in previous
sections.

The exact correspondence breaks, however, for more generic cases, such as when
the local factor of the correlator contains

$\displaystyle\nabla_{A}\nabla_{B}E_{\Delta_{1}}\nabla^{A}\nabla^{B}E_{\Delta_{2}}=\frac{1}{X^{2}}\Big{[}(D_{1}\cdot
D_{2})^{2}-(d-1)(D_{1}\cdot D_{2})\Big{]}E_{\Delta_{1}}E_{\Delta_{2}}\,,$
(5.37a)
$\displaystyle\nabla_{A}\nabla_{B}\nabla_{C}E_{\Delta_{1}}\nabla^{A}E_{\Delta_{2}}\nabla^{B}E_{\Delta_{3}}\nabla^{C}E_{\Delta_{1}}$
$\displaystyle=\frac{1}{X^{6}}\Big{[}(D_{1}\cdot D_{2})(D_{1}\cdot
D_{3})(D_{1}\cdot D_{4})-(D_{1}\cdot D_{3})(D_{2}\cdot
D_{4})\Big{]}E_{\Delta_{1}}E_{\Delta_{2}}E_{\Delta_{3}}E_{\Delta_{4}}\,.$
(5.37b)

These relations can be derived by using the commutators,

$\displaystyle[\nabla^{A},\nabla^{B}]=\frac{1}{X^{2}}D_{X}^{AB}\,,$
$\displaystyle[\nabla^{A},D_{X}^{CD}]=\frac{\eta^{AC}\nabla^{D}-\eta^{AD}\nabla^{C}}{\sqrt{2}}\,,$
$\displaystyle[\nabla_{A},G_{BC}]=-\frac{G_{AB}X_{C}+G_{AC}X_{B}}{X^{2}}\,,$
$\displaystyle[\nabla_{A},X_{B}]=G_{AB}\,.$ (5.38)

Such commutators typically reduce the number of derivatives by one, which is
the reason that terms with lower powers of $D_{i}\cdot D_{j}$ appear in eq.
(5.37).

We will prove in section 6 that a differential representation of the form
(3.1) to (3.3) exists for all scalar boundary correlators. We will see that
the terms of highest degree of $D_{i}$ are obtained by directly replacing a
bulk derivative with AdS symmetry generator acting on the same bulk-boundary
propagator. Thus, the replacement $p_{i}\cdot p_{j}\rightarrow D_{i}\cdot
D_{j}$ in the flat-space kinematic numerators will always give correctly the
terms with the highest number of $D_{i}\cdot D_{j}$. Moreover, such terms
correspond to the leading order in the large-$d$ limit of the complete local
factor of the correlator.131313We assume here that $\Delta_{i}$ scale as
$d^{n>1/2}$ for all the boundary operators. Indeed, the action of
$D_{i}E_{\Delta}\propto\Delta$, cf. eq. (5.33), $D_{i}\cdot D_{j}$ dominates
if $\lim_{d\rightarrow\infty}({d}/{\Delta^{2}})=0$.  Therefore, we conclude
that in the large-$d$ limit the differential representation of the massless
scalar AdS boundary correlators exactly agrees with the flat space amplitudes
under the replacement $p_{i}\cdot p_{j}\rightarrow D_{i}\cdot D_{j}$ (or
$s_{ij}\rightarrow\mathcal{D}_{ij}$).

It also follows that, by starting with flat-space kinematic numerators obeying
color-kinematics duality, this replacement yields a large-$d$ differential
representation for scalar correlators that obeys color-kinematics duality in
AdS space (and also the AdS analog of the fundamental BCJ relations).
Moreover, since the symmetry generators can be taken to commute in this limit,
the double-copy of scalar correlators will yield scalar correlators in the
corresponding double-copy theory. Section 5.2.2 illustrates this general
observation: by dropping terms with a single $D_{i}\cdot D_{j}$ in eq. (5.31)
that equation takes the standard flat-space double-copy form. We will return
to this point in section 8 where we will also briefly discuss various other
double-copy prescriptions at finite $d$.

We note that this conclusion is based on a term-by-term analysis of the
contact interaction. In a physical theory, which contains a combination of
terms of the form (5.32) in the local factor of correlators, it is possible
that the terms with fewer $D_{i}\cdot D_{j}$ products cancel each other out
such that the correspondence to flat space amplitudes goes beyond the
large-$d$ limit. Indeed, in the scalar correlator due to graviton exchange,
index contractions as in eq. (5.37a) can appear for a certain choice of
Feynman rules. The subleading terms in the large-$d$ limit will cancel at the
end, and the resulting correlator is related to the flat space amplitude by
the direct replacement $s_{ij}\rightarrow\mathcal{D}_{ij}$.

##  6 Differential representation for scalar correlators

In this section, we show that a differential representation exists for AdS
scalar boundary correlators in any AdS field theory. Given the separation into
nonlocal and local parts and the form of the nonlocal part discussed in
section 4 we only need to show that the local part of the correlators can be
expressed in terms of generators of the AdS symmetry group acting on a single
$D$-function. Starting from the examples considered in section 5.3, let us
consider a generic AdS scalar constructed from $2n$ covariant derivatives
acting on some number of scalar bulk-boundary propagators,

$\begin{split}\mathcal{F}_{2n}&\equiv T^{A_{1}\ldots A_{j}B_{1}\ldots
B_{k}\ldots C_{1}\ldots
C_{l}}(\nabla_{A_{1}}\ldots\nabla_{A_{j}}E_{\Delta_{1}})(\nabla_{B_{1}}\ldots\nabla_{B_{k}}E_{\Delta_{2}})\\\
&\quad\times\ldots(\nabla_{C_{1}}\ldots\nabla_{C_{l}}E_{\Delta_{m}})\,,\end{split}$
(6.1)

where the tensor $T$ is a monomial of degree $n$ in $\eta^{AB}$ (so it has
$2n$ indices) that contracts all indices of the derivatives.141414 We assume
that the tensor $T$ does not contract the indices of two derivatives acting on
the same bulk-boundary propagator. If such a contraction occurs, we can always
commute one of the derivatives to the front and move it on the other factors
through integration-by-parts, while generating additional terms with fewer
derivatives due to the nonzero commutators. Alternatively, we can also move
both derivatives next to the propagator (while again generating terms with
fewer derivatives due to the nonzero commutators) and use the equation of
motion to trade the resulting AdS Laplacian for the AdS mass parameter.

Based on the examples in section 5.3, we would like to prove that
$\mathcal{F}_{2n}$ has the manifestly local form of polynomial in $D_{i}^{AB}$
acting on (the integrand of) a single $D$-function. More specifically,

$\displaystyle\mathcal{F}_{2n}=\frac{1}{X^{2}}\Big{[}M_{2n}(D_{i})+P_{2n-2}(D_{i})+\ldots+P_{2}(D_{i})+P_{0}\Big{]}\prod_{i}E_{\Delta_{i}}\,,$
(6.2)

where $M_{2n}$ is a monomial of degree $2n$ in $D_{i}^{AB}$, and
$P_{2k\leqslant 2n-2}$ are polynomials of degree $2k$ in $D_{i}$. Compared
with the leading monomial $M_{2n}$, their degree decreases by at least two
units. The readers not interested in the details of the proof may skip the
rest of this section. We will show that $M_{2n}$ can be obtained from the
corresponding flat-space expression via the replacement $p_{i}\cdot
p_{j}\mapsto D_{i}\cdot D_{j}$ and that the sub-leading terms, $P_{2n-2i}$,
are suppressed in the large-$d$ and $\Delta$ limit.

We will prove the statement recursively, with eq. (6.2) serving as the
recursive assumption. The starting point is eq. (5.34), which is of the form
(6.2) with the monomial $M_{2}=D_{1}\cdot D_{2}$ and can be obtained through
the replacement $p_{1}\cdot p_{2}\rightarrow D_{1}\cdot D_{2}$ in the
numerator of the corresponding flat space expression in which the covariant
derivatives are simply partial derivatives and $E_{\Delta}\rightarrow
e^{ipx}$. We then consider eq. (6.1) with two additional derivatives, which
without loss of generality, we choose to act on $E_{\Delta_{1}}$ and
$E_{\Delta_{2}}$, and define

$\displaystyle\mathcal{F}_{2n+2}\equiv T^{A_{1}\ldots A_{j}B_{1}\ldots
B_{k}C_{1}\ldots
C_{l}}(\nabla_{A_{1}}\ldots\nabla_{N}\ldots\nabla_{A_{j}}E_{\Delta_{1}})(\nabla_{B_{1}}\ldots\nabla^{N}\ldots\nabla_{B_{k}}E_{\Delta_{2}})\ldots\,,$
(6.3)

where the ellipsis stand for additional factors present in eq. (6.1) that do
not depend on $E_{\Delta_{1}}$ and $E_{\Delta_{2}}$. The goal is to show that
$\mathcal{F}_{2n+2}$ has the same form (6.2) with $n\rightarrow n+1$, i.e.
that it begins with a degree-$(2n+2)$ monomial in $D_{i}$ and continues with
subleading polynomials whose degrees decrease by two units.

To evaluate $\mathcal{F}_{2n+2}$ we start by commuting $\nabla_{N}$, using eq.
(5.38), to act directly on $E_{\Delta_{1}}$ and $E_{\Delta_{2}}$,

$\displaystyle\nabla_{A_{1}}\ldots\nabla_{N}\ldots\nabla_{A_{j}}E_{\Delta_{1}}=\nabla_{A_{1}}\ldots\nabla_{A_{j}}\nabla_{N}E_{\Delta_{1}}+\mathcal{O}(D_{X}\nabla^{j-1}E_{\Delta_{1}})\,,$
$\displaystyle\nabla_{B_{1}}\ldots\nabla^{N}\ldots\nabla_{B_{k}}E_{\Delta_{2}}=\nabla_{B_{1}}\ldots\nabla_{B_{k}}\nabla^{N}E_{\Delta_{2}}+\mathcal{O}(D_{X}\nabla^{k-1}E_{\Delta_{2}})\,,$
(6.4)

where $\mathcal{O}(D_{X}\nabla^{\ldots})$ are terms resulting from the
commutator $[\nabla^{A},\nabla^{B}]=-D_{X}^{AB}$. They contain at least one
fewer derivative than the left hand side. In the product, the cross term
$\nabla_{B_{1}}\ldots\nabla_{B_{k}}\nabla^{N}E_{\Delta_{2}}\mathcal{O}(D_{X}\nabla^{j-1}E_{\Delta_{1}})$
appears to have only one fewer derivative than the leading term, with the term
with the largest number of derivatives containing a factor of the form

$\displaystyle\nabla_{A_{1}}\ldots\nabla_{A_{i-1}}D_{X}^{NA_{i}}\nabla_{A_{i+1}}\ldots\nabla_{A_{j}}E_{\Delta_{1}}\nabla_{B_{1}}\ldots\nabla_{B_{k}}\nabla_{N}E_{\Delta_{2}}$
(6.5) $\displaystyle=\frac{X^{A_{i}}}{X^{2}}D_{1}\cdot
D_{2}\nabla_{A_{1}}\ldots\nabla_{A_{i-1}}\nabla_{A_{i+1}}\ldots\nabla_{A_{j}}E_{\Delta_{1}}\nabla_{B_{1}}\ldots\nabla_{B_{k}}E_{\Delta_{2}}+(\text{commutators})\,,$

where the term written comes from the explicit evaluation of $D_{X}^{NA_{i}}$
and $\nabla_{N}$. The commutators $[D_{X},\nabla]$ and $[X,\nabla]$, arising
from pushing $D_{X}$ to $E_{\Delta_{1}}$ and from extracting $X^{A_{i}}$, lead
to lower number of derivatives. Since all indices are contracted, $X^{A_{i}}$
will be contracted with another $\nabla_{A_{i}}$. Because
$X^{A_{i}}\nabla_{A_{i}}=0$ , eq. (6.5) can contribute to the complete product
only the commutator of $X^{A_{i}}$ with another derivative,
$[\nabla^{A},X^{B}]=G^{AB}$, which further reduces the number of derivatives
by one. It therefore follows that the cross-terms
$\nabla_{B_{1}}\ldots\nabla_{B_{k}}\nabla^{M}E_{\Delta_{2}}\mathcal{O}(D_{X}\nabla^{j-1}E_{\Delta_{1}})$
also contribute terms with two derivatives fewer than the initial number. That
is,

$\displaystyle(\nabla_{A_{1}}\ldots\nabla_{M}\ldots\nabla_{A_{j}}E_{\Delta_{1}})(\nabla_{B_{1}}\ldots\nabla^{M}\ldots\nabla_{B_{k}}E_{\Delta_{2}})$
(6.6)
$\displaystyle=(\nabla_{A_{1}}\ldots\nabla_{A_{j}}\nabla_{M}E_{\Delta_{1}})(\nabla_{B_{1}}\ldots\nabla_{B_{k}}\nabla^{M}E_{\Delta_{2}})+(\text{subleading)}\,,$

where the subleading terms have the form

$\displaystyle(\text{subleading)}\sim\mathcal{O}(D_{1}\cdot
D_{2}\nabla^{2n-2}E_{\Delta_{1}}E_{\Delta_{2}}\ldots)\,,$ (6.7)

i.e., they have at least two fewer derivatives, as we expected from eq. (6.2).
We note that the lower order terms contain more generic contractions between
$\nabla^{A}$ and two $D_{i}^{AB}$ acting on different bulk-boundary
propagators. Using the same relations, we can show that the number of
derivatives is even and at most $2n-2$.

Having seen that the difference between the highest and next-to-highest number
of derivatives in ${\cal F}_{2n+2}$ is two units, we proceed to show that
${\cal F}_{2n+2}$ can be written in terms of the AdS symmetry generators. To
this end, we evaluate $\nabla^{M}E_{\Delta}$ using eq. (5.33), and collect the
result as

$\displaystyle(\nabla_{A_{1}}\ldots\nabla_{A_{j}}\nabla_{N}E_{\Delta_{1}})(\nabla_{B_{1}}\ldots\nabla_{B_{k}}\nabla^{N}E_{\Delta_{2}})$
$\displaystyle=4\Delta_{1}\Delta_{2}(P_{1}\cdot
P_{2})\nabla_{A_{1}}\ldots\nabla_{A_{j}}E_{\Delta_{1}+1}\nabla_{B_{1}}\ldots\nabla_{B_{k}}E_{\Delta_{2}+1}$
$\displaystyle\quad-\Delta_{1}\Delta_{2}\nabla_{A_{1}}\ldots\nabla_{A_{j}}E_{\Delta_{1}}\nabla_{B_{1}}\ldots\nabla_{B_{k}}E_{\Delta_{2}}+(\text{subleading})\,.$
(6.8)

The subleading terms contain various commutators $[\nabla^{A},X^{B}]=G^{AB}$
and thus have fewer derivatives. Those with the most derivatives have the form

$\displaystyle
2\Delta_{1}\Delta_{2}P_{1,N}G^{N}{}_{B_{i}}\nabla_{A_{1}}\ldots\nabla_{A_{j}}E_{\Delta_{1}+1}\nabla_{B_{1}}\ldots\nabla_{B_{i-1}}\nabla_{B_{i+1}}\ldots\nabla_{B_{k}}E_{\Delta_{2}}\,.$
(6.9)

Since the index $B_{i}$ is contracted with a derivative acting on another
bulk-boundary propagator $E_{\Delta_{\ell}}$, we can use the identity151515In
principle, there are additional derivatives acting on $E_{\Delta_{1}+1}$ and
$E_{\Delta_{\ell}}$ in eq. (6.9). To take them into account, we need to add
terms containing a commutator $[\nabla^{A},G^{BC}]$ to eq. (6.10), similar to
eq. (6.5). They all contain fewer derivatives so we omit them from our
discussion.

$\displaystyle
2\Delta_{1}P_{1,N}G^{N}{}_{B_{i}}E_{\Delta_{1}+1}\nabla^{B_{i}}E_{\Delta_{\ell}}=\frac{1}{X^{2}}D_{1}\cdot
D_{\ell}E_{\Delta_{1}}E_{\Delta_{\ell}}\,,$ (6.10)

to trade one derivative on $E_{\Delta_{\ell}}$ for $D_{\ell}$, and the $P_{1}$
factor for $D_{1}$. Therefore, the terms with the most derivatives in the
subleading term in eq. (6) have the form

$\displaystyle\mathcal{O}(\Delta_{2}D_{1}\cdot
D_{\ell}\nabla^{2n-2}E_{\Delta_{1}}E_{\Delta_{2}}\ldots)\,,$ (6.11)

plus the $1\leftrightarrow 2$ exchange. Now we consider

$\displaystyle\frac{1}{X^{2}}D_{1}\cdot
D_{2}(\nabla_{A_{1}}\ldots\nabla_{A_{j}}E_{\Delta_{1}})(\nabla_{B_{1}}\ldots\nabla_{B_{k}}E_{\Delta_{2}})$
$\displaystyle=4\Delta_{1}\Delta_{2}(P_{1}\cdot
P_{2})\nabla_{A_{1}}\ldots\nabla_{A_{j}}E_{\Delta_{1}+1}\nabla_{B_{1}}\ldots\nabla_{B_{k}}E_{\Delta_{2}+1}$
$\displaystyle\quad-\Delta_{1}\Delta_{2}\nabla_{A_{1}}\ldots\nabla_{A_{j}}E_{\Delta_{1}}\nabla_{B_{1}}\ldots\nabla_{B_{k}}E_{\Delta_{2}}+(\text{subleading})\,.$
(6.12)

The first two terms agree with the first two terms in eq. (6). For the
subleading terms in eq. (6), we again focus on those with the most
derivatives. They are of two types: one is of the form (6.11), the other one
comes from

$\displaystyle\Delta_{1}\Delta_{2}(P_{1}\cdot
P_{2})G_{A_{i}}{}^{B_{m}}\nabla_{A_{1}}..\nabla_{A_{i-1}}\nabla_{A_{i+1}}..\nabla_{A_{j}}E_{\Delta_{1}+1}\nabla_{B_{1}}..\nabla_{B_{m-1}}\nabla_{B_{m+1}}..\nabla_{B_{k}}E_{\Delta_{2}+1}\,.$
(6.13)

Up to terms with less derivative, we can trade $\Delta_{1}\Delta_{2}P_{1}\cdot
P_{2}$ with $D_{1}\cdot D_{2}$. The $G_{A_{i}}{}^{B_{m}}$ factor can bring at
most a factor of $d$ when the index $A_{i}$ and $B_{m}$ are contracted with
each other. Therefore, the terms of this second type can be formally written
as

$\displaystyle\mathcal{O}(d\,D_{1}\cdot
D_{2}\nabla^{2n-2}E_{\Delta_{1}}E_{\Delta_{2}}\ldots)\,,$ (6.14)

i.e., they can be written in terms of $D_{1}\cdot D_{2}$ acting on AdS scalars
with fewer derivatives which, following our recursion, have the form (6.2).

Applying the argument above to the subleading terms in eqs. (6.11) and (6.14)
implies that they contain at most $2n-2$ derivatives composed of $D_{i}\cdot
D_{j}$ and $\nabla^{A}$. The number of derivatives drops only in steps of two,
accompanied by a coefficient at most proportional to $\Delta$ or $d$.

Combining this with eq. (6), (6) and (6), we find that

$\displaystyle\mathcal{F}_{2n+2}$ $\displaystyle=\frac{1}{X^{2}}(D_{1}\cdot
D_{2})\mathcal{F}_{2n}+(\text{subleading})$
$\displaystyle=\frac{1}{X^{2}}\Big{[}M_{2n+2}(D_{i})+\tilde{P}_{2n}(D_{i})+\ldots+\tilde{P}_{0}\Big{]}\prod_{i}E_{\Delta_{i}}\,.$
(6.15)

It also follows that the leading monomial $M_{2n+2}(D_{i})$ is related to the
leading one $M_{2n}(D_{i})$ of $\mathcal{F}_{2n}$ as

$\displaystyle M_{2n+2}(D_{i})=(D_{1}\cdot D_{2})M_{2n}(D_{i})\,.$ (6.16)

Therefore, the leading monomial can be obtained from the corresponding flat
space amplitude numerator by the replacement $p_{i}\cdot p_{j}\rightarrow
D_{i}\cdot D_{j}$. Note that the ordering of $D_{i}\cdot D_{j}$ only affects
the subleading terms. The subleading polynomial $\tilde{P}_{2n}(D_{i})$
receives contributions from the terms in eqs. (6.7), (6.11) and (6.14). They
all have a factor of overall $D_{i}\cdot D_{j}$ and $2n-2$ covariant
derivatives $\nabla$ contracted among themselves. Then according to the
recursive assumption (6.2), they can be converted to a polynomial of degree
$2n$ in $D_{i}$, with coefficients that are at most linear in $d$ or in
$\Delta_{i}$. In addition, the generators $D_{i}$ appear only in the product
$D_{i}\cdot D_{j}$.

We note that the polynomials of degree $2n-2$ and lower in eq. (6) receive
contributions from more complicated contractions between $\nabla^{A}$ and at
most two $D_{i}^{AB}$. To offer another perspective on the argument above that
two $D_{i}^{AB}$, if they appear, will always contract with each other and
factor out of the bulk integral, allowing us to use the recursive assumption
(6.2) to write $\mathcal{F}_{2n+2}$ completely in terms of polynomials of
$D_{i}\cdot D_{j}$, let us consider another possible index contraction —
$\nabla^{A}$ contracted with $D_{1}^{AB}$. Then, the identity

$\displaystyle
D_{1}^{AB}E_{\Delta_{1}}\nabla_{A}E_{\Delta_{\ell}}=\frac{X^{B}}{X^{2}}D_{1}\cdot
D_{\ell}E_{\Delta_{1}}E_{\Delta_{\ell}}$ (6.17)

converts it into $D_{1}\cdot D_{\ell}$ and $X^{B}$. If the free index $B$ is
contracted with another derivative $\nabla$, then the $X^{B}$ factor will
effectively reduce the number of derivatives because $X^{A}\nabla_{A}=0$. If
it is contracted with the other $D^{AB}$, then we use

$\displaystyle\frac{\sqrt{2}\,X^{B}}{X^{2}}D_{AB}E_{\Delta}=\nabla_{A}E_{\Delta}$
(6.18)

to convert it into a $\nabla$. For both cases, additional $\nabla$’s acting on
$E_{\Delta_{1}}$ and $E_{\Delta_{\ell}}$ will lead to terms with fewer
derivatives by a step of two through commutators. We can then proceed to lower
order terms and systematically separate $D_{i}\cdot D_{j}$ from $\nabla$’s.
With the recursive assumption (6.2), this completes the proof that
$\mathcal{F}_{2n+2}$, as given in eq. (6), has the same form as eq. (6.2).
Importantly, as stated in section 5.3, the polynomials $\tilde{P}_{2k\leqslant
2n}$ are subleading in the large-$d$ limit compared with $M_{2n+2}(D_{i})$.

##  7 Evaluating the Differential Representation

We have seen that scalar correlators in differential representation have very
simple expressions that resemble flat space amplitudes. Now we must address
the question of how we should evaluate the Witten diagrams. For simplicity, we
first assume that the conformal weights for boundary and bulk states are
chosen such that the position-space correlators can be expressed in terms of a
finite sum of $D$-functions. To evaluate the Witten diagrams, one can of
course always go back to the conventional position-space representation, for
example, eq. (5.5), for which various techniques have been developed in the
past. In contrast, we will show that the expansion of the correlators in terms
of $D$-functions can be derived by only utilizing the action of the operator
$\mathcal{D}_{ij}$ on products of

$\displaystyle P_{ij}\equiv-2P_{i}\cdot P_{j}$ (7.1)

and $D$-functions. Our method is independent of the explicit representation
and gauge fixing of the bulk-bulk propagators, which significantly simplifies
the computation compared with existing methods.

###  7.1 Scalar Exchange

We begin with evaluating the scalar exchange diagram (3.9). Our strategy will
be to build a recursion relation for the quantity

$\displaystyle\frac{1}{\mathcal{D}_{12}+\Delta(\Delta-d)}\Big{[}P_{12}^{-k+1}D_{\Delta_{1}-k+1,\Delta_{2}-k+1,\Delta_{3},\Delta_{4}}\Big{]}\,.$
(7.2)

Once the recursion is solved, we will need only the $k=1$ term. To derive the
recursion, we start with the action of $\mathcal{D}_{12}+\Delta(\Delta-d)$ on
$P_{12}^{-k}D_{\Delta_{1}-k,\Delta_{2}-k,\Delta_{3},\Delta_{4}}$,

$\displaystyle\Big{[}\mathcal{D}_{12}+\Delta(\Delta-d)\Big{]}\Big{[}P_{12}^{-k}D_{\Delta_{1}-k,\Delta_{2}-k,\Delta_{3},\Delta_{4}}\Big{]}$
(7.3)
$\displaystyle=4(\Delta_{1}-k)(\Delta_{2}-k)P_{12}^{-k+1}D_{\Delta_{1}-k+1,\Delta_{2}-k+1,\Delta_{3},\Delta_{4}}$
$\displaystyle\quad-(\Delta_{1}+\Delta_{2}+\Delta-d-2k)(\Delta_{1}+\Delta_{2}-\Delta-2k)P_{12}^{-k}D_{\Delta_{1}+k,\Delta_{2}+k,\Delta_{3},\Delta_{4}}\,.$

We then move the operator to the right hand side. After some rearrangements,
we obtain the desired recursion:

$\displaystyle\frac{1}{\mathcal{D}_{12}+\Delta(\Delta-d)}\Big{[}P_{12}^{-k+1}D_{\Delta_{1}-k+1,\Delta_{2}-k+1,\Delta_{3},\Delta_{4}}\Big{]}=\frac{P_{12}^{-k}D_{\Delta_{1}-k,\Delta_{2}-k,\Delta_{3},\Delta_{4}}}{4(\Delta_{1}-k)(\Delta_{2}-k)}$
(7.4)
$\displaystyle+\frac{(\Delta_{1}{+}\Delta_{2}{+}\Delta{-}d{-}2k)(\Delta_{1}{+}\Delta_{2}{-}\Delta{-}2k)}{4(\Delta_{1}-k)(\Delta_{2}-k)}\frac{1}{\mathcal{D}_{12}+\Delta(\Delta-d)}\Big{[}P_{12}^{-k}D_{\Delta_{1}-k,\Delta_{2}-k,\Delta_{3},\Delta_{4}}\Big{]}\,.$

It is interesting to note that the recursion terminates if either $k_{\rm
max}=\frac{\Delta_{1}+\Delta_{2}-\Delta}{2}$ or $k_{\rm
max}=\frac{\Delta_{1}+\Delta_{2}+\Delta-d}{2}$ is an integer.161616If both are
integers, then the recursion terminates at $k_{\rm
max}=\frac{\Delta_{1}+\Delta_{2}-\Delta}{2}$ since it is the smaller one of
the two if we assume $\Delta$ is the larger root of $\Delta(\Delta-d)=m^{2}$.
For such cases, eq. (3.9) can be written as a finite sum of $D$-functions,

$\displaystyle\frac{1}{\mathcal{D}_{12}+\Delta(\Delta-d)}D_{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4}}=\sum_{l=1}^{k_{\rm
max}}\frac{(\Delta_{1})_{-l}(\Delta_{2})_{-l}P_{12}^{-l}D_{\Delta_{1}-l,\Delta_{2}-l,\Delta_{3},\Delta_{4}}}{4\\!\left(\frac{\Delta_{1}+\Delta_{2}-\Delta}{2}\right)_{1-l}\left(\frac{\Delta_{1}+\Delta_{2}+\Delta-d}{2}\right)_{1-l}}\,,$
(7.5)

which is exactly the $D$-function expansion for this Witten diagram
DHoker:1999mqo . As before, $(a)_{n}\equiv\frac{\Gamma(a+n)}{\Gamma(a)}$ is
the Pochhammer symbol.

We have explicitly restricted ourselves to correlators that can be represented
as finite sums of $D$-functions. Such expressions manifestly obey desirable
boundary conditions for the correlator. Correlators that cannot be expressed
as finite sums of $D$-functions will generically depend on multiple infinite
families $D$-functions, see Appendix C of Ref. Zhou:2018sfz  for an explicit
example.

###  7.2 Gluon Exchange

The scalar exchange computation demonstrates the general strategy for the
evaluation of Witten diagrams in differential representation. For the gluon
exchange diagram (5.2.1), we first expand the numerator in terms of
$D$-functions,

$\displaystyle\frac{1}{\mathcal{D}_{12}}(\mathcal{D}_{23}-\mathcal{D}_{13})D_{d,d,d,d}=\frac{4d^{2}}{\mathcal{D}_{12}}(P_{23}D_{d,d+1,d+1,d}-P_{13}D_{d+1,d,d+1,d})\,.$
(7.6)

Then the action of $\mathcal{D}_{12}^{-1}$ can be obtained by a similar
recursion. In fact, we only need to work out its action on
$P_{13}D_{d+1,d,d+1,d}$ since the other can be simply obtained by relabelling.
We start with

$\displaystyle\mathcal{D}_{12}\Big{[}P_{12}^{-k}P_{13}$ $\displaystyle
D_{d-k+1,d-k,d+1,d}\Big{]}=4(d-k)(d-k+1)P_{12}^{-k+1}P_{13}D_{d-k+2,d-k+1,d+1,d}$
$\displaystyle\quad+2(d-k)P_{12}^{-k}\Big{[}P_{23}D_{d-k,d-k+1,d+1,d}-P_{12}D_{d-k+1,d-k+1,d,d}\Big{]}$
$\displaystyle\quad-2(d-k)(d-2k+1)P_{12}^{-k}P_{13}D_{d-k+1,d-k,d+1,d}\,.$
(7.7)

We can eliminate the term $P_{23}D_{d-k,d-k+1,d+1,d}$ on the right-hand side
by the identity

$\displaystyle P_{23}D_{d-k,d-k+1,d+1,d}$
$\displaystyle=-\frac{d-k}{d}P_{12}D_{d-k+1,d-k+1,d,d}-P_{13}D_{d-k+1,d-k,d+1,d}$
$\displaystyle\quad-\frac{2k-3d}{2d}D_{d-k,d-k,d,d}\,,$ (7.8)

which can be derived from eq. (2.20). This gives us the desired recursion

$\displaystyle\frac{1}{\mathcal{D}_{12}}\Big{[}P_{12}^{-k+1}P_{13}D_{d-k+2,d-k+1,d+1,d}\Big{]}=\frac{1}{4(d{-}k)(d{-}k{+}1)}P_{12}^{-k}P_{13}D_{d-k+1,d-k,d+1,d}$
$\displaystyle+\frac{d{-}2k{+}2}{2(d{-}k{+}1)}\frac{1}{\mathcal{D}_{12}}\Big{[}P_{12}^{-k}P_{13}D_{d-k+1,d-k,d+1,d}\Big{]}{+}\frac{2d{-}k}{2d(d{-}k{+}1)}\frac{1}{\mathcal{D}_{12}}\Big{[}P_{12}^{-k+1}D_{d-k+1,d-k+1,d,d}\Big{]}$
$\displaystyle-\frac{3d-2k}{4d(d-k+1)}\frac{1}{\mathcal{D}_{12}}\Big{[}P_{12}^{-k}D_{d-k,d-k,d,d}\Big{]}\,.$
(7.9)

The coefficient of the second term on the right-hand side guarantees that the
recursion terminates at $k_{\rm max}=\frac{d}{2}+1$ if $d$ is even. For these
cases, the solution is

$\displaystyle\frac{1}{\mathcal{D}_{12}}P_{13}D_{d+1,d,d+1,d}$
$\displaystyle=P_{13}\frac{1}{\mathcal{D}_{12}{-}d{+}1}D_{d+1,d,d+1,d}$
$\displaystyle\quad+\sum_{k=1}^{d/2+1}\frac{(d{+}1)_{-k}(2d{-}k)}{2d(d/2{+}1)_{1-k}}\frac{1}{\mathcal{D}_{12}}P_{12}^{-k+1}D_{d+1-k,d+1-k,d,d}$
$\displaystyle\quad-\sum_{k=1}^{d/2+1}\frac{(d{+}1)_{-k}(3d{-}2k)}{4d(d/2{+}1)_{1-k}}\frac{1}{\mathcal{D}_{12}}P_{12}^{-k}D_{d-k,d-k,d,d}\,.$
(7.10)

We can pull the $\mathcal{D}_{12}^{-1}$ in the last two terms out of the
summation. Then using eq. (7.5), we see that they sum into a scalar-exchange
diagram with internal weight $\Delta=d-2$ and a contact diagram,

$\displaystyle\frac{1}{\mathcal{D}_{12}}\left[-\frac{(d-2)(d-1)^{2}}{d^{2}}\frac{1}{\mathcal{D}_{12}-2(d-2)}D_{d,d,d,d}+\frac{2d-1}{2d^{2}}D_{d,d,d,d}\right]\,.$
(7.11)

After partial fractioning the product of inverse operators, we obtain the
final result

$\displaystyle\frac{1}{\mathcal{D}_{12}}\Big{[}P_{13}D_{d+1,d,d+1,d}\Big{]}$
$\displaystyle=P_{13}\frac{1}{\mathcal{D}_{12}-d+1}D_{d+1,d,d+1,d}$
$\displaystyle\quad-\frac{(d-1)^{2}}{2d^{2}}\frac{1}{\mathcal{D}_{12}-2(d-2)}D_{d,d,d,d}+\frac{1}{2}\frac{1}{\mathcal{D}_{12}}D_{d,d,d,d}\,.$
(7.12)

Replacing this into eq. (7.6), we get

$3$$2$$1$$4$
$\displaystyle=\frac{1}{\mathcal{D}_{12}}(\mathcal{D}_{23}-\mathcal{D}_{13})D_{d,d,d,d}$
$\displaystyle=P_{23}\frac{4d^{2}}{\mathcal{D}_{12}-d+1}D_{d,d+1,d+1,d}-P_{13}\frac{4d^{2}}{\mathcal{D}_{12}-d+1}D_{d+1,d,d+1,d}\,,$
(7.13)

Then, the explicit $D$-function representation of the gluon exchange diagram
can be obtained by using eq. (7.5) in the second line of the equation above.

###  7.3 Graviton Exchange

Among the three operators in the graviton exchange diagram (5.28), the last
one, $\mathcal{D}_{12}^{2}$, commutes with the propagator. It can thus be
directly reduced to a scalar exchange diagram and a contact term. For the
first two terms, we start by carrying out the derivatives,

$\displaystyle\frac{1}{8}(\mathcal{D}_{23}^{2}+\mathcal{D}_{13}^{2})D_{d,d,d,d}$
$\displaystyle=2d^{2}(d{+}1)^{2}(P_{23}^{2}D_{d,d+2,d+2,d}+P_{13}^{2}D_{d+2,d,d+2,d})$
(7.14)
$\displaystyle\quad+\frac{2d^{2}+3d+2}{4}\mathcal{D}_{12}D_{d,d,d,d}-d^{2}(d+1)(d+2)D_{d,d,d,d}\,.$

The two terms in the second line commute with the propagator so that they
become a scalar exchange contribution and a contact term. We are thus left to
evaluate

$\displaystyle\frac{1}{\mathcal{D}_{12}+2d}(P_{23}^{2}D_{d,d+2,d+2,d}+P_{13}^{2}D_{d+2,d,d+2,d})\,.$
(7.15)

We again need to derive a recursive relation from

$\displaystyle(\mathcal{D}_{12}+2d)\Big{[}P_{12}^{-k}(P_{23}^{2}D_{d-k,d-k+2,d+2,d}+P_{13}^{2}D_{d-k+2,d-k,d+2,d})\Big{]}\,.$
(7.16)

The action of $\mathcal{D}_{12}$ will generate terms that contain
$P_{13}P_{23}$. Using eq. (2.20), the differential operators should be
eliminated in favor of $P_{12}P_{13}$ and $P_{12}P_{23}$ while maintaining the
symmetry under $1\leftrightarrow 2$ exchange. The final form of the recursion
is

$\displaystyle\frac{1}{\mathcal{D}_{12}+2d}P_{12}^{-k+1}\Big{[}P_{23}^{2}D_{d-k+1,d-k+3,d+2,d}+P_{13}^{2}D_{d-k+3,d-k+1,d+2,d}\Big{]}$
$\displaystyle=\frac{1}{4(d-k)(d-k+2)}P_{12}^{-k}\Big{[}P_{23}^{2}D_{d-k,d-k+2,d+2,d}+P_{13}^{2}D_{d-k+2,d-k,d+2,d}\Big{]}$
$\displaystyle\quad+\frac{(d-k+1)(d-2k+2)}{2(d-k)(d-k+2)}\frac{1}{\mathcal{D}_{12}{+}2d}P_{12}^{-k}\Big{[}P_{23}^{2}D_{d-k,d-k+2,d+2,d}+P_{13}^{2}D_{d-k+2,d-k,d+2,d}\Big{]}$
$\displaystyle\quad-\frac{(3d-2k)(3d-2k+2)(d-k+1)}{4d(d+1)(d-k)(d-k+2)}\frac{1}{\mathcal{D}_{12}+2d}P_{12}^{-k}D_{d-k,d-k,d,d}$
$\displaystyle\quad+\frac{(3d-2k+2)(3d-2k+3)}{2d(d+1)(d-k+2)}\frac{1}{\mathcal{D}_{12}+2d}P_{12}^{-k+1}D_{d-k+1,d-k+1,d,d}$
$\displaystyle\quad-\frac{(d-k+1)(2d-k+2)}{d(d+1)(d-k+2)}\frac{1}{\mathcal{D}_{12}+2d}P_{12}^{-k+2}D_{d-k+2,d-k+2,d,d}\,,$
(7.17)

which terminates at $k_{\rm max}=\frac{d}{2}+1$ for even $d$. We note that the
$2d$ shift in the propagator is crucial for this to happen. Should it take a
different value, the recursion would not truncate for even $d$. The solution
to the above recursion is

$\displaystyle\frac{1}{\mathcal{D}_{12}+2d}\Big{[}P_{23}^{2}D_{d,d+2,d+2,d}+P_{13}^{2}D_{d+2,d,d+2,d}\Big{]}$
$\displaystyle=P_{23}^{2}\frac{1}{\mathcal{D}_{12}}D_{d,d+2,d+2,d}+P_{13}^{2}\frac{1}{\mathcal{D}_{12}}D_{d+2,d,d+2,d}-\frac{d(d{-}1)}{2(d{+}1)^{2}}\frac{1}{\mathcal{D}_{12}{-}2d{+}4}D_{d,d,d,d}$
$\displaystyle\quad-\frac{2d+1}{4d^{2}(d+1)^{2}}\frac{1}{\mathcal{D}_{12}+2d}\mathcal{D}_{12}D_{d,d,d,d}+\frac{d^{2}{+}4d{+}2}{2(d+1)^{2}}\frac{1}{\mathcal{D}_{12}+2d}D_{d,d,d,d}\,.$
(7.18)

Together with eq. (7.14), we can now write down the result for the s-channel
graviton-exchange contribution to the four-scalar boundary correlator:

$3$$2$$1$$4$
$\displaystyle=\frac{1}{8}\frac{1}{\mathcal{D}_{12}+2d}(\mathcal{D}_{13}^{2}+\mathcal{D}_{23}^{2}-\mathcal{D}_{12}^{2})D_{d,d,d,d}$
$\displaystyle=2d^{2}(d+1)^{2}\left[P_{23}^{2}\frac{1}{\mathcal{D}_{12}}D_{d,d+2,d+2,d}+P_{13}^{2}\frac{1}{\mathcal{D}_{12}}D_{d+2,d,d+2,d}\right]$
$\displaystyle\quad-\frac{d^{3}(d-1)}{\mathcal{D}_{12}-2d+4}D_{d,d,d,d}-\frac{d^{2}}{2}P_{12}D_{d+1,d+1,d,d}+\frac{3d^{2}}{4}D_{d,d,d,d}\,.$
(7.19)

In particular, for $d=4$, it becomes

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{}{}\pgfsys@beginscope\pgfsys@invoke{
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{-15.08904pt}{-15.08904pt}\pgfsys@lineto{-8.5359pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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$\displaystyle=800P_{23}^{2}\frac{1}{\mathcal{D}_{12}}D_{4,6,6,4}+800P_{13}^{2}\frac{1}{\mathcal{D}_{12}}D_{6,4,6,4}-192\frac{1}{\mathcal{D}_{12}-4}D_{4,4,4,4}$
$\displaystyle\quad-8P_{12}D_{5,5,4,4}+12D_{4,4,4,4}\,.$ (7.20)

Using eq. (7.5) to expand every term into $D$-functions, we find that it
indeed agrees with DHoker:1999kzh  up to an overall normalization factor.

###  7.4 Constraining the graviton exchange correlators

Very interestingly, we can also use the recursion-based technique described
above to construct boundary correlators from a bootstrap point of view. We
illustrate this idea by considering the graviton exchange. Our starting point
is based on the following requirements:

  1. (R.1)

The scalar AdS correlator with $\Delta_{i}=d$ can be expressed in terms of a
finite sum of $D$-functions in the position space for even $d$.

  2. (R.2)

Every Witten diagram can be written as a set of
$\frac{1}{\mathcal{D}_{I}+\\#}$ acting on a contact interaction, where
$\mathcal{D}_{I}$’s are in one-to-one correspondence with bulk-bulk
propagators in this Witten diagram.

  3. (R.3)

The contact interaction in the previous requirement comes from pinching
internal legs of the Witten diagram. Specifically for the graviton exchange,
we assume that the minimal coupling to a massless spin-2 particle
$h_{\mu\nu}\nabla^{\mu}\varphi\nabla^{\nu}\varphi$ is part of the interaction.

The requirement (R.1) is motivated by explicit examples, such as the ones
described in previous sections as well as in certain supersymmetric theories
as discussed in e.g. DHoker:1999kzh ; Rastelli:2017udc ; Rastelli:2019gtj ,
where boundary correlators are given by a finite sum of $D$-functions. The
requirement (R.2) is motivated by an analogy with flat space amplitudes.
Remarkably, we do not need to fix a priori the shift $\\#$ to be the quadratic
Casimir. The requirement (R.3) comes from our expectations regarding the
interaction Lagrangian. We only write down the most essential interaction, but
keep an open mind that their consistency with (R.1) and (R.2) may lead to
additional contributions. We note that the Lagrangian (5.4) also contains a
coupling to the trace of the graviton,
$h^{\nu}{}_{\nu}\nabla^{\mu}\varphi\nabla_{\mu}\varphi$. We can of course add
this term to our requirement, but it will be interesting to see how far we can
go without it. Although based on slightly different theories, our approach
agrees in spirit with that in Ref. Rastelli:2017udc .

Using only the information above, we now try to construct the s-channel
diagram (5.28). From the interaction in (R.3), we know that the contact
interaction in (R.2) should take the form
$\nabla_{A}E_{1}\nabla^{A}E_{3}\nabla_{B}E_{2}\nabla^{B}E_{4}$ together with
its permutations. In particular, the contraction should be between legs at the
opposite ends of the bulk-bulk propagator. Therefore, after taking into
account Bose symmetry, the bulk integration of the contact interaction gives,

$\displaystyle\frac{1}{8}\frac{1}{\mathcal{D}_{12}+\\#}(\mathcal{D}_{23}^{2}D_{d,d+2,d+2,d}+\mathcal{D}_{13}^{2}D_{d+2,d,d+2,d})\,.$
(7.21)

While the relative coefficient is fixed by Bose symmetry, the overall
coefficient is of course unfixed; here we made an arbitrary choice for
convenience.

We can then follow eq. (7.14) and (7.3) to evaluate the action of
$\frac{1}{\mathcal{D}_{12}+\\#}$. As already noted below eq. (7.3), requiring
that the recursion truncates when $d$ is even leads to $\\#=2d$. This is
necessary for (R.1) to hold. Thus, the above consideration leads us to

$\displaystyle\frac{1}{8}\frac{1}{\mathcal{D}_{12}+2d}(\mathcal{D}_{23}^{2}+\mathcal{D}_{13}^{2})D_{d,d,d,d}$
$\displaystyle=2d^{2}(d+1)^{2}\left[P_{23}^{2}\frac{1}{\mathcal{D}_{12}}D_{d,d+2,d+2,d}+P_{13}^{2}\frac{1}{\mathcal{D}_{12}}D_{d+2,d,d+2,d}\right]-\frac{d^{3}(d-1)}{\mathcal{D}_{12}-2d+4}D_{d,d,d,d}$
$\displaystyle\quad+\frac{d^{2}}{2}D_{d,d,d,d}-\frac{d}{4}\frac{1}{\mathcal{D}_{12}+2d}\mathcal{D}_{12}D_{d,d,d,d}\,.$
(7.22)

Although the recursion does truncate, the correlator still is not a finite sum
of $D$-functions because of the last term in the above expression. The
simplest solution is to add it to the left hand side. This indicates that the
consistency with (R.1) and (R.2) naturally introduces additional interactions
other than the one in (R.3). It therefore follows that the “bootstrap”
procedure described here yields the following s-channel contribution to the
four-scalar correlator:

$\displaystyle\mathcal{M}_{\rm
bootstrap}(\text{s})=\frac{1}{8}\frac{1}{\mathcal{D}_{12}+2d}(\mathcal{D}^{2}_{23}+\mathcal{D}^{2}_{13}+2d\mathcal{D}_{12})D_{d,d,d,d}\,.$
(7.23)

It differs from the true s-channel graviton exchange (5.28) only by a contact
term,

$3$$2$$1$$4$ $\displaystyle=\mathcal{M}_{\rm
bootstrap}(\text{s})-\frac{1}{8}\mathcal{D}_{12}D_{d,d,d,d}\,.$ (7.24)

This additional contact term is exactly the contribution from the trace
$h^{\nu}{}_{\nu}\nabla^{\mu}\varphi\nabla_{\mu}\varphi$. The difference is
expected: our requirements cannot constrain polynomials of $\mathcal{D}_{ij}$.
Therefore, we have shown that based on three modest assumptions, we can
reproduce the s-channel graviton exchange up to contact terms. If we further
assume that the scalars do not have four-point contact interactions, then the
only ambiguity is $\mathcal{D}_{ij}D_{d,d,d,d}$. These terms sum up to zero if
we add up all three exchange channels. As a result, under the additional
assumption that scalar four-point interactions are absent, our bootstrap can
reproduce the AdS boundary correlator of four identical scalars mediated by a
graviton exchange,

$\displaystyle\mathcal{M}(\varphi_{1}\varphi_{2}\varphi_{3}\varphi_{4})=\mathcal{M}_{\rm
bootstrap}(\text{s})+\mathcal{M}_{\rm bootstrap}(\text{t})+\mathcal{M}_{\rm
bootstrap}(\text{u})\,.$ (7.25)

It is not difficult to verify that the sum indeed agrees with eq. (5.30) on
the support of the conformal Ward identity,
$(\mathcal{D}_{12}+\mathcal{D}_{23}+\mathcal{D}_{13})D_{d,d,d,d}=0$.

##  8 Conclusion

The differential representation of AdS boundary correlators is a new method
for both the exploration of their properties and for their evaluation. By
(conjecturally) organizing them solely in terms of generators of the AdS
symmetry group (acting on a single contact integral), it has the ability to
offer both a global perspective on all approaches to leading order AdS
boundary correlators and to expose some of their hidden properties and
unifying structures with flat space scattering amplitudes. For example, it
already provided a definition of the duality between color and kinematics and