aainstitutetext: Department of Physics, Brown University, Providence, RI
02912, USAbbinstitutetext: Perimeter Institute for Theoretical Physics,
Waterloo, ON N2L 2Y5, Canadaccinstitutetext: Mathematical Institute,
University of Oxford, Oxford, OX2 6GG, UKddinstitutetext: Department of
Physics, Harvard University, Cambridge, MA 02138, USA

# Scalar-Graviton Amplitudes and Celestial Holography

Adam Ball a Shounak De a,c Akshay Yelleshpur Srikant a,d and Anastasia
Volovich<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We compute scattering amplitudes involving one massive scalar and two, three,
or four gravitons. We show that when the conformal dimension of the massive
scalar is set to zero, the resulting celestial correlators depend only on the
coordinates of the gravitons. Such correlators of gravitons are well-defined
and do not suffer from divergences associated with the Mellin transform of
usual graviton amplitudes. Moreover, they are non-distributional and take the
form of standard CFT correlators. We show that they are consistent with the
usual OPEs but the statement of the soft theorem is modified.

##  1 Introduction

Understanding how bulk translational symmetry emerges is one of the central
problems in the celestial holography program. In addition to being constrained
by conformal invariance, correlation functions in celestial conformal field
theory (CCFT) Pasterski:2016qvg ; Pasterski:2017kqt ; Pasterski:2017ylz  must
also obey translation invariance. This acts as an “external symmetry” and
forces low point correlators to be distributional Law:2019glh . These effects
are not as stark at higher points where the correlators are non-
distributional. However, the positions of the operators are still constrained
and cannot be chosen to lie arbitrarily on the celestial sphere Mizera:2022sln
. Such correlation functions are exotic from the perspective of a 2D conformal
field theory and are one of the main roadblocks to finding a top-down
description of CCFT. Indeed many of the existing top-down constructions
Costello:2022jpg ; Costello:2023hmi ; Bittleston:2023bzp  correspond to
theories in the bulk which differ from Yang-Mills or GR in significant ways.

One way to circumvent this difficulty has been to introduce nontrivial
backgrounds in the bulk Casali:2022fro ; Fan:2022kpp ; Fan:2022vbz ;
Stieberger:2022zyk ; Gonzo:2022tjm ; Banerjee:2023rni ; Adamo:2020syc ;
Adamo:2020yzi ; Adamo:2022mev ; Adamo:2023fbj ; Adamo:2023zeh ; Bogna:2023bbd
. The backgrounds must not only be Lorentz invariant — thus preserving the 2D
conformal symmetry of the CCFT — but should also maintain the asymptotic
flatness of the 4D bulk spacetime. The celestial counterparts of scattering
amplitudes on these backgrounds typically have a number of desirable features.
Firstly, they are unconstrained by translation invariance: the background acts
as a source/sink of momentum. Consequently, they resemble traditional CFT
correlators. Secondly, they have better convergence properties and we avoid
having to deal with divergent integrals. Finally, for well-chosen backgrounds,
the OPEs and symmetry algebras of the resulting theory resemble those of the
CCFT without a background Casali:2022fro ; Banerjee:2023rni ; Melton:2022fsf .
These properties suggest that the CFT from which these correlators arise might
be closely related to the background free CCFT. Moreover, the semblance of
these correlators to those of a traditional CFT makes it easier to identify an
independent definition. We refer the reader to Stieberger:2022zyk ;
Stieberger:2023fju  for some progress in this direction in the case of Yang-
Mills theory.

In this work, we are concerned with gravity amplitudes in 4D Minkowski and
Klein spacetimes Atanasov:2021oyu . In particular, our interests lie in
generalizing the results of Melton:2022fsf , where the authors considered
gluon amplitudes on a massive scalar background, to gravity amplitudes. The
scalar was not colored and interacted with gluons only via a non-minimal
coupling which gave rise to particularly simple amplitudes. A straightforward
extension of this to gravity is impossible since a non-dynamical scalar
background breaks diffeomorphism invariance. Moreover, it is impossible to
avoid a minimal coupling of the scalar to gravitons. Thus, amplitudes
involving $n$ gravitons and one scalar will necessarily involve a scalar
exchange. Nevertheless, we will show that the celestial counterparts of these
amplitudes exhibit all of the desirable properties mentioned in the previous
paragraph.

This paper is structured as follows. In Section 2, we describe the precise
quantity of interest and its relation to scattering amplitudes. In Section 3,
we compute scattering amplitudes involving two, three, and four gravitons and
a single massive scalar. To our knowledge, these amplitudes have not appeared
in the literature before. We then use these to compute the associated
celestial correlators in Section 4. The OPE of two positive helicity gravitons
extracted from these correlators is consistent with the usual OPE as shown in
Section 5. However, the celestial avatar of the leading soft theorem appears
to be slightly different, which we expand on in Section 6. Finally, we end
with some outlook and discussions in Section 7.

##  2 Preliminaries

In this paper, we will be concerned with scattering amplitudes involving $n$
massless gravitons and a single massive scalar. We will always label the
graviton momenta by $p_{1},\dots,p_{n}$ and the scalar momentum will be
denoted by $p_{n+1}$. In order to compute the corresponding celestial
amplitudes, it is convenient to use the following parametrization for the
massless momenta:

$\displaystyle
p_{i}=\epsilon_{i}\omega_{i}\left(1+z_{i}\bar{z}_{i},z_{i}+\bar{z}_{i},z_{i}-\bar{z}_{i},1-z_{i}\bar{z}_{i}\right),\qquad
i=1,\dots,n~{}.$ (1)

This parametrization is written for a spacetime with $(2,2)$ signature
(specifically $(+,-,+,-)$). $\omega_{i}$ is a positive real number while
$z_{i},\bar{z}_{i}$ are real and independent. $\epsilon_{i}=\pm 1$ is the
direction of the momentum. The parametrization for $(3,1)$ signature is
obtained by Wick rotating the third component. The scattering amplitude will
be written in terms of spinor helicity variables, which for the momentum
parametrization above, take the form

$\displaystyle\lambda_{i}=\sqrt{2\omega_{i}}\epsilon_{i}\begin{pmatrix}1\\\
z_{i}\end{pmatrix},\qquad\tilde{\lambda}_{i}=\sqrt{2\omega_{i}}\begin{pmatrix}1\\\
\bar{z}_{i}\end{pmatrix},\qquad i=1,\dots,n~{}.$ (2)

The analogous parametrization for the massive scalar momentum (also in $(2,2)$
signature) is

$\displaystyle
p_{n+1}=\frac{m\epsilon_{n+1}}{2y}\left(1+y^{2}+z_{n+1}\bar{z}_{n+1},z_{n+1}+\bar{z}_{n+1},z_{n+1}-\bar{z}_{n+1},1-y^{2}-z_{n+1}\bar{z}_{n+1}\right)~{}.$
(3)

Here $\epsilon_{n+1}=\pm 1$ is the direction of the momentum,
$y,z_{n+1},\bar{z}_{n+1}$ are real and independent with $y>0$. The celestial
amplitude can be obtained from the momentum space scattering amplitude by
changing the basis from plane waves to conformal primary wavefunctions. For
the case of $n$ massless and one massive particle, this is implemented by
Pasterski:2016qvg ; Pasterski:2017ylz

$\tilde{\mathcal{A}}_{n+1}^{\epsilon}\left(\\{\Delta_{1},z_{1},\bar{z}_{1}\\}^{J_{1}},\dots,\\{\Delta_{n},z_{n},\bar{z}_{n}\\}^{J_{n}};\\{\Delta_{n+1},w,\bar{w}\\}\right)=\\\
\frac{m^{2}}{4}\int_{0}^{\infty}\prod_{i=1}^{n}\frac{d\omega_{i}}{\omega_{i}}\omega_{i}^{\Delta_{i}}\int_{0}^{\infty}\frac{dy}{y^{3}}\int_{-\infty}^{\infty}dz_{n+1}\,d\bar{z}_{n+1}\\\
G_{\Delta_{n+1}}\left(y,z_{n+1},\bar{z}_{n+1};w,\bar{w}\right)\mathcal{A}_{n+1}\left(p_{1}^{J_{1}},\dots,p_{n}^{J_{n}},p_{n+1}\right)~{}.$
(4)

Here $J_{1},\dots,J_{n}$ are the helicities of the massless particles and

$\displaystyle
G_{\Delta}\left(y,z_{n+1},\bar{z}_{n+1};w,\bar{w}\right)=\left(\frac{y}{y^{2}+|z_{n+1}-w|^{2}}\right)^{\Delta}$
(5)

is the bulk-to-boundary propagator. The superscript $\epsilon$ in (4) is meant
to indicate the dependence of the celestial amplitude on the directions of the
momenta. The quantity of interest in this paper is111The double brackets in
the LHS of (6) is meant to serve as a reminder of the presence of the massive
scalar profile and we use $\tilde{\mathrm{A}}$ for the celestial amplitude
with the conformal dimension of the scalar set to 0.

$\displaystyle\tilde{\mathrm{A}}_{n}$
$\displaystyle=\left\langle\langle\mathcal{O}_{\Delta_{1},J_{1}}\left(z_{1},\bar{z}_{1}\right)\dots\mathcal{O}_{\Delta_{n},J_{n}}\left(z_{n},\bar{z}_{n}\right)\right\rangle\rangle$
(6)
$\displaystyle\equiv\sum_{\epsilon_{i}}\tilde{\mathcal{A}}_{n+1}^{\epsilon}\left(\left\\{\Delta_{1},z_{1},\bar{z}_{1}\right\\}^{J_{1}},\dots,\left\\{\Delta_{n},z_{n},\bar{z}_{n}\right\\}^{J_{n}};\left\\{0,w,\bar{w}\right\\}\right)$

whose features we now explain. But first, we pause to make a comment on
notation. We will use $\langle\langle\dots\rangle\rangle$ and
$\tilde{\mathrm{A}}_{n}$ interchangeably — the former being used more often
for brevity and the latter when we wish to emphasize its nature as a CCFT
correlator. Implicit in both is the fact that we have set the conformal
dimension of the scalar, $\Delta_{n+1}$ to 0. The subscript on
$\tilde{\mathrm{A}}_{n}$ matches the number of operators in the double
brackets $\langle\langle\dots\rangle\rangle$ and is one less than the
subscript on the corresponding momentum space scattering amplitude. Returning
to the features of $\tilde{\mathrm{A}}_{n}$, firstly note that setting
$\Delta_{n+1}=0$ eliminates all dependence on the scalar coordinates
$w,\bar{w}$. The RHS of (6) behaves like an $n$-point correlator222In the
celestial amplitude wherein the conformal dimension of the scalar has been set
to zero, the subscript $n$ of $\tilde{\mathcal{A}}_{n}$ counts only the number
of gravitons. — a claim that shall be supported by the computations in this
paper. When $\phi$ is a non-dynamical field, this has the interpretation of
being the Mellin transform of an amplitude on a scalar background and has been
studied for gluons in Casali:2022fro . These amplitudes take on a particularly
simple form Dixon:2004za  due to the absence of scalar exchange contributions
to the amplitude. Upon including gravity, we must promote $\phi$ to a
dynamical field and include the resulting scalar exchange terms in the
amplitude.

Secondly, despite the absence of manifest Lorentz invariance due to the
presence of a three-dimensional integral, (6) is indeed Lorentz invariant. The
three-dimensional measure corresponds to integration over the Lorentz
invariant phase space of one massive particle:

$\displaystyle\int\frac{d^{4}Q}{\left(2\pi\right)^{4}}\,\delta^{+}\left(Q^{2}-m^{2}\right)=\int\frac{d^{3}\vec{Q}}{\left(2\pi\right)^{3}2Q^{0}}=\frac{m^{2}}{4}\int_{0}^{\infty}\frac{dy}{y^{3}}\,\int_{-\infty}^{\infty}dz\,d\bar{z}~{}.$
(7)

Here $Q$ is an off-shell momentum which is put on-shell (with positive energy)
by the $\delta$ function. Once this is on-shell, we can use a parametrization
similar to (3) and perform a change of variables as shown in the second
equality above.

Finally, we sum over the directions $\epsilon_{i}$ purely for practical
purposes since this simplifies computations, particularly in Klein space. This
simplification has already been exploited before Fan:2021isc ; Hu:2022syq ;
De:2022gjn .333This procedure computes the celestial amplitude for certain
“boost + $\mathbb{Z}_{2}$” eigenstates Jorge-Diaz:2022dmy .

##  3 Massive scalar-graviton scattering amplitudes

We must first construct the scattering amplitudes in a theory of scalars and
gravitons. We will make a choice for the three-point amplitudes motivated by
the results of Casali:2022fro . In this work, the authors considered gluons
coupled to a single massive, uncolored scalar which resulted in the following
three-point amplitudes

$\displaystyle\mathcal{A}_{3}\left(1^{+},2^{+},3^{-}\right)=\kappa_{1,1,-1}\frac{[12]^{3}}{[23][31]},\qquad\mathcal{A}_{3}(1^{+},2^{+},3^{\phi})$
$\displaystyle=\kappa_{1,1,0}[12]^{2}~{}.$ (8)

Here and in the rest of this paper, we will leave the momentum conserving
delta function implicit while writing scattering amplitudes in momentum space.
The authors of Casali:2022fro  worked in $(2,2)$ signature and this
corresponded to introducing terms proportional to
$\phi\,\text{Tr}\left(F_{+}\right)^{2}$ in the Lagrangian where

$\displaystyle
F_{+}^{a\,\mu\nu}=\frac{1}{2}\left(F^{a\,\mu\nu}+\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F^{a}_{\rho\sigma}\right)$
(9)

is the self-dual gauge field strength.444This is real in (2,2) signature.
$\kappa_{1,1,-1}$ is the Yang-Mills coupling and $\kappa_{1,1,0}$ is related
by some numerical factor to the Wilson coefficient of the non-minimal
$\phi\,\text{Tr}\left(F_{+}\right)^{2}$ interaction. The scalar is uncolored
and does not couple minimally to the gluon.

The analogous three-point amplitudes in gravity can be easily written down and
take the following form

$\displaystyle\mathcal{A}_{3}\left(1^{++},2^{++},3^{--}\right)=\kappa_{2,2,-2}\frac{[12]^{6}}{[23]^{2}[13]^{2}}~{},\qquad\mathcal{A}_{3}\left(1^{++},2^{++},3^{\phi}\right)$
$\displaystyle=\kappa_{2,2,0}\,[12]^{4}~{}.$ (10)

However, it is also necessary to introduce the scalar-graviton minimal
coupling Arkani-Hamed:2017jhn

$\displaystyle\mathcal{A}_{3}\left(1^{++},2^{\phi},3^{\phi}\right)$
$\displaystyle=\kappa_{2,0,0}\frac{\left\langle\xi_{1}|p_{2}|1\right]}{\langle\xi_{1}1\rangle}\frac{\left\langle\xi_{2}|p_{2}|1\right]}{\langle\xi_{2}1\rangle}~{},$
(11)

where $\xi_{1},\xi_{2}$ indicate arbitrary reference spinors. From the
Lagrangian perspective, the minimal coupling is unavoidable in order to
maintain diffeomorphism invariance. From a purely amplitudes perspective, as
we will show later, any four-point amplitude constructed without the minimal
coupling (11) is inconsistent with the soft graviton theorem. Finally, we note
that the amplitudes in (10, 11) arise from terms which are schematically of
the form $R^{2}\phi,R\phi^{2}$ in the Lagrangian.

These three-point amplitudes serve as seeds for the construction of four- and
higher-point ones. We utilize recursion relations, in particular those
generated by a three-line shift Risager:2005vk  for their construction, which
begins with defining

$\displaystyle\hat{\lambda}_{i}=\lambda_{i}-zw_{i}X~{},\qquad i=1,2,3$ (12)

such that $w_{i}\tilde{\lambda}_{i}=0$ and $X$ is an arbitrary reference
spinor which must drop out at the end. A particularly convenient choice is

$\displaystyle w_{1}=[23]~{},\quad w_{2}=[31]~{},\quad w_{3}=[12]~{}.$ (13)

The three shifted momenta are now complex and functions of $z$. Locality
dictates that poles can occur at

$\displaystyle z_{\star}\quad\text{such
that}\quad\hat{P}^{2}=0\quad\text{or}\quad\hat{P}^{2}=m^{2}~{},$ (14)

where $\hat{P}$ is a sum of external momenta necessarily involving at most two
of $\hat{p}_{1},\hat{p}_{2},\hat{p_{3}}$. The residues on these poles are
governed by unitarity implying that they factorize into a product of lower
point amplitudes. The amplitude can thus be written as (assuming that there is
no contribution from the pole at infinity)

$\displaystyle\mathcal{A}_{n}\equiv\mathcal{A}_{n}\left(0\right)=\sum_{z_{\star}}\left[\mathcal{A}_{m}^{(L)}\left(z_{\star}\right)\frac{1}{P^{2}}\mathcal{A}_{n-m}^{(R)}\left(z_{\star}\right)+\mathcal{A}_{m}^{(L)}\left(z_{\star}\right)\frac{1}{P^{2}-m^{2}}\mathcal{A}_{n-m}^{(R)}\left(z_{\star}\right)\right]~{}.$
(15)

We will now employ this to construct the four- and five-point graviton-scalar
amplitudes.

###  3.1 Four-point amplitudes

We first consider the amplitude
$\mathcal{A}\left(1^{++},2^{++},3^{++},4^{\phi}\right)$ which receives
contributions from both scalar and graviton exchange channels

$\mathcal{A}_{4}\left(1^{++},2^{++},3^{++},4^{\phi}\right)\equiv\mathcal{A}_{4}^{gr,s}+\mathcal{A}_{4}^{gr,t}+\mathcal{A}_{4}^{gr,u}+\mathcal{A}_{4}^{sc,s}+\mathcal{A}_{4}^{sc,t}+\mathcal{A}_{4}^{sc,u}.$
(16)

The first of these terms corresponds to an $s$-channel graviton exchange

$\displaystyle\mathcal{A}_{4}^{gr,s}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.6}]{Figs/4ptamp1part1.pdf}
\includegraphics[scale={0.6}]{Figs/4ptamp1part2.pdf}}}$ (17)
$\displaystyle=\mathcal{A}_{3}\left(\hat{1},\hat{2},P\right)\frac{1}{P^{2}}\mathcal{A}_{3}\left(\hat{3},4,-P\right)~{},$

with the value of $z_{\star}$ given by

$\displaystyle\left(\hat{p}_{1}+\hat{p}_{2}\right)^{2}=0\implies
z_{\star}=\frac{\langle 12\rangle}{w_{1}\langle X2\rangle-w_{2}\langle
X1\rangle}~{}.$ (18)

Using the three-point amplitudes in (10, 11), this evaluates to

$\displaystyle\mathcal{A}_{4}^{gr,s}$
$\displaystyle=\kappa_{2,2,-2}\left(\frac{[12]^{3}}{[2I][I1]}\right)^{2}\frac{1}{\langle
12\rangle[21]}\kappa_{2,2,0}[3I]^{4}$ (19)
$\displaystyle=-\kappa_{2,2,-2}\kappa_{2,2,0}\frac{[12]}{\langle
12\rangle}\frac{\left[3|p_{4}|X\right\rangle^{4}}{\langle 1X\rangle^{2}\langle
2X\rangle^{2}}~{}.$

Similarly, the $s$-channel scalar exchange term is

$\displaystyle\mathcal{A}_{4}^{sc,s}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.6}]{Figs/4ptamp2part1.pdf}
\includegraphics[scale={0.6}]{Figs/4ptamp2part2.pdf}}}$ (20)
$\displaystyle=\mathcal{A}_{3}\left(\hat{1},\hat{2},P\right)\frac{1}{P^{2}-m^{2}}\mathcal{A}_{3}\left(\hat{3},4,-P\right)$
$\displaystyle=\kappa_{2,2,0}[12]^{4}\frac{1}{s-m^{2}}\kappa_{2,0,0}\frac{\left[3|p_{4}|\xi_{1}\right\rangle}{\langle\hat{3}\xi_{1}\rangle}\frac{\left[3|p_{4}|\xi_{2}\right\rangle}{\langle\hat{3}\xi_{2}\rangle}~{}.$

Here $s=\left(p_{1}+p_{2}\right)^{2}$ is the usual Mandelstam invariant. The
remaining terms can be obtained by the replacements $1\leftrightarrow 3$ and
$2\leftrightarrow 3$. Adding all of these terms and setting the arbitrary
reference spinors $\xi_{1}=\xi_{2}=X$, we finally get

$\mathcal{A}_{4}\left(1^{++},2^{++},3^{++},4^{\phi}\right)=-\kappa_{2,2,-2}\kappa_{2,2,0}\frac{[12]}{\langle
12\rangle}\frac{\left[3|p_{4}|X\right\rangle^{4}}{\langle 1X\rangle^{2}\langle
2X\rangle^{2}}+\kappa_{2,0,0}\kappa_{2,2,0}\frac{[12]^{4}}{s-m^{2}}\frac{\left[3|p_{4}|X\right\rangle^{2}}{\langle
3X\rangle^{2}}\\\ \qquad\qquad+\qquad\qquad(2\leftrightarrow
3)\qquad\qquad+\qquad\qquad(1\leftrightarrow 3)~{}.$ (21)

The spinor $X$ is arbitrary and the amplitude must be independent of it. It
can be checked that the dependence on the arbitrary spinor $X$ drops out only
if $\kappa_{2,0,0}=\kappa_{2,2,-2}$ and in that case the expression simplifies
to

$\mathcal{A}_{4}\left(1^{++},2^{++},3^{++},4^{\phi}\right)=\kappa_{2,2,-2}\kappa_{2,2,0}m^{4}\frac{[12]}{\langle
12\rangle}\frac{[23]}{\langle 23\rangle}\frac{[31]}{\langle 31\rangle}\\\
\times\left(-2+\frac{s}{s-m^{2}}+\frac{t}{t-m^{2}}+\frac{u}{u-m^{2}}\right),$
(22)

with $t=\left(p_{1}+p_{4}\right)^{2}$ and $u=\left(p_{1}+p_{3}\right)^{2}$. We
pause here to point out that this expression involves a nontrivial
cancellation of the $X$ dependence between the graviton exchange terms (which
arise from the non-minimal coupling $\kappa_{2,2,0}$) and the scalar exchange
terms (arising from the minimal coupling $\kappa_{2,0,0}$). This affirms the
statement made in the beginning of this section that the presence of this
minimal coupling is crucial to have a consistent and non-zero four-point
amplitude. Furthermore, it is also necessary to have
$\kappa_{2,0,0}=\kappa_{2,2,-2}$ — a relation which is to be expected on
grounds of the equivalence principle. It is interesting to compare this to the
four-point amplitude involving one scalar and three positive helicity gluons
originally computed in Dixon:2004za  to be

$\displaystyle\mathcal{A}_{4}(1^{+},2^{+},3^{+},4^{\phi})=\kappa_{1,1,-1}\kappa_{1,1,0}\,\frac{m^{4}}{\langle
12\rangle\langle 23\rangle\langle 13\rangle}~{}.$ (23)

This amplitude is particularly simple since the absence of the gluon-scalar
minimal coupling eliminates massive exchange contributions.

Moving on to the other four-point amplitudes in this theory,
$\mathcal{A}\left(1^{++},2^{++},3^{\phi},4^{\phi}\right)$ has been computed in
Britto:2021pud  via BCFW recursion relations to be

$\displaystyle\mathcal{A}_{4}\left(1^{++},2^{++},3^{\phi},4^{\phi}\right)$
$\displaystyle=\kappa^{2}_{2,2,-2}m^{4}\frac{[12]^{4}}{s\left(t-m^{2}\right)\left(u-m^{2}\right)}~{}.$
(24)

Finally, the amplitude $\mathcal{A}\left(1^{++},2^{++},3^{--},4^{\phi}\right)$
cannot be computed by a three-line shift (or using BCFW recursion). We provide
an independent derivation of this amplitude in Appendix A and show that it is
given by

$\displaystyle\mathcal{A}_{4}\left(1^{++},2^{++},3^{--},4^{\phi}\right)=\kappa_{2,2,0}\kappa_{2,-2,-2}\frac{[12]^{6}\langle
23\rangle^{2}\langle 13\rangle^{2}}{\left(s-m^{2}\right)tu}~{}.$ (25)

###  3.2 Five-point amplitudes

The five-point computation proceeds in a similar, albeit more complicated
manner to that at four points. The three-line shift in (12) and the associated
recursion relation imply that the amplitude
$\mathcal{A}\left(1^{++},2^{++},3^{++},4^{++},5^{\phi}\right)$ receives
contributions from 18 compatible factorization channels.

$\mathcal{A}_{5}\left(1^{++},2^{++},3^{++},4^{++},5^{\phi}\right)=\sum_{i=1}^{3}\sum_{j=i+1}^{4}\left(\mathcal{A}_{5}^{gr,i,j}+\mathcal{A}_{5}^{sc,i,j}\right)+\sum_{i=1}^{3}\mathcal{A}_{5}^{sc,i,5}.$
(26)

They can be divided into five classes with
$\mathcal{A}_{5}^{gr,1,2},\mathcal{A}_{5}^{gr,3,4},\mathcal{A}_{5}^{sc,1,2},\mathcal{A}_{5}^{sc,3,4},\mathcal{A}_{5}^{gr,1,5}$
being their representatives. The rest can be obtained by making appropriate
replacements on the external legs. As we will now demonstrate, it will be
useful to set $X=\lambda_{4}$.

Evaluation of $\mathcal{A}_{5}^{gr,3,4}$ and $\mathcal{A}_{5}^{sc,3,4}$

These classes of terms arise from graviton and scalar exchanges and correspond
to the momentum $\left(\hat{p}_{1}+p_{4}\right)^{2}$ going on-shell.

$\displaystyle\mathcal{A}_{5}^{gr,3,4}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.8}]{Figs/5ptamp4part1.pdf}
\includegraphics[scale={0.8}]{Figs/5ptamp4part2.pdf}}}$ (27)
$\displaystyle=\mathcal{A}_{3}\left(\hat{3}^{++},4^{++},I^{--}\right)\frac{1}{P^{2}}\mathcal{A}_{4}\left(-I^{++},\hat{2}^{++},\hat{1}^{++},5^{\phi}\right)$
$\displaystyle=\left(\frac{[34]^{3}}{[4I][I3]}\right)^{2}\frac{1}{\langle
34\rangle[43]}\frac{[12]}{\langle\hat{1}\hat{2}\rangle}\frac{[2I]}{\langle\hat{2}I\rangle}\frac{[I1]}{\langle
I\hat{1}\rangle}$
$\displaystyle\qquad\qquad\times\left(-2+\frac{s_{\hat{1}\hat{2}}}{s_{\hat{1}\hat{2}}-m^{2}}+\frac{s_{\hat{2}\hat{3}4}}{s_{\hat{2}\hat{3}4}-m^{2}}+\frac{s_{\hat{1}\hat{3}4}}{s_{\hat{1}\hat{3}4}-m^{2}}\right)~{},$

where we’ve used the notation
$s_{ij}=\left(p_{i}+p_{j}\right)^{2},s_{ijk}=\left(p_{i}+p_{j}+p_{k}\right)^{2}$.
After some manipulation this term takes the form

$\displaystyle\mathcal{A}_{5}^{gr,3,4}$ $\displaystyle=-\frac{[34]}{\langle
34\rangle}\frac{[12]}{\langle\hat{1}\hat{2}\rangle}\frac{\left[2|\hat{3}+4|X\right\rangle\left[1|\hat{3}+4|X\right\rangle}{\langle
3X\rangle\langle
4X\rangle\langle\hat{1}\hat{3}\rangle\langle\hat{2}4\rangle}\left[-2+\frac{s_{\hat{1}\hat{2}}}{s_{\hat{1}\hat{2}}-m^{2}}+\frac{s_{\hat{2}\hat{3}4}}{s_{\hat{2}\hat{3}4}-m^{2}}+\frac{s_{\hat{1}\hat{3}4}}{s_{\hat{1}\hat{3}4}-m^{2}}\right]~{}.$
(28)

The shifted spinors are determined by first solving $\langle\hat{3}4\rangle=0$
for $z$. This gives

$\displaystyle z=\frac{\langle 34\rangle}{\langle
X4\rangle}~{},\quad\hat{\lambda}_{1}=\lambda_{1}-\frac{[23]\langle
34\rangle}{[12]\langle
X4\rangle}X~{},\quad\hat{\lambda}_{2}=\lambda_{2}-\frac{[31]\langle
34\rangle}{[12]\langle X4\rangle}X~{},\quad\hat{\lambda}_{3}=\frac{\langle
X3\rangle}{\langle X4\rangle}\lambda_{4}~{}.$ (29)

Plugging all of this into (28), we find that

$\displaystyle\mathcal{A}_{5}^{gr,3,4}\propto\langle X4\rangle~{},$ (30)

which vanishes when $X=\lambda_{4}$. A similar analysis for the scalar
exchange diagrams shows that it also vanishes. Thus all diagrams in this class
do not contribute to the amplitude and we have

$\displaystyle\mathcal{A}_{5}^{gr,1,4}=\mathcal{A}_{5}^{gr,2,4}=\mathcal{A}_{5}^{gr,3,4}=\mathcal{A}_{5}^{sc,1,4}=\mathcal{A}_{5}^{sc,2,4}=\mathcal{A}_{5}^{sc,3,4}=0~{}.$
(31)

Evaluation of $\mathcal{A}_{5}^{gr,1,2}$

This class of terms arise from the momentum
$\left(\hat{p}_{1}+\hat{p}_{2}\right)^{2}$ going on-shell and corresponds to
the exchange of a graviton. This channel cannot be obtained from (27) because
the shift does not treat $p_{4}$ on the same footing as $p_{1},p_{2},p_{3}$.

$\displaystyle\mathcal{A}_{5}^{gr,1,2}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.8}]{Figs/4ptamp1part1.pdf}
\includegraphics[scale={0.8}]{Figs/5ptamp1part2.pdf}}}$ (32)
$\displaystyle=\mathcal{A}_{3}\left(\hat{1}^{++},\hat{2}^{++},I^{--}\right)\frac{1}{P^{2}}\mathcal{A}_{4}\left(-I^{++},\hat{3}^{++},4^{++},5^{\phi}\right)$
$\displaystyle=\left(\frac{[12]^{3}}{[2I][I1]}\right)^{2}\frac{1}{\langle
12\rangle[21]}\frac{[34][4I][I3]}{\langle\hat{3}\hat{4}\rangle\langle
4I\rangle\langle I\hat{3}\rangle}$
$\displaystyle\qquad\qquad\times\left(-2+\frac{s_{\hat{3}4}}{s_{\hat{3}4}-m^{2}}+\frac{s_{\hat{1}\hat{2}4}}{s_{\hat{1}\hat{2}4}-m^{2}}+\frac{s_{\hat{1}\hat{2}\hat{3}}}{s_{\hat{1}\hat{2}\hat{3}}-m^{2}}\right)~{}.$

The kinematic invariants involving shifted spinors can be evaluated by first
determining the value of $z$ for this channel by setting
$\langle\hat{1}\hat{2}\rangle=0$ and plugging in this value of $z$ into (12).
On doing this, we get

$\displaystyle\langle\hat{3}4\rangle=\langle 34\rangle,\qquad\langle
4I\rangle=-1,\qquad\langle
I\hat{3}\rangle=-\frac{s_{123}}{\left[3|p_{1}+p_{2}|4\right\rangle}$ (33)
$\displaystyle s_{\hat{1}\hat{2}4}=s_{124}-s_{12},\qquad
s_{\hat{1}\hat{2}\hat{3}}=s_{123}~{}.$

This brings this term to the form

$\mathcal{A}_{5}^{gr,1,2}=-\frac{[12]}{\langle 12\rangle}\frac{[34]}{\langle
34\rangle}\frac{[14]}{\langle 14\rangle}\frac{[24]}{\langle
24\rangle}\left(\frac{1}{s_{24}}+\frac{1}{s_{14}}\right)\frac{\left[3|p_{5}|4\right\rangle^{2}}{s_{123}}\\\
\times\left(-2+\frac{s_{34}}{s_{34}-m^{2}}+\frac{s_{14}+s_{24}}{s_{14}+s_{24}-m^{2}}+\frac{s_{123}}{s_{123}-m^{2}}\right)~{}.$
(34)

Evaluation of $\mathcal{A}_{5}^{sc,1,5}$

This class of terms arises from the momentum
$\left(\hat{p}_{1}+p_{5}\right)^{2}$ going on-shell and corresponds to a
scalar exchange.

$\displaystyle\mathcal{A}_{5}^{sc,1,5}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.8}]{Figs/5ptamp3part1.pdf}
\includegraphics[scale={0.8}]{Figs/5ptamp3part2.pdf}}}$ (35)
$\displaystyle=\mathcal{A}_{3}\left(\hat{1}^{++},5^{\phi},I^{\phi}\right)\frac{1}{P^{2}-m^{2}}\mathcal{A}_{4}\left(-I^{\phi},\hat{3}^{++},4^{++},\hat{2}^{++}\right)$
$\displaystyle=\frac{\left[1|p_{5}|\xi_{1}\right\rangle\left[1|p_{5}|\xi_{2}\right\rangle}{\langle\hat{1}\xi_{1}\rangle\langle\hat{2}\xi_{2}\rangle}\frac{1}{s_{15}-m^{2}}\frac{[23][34][42]}{\langle\hat{2}\hat{3}\rangle\langle\hat{3}4\rangle\langle
4\hat{2}\rangle}$
$\displaystyle\qquad\qquad\times\left(-2+\frac{s_{\hat{3}4}}{s_{\hat{3}4}-m^{2}}+\frac{s_{\hat{3}\hat{2}}}{s_{\hat{3}\hat{2}}-m^{2}}+\frac{s_{\hat{2}4}}{s_{\hat{2}4}-m^{2}}\right)~{}.$

In this case, the kinematic invariants involving shifted momenta evaluate to

$\displaystyle\langle\hat{2}\hat{3}\rangle=\frac{s_{15}-s_{23}-m^{2}}{[23]}~{},\qquad
s_{\hat{2}\hat{3}}=s_{23}-s_{15}+m^{2}~{},\qquad s_{\hat{3}4}=s_{34}~{},\qquad
s_{\hat{2}4}=s_{24}~{}.$ (36)

Setting $\xi_{1}=\xi_{2}=\lambda_{4}$ for simplicity, we get

$\displaystyle\mathcal{A}_{5}^{sc,1,2}=$ $\displaystyle\frac{[23]}{\langle
23\rangle}\frac{[34]}{\langle 34\rangle}\frac{[24]}{\langle
24\rangle}\frac{[14]}{\langle
14\rangle}\frac{s_{23}}{s_{14}}\frac{\left[1|p_{5}|4\right\rangle^{2}}{\left(s_{15}-m^{2}\right)\left(s_{15}-s_{23}-m^{2}\right)}$
$\displaystyle\qquad\qquad\times\left(-2+\frac{s_{34}}{s_{34}-m^{2}}+\frac{s_{123}+s_{14}}{s_{123}+s_{14}-m^{2}}+\frac{s_{24}}{s_{24}-m^{2}}\right).$

Evaluation of $\mathcal{A}_{5}^{sc,1,2}$

This final class of terms also corresponds to a scalar exchange but arises
from the momentum $\left(\hat{p}_{1}+\hat{p}_{2}\right)^{2}$ going on-shell.

$\displaystyle\mathcal{A}_{5}^{sc,1,2}$
$\displaystyle=\vbox{\hbox{\includegraphics[scale={0.8}]{Figs/4ptamp2part1.pdf}
\includegraphics[scale={0.8}]{Figs/5ptamp2part2.pdf}}}$ (37)
$\displaystyle=\mathcal{A}_{3}\left(\hat{1}^{++},\hat{2}^{++},I^{\phi}\right)\frac{1}{P^{2}-m^{2}}\mathcal{A}_{4}\left(-I^{\phi},\hat{3}^{++},4^{++},5^{\phi}\right)$
$\displaystyle=[12]^{4}\,\frac{1}{s_{12}-m^{2}}\,\frac{[34]^{4}}{s_{\hat{3}4}\left(s_{\hat{1}\hat{2}4}-m^{2}\right)\left(s_{\hat{1}\hat{2}\hat{3}}-m^{2}\right)}~{}.$

Here the shifted momenta are determined by
$\langle\hat{1}\hat{2}\rangle[12]=m^{2}$ and this term evaluates to

$\mathcal{A}_{5}^{sc,1,2}=\frac{[12]^{4}[34]^{4}}{\left(s_{12}-m^{2}\right)s_{34}\left(s_{123}-m^{2}\right)\left(s_{14}+s_{24}\right)}~{}.$
(38)

This completes the computation of
$\mathcal{A}_{5}\left(1^{++},2^{++},3^{++},4^{++},5^{\phi}\right)$. In
computing this amplitude using the recursion relations, we have implicitly
assumed that there is no contribution from a pole at infinity in the complex
$z$ plane. In order to verify the validity of this assumption, we have checked
that the amplitude we computed has all the correct soft and collinear limits.

##  4 Celestial amplitudes

We are interested in evaluating the celestial amplitudes corresponding to the
bulk scattering amplitudes of the previous section. As explained in Section 2,
we are interested in the correlators obtained by setting $\Delta=0$, where
$\Delta$ is the conformal dimension of the massive scalar. Since all the
dependence on the scalar coordinates drops out of the correlation functions,
we will refer to the celestial counterparts of $n+1$-point bulk scattering
amplitudes as $n$-point correlators. This is further supported, as we will now
show, by the structure of these correlators.

###  4.1 Two-point function

The momentum space three-point amplitude (10), written using the
parameterization (1, 3) is given by

$\displaystyle\mathcal{A}_{3}\left(1^{++},2^{++},3^{\phi}\right)=\kappa_{2,2,0}\,m^{4}\,\frac{\bar{z}_{12}^{2}}{z_{12}^{2}}\delta^{(4)}\left(p_{1}+p_{2}+Q\right)~{}.$
(39)

We have made use of momentum conservation ($\langle 12\rangle[12]=m^{2}$) in
arriving at the above form. The corresponding celestial amplitude integrated
over the scalar phase space is given by the expression

$\displaystyle\tilde{\mathrm{A}}_{2}\left(\left\\{\Delta_{1},z_{1},\bar{z}_{1}\right\\}^{++},\left\\{\Delta_{2},z_{2},\bar{z}_{2}\right\\}^{++}\right)$
$\displaystyle=\kappa_{2,2,0}\,\sum_{\epsilon_{1},\epsilon_{3}}\int_{0}^{\infty}dy\,dz_{3}\,d\bar{z}_{3}\,\frac{m^{2}}{4y^{3}}\times\int_{0}^{\infty}d\omega_{1}\,d\omega_{2}\,\omega_{1}^{\Delta_{1}-1}\,\omega_{2}^{\Delta_{2}-1}\,\frac{\bar{z}_{12}^{2}}{z_{12}^{2}}\,\delta^{(4)}\left(p_{1}+p_{2}+Q\right)~{}.$
(40)

We can use the $\delta$-function to solve for $y,z,\bar{z},\omega_{1}$ to get

$\displaystyle\delta^{(4)}\left(p_{1}+p_{2}+Q\right)=\frac{1}{\left|J\right|}\delta\left(y-y^{\star}\right)\delta\left(\bar{z}_{3}-\bar{z}^{\star}_{3}\right)\delta\left(z_{3}-z_{3}^{\star}\right)\delta\left(\omega_{1}-\omega_{1}^{\star}\right)~{},$
(41)

where555We have not displayed the solutions for $z_{3},\bar{z}_{3}$ since they
are not required for this computation.

$\displaystyle\omega_{1}^{\star}=\frac{m^{2}}{4\epsilon_{1}\epsilon_{2}z_{12}\bar{z}_{12}\omega_{2}},\qquad
y^{\star}=\frac{-2m\epsilon_{2}\epsilon_{3}z_{12}\bar{z}_{12}\omega_{2}}{m^{2}+4z_{12}\bar{z}_{12}\omega_{2}^{2}},\qquad
J=\frac{m^{2}z_{12}\bar{z}_{12}\epsilon_{1}\epsilon_{2}\omega_{2}}{(y^{\star})^{3}}~{}.$
(42)

After integrating over $y,z_{3},\bar{z}_{3},\omega_{1}$, we get

$\tilde{\mathrm{A}}_{2}=\kappa_{2,2,0}m^{2}\left(\frac{\bar{z}_{12}}{z_{12}}\right)^{2}\sum_{\epsilon_{1},\epsilon_{3}}\int_{0}^{\infty}d\omega_{2}\,\left(\frac{m^{2}}{4\epsilon_{1}\epsilon_{2}z_{12}\bar{z}_{12}\omega_{2}}\right)^{\Delta_{1}}\,\omega_{2}^{\Delta_{2}-1}\,\Theta\left(\omega_{1}^{\star}\right)\,\Theta\left(y^{\star}\right)~{}.$
(43)

The integral over the $\delta$-function gives rise to the two $\Theta$
functions since the $y,\omega_{1}$ integrals are over $\mathbb{R}^{+}$. The
sum over $\epsilon_{3}$ eliminates $\Theta\left(y^{\star}\right)$ while the
sum over $\epsilon_{1}$ eliminates $\Theta\left(\omega_{1}^{\star}\right)$
yielding

$\displaystyle\tilde{\mathrm{A}}_{2}$
$\displaystyle=\kappa_{2,2,0}m^{2}\left|\frac{\bar{z}_{12}}{z_{12}}\right|^{2}\left|\frac{m^{2}}{4z_{12}\bar{z}_{12}}\right|^{\Delta_{1}}\int_{0}^{\infty}d\omega_{2}\,\omega_{2}^{\Delta_{2}-\Delta_{1}-1}~{}.$
(44)

Performing the $\omega_{2}$ integral, we obtain the final result for the two-
point celestial correlator

$\langle\langle\mathcal{O}^{++}_{\Delta_{1}}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}^{++}_{\Delta_{2}}\left(z_{2},\bar{z}_{2}\right)\rangle\rangle\equiv\tilde{\mathrm{A}}_{2}=\kappa_{2,2,0}m^{2}\left|\frac{\bar{z}_{12}}{z_{12}}\right|^{2}\left|\frac{m^{2}}{4z_{12}\bar{z}_{12}}\right|^{\Delta_{1}}\times
2\pi\delta\left(\Delta_{1}-\Delta_{2}\right)~{}.$ (45)

We see that the resulting object indeed resembles a two-point correlation
function as in an ordinary CFT.

It is also natural to consider correlation functions like
$\langle\mathcal{O}_{\Delta_{1}}^{++}\left(z_{1},\bar{z}_{1}\right)\mathcal{O}_{\Delta_{2}}^{0}\left(z_{2},\bar{z}_{2}\right)\mathcal{O}_{0}^{0}\left(z_{3},\bar{z}_{3}\right)\rangle$.
Interpreting it as a two-point function of a graviton and scalar, we expect it
to vanish from the 2D CFT perspective. However, it is nonzero and is
suggestive of the fact that additional insertions of the scalar cannot be
treated on the same footing as gravitons. We leave the exploration of such
correlators and their meaning to future work, noting for now that the singular
part of the graviton OPE does not involve the scalar, so the set of
correlators $\langle\langle\dots\rangle\rangle$ with only graviton insertions
forms a self-consistent sector in some sense, similar in spirit to studying a
current algebra independently from any actual CFT.666We thank Sruthi Narayanan
for bringing this to our attention.

###  4.2 Three-point functions

####  4.2.1
$\langle\langle\mathcal{O}_{\Delta_{1}}^{++}\,\mathcal{O}_{\Delta_{2}}^{++}\,\mathcal{O}_{\Delta_{3}}^{++}\rangle\rangle$

The three-point functions are computed from momentum space four-point
scattering amplitudes. We start with (22) which we write as

$\mathcal{A}_{4}\left(1^{++},2^{++},3^{++},4^{\phi}\right)=\mathcal{A}_{4}^{gr}+\mathcal{A}_{4}^{sc,s}+\mathcal{A}_{4}^{sc,t}+\mathcal{A}_{4}^{sc,u},$
(46)

with $\mathcal{A}_{4}^{gr}$ representing the contribution of graviton exchange
in all channels. As in the previous section, the $\delta$-function is used to
eliminate the scalar momentum variables $z_{4},\bar{z}_{4},y$ as well as
$\omega_{3}$

$\displaystyle\delta^{(4)}\left(p_{1}+p_{2}+p_{3}+Q\right)=\frac{1}{\left|J\right|}\delta\left(y-y^{\star}\right)\delta\left(\bar{z}-\bar{z}_{\star}\right)\delta\left(z-z_{\star}\right)\delta\left(\omega_{3}-\omega_{3}^{\star}\right)~{},$
(47)

with

$\displaystyle\omega_{3}^{\star}=\frac{\epsilon_{3}}{4}\frac{m^{2}-4\omega_{1}\omega_{2}z_{12}\bar{z}_{12}\epsilon_{1}\epsilon_{2}}{z_{13}\bar{z}_{13}\epsilon_{1}\omega_{1}+z_{23}\bar{z}_{23}\epsilon_{2}\omega_{2}}~{}~{},~{}~{}y^{\star}=-\frac{2m\epsilon_{4}\left(z_{13}\bar{z}_{13}\epsilon_{1}\omega_{1}+z_{23}\bar{z}_{23}\epsilon_{2}\omega_{2}\right)}{m^{2}+4\left(\epsilon_{1}\omega_{1}z_{13}+\epsilon_{2}\omega_{2}z_{23}\right)\left(\epsilon_{1}\omega_{1}\bar{z}_{13}+\epsilon_{2}\omega_{2}\bar{z}_{23}\right)}~{},$
$\displaystyle\hskip
142.26378ptJ=\frac{m^{2}}{y^{3}}\left(z_{13}\bar{z}_{13}\epsilon_{1}\omega_{1}+z_{23}\bar{z}_{23}\epsilon_{2}\omega_{2}\right)~{}.$
(48)

The resulting three-point celestial correlator for both the graviton and
scalar exchanges is given by

$\displaystyle\tilde{\mathrm{A}}_{3}=\frac{1}{4}\sum_{\epsilon_{i}}\int_{0}^{\infty}d\omega_{1}\,d\omega_{2}\,\omega_{1}^{\Delta_{1}-1}\,\omega_{2}^{\Delta_{2}-1}\frac{\left(\omega_{3}^{\star}\right)^{\Delta_{3}-1}}{\left|z_{13}\bar{z}_{13}\epsilon_{1}\omega_{1}+z_{23}\bar{z}_{23}\epsilon_{2}\omega_{2}\right|}\Theta\left(y^{\star}\right)\Theta\left(\omega_{3}^{\star}\right)\mathcal{A}_{4}.$
(49)

We will now compute the above integral for each term in $\mathcal{A}_{4}$
(46), whose explicit form in the momentum space is given by the relation (22).

Mellin transform of $\mathcal{A}_{4}^{gr}$

In the parametrization of (1), the graviton exchange term of the amplitude is

$\displaystyle\mathcal{A}_{4}^{gr}=-2\kappa_{2,2,-2}\kappa_{2,2,0}m^{4}\frac{\bar{z}_{12}\bar{z}_{23}\bar{z}_{31}}{z_{12}z_{23}z_{31}}.$
(50)

Plugging this into (49) and performing the sum over
$\epsilon_{3},\epsilon_{4}$ gives

$\tilde{\mathrm{A}}_{3}^{gr}=-2^{1-2\Delta_{3}}\kappa_{2,2,-2}\kappa_{2,2,0}m^{4}\frac{\bar{z}_{12}\bar{z}_{23}\bar{z}_{31}}{z_{12}z_{23}z_{31}}\sum_{\epsilon_{1},\epsilon_{2}}\int_{0}^{\infty}d\omega_{1}\,d\omega_{2}\,\omega_{1}^{\Delta_{1}-1}\,\omega_{2}^{\Delta_{2}-1}\\\
\times\left|m^{2}-4\omega_{1}\omega_{2}z_{12}\bar{z}_{12}\epsilon_{1}\epsilon_{2}\right|^{\Delta_{3}-1}\left|z_{13}\bar{z}_{13}\epsilon_{1}\omega_{1}+z_{23}\bar{z}_{23}\epsilon_{2}\omega_{2}\right|^{-\Delta_{3}}.$
(51)

Changing variables to

$\displaystyle\omega_{1}=\frac{m}{2}\left(\frac{\left|z_{23}\bar{z}_{23}\right|}{\left|z_{12}\bar{z}_{12}\right|\left|z_{13}\bar{z}_{13}\right|}\right)^{\frac{1}{2}}\sqrt{\frac{Y}{X}}~{}~{},~{}~{}\omega_{2}=\frac{m}{2}\left(\frac{\left|z_{13}\bar{z}_{13}\right|}{\left|z_{12}\bar{z}_{12}\right|\left|z_{23}\bar{z}_{23}\right|}\right)^{\frac{1}{2}}\sqrt{XY}~{}~{},~{}~{}\epsilon_{1}\epsilon_{2}=\epsilon~{},$
(52)

brings the integral to the form

$\displaystyle\tilde{\mathrm{A}}_{3}^{gr}=-\kappa_{2,2,-2}\kappa_{2,2,0}\,\mathcal{Z}\left(z_{ij},\bar{z}_{ij}\right)\,\sum_{\epsilon}f_{1}\left(\Delta_{i},\alpha_{1}\right)\,f_{2}\left(\Delta_{i},\alpha_{2}\right)~{},$
(53)

with

$\displaystyle\mathcal{Z}=2^{-\beta-3}m^{\beta+5}\frac{\bar{z}_{12}\bar{z}_{23}\bar{z}_{13}}{z_{12}z_{23}z_{13}}\left|z_{12}\bar{z}_{12}\right|^{\frac{\Delta_{3}-\Delta_{1}-\Delta_{2}}{2}}\left|z_{23}\bar{z}_{23}\right|^{\frac{\Delta_{1}-\Delta_{2}-\Delta_{3}}{2}}\left|z_{13}\bar{z}_{13}\right|^{\frac{\Delta_{2}-\Delta_{1}-\Delta_{3}}{2}}~{},$
(54)

being a standard conformal factor and $\beta=-3+\sum_{i=1}^{3}\Delta_{i}$.
$f_{1}$ and $f_{2}$ are integrals over the variables $X$ and $Y$ respectively
and depend on the $z_{ij},\bar{z}_{ij}$ only through

$\alpha_{1}=\epsilon\,\text{sgn}\left(z_{13}\bar{z}_{13}z_{23}\bar{z}_{23}\right),\quad\alpha_{2}=\epsilon\,\text{sgn}\left(z_{12}\bar{z}_{12}\right).$
(55)

The $f_{1},f_{2}$ integrals are now readily performed giving us

$f_{1}\left(\Delta_{i},\alpha_{1}\right)\equiv\int_{0}^{\infty}dX\,X^{\frac{\Delta_{2}-\Delta_{1}+\Delta_{3}}{2}-1}\,\left|X+\alpha_{1}\right|^{-\Delta_{3}}\\\
=\frac{\Gamma\left(\frac{\Delta_{2}+\Delta_{3}-\Delta_{1}}{2}\right)\Gamma\left(\frac{\Delta_{1}-\Delta_{2}+\Delta_{3}}{2}\right)}{\Gamma\left(\Delta_{3}\right)}\begin{cases}1&\alpha_{1}=1\\\
\frac{\cos\left(\frac{\pi}{2}\left(\Delta_{1}-\Delta_{2}\right)\right)}{\cos\frac{\pi}{2}\Delta_{3}}&\alpha_{1}=-1~{},\end{cases}$
(56)

and

$f_{2}\left(\Delta_{i},\alpha_{2}\right)\equiv\int_{0}^{\infty}dY\,Y^{\frac{\Delta_{1}-\Delta_{3}+\Delta_{2}}{2}-1}\left|1-\alpha_{2}Y\right|^{\Delta_{3}-1}\\\
\qquad=\frac{1}{\pi}\Gamma\left(\frac{\Delta_{1}+\Delta_{2}-\Delta_{3}}{2}\right)\Gamma\left(\frac{2-\Delta_{1}-\Delta_{2}-\Delta_{3}}{2}\right)\Gamma\left(\Delta_{3}\right)\\\
\qquad\qquad\qquad\qquad\times\begin{cases}2\sin\frac{\pi}{2}\left(\Delta_{1}+\Delta_{2}\right)\cos\frac{\pi}{2}\Delta_{3}&\alpha_{2}=1\\\
\sin\pi\Delta_{3}&\alpha_{2}=-1~{}.\end{cases}$ (57)

Putting all of this together, we finally get two regions for the graviton
exchange term

$\displaystyle\tilde{\mathrm{A}}_{3}^{gr}=-\kappa_{2,2,-2}\kappa_{2,2,0}\,\mathcal{Z}\left(z_{ij},\bar{z}_{ij}\right)\,\mathcal{G}^{gr}\left(\Delta_{i}\right)\,\,\times\,\,\begin{cases}\mathcal{S}_{+}\qquad&\left(z_{12}\bar{z}_{12}z_{13}\bar{z}_{13}z_{23}\bar{z}_{23}\right)>0\\\
\mathcal{S}_{-}\qquad&\left(z_{12}\bar{z}_{12}z_{13}\bar{z}_{13}z_{23}\bar{z}_{23}\right)<0~{},\end{cases}$
(58)

where

$\displaystyle\mathcal{G}^{gr}\left(\Delta_{i}\right)$
$\displaystyle=\frac{1}{\pi}\,\Gamma\left(\frac{2-\Delta_{1}-\Delta_{2}-\Delta_{3}}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}-\Delta_{3}}{2}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times\Gamma\left(\frac{\Delta_{2}+\Delta_{3}-\Delta_{1}}{2}\right)\Gamma\left(\frac{\Delta_{1}-\Delta_{2}+\Delta_{3}}{2}\right)~{},$
(59)

and

$\displaystyle\mathcal{S}_{+}$
$\displaystyle=\sin\frac{\pi}{2}\left(\Delta_{1}+\Delta_{2}+\Delta_{3}\right)+\sin\frac{\pi}{2}\left(\Delta_{1}+\Delta_{2}-\Delta_{3}\right)$
$\displaystyle\qquad\qquad\qquad+\sin\frac{\pi}{2}\left(\Delta_{1}-\Delta_{2}+\Delta_{3}\right)+\sin\frac{\pi}{2}\left(\Delta_{2}-\Delta_{1}+\Delta_{3}\right)~{},$
$\displaystyle\mathcal{S}_{-}$
$\displaystyle=\sin\pi\Delta_{1}+\sin\pi\Delta_{2}+\sin\pi\Delta_{3}~{}.$ (60)

Mellin transform of $\mathcal{A}_{4}^{sc,s},\mathcal{A}_{4}^{sc,t}$ and
$\mathcal{A}_{4}^{sc,u}$

We only need to evaluate the $s$-channel exchange since the other channels can
be simply obtained from this by interchanging $1\leftrightarrow 3$ and
$2\leftrightarrow 3$. Moreover, since we have the following relation between
the graviton and scalar exchange terms

$\displaystyle\mathcal{A}_{4}^{sc,s}=-\frac{1}{2}\mathcal{A}_{4}^{gr}\times\frac{4\epsilon_{1}\epsilon_{2}\omega_{1}\omega_{2}z_{12}\bar{z}_{12}}{4\epsilon_{1}\epsilon_{2}\omega_{1}\omega_{2}z_{12}\bar{z}_{12}-m^{2}}~{},$
(61)

their Mellin transforms are quite similar. Following the steps that led from
(50) to (53), we get

$\displaystyle\tilde{\mathrm{A}}_{3}^{sc,s}=\frac{1}{2}\kappa_{2,2,-2}\kappa_{2,2,0}\,\mathcal{Z}\left(z_{ij},\bar{z}_{ij}\right)\sum_{\epsilon}\epsilon\,f_{1}\left(\Delta_{i},\alpha_{1}\right)f_{3}\left(\Delta_{i},\alpha_{2}\right)~{},$
(62)

where

$f_{3}\left(\Delta_{i},\alpha_{2}\right)\equiv\int_{0}^{\infty}dY\,Y^{\frac{\Delta_{1}-\Delta_{3}+\Delta_{2}}{2}}\left|1-\alpha_{2}Y\right|^{\Delta_{3}-2}\,\text{sgn}\left(1-\alpha_{2}Y\right)\\\
=\frac{1}{\pi}\Gamma\left(\frac{2+\Delta_{1}+\Delta_{2}-\Delta_{3}}{2}\right)\Gamma\left(\frac{2-\Delta_{1}-\Delta_{2}-\Delta_{3}}{2}\right)\Gamma\left(\Delta_{3}-1\right)\\\
\times\begin{cases}-2\sin\frac{\pi}{2}\left(\Delta_{1}+\Delta_{2}\right)\cos\frac{\pi}{2}\Delta_{3}&\alpha_{2}=1\\\
\sin\pi\Delta_{3}&\alpha_{2}=-1~{}.\end{cases}$ (63)

We can now put together the celestial amplitude corresponding to the
$s$-channel exchange by summing over $\epsilon$ resulting in

$\displaystyle\tilde{\mathrm{A}}_{3}^{sc,s}=-\frac{1}{2}\kappa_{2,2,-2}\kappa_{2,2,0}\,\mathcal{Z}\left(z_{ij},\bar{z}_{ij}\right)\,\mathcal{G}^{sc,s}\left(\Delta_{i}\right)\,\,\times\begin{cases}\mathcal{S}_{+}\quad(z_{12}\bar{z}_{12}z_{13}\bar{z}_{13}z_{23}\bar{z}_{23})\,>\,0\\\
\mathcal{S}_{-}\quad(z_{12}\bar{z}_{12}z_{13}\bar{z}_{13}z_{23}\bar{z}_{23})\,<\,0\end{cases}$
(64)

where

$\displaystyle\mathcal{G}^{sc,s}\left(\Delta_{i}\right)$
$\displaystyle=\frac{1}{\pi}\,\Gamma\left(\frac{2-\Delta_{1}-\Delta_{2}-\Delta_{3}}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}-\Delta_{3}+2}{2}\right)$
(65)
$\displaystyle\qquad\qquad\qquad\times\Gamma\left(\frac{\Delta_{1}-\Delta_{2}+\Delta_{3}}{2}\right)\Gamma\left(\frac{\Delta_{2}+\Delta_{3}-\Delta_{1}}{2}\right)\frac{\Gamma\left(\Delta_{3}-1\right)}{\Gamma\left(\Delta_{3}\right)}~{},$

and $\mathcal{S}_{\pm}$ were defined in (60). The invariance of
$\mathcal{S}_{\pm}$ under cyclic permutations of the conformal dimensions
$\Delta_{i}$ allows us to write the all-plus three-point expression in a
compact manner by combining (58) and (64) as follows:

$\displaystyle\langle\langle\mathcal{O}_{\Delta_{1}}^{++}\,\mathcal{O}_{\Delta_{2}}^{++}\,\mathcal{O}_{\Delta_{3}}^{++}\rangle\rangle$
$\displaystyle=\tilde{\mathrm{A}}_{3}^{gr}+\tilde{\mathrm{A}}_{3}^{sc,s}+\tilde{\mathrm{A}}_{3}^{sc,t}+\tilde{\mathrm{A}}_{3}^{sc,u}$
(66)
$\displaystyle=-\kappa_{2,2,-2}\kappa_{2,2,0}\,\mathcal{Z}\left(z_{ij},\bar{z}_{ij}\right)\,\left[\mathcal{G}^{gr}+\frac{1}{2}\left(\mathcal{G}^{sc,s}+\mathcal{G}^{sc,t}+\mathcal{G}^{sc,u}\right)\right]$
$\displaystyle\qquad\qquad\times\begin{cases}\mathcal{S}_{+}\quad(z_{12}\bar{z}_{12}z_{13}\bar{z}_{13}z_{23}\bar{z}_{23})\,>\,0\\\
\mathcal{S}_{-}\quad(z_{12}\bar{z}_{12}z_{13}\bar{z}_{13}z_{23}\bar{z}_{23})\,<\,0~{},\end{cases}$

where $\mathcal{G}^{sc,t}(\Delta_{i})$ and $\mathcal{G}^{sc,u}(\Delta_{i})$
are obtained from $\mathcal{G}^{sc,s}(\Delta_{i})$ given in (65) by
interchanging $\Delta_{1}\leftrightarrow\Delta_{3}$ and
$\Delta_{2}\leftrightarrow\Delta_{3}$ respectively.

####  4.2.2
$\langle\langle\mathcal{O}_{\Delta_{1}}^{++}\,\mathcal{O}_{\Delta_{2}}^{++}\,\mathcal{O}_{\Delta_{3}}^{--}\rangle\rangle$

We now present the three-point function involving two positive and one
negative helicity graviton derived from the momentum-space four-point
amplitude given by (25), which in the parameterization of (1) takes the form

$\displaystyle\mathcal{A}(1^{++},2^{++},3^{--},4^{\phi})=4^{3}\kappa_{2,2,0}\kappa_{2,-2,-2}\frac{\bar{z}_{12}^{6}z_{23}z_{13}}{\bar{z}_{23}\bar{z}_{13}}\frac{\epsilon_{1}\epsilon_{2}\omega_{1}^{3}\omega_{2}^{3}}{4\epsilon_{1}\epsilon_{2}\omega_{1}\omega_{2}z_{12}\bar{z}_{12}-m^{2}}~{}.$
(67)

The computation of this Mellin transform is very similar to the previous cases
and we omit all details. The final result takes the form

$\tilde{\mathrm{A}}_{3}\left(\left\\{\Delta_{1},z_{1},\bar{z}_{1}\right\\}^{++},\left\\{\Delta_{2},z_{2},\bar{z}_{2}\right\\}^{++},\left\\{\Delta_{3},z_{3},\bar{z}_{3}\right\\}^{--}\right)\\\
=\kappa_{2,2,0}\,\kappa_{2,-2,-2}\,\tilde{\mathcal{Z}}\left(z_{ij},\bar{z}_{ij}\right)\,\sum_{\epsilon}\epsilon\,f_{1}\left(\Delta_{i},\alpha_{1}\right)\,f_{4}\left(\Delta_{i},\alpha_{2}\right)~{},$
(68)

where

$\displaystyle\tilde{\mathcal{Z}}=2^{-\beta-4}m^{\beta+5}\,\frac{\bar{z}_{12}^{6}z_{23}z_{13}}{\bar{z}_{23}\bar{z}_{13}}\left|z_{12}\bar{z}_{12}\right|^{\frac{\Delta_{3}-\Delta_{1}-\Delta_{2}-6}{2}}\left|z_{23}\bar{z}_{23}\right|^{\frac{\Delta_{1}-\Delta_{2}-\Delta_{3}}{2}}\left|z_{13}\bar{z}_{13}\right|^{\frac{\Delta_{2}-\Delta_{1}-\Delta_{3}}{2}}~{},$
(69)

is now the appropriate conformal factor,
$f_{1}\left(\Delta_{i},\alpha_{1}\right)$ is the same integral as in (56) and

$\displaystyle f_{4}\left(\Delta_{i},\alpha_{2}\right)$
$\displaystyle\equiv\int_{0}^{\infty}dY\,Y^{\frac{\Delta_{1}-\Delta_{3}+\Delta_{2}}{2}+4}\left|1-\alpha_{2}Y\right|^{\Delta_{3}-2}\,\text{sgn}\left(1-\alpha_{2}Y\right)$
(70)
$\displaystyle=\frac{1}{\pi}\Gamma\left(\frac{6+\Delta_{1}+\Delta_{2}-\Delta_{3}}{2}\right)\Gamma\left(\frac{-2-\Delta_{1}-\Delta_{2}-\Delta_{3}}{2}\right)\Gamma\left(\Delta_{3}-1\right)$
$\displaystyle\hskip
110.96574pt\times\begin{cases}-2\sin\frac{\pi}{2}\left(\Delta_{1}+\Delta_{2}\right)\cos\frac{\pi}{2}\Delta_{3}&\alpha_{2}=1\\\
\sin\pi\Delta_{3}&\alpha_{2}=-1\,.\end{cases}$

Putting all of this together, the result is

$\displaystyle\langle\langle\mathcal{O}_{\Delta_{1}}^{++}\mathcal{O}_{\Delta_{2}}^{++}\mathcal{O}_{\Delta_{3}}^{--}\rangle\rangle=-\kappa_{2,-2,-2}\kappa_{2,2,0}\,\tilde{\mathcal{Z}}\left(z_{ij},\bar{z}_{ij}\right)\,\tilde{\mathcal{G}}\left(\Delta_{i}\right)\,\,\times\begin{cases}\mathcal{S}_{+}\quad(z_{12}\bar{z}_{12}z_{13}\bar{z}_{13}z_{23}\bar{z}_{23})\,>\,0\\\
\mathcal{S}_{-}\quad(z_{12}\bar{z}_{12}z_{13}\bar{z}_{13}z_{23}\bar{z}_{23})\,<\,0~{},\end{cases}$
(71)

where

$\displaystyle\tilde{\mathcal{G}}\left(\Delta_{i}\right)$
$\displaystyle=\frac{1}{\pi}\,\Gamma\left(\frac{-2-\Delta_{1}-\Delta_{2}-\Delta_{3}}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}-\Delta_{3}+6}{2}\right)$
$\displaystyle\qquad\qquad\qquad\times\Gamma\left(\frac{\Delta_{1}-\Delta_{2}+\Delta_{3}}{2}\right)\Gamma\left(\frac{\Delta_{2}+\Delta_{3}-\Delta_{1}}{2}\right)\frac{\Gamma\left(\Delta_{3}-1\right)}{\Gamma\left(\Delta_{3}\right)}~{},$
(72)

and $\mathcal{S}_{\pm}$ were defined in (60).

###  4.3 Four-point function

The four-point function can be obtained from the five-point amplitude computed
in Section 3 starting from the definition

$\tilde{\mathrm{A}}_{4}\equiv\frac{m^{2}}{4}\int_{0}^{\infty}\frac{dy}{y^{3}}\,\int_{-\infty}^{\infty}\,dz_{5}\,d\bar{z}_{5}\int_{0}^{\infty}\prod_{i=1}^{4}\frac{d\omega_{i}}{\omega_{i}}\,\omega_{i}^{\Delta_{i}}\delta^{(4)}\left(\sum_{i=1}^{5}p_{i}\right)\mathcal{A}_{5}~{}.$
(73)

Due to the complexity of this amplitude, we will evaluate the corresponding
celestial correlator only in Minkowski space. In this case, the regions on
which the constraints imposed by the $\Theta$ functions are satisfied can be
enumerated. As we will see below, a simplification occurs since $\bar{z}_{i}$
are the complex conjugates of $z_{i}$ and we have
$\bar{z}_{i}z_{i}=\left|z_{i}\right|^{2}>0$.777Note that in this section the
symbols $z_{i},\bar{z}_{i},z,\bar{z}$ are all complex, and $\left|\right|$
denotes the modulus of the complex number. In the previous sections, they were
real and we used $\left|\right|$ to denote the absolute value of these real
numbers. As before, we can first eliminate the coordinates of the scalar and
one energy using

$\displaystyle\delta^{(4)}\left(\sum_{i=1}^{5}p_{i}\right)=\frac{1}{|J|}\delta\left(\omega_{4}-\omega_{4}^{\star}\right)\delta\left(z_{5}-z_{5}^{\star}\right)\delta\left(\bar{z}_{5}-\bar{z}_{5}^{\star}\right)\delta\left(y-y^{\star}\right)~{},$
(74)
$\displaystyle\omega_{4}^{\star}=\frac{m^{2}-4z_{12}\bar{z}_{12}\epsilon_{1}\epsilon_{2}\omega_{1}\omega_{2}-4z_{13}\bar{z}_{13}\epsilon_{1}\epsilon_{3}\omega_{1}\omega_{3}-4z_{23}\bar{z}_{23}\epsilon_{2}\epsilon_{3}\omega_{2}\omega_{3}}{4\epsilon_{4}\left(z_{14}\bar{z}_{14}\epsilon_{1}\omega_{1}+z_{24}\bar{z}_{24}\epsilon_{2}\omega_{2}+z_{34}\bar{z}_{34}\epsilon_{3}\omega_{3}\right)}~{},$
$\displaystyle
J=-\frac{m^{2}\epsilon_{5}\left(z_{14}\bar{z}_{14}\epsilon_{1}\omega_{1}+z_{24}\bar{z}_{24}\epsilon_{2}\omega_{2}+z_{34}\bar{z}_{34}\epsilon_{3}\omega_{3}\right)}{y^{3}}~{},$
$\displaystyle
y^{\star}=\frac{-2m\epsilon_{5}\left(z_{14}\bar{z}_{14}\epsilon_{1}\omega_{1}+z_{24}\bar{z}_{24}\epsilon_{2}\omega_{2}+z_{34}\bar{z}_{34}\epsilon_{3}\omega_{3}\right)}{m^{2}+4\left(z_{14}\epsilon_{1}\omega_{1}+z_{24}\epsilon_{2}\omega_{2}+z_{34}\epsilon_{3}\omega_{3}\right)\left(\bar{z}_{14}\epsilon_{1}\omega_{1}+\bar{z}_{24}\epsilon_{2}\omega_{2}+\bar{z}_{34}\epsilon_{3}\omega_{3}\right)}~{}.$

In arriving at the above result, we have made use of the parametrization

$p_{i}=\epsilon_{i}\,\omega_{i}\left(1+z_{i}\bar{z}_{i},z_{i}+\bar{z}_{i},i\left(z_{i}-\bar{z}_{i}\right),1-z_{i}\bar{z}_{i}\right)~{},$
(75)

which is appropriate for Minkowski space. Performing the
$y,z_{5},\bar{z}_{5},\omega_{4}$ integrals, we get

$\tilde{\mathrm{A}}_{4}\equiv\int\prod_{i=1}^{3}d\omega_{i}\,\omega_{i}^{\Delta_{i}-1}\frac{1}{|J|}\delta\left(\omega_{4}-\omega_{4}^{\star}\right)\delta\left(z_{5}-z_{5}^{\star}\right)\delta\left(\bar{z}_{5}-\bar{z}_{5}^{\star}\right)\delta\left(y-y^{\star}\right)\\\
\times\left(\omega_{4}^{\star}\right)^{\Delta-4-1}\Theta\left(y^{\star}\right)\Theta\left(\omega_{1}^{\star}\right)\mathcal{A}_{5}~{}.$
(76)

We do not sum over the directions as in the previous sections. We find it
easier to solve the $\Theta$ function constraints and evaluate the amplitude
for each case. Once we have these correlators, the summation can be trivially
carried out. It is easy to see that the two $\Theta$ functions in (76) reduce
to

$\displaystyle\Theta\left(y^{\star}\right)\Theta\left(\omega_{4}^{\star}\right)=$
$\displaystyle\Theta\left(-\epsilon_{5}\left(z_{14}\bar{z}_{14}\epsilon_{1}\omega_{1}+z_{24}\bar{z}_{24}\epsilon_{2}\omega_{2}+z_{34}\bar{z}_{34}\epsilon_{3}\omega_{3}\right)\right)$
(77)
$\displaystyle\Theta\left(\epsilon_{4}\epsilon_{5}\left(-m^{2}+4z_{12}\bar{z}_{12}\epsilon_{1}\epsilon_{2}\omega_{1}\omega_{2}+4z_{13}\bar{z}_{13}\epsilon_{1}\epsilon_{3}\omega_{1}\omega_{3}+4z_{23}\bar{z}_{23}\epsilon_{2}\epsilon_{3}\omega_{2}\omega_{3}\right)\right).$

The constraints imposed by these $\Theta$ functions on
$\omega_{1},\omega_{2},\omega_{3}$ contours are most easily worked out by
first changing variables to

$\displaystyle\omega_{1}=\frac{m}{2}\left|\frac{z_{34}}{z_{13}z_{14}}\right|\,\tilde{\omega}_{1},\,\,\omega_{2}=\frac{m}{2}\left|\frac{z_{14}z_{34}}{z_{13}}\right|\frac{1}{|z_{24}|^{2}}\,\tilde{\omega}_{2},\,\,\omega_{3}=\frac{m}{2}\left|\frac{z_{14}}{z_{13}z_{34}}\right|\,\tilde{\omega}_{3}~{}.$
(78)

The constraints can now be written purely in terms of the
$\tilde{\omega}_{i},\epsilon_{i}$, the conformal cross ratio
$z=\frac{z_{12}z_{34}}{z_{13}z_{24}}$ and its conjugate:

$\displaystyle\Theta\left(y^{\star}\right)\Theta\left(\omega_{4}^{\star}\right)=$
$\displaystyle\Theta\left(-\epsilon_{5}\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)\right)$
(79)
$\displaystyle\Theta\left(\epsilon_{4}\epsilon_{5}\left(-1+\epsilon_{1}\epsilon_{2}\left|z\right|\tilde{\omega}_{1}\tilde{\omega}_{2}+\epsilon_{1}\epsilon_{3}\tilde{\omega}_{1}\tilde{\omega}_{3}+\epsilon_{2}\epsilon_{3}\left|1-z\right|\tilde{\omega}_{2}\tilde{\omega}_{3}\right)\right)~{}.$

Before enumerating all the regions, it helps to note that the $\Theta$
functions are invariant under three operations

  1. 1.

$\epsilon_{1}\tilde{\omega}_{1}\leftrightarrow\epsilon_{2}\tilde{\omega}_{2}$

  2. 2.

$\epsilon_{1}\tilde{\omega}_{1}\leftrightarrow\epsilon_{3}\tilde{\omega}_{3}~{}~{}\text{and}~{}\left|z\right|\leftrightarrow\left|1-z\right|~{}$

  3. 3.

$\epsilon_{2}\tilde{\omega}_{2}\leftrightarrow\epsilon_{3}\tilde{\omega}_{3}~{}~{}\text{and}~{}\left|z\right|\leftrightarrow\left|1-z\right|~{}$.

Moreover, we regard combinations of $\epsilon_{i}$ differing by an overall
sign to be equivalent and set $\epsilon_{5}=-1$ in everything that follows.
These imply that we can obtain all possible regions from the ones shown in
Table 1. The combinations
$\epsilon_{1}=\epsilon_{2}=\epsilon_{3}=-\epsilon_{4}=1$ and
$\epsilon_{1}=\epsilon_{2}=\epsilon_{4}=-\epsilon_{3}=-1$ are not listed in
this table as they lead to empty regions.

Region | $\epsilon_{1}$ | $\epsilon_{2}$ | $\epsilon_{3}$ | $\epsilon_{4}$ | Constraints
---|---|---|---|---|---
$\mathcal{R}_{1}$ | 1 | 1 | 1 | 1 | $\tilde{\omega}_{1}>0,\quad\tilde{\omega}_{2}<\frac{1}{\left|z\right|^{2}\tilde{\omega}_{1}},\quad 0<\tilde{\omega}_{3}<\frac{1-\tilde{\omega}_{1}\tilde{\omega}_{2}\left|z\right|^{2}}{\tilde{\omega}_{1}+\tilde{\omega}_{2}\left|1-z\right|^{2}}$
$\mathcal{R}_{2}$ | 1 | 1 | -1 | 1 | $\tilde{\omega}_{1}>0,\quad\tilde{\omega}_{2}<\frac{1}{\left|z\right|^{2}\tilde{\omega}_{1}},\quad 0<\tilde{\omega}_{3}<\tilde{\omega}_{1}+\tilde{\omega}_{2}$
$\mathcal{R}_{3}$ | 1 | 1 | -1 | 1 | $\tilde{\omega}_{1}>0,\quad\tilde{\omega}_{2}>\frac{1}{\left|z\right|^{2}\tilde{\omega}_{1}},\quad\frac{\tilde{\omega}_{1}\tilde{\omega}_{2}\left|z\right|^{2}-1}{\tilde{\omega}_{1}+\tilde{\omega}_{2}\left|1-z\right|^{2}}<\tilde{\omega}_{3}<\tilde{\omega}_{1}+\tilde{\omega}_{2}$
$\mathcal{R}_{4}$ | 1 | -1 | -1 | 1 | $\tilde{\omega}_{1}>0,\quad 0<\tilde{\omega}_{2}<\tilde{\omega}_{1},\quad 0<\tilde{\omega}_{3}<\tilde{\omega}_{1}-\tilde{\omega}_{2}$
$\mathcal{R}_{5}$ | 1 | 1 | 1 | -1 | $\tilde{\omega}_{1}>0,\quad 0<\tilde{\omega}_{2}<\frac{1}{\left|z\right|^{2}\tilde{\omega}_{1}},\quad\tilde{\omega}_{3}>\frac{1-\left|z\right|^{2}\tilde{\omega}_{1}\tilde{\omega}_{2}}{\tilde{\omega}_{1}+\tilde{\omega}_{2}\left|1-z\right|^{2}}$
$\mathcal{R}_{6}$ | 1 | 1 | 1 | -1 | $\tilde{\omega}_{1}>0,\quad\tilde{\omega}_{2}>\frac{1}{\left|z\right|^{2}\tilde{\omega}_{1}},\quad\tilde{\omega}_{3}>0$
$\mathcal{R}_{7}$ | 1 | 1 | -1 | -1 | $\tilde{\omega}_{1}>0,\quad\tilde{\omega}_{2}>\frac{1}{\left|z\right|^{2}\tilde{\omega}_{1}},\quad\tilde{\omega}_{3}<\frac{\left|z\right|^{2}\tilde{\omega}_{1}\tilde{\omega}_{2}-1}{\tilde{\omega}_{1}+\left|1-z\right|^{2}\tilde{\omega}_{2}}$
Table 1: A list of regions carved out by the $\Theta$ function constraints for
different values of $\epsilon_{i}$. The regions for any combinations of
$\epsilon_{i}$ not listed here are either empty or are related to the listed
ones via symmetries. In all of these cases we have set $\epsilon_{5}=-1$.

The variable change (78) also vastly simplifies (74) to

$\displaystyle\omega_{4}^{\star}=-\frac{m\epsilon_{4}}{2}\left|\frac{z_{13}}{z_{14}z_{34}}\right|\frac{\left(1-|z|^{2}\epsilon_{1}\epsilon_{2}\tilde{\omega}_{1}\tilde{\omega}_{2}-\epsilon_{1}\epsilon_{3}\tilde{\omega}_{1}\tilde{\omega}_{3}-\left|1-z\right|^{2}\epsilon_{2}\epsilon_{3}\tilde{\omega}_{2}\tilde{\omega}_{3}\right)}{\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)}~{},$
(80) $\displaystyle
J=-\frac{m^{3}\epsilon_{5}}{2y^{3}}\left|\frac{z_{14}z_{34}}{z_{13}}\right|\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)~{}.$

and the integral for the celestial correlator

$\tilde{\mathrm{A}}_{4}=\mathcal{Z}\int_{\mathcal{R}_{i}}d\omega_{1}\,d\omega_{2}\,d\omega_{3}\,\omega_{1}^{\Delta_{1}-1}\omega_{2}^{\Delta_{2}-1}\omega_{3}^{\Delta_{3}-1}\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)^{-\Delta_{4}}\\\
\left(1-|z|^{2}\epsilon_{1}\epsilon_{2}\tilde{\omega}_{1}\tilde{\omega}_{2}-\epsilon_{1}\epsilon_{3}\tilde{\omega}_{1}\tilde{\omega}_{3}-\left|1-z\right|^{2}\epsilon_{2}\epsilon_{3}\tilde{\omega}_{2}\tilde{\omega}_{3}\right)^{\Delta_{4}-1}\mathcal{A}_{5}^{\star}~{}.$
(81)

$\mathcal{A}_{5}^{\star}$ is meant to represent the five-point amplitude after
imposing momentum conservation and the variable change in (78) and

$\mathcal{Z}=\left|z_{34}\right|^{\frac{1}{2}\left(\Delta_{1}+\Delta_{2}-\Delta_{3}-\Delta_{4}-1\right)}\,\left|z_{13}\right|^{\frac{1}{2}\left(-\Delta_{1}-\Delta_{2}-\Delta_{3}+\Delta_{4}+1\right)}\\\
\left|z_{14}\right|^{\frac{1}{2}\left(-\Delta_{1}+\Delta_{2}+\Delta_{3}-\Delta_{4}+1\right)}\left|z_{24}\right|^{1-\Delta_{2}}\left(\frac{m}{2}\right)^{-5+\sum_{i=1}^{4}\Delta_{i}}~{},$
(82)

is the standard conformal factor.

It is prudent to first identify a few classes of integrals which arise in the
computation of the correlator above. To this end, it is necessary to rewrite
the five-point amplitude $(\ref{eq:5ptgravscalar})$ using the variables
$(\ref{eq:varchange2})$ as in Appendix B. We can check that all the terms
$\mathcal{A}_{5}^{gr,1,2,(i)}$ (defined in Appendix B) lead to the following
class of integrals

$I^{(1)}_{\mathcal{R}_{i}}\left(\alpha,\beta,\gamma,\delta,\rho,a,b,c\right)=\int_{\mathcal{R}_{i}}d\tilde{\omega}_{1}\,d\tilde{\omega}_{2}\,d\tilde{\omega}_{3}\tilde{\omega}_{1}^{\alpha}\,\tilde{\omega}_{2}^{\beta}\,\tilde{\omega}_{3}^{\gamma}\left(1-\hat{s}\right)^{\delta}\hat{s}^{a}\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)^{\rho}\\\
\left(\hat{s}\epsilon_{3}\tilde{\omega}_{3}+\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}\right)^{b}\left(\hat{s}\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}\right)+\epsilon_{3}\tilde{\omega}_{3}\right)^{c}~{},$
(83)
$I^{(2)}_{\mathcal{R}_{i}}\left(\alpha,\beta,\gamma,\delta,\rho,a,b,c,d,e\right)\\\
=\int_{\mathcal{R}_{i}}d\tilde{\omega}_{1}\,d\tilde{\omega}_{2}\,d\tilde{\omega}_{3}\tilde{\omega}_{1}^{\alpha}\,\tilde{\omega}_{2}^{\beta}\,\tilde{\omega}_{3}^{\gamma}\left(1-\hat{s}\right)^{\delta}\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)^{\rho}\left(\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)^{d}\\\
\times\left(\hat{s}\left(\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)+\epsilon_{1}\tilde{\omega}_{1}\right)^{a}\left(\hat{s}\epsilon_{3}\tilde{\omega}_{3}+\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}\right)^{b}\\\
\times\left(1+\left|z\right|^{2}\tilde{\omega}_{2}^{2}+\left(z+\bar{z}\right)\epsilon_{2}\epsilon_{3}\tilde{\omega}_{2}\tilde{\omega}_{3}+\tilde{\omega}_{3}^{2}\right)^{c}\left(\epsilon_{2}\tilde{\omega}_{2}\hat{s}+\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{3}\tilde{\omega}_{3}\right)^{e}~{},$
(84)

where
$\hat{s}=|z|\epsilon_{1}\epsilon_{2}\tilde{\omega}_{1}\tilde{\omega}_{2}+\epsilon_{1}\epsilon_{3}\tilde{\omega}_{1}\tilde{\omega}_{3}+\left|1-z\right|\epsilon_{2}\epsilon_{3}\tilde{\omega}_{2}\tilde{\omega}_{3}$.
The exponents represented by Greek letters can take complex values while the
Latin ones only take on integer values. All of the terms in
$\tilde{\mathrm{A}}_{4}$ can be expressed in terms of these integrals. This
decomposition can be performed easily using Mathematica.

The integrals in (83, 84) must first be regulated with appropriate $i\epsilon$
prescriptions. Here, we must distinguish between the factors with Greek and
Latin exponents. The former are positive in all the regions $\mathcal{R}_{i}$
as they are just the arguments of the $\Theta$ functions. Consequently they do
not require $i\epsilon$ prescriptions. However, the factors with integer
exponents introduce new singularities which sometimes intersect the regions
$\mathcal{R}_{i}$. The necessary $i\epsilon$ prescription follows from the
corresponding Feynman $i\epsilon$ for the propagators in the tree amplitude.
We will now tacitly assume such a prescription and outline how we can derive a
Mellin-Barnes representation for these integrals. Applying the Mellin-Barnes
decomposition formula888Here and henceforth, we will suppress the contour of
integration. It is understood that these contours run upward along the
imaginary axis such that the poles separate into left and right ones. For more
details, please refer to Dubovyk:2022obc .

$\frac{1}{(A_{1}+\dots+A_{n})^{\lambda}}=\frac{1}{(2\pi
i)^{n-1}\Gamma(\lambda)}\int\prod_{i=1}^{n-1}\,dx_{i}\,A_{i}^{x_{i}}A_{n}^{-\lambda-
x_{1}-\dots-x_{n-1}}\\\
\times\prod_{i=1}^{n-1}\Gamma(-x_{i})\Gamma(\lambda+x_{1}+\dots+x_{n-1})~{},$
(85)

to each factor with an integer exponent yields an integral of the form

$\displaystyle\mathcal{I}^{(j)}_{\mathcal{R}_{i}}=\int\prod_{k=1}^{r}dx_{k}\,G\left(x_{1},\dots,x_{r},a,b,c,d,e,z,\bar{z}\right)\mathcal{J}_{\mathcal{R}_{i}}~{}.$
(86)

This generic form is motivated by the observation that each of the factors
$A_{i}$ in the Mellin-Barnes decomposition is a product of powers of
$\tilde{\omega}_{1},\tilde{\omega}_{2},\tilde{\omega}_{3},z$ and $\bar{z}$.
This modifies the exponents of
$\tilde{\omega}_{1},\tilde{\omega}_{2},\tilde{\omega}_{3}$ from
$\alpha,\beta,\gamma$ in (83, 84) to $A,B$ and $C$. The function
$G\left(x_{1},\dots,x_{r},a,b,c,d,e\right)$ is a universal (independent of the
regions $\mathcal{R}_{i}$) factor containing all the $\Gamma$ functions and
powers of $z,\bar{z}$ arising from (85). $r$ is an integer which represents
the total number of Mellin variables that must be introduced to achieve this
form. We have also defined

$\mathcal{J}_{\mathcal{R}_{i}}=\int_{\mathcal{R}_{i}}d\tilde{\omega}_{1}\,d\tilde{\omega}_{2}\,d\tilde{\omega}_{3}\,\tilde{\omega}_{1}^{A}\,\tilde{\omega}_{2}^{B}\,\tilde{\omega}_{3}^{C}\left(1-\hat{s}\right)^{\delta}\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)^{\rho}~{}.$
(87)

We can now complete the reduction to Mellin-Barnes form more efficiently by
performing some of the integrals and identifying them with well known
hypergeometric functions. The details of this somewhat lengthy procedure are
relegated to Appendix C. Here, we merely present the final result:

$\mathcal{J}_{\mathcal{R}_{4}}=\int\prod_{i=1}^{4}\,dx_{r+i}\,\mathcal{G}\left(x_{r+1},\dots,x_{r+4}\right)\left(z-1\right)^{x_{r+2}}\left(\bar{z}-1\right)^{x_{r+3}}\left(1-\left|z\right|^{2}\right)^{x_{r+4}},$
(88)

where $\mathcal{G}$ is defined in (134).

##  5 Graviton OPEs

We can obtain the graviton OPEs (for positive helicity gravitons) from the
three-point correlator (58, 64). Since the OPE is local and is insensitive to
insertions far away from the position of the relevant operators, we expect it
to agree with the OPE derived for correlators without a massive scalar.
Consider the limit as $z_{23}\to 0$ in which we have for the conformal factor

$\mathcal{Z}\xrightarrow{2\parallel
3}2^{2-2\Delta_{1}}m^{2\Delta_{1}+2}\frac{\bar{z}_{13}^{2}}{z_{13}^{2}}\frac{\bar{z}_{23}}{z_{23}}\left|z_{13}\bar{z}_{13}\right|^{-\Delta_{1}}\left(-i\pi\right)\delta\left(\Delta_{2}+\Delta_{3}-\Delta_{1}\right)~{}.$
(89)

In arriving at this result, we have made use of the identity Donnay:2020guq

$\lim_{\nu\to 0}\nu^{z-\Delta}\,\Gamma\left(\Delta-z\right)=-2\pi
i\delta\left(\Delta-z\right)~{}.$ (90)

The terms in the three-point correlator now become

$\tilde{\mathrm{A}}_{3}^{gr}\xrightarrow{2\parallel
3}-2i\kappa_{2,2,-2}\frac{\bar{z}_{23}}{z_{23}}\frac{1}{\pi}\Gamma\left(1-\Delta_{2}-\Delta_{3}\right)\Gamma\left(\Delta_{2}\right)\Gamma\left(\Delta_{3}\right)\\\
\times\left(\sin\pi\Delta_{2}+\sin\pi\Delta_{3}+\sin\pi\left(\Delta_{2}+\Delta_{3}\right)\right)\tilde{\mathrm{A}}_{2}\\\
=-2\kappa_{2,2,-2}\frac{\bar{z}_{23}}{z_{23}}\Big{(}B(\Delta_{2},\Delta_{3})+B(1-\Delta_{2}-\Delta_{3},\Delta_{3})+B(1-\Delta_{2}-\Delta_{3},\Delta_{2})\Big{)}\tilde{\mathrm{A}}_{2}~{},$
(91)

$\tilde{\mathrm{A}}_{3}^{sc,s}\xrightarrow{2\parallel
3}-i\kappa_{2,2,-2}\frac{\bar{z}_{23}}{z_{23}}\frac{1}{\pi}\Gamma\left(1-\Delta_{2}-\Delta_{3}\right)\Gamma\left(\Delta_{2}+1\right)\Gamma\left(\Delta_{3}-1\right)\\\
\times\left(-\sin\pi\Delta_{2}-\sin\pi\Delta_{3}-\sin\pi\left(\Delta_{2}+\Delta_{3}\right)\right)\tilde{\mathrm{A}}_{2}\\\
=-\kappa_{2,2,-2}\frac{\bar{z}_{23}}{z_{23}}\Big{(}B(\Delta_{2}+1,\Delta_{3}-1)-B(1-\Delta_{2}-\Delta_{3},\Delta_{3}-1)+B(1-\Delta_{2}-\Delta_{3},\Delta_{2}+1)\Big{)}\tilde{\mathrm{A}}_{2}~{},$
(92)

$\tilde{\mathrm{A}}_{3}^{sc,u}\xrightarrow{2\parallel
3}-i\kappa_{2,2,-2}\frac{\bar{z}_{23}}{z_{23}}\frac{1}{\pi}\Gamma\left(1-\Delta_{2}-\Delta_{3}\right)\Gamma\left(\Delta_{3}+1\right)\Gamma\left(\Delta_{2}-1\right)\\\
\times\left(-\sin\pi\Delta_{2}-\sin\pi\Delta_{3}-\sin\pi\left(\Delta_{2}+\Delta_{3}\right)\right)\tilde{\mathrm{A}}_{2}\\\
=-\kappa_{2,2,-2}\frac{\bar{z}_{23}}{z_{23}}\Big{(}B(\Delta_{2}-1,\Delta_{3}+1)-B(1-\Delta_{2}-\Delta_{3},\Delta_{2}-1)+B(1-\Delta_{2}-\Delta_{3},\Delta_{3}+1)\Big{)}\tilde{\mathrm{A}}_{2}~{},$
(93)

while $\tilde{\mathrm{A}}_{3}^{sc,t}$ is subleading in this limit and hence we
drop it. Here $B(x,y)$ is the Euler beta function. Now, if we write the OPE as

$\mathcal{O}_{\Delta_{2}}^{++}\left(z_{2},\bar{z}_{2}\right)\mathcal{O}_{\Delta_{3}}^{++}\left(z_{3},\bar{z}_{3}\right)\sim\frac{\bar{z}_{23}}{z_{23}}C_{\Delta_{2},\Delta_{3}}^{\Delta_{2}+\Delta_{3}}\,\mathcal{O}_{\Delta_{2}+\Delta_{3}}^{++}\left(z_{3},\bar{z}_{3}\right)+\dots~{},$
(94)

we can read off the OPE coefficient to be

$\displaystyle C_{\Delta_{2},\Delta_{3}}^{\Delta}=-\kappa_{2,2,-2}\Big{(}$
$\displaystyle
2B(\Delta_{2},\Delta_{3})+2B(1-\Delta_{2}-\Delta_{3},\Delta_{3})+2B(1-\Delta_{2}-\Delta_{3},\Delta_{2})$
$\displaystyle+B(\Delta_{2}+1,\Delta_{3}-1)-B(1-\Delta_{2}-\Delta_{3},\Delta_{3}-1)-B(1-\Delta_{2}-\Delta_{3},\Delta_{2}+1)$
$\displaystyle+B(\Delta_{2}-1,\Delta_{3}+1)-B(1-\Delta_{2}-\Delta_{3},\Delta_{2}-1)-B(1-\Delta_{2}-\Delta_{3},\Delta_{3}+1)\Big{)}$
$\displaystyle=-\kappa_{2,2,-2}\Big{(}$ $\displaystyle
B(\Delta_{2}-1,\Delta_{3}-1)-B(\Delta_{2}-1,3-\Delta_{2}-\Delta_{3})-B(\Delta_{3}-1,3-\Delta_{2}-\Delta_{3})\Big{)}~{},$
(95)

where we have used the following identities:

  * •

$B(\Delta_{2}-1,\Delta_{3}+1)+B(\Delta_{2}+1,\Delta_{3}-1)+2B(\Delta_{2},\Delta_{3})-B(\Delta_{2}-1,\Delta_{3}-1)=0$

  * •

$B(1-\Delta_{2}-\Delta_{3},\Delta_{2}-1)+B(1-\Delta_{2}-\Delta_{3},\Delta_{2}+1)-2B(1-\Delta_{2}-\Delta_{3},\Delta_{2})\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-B(\Delta_{2}-1,3-\Delta_{2}-\Delta_{3})=0$

  * •

$B(1-\Delta_{2}-\Delta_{3},\Delta_{3}-1)+B(1-\Delta_{2}-\Delta_{3},\Delta_{3}+1)-2B(1-\Delta_{2}-\Delta_{3},\Delta_{3})\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-B(\Delta_{3}-1,3-\Delta_{2}-\Delta_{3})=0~{}$,

to simplify the OPE coefficients in the final equality above. This OPE
coefficient cannot be directly compared with those obtained in
Stieberger:2018onx ; Pate:2019lpp  as it does not correspond to OPEs of
incoming or outgoing operators, but to specific linear combinations of them.
We can use the OPEs of Stieberger:2018onx ; Pate:2019lpp  to derive the
appropriate quantity for comparison. To do this, we start by considering999We
set $\epsilon_{2}=1$ since there are only two inequivalent choices for
$\epsilon_{1},\epsilon_{2}$ upto an overall sign.,101010We suppress all the
coordinate dependence of the operators.

$\sum_{\epsilon_{3}=\pm
1}\mathcal{O}_{\Delta_{2}}^{++,+1}\,\mathcal{O}_{\Delta_{3}}^{++,\epsilon_{3}}=\mathcal{O}_{\Delta_{2}}^{++,+1}\,\mathcal{O}_{\Delta_{3}}^{++,+1}+\mathcal{O}_{\Delta_{2}}^{++,+1}\mathcal{O}_{\Delta_{3}}^{++,-1}.$
(96)

Applying the usual OPE we get,

$\displaystyle\sum_{\epsilon_{3}=\pm
1}\mathcal{O}_{\Delta_{2}}^{++,+1}\,\mathcal{O}_{\Delta_{3}}^{++,\epsilon_{3}}\sim-\kappa_{2,2,-2}\frac{\bar{z}_{23}}{z_{23}}\Big{(}$
$\displaystyle
B(\Delta_{2}-1,\Delta_{3}-1)\,\mathcal{O}_{\Delta_{2}+\Delta_{3}}^{++,+1}$
$\displaystyle\quad-B(\Delta_{2}-1,3-\Delta_{2}-\Delta_{3})\,\mathcal{O}_{\Delta_{2}+\Delta_{3}}^{++,+1}$
$\displaystyle\qquad-B(\Delta_{3}-1,3-\Delta_{2}-\Delta_{3})\mathcal{O}_{\Delta_{2}+\Delta_{3}}^{++,-1}\Big{)}~{}.$
(97)

If we now insert this into a three-point function and sum over the directions
of the new insertion, we get

$\displaystyle\sum_{\epsilon_{1},\epsilon_{3}}\langle\langle\mathcal{O}_{\Delta_{1}}^{++,\epsilon_{1}}\,\mathcal{O}_{\Delta_{2}}^{++,+1}\,\mathcal{O}_{\Delta_{3}}^{++,\epsilon_{3}}\rangle\rangle\sim-\kappa_{2,2,-2}$
$\displaystyle\frac{\bar{z}_{23}}{z_{23}}\Big{(}B(\Delta_{2}-1,\Delta_{3}-1)-B(\Delta_{2}-1,3-\Delta_{2}-\Delta_{3})$
$\displaystyle-B(\Delta_{3}-1,3-\Delta_{2}-\Delta_{3})\Big{)}\sum_{\epsilon_{1}}\langle\langle\mathcal{O}_{\Delta_{1}}^{++,\epsilon_{1}}\mathcal{O}_{\Delta_{2}+\Delta_{3}}^{++}\rangle\rangle~{},$
(98)

matching the expression for the OPE coefficient obtained in (95). This shows
that the OPEs in the presence of a massive scalar are consistent with those in
its absence.

##  6 The conformally soft graviton theorem

The computations of the previous sections have shown that celestial amplitudes
involving $n$ gravitons and a massive scalar behave like $n$-point graviton
correlators when the conformal dimension of the scalar is taken to vanish. We
will now examine whether this similarly persists for the conformal soft
theorems Puhm:2019zbl . In this paper, we will only examine the leading soft
theorem leaving the others for future work. In momentum space, the leading
soft theorem Weinberg  for an amplitude with $n$ positive helicity gravitons
and a massive scalar is

$\displaystyle\mathcal{A}_{n+1}\left(\varepsilon
p_{1}^{++},p_{2}^{++},\dots,p_{n}^{++},p_{n+1}\right)\xrightarrow[]{\varepsilon\to
0}\,\frac{1}{\varepsilon}S\left[p_{1}^{++}\right]\mathcal{A}_{n}\left(p_{2}^{++},\dots,p_{n}^{++},p_{n+1}\right)+\mathcal{O}\left(\varepsilon^{0}\right)~{},$
(99)

where $S\left[p_{1}^{++}\right]$ is the relevant soft factor given by

$\displaystyle
S\left[p_{1}^{++}\right]=\sum_{i=2}^{n+1}\frac{\epsilon_{\mu\nu}\left(p_{1}\right)p_{i}^{\mu}p_{i}^{\nu}}{p_{i}\cdot
p_{1}}~{},$ (100)

and $\epsilon_{\mu\nu}\left(p_{1}\right)$ is the polarization tensor of the
soft graviton, which takes the form111111In general, this can be written with
two arbitrary spinors. Here, we have made a specific choice. The soft factor
is, of course, independent of this choice.

$\displaystyle\epsilon^{\mu\nu}\left(p_{1}\right)=\frac{\langle
2|\gamma^{\mu}|1]}{\langle 21\rangle}\frac{\langle 2|\gamma^{\nu}|1]}{\langle
21\rangle}~{},$ (101)

for a positive helicity graviton. Note that there is a sum over all the other
particles involved in the amplitude. For the four-point amplitude
$\mathcal{A}_{4}\left(1^{++},2^{++},3^{++},4^{\phi}\right)$ in (22), this
becomes

$\displaystyle S\left[p_{1}^{++}\right]$ $\displaystyle=\frac{[12]}{\langle
12\rangle}\frac{[13]}{\langle
13\rangle}\frac{[23]^{2}}{t-m^{2}}=\frac{\bar{z}_{12}\bar{z}_{13}z_{23}^{2}}{z_{12}z_{13}}\frac{\omega_{2}\omega_{3}}{2m\omega_{1}}\frac{y}{y^{2}+z_{14}\bar{z}_{14}}.$
(102)

It can be checked that the momentum-space amplitude (22) satisfies the soft
theorems. It is worth noting that the leading soft behavior arises entirely
from the term in the amplitude that corresponds to the exchange of the massive
scalar. Indeed, without this, the amplitude would fail to satisfy the soft
graviton theorem as already stated in Section 3. In order to examine the
celestial avatar of this statement, we must compute

$\lim_{\Delta_{1}\to
1}\left(\Delta_{1}-1\right)\tilde{\mathcal{A}}_{n+1}^{\epsilon}=\frac{m^{2}}{4}\underset{\omega_{1}=0}{\oint}d\omega_{1}\int_{0}^{\infty}\prod_{i=2}^{n}\frac{d\omega_{i}}{\omega_{i}}\omega_{i}^{\Delta_{i}}\int_{0}^{\infty}\frac{dy}{y^{3}}\\\
\times\int_{-\infty}^{\infty}dz_{n+1}\,d\bar{z}_{n+1}\,G_{\Delta_{n+1}}\left(z_{n+1},\bar{z}_{n+1},w,\bar{w}\right)\mathcal{A}_{n+1}$
(103)

where we have used the definition (4) and the identity

$\lim_{\varepsilon\to
0}\varepsilon\left|x\right|^{\varepsilon-1}=\delta\left(x\right),$ (104)

which holds for functions falling off at infinity at least as fast as $x^{-b}$
for some $b>0$. For the four-point amplitude under consideration, this turns
into

$\lim_{\Delta_{1}\to
1}(\Delta_{1}-1)\,\tilde{\mathcal{A}}_{4}=\frac{m}{8}\frac{\bar{z}_{12}\bar{z}_{13}z_{23}^{2}}{z_{12}z_{13}}\int\frac{dy}{y^{3}}\,dz_{4}\,d\bar{z}_{4}d\omega_{2}\,d\omega_{3}\,\omega_{2}^{\Delta_{2}}\,\omega_{3}^{\Delta_{3}}\\\
\times\left(\frac{y}{y^{2}+z_{45}\bar{z}_{45}}\right)^{\Delta_{4}}\left(\frac{y}{y^{2}+z_{14}\bar{z}_{14}}\right)\mathcal{A}_{3}~{}.$
(105)

We must now set $\Delta_{4}=0$ in order to obtain the conformal soft theorem
for the correlators considered in this paper. The presence of the soft factor
prevents us from interpreting this quantity as a lower point celestial
amplitude with $\Delta_{4}=0$. In fact, this can be interpreted as a new
correlator with $\Delta_{4}=1$ and the scalar coordinates being
$\left\\{z_{1},\bar{z}_{1}\right\\}$. Thus, we can write

$\lim_{\Delta_{1}\to
1}(\Delta_{1}-1)\,\tilde{\mathcal{A}}_{4}\left(\left\\{\Delta_{1},z_{1},\bar{z}_{1}\right\\},\left\\{\Delta_{2},z_{2},\bar{z}_{2}\right\\},\left\\{\Delta_{3},z_{3},\bar{z}_{3}\right\\},\left\\{\Delta_{4},w,\bar{w}\right\\}\right)\\\
=\frac{1}{2m}\frac{\bar{z}_{12}\bar{z}_{13}z_{23}^{2}}{z_{12}z_{13}}\tilde{\mathcal{A}}_{3}\left(\left\\{\Delta_{2}+1,z_{2},\bar{z}_{2}\right\\},\left\\{\Delta_{3}+1,z_{3},\bar{z}_{3}\right\\},\left\\{1,z_{1},\bar{z}_{1}\right\\}\right)~{}.$
(106)

This is also supported by the following direct computation

$\displaystyle\lim_{\Delta_{1}\to 1}(\Delta_{1}-1)\tilde{\mathrm{A}}_{3}$
$\displaystyle=\frac{m^{2}}{2}\frac{\bar{z}_{12}^{3}z_{23}^{2}}{z_{12}^{3}z_{13}^{2}}\bigg{|}\frac{m^{2}}{4z_{23}\bar{z}_{23}}\bigg{|}^{\Delta_{3}}\bigg{|}\frac{m^{2}z_{13}\bar{z}_{13}}{4z_{12}\bar{z}_{12}z_{23}\bar{z}_{23}}\bigg{|}^{\frac{\Delta_{2}-\Delta_{3}+1}{2}}$
$\displaystyle~{}\qquad\qquad\times\begin{cases}\frac{\pi}{\cos{\left(\frac{\pi}{2}(\Delta_{2}-\Delta_{3})\right)}}~{}~{};~{}~{}\textrm{sgn}(z_{12}\bar{z}_{12}z_{23}\bar{z}_{23})>0\\\
-\pi\tan{\left(\frac{\pi}{2}(\Delta_{2}-\Delta_{3})\right)}~{}~{};~{}~{}\textrm{sgn}(z_{12}\bar{z}_{12}z_{23}\bar{z}_{23})<0~{}~{}.\end{cases}$
(107)

This suggests that the leading soft theorem relates the set of correlators
with vanishing scalar conformal dimension to those with unit conformal
dimension. It would be intriguing to investigate if this has an interpretation
along the lines of Kapec:2022axw ; Kapec:2022hih . The connection between soft
theorems and Ward identities of asymptotic symmetries in the presence of
massive particles has been studied in Campiglia:2015kxa ; Campiglia:2015qka
in general. It would also be interesting to see if any lessons could be
extracted from this analysis.

##  7 Outlook and discussion

The correlators computed in this paper are well-defined and non-
distributional, resembling ordinary CFT correlators. This computation opens up
several lines of investigation. The first of these is understanding whether
the remaining soft theorems are modified, along with their associated
symmetries. Next is understanding the singularity structure of these
correlators either using the Mellin Barnes representations (88) and
generalizing the analysis in Yuan:2018qva  or using the Euler integral
representation (83, 84) and methods in Mizera:2021icv ; Caron-Huot:2021xqj . A
particularly interesting question is understanding how the massive exchange is
manifest in the singularity structure. Another question which would benefit
from further investigation is if an on-shell version of the correlators in
Sleight:2023ojm  agree with those in this paper. This would provide a
connection between the two different definitions. The final and perhaps most
interesting and challenging question is whether we can identify a 2D CFT that
reproduces the correlation functions in this paper.

## Acknowledgements

We are grateful to Shamik Banerjee, Eduardo Casali, Walker Melton, Sruthi
Narayanan, Andrzej Pokraka, Andrea Puhm, Marcus Spradlin and Andy Strominger
for helpful discussions. This work was supported in part by the US Department
of Energy under contract DE-SC0010010 Task F, by Simons Investigator Award
#376208, and by a Bershadsky Distinguished Visiting Fellowship at Harvard
(AV). AB was also supported by the Celestial Holography Initiative at the
Perimeter Institute for Theoretical Physics and the Simons Collaboration on
Celestial Holography. AYS was also supported by the STFC grant DRR00590.

##  Appendix A Derivation of the $\mathcal{A}(1^{++},2^{++},3^{--},4^{\phi})$
amplitude

This amplitude is not constructible via BCFW or three line shift recursions.
The simplest way to derive this amplitude is by appealing to consistent
factorization and locality along the lines of Arkani-Hamed:2017jhn . The basic
procedure is to compute the residues on all factorization channels by
multiplying the appropriate three-point amplitudes and then guessing an
expression that correctly reproduces all them. For the case at hand, we begin
by computing the residue in the $u$-channel corresponding to a graviton
exchange. This is

$\displaystyle\underset{[13]\to
0}{\text{Res}}\,\mathcal{A}_{4}\left(1^{++},2^{++},3^{--},4^{\phi}\right)$
$\displaystyle=\frac{1}{\langle
13\rangle}\vbox{\hbox{\includegraphics[scale={0.6}]{Figs/4ptamp3part1.pdf}
\includegraphics[scale={0.6}]{Figs/4ptamp3part2.pdf}}}$ (108)
$\displaystyle=\frac{1}{\langle 13\rangle}\kappa_{2,2,-2}\left(\frac{\langle
3I\rangle^{3}}{\langle I1\rangle\langle
13\rangle}\right)^{2}\,\kappa_{2,2,0}\,[2I]^{4}$
$\displaystyle=\kappa_{2,2,0}\kappa_{2,2,-2}\frac{[12]^{6}\langle
23\rangle^{2}\langle 13\rangle^{2}}{\langle 13\rangle t^{2}}~{}.$ (109)

The expression above has a double pole in $t$ which would be unphysical in an
amplitude. However, on the support of $u=0$, we have $s-m^{2}=-t$. An ansatz
for the amplitude is thus

$\mathcal{A}_{4}\left(1^{++},2^{++},3^{--},4^{\phi}\right)=\kappa_{2,2,0}\kappa_{2,2,-2}\frac{[12]^{6}\langle
23\rangle^{2}\langle 13\rangle^{2}}{t\,u\left(s-m^{2}\right)}~{}.$ (110)

It can be checked that this correctly reproduces the $t$\- and $u$-channel
residues and has the correct soft limits.

##  Appendix B Expressions for various terms in the five-point amplitude

This appendix contains expressions for all the terms in (26) written in terms
of the variables introduced in (78).

$\displaystyle\mathcal{A}_{5}^{gr,1,2,(1)}\equiv\frac{[12]}{\langle
12\rangle}\frac{[34]}{\langle 34\rangle}\frac{[14]}{\langle
14\rangle}\frac{[24]}{\langle
24\rangle}\frac{\left[3|p_{5}|4\right\rangle^{2}}{s_{24}}\frac{2}{s_{123}}=\frac{\bar{z}}{z}\frac{\bar{z}_{13}^{2}\bar{z}_{24}^{2}}{z_{13}^{2}z_{24}^{2}}\epsilon_{2}\epsilon_{3}\frac{\tilde{\omega}_{3}}{\tilde{\omega}_{2}}\frac{2\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}\left(1-\bar{z}\right)\right)^{2}}{\hat{s}}$
$\displaystyle\mathcal{A}_{5}^{gr,1,2,(2)}\equiv\frac{[12]}{\langle
12\rangle}\frac{[34]}{\langle 34\rangle}\frac{[14]}{\langle
14\rangle}\frac{[24]}{\langle
24\rangle}\frac{\left[3|p_{5}|4\right\rangle^{2}}{s_{14}}\frac{2}{s_{123}}=\frac{\bar{z}}{z}\frac{\bar{z}_{13}^{2}\bar{z}_{24}^{2}}{z_{13}^{2}z_{24}^{2}}\epsilon_{1}\epsilon_{3}\frac{\tilde{\omega}_{3}}{\tilde{\omega}_{1}}\frac{2\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}\left(1-\bar{z}\right)\right)^{2}}{\hat{s}}$
(111)

$\displaystyle\mathcal{A}_{5}^{gr,1,2,(3)}\equiv-\frac{[12]}{\langle
12\rangle}\frac{[34]}{\langle 34\rangle}\frac{[14]}{\langle
14\rangle}\frac{[24]}{\langle
24\rangle}\frac{\left[3|p_{5}|4\right\rangle^{2}}{s_{24}}\frac{s_{34}}{s_{34}-m^{2}}\frac{1}{s_{123}}=\frac{\bar{z}}{z}\frac{\bar{z}_{13}^{2}\bar{z}_{24}^{2}}{z_{13}^{2}z_{24}^{2}}\epsilon_{2}\frac{\tilde{\omega}_{3}^{2}}{\tilde{\omega}_{2}}\frac{\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}\left(1-\bar{z}\right)\right)^{2}}{\hat{s}\epsilon_{3}\tilde{\omega}_{3}+\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}}\frac{\hat{s}-1}{\hat{s}}$
$\displaystyle\mathcal{A}_{5}^{gr,1,2,(4)}\equiv-\frac{[12]}{\langle
12\rangle}\frac{[34]}{\langle 34\rangle}\frac{[14]}{\langle
14\rangle}\frac{[24]}{\langle
24\rangle}\frac{\left[3|p_{5}|4\right\rangle^{2}}{s_{14}}\frac{s_{34}}{s_{34}-m^{2}}\frac{1}{s_{123}}=\frac{\bar{z}}{z}\frac{\bar{z}_{13}^{2}\bar{z}_{24}^{2}}{z_{13}^{2}z_{24}^{2}}\epsilon_{2}\frac{\tilde{\omega}_{3}^{2}}{\tilde{\omega}_{1}}\frac{\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}\left(1-\bar{z}\right)\right)^{2}}{\hat{s}\epsilon_{3}\tilde{\omega}_{3}+\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}}\frac{\hat{s}-1}{\hat{s}}$
(112)

$\displaystyle\mathcal{A}_{5}^{gr,1,2,(5)}$
$\displaystyle\equiv-\frac{[12]}{\langle 12\rangle}\frac{[34]}{\langle
34\rangle}\frac{[14]}{\langle 14\rangle}\frac{[24]}{\langle
24\rangle}\frac{\left[3|p_{5}|4\right\rangle^{2}}{s_{24}}\frac{s_{14}+s_{24}}{s_{14}+s_{24}-m^{2}}\frac{1}{s_{123}}$
$\displaystyle=\frac{\bar{z}}{z}\frac{\bar{z}_{13}^{2}\bar{z}_{24}^{2}}{z_{13}^{2}z_{24}^{2}}\epsilon_{2}\epsilon_{3}\frac{\tilde{\omega}_{3}}{\tilde{\omega}_{2}}\frac{\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}\left(1-\bar{z}\right)\right)^{2}\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}\right)}{\hat{s}\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}\right)+\epsilon_{3}\tilde{\omega}_{3}}\frac{1-\hat{s}}{\hat{s}}$
$\displaystyle\mathcal{A}_{5}^{gr,1,2,(6)}$
$\displaystyle\equiv-\frac{[12]}{\langle 12\rangle}\frac{[34]}{\langle
34\rangle}\frac{[14]}{\langle 14\rangle}\frac{[24]}{\langle
24\rangle}\frac{\left[3|p_{5}|4\right\rangle^{2}}{s_{14}}\frac{s_{14}+s_{24}}{s_{14}+s_{24}-m^{2}}\frac{1}{s_{123}}$
$\displaystyle=\frac{\bar{z}}{z}\frac{\bar{z}_{13}^{2}\bar{z}_{24}^{2}}{z_{13}^{2}z_{24}^{2}}\epsilon_{2}\epsilon_{3}\frac{\tilde{\omega}_{3}}{\tilde{\omega}_{1}}\frac{\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}\left(1-\bar{z}\right)\right)^{2}\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}\right)}{\hat{s}\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}\right)+\epsilon_{3}\tilde{\omega}_{3}}\frac{1-\hat{s}}{\hat{s}}$
(113)

$\displaystyle\mathcal{A}_{5}^{gr,1,2,(7)}\equiv-\frac{[12]}{\langle
12\rangle}\frac{[34]}{\langle 34\rangle}\frac{[14]}{\langle
14\rangle}\frac{[24]}{\langle
24\rangle}\frac{\left[3|p_{5}|4\right\rangle^{2}}{s_{24}}\frac{1}{\hat{s}-1}=\frac{\bar{z}}{z}\frac{\bar{z}_{13}^{2}\bar{z}_{24}^{2}}{z_{13}^{2}z_{24}^{2}}\epsilon_{2}\epsilon_{3}\frac{\tilde{\omega}_{3}}{\tilde{\omega}_{2}}\frac{\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}\left(1-\bar{z}\right)\right)^{2}}{1-\hat{s}}$
$\displaystyle\mathcal{A}_{5}^{gr,1,2,(8)}\equiv-\frac{[12]}{\langle
12\rangle}\frac{[34]}{\langle 34\rangle}\frac{[14]}{\langle
14\rangle}\frac{[24]}{\langle
24\rangle}\frac{\left[3|p_{5}|4\right\rangle^{2}}{s_{14}}\frac{1}{\hat{s}-1}=\frac{\bar{z}}{z}\frac{\bar{z}_{13}^{2}\bar{z}_{24}^{2}}{z_{13}^{2}z_{24}^{2}}\epsilon_{2}\epsilon_{3}\frac{\tilde{\omega}_{3}}{\tilde{\omega}_{2}}\frac{\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}\left(1-\bar{z}\right)\right)^{2}}{1-\hat{s}}$
(114)

$\displaystyle\mathcal{A}_{5}^{sc,1,2(1)}$
$\displaystyle\equiv-\frac{[23][34][24][14]\left[1|p_{5}|4\right\rangle^{2}}{\langle
23\rangle\langle 34\rangle\langle 24\rangle\langle 14\rangle
s_{14}}\frac{2}{s_{15}-s_{23}-m^{2}}$
$\displaystyle=2\frac{\bar{z}_{13}^{2}\bar{z}_{24}^{2}}{z_{13}^{2}z_{24}^{2}}\frac{1-\bar{z}}{1-z}\left(\bar{z}\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)^{2}\frac{\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}}{\epsilon_{1}\tilde{\omega}_{1}+\hat{s}\left(\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)}$
$\displaystyle\mathcal{A}_{5}^{sc,1,2(2)}$
$\displaystyle\equiv\frac{[23][34][24][14]\left[1|p_{5}|4\right\rangle^{2}}{\langle
23\rangle\langle 34\rangle\langle 24\rangle\langle 14\rangle
s_{14}}\frac{2}{s_{15}-m^{2}}$
$\displaystyle=-2\frac{\bar{z}_{13}^{2}\bar{z}_{24}^{2}}{z_{13}^{2}z_{24}^{2}}\frac{1-\bar{z}}{1-z}\left(\bar{z}\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)^{2}\epsilon_{1}\tilde{\omega}_{1}\frac{\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}}{1+\left|z\right|\tilde{\omega}_{2}^{2}+\left(z+\bar{z}\right)\epsilon_{2}\epsilon_{3}\tilde{\omega}_{2}\tilde{\omega}_{3}+\tilde{\omega}_{3}^{2}}$
(115)

$\displaystyle\mathcal{A}_{5}^{sc,1,2(3)}$
$\displaystyle\equiv\frac{[23][34][24][14]\left[1|p_{5}|4\right\rangle^{2}}{\langle
23\rangle\langle 34\rangle\langle 24\rangle\langle 14\rangle
s_{14}}\frac{1}{s_{15}-s_{23}-m^{2}}\frac{s_{34}}{s_{34}-m^{2}}$
$\displaystyle=\frac{\bar{z}_{13}^{2}\bar{z}_{24}^{2}}{z_{13}^{2}z_{24}^{2}}\frac{1-\bar{z}}{1-z}\frac{\epsilon_{3}\tilde{\omega}_{3}\left(\bar{z}\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)^{2}\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)\left(1-\hat{s}\right)}{\left(\epsilon_{1}\tilde{\omega}_{1}+\hat{s}\left(\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)\right)\left(\hat{s}\epsilon_{3}\tilde{\omega}_{3}+\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}\right)}$
$\displaystyle\mathcal{A}_{5}^{sc,1,2(4)}$
$\displaystyle\equiv-\frac{[23][34][24][14]\left[1|p_{5}|4\right\rangle^{2}}{\langle
23\rangle\langle 34\rangle\langle 24\rangle\langle 14\rangle
s_{14}}\frac{1}{s_{15}-m^{2}}\frac{s_{34}}{s_{34}-m^{2}}$
$\displaystyle=-\frac{\bar{z}_{13}^{2}\bar{z}_{24}^{2}}{z_{13}^{2}z_{24}^{2}}\frac{1-\bar{z}}{1-z}\frac{\left(\bar{z}\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)^{2}\epsilon_{1}\epsilon_{3}\tilde{\omega}_{1}\tilde{\omega}_{3}\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)\left(1-\hat{s}\right)}{\left(1+\left|z\right|^{2}\tilde{\omega}_{2}^{2}+\left(z+\bar{z}\right)\epsilon_{2}\epsilon_{3}\tilde{\omega}_{2}\tilde{\omega}_{3}+\tilde{\omega}_{3}^{2}\right)\left(\hat{s}\epsilon_{3}\tilde{\omega}_{3}+\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}\right)}$
(116)

$\displaystyle\mathcal{A}_{5}^{sc,1,2(5)}$
$\displaystyle\equiv\frac{[23][34][24][14]\left[1|p_{5}|4\right\rangle^{2}}{\langle
23\rangle\langle 34\rangle\langle 24\rangle\langle 14\rangle
s_{14}}\frac{1}{s_{15}-s_{23}-m^{2}}\frac{s_{123}+s_{14}}{s_{123}+s_{14}-m^{2}}$
$\displaystyle=-\frac{\bar{z}_{13}^{2}\bar{z}_{24}^{2}}{z_{13}^{2}z_{24}^{2}}\frac{1-\bar{z}}{1-z}\left(\bar{z}\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)^{2}\frac{\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}}{\left(\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)\left(\hat{s}-1\right)}$
$\displaystyle\mathcal{A}_{5}^{sc,1,2(6)}$
$\displaystyle\equiv-\frac{[23][34][24][14]\left[1|p_{5}|4\right\rangle^{2}}{\langle
23\rangle\langle 34\rangle\langle 24\rangle\langle 14\rangle
s_{14}}\frac{1}{s_{15}-m^{2}}\frac{s_{123}+s_{14}}{s_{123}+s_{14}-m^{2}}$
$\displaystyle=\frac{\bar{z}_{13}^{2}\bar{z}_{24}^{2}}{z_{13}^{2}z_{24}^{2}}\frac{1-\bar{z}}{1-z}\frac{\left(\bar{z}\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)^{2}\epsilon_{1}\tilde{\omega}_{1}\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)\left(\epsilon_{1}\tilde{\omega}_{1}+\hat{s}\left(\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)\right)}{\left(1+\left|z\right|^{2}\tilde{\omega}_{2}^{2}+\left(z+\bar{z}\right)\epsilon_{2}\epsilon_{3}\tilde{\omega}_{2}\tilde{\omega}_{3}+\tilde{\omega}_{3}^{2}\right)\left(\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)\left(\hat{s}-1\right)}$
(117)

$\displaystyle\mathcal{A}_{5}^{sc,1,2(7)}$
$\displaystyle\equiv\frac{[23][34][24][14]\left[1|p_{5}|4\right\rangle^{2}}{\langle
23\rangle\langle 34\rangle\langle 24\rangle\langle 14\rangle
s_{14}}\frac{1}{s_{15}-s_{23}-m^{2}}\frac{s_{24}}{s_{24}-m^{2}}$
$\displaystyle=\frac{\bar{z}_{13}^{2}\bar{z}_{24}^{2}}{z_{13}^{2}z_{24}^{2}}\frac{1-\bar{z}}{1-z}\frac{\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)\left(\bar{z}\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)^{2}\epsilon_{2}\tilde{\omega}_{2}\left(1-\hat{s}\right)}{\epsilon_{1}\tilde{\omega}_{1}+\hat{s}\left(\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)\left(\epsilon_{2}\tilde{\omega}_{2}\hat{s}+\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{3}\tilde{\omega}_{3}\right)}$
$\displaystyle\mathcal{A}_{5}^{sc,1,2(8)}$
$\displaystyle\equiv-\frac{[23][34][24][14]\left[1|p_{5}|4\right\rangle^{2}}{\langle
23\rangle\langle 34\rangle\langle 24\rangle\langle 14\rangle
s_{14}}\frac{1}{s_{15}-m^{2}}\frac{s_{24}}{s_{24}-m^{2}}$
$\displaystyle=-\frac{\bar{z}_{13}^{2}\bar{z}_{24}^{2}}{z_{13}^{2}z_{24}^{2}}\frac{1-\bar{z}}{1-z}\frac{\left(\bar{z}\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)^{2}\epsilon_{1}\epsilon_{2}\tilde{\omega}_{1}\tilde{\omega}_{2}\left(\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{2}\tilde{\omega}_{2}+\epsilon_{3}\tilde{\omega}_{3}\right)\left(1-\hat{s}\right)}{1+\left|z\right|^{2}\tilde{\omega}_{2}^{2}+\left(z+\bar{z}\right)\epsilon_{2}\epsilon_{3}\tilde{\omega}_{2}\tilde{\omega}_{3}+\tilde{\omega}_{3}^{2}\left(\epsilon_{2}\tilde{\omega}_{2}\hat{s}+\epsilon_{1}\tilde{\omega}_{1}+\epsilon_{3}\tilde{\omega}_{3}\right)}~{}.$
(118)

##  Appendix C Reduction of $\mathcal{J}_{\mathcal{R}_{4}}$ to a Mellin-Barnes
integral

Start by changing variables to
$\tilde{\omega}_{1}=\tilde{\omega}_{3}X,\tilde{\omega}_{2}=\tilde{\omega}_{3}Y$.
This transforms (87) to

$\mathcal{J}_{\mathcal{R}_{4}}=\int_{0}^{\infty}dY\,\int_{1+Y}^{\infty}dX\,X^{A}\,Y^{B}\,\left(X-Y-1\right)^{\rho}\\\
\int_{0}^{\infty}d\tilde{\omega}_{3}\,\tilde{\omega}_{3}^{A+B+C+2+\rho}\left[1+\tilde{\omega}_{3}^{2}\left(\left|z\right|^{2}XY+X-Y\left|1-z\right|^{2}\right)\right]^{\delta}~{}.$
(119)

We pause here to draw attention to the fact that the integral is well-defined
as the integrand is strictly positive in the integration domain. The
$\tilde{\omega}_{3}$ integral evaluates to

$\int_{0}^{\infty}d\tilde{\omega}_{3}\,\tilde{\omega}_{3}^{A+B+C+2+\rho}\left[1+\tilde{\omega}_{3}^{2}\left(\left|z\right|^{2}XY+X-Y\left|1-z\right|^{2}\right)\right]^{\delta}\\\
=\mathcal{G}_{1}\,\left(\left|z\right|^{2}XY+X-Y\left|1-z\right|^{2}\right)^{-\frac{1}{2}\left(3+A+B+C+\rho\right)}~{},$
(120)

with

$\mathcal{G}_{1}=\frac{1}{2\Gamma\left(-\delta\right)}\Gamma\left(-\frac{1}{2}\left(3+A+B+C+\rho+2\delta\right)\right)\Gamma\left(\frac{1}{2}\left(3+A+B+C+\rho\right)\right)~{}.$
(121)

After this, (87) reads,

$\mathcal{J}_{\mathcal{R}_{4}}=\mathcal{G}_{1}\int_{0}^{\infty}dY\,\int_{1+Y}^{\infty}dX\,X^{A}\,Y^{B}\,\left(X-Y-1\right)^{\rho}\,\left(\left|z\right|^{2}XY+X-Y\left|1-z\right|^{2}\right)^{-\frac{1}{2}\left(3+A+B+C+\rho\right)}.$
(122)

The evaluation of the integral over $X$ is expedited by the further change of
variables, $t=\frac{-1+X-Y}{X-Y}$ as seen below:

$\mathcal{J}_{\mathcal{R}_{4}}=\mathcal{G}_{1}\int_{0}^{\infty}dY\,Y^{B}\left(1+Y\right)^{A}w_{2}^{-\frac{3+A+B+C+\rho}{2}}\\\
\times\int_{0}^{1}dt\,t^{\rho}\,\left(1-t\right)^{-\frac{1+A+\rho-
B-C}{2}}\left(1-\frac{Y}{Y+1}t\right)^{A}\left(1+\frac{w_{1}}{w_{2}}t\right)^{-\frac{3+A+B+C+\rho}{2}}.$
(123)

Here,
$w_{1}=Y(1-Y)\left|z\right|^{2}-2Y\,Re(z),\,w_{2}=1+Y^{2}\left|z\right|^{2}+2Y\,Re(z)$.
We can recognize the integral over $t$ as a representation of the Appell
$F_{1}$ function and write

$\int_{0}^{1}dt\,t^{\rho}\,\left(1-t\right)^{-\frac{1+A+\rho-
B-C}{2}}\left(1-\frac{Y}{Y+1}t\right)^{A}\left(1+\frac{w_{1}}{w_{2}}t\right)^{-\frac{3+A+B+C+\rho}{2}}\\\
=\mathcal{G}_{2}\,F_{1}\left(\rho+1;-A,\frac{3+A+B+C+\rho}{2};\frac{3-A+B+C+\rho}{2};\frac{Y}{Y+1},-\frac{w_{1}}{w_{2}}\right)\\\
=\mathcal{G}_{2}\left(\frac{w_{1}+w_{2}}{w_{2}}\right)^{-1-\rho}{}_{2}F_{1}\left(-A,1+\rho,\frac{1}{2}\left(3-A+B+C+\rho\right),\frac{w_{1}+Y(w_{1}+w_{2})}{(w_{1}+w_{2})(1+Y)}\right),$
(124)

with
$\mathcal{G}_{2}=\frac{\Gamma\left(\rho+1\right)\Gamma\left(\frac{1-A+B+C-\rho}{2}\right)}{\Gamma\left(\frac{3-A+B+C+\rho}{2}\right)}$
and we have used an identity of the Appell $F_{1}$ in arriving at the final
equality. Plugging in the values of $w_{1},w_{2}$, we see that the argument of
the hypergeometric ${}_{2}F_{1}$

$\displaystyle
0<\frac{w_{1}+Y(w_{1}+w_{2})}{(w_{1}+w_{2})(1+Y)}=\frac{Y\left|1-z\right|^{2}}{(1+y)(1+y\left|z\right|^{2})}<1~{}.$
(125)

This allows us to use the relevant Mellin-Barnes representation of the Gauss
hypergeometric function,

${}_{2}F_{1}\left(a,b,c,z\right)=\frac{\Gamma(c)}{2\pi
i\Gamma(a)\Gamma(b)\Gamma(c-a)\Gamma(c-b)}\\\ \times\int
ds\,\Gamma(-s)\Gamma(c-a-b-s)\Gamma(a+s)\Gamma(b+s)\left(1-z\right)^{s}.$
(126)

In applying this formula, we introduce a new Mellin variable $x_{r+1}$ in
addition to the $x_{1},\dots,x_{r}$ introduced in (86). Thus,

$\mathcal{J}_{\mathcal{R}_{4}}=\int
dx_{r+1}\,\mathcal{G}_{3}\left(x_{r+1}\right)\int_{0}^{\infty}dY\,Y^{B}\left(1+Y\right)^{A-x_{r+1}}\left(1+y\left|z\right|^{2}\right)^{-1-\rho-
x_{r+1}}\\\ \times\left(1+y\,z\right)^{-\frac{1}{2}\left(1+A+B+C-\rho-
x_{r+1}\right)}\left(1+y\,\bar{z}\right)^{-\frac{1}{2}\left(1+A+B+C-\rho-
x_{r+1}\right)}~{},$ (127)

where

$\mathcal{G}_{3}\left(x_{r+1}\right)=\Gamma(-x_{r+1})\,\Gamma\left(-x_{r+1}+\frac{1}{2}(A+B+C-\rho)\right)\Gamma(-A+x_{r+1})\\\
\times\Gamma(\rho+1+x_{r+1})\frac{\Gamma\left(\frac{1}{2}(3+A+B+C+\rho+2\delta)\right)}{2\Gamma(-A)\Gamma(-\delta)}~{}.$
(128)

A final change of variables $Y=\frac{\eta}{1-\eta}$ turns the integral into

$\mathcal{J}_{\mathcal{R}_{4}}=\int
dx_{r+1}\,\mathcal{G}_{3}\left(x_{r+1}\right)\int_{0}^{1}d\eta\,\eta^{B}\left(1-\eta\right)^{x_{r+1}+C}\left(1-\left(1-\left|z\right|^{2}\right)\eta\right)^{-1-\rho-
x_{r+1}}\\\
\times\left(1-\eta\left(1-z\right)\right)^{-\frac{1}{2}\left(1+A+B+C-\rho-
x_{r+1}\right)}\left(1-\eta\left(1-\bar{z}\right)\right)^{-\frac{1}{2}\left(1+A+B+C-\rho-
x_{r+1}\right)}~{},$ (129)

which can immediately be recognized as the integral representation of a type D
Lauricella function,

$\mathcal{J}_{\mathcal{R}_{4}}=\int
dx_{r+1}\,\mathcal{G}_{4}\left(x_{r+1}\right)F_{D}^{(3)}\left(a_{1};b_{1},b_{2},b_{3},c_{1};1-z,1-\bar{z},1-\left|z\right|^{2}\right)~{},$
(130)

where

$\displaystyle\mathcal{G}_{4}\left(x_{r+1}\right)=\mathcal{G}_{3}\left(x_{r+1}\right)\frac{\Gamma\left(B+1\right)\Gamma\left(-x_{r+1}+C+1\right)}{\Gamma\left(-x_{r+1}+B+C+2\right)}~{},$
$\displaystyle a_{1}=B+1,\quad b_{1}=b_{2}=\frac{1}{2}\left(1+A+B+C-\rho-
x_{r+1}\right)~{},$ $\displaystyle b_{3}=1+\rho-x_{r+1},\quad
c_{1}=-x_{r+1}+B+C+2~{}.$ (131)

To complete the reduction, we now use the Mellin-Barnes representation of the
Lauricella function book:3578410

$F_{D}^{(m)}\left(a_{1};b_{m},\dots,b_{m};c_{1},x_{1},\dots,x_{m}\right)\\\
=\left(\frac{1}{2\pi
i}\right)^{m}\int\prod_{i=1}^{m}dt_{i}\,\Gamma\left(-t_{i}\right)\left(-x_{i}\right)^{t_{i}}\frac{\Gamma\left(c_{1}\right)\Gamma\left(a_{1}+\sum_{i=1}^{m}t_{i}\right)\prod_{i=1}^{m}\Gamma\left(b_{i}+t_{i}\right)}{\Gamma\left(a\right)\Gamma\left(c_{1}+\sum_{i=1}^{m}t_{i}\right)\prod_{i=1}^{m}\Gamma\left(b_{i}\right)}~{},$
(132)

to get the final Mellin-Barnes form

$\mathcal{J}_{\mathcal{R}_{4}}=\int\prod_{i=1}^{4}\,dx_{r+i}\,\mathcal{G}\left(x_{r+1},\dots,x_{r+4}\right)\left(z-1\right)^{x_{r+2}}\left(\bar{z}-1\right)^{x_{r+3}}\left(1-\left|z\right|^{2}\right)^{x_{r+4}}~{},$
(133)

with

$\mathcal{G}\left(x_{r+1},\dots,x_{r+4}\right)=\left[\prod_{i=1}^{4}\Gamma\left(-x_{r+i}\right)\right]\frac{\Gamma\left(a_{1}+\sum_{i=2}^{4}x_{r+i}\right)}{\Gamma\left(c_{1}+\sum_{i=2}^{4}x_{r+i}\right)}\left[\prod_{i=2}^{4}\frac{\Gamma\left(b_{1}+x_{r+i}\right)}{\Gamma\left(b_{i}\right)}\right]\Gamma\left(c_{1}-a_{1}\right)\mathcal{G}_{3}~{}.$
(134)

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