$\displaystyle\sigma^{*}_{1}=\dfrac{1}{D}\dfrac{z_{24}\bar{z}_{34}}{z_{12}\bar{z}_{13}}\,,~{}~{}\sigma^{*}_{2}=-\dfrac{1}{D}\dfrac{z_{34}\bar{z}_{14}}{z_{23}\bar{z}_{12}}\,,~{}~{}\sigma^{*}_{3}=-\dfrac{1}{D}\dfrac{z_{24}\bar{z}_{14}}{z_{23}\bar{z}_{13}}\,,~{}~{}\sigma^{*}_{4}=\dfrac{1}{D}$
$\displaystyle
D=2\dfrac{z_{24}\bar{z}_{34}}{z_{12}\bar{z}_{13}}-2\dfrac{z_{34}\bar{z}_{14}}{z_{23}\bar{z}_{12}}$
(182)

One can show that the combination

$\sigma^{*}_{1}u_{1}+\sigma^{*}_{2}u_{2}-\sigma^{*}_{3}u_{3}-\sigma^{*}_{4}u_{4}\,,$
(183)

is proportional to $(z-\bar{z})$ under a global space-time translation
$u_{i}\to u_{i}+A+Bz_{i}+\bar{B}\bar{z}_{i}+Cz_{i}\bar{z}_{i}$. Thus, the
amplitude is invariant under global space-time translations due to the
$\delta(|z-\bar{z}|)$ constraint Banerjee:2018gce . This shows that the four
point function satisfies the Carrollian Ward identities.

####  6.5 Scalar four point function: exchange diagram

The analysis is identical to section 6.4, but the only difference is that we
have a bulk to bulk propagator in the Witten diagram. We consider the
t-channel massless exchange. This diagram has been used to illustrate the flat
space limit in Penedones:2010ue . For this diagram, we need a $\phi^{3}$ term
in the bulk.

$\begin{split}\langle
O_{\Delta_{1}}(\textbf{P}_{1})O_{\Delta_{2}}(\textbf{P}_{2})O_{\Delta_{3}}(\textbf{P}_{3})O_{\Delta_{4}}(\textbf{P}_{4})\rangle&=(ig)^{2}\int_{AdS_{d+1}}d^{d+1}\textbf{X}d^{d+1}\textbf{Y}\Pi_{\Delta}(\textbf{X},\textbf{Y})\\\
&~{}~{}\textbf{K}_{\Delta_{1}}(\textbf{P}_{1},\textbf{X})\textbf{K}_{\Delta_{3}}(\textbf{P}_{3},\textbf{X})\textbf{K}_{\Delta_{2}}(\textbf{P}_{2},\textbf{Y})\textbf{K}_{\Delta_{4}}(\textbf{P}_{4},\textbf{Y})\,.\end{split}$
(184)

The general expression of (143) evaluates to

$\begin{split}\langle
O_{\Delta_{1}}(\textbf{P}_{1})O_{\Delta_{2}}(\textbf{P}_{2})O_{\Delta_{3}}(\textbf{P}_{3})O_{\Delta_{4}}(\textbf{P}_{4})\rangle&\simeq\left(\prod_{i=1}^{4}N_{\Delta_{i}}^{d}\right)\left((ig)^{2}\int_{\mathbb{R}^{1,d}}d^{d+1}\textbf{X}d^{d+1}\textbf{Y}G(x,y)\right.\\\
&\left.\times\psi_{\Delta_{1},\tilde{q}_{1},u_{1}}^{-}(x)\psi_{\Delta_{3},\tilde{q}_{3},u_{3}}^{-}(x)\psi_{\Delta_{2},\tilde{q}_{2},u_{2}}^{+}(y)\psi_{\Delta_{4},\tilde{q}_{4},u_{4}}^{+}(y)\right)\,.\end{split}$
(185)

Due to momentum conservation, the internal momenta $k$ takes the value given
by

$\begin{split}k&=-\omega_{1}\tilde{q}_{1}-\omega_{3}\tilde{q}_{3}\\\ \implies
k^{2}&=2\omega_{1}\omega_{3}\tilde{q}_{1}\cdot\tilde{q}_{3}=-\dfrac{4\omega_{1}\omega_{3}z_{13}\bar{z}_{13}}{(1+z_{1}\bar{z}_{1})(1+z_{3}\bar{z}_{3})}=s_{13}\,,\end{split}$
(186)

where $s_{13}$ is the Mandelstam invariant. Thus, if we substitute this in
(185), we get

$\begin{split}\mathcal{A}_{(4)ex}(ig)^{2}\int
d\omega_{1}d\omega_{2}d\omega_{3}d\omega_{4}\,\omega^{\Delta_{1}-1}_{1}\omega^{\Delta_{2}-1}_{2}\omega^{\Delta_{3}-1}_{3}\omega^{\Delta_{4}-1}_{4}\,e^{i(\omega_{1}u_{1}+\omega_{3}u_{3}-\omega_{2}u_{2}-\omega_{4}u_{4})}\\\
\times\delta^{(4)}(\omega_{1}\tilde{q}_{1}+\omega_{3}\tilde{q}_{3}-\omega_{2}\tilde{q}_{2}-\omega_{4}\tilde{q}_{4})\dfrac{1}{s_{13}+i\epsilon}\,.\end{split}$
(187)

$\frac{1}{s_{13}}$ is the on-shell scattering amplitude of the t-channel
exchange process in flat space. Thus, in the large $R$ limit, we essentially
get a modified Mellin transformation of the scattering amplitude. This is what
one expects from the general proposal of Bagchi:2022emh .

Introducing simplex variables and doing a similar analysis as Appendix A.1, we
get the final result for the four point correlator as

$\mathcal{A}_{(4)ex}\dfrac{(1+z_{1}\bar{z}_{1})(1+z_{3}\bar{z}_{3})}{4z^{2}_{13}\bar{z}^{2}_{13}z_{24}\bar{z}_{24}}\delta(|z-\bar{z}|)\dfrac{(\sigma^{*}_{1})^{\Delta_{1}-2}(\sigma^{*}_{2})^{\Delta_{2}-1}(\sigma^{*}_{3})^{\Delta_{3}-2}(\sigma^{*}_{4})^{\Delta_{4}-1}}{[-i(\sigma^{*}_{1}u_{1}+\sigma^{*}_{3}u_{3}-\sigma^{*}_{2}u_{2}-\sigma^{*}_{4}u_{4})]^{\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4}-6}}\,.$
(188)

Here

$\mathcal{A}_{(4)ex}=\dfrac{(-i)^{\Delta_{1}+\Delta_{3}}(i)^{\Delta_{2}+\Delta_{4}}}{16\pi^{6}\prod_{i=1}^{4}\Gamma\left(\Delta_{i}-\frac{1}{2}\right)R^{4-(\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4})}}\,.$
(189)

The other quantities are

$\displaystyle\sigma^{*}_{1}=\dfrac{1}{D}\dfrac{z_{24}\bar{z}_{34}}{z_{12}\bar{z}_{13}}\,,~{}~{}\sigma^{*}_{2}=\dfrac{1}{D}\dfrac{z_{34}\bar{z}_{14}}{z_{23}\bar{z}_{12}}\,,~{}~{}\sigma^{*}_{3}=\dfrac{1}{D}\dfrac{z_{24}\bar{z}_{14}}{z_{23}\bar{z}_{13}}\,,~{}~{}\sigma^{*}_{4}=\dfrac{1}{D}$
$\displaystyle
D=2\dfrac{z_{24}\bar{z}_{34}}{z_{12}\bar{z}_{13}}+2\dfrac{z_{24}\bar{z}_{14}}{z_{23}\bar{z}_{13}}$
(190)

The presence of $\delta(|z-\bar{z}|)$ where $z$ is the cross ratio ensures
that the four point correlator remains invariant under global spacetime
translations. This factor in the results of (178) and (188) agrees with the
general intrinsic analysis of the four point Carroll correlators given in
section 4.3 (in particular (120)).

####  6.6 Gluon two point function

We now proceed to consider non-trivial spinning Witten diagrams. We first
consider the gluon two point function. The gluon two point function is given
by the following diagram 121212Note that here we do a bulk integral to obtain
the two point function. This is unlike the case in Witten:1998qj , where one
does a boundary integral. The difference between the two computations is just
an overall normalization constant.

$\langle
O_{\Delta_{1};\beta_{1}}(\textbf{P}_{1})O_{\Delta_{2};\beta_{2}}(\textbf{P}_{2})\rangle=\int
d\textbf{X}\,K^{\Delta_{1},1}_{\alpha_{1};\beta_{1}}(\textbf{P}_{1},\textbf{X})\,\eta^{\alpha_{1}\alpha_{2}}\,K^{\Delta_{2},1}_{\alpha_{2};\beta_{2}}(\textbf{P}_{2},\textbf{X})\,.$
(191)

$O_{\Delta_{2};\beta}$ corresponds to the operator with spin 1. Here
$\beta=z,\bar{z}$. Most importantly, $\beta\neq u$ because the bulk to
boundary propagator becomes pure gauge (150). We insert $\textbf{P}_{1}$ at
$\tau_{p}=-\frac{\pi}{2}+\frac{u_{1}}{R}$ and $\textbf{P}_{2}$ at
$\tau_{p}=+\frac{\pi}{2}+\frac{u_{2}}{R}$. In the large $R$ limit, using (153)
and (156) for $\beta_{i}=z^{a}$ we have

$\begin{split}\langle
O_{\Delta_{1};\beta_{1}}(\textbf{P}_{1})O_{\Delta_{2};\beta_{2}}(\textbf{P}_{2})\rangle&=\mathcal{D}_{(1)}\partial_{\beta_{1}}\tilde{q}_{1\alpha}\partial_{\beta_{2}}\tilde{q}^{\alpha}_{2}\int
d^{4}x\,\psi^{-}_{\Delta_{1},\tilde{q}_{1},u_{1}}(x)\,\psi^{+}_{\Delta_{2},\tilde{q}_{2},u_{2}}(x)\,,\\\
&=\mathcal{D}_{(1)}\partial_{\beta_{1}}\tilde{q}_{1\alpha}\partial_{\beta_{2}}\tilde{q}^{\alpha}_{2}\int
d^{4}x\int
d\omega_{1}d\omega_{2}\omega^{\Delta_{1}-1}_{1}\omega^{\Delta_{2}-1}_{2}\\\
&~{}~{}~{}~{}~{}~{}~{}\times
e^{i\omega_{1}u_{1}-i\omega_{2}u_{2}}e^{i(\omega_{1}\tilde{q}_{1}-\omega_{2}\tilde{q}_{2})\cdot
x}\,,\\\
&=\mathcal{D}_{(1)}\partial_{\beta_{1}}\tilde{q}_{1\alpha}\partial_{\beta_{2}}\tilde{q}^{\alpha}_{2}\dfrac{\Gamma(\Delta_{1}+\Delta_{2}-2)\,\delta^{2}(z_{12})}{[i(u_{2}-u_{1})]^{\Delta_{1}+\Delta_{2}-2}}\,.\end{split}$
(192)

Thus,

$\langle
O_{\Delta_{1};\beta_{1}}(\textbf{P}_{1})O_{\Delta_{2};\beta_{2}}(\textbf{P}_{2})\rangle=\mathcal{D}_{(1)}\partial_{\beta_{1}}\tilde{q}_{1\alpha}\partial_{\beta_{2}}\tilde{q}^{\alpha}_{2}\dfrac{\Gamma(\Delta_{1}+\Delta_{2}-2)\,\delta^{2}(z_{12})}{[i(u_{2}-u_{1})]^{\Delta_{1}+\Delta_{2}-2}}\,.$
(193)

Here

$\mathcal{D}_{(1)}=\dfrac{\mathcal{C}^{3}_{\Delta_{1};1}}{(-i)^{\Delta_{1}}\Gamma(\Delta_{1})}\dfrac{\Delta_{1}-1}{\Delta_{1}}\dfrac{\mathcal{C}^{3}_{\Delta_{2};1}}{(i)^{\Delta_{2}}\Gamma(\Delta_{2})}\dfrac{\Delta_{2}-1}{\Delta_{2}}=\dfrac{(i)^{\Delta_{1}}(-i)^{\Delta_{2}}}{4\pi^{3}\Gamma\left(\Delta_{1}-\frac{1}{2}\right)\Gamma\left(\Delta_{2}-\frac{1}{2}\right)R^{4-\Delta_{1}-\Delta_{2}}}\,.$
(194)

This result is identical to the scalar two point function of Bagchi:2022emh ;
Bagchi:2023fbj  except for normalization and a crucial factor of
$\partial_{\beta_{1}}\tilde{q}_{1\alpha}\partial_{\beta_{2}}\tilde{q}^{\alpha}_{2}$.
This crucial factor for various cases of $\nu_{i}$ is given by

$\begin{split}\partial_{\bar{z}_{1}}\tilde{q}_{1\alpha}\partial_{\bar{z}_{2}}\tilde{q}^{\alpha}_{2}&=\dfrac{-2\,z^{2}_{12}}{(1+z_{1}\bar{z}_{1})^{2}(1+z_{2}\bar{z}_{2})^{2}}\,,\\\
\partial_{z_{1}}\tilde{q}_{1\alpha}\partial_{z_{2}}\tilde{q}^{\alpha}_{2}&=\dfrac{-2\,\bar{z}^{2}_{12}}{(1+z_{1}\bar{z}_{1})^{2}(1+z_{2}\bar{z}_{2})^{2}}\,,\\\
\partial_{z_{1}}\tilde{q}_{1\alpha}\partial_{\bar{z}_{2}}\tilde{q}^{\alpha}_{2}&=\dfrac{2(1+z_{2}\bar{z}_{1})^{2}}{(1+z_{1}\bar{z}_{1})^{2}(1+z_{2}\bar{z}_{2})^{2}}\,.\end{split}$
(195)

Now, because of the delta function $\delta^{2}(z_{12})$, the two point
function for spin 1 particles will be non zero only if $\nu_{1}=z_{1}$ and
$\nu_{2}=\bar{z}_{2}$ and the result is given by

$\langle
O_{\Delta_{1};z_{1}}(\textbf{P}_{1})O_{\Delta_{2};{\bar{z}}_{2}}(\textbf{P}_{2})\rangle=\mathcal{D}_{(1)}\dfrac{2}{(1+z_{1}\bar{z}_{1})(1+z_{2}\bar{z}_{2})}\dfrac{\Gamma(\Delta_{1}+\Delta_{2}-2)\,\delta^{2}(z_{12})}{[i(u_{2}-u_{1})]^{\Delta_{1}+\Delta_{2}-2}}\,.$
(196)

We are learning that the $z$ component of the spinning primary behaves like a
helicity $1$ particle, and the $\bar{z}$ component of the spinning primary
behaves like a helicity $-1$ particle. This implies $\sigma_{1}+\sigma_{2}=0$
for a non-zero two point function. Also, another non-trivial result is that
the correlators involving the $u$-component of the field become zero because
the bulk to boundary propagator becomes pure gauge (150). (196) agrees with
the intrinsic result of (79) if we construct representations of vector
primaries with upper indices. Our result also agrees with the recently
obtained result from the embedding space approach Salzer:2023jqv  and the bulk
result of Nguyen:2023miw .

####  6.7 Scalar-gluon-scalar three point function: split signature

We now consider different cases of the three point function. The simplest
possible local cubic interaction vertex in AdS involving a spin $J$ field and
two scalars $\phi_{1}$ and $\phi_{2}$ is given by Costa:2014kfa

$g_{\phi_{1}\phi_{2}h}\int_{\text{AdS}_{d+1}}d\textbf{X}\left(\phi_{1}\nabla_{\mu_{1}}\dots\nabla_{\mu_{J}}\phi_{2}\right)h^{\mu_{1}\dots\mu_{J}}\,.$
(197)

The derivatives can act on either of the scalar fields because we consider the
spin $J$ field to have vanishing divergence.

We now consider spin 1, and the contact Witten diagram is given by

$\begin{split}\langle
O_{\Delta_{1}}(\textbf{P}_{1})O_{\Delta_{2};\beta}(\textbf{P}_{2})&O_{\Delta_{3}}(\textbf{P}_{3})\rangle\\\
&=g_{\phi_{1}\phi_{2}h}\int_{\text{AdS}_{d+1}}d\textbf{X}\,K_{\Delta_{1}}(\textbf{P}_{1},\textbf{X})\,K^{\Delta_{2},1}_{\alpha,\beta}(\textbf{P}_{2},\textbf{X})\,K_{\Delta_{3}}(\textbf{P}_{3},\textbf{X})(k^{\alpha}_{1}+k^{\alpha}_{3})\,.\end{split}$
(198)

Here $k^{\alpha}_{1}$ and $k^{\alpha}_{3}$ correspond to the ingoing and
outgoing momenta of the massless scalars. Inserting $\textbf{P}_{1}$ at
$\tau_{p}=-\frac{\pi}{2}+\frac{u_{1}}{R}$, $\textbf{P}_{2}$ at
$\tau_{p}=+\frac{\pi}{2}+\frac{u_{2}}{R}$, $\textbf{P}_{3}$ at
$\tau_{p}=+\frac{\pi}{2}+\frac{u_{3}}{R}$ and then taking the large $R$ limit,
we have using (138), (153) and (156)

$\begin{split}\langle
O_{\Delta_{1}}(\textbf{P}_{1})O_{\Delta_{2};\beta}(\textbf{P}_{2})O_{\Delta_{3}}(&\textbf{P}_{3})\rangle=\mathcal{B}_{(1)sp}\int_{\mathbb{R}^{1,3}}d^{4}x\,\psi^{-}_{\Delta_{1},\tilde{q}_{1},u_{1}}(x)\,\psi^{-}_{\Delta_{2},\tilde{q}_{2},u_{2}}(x)\,\psi^{+}_{\Delta_{3},\tilde{q}_{3},u_{3}}(x)\\\
&\hskip
85.35826pt\times\partial_{\beta}\tilde{q}_{2\alpha}(k^{\alpha}_{1}+k^{\alpha}_{3})\\\
&=\mathcal{B}_{(1)sp}\int
d\omega_{1}\,d\omega_{2}\,d\omega_{3}\,\omega^{\Delta_{1}-1}_{1}\omega^{\Delta_{2}-1}_{2}\omega^{\Delta_{3}-1}_{3}e^{i(\omega_{1}u_{1}+\omega_{2}u_{2}-\omega_{3}u_{3})}\\\
&~{}~{}~{}(\omega_{1}\tilde{q}^{\alpha}_{1}+\omega_{3}\tilde{q}^{\alpha}_{3})(\partial_{\beta}\tilde{q}_{2\alpha})\delta^{(4)}(\omega_{1}\tilde{q}_{1}+\omega_{2}\tilde{q}_{2}-\omega_{3}\tilde{q}_{3})\,.\end{split}$
(199)

Here, the normalization constant is given by

$\mathcal{B}_{(1)sp}=N^{3}_{\Delta_{1}}\dfrac{\mathcal{C}^{d}_{\Delta_{2};1}}{(i)^{\Delta_{2}}\Gamma(\Delta_{2})}\dfrac{\Delta_{2}-1}{\Delta_{2}}N^{3}_{\Delta_{3}}=\dfrac{i^{\Delta_{1}}\,(-i)^{\Delta_{2}+\Delta_{3}}}{8\pi^{\frac{9}{2}}\,\prod^{3}_{i=1}\Gamma\left(\Delta_{i}-\frac{1}{2}\right)R^{4-(\Delta_{1}+\Delta_{2}+\Delta_{3})}}\,.$
(200)

We have used (140) and (146).

We now use the split of (171) for the momentum conserving delta function. If
we use that, we have 131313 For these vertex factors, we take the momenta
$\tilde{q}_{i}$ of the polarization vectors in the bulk to boundary propagator
of (153) in the standard parametrization given by (162). For the
$\tilde{q}_{i}$ that appears in the momentum conserving delta function in the
combination $\omega_{i}\tilde{q}_{i}$, we choose
$\tilde{q}_{i}=(1+z_{i}\bar{z}_{i},z_{i}+\bar{z}_{i},-i(z_{i}-\bar{z}_{i}),1-z_{i}\bar{z}_{i})\,.$
We can always do this by suitably redefining
$\omega_{i}\to\omega_{i}(1+z_{i}\bar{z}_{i})$. This ensures that the vertex
factors are simple. This will be extremely helpful when we look at graviton
three point amplitudes.

$V_{3\,\text{gluon}(-)}=(\omega_{1}\tilde{q}^{\alpha}_{1}+\omega_{3}\tilde{q}^{\alpha}_{3})(\partial_{\bar{z_{2}}}\,\tilde{q}_{2\alpha})=\dfrac{4\,z_{32}\,\omega_{3}}{(1+z_{2}\bar{z}_{2})}\,.$
(201)

We finally get

$\begin{split}\langle
O_{\Delta_{1}}(\textbf{P}_{1})O_{\Delta_{2};\bar{z}_{2}}(\textbf{P}_{2})O_{\Delta_{3}}(\textbf{P}_{3})\rangle=32\,\mathcal{B}_{(1)sp}\,C_{z}\,(-1)^{\Delta_{2}}z^{\Delta_{3}-1}_{12}z^{\Delta_{1}-1}_{32}z^{\Delta_{2}-2}_{31}\\\
\times\dfrac{\Gamma(\Delta_{1}+\Delta_{2}+\Delta_{3}-3)\,\delta(\bar{z}_{12})\,\delta(\bar{z}_{13})}{[-i(z_{1}u_{23}+z_{2}u_{31}+z_{3}u_{12})]^{\Delta_{1}+\Delta_{2}+\Delta_{3}-3}}\,.\end{split}$
(202)

$C_{z}$ is a normalization constant that is given by

$C_{z}=\dfrac{1}{1+z_{2}\bar{z}_{2}}\,.$ (203)

(202) matches with the intrinsic result (106) once we use the fact that the
helicities are given by $\sigma_{2}=-1,\sigma_{1}=0,\sigma_{3}=0$. (202) also
agrees with Salzer:2023jqv . We can also consider the equivalent $z$ component
of (202). This would just be given by the complex conjugate of (202). The
detailed calculation resulting in (202) is given in Appendix A.3.

####  6.8 Gluon MHV three point amplitude: split signature

The gluon three point vertex with three incoming momenta $k,p,q$ is given by
schwartz_2013

$g\,f^{abc}[g^{\mu\nu}(k-p)^{\rho}+g^{\nu\rho}(p-q)^{\mu}+g^{\rho\mu}(q-k)^{\nu}]\,.$
(204)

Here $g$ is the coupling constant, and $a,b,c$ denote the colour indices. Thus
$f^{abc}$ denotes the structure constants of the $SU(N)$ gauge group. We have
two incoming momenta $q_{1},q_{2}$ and one outgoing momenta $q_{3}$. Thus, the
relevant vertex function for us is given by

$V^{\alpha_{1}\alpha_{2}\alpha_{3}}_{3g}=gf^{abc}[g^{\alpha_{1}\alpha_{2}}(q_{1}-q_{2})^{\alpha_{3}}+g^{\alpha_{2}\alpha_{3}}(q_{2}+q_{3})^{\alpha_{1}}+g^{\alpha_{3}\alpha_{1}}(-q_{3}-q_{1})^{\alpha_{2}}]\,.$
(205)

As explained in Witten:1998qj , we can take the vertex factor from the usual
QFT analysis to evaluate contact Witten diagrams. For us, in the flat limit,
the metric factors $g^{\mu\nu}\to\eta^{\mu\nu}$.

Thus, the contact three point Witten diagram is given by

$\begin{split}\langle
O_{\Delta_{1};\beta_{1}}(\textbf{P}_{1})O_{\Delta_{2};\beta_{2}}(\textbf{P}_{2})&O_{\Delta_{3};\beta_{3}}(\textbf{P}_{3})\rangle\\\
&=\int
d\textbf{X}\,K^{\Delta_{1},1}_{\alpha_{1};\beta_{1}}(\textbf{P}_{1},\textbf{X})\,K^{\Delta_{2},1}_{\alpha_{2};\beta_{2}}(\textbf{P}_{2},\textbf{X})\,K^{\Delta_{3},1}_{\alpha_{3};\beta_{3}}(\textbf{P}_{3},\textbf{X})V^{\alpha_{1}\alpha_{2}\alpha_{3}}_{3g}\,.\end{split}$
(206)

We consider the $(--+)$ MHV amplitude $\beta_{1}=\bar{z}_{1}$,
$\beta_{2}=\bar{z}_{2}$, $\beta_{3}=z_{3}$. Thus, in the large $R$ limit,
using (153) and (156) we have

$\begin{split}\langle
O_{\Delta_{1};\bar{z}_{1}}(\textbf{P}_{1})O_{\Delta_{2};\bar{z}_{2}}(\textbf{P}_{2})&O_{\Delta_{3};z_{3}}(\textbf{P}_{3})\rangle\\\
&=\mathcal{C}_{(3)g}\int
d\omega_{1}d\omega_{2}d\omega_{3}\omega^{\Delta_{1}-1}_{1}\omega^{\Delta_{2}-1}_{2}\omega^{\Delta_{3}-1}_{3}e^{i(\omega_{1}u_{1}+\omega_{2}u_{2}-\omega_{3}u_{3})}\\\
&~{}~{}~{}~{}\times\partial_{\bar{z}_{1}}\tilde{q}_{1\alpha_{1}}\partial_{\bar{z}_{2}}\tilde{q}_{2\alpha_{2}}\partial_{z_{3}}\tilde{q}_{3\alpha_{3}}V^{\alpha_{1}\alpha_{2}\alpha_{3}}_{3g}\delta^{(4)}(\omega_{1}\tilde{q}_{1}+\omega_{2}\tilde{q}_{2}-\omega_{3}\tilde{q}_{3})\,.\end{split}$
(207)

Using the momentum conserving delta function (171), we evaluate the vertex
factor according to the parametrization explained in footnote 13:

$\begin{split}\partial_{\bar{z}_{1}}\tilde{q}_{1\alpha_{1}}\partial_{\bar{z}_{2}}\tilde{q}_{2\alpha_{2}}\partial_{z_{3}}\tilde{q}_{3\alpha_{3}}V^{\alpha_{1}\alpha_{2}\alpha_{3}}_{3g}&=gf^{abc}\left[\partial_{\bar{z}_{1}}\tilde{q}_{1\mu}\partial_{\bar{z}_{2}}\tilde{q}^{\mu}_{2}\partial_{z_{3}}\tilde{q}_{3\nu}(\omega_{1}\tilde{q}_{1}+\omega_{2}\tilde{q}_{2})^{\nu}+\partial_{\bar{z}_{2}}\tilde{q}_{2\mu}\partial_{z_{3}}\tilde{q}^{\mu}_{3}\partial_{\bar{z}_{1}}\tilde{q}_{1\nu}\right.\\\
&~{}~{}~{}\left.\times(\omega_{2}\tilde{q}_{2}+\omega_{3}\tilde{q}_{3})^{\nu}+\partial_{\bar{z}_{1}}\tilde{q}_{1\mu}\partial_{z_{3}}\tilde{q}^{\mu}_{3}\partial_{\bar{z}_{2}}\tilde{q}_{2\nu}(-\omega_{3}\tilde{q}_{3}-\omega_{1}\tilde{q}_{1})^{\nu}\right]\\\
&=\dfrac{-8g\,f^{abc}\,z_{12}\,\omega_{3}}{(1+z_{1}\bar{z}_{1})(1+z_{2}\bar{z}_{2})(1+z_{3}\bar{z}_{3})}\\\
&=gf^{abc}D_{z}z_{12}\,\omega_{3}=V_{3\,\text{gloun}\,(--+)}\,.\end{split}$
(208)

Here $D_{z}$ is given by

$D_{z}=\dfrac{1}{(1+z_{1}\bar{z}_{1})(1+z_{2}\bar{z}_{2})(1+z_{3}\bar{z}_{3})}\,.$
(209)

At this point it is useful to recall the MHV $(--+)$ gluon amplitude in terms
of spinor helicity variables schwartz_2013 :

$\mathcal{A}^{\text{gloun}}_{--+}=\dfrac{\langle 12\rangle^{3}}{\langle
23\rangle\langle
31\rangle}\delta^{(4)}(p^{\mu}_{1}+p^{\mu}_{2}+p^{\mu}_{3})\,.$ (210)

Following the conventions for spinor-helicity variables of Pasterski:2017ylz
and using the momentum conserving delta function (171), we have

$\dfrac{\langle 12\rangle^{3}}{\langle 23\rangle\langle
31\rangle}\propto\omega_{3}z_{12}\propto V_{3\,\text{gloun}\,(--+)}\,.$ (211)

Thus, in the large $R$ limit, the Witten diagram (207) essentially becomes the
modified Mellin transform of the flat space scattering amplitude given by
(208).

Using (171), we can simplify (207) using similar manipulations as Appendix A.2
and A.3. The final answer is given by

$\begin{split}\langle
O_{\Delta_{1};\bar{z}_{1}}(\textbf{P}_{1})O_{\Delta_{2};\bar{z}_{2}}(\textbf{P}_{2})O_{\Delta_{3};z_{3}}(\textbf{P}_{3})\rangle&=32gf^{abc}\mathcal{C}_{(3)g}\,D_{z}\,(-1)^{\Delta_{1}+\Delta_{2}}z^{\Delta_{3}}_{12}z^{\Delta_{1}-2}_{23}z^{\Delta_{2}-2}_{31}\\\
&~{}~{}~{}\times\dfrac{\Gamma(\Delta_{1}+\Delta_{2}+\Delta_{3}-3)\,\delta(\bar{z}_{13})\delta(\bar{z}_{23})}{[-i(z_{1}u_{23}+z_{2}u_{31}+z_{3}u_{12})]^{\Delta_{1}+\Delta_{2}+\Delta_{3}-3}}\,.\end{split}$
(212)

Here $\mathcal{C}_{(3)g}$ is given by

$\begin{split}\mathcal{C}_{(3)g}&=\dfrac{\mathcal{C}^{3}_{\Delta_{1};1}}{(-i)^{\Delta_{1}}\Gamma(\Delta_{1})}\dfrac{\Delta_{1}-1}{\Delta_{1}}\dfrac{\mathcal{C}^{3}_{\Delta_{2};1}}{(-i)^{\Delta_{2}}\Gamma(\Delta_{2})}\dfrac{\Delta_{2}-1}{\Delta_{2}}\dfrac{\mathcal{C}^{3}_{\Delta_{3};1}}{(i)^{\Delta_{3}}\Gamma(\Delta_{3})}\dfrac{\Delta_{3}-1}{\Delta_{3}}\\\
&=\dfrac{i^{\Delta_{1}+\Delta_{2}}(-i)^{\Delta_{3}}}{8\pi^{\frac{9}{2}}\prod_{i=1}^{3}\Gamma\left(\Delta_{i}-\frac{1}{2}\right)R^{6-\Delta_{1}-\Delta_{2}-\Delta_{3}}}\,.\end{split}$
(213)

This answer matches with the intrinsic result (106) as we have
$\sigma_{1}=-1$, $\sigma_{2}=-1$ and $\sigma_{3}=+1$.

We can also consider the other $(++-)$ MHV amplitude. In (206), we basically
consider $\beta_{1}=z_{1},\beta_{2}=z_{2},\beta_{3}=\bar{z}_{3}$. Thus, in the
large $R$ limit, we have

$\begin{split}\langle
O_{\Delta_{1};z_{1}}(\textbf{P}_{1})O_{\Delta_{2};z_{2}}(\textbf{P}_{2})&O_{\Delta_{3};\bar{z}_{3}}(\textbf{P}_{3})\rangle\\\
&=\mathcal{C}_{(3)}\int
d\omega_{1}d\omega_{2}d\omega_{3}\omega^{\Delta_{1}-1}_{1}\omega^{\Delta_{2}-1}_{2}\omega^{\Delta_{3}-1}_{3}e^{i(\omega_{1}u_{1}+\omega_{2}u_{2}-\omega_{3}u_{3})}\\\
&~{}~{}~{}~{}\times\partial_{z_{1}}\tilde{q}_{1\alpha_{1}}\partial_{z_{2}}\tilde{q}_{2\alpha_{2}}\partial_{\bar{z}_{3}}\tilde{q}_{3\alpha_{3}}V^{\alpha_{1}\alpha_{2}\alpha_{3}}_{3g}\delta^{(4)}(\omega_{1}\tilde{q}_{1}+\omega_{2}\tilde{q}_{2}-\omega_{3}\tilde{q}_{3})\end{split}$
(214)

The vertex factor analogous to (208) essentially becomes the complex conjugate

$\begin{split}\partial_{\bar{z}_{1}}\tilde{q}_{1\alpha_{1}}\partial_{\bar{z}_{2}}\tilde{q}_{2\alpha_{2}}\partial_{z_{3}}\tilde{q}_{3\alpha_{3}}V^{\alpha_{1}\alpha_{2}\alpha_{3}}_{3g}&=\dfrac{-8g\,f^{abc}\,\bar{z}_{12}\,\omega_{3}}{(1+z_{1}\bar{z}_{1})(1+z_{2}\bar{z}_{2})(1+z_{3}\bar{z}_{3})}\\\
&=gf^{abc}D_{z}\bar{z}_{12}\,\omega_{3}=V_{3\,\text{gloun}\,(++-)}\,.\end{split}$
(215)

Now, because of this complex conjugate, we should crucially consider the
complex conjugate of the momentum conserving delta function split (171) to
ensure that we get a non-trivial result:

$\delta^{(4)}(\omega_{1}\tilde{q}_{1}+\omega_{2}\tilde{q}_{2}-\omega_{3}\tilde{q}_{3})=\dfrac{4}{\omega^{2}_{3}\bar{z}_{23}\bar{z}_{31}}\delta\left(\omega_{1}-\omega_{3}\dfrac{\bar{z}_{32}}{\bar{z}_{12}}\right)\delta\left(\omega_{2}-\omega_{3}\dfrac{\bar{z}_{31}}{\bar{z}_{21}}\right)\delta(z_{13})\delta(z_{23})\,.$
(216)

We should choose the orderings $\bar{z}_{1}<\bar{z}_{3}<\bar{z}_{2}$ or
$\bar{z}_{2}<\bar{z}_{3}<\bar{z}_{1}$ to ensure that the energies are
positive. Thus, the vertex factor of the bulk Witten diagram determines what
branch of the delta function we should take to ensure a non-trivial answer.
The final result can be obtained as

$\begin{split}\langle
O_{\Delta_{1};z_{1}}(\textbf{P}_{1})O_{\Delta_{2};z_{2}}(\textbf{P}_{2})O_{\Delta_{3};\bar{z}_{3}}(\textbf{P}_{3})\rangle&=8gf^{abc}\mathcal{C}_{(3)}\,D_{z}\,(-1)^{\Delta_{1}+\Delta_{2}}\bar{z}^{\Delta_{3}}_{12}\bar{z}^{\Delta_{1}-2}_{23}\bar{z}^{\Delta_{2}-2}_{31}\\\
&~{}~{}~{}\times\dfrac{\Gamma(\Delta_{1}+\Delta_{2}+\Delta_{3}-3)\,\delta(z_{13})\,\delta(z_{23})}{[-i(\bar{z}_{1}u_{23}+\bar{z}_{2}u_{31}+\bar{z}_{3}u_{12})]^{\Delta_{1}+\Delta_{2}+\Delta_{3}-3}}\,.\end{split}$
(217)

This answer matches the alternate branch intrinsic result (107) if we use
$\sigma_{1}=1,\sigma_{2}=1,\sigma_{3}=-1$.

We do not have MHV amplitudes for $(+++)$ and $(---)$ cases. That is reflected
from the vertex factors for both cases vanishing when we use the momentum
conserving delta function of (216) and (171) respectively i.e.,

$\begin{split}\partial_{z_{1}}\tilde{q}_{1\alpha_{1}}\partial_{z_{2}}\tilde{q}_{2\alpha_{2}}\partial_{z_{3}}\tilde{q}_{3\alpha_{3}}V^{\alpha_{1}\alpha_{2}\alpha_{3}}_{3g}&=0\,,\\\
\partial_{\bar{z}_{1}}\tilde{q}_{1\alpha_{1}}\partial_{\bar{z}_{2}}\tilde{q}_{2\alpha_{2}}\partial_{\bar{z}_{3}}\tilde{q}_{3\alpha_{3}}V^{\alpha_{1}\alpha_{2}\alpha_{3}}_{3g}&=0\,.\end{split}$
(218)

####  6.9 Gluon three point amplitude from $F^{3}$: split signature

We see that from the Carrollian field theory analysis; we do have counterparts
for correlators that have the helicity combinations $(+++)$ or $(---)$ in
(106). The little group scaling would imply that the tree level S-Matrix
vanishes for these helicity configurations if we only have renormalizable
interactions elvang_huang_2015 . From the bulk perspective, these arise from
higher derivative terms or loop corrections. For instance, in our case, they
arise from Dixon:1993xd ; Dixon:2004za

$f^{abc}F^{a\mu}_{~{}~{}\nu}F^{b\nu}_{~{}~{}\rho}F^{c\rho}_{~{}~{}\mu}\,,$
(219)

where $F^{a}_{\mu\nu}$ is the $SU(N)$ Yang-Mills field strength tensor. The
vertex factor for three ingoing momenta $p_{1},p_{2},p_{3}$ can be worked out
as

$V_{3\,g_{2}}^{\alpha_{1}\alpha_{2}\alpha_{3}}=f^{abc}\left(p^{\alpha_{1}}_{1}p^{\alpha_{2}}_{2}p^{\alpha_{3}}_{3}-p^{\alpha_{3}}_{1}p^{\alpha_{1}}_{2}p^{\alpha_{2}}_{3}\right)\,.$
(220)

Using the split signature momentum conserving delta function (171) for the
$(---)$ case, we have

$\begin{split}\partial_{\bar{z}_{1}}\tilde{q}_{1\alpha_{1}}\partial_{\bar{z}_{2}}\tilde{q}_{2\alpha_{2}}\partial_{\bar{z}_{3}}\tilde{q}_{3\alpha_{3}}V^{\alpha_{1}\alpha_{2}\alpha_{3}}_{3g}&=\dfrac{8z^{2}_{13}z^{2}_{23}\omega^{3}_{3}}{z_{12}}\dfrac{1}{(1+z_{1}\bar{z}_{1})(1+z_{2}\bar{z}_{2})(1+z_{3}\bar{z}_{3})}\\\
&=\dfrac{z^{2}_{13}z^{2}_{23}\omega^{3}_{3}}{z_{12}}D_{z}\,.\end{split}$ (221)

Here $D_{z}$ is given by (209).

It is now straightforward to work out the three point function/ amplitude by
doing similar manipulations as Appendix A.2 and A.3. The final result is given
by

$\begin{split}\langle
O_{\Delta_{1};\bar{z}_{1}}(\textbf{P}_{1})O_{\Delta_{2};\bar{z}_{2}}(\textbf{P}_{2})O_{\Delta_{3};\bar{z}_{3}}(\textbf{P}_{3})\rangle&=8gf^{abc}\mathcal{C}_{(3)}\,D_{z}\,(-1)^{\Delta_{2}}z^{\Delta_{3}}_{12}z^{\Delta_{1}}_{32}z^{\Delta_{2}}_{31}\delta(\bar{z}_{13})\delta(\bar{z}_{23})\\\
&~{}~{}~{}\times\dfrac{\Gamma(\Delta_{1}+\Delta_{2}+\Delta_{3}-1)}{[-i(z_{1}u_{23}+z_{2}u_{31}+z_{3}u_{12})]^{\Delta_{1}+\Delta_{2}+\Delta_{3}-1}}\,.\end{split}$
(222)

$\mathcal{C}_{(3)}$ is given by (213). This matches the result of (106) for
$\sigma_{1}=-1,\sigma_{2}=-1,\sigma_{3}=-1$. This was also recently worked out
in Salzer:2023jqv . The $(+++)$ case is just the complex conjugate of the
above answer with $\delta(z_{12})\delta(z_{23})$. It is instructive to note
that this amplitude has not been focussed in the celestial amplitudes
literature Pasterski:2021rjz . As we remarked before, these diagrams are
quadratically divergent in the Mellin basis. Hence, PhysRevD.96.085006 ;
Stieberger:2018edy  have mostly considered MHV amplitudes. In the Carrollian
approach, these different helicity configuration amplitudes are naturally
encoded in the holographic correlation functions, which are crucially finite.

####  6.10 Graviton two point function

We will see that features of the gluon two point function (193) are also
present for the graviton two point function. The graviton two point function
is given by

$\langle
O_{\Delta_{1};\beta_{1}\beta_{2}}(\textbf{P}_{1})O_{\Delta_{2};\beta_{3}\beta_{4}}(\textbf{P}_{2})\rangle=\int_{\text{AdS}_{4}}d\textbf{X}\,K^{\Delta_{1},2}_{\alpha_{1}\alpha_{2};\beta_{1}\beta_{2}}(\textbf{P}_{1},\textbf{X})V^{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}_{2(grav)}K^{\Delta_{1},2}_{\alpha_{3}\alpha_{4};\beta_{3}\beta_{4}}(\textbf{P}_{2},\textbf{X})\,.$
(223)

The vertex factor arises from the graviton propagator elvang_huang_2015 :

$V^{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}_{2(grav)}=g^{\alpha_{1}\alpha_{3}}g^{\alpha_{2}\alpha_{4}}+g^{\alpha_{1}\alpha_{4}}g^{\alpha_{2}\alpha_{3}}-g^{\alpha_{1}\alpha_{2}}g^{\alpha_{3}\alpha_{4}}\,.$
(224)

Due to (157), the indices $\beta_{i}\neq u$ as they will correspond to pure
gauge modes. Thus, we consider $\beta_{i}=z^{a}$, and the bulk to boundary
propagators in the flat limit have been worked in (158). The modified Mellin
gauge representative is given by

$K^{\Delta,2}_{\alpha_{1}\alpha_{2};\beta_{1}\beta_{2}}(\textbf{P},\textbf{X})=\dfrac{\mathcal{C}^{d}_{\Delta;2}}{(\pm
i)^{\Delta}\Gamma(\Delta)}\dfrac{\Delta-1}{\Delta}(\pm
P^{\rho_{1}\rho_{2}}_{\beta_{1}\beta_{2}}\,\partial_{\rho_{1}}\tilde{q}_{\alpha_{1}}\,\partial_{\rho_{2}}\tilde{q}_{\alpha_{2}})\int^{\infty}_{0}d\omega\,\omega^{\Delta-1}\,e^{\mp
i\omega\,u}e^{\mp i\omega\,\tilde{q}\cdot x}\,.$ (225)

Here $P^{\rho_{1}\rho_{2}}_{\beta_{1}\beta_{2}}$ is the projector that
projects to the transverse and traceless part of the propagator

$P^{\rho_{1}\rho_{2}}_{\beta_{1}\beta_{2}}=\delta^{\rho_{1}}_{(\beta_{1}}\delta^{\rho_{2}}_{\beta_{2})}-\dfrac{1}{2}g_{\beta_{1}\beta_{2}}g^{\rho_{1}\rho_{2}}\,.$
(226)

The graviton two point function thus becomes

$\begin{split}\langle
O_{\Delta_{1};\beta_{1}\beta_{2}}(\textbf{P}_{1})O_{\Delta_{2};\beta_{3}\beta_{4}}&(\textbf{P}_{2})\rangle\\\
&=\mathcal{A}_{(2)grav}(-P^{\rho_{1}\rho_{2}}_{\beta_{1}\beta_{2}}\,\partial_{\rho_{1}}\tilde{q}_{\alpha_{1}}\,\partial_{\rho_{2}}\tilde{q}_{\alpha_{2}})(P^{\rho_{3}\rho_{4}}_{\beta_{3}\beta_{4}}\,\partial_{\rho_{3}}\tilde{q}_{\alpha_{3}}\,\partial_{\rho_{4}}\tilde{q}_{\alpha_{4}})V^{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}_{2(grav)}\\\
&~{}~{}~{}\times\int
d^{4}x\,\psi^{-}_{\Delta_{1},\tilde{q}_{1},u_{1}}(x)\psi^{+}_{\Delta_{2},\tilde{q}_{2},u_{2}}(x)\\\
&=\mathcal{A}_{(2)grav}(-P^{\rho_{1}\rho_{2}}_{\beta_{1}\beta_{2}}\,\partial_{\rho_{1}}\tilde{q}_{\alpha_{1}}\,\partial_{\rho_{2}}\tilde{q}_{\alpha_{2}})(P^{\rho_{3}\rho_{4}}_{\beta_{3}\beta_{4}}\,\partial_{\rho_{3}}\tilde{q}_{\alpha_{3}}\,\partial_{\rho_{4}}\tilde{q}_{\alpha_{4}})V^{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}_{2(grav)}\\\
&~{}~{}~{}\times\dfrac{\Gamma(\Delta_{1}+\Delta_{2}-2)\,\delta^{2}(z_{12})}{[-i(u_{1}-u_{2})]^{\Delta_{1}+\Delta_{2}-2}}\,.\end{split}$
(227)

Here, the normalization constant is given by

$\begin{split}\mathcal{A}_{(2)grav}&=\dfrac{\mathcal{C}^{3}_{\Delta_{1};2}}{(-i)^{\Delta_{1}}\Gamma(\Delta_{1})}\dfrac{\Delta_{1}-1}{\Delta_{1}}\dfrac{\mathcal{C}^{3}_{\Delta_{2};2}}{(i)^{\Delta_{2}}\Gamma(\Delta_{2})}\dfrac{\Delta_{2}-1}{\Delta_{2}}\\\
&=\dfrac{(i)^{\Delta_{1}}(-i)^{\Delta_{2}}(\Delta_{1}+1)(\Delta_{2}+1)}{4\pi^{3}\Delta_{1}\Delta_{2}\Gamma\left(\Delta_{1}-\frac{1}{2}\right)\Gamma\left(\Delta_{2}-\frac{1}{2}\right)R^{6-\Delta_{1}-\Delta_{2}}}\,.\end{split}$
(228)

Thus, the behaviour of the two point function crucially depends on

$G_{(2)}=(-P^{\rho_{1}\rho_{2}}_{\beta_{1}\beta_{2}}\,\partial_{\rho_{1}}\tilde{q}_{\alpha_{1}}\,\partial_{\rho_{2}}\tilde{q}_{\alpha_{2}})(P^{\rho_{3}\rho_{4}}_{\beta_{3}\beta_{4}}\,\partial_{\rho_{3}}\tilde{q}_{\alpha_{3}}\,\partial_{\beta_{4}}\tilde{q}_{\alpha_{4}})V^{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}_{2(grav)}\,.$
(229)

We will evaluate $G_{(2)}$ for various cases. Let us consider the case when
$\beta_{1}=\beta_{2}=z_{1}$ and $\beta_{3}=\beta_{4}=\bar{z}_{2}$. We have

$G_{(2)}=\dfrac{8(1+z_{2}\bar{z}_{1})^{4}}{(1+z_{1}\bar{z}_{1})^{4}(1+z_{2}\bar{z}_{2})^{4}}\,.$
(230)

Thus, the two point function is non-zero.

Let us consider the case when $\beta_{1}=\beta_{2}=z_{1}$ and
$\beta_{3}=\beta_{4}=z_{2}$. We have

$G_{(2)}=\dfrac{8(\bar{z}_{12})^{4}}{(1+z_{1}\bar{z}_{1})^{4}(1+z_{2}\bar{z}_{2})^{4}}\,.$
(231)

The two point function vanishes for this case due to the presence of
$\delta^{2}(z_{12})$. A similar analysis also holds for the antiholomorphic
components, which will have a factor of $(z_{12})^{4}$, leading to a vanishing
two point function.

Finally, let us consider the case when $\beta_{1}=z_{1}$ and
$\beta_{2}=\bar{z}_{1}$. For this, however, we trivially have

$\begin{split}P^{\rho_{1}\rho_{2}}_{z_{1}\bar{z}_{1}}\partial_{\rho_{1}}\tilde{q}_{1\alpha_{1}}\partial_{\rho_{2}}\tilde{q}_{1\alpha_{2}}&=\dfrac{1}{2}(\partial_{z_{1}}\tilde{q}_{\alpha_{1}}\partial_{\bar{z}_{1}}\tilde{q}_{1\alpha_{2}}+\partial_{\bar{z}_{1}}\tilde{q}_{1\alpha_{1}}\partial_{z_{1}}\tilde{q}_{1\alpha_{2}})\\\
&~{}~{}~{}-\dfrac{1}{2}(\partial_{z_{1}}\tilde{q}_{\alpha_{1}}\partial_{\bar{z}_{1}}\tilde{q}_{1\alpha_{2}}+\partial_{\bar{z}_{1}}\tilde{q}_{1\alpha_{1}}\partial_{z_{1}}\tilde{q}_{1\alpha_{2}})=0\,.\end{split}$
(232)

This is because we project onto the transverse and traceless components only.
Thus, the two point function vanishes in this case. Thus, the only non-trivial
two point function is

$\langle
O_{\Delta_{1};z_{1}z_{2}}(\textbf{P}_{1})O_{\Delta_{2};{\bar{z}}_{3}{\bar{z}}_{4}}(\textbf{P}_{2})\rangle=\dfrac{8}{(1+z_{1}\bar{z}_{1})^{2}(1+z_{2}\bar{z}_{2})^{2}}\dfrac{\Gamma(\Delta_{1}+\Delta_{2}-2)\,\delta^{2}(z_{12})}{[-i(u_{1}-u_{2})]^{\Delta_{1}+\Delta_{2}-2}}\,.$
(233)

Our result is consistent with the standard one Bagchi:2022emh  since we
respect $\delta_{\sigma_{1}+\sigma_{2},0}$ i.e. it agrees with (82). The
graviton two point function is similar to the gluon two point function (193).
This is related to the fact that both photon and graviton have only two
polarizations corresponding to positive and negative helicity.

####  6.11 Scalar-graviton-scalar three point amplitude: split signature

We consider the following AdS Witten contact diagram arising from (197)

$\begin{split}\langle
O_{\Delta_{1}}(\textbf{P}_{1})O_{\Delta_{2};\bar{z}_{2}\bar{z}_{2}}(\textbf{P}_{2})O_{\Delta_{3}}(\textbf{P}_{3})\rangle=\int
d\textbf{X}K_{\Delta_{1}}(\textbf{P}_{1},\textbf{X})K^{\Delta_{2},2;\alpha_{1}\alpha_{2}}_{\hskip
39.83368pt;\bar{z}_{2}\bar{z}_{2}}(\textbf{P}_{2},\textbf{X})\\\ \times
K_{\Delta_{3}}(\textbf{P}_{3},\textbf{X})V_{3\,\alpha_{1}\alpha_{2}}\,.\end{split}$
(234)

The vertex factor is given by (197)

$V_{3\,\alpha_{1}\alpha_{2}}=k_{1\alpha_{1}}k_{3\alpha_{2}}+k_{3\alpha_{1}}k_{1\alpha_{2}}-\eta_{\alpha_{1}\alpha_{2}}k_{1}\cdot
k_{3}\,.$ (235)

Using the bulk to boundary propagator of the graviton in the large $R$ limit
(158) and using (171),

$V_{3\,\text{grav}(-)}=\partial_{\bar{z}_{2}}\tilde{q}^{\alpha_{1}}_{2}\partial_{\bar{z}_{2}}\tilde{q}^{\alpha_{2}}_{2}V_{3\,\alpha_{1}\alpha_{2}}=\dfrac{8z^{2}_{23}\omega^{2}_{3}}{(1+z_{2}\bar{z}_{2})^{2}}=\dfrac{\left(V_{3\,\text{gluon}(-)}\right)^{2}}{2}\,,$
(236)

where $V_{3\,\text{gluon}(-)}$ is given in (201). This is the expected result,
and one can evaluate the Witten diagram in the large $R$ limit in a
straightforward way to be

$\begin{split}\langle
O_{\Delta_{1}}(\textbf{P}_{1})&O_{\Delta_{2};\bar{z}_{2}\bar{z}_{2}}(\textbf{P}_{2})O_{\Delta_{3}}(\textbf{P}_{3})\rangle\\\
&=\dfrac{32\mathcal{C}_{(1)grav}}{(1+z_{2}\bar{z}_{2})^{2}}z^{\Delta_{3}}_{12}z^{\Delta_{1}}_{32}z^{\Delta_{2}-2}_{13}\dfrac{\Gamma(\Delta_{1}+\Delta_{2}+\Delta_{3}-2)\,\delta(\bar{z}_{12})\,\delta(\bar{z}_{23})}{[-i(z_{1}u_{23}+z_{2}u_{31}+z_{3}u_{12})]^{\Delta_{1}+\Delta_{2}+\Delta_{3}-2}}\,.\end{split}$
(237)

This agrees with the intrinsic result (106) if we set
$\sigma_{1}=0,\sigma_{2}=-2,\sigma_{3}=0$. The normalization constant
$\mathcal{C}_{(1)grav}$ is given by

$\begin{split}\mathcal{C}_{(1)grav}&=N^{3}_{\Delta_{1}}\dfrac{\mathcal{C}^{3}_{\Delta_{2};2}}{(-i)^{\Delta_{2}}\Gamma(\Delta_{2})}\dfrac{\Delta_{2}-1}{\Delta_{2}}N^{3}_{\Delta_{3}}=\dfrac{i^{\Delta_{1}+\Delta_{2}}(-i)^{\Delta_{3}}(\Delta_{2}+1)}{8\pi^{\frac{9}{2}}\prod_{i=1}^{3}\Gamma\left(\Delta_{i}-\dfrac{1}{2}\right)\,\Delta_{2}\,R^{5-\Delta_{1}-\Delta_{2}-\Delta_{3}}}\,.\end{split}$
(238)

####  6.12 Graviton MHV three point amplitude: split signature

Much of the literature has focussed on how the MHV amplitudes for graviton
scattering undergo an enormous simplification, resulting in simple formulas
for their expression Nguyen:2009jk ; Hodges:2011wm ; Hodges:2012ym . These
works were inspired by the famous Parke-Taylor formula for $n$-gloun
scattering amplitude Parke:1986gb . We now obtain these simple answers for MHV
amplitudes from the limit of AdS Witten diagrams. It is well known that the
standard Mellin transform of tree level graviton amplitudes is UV divergent
Stieberger:2018edy ; Puhm:2019zbl . However, it was pointed out that the
modified Mellin transform of MHV graviton scattering amplitudes is UV finite
Banerjee:2019prz  since $u$ acts as a UV regulator. Here, we show how to
obtain the modified Mellin transformation of the three point MHV graviton
amplitude worked out in Banerjee:2019prz  from an AdS Witten contact diagram.

The three point vertex has been derived in DeWitt:1967uc , and it is given by

$\begin{split}V^{3(\text{grav})}_{\mu\alpha,\nu\beta,\sigma\gamma}=\lambda_{3}\text{sym}\Bigg{[}-\dfrac{1}{2}P_{3}(k_{1}\cdot
k_{2}\,\eta_{\mu\alpha}\eta_{\nu\beta}\eta_{\sigma\gamma})-\dfrac{1}{2}P_{6}(k_{1\nu}k_{1\beta}\eta_{\mu\alpha}\eta_{\sigma\gamma})\\\
+\dfrac{1}{2}P_{3}(k_{1}\cdot
k_{2}\,\eta_{\mu\nu}\eta_{\alpha\beta}\eta_{\sigma\gamma})+P_{6}(k_{1}\cdot
k_{2}\,\eta_{\mu\alpha}\eta_{\nu\sigma}\eta_{\beta\gamma})+2P_{3}(k_{1\nu}k_{1\gamma}\eta_{\mu\alpha}\eta_{\beta\sigma})\\\
-P_{3}(k_{1\beta}k_{2\mu}\eta_{\alpha\nu}\eta_{\sigma\gamma})+P_{3}(k_{1\sigma}k_{2\gamma}\eta_{\mu\nu}\eta_{\alpha\beta})-P_{6}(k_{1\sigma}k_{1\gamma}\,\eta_{\mu\nu}\eta_{\alpha\beta})\\\
+2P_{6}(k_{1\nu}k_{2\gamma}\eta_{\beta\mu}\eta_{\alpha\sigma})-2P_{3}(k_{1\nu}k_{2\mu}\eta_{\beta\sigma}\eta_{\gamma\alpha})-2P_{3}(k_{1}\cdot
k_{2}\eta_{\alpha\nu}\eta_{\beta\sigma}\eta_{\gamma\mu})\Bigg{]}\,.\end{split}$
(239)

We have adopted the normalization of PhysRevD.34.1749  and
$\lambda_{3}\propto\sqrt{G_{N}}$ where $G_{N}$ is the Newton’s constant. We
follow the conventions of DeWitt:1967uc : sym implies that we must symmetrize
the three point vertex in each pair of indices
$\mu\alpha,\nu\beta,\sigma\beta$. $P_{k}$ indicates that we should sum over
the distinct permutations of the momentum index combinations
$k_{1}-\mu\alpha,k_{2}-\nu\beta,k_{3}-\sigma\gamma$ and the subscript $k$
denotes the number of such terms in each class. Altogether, we have 171 terms
that need to be summed over. One crucial difference between the expressions of
DeWitt:1967uc ; PhysRevD.34.1749  is that the 8th term and 10th term of (239)
have an overall minus sign. We need that sign to obtain the correct answer,
which was explicitly checked in Mathematica.

The graviton MHV $(--+)$ three point function is given by following the
contact AdS Witten diagram

$\begin{split}&\langle
O_{\Delta_{1};\bar{z}_{1}\bar{z}_{1}}(\textbf{P}_{1})O_{\Delta_{2};\bar{z}_{2}\bar{z}_{2}}(\textbf{P}_{2})O_{\Delta_{3};z_{3}z_{3}}(\textbf{P}_{3})\rangle=\\\
&\int d\textbf{X}K^{\Delta_{1},2;\alpha_{1}\alpha_{2}}_{\hskip
39.83368pt;\bar{z}_{1}\bar{z}_{1}}(\textbf{P}_{1},\textbf{X})K^{\Delta_{2},2;\alpha_{3}\alpha_{4}}_{\hskip
39.83368pt;\bar{z}_{2}\bar{z}_{2}}(\textbf{P}_{2},\textbf{X})K^{\Delta_{3},2;\alpha_{5}\alpha_{6}}_{\hskip
39.83368pt;z_{3}z_{3}}(\textbf{P}_{3},\textbf{X})V^{3(\text{grav})}_{\alpha_{1}\alpha_{2},\alpha_{3}\alpha_{4},\alpha_{5}\alpha_{6}}\,.\end{split}$
(240)

Here
$K^{\Delta_{2},2}_{\alpha_{1}\alpha_{2};\beta_{1}\beta_{2}}(\textbf{P}_{i},\textbf{X})$
denotes the bulk to boundary propagator of the graviton field and in the large
$R$ limit, it is given by the Mellin representative (158). The action of the
projector is analyzed in the graviton propagator calculation of Section 6.10.
Thus, in the large $R$ limit, the three point function of (240) becomes

$\begin{split}\mathcal{C}^{grav}_{(3)}\int
d\omega_{1}d\omega_{2}d\omega_{3}\omega^{\Delta_{1}-1}_{1}\omega^{\Delta_{2}-1}_{2}\omega^{\Delta_{3}-1}_{3}e^{i(\omega_{1}u_{1}+\omega_{2}u_{2}-\omega_{3}u_{3})}\\\
~{}~{}~{}\times\delta^{(4)}(\omega_{1}\tilde{q}_{1}+\omega_{2}\tilde{q}_{2}-\omega_{3}\tilde{q}_{3})V_{3\,\text{grav}(--+)}\,.\end{split}$
(241)

The normalization constant $\mathcal{C}^{grav}_{(3)}$ is given by
($\mathcal{C}^{d}_{\Delta;J}$ is given in (146))

$\begin{split}\mathcal{C}^{grav}_{(3)}&=\dfrac{\mathcal{C}^{3}_{\Delta_{1};2}}{(-i)^{\Delta_{1}}\Gamma(\Delta_{1})}\dfrac{\Delta_{1}-1}{\Delta_{1}}\dfrac{\mathcal{C}^{3}_{\Delta_{2};2}}{(-i)^{\Delta_{2}}\Gamma(\Delta_{2})}\dfrac{\Delta_{2}-1}{\Delta_{2}}\dfrac{\mathcal{C}^{3}_{\Delta_{3};2}}{(i)^{\Delta_{3}}\Gamma(\Delta_{3})}\dfrac{\Delta_{3}-1}{\Delta_{3}}\\\
&=\dfrac{i^{\Delta_{1}+\Delta_{2}}(-i)^{\Delta_{3}}(\Delta_{1}+1)(\Delta_{2}+1)(\Delta_{3}+1)}{8\pi^{\frac{9}{2}}\prod_{i=1}^{n}\Gamma\left(\Delta_{i}-\frac{1}{2}\right)\,\Delta_{1}\,\Delta_{2}\,\Delta_{3}\,R^{9-(\Delta_{1}+\Delta_{2}+\Delta_{3})}}\,.\end{split}$
(242)

The vertex factor $V_{3\,\text{grav}(--+)}$ is given by

$\begin{split}V_{3\,\text{grav}(--+)}=\partial_{\bar{z}_{1}}\tilde{q}^{\alpha_{1}}_{1}\partial_{\bar{z}_{1}}\tilde{q}^{\alpha_{2}}_{1}\partial_{\bar{z}_{2}}\tilde{q}^{\alpha_{3}}_{2}\partial_{\bar{z}_{2}}\tilde{q}^{\alpha_{4}}_{2}\partial_{z_{3}}\tilde{q}^{\alpha_{5}}_{3}\partial_{z_{3}}\tilde{q}^{\alpha_{6}}_{3}V^{3(\text{grav})}_{\alpha_{1}\alpha_{2},\alpha_{3}\alpha_{4},\alpha_{5}\alpha_{6}}\end{split}$
(243)

Using the parametrization explained in footnote 13 and employing the momentum
conserving delta function (171), we can evaluate this crucial factor using
Mathematica to be 141414The terms 1-6 of (239) vanish trivially because the
ingoing momenta are null. Terms 7-11 are non-trivial and sum together
intricately to give this result. This is where the signs of Term 8 and Term 10
become all important to give a clean result.

$V_{3\,\text{grav}(--+)}=\dfrac{-16\lambda_{3}\,z^{2}_{12}\,\omega^{2}_{3}}{(1+z_{1}\bar{z}_{1})^{2}(1+z_{2}\bar{z}_{2})^{2}(1+z_{3}\bar{z}_{3})^{2}}=-\dfrac{1}{4}\lambda_{3}D^{2}_{z}z^{2}_{12}\omega^{2}_{3}\,,$
(244)

where $D_{z}$ was defined in (208). Thus, we get

$V_{3\,\text{grav}(--+)}=-\dfrac{\lambda_{3}}{4g^{2}(f^{abc})^{2}}\left(V_{3\,\text{gluon}(--+)}\right)^{2}\,.$
(245)

This is the standard result one obtains using spinor helicity variables
Hodges:2011wm ; elvang_huang_2015  (i.e.) the amplitude is given by

$\mathcal{A}^{\text{grav}}_{--+}=\dfrac{\langle 12\rangle^{6}}{\langle
23\rangle^{2}\langle
31\rangle^{2}}\delta^{(4)}(p^{\mu}_{1}+p^{\mu}_{2}+p^{\mu}_{3})\propto\left(\mathcal{A}^{\text{gluon}}_{--+}\right)^{2}\,,$
(246)

where $\mathcal{A}^{\text{gluon}}_{--+}$ is given in (210). The existence of
this squared structure through the Feynman diagrams has not been explicitly
worked out before Brandhuber:2022qbk  151515In order to get to this standard
result, one needs to incorporate the crucial overall negative signs in Term 8
and Term 10 of (239). The parameterisation of footnote 13 is also crucial. We
cannot comment on whether these signs will change the analysis of
PhysRevD.34.1749 .. From (246), we learn that (241) essentially becomes the
Modified Mellin transformation of the scattering amplitude as expected.

Thus, using (244) in (241), we can simplify it using (171) to obtain the final
result to be

$\begin{split}&\langle
O_{\Delta_{1};\bar{z}_{1}\bar{z}_{1}}(\textbf{P}_{1})O_{\Delta_{2};\bar{z}_{2}\bar{z}_{2}}(\textbf{P}_{2})O_{\Delta_{3};z_{3}z_{3}}(\textbf{P}_{3})\rangle\\\
&=\mathcal{C}^{grav}_{(3)}(-1)^{\Delta_{1}+\Delta_{2}+2}z^{\Delta_{3}+2}_{12}z^{\Delta_{1}-2}_{23}z^{\Delta_{2}-2}_{31}\dfrac{\Gamma(\Delta_{1}+\Delta_{2}+\Delta_{3}-2)\,\delta(\bar{z}_{12})\,\delta(\bar{z}_{23})}{[-i(z_{1}u_{23}+z_{2}u_{31}+z_{3}u_{12})]^{\Delta_{1}+\Delta_{2}+\Delta_{3}-2}}\,.\end{split}$
(247)

This matches with (4.20) of Banerjee:2019prz . This also matches the expected
intrinsic result (106) once we use $\sigma_{1}=-2,\sigma_{2}=-2,\sigma_{3}=2$.

For the $(++-)$ case, the vertex factor involved will just be the complex
conjugate, and it is given by

$V_{3\,\text{grav}(++-)}=-\dfrac{\lambda_{3}}{4g^{2}(f^{abc})^{2}}\left(V_{3\,\text{gluon}(++-)}\right)^{2}\,,$
(248)

where $V_{3\,\text{gluon}(++-)}$ is given by (215). As for the final
expression of the amplitude, it is just given by the complex conjugate of
(247) and that matches with the alternate branch intrinsic result (107) if we
use $\sigma_{1}=2,\sigma_{2}=2,\sigma_{3}=-2$.

Similar to the gluon case, there is no $(+++)$ or $(---)$ amplitudes because
the vertex factor (239) vanishes for the combinations. This is consistent with
the expectation that there are no MHV amplitudes of the form $(+++)$ or
$(---)$. We must consider the vertex from a higher derivative term like
$R^{3}$, where $R$ is the Ricci scalar to get a non-zero result. However, from
the Carrollian perspective, there will be a non-zero result for the $(+++)$
and $(---)$ correlators from (107) and (106) respectively. They will
correspond to holographic higher derivative terms in the bulk. For the Witten
diagram, we can only predict that the vertex factor will be the square of the
gluon $(---)$ vertex factor given in (221). The computation of the vertex
factor from the higher derivative term like $R^{3}$ will be tedious.

####  6.13 Graviton-gluon-gluon three point function: split signature

There is an interesting three point diagram involving two gluons and one
graviton that arises when Yang-Mills theory is minimally coupled to Einstein
gravity PhysRevLett.84.3531 ; Fu:2017uzt ; Plefka:2018zwm . Einstein-Yang-
Mills theory serves as a theoretical laboratory to explore the deep
connections between graviton and gauge amplitudes following upon the
establishment of the KLT relations Kawai:1985xq . The three point vertex
factor for two gluons and one graviton is given by Plefka:2018zwm :

$\begin{split}V^{\alpha\beta\mu_{a}\mu_{b}}_{\text{gl-gl-
grav}}=\lambda_{3}\left[\dfrac{1}{2}\eta^{\alpha\mu_{a}}\eta^{\beta\mu_{b}}p_{a}\cdot
p_{b}+\eta^{\mu_{a}\mu_{b}}p^{(\alpha}_{b}p^{\beta)}_{a}\right.\left.-\eta^{\mu_{a}(\alpha}p^{\beta)}_{b}p^{\mu_{b}}_{a}-\eta^{\mu_{b}(\alpha}p^{\beta)}_{a}p^{\mu_{a}}_{b}+\dfrac{1}{2}\eta^{\alpha\beta}p^{\mu_{a}}_{a}p^{\mu_{b}}_{b}\right]\end{split}$
(249)

Here $\alpha$ and $\beta$ are the graviton indices and $\mu_{i}$ are the gauge
indices (the subscript $a,b$ denote the colour indices).

The contact Witten diagram is given by

$\begin{split}&\langle
O_{\Delta_{1};\beta_{1}}(\textbf{P}_{1})O_{\Delta_{2};\beta_{2}}(\textbf{P}_{2})O_{\Delta_{3};\beta_{3}\beta_{4}}(\textbf{P}_{3})\rangle\\\
&=\int
d\textbf{X}\,K^{\Delta_{1},1}_{\alpha_{1};\beta_{1}}(\textbf{P}_{1},\textbf{X})\,K^{\Delta_{2},1}_{\alpha_{2};\beta_{2}}(\textbf{P}_{2},\textbf{X})\,K^{\Delta_{3},2}_{\alpha_{3}\alpha_{4};\beta_{3}\beta_{4}}(\textbf{P}_{3},\textbf{X})\,V^{\alpha_{3}\alpha_{4}\alpha_{1}\alpha_{2}}_{\text{gl-
gl-grav}}\,,\end{split}$ (250)

where the gluon and graviton bulk to boundary propagators in the large $R$
limit are given by (153) and (158). For the diagram (250) to result in a non-
trivial answer, the gluons should be of opposite helicities. Let us consider
the $(+--)$ case where the graviton is of negative helicity. In the large $R$
limit, (250) results in

$\begin{split}&\langle
O_{\Delta_{1};z_{1}}(\textbf{P}_{1})O_{\Delta_{2};\bar{z}_{2}}(\textbf{P}_{2})O_{\Delta_{3};\bar{z}_{3}\bar{z}_{4}}(\textbf{P}_{3})\rangle\\\
&=\mathcal{C}^{gl-gl-grav}_{(3)}\int
d\omega_{1}d\omega_{2}d\omega_{3}\omega^{\Delta_{1}-1}_{1}\omega^{\Delta_{2}-1}_{2}\omega^{\Delta_{3}-1}_{3}e^{i(\omega_{1}u_{1}+\omega_{2}u_{2}-\omega_{3}u_{3})}\\\
&~{}~{}~{}\times\partial_{z_{1}}q_{1\alpha_{1}}\partial_{\bar{z}_{2}}q_{2\alpha_{2}}\partial_{\bar{z}_{3}}q_{3\alpha_{3}}\partial_{\bar{z}_{4}}q_{4\alpha_{4}}V^{\alpha_{3}\alpha_{4}\alpha_{1}\alpha_{2}}_{\text{gl-
gl-
grav}}\delta^{(4)}(\omega_{1}\tilde{q}_{1}+\omega_{2}\tilde{q}_{2}-\omega_{3}\tilde{q}_{3})\,,\end{split}$
(251)

where $\mathcal{C}^{gl-gl-grav}_{(3)}$ is given by

$\begin{split}\mathcal{C}^{gl-gl-
grav}_{(3)}&=\dfrac{\mathcal{C}^{3}_{\Delta_{1};1}}{(-i)^{\Delta_{1}}\Gamma(\Delta_{1})}\dfrac{\Delta_{1}-1}{\Delta_{1}}\dfrac{\mathcal{C}^{3}_{\Delta_{2};1}}{(-i)^{\Delta_{2}}\Gamma(\Delta_{2})}\dfrac{\Delta_{2}-1}{\Delta_{2}}\dfrac{\mathcal{C}^{3}_{\Delta_{3};1}}{(i)^{\Delta_{3}}\Gamma(\Delta_{3})}\dfrac{\Delta_{3}-1}{\Delta_{3}}\\\
&=\dfrac{(i)^{\Delta_{1}+\Delta_{2}}(-i)^{\Delta_{3}}(\Delta_{3}+1)}{8\pi^{\frac{9}{2}}\Delta_{3}R^{7-(\Delta_{1}+\Delta_{2}+\Delta_{3})}\prod_{i=1}^{3}\Gamma\left(\Delta_{i}-\frac{1}{2}\right)}\,.\end{split}$
(252)

Now

$\begin{split}V^{\text{gl-gl-
grav}}_{(3)}&=\partial_{z_{1}}q_{1\alpha_{1}}\partial_{\bar{z}_{2}}q_{2\alpha_{2}}\partial_{\bar{z}_{3}}q_{3\alpha_{3}}\partial_{\bar{z}_{4}}q_{4\alpha_{4}}V^{\alpha_{3}\alpha_{4}\alpha_{1}\alpha_{2}}_{\text{gl-
gl-grav}}\\\
&=\dfrac{8\omega_{1}\omega_{2}z^{2}_{23}(1+2\bar{z}_{1}z_{2}+z_{1}\bar{z}^{2}_{1}z_{2})}{(1+z_{1}\bar{z}_{1})^{2}(1+z_{2}\bar{z}_{2})^{2}}\\\
&=\omega_{1}\omega_{2}z^{2}_{23}\,G_{z}\end{split}$ (253)

Using (253) in (251) and simplifying the integrals following analogous steps
of Appendix A.3, we get

$\begin{split}\langle
O_{\Delta_{1};z_{1}}(\textbf{P}_{1})&O_{\Delta_{2};\bar{z}_{2}}(\textbf{P}_{2})O_{\Delta_{3};\bar{z}_{3}\bar{z}_{4}}(\textbf{P}_{3})\rangle\\\
&=\mathcal{C}^{gl-gl-
grav}_{(3)}G_{z}z^{\Delta_{3}-2}_{12}z^{\Delta_{1}+1}_{23}z^{\Delta_{2}-1}_{31}\dfrac{\Gamma(\Delta_{1}+\Delta_{2}+\Delta_{3}-2)\,\delta(\bar{z}_{12})\delta(\bar{z}_{13})}{[-i(z_{1}u_{23}+z_{2}u_{31}+z_{3}u_{12})]^{\Delta_{1}+\Delta_{2}+\Delta_{3}-2}}\end{split}$
(254)

This agrees with the intrinsic result eq.(106) if we set
$\sigma_{1}=1,\sigma_{2}=-1,\sigma_{3}=-2$. We again see the Carrollian
correlators that encode gravitational scattering in the bulk are finite.

This completes our main section of computing various non-trivial Witten
diagrams, which form the bulk analysis. The expressions non-trivially match
the correlators obtained through intrinsic field theory analysis. This
completes the check of holography as seen through the lens of the limit of
AdS/CFT.

###  7 Conclusions

####  7.1 Summary of our results

In this paper, we have tried to highlight the various advantages of working
with the Carrollian approach to holography in asymptotically flat spacetimes.
After an extensive review of our approach in the first part of the paper and
comparing and contrasting it first to the Celestial holography programme and
later to a different approach in the Carrollian programme, in the second part,
our primary focus has been to make a case for Carroll stronger by working out
various non-trivial examples. In particular, we showed how the co-dimension
one Carrollian correlators that encode bulk scattering of spinning particles
naturally arise from the large AdS radius limit of Witten diagrams. One
crucial consequence of working with the co-dimension one Carrollian
correlators is that one does not need to artificially regulate the UV
divergences of the Mellin transform (4) by resorting to distributional delta
functions. The $u$ of the modified Mellin transform (2.2) naturally acts as a
UV regulator, resulting in correlation functions ultra localized across the
entire future null infinity. This serves as an indication that it is perhaps
the more appropriate way to go about flat holography.

Our work is also an extension of the results of Bagchi:2023fbj  to spinning
particles. In this regard, we looked at some new scalar Witten diagrams not
considered in Bagchi:2023fbj . These are the four point function, the soft
limit (non-collinear case), and the split signature cases of the three point
function. The soft limit of the scalar three point function was crucially
captured by the $\Delta\to 1$ subsector of Carrollian CFT. The split signature
result was a non-trivial branch of the three point function where one works
with the Celestial torus in which $z$ and $\bar{z}$ were independent.

In the context of spinning particles, we looked at the canonical free particle
propagation and also non-trivial three point amplitudes. We obtained the gluon
and graviton MHV amplitudes Pasterski:2017ylz ; Stieberger:2018edy  from
contact Witten diagrams and showed how they become finite in the modified
Mellin basis. We worked out the amplitude from a cubic interaction vertex
involving a spinning field and two scalar fields. We then proceeded to look at
Einstein-Yang-Mills theory and worked out the three point amplitude involving
one graviton and two gluons. We also looked at a non-trivial example of an
amplitude that arises from loop corrections of the form $F^{3}$ where $F$
represents the gauge field strength tensor. This was not previously considered
in the Celestial approach. In all the above examples, the large AdS limit of
bulk Witten diagrams matched the Carrollian correlators obtained from the
intrinsic Ward identities of the co-dimension one field theory.

The field theory analysis involved working out the highest weight
representations of the spinning Carrollian primaries. These primaries were
fields with upper indices. One has to distinguish fields with upper and lower
indices because the Carrollian metric is degenerate. This distinction ensured
that the correlators with time indices turned out to be zero in the leading
order. This matched with the bulk result in that the spinning bulk to boundary
propagator with one of the boundary indices being the time index becomes a
pure gauge at the leading order of the large AdS radius limit. The highest
weight representations then manifest as Ward identities that completely
constrain the two and three point correlation function for spinning primaries.
It is well known that the little group scaling completely fixes the three
point scattering amplitude for particles of any helicity in Minkowski space
elvang_huang_2015 . For the first time in the literature, we derive this
result directly from the Ward identities of the co-dimension one Carrollian
field theory. All these results validate our claims for seriously considering
the Carrollian approach to flat space holography.

####  7.2 Discussions

We list out some things that we still need to understand fully and hope to
return to in the near future and clarify.

Normalization constant: The normalization constants in the conformal primary
wavefunctions (140) and (146) diverge as $R\to\infty$ depending on the value
of $\Delta_{i}$. These constants arise in the computation of the correlators
through Witten diagrams. We expect that these divergences will be resolved
through holographic renormalization.

Sub-leading corrections to the bulk to boundary propagator: We don’t fully
understand the sub-leading large $R$ corrections to the bulk to boundary
propagator (144). These sub-leading corrections are crucial if we are looking
at operators with raised indices. Initial efforts at calculating these
corrections have run into difficulties in comparison with boundary answers.

Precise meaning of $\Delta$: $\Delta$ labels the boundary conformal field. It
arises because we have chosen a unitary representation that diagonalizes the
action of $L_{0}$ and $\bar{L}_{0}$. However, purely from the point of view of
the boundary theory, it arises from the Casimir of the boundary algebra. It
would be interesting to make this notion more precise from the point of view
of the representation theory of the Conformal Carroll theory.

####  7.3 Future Directions

There are, of course, numerous avenues of immediate research. The above-listed
points serve as the first things to clear up in our current programme. But we
hope to do much more.

As we pointed out before, there seems to be some tension between the two
different approaches of Carrollian road to flat space holography, the one
motivated by an asymptotic expansion Donnay:2022aba ; Donnay:2022wvx  and our
approach where the modified Mellin transformation plays a central role. We
have pointed out some reservations we have with the former in Sec. 2. But
clearly, something deeper is at play as the approach advocated in
Donnay:2022aba ; Donnay:2022wvx  apriori seems perfectly fine. It would be
good to clarify this apparent tension between Carrollian methods going
forward.

Understanding the soft sector in the light of modified Mellin transformation
from the perspective of a purely Carrollian theory is a priority. This has
already been done from the point of view of the modified Mellin in
Banerjee:2019prz ; Banerjee:2020zlg . We need to just have Carrollian
interpretations for the same.

In AdS/CFT, the boundary EM tensor sources the bulk graviton; hence, EM tensor
correlations are related to graviton amplitudes. Graviton amplitudes should
carry over from AdS to flat space in the infinite radius limit, and there
should be a similar story relating Carrollian EM tensors and flat space
graviton amplitudes. However, in flat spacetime, there is the additional
complication of radiation leaking out of null infinity. This is what
instigated the use of sourced Carroll Ward identities to match with soft
theorems in Donnay:2022wvx  161616See however Saha:2023hsl  for a discussion
of the same without using sources.. It would be interesting to understand how
to take the limit on AdS/CFT analysis in this context and get to graviton
amplitudes in asymptotically flat spacetimes. We hope to report on this in the
near future.

It is of interest to understand massive scattering and relate them to
Carrollian structures. For this, the Carrollian nature of $i^{\pm}$ is
important to understand. The limit from AdS/CFT in this case would also be
very interesting to understand.

To conclude, we re-emphasise that it seems the Carrollian co-dimension one
approach to holography in asymptotically flat spacetimes has several
advantages over the co-dimension two Celestial picture. We have highlighted
that in the first part of our paper and, in this second half, given several
examples from the flat limit of AdS/CFT to strengthen our point.

#### Acknowledgements

We would like to thank Daniel Grumiller, Shahin Sheikh-Jabbari, Romain
Ruzziconi and especially Shamik Banerjee for helpful discussions.

PD also thanks Tim Adamo, Dionysios Anninos, Alejandra Castro, Chandramouli
Chowdhury, Jelle Hartong, Nabil Iqbal, Arthur Lipstein, Prahar Mitra, Silvia
Nagy, Gerben Oling, Simon Ross, Joan Simon, Marika Taylor and Mritunjay Verma
for comments on the work during and after it was presented in places mentioned
below. PD acknowledges the warm hospitality of the University of Cambridge,
University of Southampton, Durham University, King’s College London, and the
University of Edinburgh during the course of this work.

AB is partially supported by a Swarnajayanti Fellowship from the Science and
Engineering Research Board (SERB) under grant SB/SJF/2019-20/08 and SERB grant
CRG/2020/002035 and further by a Royal Society of London international
exchange grant with the University of Edinburgh. SD thanks partial support
from grant SB/SJF/2019-20/08 of AB. PD would like to duly acknowledge the
Council of Scientific and Industrial Research (CSIR), New Delhi, for financial
assistance through the Senior Research Fellowship (SRF) scheme and the partial
support from the Royal Society of London international exchange grant with the
University of Edinburgh.

### APPENDICES

###  Appendix A Details of computations

In the various subsections of this appendix, we present details of
computations that we have omitted in the main text for ease of reading.

####  A.1 Scalar four point function: contact diagram analysis

We will need the following integral for most of the computations

$\int d\omega\,\omega^{\Delta-k}e^{i\omega
u}=(-iu)^{-1+k-\Delta}\Gamma(1-k+\Delta)\hskip
28.45274pt\text{if}~{}~{}\text{Re}(k-\Delta)<1\,,\text{Im}(u)>0\,.$ (255)

The $i\epsilon$ prescription always ensures $\text{Im}(u)>0$. We will now
simplify (177) to (178). To evaluate the Mellin integrals, we will use a
convenient change of variables. We will change the variables to
$s=\sum_{i}\omega_{i}$ and a set of “simplex” variables
$\sigma_{i}=s^{-1}\omega_{i}\in[0,1]$ with $\sum_{i=1}^{n}\sigma_{i}=1$
Pasterski:2017ylz ; Banerjee:2019prz . We thus have

$\prod_{i=1}^{n}\int^{\infty}_{0}d\omega_{i}\omega^{\Delta_{i}-1}_{i}[\dots]=\int
dss^{n-1+\sum_{i=1}^{n}(\Delta_{i}-1)}\prod_{i=1}^{n}\int
d\sigma_{i}\sigma^{\Delta_{i}-1}_{i}\delta\left(\sum_{i=1}^{n}\sigma_{i}-1\right)[\dots]\,.$
(256)

We can now combine this integral with the delta function in momentum
conservation ($\varepsilon_{i}=\pm 1$ is used to denote incoming and outgoing
particles respectively) given by

$\delta^{(4)}\left(\sum_{i}\varepsilon_{i}\omega_{i}\tilde{q}_{i}\right)=\delta^{(4)}\left(\sum_{i}\varepsilon_{i}\sigma_{i}s\tilde{q}_{i}\right)=\dfrac{1}{s^{4}}\delta^{(4)}\left(\sum_{i}\varepsilon_{i}\sigma_{i}\tilde{q}_{i}\right)\,.$
(257)

We have used the fact that $d=3$. Now

$\delta^{(4)}\left(\sum_{i}\varepsilon_{i}\sigma_{i}\tilde{q}_{i}\right)\delta\left(\sum_{i}\sigma_{i}-1\right)=C(z_{i},\bar{z}_{i})\prod_{i=1}^{n\leq
5}\delta(\sigma_{i}-\sigma^{*}_{i})\,.$ (258)

The form of $C(z_{i},\bar{z}_{i})$ and $\sigma^{*}_{i}$ for three point (for
split signature) and four point amplitudes is given in Pasterski:2017ylz . For
the four point amplitude in Minkowski signature, we have

$\delta^{(4)}\left(\sum_{i=1}^{4}\varepsilon_{i}\sigma_{i}\tilde{q}_{i}\right)\delta\left(\sum_{i=1}^{4}\sigma_{i}-1\right)=C(z_{i},\bar{z}_{i})\prod_{i=1}^{n=4}\delta(\sigma_{i}-\sigma^{*}_{i})\,,$
(259)

where the various quantities are given by

$C(z_{i},\bar{z}_{i})=\dfrac{1}{4}\delta(|z_{12}z_{34}\bar{z}_{13}\bar{z}_{24}-\bar{z}_{12}\bar{z}_{34}z_{13}z_{24}|)=\dfrac{\delta(|z-\bar{z}|)}{4z_{12}\bar{z}_{13}z_{24}\bar{z}_{24}}\,,$
(260)

where $z$ and $\bar{z}$ are the conformal cross ratios

$z=\dfrac{z_{12}z_{34}}{z_{13}z_{24}}\,,~{}~{}~{}~{}~{}\bar{z}=\dfrac{\bar{z}_{12}\bar{z}_{34}}{\bar{z}_{13}\bar{z}_{24}}\,.$
(261)
$\displaystyle\sigma^{*}_{1}=-\dfrac{\varepsilon_{1}\varepsilon_{4}}{D}\dfrac{z_{24}\bar{z}_{34}}{z_{12}\bar{z}_{13}}\,,~{}~{}\sigma^{*}_{2}=\dfrac{\varepsilon_{2}\varepsilon_{4}}{D}\dfrac{z_{34}\bar{z}_{14}}{z_{23}\bar{z}_{12}}\,,~{}~{}\sigma^{*}_{3}=-\dfrac{\varepsilon_{3}\varepsilon_{4}}{D}\dfrac{z_{24}\bar{z}_{14}}{z_{23}\bar{z}_{13}}\,,~{}~{}\sigma^{*}_{4}=\dfrac{1}{D}$
$\displaystyle
D=(1-\varepsilon_{1}\varepsilon_{4})\dfrac{z_{24}\bar{z}_{34}}{z_{12}\bar{z}_{13}}+(\varepsilon_{2}\varepsilon_{4}-1)\dfrac{z_{34}\bar{z}_{14}}{z_{23}\bar{z}_{12}}+(1-\varepsilon_{3}\varepsilon_{4})\dfrac{z_{24}\bar{z}_{14}}{z_{23}\bar{z}_{13}}$
(262)

Note that the parametriation for the null momenta used in Pasterski:2017ylz
differs from our parametrization (162) by an overall factor:

$q^{\mu}=(1+z\bar{z},z+\bar{z},-i(z-\bar{z}),1-z\bar{z})=(1+z\bar{z})\tilde{q}^{\mu}\,.$
(263)

But this won’t affect the above results of the product of delta functions
since we can always suitably redefine the energy $\omega_{i}$.

We will now use (256) in (177). We get

$\begin{split}\langle
O_{\Delta_{1}}(\textbf{P}_{1})&O_{\Delta_{2}}(\textbf{P}_{2})O_{\Delta_{3}}(\textbf{P}_{3})O_{\Delta_{4}}(\textbf{P}_{4})\rangle\\\
&=\mathcal{A}_{(4)c}\int^{\infty}_{0}dss^{3+\sum_{i=1}^{4}(\Delta_{i}-1)}\prod_{i=1}^{4}\int
d\sigma_{i}\sigma^{\Delta_{i}-1}_{i}e^{i\sigma_{1}su_{1}+i\sigma_{2}su_{2}-i\sigma_{3}su_{3}-i\sigma_{3}su_{4}}\\\
&~{}~{}~{}\times\dfrac{1}{s^{4}}\delta^{(4)}(\sigma_{1}\tilde{q}_{1}+\sigma_{2}\tilde{q}_{2}-\sigma_{3}\tilde{q}_{3}-\sigma_{4}\tilde{q}_{4})\delta\left(\sum_{i=1}^{4}\sigma_{i}-1\right)\,.\end{split}$
(264)

Thus, we can now use (259) to solve the integrals over $\sigma_{i}$ from the
product of delta functions. Thus, we finally get

$\begin{split}\langle
O_{\Delta_{1}}(\textbf{P}_{1})O_{\Delta_{2}}(\textbf{P}_{2})O_{\Delta_{3}}(\textbf{P}_{3})O_{\Delta_{4}}(\textbf{P}_{4})\rangle&=\int^{\infty}_{0}dss^{\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4}-5}e^{i(\sigma^{*}_{1}u_{1}+\sigma^{*}_{2}u_{2}-\sigma^{*}_{3}u_{3}-\sigma^{*}_{4}u_{4})s}\\\
&~{}~{}~{}\times\prod_{i=1}^{4}(\sigma^{*}_{i})^{\Delta_{i}-1}\dfrac{\delta(|z-\bar{z}|)}{4z_{12}\bar{z}_{13}z_{24}\bar{z}_{24}}\prod_{i=1}^{4}\mathbbm{1}_{[0,1]}(\sigma^{*}_{i})\end{split}$
(265)

Doing the $s$ integral using (255), we get (178) in a straight-forward way.

####  A.2 Scalar three point function: split signature

In this subsection, we will derive (174) from (172). We start by substituting
(171) in (172):

$\begin{split}\dfrac{\mathcal{A}_{3}\,\delta(\bar{z}_{12})\,\delta(\bar{z}_{23})}{z_{23}z_{31}}&\int^{\infty}_{0}d\omega_{3}\,\omega^{\Delta_{3}-3}_{3}e^{-i\omega_{3}\,u_{3}}\int^{\infty}_{0}d\omega_{1}\,\omega^{\Delta_{1}-1}_{1}e^{i\omega_{1}\,u_{1}}\delta\left(\omega_{1}-\omega_{3}\dfrac{z_{32}}{z_{12}}\right)\\\
&\times\int^{\infty}_{0}d\omega_{2}\,\omega^{\Delta_{2}-1}_{2}e^{i\omega_{2}\,u_{2}}\delta\left(\omega_{2}-\omega_{3}\dfrac{z_{31}}{z_{21}}\right)\\\
&=\dfrac{\mathcal{A}_{3}\delta(\bar{z}_{12})\delta(\bar{z}_{23})}{z_{23}z_{31}}\int^{\infty}_{0}d\omega_{3}\,\omega^{\Delta_{3}-3}_{3}e^{-i\omega_{3}\,u_{3}}\left(\omega_{3}\dfrac{z_{32}}{z_{12}}\right)^{\Delta_{1}-1}\text{exp}\left[i\omega_{3}\dfrac{z_{32}}{z_{12}}u_{1}\right]\\\
&\hskip
170.71652pt\times\left(\omega_{3}\dfrac{z_{31}}{z_{21}}\right)^{\Delta_{2}-1}\text{exp}\left[i\omega_{3}\dfrac{z_{31}}{z_{21}}u_{2}\right]\\\
&=\int^{\infty}_{0}d\omega_{3}\,\omega^{\Delta_{1}+\Delta_{2}+\Delta_{3}-5}_{3}\text{exp}\left[i\omega_{3}\left(\dfrac{z_{32}u_{1}+z_{13}u_{2}+z_{21}u_{3}}{z_{12}}\right)\right]\\\
&~{}~{}~{}\times\dfrac{(-1)^{\Delta_{1}+\Delta_{2}-2}\mathcal{A}_{3}\,\delta(\bar{z}_{12})\,\delta(\bar{z}_{23})\,z^{\Delta_{1}-2}_{32}z^{\Delta_{2}-2}_{31}}{z_{12}^{\Delta_{1}+\Delta_{2}-2}}\,.\end{split}$
(266)

Now we have

$z_{32}u_{1}+z_{13}u_{2}+z_{21}u_{3}=z_{1}u_{23}+z_{2}u_{31}+z_{3}u_{12}\,.$
(267)

Thus, using (255), we get (174) in a straight-forward way.

####  A.3 Scalar-gluon-scalar three point function: Split signature

In this subsection, we will derive (202) from (199). If we substitute (171) in
(199), we get

$\begin{split}&=\mathcal{B}_{(1)sp}\int
d\omega_{1}d\omega_{2}d\omega_{3}\omega^{\Delta_{1}-1}_{1}\omega^{\Delta_{2}-1}_{2}\omega^{\Delta_{3}-1}_{3}e^{i\omega_{1}\,u_{1}}e^{i\omega_{2}\,u_{2}}e^{-i\omega_{3}\,u_{3}}\\\
&~{}~{}~{}\times(\omega_{1}\tilde{q}^{\mu}_{1}+\omega_{3}\tilde{q}^{\mu}_{3})(\partial_{\bar{z}_{2}}\tilde{q}_{2\mu})\dfrac{4}{\omega^{2}_{3}z_{23}z_{31}}\delta\left(\omega_{1}-\omega_{3}\dfrac{z_{32}}{z_{12}}\right)\delta\left(\omega_{2}-\omega_{3}\dfrac{z_{31}}{z_{21}}\right)\delta(\bar{z}_{12})\delta(\bar{z}_{13})\\\
&=\dfrac{4\mathcal{B}_{(1)sp}}{z_{23}z_{31}}\left(\dfrac{z_{31}}{z_{21}}\right)^{\Delta_{2}-1}\left(\dfrac{z_{32}}{z_{12}}\right)^{\Delta_{1}-1}\int
d\omega_{3}\omega^{\Delta_{1}+\Delta_{2}+\Delta_{3}-5}_{3}\,\delta(\bar{z}_{12})\delta(\bar{z}_{13})\\\
&~{}~{}~{}\times\left[\omega_{3}\dfrac{{z}_{32}}{z_{12}}\tilde{q}^{\mu}_{1}+\omega_{3}\tilde{q}^{\mu}_{3}\right]\partial_{\bar{z}_{2}}\tilde{q}_{2\mu}\,\text{exp}\left[i\omega_{3}u_{1}\dfrac{z_{32}}{z_{12}}+i\omega_{3}u_{2}\dfrac{z_{31}}{z_{21}}-i\omega_{3}u_{3}\right]\,.\end{split}$
(268)

Using the parametrization of footnote 13, we have

$\left[\dfrac{z_{32}}{z_{12}}\tilde{q}^{\mu}_{1}+\tilde{q}^{\mu}_{3}\right]\partial_{\bar{z}_{2}}\tilde{q}_{2\mu}=\dfrac{4\,z_{32}\,\omega_{3}}{(1+z_{2}\bar{z}_{2})}=4z_{32}\,C_{z}\,.$
(269)

Doing the $\omega_{3}$ integral, we straightforwardly get (202). This is
similar to the calculations of Appendix A.2.

####  A.4 Lorentz invariance of delta functions

We check the Lorentz invariance of the split given in (171)

$\delta^{(4)}(\omega_{1}\tilde{q}_{1}+\omega_{2}\tilde{q}_{2}-\omega_{3}\tilde{q}_{3})=\dfrac{4}{\omega^{2}_{3}z_{23}z_{31}}\delta\left(\omega_{1}-\omega_{3}\dfrac{z_{32}}{z_{12}}\right)\delta\left(\omega_{2}-\omega_{3}\dfrac{z_{31}}{z_{21}}\right)\delta(\bar{z}_{13})\delta(\bar{z}_{23})\,.$
(270)

Let us consider a non-trivial case of the SL$(2,\mathbb{C})$ transformation
given by

$z_{i}\to-\dfrac{1}{z_{i}}\,.$ (271)

Thus after using $\bar{z}_{1}=\bar{z}_{2}=\bar{z}_{3}$, we have

$\begin{split}\delta\left(\omega_{1}-\omega_{3}\dfrac{z_{32}}{z_{12}}\right)&\to\dfrac{1}{z_{1}\bar{z}_{1}}\delta\left(\omega_{1}-\omega_{3}\dfrac{z_{32}}{z_{12}}\right)\\\
\delta\left(\omega_{2}-\omega_{3}\dfrac{z_{31}}{z_{21}}\right)&\to\dfrac{1}{z_{2}\bar{z}_{2}}\delta\left(\omega_{2}-\omega_{3}\dfrac{z_{31}}{z_{21}}\right)\\\
\delta(\bar{z}_{13})&\to\bar{z}_{1}\bar{z}_{3}\delta(\bar{z}_{13})\\\
\delta(\bar{z}_{23})&\to\bar{z}_{2}\bar{z}_{3}\delta(\bar{z}_{23})\\\
\dfrac{1}{\omega^{2}_{3}z_{23}z_{31}}&\to\dfrac{z_{1}z_{2}}{\bar{z}^{2}_{3}}\dfrac{1}{\omega^{2}_{3}z_{23}z_{31}}\,.\end{split}$
(272)

Combining them we see that the split of the delta function given in (171) is
indeed Lorentz invariant.

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