††institutetext: Indian Institute of Technology Kanpur, Kanpur 208016, India.

# Holography in Flat Spacetimes: the case for Carroll

Arjun Bagchi  Prateksh Dhivakar  and Sudipta Dutta
(abagchi, prateksh<EMAIL_ADDRESS>
###### Abstract

We compare and contrast the two approaches of holography in asymptotically
flat spacetimes, viz. the co-dimension two Celestial approach based on the
Mellin transformation and the co-dimension one Carrollian approach based on
the modified Mellin and elucidate how some of the problems of the Celestial
approach can be rectified by the Carrollian one. Considering flat holography
as a limit from AdS/CFT makes a co-dimension one dual more plausible, and our
previous construction of Carrollian correlations from AdS Witten diagrams is
testimony to this. In this paper, we show how to generalize our earlier
analysis for operators with spin. We work out a large number of explicit non-
trivial examples (twelve) and show matching between the limit of AdS4 Witten
diagrams and 3d boundary symmetry considerations, thus making the case for the
Carrollian dual even stronger.

###  1 Introduction

Holography has been our best tool to probe the mysteries of quantum gravity.
While progress in understanding holography in the context of Anti de Sitter
(AdS) spacetimes has been spectacularly successful through the celebrated
AdS/CFT correspondence, contraction of the hologram for spacetimes different
from ads has proven to be challenging. This paper aims to address the question
of building holography for asymptotically flat spacetimes and build on recent
progress in this direction.

####  1.1 Celestial and Carrollian dual theories

Following the lead of Strominger Strominger:2013jfa ; He:2014laa ;
Cachazo:2014fwa ; Kapec:2014opa ; Strominger:2014pwa ; He:2014cra ;
Strominger:2017zoo , there has been a recent resurgence in activities in
investigating aspects of flat space physics. There has been a renewed
understanding of various aspects of infrared physics connecting asymptotic
symmetries, soft theorems, and memory effects in a triangle of relations
nowadays called the infrared triangle. A proposal of a hologram of flat space
has emerged out of this and is called Celestial holography 111For a review of
this approach with a comprehensive literature guide, the reader is referred to
Pasterski:2021rjz ; Raclariu:2021zjz ; Pasterski:2021raf .. This relates bulk
physics in 4d asymptotically flat spacetimes to a 2d conformal field theory
(CFT) living on the celestial sphere at the null boundary of flatspace. The
principle observation here is that the Lorentz group in 4d acts as the global
conformal group on the sphere at the boundary, and through Mellin
transformations, one can map between bulk S-matrix elements in flat spacetimes
to $n$-point correlation functions in 2d CFT Pasterski:2016qvg ;
Pasterski:2017kqt ; Pasterski:2017ylz . The Celestial programme has had many
successes discovering new results in asymptotic symmetries and scattering
amplitudes.

There are attempts at holography in asymptotically flat spacetimes, which
predates the Celestial programme deBoer:2003vf ; Barnich:2010eb ;
Bagchi:2010zz . We will especially be interested in the approach that is
nowadays called the Carrollian approach and originates from the observation of
the isomorphism between the asymptotic symmetries of flat spacetimes and non-
Lorentzian conformal symmetries in one lower dimension Bagchi:2010zz ;
Bagchi:2012cy ; Duval:2014uva ; Bagchi:2016bcd . While the initial thrust in
this direction was understanding the case of 3d asymptotically flat spacetimes
in terms of 2d field theories with Carrollian conformal or BMS3 symmetry
Bagchi:2012yk ; Bagchi:2012xr ; Afshar:2013vka ; Gonzalez:2013oaa ;
Bagchi:2014iea ; Hartong:2015usd ; Bagchi:2015wna ; Jiang:2017ecm ;
Hijano:2017eii ; Apolo:2020bld , of late there is an emerging body of
interesting work which relates to 4d asymptotically flat spacetimes
Bagchi:2016bcd ; Banerjee:2018gce ; Bagchi:2022emh ; Bagchi:2023fbj ;
Donnay:2022aba ; Donnay:2022wvx ; Nguyen:2023vfz .

In this paper, our aim is two-fold. We first begin by comparing and
contrasting the Carrollian and Celestial approaches to flat holography. We
will justify why we think the Carrollian co-dimension one approach, which
explicitly keeps track of the null nature of the asymptotic boundary of
Minkowski spacetimes, is more appropriate for constructing a hologram of
asymptotically flat spacetimes. The modifed Mellin transformation, introduced
in Banerjee:2018gce , would be central to our arguments and analysis.

We will then focus on our recent construction of the flat limit of Witten
diagrams in AdS4, which yields Carroll conformal correlation functions on the
boundary Bagchi:2023fbj . In the earlier paper Bagchi:2023fbj , we focussed on
spinless two and three-point correlations. We show in this paper that our
construction generalises to more involved correlation functions, e.g., four-
point scalar correlators, correlators in split signature, and spinning
correlators, and we explore a whole range of examples (a total of 12 explicit
cases) and show matching between bulk results obtained in the limit from AdS
and boundary results analyzed with the help symmetry structures.

Of these cases, we also examine a particular one that does not have a
Celestial analogue. This is a non-MHV $(+++)/(---)$ three point amplitude.
Such amplitudes arise from loop corrections or higher dimensional operators
Dixon:1993xd ; Dixon:2004za . These amplitudes are UV divergent in the Mellin
basis. Thus, it is natural that these amplitudes were not focused in the
Celestial literature PhysRevD.96.085006 ; Schreiber:2017jsr . We will show
that such amplitudes are UV finite in the modified Mellin basis, and moreover,
they can be captured by the finite holographic Carrollian CFT correlators.
This enhances the validity of our analysis first put forward in Bagchi:2023fbj
and shows that the Carrollian picture emerges naturally out of a careful limit
of AdS Witten diagrams.

####  1.2 Advantages of the Carrollian approach

There are various reasons why we believe that the Carrollian perspective is
more appropriate for Minkowskian holography. For the reader wanting to take
away the main message without going through the whole of the paper, we provide
a list below with brief descriptions. We will be elaborating on (most of)
these points in the next section.

  * •

Symmetries: The most rudimentary check of a holographic correspondence is the
matching of symmetries between the bulk and the boundary theories. In
AdS5/CFT4, this is the isomorphism between the isometry algebra $so(4,2)$ of
the bulk and the conformal algebra, which is also $so(4,2)$ in $d=4$. Even
without going into the intricacies of asymptotic symmetries, the isometry of
Minkowski spacetime is the Poincare algebra $iso(3,1)$, and not just its
Lorentz sub-algebra $so(3,1)$. We believe that there is no reason to treat the
translation sub-algebra differently and disregard this from the global
symmetry algebra of the boundary theory. The celestial CFT, which is based on
the observation of the realisation of the Lorentz algebra as the global
conformal algebra, unfortunately, does this. The Carrollian programme, on the
other hand, is based on the isomorphism between the 4d Poincare algebra and
the global symmetries of 3d Carroll CFTs, which is also $iso(3,1)$ and treats
translations democratically.

  * •

Changing of scaling dimensions under translation: As a direct consequence of
not considering translations in the symmetries of the dual field theory,
Celestial CFTs inherit undesirable properties. The operators in the 2d dual
Celestial CFT are labeled by weights under $L_{0},\bar{L}_{0}$, which are
$(h,\bar{h})$. If we perform spacetime translations on these operators, these
weights shift. Assuming that there is a map between bulk fields and boundary
operators, this means that if you translate the bulk field, it would map into
a different operator on the boundary. This is very strange from the point of a
dual field theory because translations are very natural transformations in the
bulk, and this manifestation of shifting weights is a direct consequence of
not taking these symmetries properly into account in the putative dual field
theory. The Carrollian higher dimensional dual theory has fields that
transform properly under spacetime translations, thus eradicating this
problem.

  * •

Mellin transformation and divergences: As indicated above and will be detailed
later, the Mellin transformation forms the basis of the relation between 2d
Celestial CFTs and 4d asymptotically flat spacetimes by relating S-matrix
elements to correlation functions of the 2d field theory through the integral
transformation. However, Mellin transformations on, e.g., graviton amplitudes
yield infinite answers. Although it is generally argued that this is not a
problem since general relativity is not an ultra-violet complete theory, we
find this argument unconvincing. Graviton amplitudes are connected to
observables and, hence, in the age of gravitational wave detection, of prime
experimental relevance.

In Banerjee:2018gce , it was shown that by considering covariance under the
whole of the Poincare group and, in particular, spacetime translations, the
Mellin transformation can be improved to a Modified Mellin transformation. It
was later shown in Banerjee:2019prz  that this transformation leads to finite
graviton amplitudes in the bulk. The map between the correlation function of
Carrollian primaries and S-matrix elements utlilise the modified Mellin
transformation instead of the usual Mellin transformation and hence also
addresses the problem of the ultraviolet divergence of original Mellin
transformations.

  * •

Limit from AdS: Holography is, of course, best understood in its AdS avatar,
where the dual theory is a co-dimension one conformal field theory living on
the boundary of AdS. The infinite radius limit of AdS leads to flat
spacetimes. The bulk theory does not reduce in dimension while the limit is
taken. It is natural to think of flat holography as a singular limit of AdS
holography. Perhaps all features of flat spacetimes would not appear in this
limit, but given the wealth of information we have assembled in AdS/CFT,
taking limits, albeit singular, is an attractive proposition. It is thus also
natural that on the dual side, the infinite radius limit is perceived as a
singular limit on the CFT but does not reduce the dimension.

It was shown in Bagchi:2012cy  that this limit on the dual side actually sends
the speed of light to zero on the field theory side, which is a natural
consequence of a timelike boundary being boosted to a null boundary. This
process contracts the CFT algebra in $(d-1)$ dimensions to a Carroll conformal
algebra in $(d-1)$ and not just a $(d-2)$ dimensional relativistic conformal
algebra. The $(d-2)$-dimensional CFT algebra forms a subalgebra of the
$(d-1)$-dimensional Carroll conformal algebra. This sub-algebra maps to the
Lorentz subalgebra of the $d$ dimensional Minkowski spacetime. But, crucially,
the translation generators are missed. It is thus important to take the whole
of the conformal Carroll algebra, which appears in the limit of the vanishing
speed of light.

  * •

The case of three bulk dimensions: The asymptotic symmetry algebra for 3d
asymptotically flat spacetimes is the BMS3 algebra. For Einstein gravity, this
involves a central term between the superrotations and supertranslations. The
superrotations themselves form a Virasoro algebra, which should form the basis
of a celestial construction. This Virasoro algebra does not contain a central
term for Einstein gravity and hence cannot account for the entropy of Flat
Space Cosmologies (FSCs), which, on the other hand, is captured perfectly by
the BMS-Cardy analysis that follows from a Carrollian approach Bagchi:2012xr
222See Bhattacharjee:2023sfd  for a recent attempt at an effective 1d field
theory approach reproducing the FSC entropy, although this seems to be a
restriction of the 2d dual theory to the $u=0$ slice, where $u$ is the null
time direction on $\mathscr{I}^{\pm}$.. The co-dimension one Carrollian dual
theory has also had many other successes, e.g., in reproducing stress-tensor
correlators Bagchi:2015wna , entanglement entropy Bagchi:2014iea ;
Jiang:2017ecm ; Hijano:2017eii . It seems unlikely that the Celestial
approach, without the proper inclusion of the supertranslations and, hence,
the crucial central term, would be able to reproduce bulk answers.

Apart from all of these specific points listed above, the holographic
principle itself was historically based on the observation that the black hole
entropy is proportional to the area of the event horizon, and hence, from the
extensivity of entropy, this leads to the proposal that a theory of quantum
gravity which describes the black hole is equivalent to a quantum field theory
in one lower dimension where this entropy becomes extensive. It is thus in the
spirit of the original holographic principle to have a co-dimension one dual
for flat spacetimes.

####  1.3 What is in this paper?

Our paper, as advertised above, is made of two parts.

The first part, which is essentially Sec. 2, is a comparison between the
Celestial and Carrollian approaches to flat holography. We give the reader a
quick overview of Celestial holography, based principally on the observation
that S-matrix elements can be related to 2d Celestial CFT correlation
functions by the Mellin transformation. We then show that by taking care of
translations properly, we arrive at a three-dimensional boundary theory whose
correlators are also related to S-matrix elements, now by the modified Mellin
transformation. We also comment on a different approach for Carrollian
holography advocated in Donnay:2022aba ; Donnay:2022wvx .

The second part of the paper addresses the flat limit of AdS/CFT and
specifically how to obtain Carrollian correlation functions from Witten
diagrams, generalizing our construction Bagchi:2023fbj  to operators with
spin. The time-dependent correlation functions of Carrollian CFTs emerge
naturally as the leading order term under a large $R$ series expansion ($R$
being the AdS radius) for the scattering amplitudes of Witten diagrams
Bagchi:2023fbj . This result, though generic, was just established for
massless scalar particles. In this paper, we extend it to massless spinning
particles.

Sec. 3 and Sec. 4 are devoted to the field theory analysis of Carrollian CFTs
where various correlation functions are computed from symmetries. Sec. 5
contains a review of the construction of Bagchi:2023fbj  of scalar Witten
diagrams and the flat limit, and we then set up the formulation for more
complicated operators with non-zero spin in the latter half of this section.

In Sec. 6, we provide explicit examples of our construction and show matching
between the limiting construction of Sec. 5 and the symmetry analysis of Sec.
4. We have a total of twelve examples. The reader is pointed to Sec. 6.1 for a
quick summary of all the computations. We end with some discussions in Sec. 7.

##  Part I Carrollian Holography: An overview

In this part, which essentially consists of the following section, we give an
extensive overview of Carrollian holography and highlight the advantages over
its Celestial counterpart.

The version of Carrollian holography that we will discuss is the one which is
intimately linked to the Modified Mellin transformation. The principal
references for this are Banerjee:2018gce ; Bagchi:2022emh ; Bagchi:2023fbj .
There exists another school of research on Carrollian holography advocated in
Donnay:2022aba ; Donnay:2022wvx . (See also Salzer:2023jqv ; Nguyen:2023vfz ;
Saha:2023hsl ; Saha:2023abr .) We will briefly compare and contrast this to
our approach at the end of Sec. 2.

###  2 Flat holography and integral transformations

The main observable for quantum gravity in asymptotically flat spacetimes are
the S-matrix elements. Thus, it is mandatory that the holographic dual should
be able to capture these S-matrix elements in terms of its correlation
functions. Below, we detail how two different integral transformations on the
S-matrix elements lead to two different dual pictures of holography in
asymptotically flat spacetime.

####  2.1 Mellin transformations and 2d Celestial CFT

The Celestial approach to Minkowskian holography proposes 2d CFT on the
celestial sphere as a dual to 4d gravity in asymptotically flat spacetimes.
The essence of this proposal emerges from the observation that the double
cover of the Lorentz group in four spacetime dimensions is SL$(2,\mathbb{C})$,
which is the group of global conformal transformation in $d=2$. Furthermore,
the existence of the subleading soft graviton theorems suggests that this
global group should be extended to include the infinite local conformal
transformations as well. The enhancement of these symmetries indicates a 2d
CFT dual. Let us elaborate on these statements by considering the scattering
of massless particles. The momenta of a massless particle $p^{\mu}$ can be
parametrized as

$p^{\mu}=\omega(1-z\bar{z},z+\bar{z},-i(z-\bar{z}),1+z\bar{z})$ (1)

Here $\omega$ denotes the energy of the particle and $z,\bar{z}$ are
coordinates of the celestial sphere on null infinity. A generic Lorentz
transformation or SL$(2,\mathbb{C})$ transforms the massless momenta as

$\displaystyle\mathbb{P}$
$\displaystyle\to\Lambda\mathbb{P}\Lambda^{-1},\quad\mathbb{P}=p_{\mu}\sigma^{\mu}$
(2) $\displaystyle i.e.\quad\omega$ $\displaystyle\to\omega|cz+d|,\quad
z\to\frac{az+b}{cz+d},\quad\bar{z}\to\frac{\bar{a}\bar{z}+\bar{b}}{\bar{c}\bar{z}+\bar{d}}$

Here $\sigma^{\mu}=(1,\sigma^{i})$ are the Pauli matrices and $a,b,c,d$
parametrise Lorentz transformations:

$\Lambda=\begin{bmatrix}a\quad b\\\ c\quad d\end{bmatrix}\quad\text{such
that}\quad ad-bc=1.$ (3)

From (2), it is evident that the Lorentz group acts as global conformal
transformations on the celestial sphere. Now let us consider the particle
states with definite momentum, which can be created or annihilated by the
standard momentum space operators $a(\omega,z,\bar{z},\sigma)$ and
$a^{\dagger}(\omega,z,\bar{z},\sigma)$. The additional level $\sigma$ of
$(a,a^{\dagger})$ denotes the helicity of the corresponding massless particle.
We shall use a compact notation $a(\epsilon\omega,z,\bar{z},\sigma)$, where
$\epsilon=\pm 1$ would decide if the operator is a creation or an annihilation
operator.

Using these momentum space operators, one can define local operators on the
celestial sphere via a Mellin transformation as

$\displaystyle\mathcal{O}^{\epsilon}_{h,\bar{h}}(z,\bar{z})=\int_{0}^{\infty}d\omega\,\omega^{\Delta-1}a(\epsilon\omega,z,\bar{z},\sigma),\quad
h,\bar{h}=\frac{\Delta\pm\sigma}{2}$ (4)

Essentially, this Mellin transformation amounts to a change of basis in which
the action of the boost generators is diagonalized. The transformation trades
the energy $\omega$ with the boost weights $\Delta$. One advantage of choosing
this basis is a manifestation of the conformal properties in the wave
functions defined in (4). As suggested by the notation, the local operator on
celestial sphere $\mathcal{O}_{h,\bar{h}}(z,\bar{z})$ now transforms like a
primary field of weights $h$ and $\bar{h}$ with respect to the
SL$(2,\mathbb{C})$.

$\displaystyle\mathcal{O}^{\epsilon}_{h,\bar{h}}(z,\bar{z})\to\mathcal{O}^{\epsilon}_{h,\bar{h}}(z^{\prime},\bar{z^{\prime}})=\frac{1}{|cz+d|^{2h}}\frac{1}{|\bar{c}\bar{z}+\bar{d}|^{2\bar{h}}}\mathcal{O}^{\epsilon}_{h,\bar{h}}(z,\bar{z})$
(5)

Having this new basis at hand, the scattering amplitude now readily translates
into the conformal correlators on the celestial sphere,

$\displaystyle\langle\mathcal{O}_{1}(z_{1},\bar{z}_{1})\mathcal{O}_{2}(z_{2},\bar{z}_{2})....\mathcal{O}_{n}(z_{n},\bar{z}_{n})\rangle=\prod_{i=1}^{n}\int_{0}^{\infty}d\omega_{i}\omega_{i}^{\Delta_{1}-1}\mathcal{S}_{n}(\omega_{i},z_{i},\bar{z}_{i},\sigma_{i})$
(6)

In the above, $\mathcal{O}_{i}(z_{i},\bar{z}_{i})$ is the shorthand for
$\mathcal{O}^{\epsilon}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})$. While doing
this transformation, we go from the basis of plane waves to the basis of
conformal primary wavefunctions. These wavefunctions are given by

$\Phi(x|z,\bar{z})=\int d\omega\,\omega^{\Delta-1}\,e^{\pm i\tilde{q}\cdot
x}e^{-\epsilon\omega}=\dfrac{(\mp i)^{\Delta}\Gamma(\Delta)}{(-\tilde{q}\cdot
x\mp i\epsilon)^{\Delta}}.$ (7)

$\Phi(x|z\bar{z})$ satisfies the massless Klein-Gordon equation and it
transforms like (5) under an SL$(2,\mathbb{C})$ transformation.

Although the conformal properties are explicit in this basis, the Mellin
transformation runs into some peculiarities while dealing with the
translations. For example, under the translation by $P^{3}$, the the boundary
operator $\mathcal{O}_{h,\bar{h}}(z,\bar{z})$ changes as

$\displaystyle\delta_{P^{3}}\mathcal{O}_{h,\bar{h}}(z,\bar{z})$
$\displaystyle=\int_{0}^{\infty}d\omega\omega^{\Delta-1}[P^{3},a(\epsilon\omega,z,\bar{z},\sigma)]$
(8)
$\displaystyle=(1-z\bar{z})\int_{0}^{\infty}d\omega\omega\omega^{\Delta-1}a(\epsilon\omega,z,\bar{z},\sigma)$
$\displaystyle=(1-z\bar{z})\mathcal{O}_{h+\frac{1}{2},\bar{h}+\frac{1}{2}}(z,\bar{z})$

The above equation shows that the bulk translations shift the conformal
dimensions of the boundary fields by $\frac{1}{2}$. This is a feature we find
undesirable from the point of view of a boundary dual theory. For us, this is
an indication that the symmetry algebra of the boundary theory was not taken
into account appropriately, and there is something amiss with this 2d
Celestial picture of holography in flat spacetimes.

As discussed in the introduction, the Mellin transformation comes with the
inherent problem of being UV divergent. This, e.g., shows up in the
transformations of flat space graviton amplitudes. We will see below that
there is a single solution to both problems above when we take translations
into account properly.

####  2.2 Covariance under Poincare and Modified Mellin Transformations

Our aim now is to take translations into account properly so that the weights
of the boundary fields do not shift under this subgroup of the Poincare group.
In a sense, this is demanding covariance of the theory under the whole
Poincare group and not its Lorentz subgroup. To do this, we do the following
time translation:

$\mathcal{O}^{\epsilon}_{h,\bar{h}}(z,\bar{z})\mapsto
e^{-iHU}\mathcal{O}^{\epsilon}_{h,\bar{h}}(z,\bar{z})e^{iHU}.$ (9)

Following the Mellin transformation (4), this leads to some three dimensional
fields:

$\displaystyle\Phi^{\epsilon}_{h,\bar{h}}(u,z,\bar{z})$
$\displaystyle=e^{-iHU}\mathcal{O}^{\epsilon}_{h,\bar{h}}(z,\bar{z})e^{iHU}=\int_{0}^{\infty}d\omega\omega^{\Delta-1}e^{-iHU}a(\epsilon\omega,z,\bar{z},\sigma)e^{iHU}$
$\displaystyle\Rightarrow\Phi^{\epsilon}_{h,\bar{h}}(u,z,\bar{z})$
$\displaystyle=\int_{0}^{\infty}d\omega\omega^{\Delta-1}e^{i\epsilon\omega
u}a(\epsilon\omega,z,\bar{z},\sigma).$ (10)

In the above equation, we define $u=U(1+z\bar{z})$, where $U$ and $u$ are the
time in the bulk and the boundary, which are related by a conformal factor
$(1+z\bar{z})$. This modification now gives rise to a field that lives on the
whole of null infinity ($\mathscr{I}^{\pm}$) instead of just the celestial
sphere, and the relation (2.2) was called the Modified Mellin transformation
in Banerjee:2018gce , where it was first introduced.

The construction above gives rise to a field on $\mathscr{I}^{\pm}$, and we
briefly study the transformation properties of
$\Phi^{\epsilon}_{h,\bar{h}}(u,z,\bar{z})$ under the global Poincaré group.
These transformation rules directly follow from the above integral
transformation of creation and annihilation operators Banerjee:2018gce . For
simplicity, let us consider scalar particles in the bulk (i.e, $h-\bar{h}=0$).
$\Phi^{\epsilon}_{h,\bar{h}}(u,z,\bar{z})$ then transforms under the Lorentz
group as

$\displaystyle\Phi^{\epsilon}_{h,\bar{h}}(u,z,\bar{z})\to\Phi^{\epsilon}_{h,\bar{h}}(u^{\prime},z^{\prime},\bar{z^{\prime}})$
$\displaystyle=\int_{0}^{\infty}d\omega\omega^{\Delta-1}e^{i\epsilon\omega
u}\Lambda^{-1}a(\epsilon\omega,z,\bar{z})\Lambda$ (11)
$\displaystyle=\int_{0}^{\infty}d\omega\omega^{\Delta-1}e^{i\epsilon\omega
u}a\Big{(}\epsilon\omega|cz+d|^{2},\frac{az+b}{cz+d},\frac{\bar{a}\bar{z}+\bar{b}}{\bar{c}\bar{z}+\bar{d}}\Big{)}$
$\displaystyle=\frac{1}{|cz+d|^{2\Delta}}\Phi^{\epsilon}_{h,\bar{h}}\Big{(}\frac{u}{|cz+d|^{2}},\frac{az+b}{cz+d},\frac{a\bar{z}+\bar{b}}{\bar{c}\bar{z}+\bar{d}}\Big{)}$

We performed the last step by redefining the integration variable $\omega$.
Following similar arguments, it can be shown that the generic transformation
rules for the spinning primaries under the Lorentz group are given by

$\displaystyle\Phi^{\epsilon}_{h,\bar{h}}(u,z,\bar{z})\to\Phi^{\epsilon}_{h,\bar{h}}(u^{\prime},z^{\prime},\bar{z^{\prime}})=\frac{1}{|cz+d|^{2h}}\frac{1}{|\bar{c}\bar{z}+\bar{d}|^{2\bar{h}}}\Phi^{\epsilon}_{h,\bar{h}}\Big{(}\frac{u}{|cz+d|^{2}},\frac{az+b}{cz+d},\frac{a\bar{z}+\bar{b}}{\bar{c}\bar{z}+\bar{d}}\Big{)}$
(12)

Further, the action of bulk translations can also be derived following similar
prescriptions. For example, the action of the generator $P^{3}$ is given by

$\displaystyle\delta_{P^{3}}\Phi^{\epsilon}_{h,\bar{h}}(u,z,\bar{z})$
$\displaystyle=\int_{0}^{\infty}d\omega\omega^{\Delta-1}e^{i\epsilon\omega
u}[P^{3},a(\epsilon\omega,z,\bar{z})]$ (13)
$\displaystyle=\int_{0}^{\infty}d\omega\omega^{\Delta-1}e^{i\epsilon\omega
u}\omega(1-z\bar{z})a(\epsilon\omega,z,\bar{z})$
$\displaystyle=(1-z\bar{z})\partial_{u}\Phi^{\epsilon}_{h,\bar{h}}(u,z,\bar{z})$

Generically $\Phi^{\epsilon}_{h,\bar{h}}(u,z,\bar{z})$ transforms under
translations as

$\displaystyle\Phi^{\epsilon}_{h,\bar{h}}(u,z,\bar{z})\to\Phi^{\epsilon}_{h,\bar{h}}(u^{\prime},z^{\prime},\bar{z^{\prime}})=\Phi^{\epsilon}_{h,\bar{h}}(u+p+qz+r\bar{z}+sz\bar{z},z,\bar{z})$
(14)

where $p,q,r$ and $s$ parametrises the four spacetime translations.

Now, the basis of plane waves becomes modified under the modified Mellin
transformation (2.2). They take up the form

$\psi(x|u,z,\bar{z})=\int d\omega\omega^{\Delta-1}e^{\pm i\omega\,u}e^{\pm
i\tilde{q}\cdot x}e^{-\epsilon\omega}=\dfrac{(\mp
i)^{\Delta}\Gamma(\Delta)}{(-u-\tilde{q}\cdot x\mp i\epsilon)^{\Delta}}\,.$
(15)

This Carrollian primary wavefunction is a modified avatar of (7)
Banerjee:2018gce . (15) satisfies the massless Klein-Gordon equation for a
fixed value of $u$ i.e.,

$\partialderivative{x_{\mu}}\partialderivative{x^{\mu}}\psi(x|u,z,\bar{z})=0~{}~{}~{}\text{for
a fixed $u$}\,.$ (16)

They also have the crucial feature that they transform covariantly under the
complete set of bulk isometries: bulk Lorentz transformations and bulk
translations i.e., they transform like (12) and (14). In doing the modified
Mellin transform, we go from the momentum eigenstate (plane wave) basis to the
boost eigenstate basis.

####  2.3 BMS and Carrollian holography

Interestingly, the asymptotic symmetries of asymptotically flat spacetimes are
not only the Poincare group but an infinite dimensional group called the BMS4
group, which includes supertranslations and superrotations extending usual
translations and rotations of the Poincare group. Historically, Bondi, van der
Burg, Metzner Bondi:1962px  and Sachs Sachs:1962wk  discovered the
supertranslations while studying the asymptotic structures of flat spacetimes.
Later on, it was realised that the asymptotic symmetry group could further be
extended to include the superrotations Barnich:2010eb . Gravitational
scattering in the bulk respects not only global Poincare symmetries but these
infinite dimensional supertranslations and superrotations as well. The leading
and subleading soft graviton theorems are direct consequences of the existence
of these infinite dimensional asymptotic symmetries.

The most celebrated example of asymptotic symmetry is the case of AdS3 by
Brown and Henneaux Brown:1986nw , who discovered two copies of the Virasoro
algebra arising from their canonical analysis extending the finite dimensional
$so(2,2)$ isometry algebra of AdS3. This was, of course, also realised as the
symmetries of a 2d relativistic CFT and is seen as one of the principle
precursors of the AdS/CFT correspondence.

BMS symmetries can also be realised in a similar holographic manner in a lower
dimensional theory Bagchi:2010zz ; Bagchi:2012cy ; Duval:2014uva ;
Bagchi:2016bcd . These symmetries are the same as Carrollian conformal
symmetries in one lower dimension Duval:2014uva . The proposal is thus:

${\boxed{\textit{Gravity in D-dim asymptotically flat
spacetime}\equiv\textit{Carrollian CFT in D-1 dim}.}}$

The dual theory resides on the null boundary of the asymptotically flat
spacetime. This is the foundation of the Carrollian approach to flat
holography. The most rudimentary check of holography, the matching of
symmetries between the bulk and boundary theory, is inbuilt into this
approach, unlike Celestial holography where translations and, as a
consequence, supertranslations are not included as symmetries of the dual
field theory.

We will focus on the relation between 4d asymptotically flat spacetimes and 3d
Carrollian CFTs. The algebra in question is thus the BMS4 algebra:

$\displaystyle[L_{n},L_{m}]$
$\displaystyle=(n-m)L_{n+m},\quad[\bar{L}_{n},\bar{L}_{m}]=(n-m)\bar{L}_{n+m}$
(17) $\displaystyle[L_{n},M_{r,s}]$
$\displaystyle=\left(\frac{n+1}{2}-r\right)M_{n+r,s},\quad[\bar{L}_{n},M_{r,s}]=\left(\frac{n+1}{2}-s\right)M_{r,n+s}$
$\displaystyle[M_{r,s},M_{p,q}]$ $\displaystyle=0.$

This is the asymptotic symmetry algebra at null infinity
($\mathscr{I}^{\pm}$), which is topologically $\mathbb{R}_{u}\times$ S2. In
the above, $L_{n}$ and $\bar{L}_{n}$ are the generators of superrotations and
form the Virasoro sub-algebra. $M_{r,s}$ are the supertranslations, the angle
dependent translations of the null direction.
$\\{L_{a},\bar{L}_{a};M_{i,j}\\}$ where $a=0,\pm 1$ and $i,j=0,1$ forms the
10-parameter Poincare subalgebra with $\\{L_{a},\bar{L}_{a}\\}$ forming the
Lorentz part. The vector field representation of the generators of this
algebra, which also can be obtained as asymptotic Killing vectors projected on
null infinity, is given by

$\displaystyle
L_{n}=z^{n+1}\partial_{z}+\frac{1}{2}(n+1)z^{n}u\partial_{u},\quad\bar{L}_{n}=\bar{z}^{n+1}\partial_{\bar{z}}+\frac{1}{2}(n+1)\bar{z}^{n}u\partial_{u},$
(18a) $\displaystyle M_{r,s}=z^{r}\bar{z}^{s}\partial_{u}\qquad\forall\quad
n,r,s\in Z$ (18b)

Here $z,{\bar{z}}$ are the stereographic coordinates on S2, and $u$ is the
null direction on $\mathscr{I}^{\pm}$.

We define our putative dual 3d Carrollian CFT to live on $\mathscr{I}^{\pm}$
and the generators (18) reflect the $\mathbb{R}_{u}\times$ S2 topology of the
background manifold. We will now build the quantum field theory living on
$\mathscr{I}^{\pm}$ with (17) as its underlying symmetry. We label states
$\Phi(0,0,0)$ by their weights under $L_{0},\bar{L}_{0}$:

$\displaystyle[L_{0},\Phi(0,0,0)]=h\Phi(0,0,0),\quad[\bar{L}_{0},\Phi(0,0,0)]=\bar{h}\Phi(0,0,0)$
(19)

Taking a cue from the usual relativistic CFT, we propose a class of Carrollian
primary states $\Phi_{p}(0,0,0)$, which are given by:

$\displaystyle[L_{n},\Phi_{p}(0,0,0)]$
$\displaystyle=[\bar{L}_{n},\Phi_{p}(0,0,0)]=0,\quad\forall n>0$ (20a)
$\displaystyle[M_{r,s},\Phi_{p}(0,0,0)]$ $\displaystyle=0,\quad\forall r,s>0$
(20b)

Notice that on top of the relations (20a) that are identical to a 2d
relativistic CFT, we also impose (20b), which is crucial to a 3d Carroll CFT.
These conditions induce the following transformation rules at arbitrary
spacetime point

$\displaystyle[L_{n},\Phi_{p}(u,z,\bar{z})]$
$\displaystyle=z^{n+1}\partial_{z}\Phi_{p}(u,z,\bar{z})+\Big{(}h+\frac{u}{2}\partial_{u}\Big{)}z^{n}\Phi_{p}(u,z,\bar{z})$
(21a) $\displaystyle[\bar{L}_{n},\Phi_{p}(u,z,\bar{z})]$
$\displaystyle=\bar{z}^{n+1}\partial_{\bar{z}}\Phi_{p}(u,z,\bar{z})+\Big{(}\bar{h}+\frac{u}{2}\partial_{u}\Big{)}\bar{z}^{n}\Phi_{p}(u,z,\bar{z})$
(21b) $\displaystyle[M_{r,s},\Phi_{p}(u,z,\bar{z})]$
$\displaystyle=z^{r}\bar{z}^{s}\partial_{u}\Phi_{p}(u,z,\bar{z})$ (21c)

These transformation rules define conformal Carroll primaries. The Carrollian
operators that obey the primary conditions (21) only for the global Poincare
subgroup would be called quasi-primary operators.

####  2.4 Modified Mellin and 3d Carroll CFT

From our discussions above, it is now straight-forward to see that the
transformation rules (21) of the Carrollian quasi-primary operators are the
infinitesimal versions of the ones we derived from the modified Mellin
transformations earlier in (12) and (14). So we see that the fields earlier
defined from demanding covariance of the Mellin transformed operators under
spacetime translation are identical to 3d Carrollian quasi-primary operators
Bagchi:2022emh . This confirms that the modified Mellin transformations of the
momentum space operators in the bulk indeed give rise to Carrollian conformal
fields that live on the null boundary.

Having this framework at hand, one can relate the scattering amplitudes in the
bulk with the correlation functions of Carrollian primaries defined by (21).
This is given by Bagchi:2022emh :

$\displaystyle\langle\prod_{i=1}^{n}\Phi^{\epsilon_{i}}_{h_{i},\bar{h}_{i}}(u_{i},z_{i},\bar{z}_{i})\rangle=\prod_{i}\int_{0}^{\infty}d\omega_{i}~{}\omega_{i}^{\Delta_{i}-1}e^{i\omega_{i}u_{i}}\mathcal{S}_{n}(\omega_{i},z_{i},\bar{z}_{i},\sigma_{i})$
(22)

Here $\mathcal{S}_{n}$ are the $n$-point massless scattering amplitudes in the
bulk. This can equivalently be thought of as doing a basis change from plane
waves to Carrollian primary wavefunctions of (15).

In Bagchi:2022emh , after laying out the basic construction, the two point
function was verified from the free propagation amplitude of a single massless
particle. The three point scattering amplitudes of massless particles in
Minkowski spacetime are zero apart from a few specific scenarios. These cases
were also shown to be consistent with (22) in Bagchi:2023fbj . Later in this
paper, we will put this formula to further tests by considering various
scattering amplitudes involving spinning particles and in spacetimes of
signature (2,2).

###### Bulk energy conservation from Carroll.

The modified Mellin transformation has several very nice features. Here, we
highlight an important connection between bulk energy conservation and
boundary Carroll time translations. Acting $M_{00}$ of (18) on both sides of
(28), we get

$-i\sum_{i=1}^{n}\partial_{u_{i}}\langle\prod_{i=1}^{n}\phi^{\epsilon_{i}}_{h,\bar{h}}(u_{i},z_{i},\bar{z}_{i})\rangle=\prod_{i=1}^{n}\int
d\omega_{i}\,\omega^{\Delta_{i}-1}_{i}\,e^{-i\epsilon_{i}\omega_{i}u_{i}}\,\left(-i\sum_{j=1}^{n}\epsilon_{j}\omega_{j}\right)S(\\{\omega_{i},z_{i},\bar{z}_{i},\sigma_{i}\\})$
(23)

From the Ward identities of Carroll time translations, we know that the LHS of
(23) is zero. Thus, we must have

$\sum_{j=1}^{n}\epsilon_{j}\omega_{j}=0\,.$ (24)

This is just the energy conservation between the incoming and outgoing
particles ($\epsilon_{i}=\pm 1$ denotes the incoming or outgoing particles).

###### Limit from AdS and Witten diagrams.

We will also show that the correlation functions associated with these
amplitudes can also be obtained by implementing a suitable flat limit of AdS-
Witten diagrams. An essential component of the Witten diagrams is the bulk to
boundary propagator $\textbf{K}_{\Delta}(\textbf{P},\textbf{X})$. In the
embedding space representation, it is given by Penedones:2010ue :

$\textbf{K}_{\Delta}(\textbf{P},\textbf{X})=\frac{C_{\Delta}^{d}}{(-P\cdot
X+i\epsilon)^{\Delta}}\,.$ (25)

Our flat limit, which contracts the symmetries of both the bulk and the
boundary, ensures that $\textbf{K}_{\Delta}(\textbf{P},\textbf{X})$ reduces to
the Carrollian primary wavefunction (15):

$\lim_{R\to\infty}\textbf{K}_{\Delta}(\textbf{P},\textbf{X})\propto\dfrac{(\mp
i)^{\Delta}\Gamma(\Delta)}{(-u-\tilde{q}\cdot x\mp i\epsilon)^{\Delta}}$ (26)

Here, $R$ is the AdS radius. We have implemented a careful large AdS radius
limit that keeps track of the emergent null direction. This is what one should
expect from the point of view of flat holography Bagchi:2022emh . In our
limit, flat space emerges in the center of AdS. We expect the S-Matrix to
arise from CFT data Penedones:2010ue ; Fitzpatrick:2011hu ; Hijano:2019qmi
333See Section 3.1 of Bagchi:2023fbj  for a comprehensive review of the
literature utilizing this approach.. Thus, it is satisfying to see that
Carrollian primary wavefunctions (the basis functions encoding Carrollian
primary fields) emerge in a carefully constructed limit. We give the details
of this construction in Section 5.

####  2.5 Modified Mellin as the superstructure

We begin by claiming that the above version of the holographic principle for
asymptotically flat spacetimes is the superstructure from which other
formulations, including Celestial holography and other approaches to
Carrollian holography, Donnay:2022aba ; Donnay:2022wvx  appear as subsectors.
To substantiate our claims, we consider again the modified Mellin
transformation Banerjee:2018gce

$\langle\Phi_{1}(u_{1},{z}_{1})\ldots\Phi_{n}(u_{n},z_{n})\rangle=\prod_{i}\int_{0}^{\infty}d\omega_{i}\
\omega_{i}^{\Delta_{i}-1}e^{-i\epsilon_{i}\omega_{i}u_{i}}S(\\{\epsilon_{i}\omega_{i},z_{i},\sigma_{i}\\}).$
(27)

As we have reviewed in Bagchi:2023fbj , this relates the 4d flat space
scattering amplitudes to 3d Carrollian primary ($\Phi$) correlators which
depend on all three boundary directions $(u,z,\bar{z})$ of $\mathscr{I}^{+}$.
(We have suppressed the $\bar{z}$ dependence in the above and also will do so
below.)

###### Carroll and Celestial duals.

It is clear that the $u\to 0$ limit of the modified Mellin transformation
lands one up in the usual Mellin transformation:

$\langle\phi_{1}({z}_{1})\ldots\phi_{n}(z_{n})\rangle=\prod_{i}\int_{0}^{\infty}d\omega_{i}\
\omega_{i}^{\Delta_{i}-1}S(\\{\epsilon_{i}\omega_{i},z_{i},\sigma_{i}\\})$
(28)

Here, we are relating 4d flat space scattering amplitudes to 2d Celestial
primary ($\phi$) correlators. The usual Mellin transformation has the problem
that this leads to non-finite amplitudes in the bulk. E.g., four point
graviton amplitudes in the bulk 4d asymptotically flat spacetime diverge.

The modified Mellin transformation, on the other hand, helps eradicate this
problem, and it has been shown in Banerjee:2019prz  that plugging in the extra
$u$ dependent term renders the graviton amplitudes finite by acting like a
regulator.

It was shown in Banerjee:2018gce  that $u\to 0$ also makes sense on the field
theory side. From the point of view of the Celestial CFT, $u$ acts as a UV
regulator, which can be taken to zero at the end of the calculation. It is
plausible that one would be able to understand a 2d Celestial CFT as the
reduction of a 3d Carroll CFT, which lives on a particular $u=0$ slice of the
3d Carroll manifold.

######  $\Delta=1$ Carrollian CFTs.

In Donnay:2022aba ; Donnay:2022wvx , a different proposal for Carrollian
holography was put forward444See also Salzer:2023jqv ; Nguyen:2023vfz  from
asymptotic expansions of fields in the bulk spacetime and IR physics in
asymptotically flat spacetimes was related to sourced Ward identities on the
Carroll side. While this is an interesting approach and seems to yield results
consistent with some 4d bulk analysis, below we point out some possible issues
with this in light of our formulation.

We begin by observing that if one takes $\Delta_{i}=1$ in the modified Mellin
transformation, one reduces this to a Fourier transform:

$\langle\chi_{1}(u_{1},{z}_{1})\ldots\chi_{n}(u_{n},z_{n})\rangle=\prod_{i}\int_{0}^{\infty}d\omega_{i}\
e^{-i\epsilon_{i}\omega_{i}u_{i}}S(\\{\epsilon_{i}\omega_{i},z_{i},\sigma_{i}\\})$
(29)

This relation maps 3d Carroll primaries $\chi_{i}$ with $\Delta_{i}=1$ to 4d
scattering in flatspace. The two transformations Eq.(28) and Eq.(29) are
related by the map considered in Donnay:2022aba ; Donnay:2022wvx . Both of
these are clearly sub-structures to the super-structure Eq.(27).

The approach advocated in Donnay:2022aba ; Donnay:2022wvx  purports that the
3d Carrollian CFT dual to 4d asymptotically flat spacetime should only have
Carroll primaries with weights $\Delta_{i}=1$. We believe that there are
reasons why this subsector of $\Delta_{i}=1$ may not be consistent by itself.
Below, we list a couple of these reasons.

IR divergences: The way to arrive at the Carrollian CFTs Donnay:2022aba ;
Donnay:2022wvx  is to restrict the more generic Carroll CFTs with arbitrary
$\Delta_{i}$ to $\Delta_{i}=1$. However, taking the limit of $\Delta_{i}\to 1$
in the modified Mellin transformation is fraught with danger as there are
poles at $\Delta_{i}=1$. These are connected to the leading soft graviton
theorems, as we elaborate below. The problem with divergences at
$\Delta_{i}=1$ manifests itself explicitly in the Ward identities of
Carrollian CFTs as well, as the correlation functions blow up Salzer:2023jqv ;
Nguyen:2023miw . A complimentary way to see this divergences is by taking the
bulk correlation functions to the null boundary. This approach was taken
recently in Nguyen:2023miw .

As is well documented, S-matrices in asymptotically flat spacetimes are
meddled with IR divergences. These divergences are readily manifest in the
expressions of the soft expansions of scattering amplitudes. The implications
of this expansion on three-dimensional fields have been studied in
Banerjee:2020kaa . Let us review the basic arguments here. The map that
relates bulk and boundary operators is the modified Mellin transformation
(2.2). As this is an integral transformation on the energy variable $\omega$,
one cannot directly take $\omega\to 0$ limit. It turns out that, analogous to
celestial holography, the poles in the energy variable of the bulk operators
map to the poles in the scaling dimensions in the boundary theory. Consider
the following soft expansion of a momentum space annihilation operator
$a(\omega,z,\bar{z},\sigma)$

$\displaystyle
a(\omega,z,\bar{z},\sigma)=\sum_{n}\frac{S_{1-n}(z,\bar{z},\sigma)}{\omega^{n}}.$
(30)

Here the coefficients $S_{0}(z,\bar{z},\sigma)$, $S_{1}(z,\bar{z},\sigma)...$
are respectively leading and subleading soft contributions and so on. Now, if
we take this formula and plug it in (2.2), then we shall have an expansion in
$\Delta$, i.e.

$\displaystyle\Phi_{h,\bar{h}}(u,z,\bar{z})=\sum_{n}\frac{\Gamma(2-\Delta-n)}{2-\Delta-n}S_{1-n}(z,\bar{z},\sigma).$
(31)

The presence of these Gamma functions in conformal dimension $\Delta$ is the
manifestation of IR divergences. For example, the coefficient of the leading
soft contribution is divergent at $\Delta=1$. Thus in order to extract the
leading soft operator, one needs to take the residue of
$\Phi_{\Delta,\sigma}(u,z,\bar{z})$ around $\Delta=1$, i.e.

$\displaystyle S_{0}(z,\bar{z},\sigma)=-\lim_{\Delta\to
1}(\Delta-1)[(iu)^{\Delta-1}\Phi_{\Delta,\sigma}(u,z,\bar{z})]$ (32)

In a similar fashion, the subleading soft contribution can be expressed as

$\displaystyle S_{1}(z,\bar{z},\sigma)=\lim_{\Delta\to
0}\Delta(1-u\partial_{u})[(iu)^{\Delta}\Phi_{\Delta,\sigma}(u,z,\bar{z})]$
(33)

The insertions of these soft operators in the correlation functions lead to
the standard Ward identities of the boundary theory. The reader is directed to
Banerjee:2020kaa  for further details. However, the main point we would like
to emphasize here is that in order to implement the soft expansion of massless
particles in boundary theory, it is crucial that we should keep the scaling
dimension of the boundary operators arbitrary.

Carroll Stress Tensor: Any translationally invariant quantum field theory
contains an energy-momentum tensor. Carrollian CFTs are, of course
translationally invariant by construction, and hence, any Carroll CFT must
have an EM tensor. Below, we argue that the stress tensor in a 3d Carrollian
CFT is a primary of weight $\Delta=3$ and hence, a Carroll CFT with
$\Delta_{i}=1$ would not have an EM tensor and be inconsistent.

A generic way of obtaining Carroll CFTs is taking the Carroll limit of a 3d
relativistic CFT. Now, consider a $d=3$ Carroll CFT that is obtained as a
$c\to 0$ limit of a 3d relativistic CFT. This limit does not change the
conformal weight $\Delta$ of an operator since the Dilatation operator
$D=t\partial_{t}+x^{i}\partial_{i}$ does not change under the limit.

The EM tensor of a Carroll CFT can be systematically obtained in the $c\to 0$
limit of a relativistic CFT. In a $d=3$ relativistic CFT, the EM tensor is a
primary of weight $\Delta=3$. In the 3d Carroll CFT, the Carrollian EM tensor
is also a weight $\Delta=3$ primary. The above statement can also be made from
intrinsic Carrollian symmetry arguments without invoking any limiting
procedure and can be checked in explicit examples of Carroll CFTs, e.g., in
the Carrollian scalar theory Dutta:2022vkg . Clearly, the EM tensor is a
$\Delta\neq 1$ primary that has to exist in every Carrollian CFT, and hence,
it is implausible that the $\Delta=1$ subsector is consistent on its own.

Possible resolutions: We now speculate about possible resolutions to the above
points. Regarding the EM tensor, the problem may be circumvented if the BMS4
algebra generates Virasoro central terms, and the EM tensor no longer remains
a primary. Also, there have been recent investigations of logarithmic
structures in 2d Celestial CFTs Fiorucci:2023lpb . If these have Carrollian
counterparts, perhaps one might be able to find a way around the arguments we
presented above.

The $\Delta_{i}\to 1$ subsector, although possibly not a self-consistent
Carrollian CFT, is expected to play a significantly important role in
understanding structures related to the soft sector. We find indications of
this in the explicit amplitudes we compute later in the paper.

###### Conclusions.

We conclude this rather important section with a summary of our arguments
above as to why (our version of) Carrollian holography provides a natural
hologram of asymptotically flat spacetimes.

If we believe that we would be able to recover (at least some aspects of) flat
holography as an infinite radius limit of AdS/CFT, it is natural that the
boundary theory does not suddenly reduce in dimension in the limit, nor is it
expected to get restricted to just a particular set of operators which have a
single scaling dimension. It is thus satisfying that the Carroll
wavefunctions, which are directly related to the modified Mellin
transformation, are the ones we get by looking carefully at the AdS Witten
diagrams in our work. This modified Mellin basis is at the heart of the
connection of Carrollian theories and holography in flat spacetimes. We will
illustrate this further in the next part of our paper.

We think we have convincingly argued that the co-dimension one holography of
asymptotically flat spacetimes provided by Carrollian holography is perhaps
the better way of constructing flat holography. This approach seems more
natural than the Celestial one, given that the translations (and
supertranslations) are naturally encoded on the dual field theory, and the
null direction of the boundary of the gravitational theory is taken into
account. The Carrollian picture is also what comes about when we take the
limit from AdS/CFT, making sure we do not neglect the null direction.

The mapping between Carrollian and Celestial holography described in
Donnay:2022wvx  is a mapping between a particular subsector of the Carroll
theory and a Celestial CFT. One can only “trade” the null direction $u$ and
the conformal dimension $\Delta$ in this case as $\Delta$ is fixed to a
particular value $\Delta=1$ in this sector of the Carroll theory. A Carroll
CFT has operators with $\Delta\neq 1$. So, in general, this seems to suggest
that the Carrollian and Celestial pictures of flat holography are not
identical and not just a change of basis as has been claimed elsewhere. It
seems that the Carrollian version may contain more information as it keeps
track of the null direction of the boundary theory555In the lower dimensional
3d-2d duality, entanglement entropy Bagchi:2014iea  is such an observable of
the theory which depends explicitly on the null direction and is zero
otherwise in Einstein gravity..

We now present a summary of our claims so that the reader can take away the
main points from this subsection.

  * $\square$

The integral transformations indicate that 2d Celestial CFTs may arise as a
certain $u\to 0$ limit of 3d Carroll CFTs, where $u$ is the null direction of
$\mathscr{I}^{+}$.

  * $\square$

The Carroll CFTs described in Donnay:2022aba ; Donnay:2022wvx  are the
$\Delta=1$ (where $\Delta$ is the weight of a Carroll primary) sub-sector of a
generic Carroll CFT.

  * $\square$

There exists a map between the $\Delta=1$ subsector of a Carroll CFT and a
Celestial CFT Donnay:2022aba  666This has been explicitly shown to work for
the $\Delta=1$ subsector of the Celestial CFT and conjectured for arbitrary
$\Delta$., and not between a generic Carroll CFT and a Celestial CFT.

  * $\square$

A Carroll CFT generically contains primary operators with dimensions
$\Delta\neq 1$ in its spectrum. If the mapping between $\Delta=1$ Carroll CFTs
and Celestial CFTs constructed in Donnay:2022wvx  holds, this points to the
fact that a Carroll CFT would contain more operators (the ones with
$\Delta\neq 1$) than a Celestial CFT. The mapping between the two would not be
one-to-one. So, in general, it seems that the Carrollian and Celestial
pictures of flat holography are not equivalent.

  * $\square$

It is likely that the truncation of Carroll CFTs to just the $\Delta=1$
subsector is not consistent since, e.g., the EM tensor of a 3d Carroll CFT is
not contained in this sector. A Carroll CFT must always contain an EM tensor,
and hence, it is likely that just the $\Delta=1$ subsector cannot describe a
consistent Carroll CFT.

While it seems that the $\Delta=1$ Carroll CFTs may not be consistent by
themselves from what we have discussed above, the approach taken in
Donnay:2022aba ; Donnay:2022wvx  of building an extrapolate dictionary seems
intriguing and a potentially correct way of approaching holography in
asymptotically flat spacetimes from a co-dimension one viewpoint. It would be
very interesting to reconsider the asymptotic expansions to see how to allow
for generic values of $\Delta$ that, on general grounds, should be allowed for
holographic Carroll CFTs. Some of these aspects are investigations in
progress. For the rest of the paper, we will focus on the flat limit of
AdS/CFT and, in particular, recovering explicit 3d Carrollian correlators (for
arbitrary $\Delta$) from AdS4 Witten diagrams.

##  Part II Spinning Witten Diagrams and Carroll correlators

In this part, we focus on the limit of AdS, which lands us up in flat
spacetimes. We show how to recover 3d Carrollian correlations from Witten
diagrams in AdS4. This part is a direct follow up of the formulation outlined
earlier in Bagchi:2023fbj . We will generalise our earlier construction to
include operators with non-zero spin. We start with a boundary analysis
including spin in the Carrollian field theoretic set-up in Sec. 3. The highest
weight representations would translate to Ward identities constraining
Carrollian correlators in the null boundary. These correlators will be worked
out in Sec. 4. We then move to the bulk in Sec. 5 and review the construction
of the limit of scalar Witten diagrams before adding spin to the game and
refining our previous formulation to include these more general structures. We
then move to explicit examples in Sec. 6 and work out a total of twelve
examples, which are matched with the boundary analysis of Sec. 4.

###  3 Carrollian symmetries and field theory analysis

In this section, we return to the field theoretic discussion of the previous
section and have a closer look at the primary operators. We have outlined the
primary transformation rules in (21). Here, we shall elaborate on this formula
emphasizing the spinning, i.e., vector and tensor primaries, and derive their
correlation function using symmetry arguments. For this purpose, we first need
to define the Carrollian quasi primaries based on the global subgroup and
consider its finite dimensional representations induced from the little group.

####  3.1 Conformal Carrollian symmetries

The Carroll group LevyLeblond ; NDS  in arbitrary dimensions comprises the
following generators:

$\displaystyle P_{i}=-i\partial_{i}\,,\quad H=-i\partial_{u}\,,$ (34)
$\displaystyle B_{i}=-ix_{i}\partial_{u}\,,\quad
J_{ij}=i(x_{i}\partial_{j}-x_{j}\partial_{i})\,.$

Here, we have denoted $u$ as the Carroll time and $x_{i}$ as the spatial
coordinates. The generators $J_{ij}$, $P_{i}$, $B_{i}$, and $H$ are spatial
rotations, translations, Carroll boosts, and the Hamiltonian, respectively.
They can be obtained from the relativistic generators in the same dimensions
via the $c\to 0$ contractions. The non-trivial commutation relations of the
Carroll algebra are given by

$\displaystyle[J_{ij},B_{k}]=-i\delta_{k[i}B_{j]},\quad[J_{ij},P_{k}]=-i\delta_{k[i}P_{j]},\quad[B_{i},P_{j}]=i\delta_{ij}H$
(35)

One noteworthy difference from the Poincare group is that the Carroll boosts,
unlike the Lorentz boosts, commute among themselves. Also, notice that the
Hamiltonian only enters the algebra as a central element. The conformal
extension of the Carroll group in arbitrary spacetime dimensions can be
obtained by further adding the dilatation $D$ and the Carrollian special
conformal generators $K$ and $K_{i}$. These generators would also follow
similarly by contractions from the relativistic conformal generators. The
vector fields associated with the conformal generators are given by

$\displaystyle D=-i(u\partial_{u}+x^{i}\partial_{i}),\quad
K=-ix^{2}\partial_{u},\quad
K_{i}=-i(x^{2}\partial_{i}-2x_{i}x^{j}\partial_{j}-2x_{i}u\partial_{u})\,,$
(36)

and the extension of the algebra is as follows

$\displaystyle[B_{i},K_{j}]=i\delta_{ij}K\,,\quad[D,K]=-iK\,,\quad[K,P_{i}]=2iB_{i}\,,$
(37)
$\displaystyle[K_{i},P_{j}]=-2i(D\delta_{ij}-J_{ij})\,,\quad[H,K_{i}]=2iB_{i}\,,\quad[D,H]=iH\,,$
$\displaystyle[D,P_{i}]=iP_{i}\,,\quad[D,K_{i}]=-iK_{i}\,.$

Although the contraction from the relativistic conformal algebra yields only a
finite dimensional algebra, in the Carrollian regime, this group of conformal
transformations also admits an infinite extension, just like the 2d
relativistic CFTs. This is a peculiar feature of Carrollian conformal field
theories that continues to hold for higher dimensional cases as well.
Particularly in $d=3$, this infinite extension is isomorphic with the
superrotation extended BMS4 algebra (17) generated by the vector fields (18).

The global part of the algebra, spanned by $n=0,\pm 1$ and $r,s=0,1$, forms
the Poincare subgroup of the infinite dimensional BMS group. These generators
can be identified in terms of finite conformal Carroll generators as

$\displaystyle M_{00}=H,\quad M_{10}=B_{x}+iB_{y},\quad
M_{01}=B_{x}-iB_{y},\quad M_{11}=K$ (38) $\displaystyle
L_{-1}=\frac{1}{2}(P_{x}-iP_{y}),\quad L_{0}=\frac{1}{2}(D+iJ),\quad
L_{1}=-\frac{1}{2}(K_{x}+iK_{y})$
$\displaystyle\bar{L}_{-1}=\frac{1}{2}(P_{x}+iP_{y}),\quad\bar{L}_{0}=\frac{1}{2}(D-iJ),\quad\bar{L}_{1}=-\frac{1}{2}(K_{x}-iK_{y})$

In order to match these generators, we have redefined the spatial coordinates
as $z=x+iy$ and $\bar{z}=x-iy$. Field theories respecting the Carroll
symmetries and its above mentioned conformal extension have been previously
addressed in Bagchi:2019xfx ; Bagchi:2019clu ; Henneaux:2021yzg ;
deBoer:2021jej ; Barnich:2022bni ; deBoer:2023fnj ; Chen:2021xkw .

####  3.2 Carrollian quasi-primary fields

The representation of the Carrollian primary fields can be obtained by
implementing similar methods to the relativistic conformal case. We first
identify the little group that leaves the origin, $\textbf{x}=0$ invariant.
These transformations are generated by the rotations, Carroll boosts,
dilatations, and Carroll SCTs. Thus, at the origin, the transformation rules
are given by

$\displaystyle[J_{ij},\Phi(0)]=\mathcal{S}_{ij}\Phi(0),\quad[B_{i},\Phi(0)]=\mathcal{B}_{i}\Phi(0),\quad[D,\Phi(0)]=-i\mathbf{{\Delta}}\Phi(0)$
(39)
$\displaystyle[K,\Phi(0)]=\mathit{k}\Phi(0),\quad[K_{i},\Phi(0)]=\mathit{k}_{i}\Phi(0).$

The generators of the little group at the origin should form a matrix
representation of the corresponding algebra. Now, using Schur’s Lemma, it is
possible to diagonalize the action of $\mathbf{\Delta}$, as they commute with
the $\mathcal{S}_{ij}$ and $\mathcal{B}_{i}$s. Further, $\mathit{k}$ and
$\mathit{k}_{i}$ can be set to zero as a consequence of the algebra. The
transformation rules at arbitrary points can be figured out by translating
these relations at the origin (39). These are Bagchi:2016bcd 777Reader is
referred to eq (5.8) in this paper.:

$\displaystyle[H,\Phi(u,x^{i}]=-i\partial_{u}\Phi(u,x^{i})$ (40a)
$\displaystyle[P_{i},\Phi(u,x^{i})]=-i\partial_{i}\Phi(u,x^{i})$ (40b)
$\displaystyle[J_{ij},\Phi(u,x^{i})]=-i(i\mathcal{S}_{ij}-x_{i}\partial_{j}+x_{j}\partial_{i})\Phi(u,x^{i})$
(40c)
$\displaystyle[D,\Phi(u,x^{i})]=-i(\Delta+u\partial_{u}+x^{i}\partial_{i})\Phi(u,x^{i})$
(40d)
$\displaystyle[K,\Phi(u,x^{i})]=(-ix^{2}\partial_{u}+2x^{i}\mathcal{B}_{i})\Phi(u,x^{i})$
(40e)
$\displaystyle[B_{i},\Phi(u,x^{i})]=(-ix_{i}\partial_{u}+\mathcal{B}_{i})\Phi(u,x^{i})$
(40f)
$\displaystyle[K_{i},\Phi(u,x^{i})]=-i(-2x_{i}\Delta+2ix^{j}\mathcal{S}_{ij}-2iu\mathcal{B}_{i}-2ux_{i}\partial_{u}-2x_{i}x^{j}\partial_{j}+x^{2}\partial_{j})\Phi(u,x^{i})$
(40g)

These transformation rules define the Carroll primary in arbitrary dimensions.
However, we shall be interested in only 3d Carroll CFTs that concern
holography of 4 dimensional flat spacetimes. Specifically, in $d=3$, the
Carroll group is generated by one rotation, two Carroll boosts, and the
spatial translations. The algebra of the spin matrices associated to rotations
($\mathcal{J}$) and Carroll boosts ($\mathcal{B}_{x},\mathcal{B}_{y}$) in this
case is given by

$\displaystyle[\mathcal{J},\mathcal{B}_{x}]=-i\mathcal{B}_{y},\quad[\mathcal{J},\mathcal{B}_{y}]=i\mathcal{B}_{x},\quad[\mathcal{B}_{x},\mathcal{B}_{y}]=0$
(41)

#### Spin 0

The spin 0 or the scalar representation of the conformal Carroll fields can be
obtained by trivially setting

$\mathcal{J}=\mathcal{B}_{x}=\mathcal{B}_{y}=0$ (42)

With this input, the transformation of the quasi primaries defined in (40) can
be cast into the standard form

$\displaystyle[L_{n},\Phi(u,z,\bar{z})]=-i[z^{n+1}\partial_{z}+(\frac{\Delta}{2}+\frac{u}{2}\partial_{u})z^{n}]\Phi(u,z,\bar{z})$
(43)
$\displaystyle[\bar{L}_{n},\Phi(u,z,\bar{z})]=-i[\bar{z}^{n+1}\partial_{\bar{z}}+(\frac{\Delta}{2}+\frac{u}{2}\partial_{u})\bar{z}^{n}]\Phi(u,z,\bar{z})\quad\forall
n\in 0,\pm 1$
$\displaystyle\text{and}\quad[M_{r,s},\Phi(u,z,\bar{z})]=-iz^{r}\bar{z}^{s}\partial_{u}\Phi(u,z,\bar{z})\quad\forall
r,s\in 0,1$

These transformation properties expectedly match up with the previously
defined Carroll primaries in (21) with $h=\bar{h}=\frac{\Delta}{2}$.

#### Spin 1

The spin 1 representation of the rotation generator is given by

$\displaystyle\mathcal{J}_{s=1}=\begin{bmatrix}0&0&0\\\ 0&0&-i\\\
0&i&0\end{bmatrix}$ (44)

This is just the fundamental representation of $SO(2)$. We should choose the
boost matrices in a way that is consistent with the commutation relations
given by (41). There are two possible scenarios. One choice would be to set
these matrices equal to zero, i.e., simply

$\mathcal{B}_{x}=\mathcal{B}_{y}=0\,.$ (45)

For this choice, the required commutators are trivially satisfied. However,
there is another non-trivial choice consistent with (41). We shall stick to
(45) for our purposes as they induce the transformation rules that agree with
the bulk analysis. We discuss this choice and another possible choice below
around equation (51). Once this choice is assumed, we can readily derive the
transformations of the vector primaries. They are given by

  * •

$\mathbf{\Phi^{u}}(u,z,\bar{z})$

$\displaystyle[L_{n},\Phi^{u}(u,z,\bar{z})]=-i[z^{n+1}\partial_{z}+(\frac{\Delta}{2}+\frac{u}{2}\partial_{u})z^{n}]\Phi^{u}(u,z,\bar{z})$
(46)
$\displaystyle[\bar{L}_{n},\Phi^{u}(u,z,\bar{z})]=-i[\bar{z}^{n+1}\partial_{\bar{z}}+(\frac{\Delta}{2}+\frac{u}{2}\partial_{u})\bar{z}^{n}]\Phi^{u}(u,z,\bar{z})\quad\forall
n\in 0,\pm 1$
$\displaystyle[M_{r,s},\Phi^{u}(u,z,\bar{z})]=-iz^{r}\bar{z}^{s}\partial_{u}\Phi^{u}(u,z,\bar{z})$

  * •

$\mathbf{\Phi^{z}}(u,z,\bar{z})$

$\displaystyle[L_{n},\Phi^{z}(u,z,\bar{z})]=-i[z^{n+1}\partial_{z}+(\frac{\Delta-1}{2}+\frac{u}{2}\partial_{u})z^{n}]\Phi^{z}(u,z,\bar{z})$
(47)
$\displaystyle[\bar{L}_{n},\Phi^{z}(u,z,\bar{z})]=-i[\bar{z}^{n+1}\partial_{\bar{z}}+(\frac{\Delta+1}{2}+\frac{u}{2}\partial_{u})\bar{z}^{n}]\Phi^{z}(u,z,\bar{z})\quad\forall
n\in 0,\pm 1$
$\displaystyle[M_{r,s},\Phi^{z}(u,z,\bar{z})]=-iz^{r}\bar{z}^{s}\partial_{u}\Phi^{z}(u,z,\bar{z})$

  * •

$\mathbf{\Phi^{\bar{z}}}(u,z,\bar{z})$

$\displaystyle[L_{n},\Phi^{\bar{z}}(u,z,\bar{z})]=-i[z^{n+1}\partial_{z}+(\frac{\Delta+1}{2}+\frac{u}{2}\partial_{u})z^{n}]\Phi^{\bar{z}}(u,z,\bar{z})$
(48)
$\displaystyle[\bar{L}_{n},\Phi^{\bar{z}}(u,z,\bar{z})]=-i[\bar{z}^{n+1}\partial_{\bar{z}}+(\frac{\Delta-1}{2}+\frac{u}{2}\partial_{u})\bar{z}^{n}]\Phi^{\bar{z}}(u,z,\bar{z})\quad\forall
n\in 0,\pm 1$
$\displaystyle[M_{r,s},\Phi^{\bar{z}}(u,z,\bar{z})]=-iz^{r}\bar{z}^{s}\partial_{u}\Phi^{\bar{z}}(u,z,\bar{z})$

For these vector primaries, we have redefined the components as
$\Phi^{z}=\Phi^{x}-i\Phi^{y}$ and $\Phi^{\bar{z}}=\Phi^{x}+i\Phi^{y}$. Looking
at the transformation rules of these components, it can be immediately figured
out that each component of the vector primary again transforms like (21).

As a consequence of the trivial choice of the Carroll boost matrices, these
components do not transform into each other. However, the information about
the spin is still manifested in the holomorphic and anti-holomorphic weights
($h$ and $\bar{h}$) of these components. Here we should remind the reader that
$\Delta$ is the three dimensional scaling eigenvalue (as evident from the
Killing vector fields), as opposed to celestial CFT. However, because of the
trivial choice of these boost matrices, the spin $\sigma$ effectively reduces
to a 2d spin. The weights of the components are organised as in Table 1.

weights | $\Phi^{u}$ | $\Phi^{z}$ | $\Phi^{\bar{z}}$
---|---|---|---
$h$ | ${\Delta}/{2}$ | $\frac{\Delta-1}{2}$ | $\frac{\Delta+1}{2}$
$\bar{h}$ | ${\Delta}/{2}$ | $\frac{\Delta+1}{2}$ | $\frac{\Delta-1}{2}$
Table 1: Weights of vector primaries with upper indices

We have considered Carrollian vector primaries, which are naturally defined
with the upper indices. However, we need vector fields with lower indices for
several practical purposes, specifically for calculating flat space scattering
amplitudes later in the paper. We will elaborate on why it is important to
have lower indices in our bulk analysis. If we lower these indices using the
degenerate metric, we shall get

$\Phi_{u}=g_{uu}\Phi^{u}=0\,,~{}~{}~{}~{}~{}\Phi_{z}=g_{z\bar{z}}\Phi^{{\bar{z}}}\,,~{}~{}~{}~{}~{}\Phi_{\bar{z}}=g_{{\bar{z}}z}\Phi^{z}\,.$
(49)

We see that the $\Phi_{u}$ component is identically zero if we work with the
lower index instead of the upper one

$\Phi_{u}=0\,.$ (50)

The holomorphic and anti-holomorphic weights for the non-zero $\Phi_{z}$ and
$\Phi_{\bar{z}}$ are given by Table 2.

weights | $\Phi_{z}$ | $\Phi_{\bar{z}}$
---|---|---
$h$ | $\frac{\Delta+1}{2}$ | $\frac{\Delta-1}{2}$
$\bar{h}$ | $\frac{\Delta-1}{2}$ | $\frac{\Delta+1}{2}$
Table 2: Weights of vector primaries with lower indices

The weights for these spatial components swap simply because of the off
diagonal nature of the metric component $g_{z\bar{z}}$. These results also
agree with that of Salzer:2023jqv ; Nguyen:2023vfz .

We mentioned above that in our analysis, the boost matrices were trivially set
to zero. There is another set of matrices forming a finite dimensional irrep
of (41). These matrices are given by

$\displaystyle\mathcal{B}_{x}=\begin{bmatrix}0&0&0\\\ 1&0&0\\\
0&0&0\end{bmatrix},\quad\mathcal{B}_{y}=\begin{bmatrix}0&0&0\\\ 0&0&0\\\
1&0&0\end{bmatrix}$ (51)

These non-trivial boost matrices are realised on various Carrollian models
Bagchi:2016bcd . As these matrices described above are non-diagonalisable, the
components of spinning primaries would mix under boost transformations when
they are turned on. However, in the holographic context, the presence of the
boost matrices is undesirable because the Carrollian boosts are isomorphic to
bulk translations. Thus, when turned on, they would lead to non-trivial
transformations of these primaries. For example,using the mapping of
generators in (38), one can show that for the vector primary,

$[M_{10},\Phi_{z}(u,z,{\bar{z}})]=-iz\partial_{u}\Phi_{z}+2\Phi_{u}\,.$ (52)

Thus, the fields will not transform covariantly under bulk translations
anymore. Hence, if we consider holographic Carrollian CFTs, we should choose
the representation with the trivial boost matrices. We leave a careful
analysis of non-trivial boost matrices and their possible implications on flat
holography for future work.

#### Spin 2

The spin 2 representation of the rotation generator is given by

$\displaystyle\mathcal{J}_{s=2}=\begin{bmatrix}0&0&0&0&0\\\ 0&0&-i&0&0\\\
0&i&0&0&0\\\ 0&0&0&0&-2i\\\ 0&0&0&2i&0\end{bmatrix}$ (53)

The representation of (53) for the 5 dimensional traceless and symmetric spin
2 of $SO(2)$ can be built out of the three dimensional spin 1 representation
(44) of $SO(2)$ through the direct product as follows:

$\mathbf{3}\otimes\mathbf{3}=\mathbf{1}\oplus\mathbf{5}\oplus\mathbf{3}\,.$
(54)

Here $\mathbf{1}$ denotes the one dimensional trace, $\mathbf{3}$ denotes the
three dimensional anti-symmetric representation and $\mathbf{5}$ denotes the
required five dimensional symmetric and traceless representation. Thus, to
build the required representation, we exponentiate the spin 1 generator in(44)
and then take the direct product ($\theta$ is a real parameter):

$e^{-i\theta\mathcal{J}_{s=1}}\otimes e^{-i\theta\mathcal{J}_{s=1}}\,.$ (55)

One can block diagonalize this $9\times 9$ matrix and extract the five
dimensional spin 2 representation. This is same as (53) because

$\text{Diag}_{5\times 5}\left[e^{-i\theta\mathcal{J}_{s=1}}\otimes
e^{-i\theta\mathcal{J}_{s=1}}\right]=e^{-i\theta\mathcal{J}_{s=2}}\,.$ (56)

This five dimensional representation (53) is for the traceless and symmetric
spin 2 primary field with the following vector representation:

$\displaystyle\Phi_{s=2}(u,z,\bar{z})=\begin{bmatrix}\Phi^{uu}(u,z,\bar{z})\\\
\Phi^{ux}(u,z,\bar{z})\\\ \Phi^{uy}(u,z,\bar{z})\\\ \Phi^{xx}(u,z,\bar{z})\\\
\Phi^{xy}(u,z,\bar{z})\end{bmatrix}$ (57)

We have chosen a symmetric and traceless representation for the spin 2 field
since we want to holographically describe the bulk graviton field. Using the
tracelessness condition we have eliminated $\Phi^{yy}$ as

$g_{\mu\nu}\Phi^{\mu\nu}=\Phi^{xx}+\Phi^{yy}=0\,,$ (58)

where we used $g_{uu}=0$ and $g_{ij}=\delta_{ij}$. Clearly, to impose this
tracelessness, which selects only the spatial components, we had to work with
tensor primaries with upper indices. This also justifies why we worked with
upper indices for the vector primaries, which ultimately resulted in the
crucial (50). From (53), we have

$\displaystyle\mathcal{J}\Phi_{s=2}(u,z,\bar{z})=\begin{bmatrix}0\\\
-i\Phi^{uy}(u,z,\bar{z})\\\ i\Phi^{ux}(u,z,\bar{z})\\\
-2i\Phi^{xy}(u,z,\bar{z})\\\ 2i\Phi^{xx}(u,z,\bar{z})\end{bmatrix}$ (59)

To build the transformation rules for the spin 2 Carrollian primary field from
(40), we use the following combination to work out an example of $L_{0}$

$\displaystyle\Delta\Phi_{s=2}(u,z,\bar{z})-\mathcal{J}\Phi_{s=2}(u,z,\bar{z})=\begin{bmatrix}\Delta\Phi^{uu}\\\
\Delta\Phi^{ux}+i\Phi^{uy}\\\ \Delta\Phi^{uy}-i\Phi^{ux}\\\
\Delta\Phi^{xx}+2i\Phi^{xy}\\\ \Delta\Phi^{xy}-2i\Phi^{xx}\end{bmatrix}$ (60)

We now define the fields in terms of the $z$ and $\bar{z}$ components as

$\begin{split}\Phi^{zz}&=2\Phi^{xx}-2i\Phi^{xy}\,,\\\
\Phi^{\bar{z}\bar{z}}&=2\Phi^{xx}+2i\Phi^{xy}\,,\\\
\Phi^{uz}&=\Phi^{ux}-i\Phi^{uy}\,,\\\
\Phi^{u\bar{z}}&=\Phi^{ux}+i\Phi^{uy}\,.\end{split}$ (61)

Once we have the (61), we can straightforwardly work out the transformation
properties of the various components of (57). We set the Boost matrices of
(41) zero. The primary transformation rules thus become

  * •

$\mathbf{\Phi^{uu}}(u,z,\bar{z})$

$\displaystyle[L_{n},\Phi^{uu}(u,z,\bar{z})]=-i[z^{n+1}\partial_{z}+(\frac{\Delta}{2}+\frac{u}{2}\partial_{u})z^{n}]\Phi^{uu}(u,z,\bar{z})$
(62)
$\displaystyle[\bar{L}_{n},\Phi^{uu}(u,z,\bar{z})]=-i[\bar{z}^{n+1}\partial_{\bar{z}}+(\frac{\Delta}{2}+\frac{u}{2}\partial_{u})\bar{z}^{n}]\Phi^{uu}(u,z,\bar{z})\quad\forall
n\in 0,\pm 1$
$\displaystyle[M_{r,s},\Phi^{uu}(u,z,\bar{z})]=-iz^{r}\bar{z}^{s}\partial_{u}\Phi^{uu}(u,z,\bar{z})$

  * •

$\mathbf{\Phi^{uz}}(u,z,\bar{z})$

$\displaystyle[L_{n},\Phi^{uz}(u,z,\bar{z})]=-i[z^{n+1}\partial_{z}+(\frac{\Delta-1}{2}+\frac{u}{2}\partial_{u})z^{n}]\Phi^{uz}(u,z,\bar{z})$
(63)
$\displaystyle[\bar{L}_{n},\Phi^{uz}(u,z,\bar{z})]=-i[\bar{z}^{n+1}\partial_{\bar{z}}+(\frac{\Delta+1}{2}+\frac{u}{2}\partial_{u})\bar{z}^{n}]\Phi^{uz}(u,z,\bar{z})\quad\forall
n\in 0,\pm 1$
$\displaystyle[M_{r,s},\Phi^{uz}(u,z,\bar{z})]=-iz^{r}\bar{z}^{s}\partial_{u}\Phi^{uz}(u,z,\bar{z})$

  * •

$\mathbf{\Phi^{u\bar{z}}}(u,z,\bar{z})$

$\displaystyle[L_{n},\Phi^{u\bar{z}}(u,z,\bar{z})]=-i[z^{n+1}\partial_{z}+(\frac{\Delta+1}{2}+\frac{u}{2}\partial_{u})z^{n}]\Phi^{u\bar{z}}(u,z,\bar{z})$
(64)
$\displaystyle[\bar{L}_{n},\Phi^{u\bar{z}}(u,z,\bar{z})]=-i[\bar{z}^{n+1}\partial_{\bar{z}}+(\frac{\Delta-1}{2}+\frac{u}{2}\partial_{u})\bar{z}^{n}]\Phi^{u\bar{z}}(u,z,\bar{z})\quad\forall
n\in 0,\pm 1$
$\displaystyle[M_{r,s},\Phi^{u\bar{z}}(u,z,\bar{z})]=-iz^{r}\bar{z}^{s}\partial_{u}\Phi^{u\bar{z}}(u,z,\bar{z})$

  * •

$\mathbf{\Phi^{zz}}(u,z,\bar{z})$

$\displaystyle[L_{n},\Phi^{zz}(u,z,\bar{z})]=-i[z^{n+1}\partial_{z}+(\frac{\Delta-2}{2}+\frac{u}{2}\partial_{u})z^{n}]\Phi^{zz}(u,z,\bar{z})$
(65)
$\displaystyle[\bar{L}_{n},\Phi^{zz}(u,z,\bar{z})]=-i[\bar{z}^{n+1}\partial_{\bar{z}}+(\frac{\Delta+2}{2}+\frac{u}{2}\partial_{u})\bar{z}^{n}]\Phi^{zz}(u,z,\bar{z})\quad\forall
n\in 0,\pm 1$
$\displaystyle[M_{r,s},\Phi^{zz}(u,z,\bar{z})]=-iz^{r}\bar{z}^{s}\partial_{u}\Phi^{zz}(u,z,\bar{z})$

  * •

$\mathbf{\Phi^{\bar{z}\bar{z}}}(u,z,\bar{z})$

$\displaystyle[L_{n},\Phi^{\bar{z}\bar{z}}(u,z,\bar{z})]=-i[z^{n+1}\partial_{z}+(\frac{\Delta+2}{2}+\frac{u}{2}\partial_{u})z^{n}]\Phi^{\bar{z}\bar{z}}(u,z,\bar{z})$
(66)
$\displaystyle[\bar{L}_{n},\Phi^{\bar{z}\bar{z}}(u,z,\bar{z})]=-i[\bar{z}^{n+1}\partial_{\bar{z}}+(\frac{\Delta-2}{2}+\frac{u}{2}\partial_{u})\bar{z}^{n}]\Phi^{\bar{z}\bar{z}}(u,z,\bar{z})\quad\forall
n\in 0,\pm 1$
$\displaystyle[M_{r,s},\Phi^{\bar{z}\bar{z}}(u,z,\bar{z})]=-iz^{r}\bar{z}^{s}\partial_{u}\Phi^{\bar{z}\bar{z}}(u,z,\bar{z})$

The weights of the various components are summarized in Table 3. The
tracelessness condition on the spin 2 primaries is manifested as

$\Phi^{z\bar{z}}=0\,.$ (67)

weights | $\Phi^{uu}$ | $\Phi^{uz}$ | $\Phi^{u\bar{z}}$ | $\Phi^{zz}$ | $\Phi^{\bar{z}\bar{z}}$
---|---|---|---|---|---
$h$ | ${\Delta}/{2}$ | $\frac{\Delta-1}{2}$ | $\frac{\Delta+1}{2}$ | $\frac{\Delta-2}{2}$ | $\frac{\Delta+2}{2}$
$\bar{h}$ | ${\Delta}/{2}$ | $\frac{\Delta+1}{2}$ | $\frac{\Delta-1}{2}$ | $\frac{\Delta+2}{2}$ | $\frac{\Delta-2}{2}$
Table 3: Weights of tensor primaries with upper indices

We can now lower the indices similar to (49) to get

$\begin{split}\Phi_{uu}=g_{uu}g_{uu}\Phi^{uu}=0\,,~{}~{}~{}~{}~{}\Phi_{uz}=g_{uu}g_{z{\bar{z}}}\Phi^{u{\bar{z}}}=0\,,~{}~{}~{}~{}~{}\Phi_{u{\bar{z}}}=g_{uu}g_{{\bar{z}}z}\Phi^{uz}=0\,,\\\
\Phi_{zz}=g_{z{\bar{z}}}g_{z{\bar{z}}}\Phi^{{\bar{z}}{\bar{z}}}\,,~{}~{}~{}~{}~{}\Phi_{{\bar{z}}{\bar{z}}}=g_{{\bar{z}}z}g_{{\bar{z}}z}\Phi^{zz}\,,~{}~{}~{}~{}~{}\Phi_{z{\bar{z}}}=g_{{\bar{z}}z}g_{z{\bar{z}}}\Phi^{z{\bar{z}}}=0\,.\end{split}$
(68)

Thus, the $u$ components of the field are similar to the spin 1 case (50):

$\Phi_{uu}=\Phi_{uz}=\Phi_{u{\bar{z}}}=0\,.$ (69)

The weights of the non-trivial tensor primaries with lower indices $\Phi_{zz}$
and $\Phi_{{\bar{z}}{\bar{z}}}$ are given by Table 4.

weights | $\Phi_{zz}$ | $\Phi_{{\bar{z}}\bar{z}}$
---|---|---
$h$ | $\frac{\Delta+2}{2}$ | $\frac{\Delta-2}{2}$
$\bar{h}$ | $\frac{\Delta-2}{2}$ | $\frac{\Delta+2}{2}$
Table 4: Weights of vector primaries with lower indices

If we were to work with the non-trivial choice of boost matrices of (41), this
would mix up various components of (57), and the analysis of the correlations
functions would thus change. This and consequences for holography in flat
spacetimes will be addressed elsewhere.

###  4 Carrollian correlation functions

Now, we derive the correlation functions of the Carroll primary operators
whose transformation rules we derived in the previous section. The two and
three point correlators of these primaries can be exactly determined by
solving the Ward identities associated with the global generators. 888Spinning
Carrollian correlators have been previously addressed in Salzer:2023jqv  and
more recently in Nguyen:2023miw  when this paper was being readied for
submission. While Salzer:2023jqv  computes some specific cases, Nguyen:2023miw
focusses on more generic three-point functions. Their answers match with our
previous construction in Bagchi:2023fbj  and the answers presented later. We
also have more general three-point functions on the celestial torus and four
point functions in this paper. In order to determine these correlators, let us
first study the correlation function of Carroll primaries of arbitrary spin in
(21). For the reader’s convenience, we repeat these expressions below

$\displaystyle\delta_{L_{n}}\Phi(u,z,\bar{z})=-i[z^{n+1}\partial_{z}+(n+1)(h+\frac{u}{2}\partial_{u})z^{n}]\Phi(u,z,\bar{z})\,,$
(70)
$\displaystyle\delta_{\bar{L}_{n}}\Phi(u,z,\bar{z})=-i[\bar{z}^{n+1}\partial_{\bar{z}}+(n+1)(\bar{h}+\frac{u}{2}\partial_{u})\bar{z}^{n}]\Phi(u,z,\bar{z})\,,$
$\displaystyle\delta_{M_{r,s},\Phi(u,z,\bar{z}}=-iz^{r}\bar{z}^{s}\partial_{u}\Phi(u,z,\bar{z})\,.$

####  4.1 Two point function

The two point correlation function of these objects was determined, and its
relationship with scattering amplitudes was established in Bagchi:2022emh .
Here, we review the basic arguments. It turns out that the Ward identities
associated with these global supertranslations have two classes of solutions
depending on whether the solutions have any $u$ coordinate dependence. For
e.g.

$\displaystyle\delta_{M_{10}}G^{(2)}(u,u^{\prime},z,z^{\prime},\bar{z},\bar{z}^{\prime})=(z\partial_{u}+z^{\prime}\partial_{u^{\prime}})G^{(2)}(u,u^{\prime},z,z^{\prime},\bar{z},\bar{z}^{\prime})=0\,.$
(71)

One way to solve the equation is to drop the $u$ dependence. The rest of the
supertranslation equations would also be automatically satisfied. Further, the
global superrotations $L_{0,\pm 1},\bar{L}_{0,\pm 1}$ would fix it to be 2d
CFT primary correlator Bagchi:2016bcd :

$\displaystyle
G^{(2)}(u,z,\bar{z})=\frac{\delta_{h,h^{\prime}}\delta_{\bar{h},\bar{h}^{\prime}}}{(z-z^{\prime})^{2h}(\bar{z}-\bar{z}^{\prime})^{2\bar{h}}}\,.$
(72)

On the other hand, (71) can also be solved by a $u$ dependent branch. This is
possible only when the spatial part is a contact term. In this case, the
supertranslation equations force the ansatz to be of the form,

$\displaystyle
G^{(2)}(u,u^{\prime},z,z^{\prime},\bar{z},\bar{z}^{\prime})=f(u-u^{\prime})\delta^{2}(z-z^{\prime})\,.$
(73)

Invariance under $L_{0,\pm 1}$ and $\bar{L}_{0,\pm 1}$ would further fix it to
be

$\displaystyle
G^{(2)}(u,z,\bar{z},u^{\prime},z^{\prime},\bar{z}^{\prime})=\delta_{\sigma+\sigma^{\prime},0}\frac{\delta^{(2)}(z-z^{\prime},\bar{z}-\bar{z}^{\prime})}{(u-u^{\prime})^{\Delta+\Delta^{\prime}-2}}\,.$
(74)

Notice that unlike the CFT branch, this branch of the correlator is non-
trivial even if the primaries have unequal scaling dimensions. The constraint
restricts only the spins. We would now use this construction to compute the
correlators of different components of spinning Carroll primaries discussed
above. For vectors, the ansatz associated with the delta-branch for the two
point correlator can be taken as

$\displaystyle
G^{(2)\,IJ}(u,u^{\prime},z,z^{\prime},\bar{z},\bar{z}^{\prime})=f^{IJ}(u-u^{\prime})\delta^{2}(z-z^{\prime})\,.$
(75)

However now the invarince under the spatial rotations ($L_{0}-\bar{L}_{0}$)
would imply

$\sigma^{I}+\sigma^{J}=0\,.$ (76)

From the weights of these spinning primaries in Table 1, we have
$\sigma^{u}=0$, $\sigma^{z}=-1$ and $\sigma^{\bar{z}}=1$. It is immediately
obvious from the above equation that only non-trivial $f^{IJ}$ s are $f^{uu}$
and $f^{z\bar{z}}$.

$f^{uu}=f^{z\bar{z}}=\frac{1}{(u-u^{\prime})^{\Delta+\Delta^{\prime}-2}}\,.$
(77)

These are the only non-trivial correlators for spinning primaries when the
boost matrices are zero. However, we shall see in the next section that we
need to work with primaries of lower indices in order to match up with the
bulk analysis. It is evident from our discussion above that if we work with
lower indices, the $u$ component vanishes, and as a consequence, any
correlation function involving $u$-component fields also vanishes. Thus, we
should have

$f_{uu}=0\,.$ (78)

However, $f_{z{\bar{z}}}$ remains non zero and is given by

$f_{z{\bar{z}}}=\big{(}g_{z{\bar{z}}}(z)g_{z{\bar{z}}}(z^{\prime})\big{)}\frac{1}{(u-u^{\prime})^{\Delta+\Delta^{\prime}-2}}\,.$
(79)

A similar analysis can be done for the spin 2 primaries using the weights of
Table 3. The invariance under spatial rotations $(L_{0}-\bar{L}_{0})$ would
imply that the only non-trivial correlators are

$f^{uu,uu}=f^{uz,u\bar{z}}=f^{zz,\bar{z}\bar{z}}=\frac{1}{(u-u^{\prime})^{\Delta+\Delta^{\prime}-2}}\,.$
(80)

Lowering the indices (analogous to how we arrived at (78)) and using (69), we
have999We will see later in our bulk analysis that (81) is consistent with the
fact that the $uu$ component of the bulk to boundary propagator becomes pure
diffeomorphism in (157).

$f_{uu,uu}=f_{uz,u\bar{z}}=0\,.$ (81)

Thus, the only non-trivial correlator is that of $f_{zz,\bar{z}\bar{z}}$ in
(80):

$f_{zz,\bar{z}\bar{z}}=\big{(}g_{z{\bar{z}}}(z)^{2}g_{z{\bar{z}}}(z^{\prime})^{2}\big{)}\frac{1}{(u-u^{\prime})^{\Delta+\Delta^{\prime}-2}}\,.$
(82)

####  4.2 Three point function

We now focus on three point functions. The three point function also admits
two different branches of solutions. The time independent branch again
coincides with the 2d CFT three point correlators, i.e.

$\displaystyle
G^{(3)}(u,z,\bar{z})=\frac{1}{(z_{12})^{h_{1}+h_{2}-h_{3}}(z_{23})^{h_{2}+h_{3}-h_{1}}(z_{13})^{h_{1}+h_{3}-h_{2}}}$
(83)

However the more relevant delta function branch has intricate substructures,
which computes different three point scattering amplitudes in the bulk. For
generic momenta, the three point function is trivially zero and this can be
thought of as a consequence of momentum conservation applied to massless
particles in 4d Minkowski spacetimes Banerjee:2018gce ; Bagchi:2022emh . We
addressed this earlier in Bagchi:2023fbj , where we also considered a non-
trivial three point solution. This solutions corresponds to a case where all
three massless momenta are collinear. This subclass of solutions is given by

$\displaystyle
G^{(3)}(u,z,\bar{z})=u_{12}^{a}u_{23}^{b}u_{31}^{c}\delta^{(2)}(z_{12})\delta^{(2)}(z_{23}),$
(84) $\displaystyle\text{subjected to}\quad
a+b+c=\sum_{i}\Delta_{i},\quad\sum_{i}\sigma_{i}=0$

Below we consider other non-trivial cases.

#####  4.2.1 Non collinear case

For this case, we assume only one delta function, which imposes collinearity
in two particle momenta, unlike the previous case considered in Bagchi:2023fbj
, where all three particles were collinear. The ansatz we start off with is

$\displaystyle
G^{(3)}(u_{i},z_{i},\bar{z}_{i})=F(u_{i})\,{z_{12}}^{p}\,{\bar{z}_{12}}^{q}\delta^{2}(z_{13})\,.$
(85)

The lowest supertranslation generator fixes $F(u_{i})$ as
$F(u_{12},u_{23},u_{31})$ as usual. However, higher supertranslation modes
have non-trivial consequences on this type of correlation function. For
example, consider the variation with respect to the generator $M_{10}$

$\displaystyle\delta_{M_{10}}G^{(3)}(u_{i},z_{i},\bar{z}_{i})=0\implies[z_{1}(\partial_{u_{1}}+\partial_{u_{3}})+z_{2}\partial_{u_{2}}]F(u_{i})=0\,.$
(86)

This equation only allows terms which are independent of $u_{2}$. Hence

$\displaystyle F(u_{i})\equiv F(u_{13})=\sum_{c}f_{c}\,{u_{13}}^{c}\,.$ (87)

SL$(2,\mathbb{C})$ or Lorentz symmetries in the bulk would constrain the
exponents. Again, the ansatz is already invariant under the translations along
$z$ and $\bar{z}$. The scale transformation generators $L_{0}$ and
$\bar{L}_{0}$ constraint the exponents in the following way

$\displaystyle\sum_{i}h_{i}=1-p-\frac{c}{2}\,,\qquad\sum_{i}\bar{h}_{i}=1-q-\frac{c}{2}\,.$
(88)

However, unlike the collinear case Bagchi:2023fbj , the special conformal
transformations play a role and further fix these coefficients. Consider the
$L_{1}$ equation

$\displaystyle\sum_{i}z_{i}^{2}\partial_{z_{i}}G^{3}(u_{i},z_{i},\bar{z}_{i})+2\sum_{i}z_{i}(h_{i}+\frac{u_{i}}{2}\partial_{u_{i}})G^{3}(u_{i},z_{i},\bar{z}_{i})=0\,,$
(89)
$\displaystyle\Big{[}(-2+2(h_{1}+h_{3})+c+p)z_{1}+(2h_{2}+p)z_{2}\Big{]}G^{3}(u_{i},z_{i},\bar{z}_{i})=0\,,$
$\displaystyle\implies h_{1}+h_{3}=\frac{1}{2}(2-c-p),\quad
h_{2}=-\frac{p}{2}\,.$

Similarly from $\bar{L}_{1}$ we have

$\bar{h}_{1}+\bar{h}_{3}=\frac{1}{2}(2-c-q),\quad\bar{h}_{2}=-\frac{q}{2}\,.$
(90)

From these equations, we have

$s_{1}+s_{3}=\dfrac{q-p}{2}\,,~{}~{}~{}s_{2}=\dfrac{q-p}{2}$ (91)

It is clear that a scattering process with two massless scalars and one photon
can never satisfy $s_{1}+s_{2}=s_{3}$. We will, however, see that the
scattering process with three massless scalars has non-trivial contributions
from the soft sector. Thus, we have the result:

$G^{(3)}(u_{i},z_{i},\bar{z}_{i})=u^{2-\Delta_{1}-\Delta_{3}-\frac{p+q}{2}}_{13}\,{z_{12}}^{p}\,{\bar{z}_{12}}^{q}\delta^{2}(z_{13})\,.$
(92)

#####  4.2.2  On $\mathbb{R}_{u}\times\mathbb{C}^{2}$

The three point function has a non-trivial contribution if we treat $z$ and
$\bar{z}$ as independent coordinates. This amounts to working on a celestial
torus Stieberger:2018edy  instead of a sphere. This is relevant for the
scattering processes in the bulk when the momenta of the particles are either
complexified or the bulk spacetime has a (2,2) signature.

To proceed with this set up, we set $\bar{z}_{1}=\bar{z}_{2}=\bar{z}_{3}$,
(i.e.) our ansatz is

$G^{3}(u_{i},z_{i},\bar{z}_{i})=F(u_{i},z_{i})\delta(\bar{z}_{12})\delta(\bar{z}_{23})\,.$
(93)

The variation with respect to the supertranslations is given by

$\displaystyle\delta_{M_{r,s}}G^{3}(u_{i},z_{i},\bar{z}_{i})=\sum_{i=1}^{3}z_{i}^{r}\bar{z}_{i}^{s}\partial_{u_{i}}G^{3}(u_{i},z_{i},\bar{z}_{i})\,.$
(94)

We shall demand that the correlation function is invariant under the following
generators: $M_{00},M_{10},M_{01},M_{11}$. These generators, as mentioned
previously, are isomorphic to the bulk translations. Now the equations
corresponding to $M_{00}$ and $M_{10}$ symmetries lead to

$\displaystyle\sum_{i=1}^{3}\partial_{u_{i}}F(u_{i},z_{i})=0,\quad\sum_{i=1}^{3}z_{i}\partial_{u_{i}}F(u_{i},z_{i})=0\,.$
(95)

Due to the presence of delta function in $\bar{z}$, the other two equations
($M_{01},M_{11}$) become the same as (95). These two above equations can be
solved if

$\displaystyle F(u_{i},z_{i})\cong
F(U,z_{i})=f(U)g(z_{i}),\quad\text{where}\quad
U=z_{1}u_{23}+z_{2}u_{31}+z_{3}u_{12}\,.$ (96)

There are still six more equations that would impose global superrotation
invariance, which is isomorphic to the bulk Lorentz group. These equations
would completely fix the form of $F(U,z_{i})$. Variations with respect to the
holomorphic and anti-holomorphic generators are given by

$\begin{split}\delta_{L_{n}}G^{3}(u_{i},z_{i},\bar{z}_{i})&=\sum_{i=1}^{3}\Big{[}z_{i}^{n+1}\partial_{z_{i}}+(n+1)z_{i}^{n}(h_{i}+\frac{1}{2}u_{i}\partial_{u_{i}})\Big{]}G^{3}(u_{i},z_{i},\bar{z}_{i})\,,\\\
\delta_{\bar{L}_{n}}G^{3}(u_{i},z_{i},\bar{z}_{i})&=\sum_{i=1}^{3}\Big{[}\bar{z_{i}}^{n+1}\partial_{\bar{z}_{i}}+(n+1)\bar{z_{i}}^{n}(\bar{h}_{i}+\frac{1}{2}u_{i}\partial_{u_{i}})\Big{]}G^{3}(u_{i},z_{i},\bar{z}_{i})\,,\end{split}$
(97)

and we need to solve these equations for $n=0,\pm 1$.

Now, $n=-1$ imposes translational invariance on the spatial direction. Hence,
the anti holomorphic equation $\bar{L}_{-1}$ is already solved by our ansatz
(93). The holomorphic equation $L_{-1}$ would be solved if

$g(z_{i})\cong g(z_{12},z_{23},z_{31})\,.$ (98)

For $n=0$, the anti-holomorphic equation $\bar{L}_{0}$ would gives us

$\displaystyle\sum_{i=1}^{3}(\bar{h}_{i}-2)+\frac{1}{2}U\partial_{U}f(U)=0\quad\text{thus}\quad
f(U)=U^{-2\Big{(}\sum_{i=1}^{3}\bar{h}_{i}-2\Big{)}}\,.$ (99)

Using (99), the holomorphic equation $L_{0}$ also simplifies to

$\sum_{i=1}^{3}z_{i}\partial_{z_{i}}g(z_{i})+\Big{(}3(\bar{h}_{i}-2)+h_{i}\Big{)}g(z_{i})=0\,.$
(100)

Now from (98), we can assume $g(z_{i})$ would be of the following form

$g(z_{i})=z_{12}^{a}\,z_{23}^{b}\,z_{31}^{c}\,.$ (101)

Plugging (101) back in (100) would yield

$a+b+c=-\sum_{i=1}^{3}[3(2-\bar{h}_{i})-h_{i}]\,.$ (102)

This leaves two more equations from $L_{1}$ and $\bar{L}_{1}$. Plugging in the
form of solution from (99) and (101) we have obtained so far,

$G^{3}(u_{i},z_{i},\bar{z}_{i})=f(U)g(z_{i})\delta(\bar{z}_{12})\delta(\bar{z}_{23})\,,$
(103)

in the $L_{1}$ equation, we would obtain the following equation

$\displaystyle
z_{1}(a+c+2h_{1})+z_{2}(a+b+2h_{2})+z_{3}(b+c+2h_{3})-2\Big{(}\sum_{i=1}^{3}\bar{h}_{i}-2\Big{)}\sum_{i=1}z_{i}=0\,.$
(104)

Now (102) and (104) can be solved if we choose

$\displaystyle
a=h_{3}-h_{1}-h_{2}-(2-\sum_{i=1}^{3}\bar{h}_{i})=\Delta_{3}-\sigma_{1}-\sigma_{2}-2\,,$
(105a) $\displaystyle
b=h_{1}-h_{2}-h_{3}-(2-\sum_{i=1}^{3}\bar{h}_{i})=\Delta_{1}-\sigma_{2}-\sigma_{3}-2\,,$
(105b) $\displaystyle
c=h_{2}-h_{1}-h_{3}-(2-\sum_{i=1}^{3}\bar{h}_{i})=\Delta_{2}-\sigma_{1}-\sigma_{3}-2\,.$
(105c)

This fixes the 3-pt function exactly. Given this solution, the equation that
arises from $\bar{L}_{1}$ is trivially solved. Thus, we have

$G^{3}(u_{i},z_{i},\bar{z}_{i})=(z_{1}u_{23}+z_{2}u_{31}+z_{3}u_{12})^{-2\Big{(}\sum_{i=1}^{3}\bar{h}_{i}-2\Big{)}}z_{12}^{a}\,z_{23}^{b}\,z_{31}^{c}\delta(\bar{z}_{12})\delta(\bar{z}_{23})\,,$
(106)

where $a,b,c$ are given by (105). Though the final result depends only on the
weights $h_{i},\bar{h}_{i}$ and the helicities $\sigma_{i}$, it should be
noted that when applied for vector and tensor primaries, there are no $u$
components because of (50) and (69). Thus, the three point function receives
contributions only from the spatial components of the vector and tensor
primaries.

We can also consider the other holomorphic branch by setting
$z_{1}=z_{2}=z_{3}$. This is just a complex conjugation and can be implemented
by switching $\sigma_{i}\to-\sigma_{i}$ in (106). This would result in

$G^{3}(u_{i},z_{i},\bar{z}_{i})=(\bar{z}_{1}u_{23}+\bar{z}_{2}u_{31}+\bar{z}_{3}u_{12})^{-2\Big{(}\sum_{i=1}^{3}h_{i}-2\Big{)}}\bar{z}_{12}^{\bar{a}}\,\bar{z}_{23}^{\bar{b}}\,\bar{z}_{31}^{\bar{c}}\delta(z_{12})\delta(z_{23})\,,$
(107)

where

$\displaystyle\bar{a}=\bar{h}_{3}-\bar{h}_{1}-\bar{h}_{2}-(2-\sum_{i=1}^{3}h_{i})=\Delta_{3}+\sigma_{1}+\sigma_{2}-2\,,$
(108a)
$\displaystyle\bar{b}=\bar{h}_{1}-\bar{h}_{2}-\bar{h}_{3}-(2-\sum_{i=1}^{3}h_{i})=\Delta_{1}+\sigma_{2}+\sigma_{3}-2\,,$
(108b)
$\displaystyle\bar{c}=\bar{h}_{2}-\bar{h}_{1}-\bar{h}_{3}-(2-\sum_{i=1}^{3}h_{i})=\Delta_{2}+\sigma_{1}+\sigma_{3}-2\,.$
(108c)

At this point, it is useful to mention that we should also expect this result
from the point of view of scattering amplitudes. If one expresses the three
point scattering amplitude in terms of spinor helicity variables, the little
group scaling completely fixes the amplitude up to normalization
elvang_huang_2015 . If one now applies a Modified Mellin transform on this
amplitude, one should get (106) and (107).

####  4.3 Four point function

For this analysis, we basically follow Banerjee:2018gce . We use the
$SL(2,\mathbb{C})$ invariance to first map the four points
$z_{1},z_{2},z_{3},z_{4}$ at the celestial spheres at null infinity to the
following points:

$(z_{1},z_{2},z_{3},z_{4})\to(z,1,0,\infty)\,,$ (109)

where $z$ is the conformal cross ratio given by

$z=\dfrac{z_{12}z_{34}}{z_{13}z_{24}}\,.$ (110)

The four point correlator of Carrollian primary fields with weights
$(h_{i},\bar{h}_{i})$ is thus given by

$\begin{split}G_{4}(u_{1},u_{2},&u_{3},u_{4},z,\bar{z})\\\
&=\langle\Phi_{(h_{1},\bar{h}_{1})}(u_{1},z,\bar{z})\Phi_{(h_{2},\bar{h}_{2})}(u_{2},1,1)\Phi_{(h_{3},\bar{h}_{3})}(u_{3},0,0)\Phi_{(h_{4},\bar{h}_{4})}(u_{4},\infty,\infty)\rangle\,.\end{split}$
(111)

The lowest supertranslation generator $M_{00}$ (Carroll time translations)
imposes the following constraint on the four point correlator:

$\left(\dfrac{\partial}{\partial u_{1}}+\dfrac{\partial}{\partial
u_{2}}+\dfrac{\partial}{\partial u_{3}}+\dfrac{\partial}{\partial
u_{4}}\right)G_{4}(u_{1},u_{2},u_{3},u_{4},z,\bar{z})=0\,.$ (112)

The higher supertranslation modes $M_{10}$ and $M_{01}$ (Carroll boosts)
impose the following constraints

$\begin{split}\left(z\dfrac{\partial}{\partial
u_{1}}+\dfrac{\partial}{\partial u_{2}}+\alpha\dfrac{\partial}{\partial
u_{4}}\right)G_{4}(u_{1},u_{2},u_{3},u_{4},z,\bar{z})&=0\,,\\\
\left(\bar{z}\dfrac{\partial}{\partial u_{1}}+\dfrac{\partial}{\partial
u_{2}}+\alpha\dfrac{\partial}{\partial
u_{4}}\right)G_{4}(u_{1},u_{2},u_{3},u_{4},z,\bar{z})&=0\,.\end{split}$ (113)

Here $\alpha\to\infty$. These two equations can be combined together (by
suitably regulating the infinity by a limiting procedure) to obtain

$(z-\bar{z})\dfrac{\partial}{\partial
u_{1}}G_{4}(u_{1},u_{2},u_{3},u_{4},z,\bar{z})=0\,.$ (114)

If we choose the branch

$\dfrac{\partial}{\partial
u_{1}}G_{4}(u_{1},u_{2},u_{3},u_{4},z,\bar{z})=0\,,$ (115)

then (113) implies

$\left(\dfrac{\partial}{\partial u_{2}}+\alpha\dfrac{\partial}{\partial
u_{4}}\right)G_{4}(u_{1},u_{2},u_{3},u_{4},z,\bar{z})=0\,.$ (116)

Thus, the only possible solution is

$\dfrac{\partial\,G_{4}}{\partial
u_{2}}=0\,,~{}~{}~{}~{}~{}~{}\dfrac{\partial\,G_{4}}{\partial u_{4}}=0\,.$
(117)

Thus, (112) implies

$\dfrac{\partial\,G_{4}}{\partial u_{3}}=0\,.$ (118)

Hence, we essentially end up on the $u$-independent CFT branch four point
correlator that depends only on the conformal cross ratio. To get the desired
$u$-dependent branch of the Carrollian correlator, we must choose the other
branch in (114) given by

$z-\bar{z}=0\,.$ (119)

Thus, the expression for the four point Carrollian correlator becomes

$G_{4}(u_{1},u_{2},u_{3},u_{4},z,\bar{z})\propto\delta(|z-\bar{z}|)\,.$ (120)

This is the only constraint on the four point Carrollian correlator.

###  5 Flat limit of Witten diagrams

In part I, we have argued why it is more natural to consider the Carrollian
approach to flat holography as opposed to the Celestial one. We will now show
how these non-trivial Carroll correlators we constructed in the previous
section that encode bulk scattering arise in a carefully constructed limit of
AdS/CFT correlation functions. If one keeps track of the null direction while
implementing the limit, we will show that Carrollian CFT correlation functions
naturally emerge in the large AdS radius limit. Before that, in this section,
we will analyse the elements of the Witten diagrams. We will review AdS/CFT
Witten diagrams for scalar fields in the large AdS radius limit and then
proceed to work out the bulk to boundary propagator for spinning particles in
the large AdS radius limit.

####  5.1 AdS in the embedding space representation

We will first quickly recap the embedding representation that is typically
used in the computation of AdS Witten diagrams Dirac:1936fq ; Penedones:2007ns
; Penedones:2010ue . We choose to work with a particular choice of the
embedding space representation. The $d+1$-dimensional AdS solution is given by
:

$-(X^{0})^{2}+\sum_{i=1}^{d}(X^{i})^{2}-(X^{d+1})^{2}=-R^{2}\,,$ (121)

which is embedded in a $\mathbb{R}^{2,d}$ manifold with the metric:

$ds^{2}=-(dX^{0})^{2}+\sum_{i=1}^{d}(dX^{i})^{2}-(dX^{d+1})^{2}\,.$ (122)

This is a slightly different version of the embedding manifold when compared
to the lightcone coordinates used in Bagchi:2023fbj . It turns out that in
order to discuss the flat limit of spinning propagators, it is convenient to
work with this form of embedding manifold.

The AdS solution can be written in a parametric form as follows:

$\begin{split}X^{0}=R\dfrac{\sin\tau}{\cos\rho}\,,~{}~{}~{}X^{d+1}=R\dfrac{\cos\tau}{\cos\rho}\,,~{}~{}~{}X^{i}=R\tan\rho\,\Omega^{i}\,,~{}~{}i=1,\dots,d\end{split}$
(123)

where $\sum_{i}\Omega^{2}_{i}=1$ represents $S^{d-1}$. In these new global
coordinates, the AdS metric is given by:

$ds^{2}=\frac{R^{2}}{\cos^{2}\rho}\left(-d\tau^{2}+d\rho^{2}+\sin^{2}\rho\,d\Omega_{S^{d-1}}^{2}\right)\,,$
(124)

where $\tau\in[-\pi,\pi]$ and $\rho\in[0,\frac{\pi}{2}]$. The AdS boundary is
approached by taking the limit $\rho\rightarrow\frac{\pi}{2}$ and it is given
by:

$\textbf{P}=\lim_{\rho\rightarrow\frac{\pi}{2}}R^{-1}\cos\rho\textbf{X}\,.$
(125)

with $\textbf{P}^{2}=0$. The boundary coordinates are given by

$P^{0}=\sin\tau_{p}\,,~{}~{}~{}P^{d+1}=\cos\tau_{p}\,,~{}~{}~{}P^{i}=\Omega^{i}_{p}\,.$
(126)

Building on deGioia:2022fcn , in Bagchi:2023fbj , we established a general
relation between AdS Witten diagrams in $d+1$ dimensions and Carrollian
correlation functions in $d$ dimensions. In Bagchi:2023fbj , we showed that
AdS Witten diagrams with boundary operators inserted at specific time slices
reduce to the Carrollian correlators in a carefully constructed large AdS
radius (denoted by $R$ here) limit. The main assumptions were

  * •

The boundary CFT operators are inserted at global time slices
$\tau_{p}=\pm\frac{\pi}{2}+\frac{u}{R}$

  * •

The spheres at the slices $\tau_{p}=\frac{\pi}{2}+\frac{u}{R}$ and
$\tau_{p}=-\frac{\pi}{2}+\frac{u}{R}$ are antipodally identified.

We implement the flat limit in the bulk by the following rescaling
Giddings:1999jq

$\tau=\dfrac{t}{R}\,,~{}~{}~{}~{}~{}\rho=\dfrac{r}{R}\,.$ (127)

When the flat limit of the boundary theory is performed, we crucially track
the null direction of the boundary. To facilitate this, we look at what
happens to the retarded time $u$ in this limit. For instance, consider

$u=t-r=R(\tau-\rho)\,.$ (128)

Carrollian correlators will be functions of $u$, and thus, we first relate $u$
to the boundary CFT time. In AdS, the boundary is reached as
$\rho\to\frac{\pi}{2}$. Labeling the boundary time as $\tau_{p}$ and
substituting $\rho=\frac{\pi}{2}$ above, we get

$u=R(\tau_{p}-\frac{\pi}{2})\implies\tau_{p}=\frac{\pi}{2}+\frac{u}{R}\,.$
(129)

We use this input to implement the flat limit in the boundary as
Bagchi:2023fbj

$\tau_{p}=\pm\dfrac{\pi}{2}+\dfrac{u}{R}\,.$ (130)

To see how the null direction emerges, we substitute this expression for
$\tau_{p}$ back into the boundary metric:

$ds^{2}_{\text{bdy}}=-d\tau^{2}+d\Omega^{2}=-\dfrac{1}{R^{2}}du^{2}+d\Omega^{2}\to
0.du^{2}+d\Omega^{2}\,.$ (131)

Thus, under (130), the metric reduces to the null/Carrollian metric at
$\mathscr{I}^{+}$. We see that if we keep track of all the symmetries, the
boundary theory does not reduce in dimension. The large AdS radius limit in
the bulk is equivalent to a $c\to 0$ ultra-relativistic limit in the boundary
Bagchi:2012cy . This feature is captured by the limit implemented using (130).

Let us specialize to $d=3$. Thus implementing (127) and (130) in (123) and
(125), we have:

$\begin{split}X&=\left(X^{0},X^{i},X^{d+1}\right)\\\
&=\left(t+\mathcal{O}(\dfrac{1}{R^{2}}),\,r\Omega_{1}+\mathcal{O}(\dfrac{1}{R^{2}}),\,r\Omega_{2}+\mathcal{O}(\dfrac{1}{R^{2}}),\,r\Omega_{3}+\mathcal{O}(\dfrac{1}{R^{2}}),\,R+\dfrac{r^{2}-t^{2}}{2R}+\mathcal{O}(\dfrac{1}{R^{2}})\right)\,,\\\
P&=\left(P^{0},P^{i},P^{d+1}\right)=\left(\pm
1+\mathcal{O}(\dfrac{1}{R^{2}}),\,\Omega^{1}_{p},\,\Omega^{2}_{p},\,\Omega^{3}_{p},\,\mp\dfrac{u}{R}+\mathcal{O}(\dfrac{1}{R^{2}})\right)\,.\end{split}$
(132) $P\cdot X=\mp t+r\Omega_{i}\Omega^{i}_{p}\pm u\,.$ (133)

Thus, we have two cases corresponding to incoming and outgoing particles:

$P\cdot X=\begin{cases}-t+r\Omega_{i}\Omega^{i}_{p}+u=u+\tilde{q}\cdot
x&\text{if $\tau_{p}=\frac{\pi}{2}+\frac{u}{R}$}\\\
+t+r\Omega_{i}\Omega^{i}_{p}-u=-u-\tilde{q}^{A}\cdot x&\text{if
$\tau_{p}=-\frac{\pi}{2}+\frac{u}{R}$}\end{cases}$ (134)

Here $\tilde{q}=(1,\mathbf{\Omega}_{p})$ denotes the direction of the boundary
towards which the outgoing massless particle propagates.
$\tilde{q}^{A}=(1,\mathbf{\Omega}^{A}_{p})$ denotes the direction of the
antipodal point from which the incoming particle propagates. $x=(t,r\Omega)$.
As considered in Bagchi:2023fbj , the spheres at
$\tau_{p}=-\frac{\pi}{2}+\frac{u}{R}$ and $\tau_{p}=\frac{\pi}{2}+\frac{u}{R}$
are antipodally matched, thus

$\mathbf{\Omega}^{A}_{p}=-\mathbf{\Omega}_{p}\,.$ (135)

This generalized antipodal matching condition was recently proved in
deGioia:2023cbd  by considering the symmetry algebra of the CFT localized in
$\mathcal{O}(R^{-1})$ at different $\tau_{p}$’s.

####  5.2 A review of scalar Witten diagrams

A major component of the Witten diagram is the bulk to boundary propagator.
The bulk to boundary propagator for scalar fields is given by Penedones:2010ue

$\textbf{K}_{\Delta}(\textbf{P},\textbf{X})=\frac{C_{\Delta}^{d}}{(-P\cdot
X+i\epsilon)^{\Delta}}\,,$ (136) 101010There is no factor of 2 multiplying
$P.X$ in the denominator because we have chosen to rescale the boundary
coordinates as in (126).

where $C^{d}_{\Delta}$ with factors of $R$ is given by

$C_{\Delta}^{d}=\frac{\Gamma(\Delta)}{2\pi^{\frac{d}{2}}\Gamma(\Delta-\frac{d}{2}+1)R^{\frac{(d-1)}{2}-\Delta}}\,.$
(137)

Here we have parametrized the bulk and boundary points by
$x\in(\tau,\rho,\Omega)$ and $p\in(\tau_{p},\Omega_{p})$ respectively.

The explicit form of the propagators in the large $R$ limit can be derived
from (134) Bagchi:2023fbj :

$\begin{split}\text{incoming}\quad\textbf{K}_{\Delta_{1}}(\textbf{P}_{1},\textbf{X})&=N_{\Delta_{1}}^{d}\psi_{\Delta_{1},\tilde{q}_{1},u_{1}}^{-}(x)+\mathcal{O}(R^{-1})\,,\\\
\text{outgoing}\quad\textbf{K}_{\Delta_{2}}(\textbf{P}_{2},\mathbf{Y})&=N_{\Delta_{2}}^{d}\psi_{\Delta_{2},\tilde{q}_{2},u_{2}}^{+}(y)+{\mathcal{O}{(R^{-1})}}\,,\\\
\text{bulk-
bulk}\quad\Pi_{\Delta}(\textbf{X},\textbf{Y})&=G(x,y)+\mathcal{O}(R^{-2})\,,\end{split}$
(138)

where,

$\displaystyle\psi_{\Delta_{1},\tilde{q}_{1},u_{1}}^{-}(x)$
$\displaystyle=\int_{0}^{\infty}d\omega_{1}\,\omega^{\Delta_{1}-1}_{1}e^{i\omega_{1}(\tilde{q}_{1}.x+u_{1})}e^{-\epsilon\omega_{1}}\,,$
(139a) $\displaystyle\psi_{\Delta_{2},\tilde{q}_{2},u_{2}}^{+}(y)$
$\displaystyle=\int_{0}^{\infty}d\omega_{2}\,\omega^{\Delta_{2}-1}_{2}e^{-i\omega_{2}(\tilde{q}_{2}.y+u_{2})}e^{-\epsilon\omega_{2}}\,,$
(139b) $\displaystyle G(x,y)$
$\displaystyle=\int\frac{d^{d+1}k}{(2\pi)^{d+1}}\frac{e^{ik\cdot(x-y)}}{k^{2}+m^{2}+i\epsilon}\,,$
(139c)

and the normalisation constant $N_{\Delta_{i}}^{d}$ is given by

$N_{\Delta_{i}}^{d}=\frac{(\mp
i)^{\Delta_{i}}}{2\pi^{\frac{d}{2}}\Gamma(\Delta_{i}-\frac{d}{2}+1)R^{\frac{(d-1)}{2}-\Delta_{i}}}\,.$
(140)

We see that $\psi^{\pm}_{\Delta_{1},\tilde{q}_{1},u_{1}}(x)$ are just the
Carrollian primary wavefunctions associated with the modified Mellin transform
(15) Banerjee:2018gce . Thus, the S-matrix is covariantly encoded in terms of
the basis of Carrollian primary wavefunctions. In the large $R$ limit, the
bulk to boundary propagator (138) reduces to the Carrollian primary
wavefunction (15) up to some overall normalization constant.

The works of Giddings:1999jq ; Penedones:2010ue ; Fitzpatrick:2011hu ;
Fitzpatrick:2011dm ; Hijano:2019qmi ; Hijano:2020szl ; Li:2021snj  give an
algorithm to extract the bulk S-Matrix in the centre of AdS from AdS/CFT
correlators. Following the lead of deGioia:2022fcn ; Bagchi:2023fbj , we see
that if one keeps track of the boundary symmetries in this limit, we end up
with the correlators relevant for flat space holography. The key to this
construction is the fact that the bulk to boundary propagator (restricted to
specific time slices of the CFT) reduces to the Carrollian primary
wavefunction that encodes flat space scattering in the center of AdS. With
this information, we can now look at Witten diagrams in the large AdS radius
limit.

A general Witten diagram with appropriate vertices, internal lines, and
external lines will be of the form

$\begin{split}\langle
O_{\Delta_{1}}(\textbf{P}_{1})O_{\Delta_{2}}(\textbf{P}_{2})\dots
O_{\Delta_{i}}(\textbf{P}_{i})O_{\Delta_{i+1}}(\textbf{P}_{i+1})O_{\Delta_{i+2}}(\textbf{P}_{i+2})\dots
O_{\Delta_{j}}(\textbf{P}_{j})\rangle\\\
=(ig)^{V}\int_{\text{AdS}_{d+1}}d^{d+1}\textbf{X}_{1}\dots
d^{d+1}\textbf{X}_{V}\Pi_{\Delta}(\textbf{X}_{k},\textbf{X}_{l})\dots\Pi_{\Delta}(\textbf{X}_{k}^{\prime},\textbf{X}_{l}^{\prime})\\\
\textbf{K}_{\Delta_{1}}(\textbf{P}_{1},\textbf{X}_{k})\dots\textbf{K}_{\Delta_{i}}(\textbf{P}_{i},\textbf{X}_{k}^{\prime})\,\textbf{K}_{\Delta_{i+1}}(\textbf{P}_{i+1},\textbf{X}_{l})\dots\textbf{K}_{\Delta_{j}}(\textbf{P}_{j},\textbf{X}_{l}^{\prime})\,,\end{split}$
(141)

where $V$ denotes the number of vertices, and there are an appropriate number
of external lines connected to the bulk to bulk propagator. For example, a
four-point function with the $\phi^{3}$ interaction will be of the form

$\begin{split}\langle
O_{\Delta_{1}}(\textbf{P}_{1})O_{\Delta_{2}}(\textbf{P}_{2})O_{\Delta_{3}}(\textbf{P}_{3})O_{\Delta_{4}}(\textbf{P}_{4})\rangle=&(ig)^{2}\int_{\text{AdS}_{d+1}}d^{d+1}\textbf{X}\,d^{d+1}\textbf{Y}\,\Pi_{\Delta}(\textbf{X},\textbf{Y})\\\
&\textbf{K}_{\Delta_{1}}(\textbf{P}_{1},\textbf{X})\textbf{K}_{\Delta_{3}}(\textbf{P}_{3},\textbf{Y})\textbf{K}_{\Delta_{2}}(\textbf{P}_{2},\textbf{X})\textbf{K}_{\Delta_{4}}(\textbf{P}_{4},\textbf{Y})\,.\end{split}$
(142)

Now suppose, $\textbf{P}_{m}$ are inserted at
$\tau=-\frac{\pi}{2}+\frac{u_{m}}{R}$ for $m=1,\dots,i$ and at
$\tau=\frac{\pi}{2}+\frac{u_{m}}{R}$ for $m=i+1,\dots,j$, then using the form
of the incoming and outgoing wavefunctions given by (138); and also the form
of the bulk to bulk propagator, we can simplify (141) in the large $R$ limit
as

$\begin{split}\langle O_{\Delta_{1}}(\textbf{P}_{1})\dots
O_{\Delta_{i}}(\textbf{P}_{i})O_{\Delta_{i+1}}(\textbf{P}_{i+1})\dots
O_{\Delta_{j}}(\textbf{P}_{j})\rangle=(\prod^{j}_{a=1}N^{d}_{\Delta_{a}})(ig)^{V}\int_{\mathbb{R}^{1,d}}d^{d+1}x_{1}\dots
d^{d+1}x_{V}\\\ \times G(x_{k},,x_{l})\dots
G(x_{k}^{\prime},x_{l}^{\prime})\,\psi^{-}_{\Delta_{1},\tilde{q}_{1},u_{1}}(x_{k})\dots\psi^{-}_{\Delta_{i},\tilde{q}_{i},u_{i}}(x_{k}^{\prime})\\\
\times\psi^{+}_{\Delta_{i+1},\tilde{q}_{i+1},u_{i+1}}(x_{l})\dots\psi^{+}_{\Delta_{j},\tilde{q}_{j},u_{j}}(x_{l}^{\prime})+O(R^{-1})\,.\end{split}$
(143)

####  5.3 Spinning propagators

We now consider spinning fields, and we are interested in studying the large
AdS radius limit of spinning Witten diagrams. The embedding space
representation of AdS is crucial to our flat limit, and the spinning
propagators have been worked out in AdS embedding space representation in
Costa:2014kfa . We borrow those expressions and then implement the flat limit
through (127) and (130). We will consider the bulk to boundary propagator and
the bulk to bulk propagator in the large AdS radius limit. The analysis of
vertices is similar to deGioia:2022fcn ; Bagchi:2023fbj .

#####  5.3.1 Bulk to boundary propagator

The bulk to boundary propagator of a spin $J$ particle with dimension $\Delta$
is given by Costa:2014kfa ; deGioia:2023cbd

$\begin{split}K^{\Delta,J}_{\vec{\alpha},\vec{\beta}}\left(\textbf{P},\textbf{X}\right)=\mathcal{C}_{\Delta;J}\,\partial_{\alpha_{1}}X^{A_{1}}\dots\partial_{\alpha_{J}}X^{A_{J}}\,\partial_{\beta_{1}}P^{B_{1}}\dots\partial_{\beta_{J}}P^{B_{J}}\\\
~{}~{}\times\dfrac{L_{\lfloor A_{1};\lfloor B_{1}}\left(X;P\right)\dots
L_{A_{J}\rfloor;B_{J}\rfloor}\left(X;P\right)}{\left(-P\cdot
X+i\epsilon\right)^{\Delta}}\,,\end{split}$ (144)

where

$L_{A;B}\left(X;P\right)=\dfrac{-P\cdot X\eta_{AB}+P_{A}X_{B}}{-P\cdot
X+i\epsilon}\,,$ (145)

and the normalization constant is given by

$\mathcal{C}^{d}_{\Delta;J}=\dfrac{(J+\Delta-1)\,\Gamma(\Delta)}{2\pi^{\frac{d}{2}}\,(\Delta-1)\,\Gamma\left(\Delta+1-\frac{d}{2}\right)R^{\frac{d-1}{2}-\Delta+J}}\,.$
(146)

The notations used are:

$\begin{split}A_{i},B_{i}&\to\text{embedding space indices}\,,\\\
\alpha_{i}&\to\text{bulk coordinates}\,,\\\ \beta_{i}&\to\text{boundary
coordinates}\,,\\\
\partial_{\alpha_{i}}X^{A_{i}},\partial_{\beta_{i}}P^{B_{i}}&\to\text{projectors
onto the corresponding bulk and boundary tensors}\,,\\\
\lfloor.\rfloor&\to\text{symmetric traceless component}\,.\end{split}$ (147)

Thus, (144) is just the scalar bulk to boundary propagator dressed with
factors of $L_{A;B}(X;P)$.

To implement the flat limit, we first rescale the $\alpha_{i}$ and $\beta_{i}$
such that $\alpha_{i}$ runs over $(t,r\Omega)$ and $\beta_{i}$ runs over
$(u,\Omega)$. One can straightforwardly verify that under a large $R$
$\left(\tau_{p}=\pm\frac{\pi}{2}+\frac{u}{R}\right)$ limit, if we substitute
(132) in (144) we have two structures

$\begin{split}(i)~{}\eta_{AB}\partial_{\alpha}X^{A}\partial_{\beta}P^{B}&=\begin{cases}\mathcal{O}(R^{-2})&\text{$\beta=u$
\, ,}\\\
\pm\partial_{a}\tilde{q}_{\alpha}+\mathcal{O}(R^{-2})&\text{$\beta=z^{a}=z,\bar{z}$
\, ,}\end{cases}\\\
(ii)~{}P_{A}X_{B}\partial_{\alpha}X^{A}\partial_{\beta}P^{B}&=\begin{cases}\pm\tilde{q}_{\alpha}+\mathcal{O}(R^{-2})&\text{$\beta=u$
\, ,}\\\ \pm\left(\partial_{a}\tilde{q}\cdot
x\right)\tilde{q}_{\alpha}+\mathcal{O}(R^{-2})&\text{$\beta=z^{a}=z,\bar{z}$
\, .}\end{cases}\end{split}$ (148)

One should consider the antipodal map (135) for the other $\tau_{p}$. It is
also crucial to rescale the coordinates $\mu_{i}$ and $\nu_{i}$ before
implementing the flat limit, or else terms diverge as $\mathcal{O}(R)$.

From (148), we can basically compute
$\partial_{\alpha}X^{A}\partial_{\beta}P^{B}L_{A;B}$. For $\nu=u$, we have
from (134)

$L_{A;B}\partial_{\alpha}X^{A}\partial_{u}P^{B}=\dfrac{\pm\tilde{q}_{\alpha}}{(\mp
u\mp\tilde{q}\cdot x+i\epsilon)}\,.$ (149)

Thus, for $\beta=u$, the bulk to boundary propagator for spin 1 (144) becomes
pure gauge because

$K^{\Delta,1}_{\alpha,u}=\mathcal{C}^{d}_{\Delta;1}\dfrac{\pm\tilde{q}_{\mu}}{(\mp
u\mp\tilde{q}\cdot
x+i\epsilon)^{\Delta+1}}=-\dfrac{\mathcal{C}^{d}_{\Delta;1}}{\Delta}\partial_{\alpha}\left[\dfrac{1}{(\mp
u\mp\tilde{q}\cdot x+i\epsilon)^{\Delta}}\right]\,.$ (150)

This exactly matches the result of (78).

This is also consistent with the fact that if we take a bulk $U(1)$ gauge
field $A_{\alpha}$ and push it to the boundary $\mathscr{I}^{+}$, the $u$
component drops out as $A_{u}\sim\mathcal{O}(\frac{1}{r})$ but the $z,\bar{z}$
components are non-trivial as $A_{z}\sim A_{\bar{z}}\sim\mathcal{O}(1)$
111111Here $r$ is the usual radial coordinate. The limit to $\mathscr{I}^{+}$
is achieved by keeping $u$ constant and letting $r\to\infty$. Thus, one has to
do a mode expansion of the bulk field and evaluate the boundary limit by a
saddle point approximation. Nguyen:2023vfz . Hence, the scattering data is
solely present in the spatial components. We will see that the
$f_{z{\bar{z}}}$ correlator of (79) agrees with the bulk result (196). We
remark that there might be sub-leading corrections to the bulk to boundary
propagator in the large $R$ limit. We leave a careful analysis for future
work.

For $\beta=z^{a}$,

$L_{A;B}\partial_{\alpha}X^{A}\partial_{a}P^{B}=\pm\partial_{a}\tilde{q}_{\alpha}+\dfrac{\pm(\partial_{a}\tilde{q}\cdot
x)\tilde{q}_{\alpha}}{(\mp u\mp\tilde{q}\cdot x+i\epsilon)}\,.$ (151)

Thus, the bulk to boundary propagator for spin 1 (144) with the boundary index
being that of the points on the Celestial sphere is given by

$K^{\Delta,1}_{\alpha,a}=\mathcal{C}^{d}_{\Delta;1}\left[\dfrac{\pm\partial_{a}\tilde{q}_{\alpha}}{(\mp
u\mp\tilde{q}\cdot
x+i\epsilon)^{\Delta}}+\dfrac{\pm(\partial_{a}\tilde{q}\cdot
x)\tilde{q}_{\alpha}}{(\mp u\mp\tilde{q}\cdot
x+i\epsilon)^{\Delta+1}}\right]\,.$ (152)

This is the modified Mellin avatar of the conformal primary wavefunctions
worked out in Pasterski:2017kqt . This is the crucial difference when compared
to deGioia:2023cbd . This is due to the difference in the large $R$ limit
(130) used, and as we will see, this will significantly change the resulting
Witten diagrams.

The form of the propagator in (152) is difficult to work with in Witten
diagrams due to the explicit appearance of the bulk point $x$. Fortunately, as
explained in Pasterski:2017kqt , one can use the following gauge
representative

$K^{\Delta,1}_{\alpha,a}=\mathcal{C}^{d}_{\Delta;1}\dfrac{\Delta-1}{\Delta}\dfrac{\pm\partial_{a}\tilde{q}_{\alpha}}{(\mp
u\mp\tilde{q}\cdot x+i\epsilon)^{\Delta}}\,.$ (153)

This is gauge equivalent to (152) because we have Donnay:2018neh

$\dfrac{\pm\partial_{a}\tilde{q}_{\alpha}}{(\mp u\mp\tilde{q}\cdot
x+i\epsilon)^{\Delta}}+\dfrac{\pm(\partial_{a}\tilde{q}\cdot
x)\tilde{q}_{\alpha}}{(\mp u\mp\tilde{q}\cdot
x+i\epsilon)^{\Delta+1}}=\dfrac{\Delta-1}{\Delta}\dfrac{\pm\partial_{a}\tilde{q}_{\alpha}}{(\mp
u\mp\tilde{q}\cdot
x+i\epsilon)^{\Delta}}+\partial_{\alpha}\gamma^{\Delta,\pm}_{a}\,,$ (154)

with

$\gamma^{\Delta,\pm}_{a}=\dfrac{\pm\partial_{a}\tilde{q}\cdot x}{\Delta(\mp
u\mp\tilde{q}\cdot x+i\epsilon)^{\Delta}}\,.$ (155)

This is advantageous to us because we have a modified Mellin integral
representation

$K^{\Delta,1}_{\alpha,a}=\dfrac{\mathcal{C}^{d}_{\Delta;1}}{(\pm
i)^{\Delta}\Gamma(\Delta)}\dfrac{\Delta-1}{\Delta}\int^{\infty}_{0}d\omega\,\omega^{\Delta-1}\,e^{\mp
i\omega\,u}e^{\mp i\omega\,\tilde{q}\cdot
x}e^{-\epsilon\,\omega}(\partial_{a}\tilde{q}_{\alpha})\,.$ (156)

From (148), one can also straightforwardly obtain the spin 2 graviton primary
wavefunctions from the large $R$ limit. Here, we should take into account the
fact that the projections in (144) only pick out the symmetric and traceless
component of the dressing functions $L_{A;B}\left(X;P\right)$. Now, the
$\beta_{1}=u,\beta_{2}=u$ component follows from (149):

$\begin{split}&K^{\Delta,2}_{\alpha_{1}\alpha_{2},uu}=\mathcal{C}^{d}_{\Delta;2}\dfrac{\pm\tilde{q}_{\alpha_{1}}\tilde{q}_{\alpha_{2}}}{(\mp
u\mp\tilde{q}\cdot x+i\epsilon)^{\Delta+2}}\\\
&=-\dfrac{\mathcal{C}^{d}_{\Delta;2}}{2(\Delta+1)}\left(\partial_{\alpha_{1}}\left[\dfrac{\tilde{q}_{\alpha_{2}}}{(\mp
u\mp\tilde{q}\cdot
x+i\epsilon)^{\Delta+1}}\right]+\partial_{\alpha_{2}}\left[\dfrac{\tilde{q}_{\alpha_{1}}}{(\mp
u\mp\tilde{q}\cdot x+i\epsilon)^{\Delta+1}}\right]\right)\\\
&=\partial_{\alpha_{1}}\xi_{\alpha_{2}}+\partial_{\alpha_{2}}\xi_{\alpha_{1}}\,.\end{split}$
(157)

Thus, analogous to (150), the $\beta_{1}=u,\beta_{2}=u$ component becomes pure
diffeomorphism. Also, one can use (149), (151) and (154) to argue that the
$\beta_{1}=u,\beta_{2}=z^{a}$ component becomes pure diffeomorphsim analogous
to (157). This can be generalized to higher spin by noting that because of
(149), the bulk to boundary propagator (144) becomes pure gauge when one of
the boundary indices is taken to be $u$.

For the $\beta_{1}=z^{a},\beta_{2}=z^{a}$ term, we can follow a similar
analysis as (154) Pasterski:2017kqt  to show that we can use the Mellin gauge
representative (up to terms that are pure diffeomorphism) given by

$K^{\Delta,2}_{\alpha_{1}\alpha_{2},a_{1}a_{2}}=\mathcal{C}^{d}_{\Delta;2}\dfrac{\Delta-1}{\Delta}\dfrac{\pm
P^{b_{1}b_{2}}_{a_{1}a_{2}}\,\partial_{b_{1}}\tilde{q}_{\alpha_{1}}\,\partial_{b_{2}}\tilde{q}_{\alpha_{2}}}{(\mp
u\mp\tilde{q}\cdot x+i\epsilon)^{\Delta}}\,.$ (158)

Here $P^{b_{1}b_{2}}_{a_{1}a_{2}}$ is the projector given by

$P^{b_{1}b_{2}}_{a_{1}a_{2}}=\delta^{b_{1}}_{(a_{1}}\delta^{b_{2}}_{a_{2})}-\dfrac{1}{d-1}g_{a_{1}a_{2}}g^{b_{1}b_{2}}\,.$
(159)

This projector ensures that the bulk to boundary propagator is transverse and
traceless.

#####  5.3.2 Bulk to bulk propagator

For the bulk to bulk propagator, we simply follow the analysis of
deGioia:2023cbd . The spin $J$ bulk-bulk propagator satisfies the following
equations Costa:2014kfa :

$\begin{split}\left(\Box_{\text{AdS}}-\dfrac{\Delta(\Delta-d)}{R^{2}}+\dfrac{J}{R^{2}}\right)\Pi_{\alpha_{1}\dots\alpha_{J},\beta_{1}\dots\beta_{J}}(X,\bar{X})&=-g_{\alpha_{1}\lfloor\beta_{1}}\dots
g_{|\alpha_{J}|\beta_{J}\rfloor}\delta_{\text{AdS}}\left(X,\bar{X}\right)\,,\\\
\nabla^{\alpha_{1}}\Pi_{\alpha_{1}\dots\alpha_{J},\beta_{1}\dots\beta_{J}}(X,\bar{X})&=0\,.\end{split}$
(160)

One can follow a similar analysis of deGioia:2022fcn ; Bagchi:2023fbj  to
arrive at

$\Pi_{\alpha_{1}\dots\alpha_{J},\beta_{1}\dots\beta_{J}}(X,\bar{X})=G_{\alpha_{1}\dots\alpha_{J},\beta_{1}\dots\beta_{J}}(x,\bar{x})+\mathcal{O}(R^{-2})\,,$
(161)

where $G_{\alpha_{1}\dots\alpha_{J},\beta_{1}\dots\beta_{J}}(x,\bar{x})$ is
the Feynman propagator for a symmetric traceless tensor of spin $J$ in
$\mathbb{R}^{1,d}$.

###  6 Explicit calculations of Witten diagrams

In this section, we will use the analysis of Sec. 5 to construct the AdS
Witten diagrams focussing on spinning particles and obtain a large $R$ series
expansion. We show that the spinning Carrollian CFT correlators of Sec. 4
arise naturally as the leading order term in the spinning equivalent of (143).
To compare with the intrinsic Carrollian CFT correlators, we work with $3+1$
dimensions because $\text{BMS}_{4}$ symmetry group was used to constrain the
correlators living on the null boundary of $\mathbb{R}^{1,3}$ in Sec. 4.
However, we emphasize that the formalism is valid for arbitrary spacetime
dimensions.

In $3+1$ dimensions, we use the following parametrization for the null vector
in the direction of $\mathbf{\Omega}_{p}$:

$\tilde{q}^{\mu}=\left[1,\dfrac{z+\bar{z}}{1+z\bar{z}},\dfrac{-i(z-\bar{z})}{1+z\bar{z}},\dfrac{1-z\bar{z}}{1+z\bar{z}}\right]\,.$
(162)

The topology of the null boundary is $\mathbb{R}\times S^{2}$ and
$(z,\bar{z})$, which characterize the coordinates on the celestial sphere. As
considered in Bagchi:2023fbj , we must carefully ensure that the split of the
momentum conserving delta functions is Lorentz invariant to evaluate Witten
diagrams. In the below subsection, we will summarize the diagrams that we have
computed. We also highlight the salient features of each diagram.

####  6.1 Summary of the diagrams computed

We list the various bulk Witten diagrams (twelve in total) that we have
computed and matched with the field theory analysis of Section 4:

  1. 1.

Scalar three point function: non collinear case
A non-trivial branch of the three point scattering where one of the particles
is soft and the remaining two scatter collinearly. This branch was not
considered in Bagchi:2023fbj . Here, we see how the $\Delta\to 1$ subsector
captures the soft sector.

  2. 2.

Scalar three point function: split signature
Three point scattering amplitude is non-zero for bulk $(2,2)$ split signature
spacetimes. We get a non-trivial result by working in the celestial torus by
treating $z$ and $\bar{z}$ independently.

  3. 3.

Scalar four point function: contact diagram
A contact Witten diagram arising from a $\phi^{4}$ interaction in the bulk
effective theory.

  4. 4.

Scalar four point function: exchange diagram
A t-channel scalar exchange Witten diagram that arises through a $\phi^{3}$
interaction.

  5. 5.

Gluon two point function: The diagram that captures free gluon propagation.

  6. 6.

Scalar-gluon-scalar three point function: split signature
A contact Witten diagram that arises through a cubic interaction vertex
involving a spin 1 field and two scalar fields.

  7. 7.

Gluon MHV three point amplitude: split signature
A contact Witten diagram that arises through a cubic gluon interaction vertex.
We get a finite result, which should be contrasted with the Celestial case
Pasterski:2017ylz ; Stieberger:2018edy , where one has distributional delta
functions in the conformal weights of primaries.

  8. 8.

Gluon three point amplitude from $F^{3}$: split signature
A non-MHV $(+++)/(---)$ amplitude arising from bulk loop corrections. These
amplitudes cannot be captured by the 2D Celestial CFT because they are
quadratically divergent (as they are loop amplitudes) in the Mellin basis.
This is even more severe than the linearly divergent MHV graviton scattering
amplitude Stieberger:2018edy . We capture this scattering in a finite Carroll
CFT correlator.

  9. 9.

Graviton two point function: The diagram that captures free graviton
propagation.

  10. 10.

Scalar-graviton-scalar three point function: split signature
A contact Witten diagram that arises through a cubic interaction vertex
involving a spin 2 field and two scalar fields.

  11. 11.

Graviton MHV three point amplitude: split signature
A contact Witten diagram that arises through a cubic graviton interaction
vertex. Graviton amplitudes are well known to be divergent in the Mellin basis
Stieberger:2018edy . We show that such scattering processes could be captured
by a finite Carroll CFT correlator Banerjee:2019prz .

  12. 12.

Graviton-gluon-gluon three point function: split signature
A contact Witten diagram that arises in Einstein-Yang-Mills theory with a
vertex involving a spin 2 field and two spin 1 fields.

We now proceed to evaluate all the diagrams.

####  6.2 Scalar three point function: non collinear case

We consider the contact Witten diagram in $\phi^{3}$ theory:

$\langle
O_{\Delta_{1}}(\textbf{P}_{1})O_{\Delta_{2}}(\textbf{P}_{2})O_{\Delta_{3}}(\textbf{P}_{3})\rangle=\int_{\text{AdS}_{4}}d\textbf{X}\,\textbf{K}_{\Delta_{1}}(\textbf{P}_{1},\textbf{X})\,\textbf{K}_{\Delta_{2}}(\textbf{P}_{2},\textbf{X})\,\textbf{K}_{\Delta_{3}}(\textbf{P}_{3},\textbf{X})\,.$
(163)

In the large $R$ limit, with $\textbf{P}_{1}$ inserted at
$\tau_{p}=-\frac{\pi}{2}+\frac{u_{1}}{R}$ and $\textbf{P}_{2}$ and
$\textbf{P}_{3}$ inserted at $\tau_{p}=\frac{\pi}{2}+\frac{u_{i}}{R}$, we have

$\begin{split}\langle
O_{\Delta_{1}}(\textbf{P}_{1})O_{\Delta_{2}}(\textbf{P}_{2})O_{\Delta_{3}}(\textbf{P}_{3})\rangle&=\mathcal{A}_{(3)s}\int_{\mathbb{R}^{1,3}}d^{4}x\,\psi^{-}_{\Delta_{1},\tilde{q}_{1},u_{1}}(x)\,\psi^{+}_{\Delta_{2},\tilde{q}_{2},u_{2}}(x)\,\psi^{+}_{\Delta_{3},\tilde{q}_{3},u_{3}}(x)\\\
&=\mathcal{A}_{(3)s}\int^{\infty}_{0}d\omega_{1}d\omega_{2}d\omega_{3}\,\omega^{\Delta_{1}-1}_{1}\omega^{\Delta_{2}-1}_{2}\omega^{\Delta_{3}-1}_{3}e^{i\omega_{1}\,u_{1}-i\omega_{2}\,u_{2}-i\omega_{3}\,u_{3}}\\\
&\hskip
142.26378pt\times\delta^{(4)}(\omega_{1}\tilde{q}_{1}-\omega_{2}\tilde{q}_{2}-\omega_{3}\tilde{q}_{3})\,.\end{split}$
(164)

The normalization constant is given by

$\mathcal{A}_{(3)s}=(i\mu)N^{3}_{\Delta_{1}}N^{3}_{\Delta_{2}}N^{3}_{\Delta_{3}}=\dfrac{(i\mu)\,(i)^{\Delta_{1}}(-i)^{\Delta_{2}+\Delta_{3}}}{8\pi^{\frac{9}{2}}\prod^{3}_{i=1}\Gamma\left(\Delta_{i}-\frac{1}{2}\right)R^{3-\Delta_{1}-\Delta_{2}-\Delta_{3}}}\,.$
(165)

We will use a soft factorization of the momentum conserving delta function to
get a non zero answer. This branch was not considered in Bagchi:2023fbj . The
split of the delta function is given by

$\delta^{(4)}(\omega_{1}\tilde{q}_{1}-\omega_{2}\tilde{q}_{2}-\omega_{3}\tilde{q}_{3})=\dfrac{1}{\omega^{2}_{3}}\dfrac{1}{z_{12}\bar{z}_{12}}\delta(\omega_{2})\delta(\omega_{1}-\omega_{3})\delta^{2}(z_{13})\,.$
(166)

Because of the crucial soft factor $\delta(\omega_{2})$, we have

$\int^{\infty}_{0}d\omega_{2}\,\omega^{\Delta_{2}-1}_{2}e^{i\omega_{2}\,u_{2}}\delta(\omega_{2})=\delta_{\Delta_{2},1}\,.$
(167)

Due to $\delta(\omega_{2})$, we need to regularize the integral by choosing a
small negative value for the lower bound as $-\epsilon$ and then let
$\epsilon\to 0$. The remaining integrals of (164) can be straightforwardly
simplified to

$\langle
O_{\Delta_{1}}(\textbf{P}_{1})O_{\Delta_{2}}(\textbf{P}_{2})O_{\Delta_{3}}(\textbf{P}_{3})\rangle=\mathcal{A}_{(3)s}\dfrac{\Gamma(\Delta_{1}+\Delta_{3}-3)\delta_{\Delta_{2},1}\,\delta^{2}(z_{13})}{\left(i(u_{1}-u_{3})^{\Delta_{1}+\Delta_{3}-3}\right)\,z_{12}\,\bar{z}_{12}}\,.$
(168)

If we compare comparing with the intrinsic result given by (92), we have

$c=3-\Delta_{1}-\Delta_{3}\,,~{}~{}~{}p=-1\,,~{}~{}~{}q=-1\,.$ (169)

This exactly matches with the intrinsic result (92) (as $\Delta_{2}=1$) from
which we have

$\Delta_{2}=-\dfrac{q+p}{2}\,,~{}~{}~{}\Delta_{1}+\Delta_{3}=2-c-p\,.$ (170)

####  6.3 Scalar three point function: split signature

The three point function is also non-zero when we consider the split signature
for the space-time Pasterski:2017ylz . This branch of the three point function
was not considered in Bagchi:2023fbj . We consider the following split of the
momentum conserving delta function that is suited for the celestial torus
Stieberger:2018edy

$\delta^{(4)}(\omega_{1}\tilde{q}_{1}+\omega_{2}\tilde{q}_{2}-\omega_{3}\tilde{q}_{3})=\dfrac{4}{\omega^{2}_{3}z_{23}z_{31}}\delta\left(\omega_{1}-\omega_{3}\dfrac{z_{32}}{z_{12}}\right)\delta\left(\omega_{2}-\omega_{3}\dfrac{z_{31}}{z_{21}}\right)\delta(\bar{z}_{13})\delta(\bar{z}_{23})\,.$
(171)

As explained in Pasterski:2017ylz ; Stieberger:2018edy , we must choose one of
the orderings $z_{1}<z_{3}<z_{2}$ or $z_{2}<z_{3}<z_{1}$ to ensure that the
energies are positive. The proof of the Lorentz invariance of the split is
given in Appendix A.4. Consider the three point contact Witten diagram in
$\phi^{3}$ theory:

$\langle
O_{\Delta_{1}}(\textbf{P}_{1})O_{\Delta_{2}}(\textbf{P}_{2})O_{\Delta_{3}}(\textbf{P}_{3})\rangle=\int_{\text{AdS}_{4}}d\textbf{X}\,\textbf{K}_{\Delta_{1}}(\textbf{P}_{1},\textbf{X})\,\textbf{K}_{\Delta_{2}}(\textbf{P}_{2},\textbf{X})\,\textbf{K}_{\Delta_{3}}(\textbf{P}_{3},\textbf{X})\,.$
(172)

In the large $R$ limit, with $\textbf{P}_{1}$ and $\textbf{P}_{2}$ inserted at
$\tau_{p}=-\frac{\pi}{2}+\frac{u_{i}}{R}$ and $\textbf{P}_{3}$ inserted at
$\tau_{p}=\frac{\pi}{2}+\frac{u_{3}}{R}$, we have

$\begin{split}\langle
O_{\Delta_{1}}(\textbf{P}_{1})O_{\Delta_{2}}(\textbf{P}_{2})O_{\Delta_{3}}(\textbf{P}_{3})\rangle&=\mathcal{A}_{(3)sp}\int_{\mathbb{R}^{1,3}}d^{4}x\,\psi^{-}_{\Delta_{1},\tilde{q}_{1},u_{1}}(x)\,\psi^{-}_{\Delta_{2},\tilde{q}_{2},u_{2}}(x)\,\psi^{+}_{\Delta_{3},\tilde{q}_{3},u_{3}}(x)\\\
&=\mathcal{A}_{(3)sp}\int^{\infty}_{0}d\omega_{1}d\omega_{2}d\omega_{3}\,\omega^{\Delta_{1}-1}_{1}\omega^{\Delta_{2}-1}_{2}\omega^{\Delta_{3}-1}_{3}e^{i\omega_{1}u_{1}+i\omega_{2}u_{2}-i\omega_{3}u_{3}}\\\
&\hskip
142.26378pt\times\delta^{(4)}(\omega_{1}\tilde{q}_{1}+\omega_{2}\tilde{q}_{2}-\omega_{3}\tilde{q}_{3})\,.\end{split}$
(173)

Using the celestial torus split of the delta function and evaluating the
resulting integrals, we get

$\begin{split}\langle
O_{\Delta_{1}}(\textbf{P}_{1})O_{\Delta_{2}}(\textbf{P}_{2})O_{\Delta_{3}}(\textbf{P}_{3})\rangle&=\mathcal{A}_{(3)sp}(-1)^{\Delta_{1}+\Delta_{2}-2}\Gamma(\Delta_{1}+\Delta_{2}+\Delta_{3}-4)\,\delta(\bar{z}_{12})\,\delta(\bar{z}_{23})\\\
&\times\left[-i(z_{1}u_{23}+z_{2}u_{31}+z_{3}u_{12})\right]^{4-\Delta_{1}-\Delta_{2}-\Delta_{3}}z^{\Delta_{3}-2}_{12}z^{\Delta_{1}-2}_{32}z^{\Delta_{2}-2}_{31}\,.\end{split}$
(174)
$\mathcal{A}_{(3)sp}=(i\mu)N^{3}_{\Delta_{1}}N^{3}_{\Delta_{2}}N^{3}_{\Delta_{3}}=\dfrac{(i\mu)\,(i)^{\Delta_{1}+\Delta_{2}}(-i)^{\Delta_{3}}}{8\pi^{\frac{9}{2}}\prod^{3}_{i=1}\Gamma\left(\Delta_{i}-\frac{1}{2}\right)R^{3-\Delta_{1}-\Delta_{2}-\Delta_{3}}}\,.$
(175)

(174) matches the intrinsic result (106) once we use the fact that we are
working with scalars with zero helicities ($\sigma_{i}=0$). The calculations
leading to (174) are given in Appendix A.2. We will see that one can extend
this by considering three point gluon and graviton amplitudes. In that case,
non-trivial helicities will play a role.

Note that the signature of AdS4 for the bulk spacetime is still $(-,+,+,+)$.
The three point function is non-zero because the momenta of the particles
become complex valued Pasterski:2017ylz .

####  6.4 Scalar four point function: contact diagram

We now consider a couple of four point scalar diagrams. Let us first evaluate
the contact Witten diagram for the four point function given by

$\begin{split}\langle
O_{\Delta_{1}}(\textbf{P}_{1})O_{\Delta_{2}}(\textbf{P}_{2})&O_{\Delta_{3}}(\textbf{P}_{3})O_{\Delta_{4}}(\textbf{P}_{4})\rangle\\\
&=(ig_{2})\int_{AdS_{4}}d^{4}\textbf{X}\,\textbf{K}_{\Delta_{1}}(\textbf{P}_{1},\textbf{X})\textbf{K}_{\Delta_{2}}(\textbf{P}_{2},\textbf{X})\textbf{K}_{\Delta_{3}}(\textbf{P}_{3},\textbf{X})\textbf{K}_{\Delta_{4}}(\textbf{P}_{4},\textbf{X})\,.\end{split}$
(176)

We need a $\phi^{4}$ interaction in the bulk for this contact diagram. We will
now insert $\textbf{P}_{1}$ and $\textbf{P}_{2}$ at
$\tau=-\frac{\pi}{2}+\frac{u_{i}}{R}$ and $\textbf{P}_{3}$ and
$\textbf{P}_{4}$ at $\tau=\frac{\pi}{2}+\frac{u_{i}}{R}$. Thus, we have

$\begin{split}\langle
O_{\Delta_{1}}(\textbf{P}_{1})&O_{\Delta_{2}}(\textbf{P}_{2})O_{\Delta_{3}}(\textbf{P}_{3})O_{\Delta_{4}}(\textbf{P}_{4})\rangle\\\
&=\mathcal{A}_{(4)c}\int
d\omega_{1}\,d\omega_{2}\,d\omega_{3}\,d\omega_{4}\,e^{i(\omega_{1}u_{1}+\omega_{2}u_{2}-\omega_{3}u_{3}-\omega_{4}u_{4})}(ig_{2})\\\
&\times\delta^{4}(\omega_{3}\tilde{q}_{3}+\omega_{4}\tilde{q}_{4}-\omega_{1}\tilde{q}_{1}-\omega_{2}\tilde{q}_{2})\,\omega^{\Delta_{1}-1}_{1}\omega^{\Delta_{2}-1}_{2}\omega^{\Delta_{3}-1}_{3}\omega^{\Delta_{4}-1}_{4}e^{-\epsilon(\omega_{1}+\omega_{2}+\omega_{3}+\omega_{4})}\,.\end{split}$
(177)

This just becomes the modified Mellin transform of the flat space scattering
amplitude $ig_{2}$. The integrals in $\omega_{i}$ can be simplified by
changing variables to simplex variables as described in Pasterski:2017ylz ;
Banerjee:2019prz . This is reviewed in Appendix A.1. The expression of (177)
can be simplified to

$\begin{split}\dfrac{\mathcal{A}_{(4)c}\,ig_{2}\,\Gamma(\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4}-4)}{\left[-i(\sigma^{*}_{1}u_{1}+\sigma^{*}_{2}u_{2}-\sigma^{*}_{3}u_{3}-\sigma^{*}_{4}u_{4})\right]^{\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4}-4}}\prod_{i=1}^{4}(\sigma^{*}_{i})^{\Delta_{i}-1}\dfrac{\delta(|z-\bar{z}|)}{4z_{12}\bar{z}_{13}z_{24}\bar{z}_{24}}\prod_{i=1}^{4}\mathbbm{1}_{[0,1]}(\sigma^{*}_{i})\,.\end{split}$
(178)

If we had set $u=0$ (the Celestial case deGioia:2022fcn ) before the
calculation, we would not have gotten the $u$ dependent factors. The $u$
independent factors agree with the equation above eq(3.11) of Banerjee:2017jeg
. The constant of proportionality is given by

$\mathcal{A}_{(4)c}=\dfrac{(-i)^{\Delta_{1}+\Delta_{2}}(i)^{\Delta_{3}+\Delta_{4}}}{16\pi^{6}\prod_{i=1}^{4}\Gamma\left(\Delta_{i}-\frac{1}{2}\right)R^{4-(\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4})}}\,.$
(179)

The indicator function is defined as

$\mathbbm{1}_{[0,1]}(x)=\begin{cases}1\,,&\text{if $x\in[0,1]$}\\\
0\,,&\text{otherwise}\end{cases}$ (180)

$z$ and $\bar{z}$ are the cross ratios given by

$z=\dfrac{z_{12}z_{34}}{z_{13}z_{24}}\,,~{}~{}~{}~{}~{}\bar{z}=\dfrac{\bar{z}_{12}\bar{z}_{34}}{\bar{z}_{13}\bar{z}_{24}}\,.$
(181)

The remaining terms in (178) are given by