# Massive celestial amplitudes and celestial amplitudes beyond four points

Reiko Liua, Wen-Jie Mab,c

###### Abstract

We compute scalar three-point celestial amplitudes involving two and three
massive scalars. The three-point coefficient of celestial amplitudes with two
massive scalars contains a hypergeometric function, and the one with three
massive scalars can be represented as a triple Mellin-Barnes integral. Using
these three-point celestial amplitudes, we investigate the conformal block
expansions of five- and six-point scalar celestial amplitudes in the comb
channel. We observe the presence of two-particle operators in the conformal
block expansion of five-point celestial amplitudes, which confirms the
previous analysis by taking multi-collinear limit. Moreover, we find that
there are new three-particle operators in the conformal block expansion of
six-point celestial amplitudes. Based on these findings, we conjecture that
exchanges of $n$-particle operators can be observed by considering the comb
channel conformal block expansion of $(n+3)$-point massless celestial
amplitudes. Finally, we show that a new series of operators appears when
turning on the mass of the first incoming particle. The leading operator in
this series can be interpreted as a two-particle exchange in the OPE of one
massive and one massless scalars.

aYau Mathematical Sciences Center (YMSC), Tsinghua University, Beijing,
100084, China

bShanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS),
Shanghai, 200433, China

cFudan Center for Mathematics and Interdisciplinary Study, Fudan University,
Shanghai, 200433, China

<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Notations and conventions
    2. 2.2 Shadow formalism
    3. 2.3 Conformal primary wavefunctions and celestial amplitudes
    4. 2.4 Split representation of Feynman propagators
  3. 3 Conformal partial wave expansions
    1. 3.1 Four-point conformal partial waves
    2. 3.2 Higher-point conformal partial waves
  4. 4 Three-point celestial amplitudes
    1. 4.1 Massive-massless-massive scalars
      1. 4.1.1 Sanity checks
    2. 4.2 Massive-massive-massive scalars
      1. 4.2.1 Mass threshold expansion
      2. 4.2.2 Massless limit
  5. 5 Four-point celestial amplitudes
    1. 5.1 Warm up: four massless scalars with massless exchange
    2. 5.2 Two massive and two massless scalars with massive exchange
  6. 6 Higher-point celestial amplitudes
    1. 6.1 Five massless scalars with massless exchange
    2. 6.2 One massive and four massless scalars
      1. 6.2.1 Massless exchanges
      2. 6.2.2 One massive exchange
    3. 6.3 Six massless scalars with massless exchange
  7. 7 Conclusion
  8. A Useful formulas
    1. A.1 EAdS integrals
    2. A.2 Analytic functional
  9. B Tables of operator spectra

##  1 Introduction

The celestial holography program suggests that a $(d+2)$-dimensional quantum
gravity in asymptotic Minkowski spacetime is dual to a hypothetical
$d$-dimensional celestial conformal field theory (celestial CFT, CCFT) on the
celestial sphere [1, 2, 3]. In this duality, the celestial amplitudes - the
bulk scattering amplitudes expanded in the conformal basis - should serve as
correlation functions in the boundary CCFT.

As a CFT, it is expected that all the information of CCFT is encoded in the
operator product expansion (OPE), which can be extracted from the three-point
celestial amplitudes and the conformal block expansions of four- and higher-
point celestial amplitudes. The three-point celestial amplitudes of three
massless scalars were computed in [4] and the ones involving one massive
scalar were computed in [5]. In CCFT2, the three-point celestial amplitudes of
three massive scalars with mass $m$, $m$, and $2m(1+\epsilon)$ were obtained
in [1] to the leading order of $\epsilon$.111By performing the integral in
position space, authors of [6] computed scalar three-point celestial
amplitudes of three massive scalars with generic mass $m_{1}$, $m_{2}$ and
$m_{3}$ in CCFTd. However, when setting $m_{1}=m_{2}=m$,
$m_{3}=2m(1+\epsilon)$ and $d=2$, their results differ from the results of [1]
by a term that is regular in $\epsilon$. However, the complete results of
three-point celestial amplitudes involving more than one massive scalar remain
unknown.

For the conformal block expansions, the authors of [5, 7, 8, 9, 10, 11, 12,
13, 14, 15] have investigated the conformal partial wave expansions and/or
conformal block expansions of four-point massless scalar amplitudes. It is
worth mentioning that four-point massless celestial amplitudes, which are
expanded in the conformal primary basis given by a Mellin transform on the
plane waves, do not have proper conformal block expansions [8]. This issue can
be solved by choosing the conformal basis and shadow conformal basis for the
incoming and outgoing particles respectively [4]. Furthermore, in order to
obtain the complete OPE, it is necessary to further explore the conformal
block expansions of celestial amplitudes involving massive particles.
Unfortunately, such conformal block expansions are still absent in the
literature since the computations are more involved.

Moreover, recent studies have revealed the presence of two-particle operators
in the leading order OPE of two massless operators, in addition to the single-
particle operators [16, 17]. These two-particle exchanges can only be observed
when higher-point celestial amplitudes are considered. Following the work of
[16, 17], two questions arise. Firstly, are there other types of exchanged
operators appearing in the OPE of two massless operators? Secondly, do the
two-particle exchanges also appear in the OPE of massive operators? The
answers to these questions can be obtained by studying the conformal block
expansions of higher-point celestial amplitudes involving massless and/or
massive particles. Unfortunately, obtaining these expansions is a challenging
task, and no results are currently available in the literature.

The split representation introduced in [13] is a powerful tool to
systematically compute the conformal block expansions of the celestial
amplitudes. The split representation can be derived by dividing the Fourier
integral in momentum space into integrals over the regions inside and outside
the lightcone at the origin, where the regions inside the past or future
lightcones are foliated by $(d+1)$-dimensional Euclidean anti-de Sitter (EAdS)
slices, and the region outside the lightcones is foliated by
$(d+1)$-dimensional de Sitter (dS) slices.222Similar foliation has been
recently discussed in [18, 19] to obtain the leaf amplitudes. Using the split
representation, the Feynman propagator can be factorized into a product of two
conformal primary wavefunctions with a common boundary point integrated over
the celestial sphere. This decomposition enables the direct derivation of the
conformal partial wave expansions for celestial amplitudes. Subsequently, it
is straightforward to obtain the conformal block expansions by the standard
techniques in CFT.

In this paper, we focus on the scattering process of scalar particles.
Firstly, we calculate the three-point celestial amplitudes involving more than
one massive scalar. Using these three-point celestial amplitudes and the split
representation, we then investigate the conformal block expansion of the four-
point celestial amplitude involving two massive scalars. Additionally, we
obtain the conformal block expansions of five- and six-point massless
celestial amplitudes by choosing the conformal basis and shadow conformal
basis for the incoming and outgoing particles, respectively. Finally, we
determine the conformal block expansions of five-point celestial amplitudes
involving one massive scalar.

The rest of this paper is organized as follows. Section 2 sets up conventions,
and reviews the shadow formalism, the dictionary of celestial holography and
the split representation. Section 3 derives higher-point conformal partial
wave expansions in the comb channel. Section 4 evaluates the three-point
coefficients of massive celestial amplitudes. Section 5 and Section 6 discuss
conformal block expansions of four- and higher-point celestial amplitudes.
Section 7 ends with concluding remarks. Appendix A collects useful formulas.
Appendix B contains tables of operator spectra.

##  2 Preliminaries

###  2.1 Notations and conventions

Throughout this paper we adopt the following notations and conventions. We
choose the most positive signature for the $(d+2)$-dimensional Minkowski
spacetime $\mathbb{R}^{1,d+1}$, _i.e_.,

$\displaystyle g_{\mu\nu}=\operatorname{diag}(-,+,\dots,+)\;.$ (2.1)

The bulk point in the Minkowski spacetime is denoted as
$X\in\mathbb{R}^{1,d+1}$, while the boundary point on the celestial sphere is
denoted as $x\in\mathbb{R}^{d}$. The difference between two boundary points
$x_{i}$ and $x_{j}$ is denoted as

$\displaystyle x_{ij}\equiv x_{i}-x_{j}\;.$ (2.2)

For a $(d+2)$-dimensional scattering process, we denote the $i$-th, mass $m$,
helicity/spin $s$ particle as $i^{s}_{m}$. If taking the shadow transform on
the $i$-th particle, we denote it as $\mathcal{S}[i^{s}_{m}]$. For example,
the $2\to 2$ celestial amplitude of four massless scalars is denoted as

$\mathcal{A}_{1^{0}_{0}+2^{0}_{0}\to 3^{0}_{0}+4^{0}_{0}}\;,$ (2.3)

and if choosing the shadow basis for outgoing states, it is denoted as

$\mathcal{A}_{1^{0}_{0}+2^{0}_{0}\to\mathcal{S}[3^{0}_{0}]+\mathcal{S}[4^{0}_{0}]}\;.$
(2.4)

For a $d$-dimensional CFT, we denote the coordinate-dependence of the standard
conformal three-point functions as

$\displaystyle\langle\mathcal{O}_{i}\mathcal{O}_{j}\mathcal{O}_{k}\rangle=|x_{12}|^{\Delta_{3,12}}|x_{13}|^{\Delta_{2,13}}|x_{23}|^{\Delta_{1,23}}\;.$
(2.5)

Here the default position and conformal dimension of the primary operator
$\mathcal{O}_{i}$ are $x_{i}$ and $\Delta_{i}$ respectively, _i.e_.,
$\mathcal{O}_{i}\equiv\mathcal{O}_{\Delta_{i}}(x_{i}),\,\widetilde{\mathcal{O}}_{i}\equiv\mathcal{O}_{d-\Delta_{i}}(x_{i})$.
Moreover, we introduce the shadow and/or multiple conformal dimensions as

$\displaystyle\widetilde{\Delta}_{a}\equiv d-\Delta_{a}\;,$ (2.6)
$\displaystyle\Delta_{a_{1}\cdots a_{n}}\equiv\sum_{i=1}^{n}\Delta_{a_{i}}\;,$
(2.7) $\displaystyle\Delta_{a_{1}\cdots a_{n},b_{1}\cdots
b_{m}}\equiv\sum_{i=1}^{n}\Delta_{a_{i}}-\sum_{j=1}^{m}\Delta_{b_{j}}\;,$
(2.8)

and the multiple $\Gamma$-symbols as

$\displaystyle\Gamma[a_{1},\cdots,a_{n}]\equiv\prod_{i=1}^{n}\Gamma[a_{i}]\;,$
(2.9)
$\displaystyle\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{a_{1},\cdots,a_{n}}{b_{1},\cdots,b_{m}}\biggr{]}\equiv\left.\prod_{i=1}^{n}\Gamma[a_{i}]\middle/\prod_{j=1}^{m}\Gamma[b_{j}]\right.\;.$
(2.10)

###  2.2 Shadow formalism

The shadow formalism is a widely used technique in CFT. It’s based on the
representation theory and harmonic analysis of conformal groups, and allows us
to conveniently compute conformal partial waves and conformal blocks [20, 21,
22, 23, 24, 25, 26].

In Euclidean CFTd, the shadow transform of a (scalar) primary operator is
defined as

$\mathcal{S}[\mathcal{O}_{\Delta}](x)=N_{\Delta}\int
d^{d}x^{\prime}\,|x-x^{\prime}|^{-2\widetilde{\Delta}}\mathcal{O}_{\Delta}(x^{\prime})\;,$
(2.11)

where the normalization factor $N_{\Delta}$ is to ensure that the double
shadow transform is equal to the identity
$\mathcal{S}^{2}[\mathcal{O}_{\Delta}]=\mathcal{O}_{\Delta}$,

$N_{\Delta}=\pi^{-\frac{d}{2}}\frac{\Gamma[\Delta]}{\Gamma[\Delta-\frac{d}{2}]}\;.$
(2.12)

For a local primary operator $\mathcal{O}_{\Delta}(x)$, the non-local operator
$\mathcal{S}[\mathcal{O}_{\Delta}](x)$ transforms as a primary operator with
conformal dimension $\widetilde{\Delta}$. This implies that
$\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{S}[\mathcal{O}_{3}]\rangle$ is
proportional to
$\langle\mathcal{O}_{1}\mathcal{O}_{2}\widetilde{\mathcal{O}}_{3}\rangle$:

$\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{S}[\mathcal{O}_{3}]\rangle=N_{\Delta}S^{\Delta_{1},\Delta_{2}}_{\Delta_{3}}\langle\mathcal{O}_{1}\mathcal{O}_{2}\widetilde{\mathcal{O}}_{3}\rangle\;,$
(2.13)

where the shadow coefficient $S^{\Delta_{1},\Delta_{2}}_{\Delta_{3}}$ is

$S^{\Delta_{1},\Delta_{2}}_{\Delta_{3}}=\pi^{\frac{d}{2}}\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\Delta_{3}-\frac{d}{2},\frac{d+\Delta_{1,23}}{2},\frac{d+\Delta_{2,31}}{2}}{d-\Delta_{3},\frac{\Delta_{23,1}}{2},\frac{\Delta_{31,2}}{2}}\biggr{]}\;.$
(2.14)

###  2.3 Conformal primary wavefunctions and celestial amplitudes

In a $d$-dimensional CCFT, the celestial amplitudes are defined by expanding
the scattering amplitudes in the conformal basis [1, 27], instead of in the
usual plane-wave basis. The conformal basis can be thought as a bulk-to-
boundary propagator in CCFTd since it connects a bulk point $X$ in the
$(d+2)$-dimensional Minkowski spacetime and a boundary point $x$ on the
$d$-dimensional celestial sphere [13].

Given a $n\to m$ amputated scattering amplitude $\mathcal{M}(X_{i})$ of
scalars in the position space, the celestial amplitude $\mathcal{A}(x_{i})$ is
defined as

$\displaystyle\mathcal{A}(x_{i})=\bigg{(}\prod_{i=1}^{n+m}\int
d^{d+2}X_{i}\bigg{)}\bigg{(}\prod_{i=1}^{n}\phi_{\Delta_{i}}^{+}(x_{i};X_{i})\bigg{)}\bigg{(}\prod_{i=n+1}^{n+m}\phi^{-}_{\Delta_{i}}(x_{i};X_{i})\bigg{)}\mathcal{M}(X_{i})\;,$
(2.15)

where $\phi^{\pm}_{\Delta}(x;X)$-s are the incoming $(+)$ and outgoing $(-)$
scalar conformal primary wavefunctions with the conformal dimension $\Delta$.
For example, the $2\to 2$ contact celestial amplitude of massless scalars is
given by

$\displaystyle\mathcal{A}^{\Delta_{i}}_{1^{0}_{0}+2^{0}_{0}\to
3^{0}_{0}+4^{0}_{0}}(x_{i})=\int
d^{d+2}X\,\phi^{+}_{\Delta_{1}}(x_{1},X)\phi^{+}_{\Delta_{2}}(x_{2},X)\phi^{-}_{\Delta_{3}}(x_{3},X)\phi^{-}_{\Delta_{4}}(x_{4},X)\;.$
(2.16)

The conformal primary wavefunctions $\phi^{\pm}_{\Delta,m}(x;X)$ and
$\phi^{\pm}_{\Delta}(x;X)$ for massive and massless scalars are

$\displaystyle\phi^{\pm}_{\Delta,m}(x;X)=\int\frac{d^{d+1}\widehat{p}}{\widehat{p}^{0}}\,(-\widehat{q}(x)\cdot\widehat{p})^{-\Delta}e^{\pm
im\widehat{p}\cdot X}\;,$ (2.17)
$\displaystyle\phi^{\pm}_{\Delta}(x;X)=\int_{0}^{+\infty}d\omega\,\omega^{\Delta-1}e^{\pm
i\omega\widehat{q}(x)\cdot X-\epsilon\omega}=(\mp
i)^{\Delta}\Gamma[\Delta](-\widehat{q}\cdot X\mp i\epsilon)^{-\Delta}\;,$
(2.18)

where $m$ is the mass of the massive scalar, and $p^{\mu}$ and $q^{\mu}$ are
massive and massless on-shell momenta with the parameterizations

$\displaystyle
p^{\mu}(m,y,x)=m\widehat{p}^{\mu}(y,x)=\frac{m}{2y}(1+y^{2}+x^{2},2x,1-y^{2}-x^{2})\;,$
(2.19) $\displaystyle
q^{\mu}(\omega,x)=\omega\widehat{q}^{\mu}(x)=\omega(1+x^{2},2x,1-x^{2})\;.$
(2.20)

There is another set of conformal primary wavefunctions given by the shadow
transform of the wavefunctions in (2.17) and (2.18). As shown in [27], the
shadow transform of the massive scalar conformal primary wavefunction
$\phi^{\pm}_{\Delta,m}(x;X)$ is simply
$\phi^{\pm}_{\widetilde{\Delta},m}(x;X)$, _i.e_.,

$\displaystyle\mathcal{S}[\phi^{\pm}_{\Delta,m}](x;X)=\phi^{\pm}_{\widetilde{\Delta},m}(x;X)\;.$
(2.21)

The shadow transform of the massless scalar conformal primary wavefunction
$\phi^{\pm}_{\Delta}(x;X)$ is

$\displaystyle\mathcal{S}[\phi_{\Delta}^{\pm}](x;X)=N_{\Delta}\int
d^{d}x^{\prime}|x-x^{\prime}|^{-2\widetilde{\Delta}}\phi_{\Delta}^{\pm}(x^{\prime};X)=\frac{(\mp
i)^{\Delta}\Gamma[\Delta]}{(\mp
i)^{\widetilde{\Delta}}\Gamma[\widetilde{\Delta}]}(-X^{2})^{\frac{d}{2}-\Delta}\phi_{\widetilde{\Delta}}^{\pm}(x;X)\;.$
(2.22)

where the normalization factor $N_{\Delta}$ is given in (2.12).

Starting with the massive conformal primary wavefunctions
$\phi^{\pm}_{\Delta,m}$, one can reproduce either the massless conformal
primary wavefunctions $\phi^{\pm}_{\Delta}$ or the massless shadow conformal
primary wavefunctions $\mathcal{S}[\phi^{\pm}_{\widetilde{\Delta}}]$ by taking
the massless limit [4]:

$\displaystyle\phi_{\Delta}^{\pm}=\lim_{m\to
0}2^{-\Delta}\pi^{-\frac{d}{2}}m^{\Delta}\frac{\Gamma[\Delta]}{\Gamma[\Delta-\frac{d}{2}]}\phi_{\Delta,m}^{\pm}\;,$
(2.23) $\displaystyle\mathcal{S}[\phi_{\widetilde{\Delta}}^{\pm}]=\lim_{m\to
0}2^{-\widetilde{\Delta}}\pi^{-\frac{d}{2}}m^{\widetilde{\Delta}}\frac{\Gamma[\widetilde{\Delta}]}{\Gamma[\widetilde{\Delta}-\frac{d}{2}]}\phi_{\Delta,m}^{\pm}\;.$
(2.24)

###  2.4 Split representation of Feynman propagators

The split representation of the Feynman propagator in the Minkowski spacetime
has been derived in [13]. It enables us to rewrite the Feynman propagator as a
product of a pair of (tachyonic) massive conformal bases with conformal
dimensions $\Delta$ and $\widetilde{\Delta}$, and with the common boundary
point, the conformal dimension $\Delta$ and the (imaginary) mass integrated.
For example, the scalar Feynman propagator in position space

$\displaystyle\mathcal{K}_{m}(X_{1},X_{2})=\int\frac{d^{d+2}p}{(2\pi)^{d+2}}\frac{i}{p^{2}+m^{2}-i\epsilon}e^{-ip\cdot(X_{1}-X_{2})}$
(2.25)

can be rewritten as

$\displaystyle\begin{split}\mathcal{K}_{m}(X_{1},X_{2})&=\int_{0}^{+\infty}\frac{dM}{(2\pi)^{d+2}}\frac{iM^{d+1}}{-M^{2}+m^{2}}\int_{\frac{d}{2}}^{\frac{d}{2}+i\infty}\frac{d\Delta}{2\pi
i\,\mu_{\Delta}}\int
d^{d}x\,\left(\phi^{-}_{\Delta,M}\phi^{+}_{\widetilde{\Delta},M}+\phi^{+}_{\Delta,M}\phi^{-}_{\widetilde{\Delta},M}\right)\\\
&\phantom{{}={}}+\frac{1}{2}\int_{0}^{\infty}\frac{dM}{(2\pi)^{d+2}}\frac{iM^{d+1}}{M^{2}+m^{2}}\bigg{(}\sum_{\epsilon=0,1}\int_{\frac{d}{2}}^{\frac{d}{2}+i\infty}\frac{(-1)^{\epsilon}d\Delta}{2\pi
i\,\mu_{\Delta}}\int
d^{d}x\,\phi_{\Delta,iM,\epsilon}\phi_{\widetilde{\Delta},iM,\epsilon}\\\
&\phantom{{}={}}+\alpha_{d}\underset{(\Delta,\epsilon)\in
D_{\text{dS}}}{\sum\Res}\frac{(-1)^{\epsilon}}{\mu_{\Delta}}\int
d^{d}x\,\phi_{\Delta,iM,\epsilon}\phi_{d-\Delta,iM,\epsilon}\bigg{)}\;.\end{split}$
(2.26)

Here $\phi_{\Delta,iM,\epsilon}$-s are the conformal primary wavefunctions of
tachyonic scalars, $\alpha_{d}=1$ for odd $d$ and $\alpha_{d}=\frac{1}{2}$ for
even $d$, the summation range of the discrete series is

$D_{\text{dS}}=\\{(\Delta\in\mathbb{Z},\epsilon):\,\Delta>\frac{d}{2}\,\text{
and }\,\epsilon=1+d+\Delta\,\operatorname{mod}\mathbb{Z}_{2}\\}\;,$ (2.27)

and $\mu^{-1}_{\Delta}$ is the Plancherel measure given by

$\displaystyle\mu_{\Delta}=\pi^{d}\,\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\Delta-\frac{d}{2},\frac{d}{2}-\Delta}{\Delta,d-\Delta}\biggr{]}\;.$
(2.28)

The split representation is a powerful tool for efficiently calculating the
conformal partial wave expansion of celestial amplitudes. For example, by
employing the split representation, it has been demonstrated that in the case
of tree-level scalar celestial amplitudes, the coefficient, apart from the
Plancherel measure, in the conformal partial wave expansion can be factorized
into a product of three-point coefficients involving massive and/or tachyonic
scalars [13].

In this paper, we will utilize the split representation (2.26) to compute the
conformal block expansion of four- and higher-point celestial amplitudes.
Since all the examples discussed in this paper are $s$-channel amplitudes,
only the first line in (2.26) contributes to the final results.

##  3 Conformal partial wave expansions

The conformal block expansions of four- and higher-point celestial amplitudes
can be obtained from the conformal partial wave expansions of the
corresponding celestial amplitudes. In this section, we will start with a
brief review of the conformal partial wave expansion of four-point celestial
amplitudes. Then, we will derive the relations between the five- and six-point
conformal partial waves and the corresponding conformal blocks. Finally, we
will establish the conformal partial wave expansions of five- and six-point
celestial amplitudes, with a focus on the comb channel. For further results of
higher-point conformal blocks, see [28, 29, 30, 31, 32, 33, 34, 35, 36, 37,
38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48].

###  3.1 Four-point conformal partial waves

The (scalar) four-point conformal partial wave is

$\Psi^{\Delta_{i}}_{\Delta_{a}}=\int
d^{d}x_{a}\,\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{a}\rangle\langle\widetilde{\mathcal{O}}_{a}\mathcal{O}_{3}\mathcal{O}_{4}\rangle\;,$
(3.1)

and is related to the conformal block by

$\displaystyle\begin{split}&\Psi^{\Delta_{i}}_{\Delta_{a}}=S^{\Delta_{3},\Delta_{4}}_{\widetilde{\Delta}_{a}}G^{\Delta_{i}}_{\Delta_{a}}+S^{\Delta_{1},\Delta_{2}}_{\Delta_{a}}G^{\Delta_{i}}_{\widetilde{\Delta}_{a}}\;.\end{split}$
(3.2)

The shadow symmetry of the four-point conformal partial wave is

$\Psi_{\widetilde{\Delta}_{a}}^{\Delta_{i}}=\frac{S_{\Delta_{a}}^{\Delta_{3},\Delta_{4}}}{S_{\Delta_{a}}^{\Delta_{1},\Delta_{2}}}\Psi_{\Delta_{a}}^{\Delta_{i}}\;.$
(3.3)

For a (scalar) four-point celestial amplitude $\mathcal{A}^{\Delta_{i}}_{4}$,
the conformal partial wave expansion is333 For generic celestial amplitudes,
spinning conformal partial waves $\Psi^{\Delta_{i}}_{\Delta_{a},J_{a}}$ with
spin $J_{a}$ can appear in the conformal partial wave expansion. However for
the examples in this paper, it turns out that there appear only scalar
exchanges with $J_{a}=0$. Thus we omit the spinning contributions and write
the conformal partial wave expansion as (3.4).

$\displaystyle\mathcal{A}^{\Delta_{i}}_{4}=\int_{\frac{d}{2}}^{\frac{d}{2}+i\infty}\frac{d\Delta_{a}}{2\pi
i\,\mu_{\Delta_{a}}}I_{\Delta_{a}}\Psi^{\Delta_{i}}_{\Delta_{a}}\;,$ (3.4)

where $\mu_{\Delta_{a}}^{-1}$ is the Plancherel measure, given in (2.28). The
density function $I_{\Delta_{a}}$ contains all the theory-specific information
in the four-point function $\mathcal{A}^{\Delta_{i}}_{4}$, and can be obtained
by the Euclidean inversion formula

$I_{\Delta_{a}}=\int\frac{d^{d}x_{1}d^{d}x_{2}d^{d}x_{3}d^{d}x_{4}}{\operatorname{Vol}{\operatorname{SO}(1,d+1)}}\mathcal{A}^{\Delta_{i}}_{4}(x_{i})\left(\Psi^{\Delta_{i}}_{\Delta_{a}}(x_{i})\right)^{*}\;.$
(3.5)

From the inversion formula the density function satisfies the shadow symmetry
automatically,

$I_{\widetilde{\Delta}_{a}}=\frac{S_{\Delta_{a}}^{\Delta_{1},\Delta_{2}}}{S_{\Delta_{a}}^{\Delta_{3},\Delta_{4}}}I_{\Delta_{a}}\;.$
(3.6)

Under the shadow symmetries (3.3) and (3.6), substituting the conformal
partial wave (3.2) into the expansion (3.4), we can rewrite (3.4) as

$\displaystyle\mathcal{A}^{\Delta_{i}}_{4}=\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d\Delta_{a}}{2\pi
i\,\mu_{\Delta_{a}}}I_{\Delta_{a}}S^{\Delta_{3},\Delta_{4}}_{\widetilde{\Delta}_{a}}G^{\Delta_{i}}_{\Delta_{a}}\;.$
(3.7)

Then enclosing the contour into the right half $\Delta_{a}$-plane and picking
up the $\Delta_{a}$-poles give the conformal block expansion of the celestial
amplitude $\mathcal{A}^{\Delta_{i}}_{4}$.

###  3.2 Higher-point conformal partial waves

In this subsection, we will establish the conformal partial wave expansions
for the scalar five- and six-point celestial amplitudes with scalar exchanges
in comb channel. Similar to (3.2), we will first express the conformal partial
waves as a linear combination of conformal blocks and shadow conformal blocks.

By the shadow formalism, the five-point comb channel conformal partial wave is

$\Psi^{\Delta_{i}}_{\Delta_{a},\Delta_{b}}=\int d^{d}x_{a}\int
d^{d}x_{b}\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{a}\rangle\langle\widetilde{\mathcal{O}}_{a}\mathcal{O}_{3}\mathcal{O}_{b}\rangle\langle\widetilde{\mathcal{O}}_{b}\mathcal{O}_{4}\mathcal{O}_{5}\rangle\;.$
(3.8)

The shadow symmetry of the four-point conformal partial wave (3.3) implies
that $\Psi^{\Delta_{i}}_{\Delta_{a},\Delta_{b}}$ satisfies

$\displaystyle\Psi^{\Delta_{i}}_{\widetilde{\Delta}_{a},\Delta_{b}}=\frac{S_{\Delta_{a}}^{\Delta_{3},\Delta_{b}}}{S_{\Delta_{a}}^{\Delta_{1},\Delta_{2}}}\Psi^{\Delta_{i}}_{\Delta_{a},\Delta_{b}}\;,$
(3.9)
$\displaystyle\Psi^{\Delta_{i}}_{\Delta_{a},\widetilde{\Delta}_{b}}=\frac{S_{\Delta_{b}}^{\Delta_{4},\Delta_{5}}}{S_{\Delta_{b}}^{\widetilde{\Delta}_{a},\Delta_{3}}}\Psi^{\Delta_{i}}_{\Delta_{a},\Delta_{b}}\;.$
(3.10)

We note that the conformal partial wave (3.8) can be written as a linear
combination of four conformal blocks since they satisfy the same set of
conformal Casimir equations. In other words, we have

$\displaystyle\begin{split}\Psi^{\Delta_{i}}_{\Delta_{a},\Delta_{b}}&=a_{1}G^{\Delta_{i}}_{\Delta_{a},\Delta_{b}}+a_{2}G^{\Delta_{i}}_{\Delta_{a},\widetilde{\Delta}_{b}}+a_{3}G^{\Delta_{i}}_{\widetilde{\Delta}_{a},\Delta_{b}}+a_{4}G^{\Delta_{i}}_{\widetilde{\Delta}_{a},\widetilde{\Delta}_{b}}\;.\end{split}$
(3.11)

Using the $\mathcal{O}_{1}\times\mathcal{O}_{2}$ OPE, we can fix the first two
coefficients as

$a_{1}=S^{\Delta_{3},\Delta_{b}}_{\widetilde{\Delta}_{a}}S^{\Delta_{4},\Delta_{5}}_{\widetilde{\Delta}_{b}}\;,\quad
a_{2}=S^{\Delta_{3},\Delta_{b}}_{\widetilde{\Delta}_{a}}S^{\Delta_{a},\Delta_{3}}_{\Delta_{b}}\;,$
(3.12)

and similarly using $\mathcal{O}_{4}\times\mathcal{O}_{5}$ OPE, we have

$a_{4}=S^{\Delta_{1},\Delta_{2}}_{\Delta_{a}}S^{\widetilde{\Delta}_{a},\Delta_{3}}_{\Delta_{b}}\;.$
(3.13)

Finally, the coefficient $a_{3}$ can be fixed by comparing the coefficients on
both sides of (3.9), leading to

$a_{3}=\left(\frac{S_{\Delta_{a}}^{\Delta_{3},\Delta_{b}}}{S_{\Delta_{a}}^{\Delta_{1},\Delta_{2}}}a_{1}\right)\bigg{|}_{\Delta_{a}\to\widetilde{\Delta}_{a}}=S^{\Delta_{1},\Delta_{2}}_{\Delta_{a}}S^{\Delta_{4},\Delta_{5}}_{\widetilde{\Delta}_{b}}\;.$
(3.14)

This fixes all of coefficients in (3.11) and gives

$\displaystyle\begin{split}\Psi^{\Delta_{i}}_{\Delta_{a},\Delta_{b}}&=S^{\Delta_{3},\Delta_{b}}_{\widetilde{\Delta}_{a}}S^{\Delta_{4},\Delta_{5}}_{\widetilde{\Delta}_{b}}G^{\Delta_{i}}_{\Delta_{a},\Delta_{b}}+S^{\Delta_{3},\Delta_{b}}_{\widetilde{\Delta}_{a}}S^{\Delta_{a},\Delta_{3}}_{\Delta_{b}}G^{\Delta_{i}}_{\Delta_{a},\widetilde{\Delta}_{b}}\\\
&\phantom{{}={}}+S^{\Delta_{1},\Delta_{2}}_{\Delta_{a}}S^{\Delta_{4},\Delta_{5}}_{\widetilde{\Delta}_{b}}G^{\Delta_{i}}_{\widetilde{\Delta}_{a},\Delta_{b}}+S^{\Delta_{1},\Delta_{2}}_{\Delta_{a}}S^{\widetilde{\Delta}_{a},\Delta_{3}}_{\Delta_{b}}G^{\Delta_{i}}_{\widetilde{\Delta}_{a},\widetilde{\Delta}_{b}}\;.\end{split}$
(3.15)

The six-point comb channel conformal partial wave is

$\Psi^{\Delta_{i}}_{\Delta_{a},\Delta_{b},\Delta_{c}}=\int d^{d}x_{a}\int
d^{d}x_{b}\int
d^{d}x_{c}\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{a}\rangle\langle\widetilde{\mathcal{O}}_{a}\mathcal{O}_{3}\mathcal{O}_{b}\rangle\langle\widetilde{\mathcal{O}}_{b}\mathcal{O}_{4}\mathcal{O}_{c}\rangle\langle\widetilde{\mathcal{O}}_{c}\mathcal{O}_{5}\mathcal{O}_{6}\rangle\;,$
(3.16)

which respects the following shadow symmetries,

$\displaystyle\Psi^{\Delta_{i}}_{\widetilde{\Delta}_{a},\Delta_{b},\Delta_{c}}=\frac{S_{\Delta_{a}}^{\Delta_{3},\Delta_{b}}}{S_{\Delta_{a}}^{\Delta_{1},\Delta_{2}}}\Psi^{\Delta_{i}}_{\Delta_{a},\Delta_{b},\Delta_{c}}\;,$
(3.17)
$\displaystyle\Psi^{\Delta_{i}}_{\Delta_{a},\widetilde{\Delta}_{b},\Delta_{c}}=\frac{S_{\Delta_{b}}^{\Delta_{4},\Delta_{c}}}{S_{\Delta_{b}}^{\widetilde{\Delta}_{a},\Delta_{3}}}\Psi^{\Delta_{i}}_{\Delta_{a},\Delta_{b},\Delta_{c}}\;,$
(3.18)
$\displaystyle\Psi^{\Delta_{i}}_{\Delta_{a},\Delta_{b},\widetilde{\Delta}_{c}}=\frac{S_{\Delta_{c}}^{\Delta_{5},\Delta_{6}}}{S_{\Delta_{c}}^{\widetilde{\Delta}_{b},\Delta_{4}}}\Psi^{\Delta_{i}}_{\Delta_{a},\Delta_{b},\Delta_{c}}\;.$
(3.19)

Following the steps in the five-point case, we get the relation between the
six-point conformal partial wave and the conformal blocks, given by

$\displaystyle\begin{split}\Psi^{\Delta_{i}}_{\Delta_{a},\Delta_{b},\Delta_{c}}&=S_{\widetilde{\Delta}_{c}}^{\Delta_{5},\Delta_{6}}S_{\widetilde{\Delta}_{a}}^{\Delta_{3},\Delta_{b}}S_{\widetilde{\Delta}_{b}}^{\Delta_{4},\Delta_{c}}G^{\Delta_{i}}_{\Delta_{a},\Delta_{b},\Delta_{c}}+S_{\Delta_{c}}^{\Delta_{b},\Delta_{4}}S_{\widetilde{\Delta}_{a}}^{\Delta_{3},\Delta_{b}}S_{\widetilde{\Delta}_{b}}^{\Delta_{4},\Delta_{c}}G^{\Delta_{i}}_{\Delta_{a},\Delta_{b},\widetilde{\Delta}_{c}}\\\
&\phantom{{}={}}+S_{\Delta_{b}}^{\Delta_{a},\Delta_{3}}S_{\widetilde{\Delta}_{c}}^{\Delta_{5},\Delta_{6}}S_{\widetilde{\Delta}_{a}}^{\Delta_{3},\Delta_{b}}G^{\Delta_{i}}_{\Delta_{a},\widetilde{\Delta}_{b},\Delta_{c}}+S_{\Delta_{b}}^{\Delta_{a},\Delta_{3}}S_{\widetilde{\Delta}_{a}}^{\Delta_{3},\Delta_{b}}S_{\Delta_{c}}^{\widetilde{\Delta}_{b},\Delta_{4}}G^{\Delta_{i}}_{\Delta_{a},\widetilde{\Delta}_{b},\widetilde{\Delta}_{c}}\\\
&\phantom{{}={}}+S_{\Delta_{a}}^{\Delta_{1},\Delta_{2}}S_{\widetilde{\Delta}_{c}}^{\Delta_{5},\Delta_{6}}S_{\widetilde{\Delta}_{b}}^{\Delta_{4},\Delta_{c}}G^{\Delta_{i}}_{\widetilde{\Delta}_{a},\Delta_{b},\Delta_{c}}+S_{\Delta_{a}}^{\Delta_{1},\Delta_{2}}S_{\widetilde{\Delta}_{b}}^{\Delta_{4},\widetilde{\Delta}_{c}}S_{\Delta_{c}}^{\widetilde{\Delta}_{b},\Delta_{4}}G^{\Delta_{i}}_{\widetilde{\Delta}_{a},\Delta_{b},\widetilde{\Delta}_{c}}\\\
&\phantom{{}={}}+S_{\Delta_{a}}^{\Delta_{1},\Delta_{2}}S_{\widetilde{\Delta}_{c}}^{\Delta_{5},\Delta_{6}}S_{\Delta_{b}}^{\Delta_{3},\widetilde{\Delta}_{a}}G^{\Delta_{i}}_{\widetilde{\Delta}_{a},\widetilde{\Delta}_{b},\Delta_{c}}+S_{\Delta_{a}}^{\Delta_{1},\Delta_{2}}S_{\Delta_{b}}^{\widetilde{\Delta}_{a},\Delta_{3}}S_{\Delta_{c}}^{\widetilde{\Delta}_{b},\Delta_{4}}G^{\Delta_{i}}_{\widetilde{\Delta}_{a},\widetilde{\Delta}_{b},\widetilde{\Delta}_{c}}\;.\end{split}$
(3.20)

Similar to (3.4), five- and six-point celestial amplitudes can be expanded in
term of the conformal partial waves:

$\displaystyle\mathcal{A}^{\Delta_{i}}_{5}=\int_{\frac{d}{2}}^{\frac{d}{2}+i\infty}\frac{d\Delta_{a}}{2\pi
i\,\mu_{\Delta_{a}}}\int_{\frac{d}{2}}^{\frac{d}{2}+i\infty}\frac{d\Delta_{b}}{2\pi
i\,\mu_{\Delta_{b}}}I_{\Delta_{a},\Delta_{b}}\Psi^{\Delta_{i}}_{\Delta_{a},\Delta_{b}}\;,$
(3.21)

and

$\displaystyle\mathcal{A}^{\Delta_{i}}_{6}=\int_{\frac{d}{2}}^{\frac{d}{2}+i\infty}\frac{d\Delta_{a}}{2\pi
i\,\mu_{\Delta_{a}}}\int_{\frac{d}{2}}^{\frac{d}{2}+i\infty}\frac{d\Delta_{b}}{2\pi
i\,\mu_{\Delta_{b}}}\int_{\frac{d}{2}}^{\frac{d}{2}+i\infty}\frac{d\Delta_{c}}{2\pi
i\,\mu_{\Delta_{c}}}I_{\Delta_{a},\Delta_{b},\Delta_{c}}\Psi^{\Delta_{i}}_{\Delta_{a},\Delta_{b},\Delta_{c}}\;.$
(3.22)

Together with the shadow symmetries of the conformal partial waves and the
density functions, the integration contour of exchanged conformal dimensions
can be extended to the whole principle series, leading to

$\displaystyle\mathcal{A}^{\Delta_{i}}_{5}=\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d\Delta_{a}}{2\pi
i\,\mu_{\Delta_{a}}}\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d\Delta_{b}}{2\pi
i\,\mu_{\Delta_{b}}}I_{\Delta_{a},\Delta_{b}}S^{\Delta_{3},\Delta_{b}}_{\widetilde{\Delta}_{a}}S^{\Delta_{4},\Delta_{5}}_{\widetilde{\Delta}_{b}}G^{\Delta_{i}}_{\Delta_{a},\Delta_{b}}\;,$
(3.23)

and

$\displaystyle\mathcal{A}^{\Delta_{i}}_{6}=\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d\Delta_{a}}{2\pi
i\,\mu_{\Delta_{a}}}\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d\Delta_{b}}{2\pi
i\,\mu_{\Delta_{b}}}\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d\Delta_{c}}{2\pi
i\,\mu_{\Delta_{c}}}I_{\Delta_{a},\Delta_{b},\Delta_{c}}S_{\widetilde{\Delta}_{a}}^{\Delta_{3},\Delta_{b}}S_{\widetilde{\Delta}_{b}}^{\Delta_{4},\Delta_{c}}S_{\widetilde{\Delta}_{c}}^{\Delta_{5},\Delta_{6}}G^{\Delta_{i}}_{\Delta_{a},\Delta_{b},\Delta_{c}}\;.$
(3.24)

Evaluating the integrals of exchanged conformal dimensions by enclosing the
contours into the right half planes and picking up the poles, we obtain the
conformal block expansions of the five- and six-point celestial amplitudes.

##  4 Three-point celestial amplitudes

To obtain the conformal block expansions of four- and higher-point celestial
amplitudes, it is necessary to compute scalar three-point celestial amplitudes
containing more than one massive scalar. The scalar three-point celestial
amplitudes with one massive scalar have been obtained in [5]. The scalar
three-point celestial amplitudes of three massive scalars with mass $m$, $m$,
and $2m(1+\epsilon)$ in CCFT2 have been obtained in [1] to the leading order
of $\epsilon$. In this section, we will compute scalar three-point celestial
amplitudes containing more than one massive scalar. It turns out that the
three-point coefficient of scalar three-point celestial amplitudes with two
massive scalars contains a hypergeometric function (4.8), while the three-
point coefficient of celestial amplitudes with three massive scalars can be
written as a triple Mellin-Barnes integral (4.25).

One of the obstacles to evaluate the celestial amplitudes with more than one
massive scalar is to solve the $(d+2)$-dimensional momentum conservation
$\delta$-function. To overcome this, we rewrite the integral over the on-shell
hypersurface $\widehat{p}^{2}=-1$ as an integral over the entire momentum
space by virtue of the identity

$\displaystyle\int\frac{d^{d+1}\widehat{p}}{\widehat{p}^{0}}=\int
d^{d+2}\widehat{p}\;2\delta(\widehat{p}^{2}+1)\theta(\widehat{p}^{0})\;.$
(4.1)

The $(d+2)$-dimensional momentum conservation $\delta$-function then can be
removed by performing the integral over $\widehat{p}$, leaving with a one-
dimensional $\delta$-function, which is easier to be solved.

As an example, we use our strategy to re-compute the celestial amplitudes with
one massive scalar. The celestial amplitude of massless-massless-massive
scalars $1_{0}^{0}+2_{0}^{0}\to 3_{m_{3}}^{0}$ is given by

$\displaystyle\mathcal{A}_{1_{0}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}=\int_{0}^{+\infty}d\omega_{1}\,\omega_{1}^{\Delta_{1}-1}\int_{0}^{+\infty}d\omega_{2}\,\omega_{2}^{\Delta_{2}-1}\int\frac{d^{d+1}\widehat{p}_{3^{\prime}}}{\widehat{p}^{0}_{3^{\prime}}}\delta^{(d+2)}(q_{1}+q_{2}-p_{3^{\prime}})\;.$
(4.2)

Applying (4.1) and performing the integral over $\widehat{p}_{3^{\prime}}$
lead to

$\displaystyle\mathcal{A}_{1_{0}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}$
$\displaystyle=m_{3}^{-d-2}\int_{0}^{+\infty}d\omega_{1}\,\omega_{1}^{\Delta_{1}-1}\int_{0}^{+\infty}d\omega_{2}\,\omega_{2}^{\Delta_{2}-1}2\delta(1-\frac{4x_{12}^{2}\omega_{1}\omega_{2}}{m_{3}})\left(2\frac{\omega_{1}x_{13}^{2}+\omega_{2}x_{23}^{2}}{m_{3}}\right)^{-\Delta_{3}}$
$\displaystyle=C_{1_{0}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}\;\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\rangle\;,$
(4.3)

where the three-point coefficient $C_{1_{0}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}$ is

$C_{1_{0}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}=2^{-\Delta_{12}}m_{3}^{\Delta_{12}-d-2}\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\frac{\Delta_{13,2}}{2},\frac{\Delta_{23,1}}{2}}{\Delta_{3}}\biggr{]}\;.$
(4.4)

This matches the result of [5, Section 3.1]. Now, we are ready to compute the
scalar three-point celestial amplitudes with more than one massive scalar by
the same strategy.

###  4.1 Massive-massless-massive scalars

The celestial three-point amplitude of massive-massless-massive scalars
$1_{m_{1}}^{0}+2_{0}^{0}\to 3_{m_{3}}^{0}$ takes the form as

$\displaystyle\mathcal{A}_{1_{m_{1}}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}$
$\displaystyle=\int_{0}^{+\infty}d\omega_{2}\,\omega_{2}^{\Delta_{2}-1}\int\frac{d^{d+1}\widehat{p}_{1^{\prime}}}{\widehat{p}_{1^{\prime}}^{0}}\,\int\frac{d^{d+1}\widehat{p}_{3^{\prime}}}{\widehat{p}^{0}_{3^{\prime}}}\,$
(4.5)
$\displaystyle\phantom{{}={}}\times(-\widehat{q}_{1}\cdot\widehat{p}_{1^{\prime}})^{-\Delta_{1}}(-\widehat{q}_{3}\cdot\widehat{p}_{3^{\prime}})^{-\Delta_{3}}\delta^{(d+2)}(p_{1^{\prime}}+q_{2}-p_{3^{\prime}}^{2})\;.$
$\displaystyle=C_{1_{m_{1}}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\rangle\;,$

and in the following we will compute the three-point coefficient
$C_{1_{m_{1}}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}$.

Applying (4.1) and performing the integral over $\widehat{p}_{3^{\prime}}$
then lead to

$\displaystyle\mathcal{A}_{1_{m_{1}}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}$
$\displaystyle=m_{3}^{-d-2}\int_{0}^{+\infty}d\omega_{2}\,\omega_{2}^{\Delta_{2}-1}\int\frac{d^{d+1}\widehat{p}_{1^{\prime}}}{\widehat{p}_{1^{\prime}}^{0}}\,2\delta\left(\frac{m_{3}^{2}-m_{1}^{2}+2m_{1}\omega_{2}\widehat{p}_{1^{\prime}}\cdot\widehat{q}_{2}}{m_{3}^{2}}\right)$
(4.6)
$\displaystyle\phantom{{}={}}\times(-\widehat{q}_{1}\cdot\widehat{p}_{1^{\prime}})^{-\Delta_{1}}\left(-\frac{m_{1}\widehat{q}_{3}\cdot\widehat{p}_{1^{\prime}}+\omega_{2}\widehat{q}_{3}\cdot\widehat{q}_{2}}{m_{3}}\right)^{-\Delta_{3}}\;.$

Integrating out $\omega_{2}$ leads to

$\displaystyle\mathcal{A}_{1_{m_{1}}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}$
$\displaystyle=2^{\Delta_{3}-\Delta_{2}+1}m_{3}^{\Delta_{3}-d}m_{1}^{\Delta_{3}-\Delta_{2}}(m_{3}^{2}-m_{1}^{2})^{\Delta_{2}-1}\int\frac{d^{d+1}\widehat{p}_{1^{\prime}}}{\widehat{p}_{1^{\prime}}^{0}}\,$
(4.7)
$\displaystyle\phantom{{}={}}\times(-\widehat{q}_{1}\cdot\widehat{p}_{1^{\prime}})^{-\Delta_{1}}(-\widehat{q}_{2}\cdot\widehat{p}_{1^{\prime}})^{\Delta_{3}-\Delta_{2}}\left(2m_{1}^{2}\widehat{q}_{2}\cdot\widehat{p}_{1^{\prime}}\widehat{q}_{3}\cdot\widehat{p}_{1^{\prime}}-(m_{3}^{2}-m_{1}^{2})\widehat{q}_{2}\cdot\widehat{q}_{3}\right)^{-\Delta_{3}}\;.$

Since $m_{1}>0,\,\omega_{2}\geqslant 0$, the $\delta$-function can be solved
by
$\omega_{2}=\frac{m_{3}^{2}-m_{1}^{2}}{-2m_{1}\widehat{p}_{1^{\prime}}\cdot\widehat{q}_{2}}$
if and only if $m_{3}\geqslant m_{1}$, which manifests the mass threshold.

Now we expand the factor $(\cdots)^{-\Delta_{3}}$ into a Taylor series of
$(-\widehat{q}_{2}\cdot\widehat{q}_{3})$, and the remaining integral is the
tree-level three-point Witten diagram on EAdS (A.8). Then we can read off the
three-point coefficient as

$\displaystyle C_{1_{m_{1}}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}$
$\displaystyle=d_{\Delta_{1},\Delta_{2},\Delta_{3}}2^{1-\Delta_{2}}m_{1}^{-\Delta_{23}}m_{3}^{-d+\Delta_{3}}(m_{3}^{2}-m_{1}^{2})^{\Delta_{2}-1}\,{}_{2}F_{1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{\frac{\Delta_{23,1}}{2},\frac{\Delta_{123}-d}{2}}{\Delta_{2}};\frac{m_{1}^{2}-m_{3}^{2}}{m_{1}^{2}}\biggr{)}\;,$
(4.8)

where $d_{\Delta_{1},\Delta_{2},\Delta_{3}}$ is the coefficient of the three-
point Witten diagram on EAdS:

$d_{\Delta_{1},\Delta_{2},\Delta_{3}}=\frac{1}{2}\pi^{d/2}\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\frac{\Delta_{12,3}}{2},\frac{\Delta_{23,1}}{2},\frac{\Delta_{31,2}}{2},\frac{\Delta_{123}-d}{2}}{\Delta_{1},\Delta_{2},\Delta_{3}}\biggr{]}\;.$
(4.9)

We stress that if the masses leave from the physical region $m_{3}\geqslant
m_{1}>0$, there are no solutions for the $\delta$-function, and then the
three-point coefficient vanishes.

####  4.1.1 Sanity checks

We check the result (4.8) by the shadow symmetry and the massless limit. From
the shadow symmetry between the massive conformal primary wavefunction and its
shadow (2.17), the three-point coefficient (4.8) should respect the following
shadow symmetries:

$C_{\mathcal{S}[1_{m_{1}}^{0}]+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\widetilde{\Delta}_{1},\Delta_{2},\Delta_{3}}=C_{1_{m_{1}}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\widetilde{\Delta}_{1},\Delta_{2},\Delta_{3}}\;,\qquad
C_{1_{m_{1}}^{0}+2_{0}^{0}\to\mathcal{S}[3_{m_{3}}^{0}]}^{\Delta_{1},\Delta_{2},\widetilde{\Delta}_{3}}=C_{1_{m_{1}}^{0}+2_{0}^{0}\to{3}_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\widetilde{\Delta}_{3}}\;.$
(4.10)

This implies that

$\displaystyle C_{1_{m_{1}}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\widetilde{\Delta}_{1},\Delta_{2},\Delta_{3}}=C_{1_{m_{1}}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}S^{\Delta_{2},\Delta_{3}}_{\Delta_{1}}N_{\Delta_{1}}\;,$
(4.11) $\displaystyle C_{1_{m_{1}}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\widetilde{\Delta}_{3}}=C_{1_{m_{1}}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}S^{\Delta_{1},\Delta_{2}}_{\Delta_{3}}N_{\Delta_{3}}\;,$
(4.12)

which can be directly verified by the property of the hypergeometric function
(A.5).

To analyze the $m_{1}\to 0$ limit of the three-point coefficient
$C_{1_{m_{1}}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}$, by the property of the
hypergeometric function (A.6) we rewrite it as

$\displaystyle\frac{C_{1_{m_{1}}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}}{d_{\Delta_{1},\Delta_{2},\Delta_{3}}}$
$\displaystyle=2^{1-\Delta_{2}}m_{1}^{-\Delta_{1}}m_{3}^{\Delta_{3}-d}(m_{3}^{2}-m_{1}^{2})^{\frac{\Delta_{12,3}}{2}-1}\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\Delta_{2},\Delta_{1}-\frac{d}{2}}{\frac{\Delta_{12,3}}{2},\frac{\Delta_{123}-d}{2}}\biggr{]}\,{}_{2}F_{1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{\frac{\Delta_{3,12}+2}{2},\frac{\Delta_{23,1}}{2}}{\Delta_{1}-\frac{d}{2}+1};\frac{m_{1}^{2}}{m_{1}^{2}-m_{3}^{2}}\biggr{)}$
$\displaystyle\phantom{{}={}}+(\Delta_{1}\to\widetilde{\Delta}_{1})\;.$ (4.13)

After multiplied by the factor in (2.23), the first term is convergent as
$m_{1}\to 0$; the second term is proportional to $m_{1}^{2\Delta_{1}-d}$, and
is convergent to $0$ (divergent to $\infty$) if
$\operatorname{Re}{\Delta_{1}}>\frac{d}{2}$
($\operatorname{Re}{\Delta_{1}}<\frac{d}{2}$). Hence under the condition
$\operatorname{Re}{\Delta_{1}}>\frac{d}{2}$, we have

$C_{1_{0}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}=\lim_{m_{1}\to
0}2^{-\Delta_{1}}\pi^{-\frac{d}{2}}m_{1}^{\Delta_{1}}\frac{\Gamma[\Delta_{1}]}{\Gamma[\Delta_{1}-\frac{d}{2}]}C_{1_{m_{1}}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}\;.$ (4.14)

Similar analysis shows that if $\operatorname{Re}{\Delta_{1}}<\frac{d}{2}$,
the $m_{1}\to 0$ limit of $C_{1_{m_{1}}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}$ recovers the three-point
coefficient under the shadow conformal basis
$C_{\mathcal{S}[1_{0}^{0}]+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\widetilde{\Delta}_{1},\Delta_{2},\Delta_{3}}$.

###  4.2 Massive-massive-massive scalars

In this section we evaluate the three-point coefficient
$C_{1_{m_{1}}^{0}+2_{m_{2}}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}$ of massive-massive-massive
scalars. The celestial amplitude is

$\displaystyle\mathcal{A}_{1_{m_{1}}^{0}+2_{m_{2}}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}$
$\displaystyle=\int\frac{d^{d+1}\widehat{p}_{1^{\prime}}}{\widehat{p}_{1^{\prime}}^{0}}\,\int\frac{d^{d+1}\widehat{p}_{2^{\prime}}}{\widehat{p}_{2^{\prime}}^{0}}\,\int\frac{d^{d+1}\widehat{p}_{3^{\prime}}}{\widehat{p}_{3^{\prime}}^{0}}\,\delta^{(d+2)}(m_{1}\widehat{p}_{1^{\prime}}+m_{2}\widehat{p}_{2^{\prime}}-m_{3}\widehat{p}_{3^{\prime}})$
(4.15)
$\displaystyle\phantom{{}={}}\times(-\widehat{q}_{1}\cdot\widehat{p}_{1^{\prime}})^{-\Delta_{1}}(-\widehat{q}_{2}\cdot\widehat{p}_{2^{\prime}})^{-\Delta_{2}}(-\widehat{q}_{3}\cdot\widehat{p}_{3^{\prime}})^{-\Delta_{3}}$
$\displaystyle=C_{1_{m_{1}}^{0}+2_{m_{2}}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\rangle\;.$
(4.16)

Applying (4.1) and performing the $\widehat{p}_{3^{\prime}}$-integral, we get

$\displaystyle\mathcal{A}_{1_{m_{1}}^{0}+2_{m_{2}}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}$
$\displaystyle=m_{1}^{-\Delta_{3}-1}m_{2}^{-1}m_{3}^{-d+\Delta_{3}}\int\frac{d^{d+1}\widehat{p}_{1^{\prime}}}{\widehat{p}_{1^{\prime}}^{0}}\,\int\frac{d^{d+1}\widehat{p}_{2^{\prime}}}{\widehat{p}_{2^{\prime}}^{0}}\,\delta\left(\frac{m_{3}^{2}-(m_{1}+m_{2})^{2}}{2m_{1}m_{2}}+1+\widehat{p}_{1^{\prime}}\cdot\widehat{p}_{2^{\prime}}\right)$
$\displaystyle\phantom{{}={}}\times(-\widehat{q}_{1}\cdot\widehat{p}_{1^{\prime}})^{-\Delta_{1}}(-\widehat{q}_{2}\cdot\widehat{p}_{2^{\prime}})^{-\Delta_{2}}\left(-\widehat{q}_{3}\cdot\widehat{p}_{1^{\prime}}-\frac{m_{2}}{m_{1}}\widehat{q}_{3}\cdot\widehat{p}_{2^{\prime}}\right)^{-\Delta_{3}}\;.$
(4.17)

If the masses do not satisfy the physical condition $m_{3}\geqslant
m_{1}+m_{2}$, there are no solutions for the $\delta$-function, and then the
three-point coefficient vanishes. To manifest this mass threshold, we
introduce the new mass parameter
$A=\frac{m_{3}^{2}-(m_{1}+m_{2})^{2}}{2m_{1}m_{2}}\geqslant 0$ in the physical
region $m_{1}>0\,,m_{2}>0,\,m_{3}\geqslant m_{1}+m_{2}$. Inserting an identity
of the Mellin transform

$f(A)=\int_{-i\infty}^{+i\infty}\frac{ds}{2\pi
i}\,A^{-s}\int_{0}^{+\infty}dA^{\prime}\,A^{\prime s-1}f(A^{\prime})\;,$
(4.18)

we have

$\displaystyle\mathcal{A}_{1_{m_{1}}^{0}+2_{m_{2}}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}$
$\displaystyle=m_{1}^{-\Delta_{3}-1}m_{2}^{-1}m_{3}^{-d+\Delta_{3}}\int_{-i\infty}^{+i\infty}\frac{ds}{2\pi
i}\,A^{-s}\int\frac{d^{d+1}\widehat{p}_{1^{\prime}}}{\widehat{p}_{1^{\prime}}^{0}}\,\int\frac{d^{d+1}\widehat{p}_{2^{\prime}}}{\widehat{p}_{2^{\prime}}^{0}}\,\int_{0}^{+\infty}dA^{\prime}\,A^{\prime
s-1}$ (4.19)
$\displaystyle\phantom{{}={}}\times\delta(A^{\prime}+1+\widehat{p}_{1^{\prime}}\cdot\widehat{p}_{2^{\prime}})(-\widehat{q}_{1}\cdot\widehat{p}_{1^{\prime}})^{-\Delta_{1}}(-\widehat{q}_{2}\cdot\widehat{p}_{2^{\prime}})^{-\Delta_{2}}\left(-\widehat{q}_{3}\cdot\widehat{p}_{1^{\prime}}-\frac{m_{2}}{m_{1}}\widehat{q}_{3}\cdot\widehat{p}_{2^{\prime}}\right)^{-\Delta_{3}}\;.$

After integrating out $A^{\prime}$, we use the Mellin-Barnes relation (A.1) to
separate the sum
$(-\widehat{q}_{3}\cdot\widehat{p}_{1^{\prime}}-\frac{m_{2}}{m_{1}}\widehat{q}_{3}\cdot\widehat{p}_{2^{\prime}})$
into a product, then the integrand is proportional to

$(\cdots)(-1-\widehat{p}_{1^{\prime}}\cdot\widehat{p}_{2^{\prime}})^{s-1}(-\widehat{q}_{1}\cdot\widehat{p}_{1^{\prime}})^{-\Delta_{1}}(-\widehat{q}_{2}\cdot\widehat{p}_{2^{\prime}})^{-\Delta_{2}}(-\widehat{q}_{3}\cdot\widehat{p}_{1^{\prime}})^{-\Delta_{3}-t}(-\widehat{q}_{3}\cdot\widehat{p}_{2^{\prime}})^{t}\;.$
(4.20)

The integral over $\widehat{p}_{1^{\prime}}$ can be computed as follows. By
the Feynman-Schwinger parameterization (A.3) and the EAdS integral (A.10), we
have

$\displaystyle\phantom{{}={}}\int\frac{d^{d+1}\widehat{p}_{1^{\prime}}}{\widehat{p}_{1^{\prime}}^{0}}\,(-1-\widehat{p}_{1^{\prime}}\cdot\widehat{p}_{2^{\prime}})^{s-1}(-\widehat{q}_{1}\cdot\widehat{p}_{1^{\prime}})^{-\Delta_{1}}(-\widehat{q}_{3}\cdot\widehat{p}_{1^{\prime}})^{-t-\Delta_{3}}$
(4.21) $\displaystyle=(\cdots)\int_{0}^{+\infty}d\alpha
d\beta\,\alpha^{\Delta_{1}-1}\beta^{t+\Delta_{3}-1}\int\frac{d^{d+1}\widehat{p}_{1^{\prime}}}{\widehat{p}_{1^{\prime}}^{0}}\,\left(-1-\widehat{p}_{1^{\prime}}\cdot(\widehat{p}_{2^{\prime}}+\alpha\widehat{q}_{1}+\beta\widehat{q}_{3})\right)^{-\Delta_{13}+s-t-1}$
$\displaystyle=(\cdots)\int_{-i\infty}^{+i\infty}\frac{du}{2\pi
i}\,\int_{0}^{+\infty}d\alpha
d\beta\,\alpha^{\Delta_{1}-1}\beta^{t+\Delta_{3}-1}(1-2\alpha\widehat{p}_{2^{\prime}}\cdot\widehat{q}_{1}-2\beta\widehat{p}_{2^{\prime}}\cdot\widehat{q}_{3}-2\alpha\beta\widehat{q}_{1}\cdot\widehat{q}_{3})^{\frac{-\Delta_{13}+s-t-u-1}{2}}\;.$

The $\alpha$-integral gives

$(\cdots)\beta^{t+\Delta_{3}-1}(-\widehat{p}_{2^{\prime}}\cdot\widehat{q}_{1}-\beta\widehat{q}_{1}\cdot\widehat{q}_{3})^{-\Delta_{1}}(1-2\beta\widehat{p}_{2^{\prime}}\cdot\widehat{q}_{3})^{\frac{\Delta_{1,3}+s-t-u-1}{2}}\;,$
(4.22)

then using the Mellin-Barnes relation (A.1) we separate the two sums
$(-\widehat{p}_{2^{\prime}}\cdot\widehat{q}_{1}-\beta\widehat{q}_{1}\cdot\widehat{q}_{3})$
and $(1-2\beta\widehat{p}_{2^{\prime}}\cdot\widehat{q}_{3})$ into products
with two new Mellin parameters $\gamma$ and $v$. After performing the
$\beta$\- and $\gamma$-integrals, we find that the $\widehat{p}_{1^{\prime}}$
-integral (4.21) is written as a double Mellin-Barnes integral over $u$ and
$v$, and the integrand is proportional to

$(\cdots)(-\widehat{p}_{2^{\prime}}\cdot\widehat{q}_{1})^{-v-\Delta_{1}}(-\widehat{p}_{2^{\prime}}\cdot\widehat{q}_{3})^{-v-\Delta_{3}}(-\widehat{q}_{1}\cdot\widehat{q}_{3})^{v}\;.$
(4.23)

The remaining integral over $\widehat{p}_{2^{\prime}}$ is the tree-level
three-point Witten diagram on EAdS (A.8). Then we can read off the three-point
coefficient as

$\displaystyle C_{1_{m_{1}}^{0}+2_{m_{2}}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}$
$\displaystyle=\int_{-i\infty}^{+i\infty}\frac{dsdtdv}{(2\pi
i)^{3}}\,\pi^{d/2}m_{1}^{-\Delta_{3}+s-t-1}m_{2}^{s+t-1}m_{3}^{\Delta_{3}-d}(m_{3}^{2}-(m_{1}+m_{2})^{2})^{-s}$
(4.24)
$\displaystyle\phantom{{}={}}\times\int_{-i\infty}^{+i\infty}\frac{du}{2\pi
i}\,(-2)^{u}\Gamma\left[\textstyle{-u,\frac{-\Delta_{13}-s-t+u-2v+1}{2},\frac{\Delta_{13}-d-s+t+u+1}{2}}\right]$
$\displaystyle\phantom{{}={}}\times
d_{\Delta_{1},\Delta_{2},\Delta_{3}}\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{-t,-v,\Delta_{3}+t+v,\frac{\Delta_{13,2}+2v}{2},\frac{\Delta_{123}-d+2v}{2}}{1-s,\Delta_{3}+v,\frac{\Delta_{13,2}}{2},\frac{\Delta_{123}-d}{2}}\biggr{]}\;.$

The $u$-integral can be performed as follows. We first rewrite
$(-2)^{u}\Gamma[-u]$ as
$\frac{(-1)^{u}}{2\sqrt{\pi}}\Gamma[\frac{1-u}{2}]\Gamma[-\frac{u}{2}]$, which
gives two series of $u$-poles in the right half plane. For each series of
$u$-poles, the residues can be resummed as a hypergeometric function with
argument unity (A.7), leading to

$\displaystyle C_{1_{m_{1}}^{0}+2_{m_{2}}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}$
$\displaystyle=\int_{-i\infty}^{+i\infty}\frac{dsdtdv}{(2\pi
i)^{3}}\,\pi^{\frac{d+1}{2}}m_{1}^{-\Delta_{3}+s-t-1}m_{2}^{s+t-1}m_{3}^{\Delta_{3}-d}(m_{3}^{2}-(m_{1}+m_{2})^{2})^{-s}$
(4.25) $\displaystyle\phantom{{}={}}\times
d_{\Delta_{1},\Delta_{2},\Delta_{3}}\left(C_{1}+C_{2}\right)\;,$

where

$\displaystyle
C_{1}=\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{-t,-v,\frac{d}{2}+s+v-\frac{1}{2},\frac{-\Delta_{13}-s-t-2v+1}{2},\Delta_{3}+t+v,\frac{\Delta_{13}-d-s+t+1}{2},\frac{\Delta_{13,2}+2v}{2},\frac{\Delta_{123}-d+2v}{2}}{1-s,\frac{-\Delta_{13}+d+s-t}{2},\Delta_{3}+v,\frac{\Delta_{13}+s+t+2v}{2},\frac{\Delta_{13,2}}{2},\frac{\Delta_{123}-d}{2}}\biggr{]}\;,$
(4.26) $\displaystyle
C_{2}=\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{-t,-v,\frac{d}{2}+s+v-\frac{1}{2},\frac{-\Delta_{13}-s-t-2v+2}{2},\Delta_{3}+t+v,\frac{\Delta_{13}-d-s+t+2}{2},\frac{\Delta_{13,2}+2v}{2},\frac{\Delta_{123}-d+2v}{2}}{1-s,\frac{-\Delta_{13}+d+s-t+1}{2},\Delta_{3}+v,\frac{\Delta_{13}+s+t+2v+1}{2},\frac{\Delta_{13,2}}{2},\frac{\Delta_{123}-d}{2}}\biggr{]}\;.$
(4.27)

We stress that if the masses leave from the physical region $m_{1},m_{2}>0$
and $m_{3}\geqslant m_{1}+m_{2}>0$, there are no solutions for the
$\delta$-function, and then the three-point coefficient vanishes.

####  4.2.1 Mass threshold expansion

Near the mass threshold $m_{3}=m_{1}+m_{2}$, we can enclose the $s$-contour to
the left half plane. After performing the $s$-integral, we enclose the $v$\-
and $t$-contours to the right half planes successively. In each of the steps
there is only one series of poles contributing to the result. This leads to
the following series representation of $C_{1_{m_{1}}^{0}+2_{m_{2}}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}$ near the mass threshold,

$\displaystyle C_{1_{m_{1}}^{0}+2_{m_{2}}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}$
$\displaystyle=\sum_{n_{1},n_{2}=0}^{+\infty}(-1)^{n_{1}+n_{2}}\pi^{\frac{d+1}{2}}m_{1}^{-\frac{d+1}{2}-\Delta_{3}-n_{1}-n_{2}}m_{2}^{-\frac{d+1}{2}-n_{1}+n_{2}}m_{3}^{\Delta_{3}-d}(m_{3}^{2}-(m_{1}+m_{2})^{2})^{\frac{d-1}{2}+n_{1}}$
$\displaystyle\phantom{{}={}}\times
d_{\Delta_{1},\Delta_{2},\Delta_{3}}(D_{1}+D_{2})\,\,{}_{4}F_{3}\biggl{(}\genfrac{.}{.}{0.0pt}{}{-n_{1},\Delta_{3}+n_{2},\frac{\Delta_{13,2}}{2},\frac{\Delta_{123}-d}{2}}{\Delta_{3},\frac{-d+2\Delta_{13}-2n_{1}+2n_{2}+1}{4},\frac{-d+2\Delta_{13}-2n_{1}+2n_{2}+3}{4}};1\biggr{)}\;,$
(4.28)

where

$\displaystyle
D_{1}=\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\Delta_{3}+n_{2},\frac{d-2\Delta_{13}+2n_{1}-2n_{2}+1}{4},\frac{-d+2\Delta_{13}+2n_{1}+2n_{2}+1}{4}}{n_{1}+1,n_{2}+1,\Delta_{3},\frac{d+1}{2}+n_{1},\frac{d-2\Delta_{13}-2n_{1}-2n_{2}+1}{4},\frac{-d+2\Delta_{13}-2n_{1}+2n_{2}+1}{4}}\biggr{]}\;,$
(4.29) $\displaystyle
D_{2}=\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\Delta_{3}+n_{2},\frac{d-2\Delta_{13}+2n_{1}-2n_{2}+3}{4},\frac{-d+2\Delta_{13}+2n_{1}+2n_{2}+3}{4}}{n_{1}+1,n_{2}+1,\Delta_{3},\frac{d+1}{2}+n_{1},\frac{d-2\Delta_{13}-2n_{1}-2n_{2}+3}{4},\frac{-d+2\Delta_{13}-2n_{1}+2n_{2}+3}{4}}\biggr{]}\;.$
(4.30)

To compare with the result in [1, Section 3], we set
$m_{3}=(1+\epsilon)(m_{1}+m_{2})$ with $\epsilon\rightarrow 0^{+}$, and in
this limit we have the estimate

$(m_{3}^{2}-(m_{1}+m_{2})^{2})^{\frac{d-1}{2}+n_{1}}\sim\epsilon^{\frac{d-1}{2}+n_{1}}\;.$
(4.31)

This implies the leading term of (4.2.1) can be obtained by setting $n_{1}=0$,
then the remaining sum of $n_{2}$ can be performed, which leads to

$C_{1_{m_{1}}^{0}+2_{m_{2}}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}=d_{\Delta_{1},\Delta_{2},\Delta_{3}}\frac{(2\pi)^{\frac{d+1}{2}}(m_{1}m_{2})^{-\frac{d+1}{2}}}{(m_{1}+m_{2})\Gamma[\frac{d+1}{2}]}\epsilon^{\frac{d-1}{2}}+O(\epsilon^{\frac{d+1}{2}})\;.$
(4.32)

Notice that the leading term does not depend on the external conformal
dimensions $\Delta_{i}$. Setting $d=2$ and $m_{1}=m_{2}=m$ then reproduce the
result in [1, Section 3] upto a factor $(2\pi)^{4}i$ due to the different
conventions of the scattering amplitude.

####  4.2.2 Massless limit

We further check our results by taking the massless limit (2.23). To take the
$m_{2}\rightarrow 0$ limit, we should enclose the $s$-contour into the right
half plane after shifting $s$ by $s\rightarrow s-t$. Using the method of pole
pinching, one can check that the $C_{2}$-term in (4.25) does not survive in
the limit $m_{2}\to 0$. The regular term can be obtained from the $C_{1}$-term
by picking up poles at $t=n_{1}$, $v=\frac{-\Delta_{13}-s+1}{2}+n_{2}$, and
$s=1-\Delta_{2}+2n_{2}+2n_{3}$ with $n_{1},n_{2},n_{3}\in\mathbb{N}$,
successively. For each triplet $(n_{1},n_{2},n_{3})$, the residue is
proportional to the factor $m_{2}^{2n_{2}+2n_{3}}$, and taking
$m_{1}\rightarrow 0$ forces $n_{2}=n_{3}=0$. This leads to

$\displaystyle\phantom{{}={}}\lim_{m_{2}\rightarrow
0}2^{-\Delta_{2}}\pi^{-\frac{d}{2}}m_{2}^{\Delta_{2}}\frac{\Gamma[\Delta_{2}]}{\Gamma[\Delta_{2}-\frac{d}{2}]}C_{1_{m_{1}}^{0}+2_{m_{2}}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}$ (4.33)
$\displaystyle=\sum_{n_{1}=0}^{+\infty}(-1)^{n_{1}}2^{1-\Delta_{2}}m_{1}^{-\Delta_{23}-2n_{1}}m_{3}^{\Delta_{3}-d}(m_{3}^{2}-m_{1}^{2})^{\Delta_{2}+n_{1}-1}\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\Delta_{2},\frac{\Delta_{23,1}+2n_{1}}{2},\frac{\Delta_{123}-d+2n_{1}}{2}}{n_{1}+1,\Delta_{2}+n_{1},\frac{\Delta_{23,1}}{2},\frac{\Delta_{123}-d}{2}}\biggr{]}$
$\displaystyle=C_{1_{m_{1}}^{0}+2_{0}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}\;,$

which reproduces the result in (4.8). We stress that to get (4.33), we have
ignored the terms that may not converge in the limit $m_{2}\to 0$. The
situation here is similar to that in Section 4.1.1, see the discussions after
(4.13).

Similarly, to take the $m_{1}\rightarrow 0$ limit, we should enclose the
$s$-contour into the right half plane after shifting $s$ by $s\rightarrow
s+t$. Again, the $C_{2}$-term in (4.25) does not survive in the limit
$m_{1}\to 0$. The regular term can be obtained from the $C_{1}$-term by
picking up poles at $t=-\Delta_{3}-v-n_{1}$,
$s=-\Delta_{1,3}+1+2n_{1}+2n_{2}$, and $v=n_{3}$ with
$n_{1},n_{2},n_{3}\in\mathbb{N}$, successively. For each triplet
$(n_{1},n_{2},n_{3})$, the residue is proportional to the factor
$m_{1}^{2n_{1}+2n_{2}}$, and taking $m_{1}\rightarrow 0$ forces
$n_{2}=n_{3}=0$. This leads to

$\displaystyle\phantom{{}={}}\lim_{m_{1}\rightarrow
0}2^{-\Delta_{1}}\pi^{-\frac{d}{2}}m_{1}^{\Delta_{1}}\frac{\Gamma[\Delta_{1}]}{\Gamma[\Delta_{1}-\frac{d}{2}]}C_{1_{m_{1}}^{0}+2_{m_{2}}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}$ (4.34)
$\displaystyle=\sum_{n_{3}=0}^{+\infty}(-1)^{n_{3}}2^{1-\Delta_{1}}m_{2}^{-\Delta_{13}-2n_{3}}m_{3}^{\Delta_{3}-d}(m_{3}^{2}-m_{2}^{2})^{\Delta_{1}+n_{3}-1}\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\Delta_{1},\frac{\Delta_{13,2}+2n_{3}}{2},\frac{\Delta_{123}-d+2n_{3}}{2}}{n_{3}+1,\Delta_{1}+n_{3},\frac{\Delta_{13,2}}{2},\frac{\Delta_{123}-d}{2}}\biggr{]}$
$\displaystyle=C_{1_{0}^{0}+2_{m_{2}}^{0}\to
3_{m_{3}}^{0}}^{\Delta_{1},\Delta_{2},\Delta_{3}}\;.$

##  5 Four-point celestial amplitudes

In this section, we will utilize the split representation (2.26) to analyze
four-point celestial amplitudes involving massive scalars. The split
representation is a useful technique that allows us to easily obtain the
conformal partial wave expansion and the conformal block expansion of
celestial amplitudes. To illustrate this, we will warm up with the simple case
of massless four-point celestial amplitudes.

###  5.1 Warm up: four massless scalars with massless exchange

Let us consider the $s$-channel tree-level scattering amplitude

$\displaystyle\mathcal{M}_{1^{0}_{0}+2^{0}_{0}\to
3^{0}_{0}+4^{0}_{0}}=\frac{i}{(q_{1}+q_{2})^{2}}\delta^{(d+2)}(q_{1}+q_{2}-q_{3}-q_{4})\;,$
(5.1)

where the external and the exchanged particles are massless scalars. In this
example, the internal momentum $(q_{1}+q_{2})$ is timelike or lightlike, thus
only the first line in the the split representation (2.26) contributes. Then
the celestial amplitude is

$\begin{split}\mathcal{A}^{\Delta_{i}}_{1_{0}^{0}+2_{0}^{0}\rightarrow
3_{0}^{0}+4_{0}^{0}}&=\int_{0}^{\infty}\frac{dm_{a}}{(2\pi)^{d+2}}\frac{im_{a}^{d+1}}{-m_{a}^{2}}\int_{\frac{d}{2}}^{\frac{d}{2}+i\infty}\frac{d\Delta_{a}}{2\pi
i\,\mu_{\Delta_{a}}}\int
d^{d}x_{a}\;\mathcal{A}^{\Delta_{1},\Delta_{2},\Delta_{a}}_{1_{0}^{0}+2_{0}^{0}\rightarrow
a_{m_{a}}^{0}}\mathcal{A}^{\widetilde{\Delta}_{a},\Delta_{3},\Delta_{4}}_{a_{m_{a}}^{0}\rightarrow
3_{0}^{0}+4_{0}^{0}}\;,\end{split}$ (5.2)

where
$\mathcal{A}^{\Delta_{1},\Delta_{2},\Delta_{a}}_{1_{0}^{0}+2_{0}^{0}\rightarrow
a_{m_{a}}^{0}}$ and
$\mathcal{A}^{\widetilde{\Delta}_{a},\Delta_{3},\Delta_{4}}_{a_{m_{a}}^{0}\rightarrow
3_{0}^{0}+4_{0}^{0}}$ are three-point celestial amplitudes containing one
massive scalar. Using (3.1) we obtain the conformal partial wave expansion of
the celestial amplitude,

$\displaystyle\begin{split}\mathcal{A}^{\Delta_{i}}_{1_{0}^{0}+2_{0}^{0}\rightarrow
3_{0}^{0}+4_{0}^{0}}&=\int_{0}^{\infty}\frac{dm_{a}}{(2\pi)^{d+2}}\frac{im_{a}^{d+1}}{-m_{a}^{2}}\int_{\frac{d}{2}}^{\frac{d}{2}+i\infty}\frac{d\Delta_{a}}{2\pi
i\,\mu_{\Delta_{a}}}I_{\Delta_{a}}\Psi^{\Delta_{i}}_{\Delta_{a}}\;.\end{split}$
(5.3)

where the density function is

$I_{\Delta_{a}}=C^{\Delta_{1},\Delta_{2},\Delta_{a}}_{1^{0}_{0}+2^{0}_{0}\rightarrow
a_{m_{a}}^{0}}C^{\widetilde{\Delta}_{a},\Delta_{3},\Delta_{4}}_{a^{0}_{m_{a}}\rightarrow
3_{0}^{0}+4_{0}^{0}}\;.$ (5.4)

Armed with the shadow symmetry of the three-point coefficient (4.11), it is
easy to check that the density function satisfies

$I_{\widetilde{\Delta}_{a}}=\frac{S_{\Delta_{a}}^{\Delta_{1},\Delta_{2}}}{S_{\Delta_{a}}^{\Delta_{3},\Delta_{4}}}I_{\Delta_{a}}\;.$
(5.5)

Together with the shadow symmetry of the conformal partial wave (3.3) and the
relation to the conformal blocks (3.2), we can rewrite (5.3) as

$\mathcal{A}_{1^{0}_{0}+2^{0}_{0}\to
3^{0}_{0}+4^{0}_{0}}^{\Delta_{i}}=\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d\Delta_{a}}{2\pi
i}\,\rho_{\Delta_{a}}G^{\Delta_{i}}_{\Delta_{a}}\;.$ (5.6)

Here $\rho_{\Delta_{a}}$ takes the form as

$\displaystyle\rho_{\Delta_{a}}$
$\displaystyle=\int_{m}^{+\infty}\frac{dm_{a}}{(2\pi)^{d+2}}\frac{im_{a}^{d+1}}{-m_{a}^{2}}\mu_{\Delta_{a}}^{-1}I_{\Delta_{a}}S^{\Delta_{3},\Delta_{4}}_{\widetilde{\Delta}_{a}}$
(5.7)
$\displaystyle=-i\,\pi^{-\frac{3d}{2}-2}2^{-\Delta_{1234}-d-2}2\pi\delta_{c}\left(i(\Delta_{1234}-d-4)\right)\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\frac{\Delta_{1a,2}}{2},\frac{\Delta_{2a,1}}{2},\frac{\Delta_{3a,4}}{2},\frac{\Delta_{4a,3}}{2}}{\Delta_{a},\Delta_{a}-\frac{d}{2}}\biggr{]}\;,$
(5.8)

where in the second line we have performed the integral over $m_{a}$ by the
$\delta$-function with complex argument (A.16).

We find that $\rho_{\Delta_{a}}$ does not have $\Delta_{a}$-poles to the right
of the principal series. This implies if the conformal cross-ratio $\chi$
satisfies $|\chi|<1$, enclosing the contour to the right half
$\Delta_{a}$-plane leads to a vanishing conformal block expansion. While if
$|\chi|>1$ the integrand in (5.6) does not decay to zero as
$\operatorname{Re}(\Delta_{a})\to+\infty$, and we cannot enclose the contour.
These two aspects reflect the fact that four-point massless celestial
amplitudes contain a nonanalytic factor $\theta(|\chi|-1)$, and the celestial
amplitude $\mathcal{A}^{\Delta_{i}}_{1^{0}_{0}+2^{0}_{0}\rightarrow
3^{0}_{0}+4^{0}_{0}}$ does not have a proper conformal block expansion [12].

To fix this issue, we will use the shadow conformal basis for outgoing states
[12]. Following the same steps, we find that the celestial amplitude in this
basis takes the form as

$\mathcal{A}_{1^{0}_{0}+2^{0}_{0}\to\mathcal{S}[3^{0}_{0}]+\mathcal{S}[4^{0}_{0}]}^{\Delta_{1},\Delta_{2},\widetilde{\Delta}_{3},\widetilde{\Delta}_{4}}=\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d\Delta_{a}}{2\pi
i}\,\sigma_{\Delta_{a}}G^{\Delta_{1},\Delta_{2},\widetilde{\Delta}_{3},\widetilde{\Delta}_{4}}_{\Delta_{a}}\;,$
(5.9)

where $\sigma_{\Delta_{a}}$ is given by

$\displaystyle\sigma_{\Delta_{a}}$
$\displaystyle=-i\,\pi^{-\frac{3d}{2}-2}2^{-\Delta_{1234}-d-2}2\pi\delta_{c}\left(i(\Delta_{1234}-d-4)\right)$
(5.10)
$\displaystyle\phantom{{}={}}\times\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\Delta_{3},\Delta_{4},\mathcolor{red}{\frac{2d-\Delta_{34a}}{2}},\frac{\Delta_{1a,2}}{2},\frac{\Delta_{2a,1}}{2},\frac{\Delta_{a,34}+d}{2},\frac{\Delta_{3a,4}}{2},\frac{\Delta_{4a,3}}{2}}{d-\Delta_{3},d-\Delta_{4},\mathcolor{red}{\frac{\Delta_{34,a}}{2}},\Delta_{a},\Delta_{a}-\frac{d}{2},\frac{\Delta_{34a}-d}{2}}\biggr{]}\;$

Then there are double-trace $\Delta_{a}$-poles from the factor
$\Gamma[\frac{2d-\Delta_{34a}}{2}]$ at

$\displaystyle\Delta_{a}=\widetilde{\Delta}_{3}+\widetilde{\Delta}_{4}+2N\;,\quad
N\in\mathbb{N}\;.$ (5.11)

In the rest of this paper, unless specified otherwise we will use the shadow
conformal basis for outgoing states to discuss the conformal block expansion
of celestial amplitudes.

###  5.2 Two massive and two massless scalars with massive exchange

We consider a theory containing the interaction vertex
$\frac{1}{2}\phi\Phi^{2}$, where $\phi$ is a massless scalar and $\Phi$ is a
massive scalar with mass $m$. In this theory, the $s$-channel tree-level
scattering amplitude of $\Phi+\phi\to\Phi+\phi$ is given by

$\displaystyle\mathcal{M}_{1^{0}_{m}+2^{0}_{0}\to
3^{0}_{m}+4^{0}_{0}}=\frac{i}{(p_{1}+q_{2})^{2}+m^{2}}\delta^{(d+2)}(p_{1}+q_{2}-p_{3}-q_{4})\;.$
(5.12)

Using the split representation (2.26) and the integral expression of the
conformal partial wave (3.1), the corresponding celestial amplitude of (5.12)
can be written as

$\displaystyle\begin{split}\mathcal{A}_{1^{0}_{m}+2^{0}_{0}\to\mathcal{S}[3^{0}_{m}]+\mathcal{S}[4^{0}_{0}]}^{\Delta_{1},\Delta_{2},\widetilde{\Delta}_{3},\widetilde{\Delta}_{4}}&=\int_{m}^{+\infty}\frac{dm_{a}}{(2\pi)^{d+2}}\frac{im_{a}^{d+1}}{-m_{a}^{2}+m^{2}}\int_{\frac{d}{2}}^{\frac{d}{2}+i\infty}\frac{d\Delta_{a}}{2\pi
i\,\mu_{\Delta_{a}}}I_{\Delta_{a}}\Psi^{\Delta_{1},\Delta_{2},\widetilde{\Delta}_{3},\widetilde{\Delta}_{4}}_{\Delta_{a}}\;,\end{split}$
(5.13)

where the density function is

$I_{\Delta_{a}}=C_{1^{0}_{m}+2^{0}_{0}\to
a^{0}_{m_{a}}}^{\Delta_{1},\Delta_{2},\Delta_{a}}C_{a^{0}_{m_{a}}\to\mathcal{S}[3^{0}_{m}]+\mathcal{S}[4^{0}_{0}]}^{\widetilde{\Delta}_{a},\widetilde{\Delta}_{3},\widetilde{\Delta}_{4}}\;.$
(5.14)

Following the same steps as in the previous subsection, we can rewrite (5.13)
as

$\mathcal{A}_{1^{0}_{m}+2^{0}_{0}\to\mathcal{S}[3^{0}_{m}]+\mathcal{S}[4^{0}_{0}]}^{\Delta_{1},\Delta_{2},\widetilde{\Delta}_{3},\widetilde{\Delta}_{4}}=\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d\Delta_{a}}{2\pi
i}\,\sigma_{\Delta_{a}}G^{\Delta_{1},\Delta_{2},\widetilde{\Delta}_{3},\widetilde{\Delta}_{4}}_{\Delta_{a}}\;,$
(5.15)

where $\sigma_{\Delta_{a}}$ takes the form as

$\displaystyle\sigma_{\Delta_{a}}$
$\displaystyle=\int_{m}^{+\infty}\frac{dm_{a}}{(2\pi)^{d+2}}\frac{im_{a}^{d+1}}{-m_{a}^{2}+m^{2}}\mu_{\Delta_{a}}^{-1}I^{\Delta_{i}}_{\Delta_{a}}S_{\widetilde{\Delta}_{a}}^{\widetilde{\Delta}_{3},\widetilde{\Delta}_{4}}$
(5.16)
$\displaystyle=(\cdots)\int_{m}^{+\infty}dm_{a}\,m_{a}(m_{a}^{2}-m^{2})^{\Delta_{24}-3}\,{}_{2}F_{1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{\frac{\Delta_{2a,1}}{2},\frac{\Delta_{12a}-d}{2}}{\Delta_{2}};1-\frac{m_{a}^{2}}{m^{2}}\biggr{)}\,{}_{2}F_{1}\biggl{(}\genfrac{.}{.}{0.0pt}{}{\frac{d+\Delta_{4,3a}}{2},\frac{\Delta_{34,a}}{2}}{\Delta_{4}};1-\frac{m_{a}^{2}}{m^{2}}\biggr{)}\;.$

Using the Mellin-Barnes representation of the hypergeometric function (A.4)
with Mellin parameters $s$ and $t$, and changing the integration variable from
$m_{a}$ to $\alpha=m_{a}^{2}-m^{2}$, the mass integral can be separated as the
$\delta$-function with complex argument (A.16),
$\int_{0}^{+\infty}d\alpha\,\alpha^{s+t+\Delta_{24}-3}$. Then we can evaluate
the integral over $s$, leading to

$\displaystyle\sigma_{\Delta_{a}}$
$\displaystyle=-i\,\pi^{-\frac{d}{2}-2}2^{-\Delta_{24}-d-3}m^{\Delta_{24}-d-4}\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\Delta_{4},\mathcolor{red}{\frac{\Delta_{12,a}}{2}},\mathcolor{red}{\frac{2d-\Delta_{34a}}{2}},\frac{\Delta_{1a,2}}{2},\frac{\Delta_{a,34}+d}{2},\frac{\Delta_{3a,4}}{2},\frac{\Delta_{4a,3}}{2}}{\Delta_{1},d-\Delta_{3},d-\Delta_{4},\mathcolor{red}{\frac{\Delta_{4,3a}+d}{2}},\mathcolor{red}{\frac{\Delta_{34,a}}{2}},\Delta_{a},\Delta_{a}-\frac{d}{2}}\biggr{]}$
$\displaystyle\phantom{{}={}}\times\int_{-i\infty}^{+i\infty}\frac{dt}{2\pi
i}\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{-t,\Delta_{24}+t-2,\mathcolor{red}{\frac{\Delta_{4,3a}+d}{2}+t},\mathcolor{red}{\frac{\Delta_{34,a}}{2}+t},\frac{\Delta_{a,12}-2\Delta_{4}+4}{2}-t,\frac{\Delta_{1a,2}-d-2\Delta_{4}+4}{2}-t}{-\Delta_{4}-t+2,\Delta_{4}+t}\biggr{]}\;.$

By the method of pole pinching, we note that performing the $t$-integral does
not produce $\Delta_{a}$-poles. The $\Delta_{a}$-poles from the factors
$\Gamma[-t]$ and $\Gamma[\frac{\Delta_{4,3a}+d}{2}+t]$ get canceled by
$\Gamma[\frac{\Delta_{4,3a}+d}{2}]$ in the denominator, and the
$\Delta_{a}$-poles from the factors $\Gamma[-t]$ and
$\Gamma[\frac{\Delta_{34,a}}{2}+t]$ get canceled by
$\Gamma[\frac{\Delta_{34,a}}{2}]$ in the denominator. Thus, only the factors
$\Gamma[\frac{\Delta_{12,a}}{2}]$ and $\Gamma[\frac{2d-\Delta_{34a}}{2}]$
provide two series of double-trace $\Delta_{a}$-poles at

$\displaystyle\Delta_{a}=\Delta_{12}+2N\;,\quad N\in\mathbb{N}\;.$ (5.17)
$\displaystyle\Delta_{a}=\widetilde{\Delta}_{3}+\widetilde{\Delta}_{4}+2N\;,\quad
N\in\mathbb{N}\;.$ (5.18)

We note that the feature of double-trace exchange is similar to that of AdS.
But unlike AdS/CFT, the conformal dimension here is not related to the mass in
the bulk.

##  6 Higher-point celestial amplitudes

Following the previous section, we will use the split representation (2.26) to
analyze higher-point celestial amplitudes. Compared to the four-point
celestial amplitudes, we observe that there are new exchanged operator
appearing in the conformal block expansions of higher-point ones.
Specifically, in the $2d$ five-point case, the conformal dimensions of the
dominant new operators are given by $(\Delta_{12}-2)$ and
$(\Delta_{12}+2\Delta_{3}-4)$, which supports the findings of two-particle
operators in [16, Section 4].

For convenience, we introduce the following naming of poles:

type-1: | $\Delta=\Delta_{i_{1}i_{2}\cdots}+\text{integer}\;,$
---|---
type-2: | $\Delta=\Delta_{i_{1}i_{2}\cdots}+2\Delta_{j}+\text{integer}\;,$
type-3: | $\Delta=\Delta_{i_{1}i_{2}\cdots}+2\Delta_{j}+2\Delta_{k}+\text{integer}\;,$
$\cdots$ |
(6.5)

###  6.1 Five massless scalars with massless exchange

$1_{0}^{0}$$2_{0}^{0}$$3_{0}^{0}$$4_{0}^{0}$$5_{0}^{0}$ Figure 1: A tree-level
diagram for $1_{0}^{0}+2_{0}^{0}+3_{0}^{0}\to 4_{0}^{0}+5_{0}^{0}$. The
external and internal particles are all massless scalars.

In the $\phi^{3}$ theory, the contribution of the tree-level diagram 1 to the
scattering amplitude is

$\displaystyle\mathcal{M}_{1_{0}^{0}+2_{0}^{0}+3_{0}^{0}\to
4_{0}^{0}+5_{0}^{0}}=i^{2}\frac{\delta^{(d+2)}(q_{1}+q_{2}+q_{3}-q_{4}-q_{5})}{(q_{1}+q_{2})^{2}(q_{1}+q_{2}+q_{3})^{2}}\;.$
(6.6)

Using the split representation (2.26) and the integral expression of the five-
point conformal partial wave (3.8), we get the celestial amplitude of (6.6) as

$\displaystyle\mathcal{A}_{1_{0}^{0}+2_{0}^{0}+3_{0}^{0}\to
4_{0}^{0}+5_{0}^{0}}^{\Delta_{i}}$
$\displaystyle=\int_{0}^{+\infty}\frac{dm_{a}}{(2\pi)^{d+2}}(-im_{a}^{d-1})\int_{m_{a}}^{+\infty}\frac{dm_{b}}{(2\pi)^{d+2}}(-im_{b}^{d-1})$
(6.7)
$\displaystyle\phantom{{}={}}\times\int_{\frac{d}{2}}^{\frac{d}{2}+i\infty}\frac{d\Delta_{a}}{2\pi
i\,\mu_{\Delta_{a}}}\int_{\frac{d}{2}}^{\frac{d}{2}+i\infty}\frac{d\Delta_{b}}{2\pi
i\,\mu_{\Delta_{b}}}I_{\Delta_{a},\Delta_{b}}\Psi^{\Delta_{i}}_{\Delta_{a},\Delta_{b}}\;,$
(6.8)

where the density function is

$I_{\Delta_{a},\Delta_{b}}=C_{1^{0}_{0}+2^{0}_{0}\to
a^{0}_{m_{a}}}^{\Delta_{1},\Delta_{2},\Delta_{a}}C_{a^{0}_{m_{a}}+3^{0}_{0}\to
b^{0}_{m_{b}}}^{\widetilde{\Delta}_{a},\Delta_{3},\Delta_{b}}C_{b^{0}_{m_{b}}\to
4^{0}_{0}+5^{0}_{0}}^{\widetilde{\Delta}_{b},\Delta_{4},\Delta_{5}}\;.$ (6.9)

From (4.11), the density function satisfies

$I_{\widetilde{\Delta}_{a},\Delta_{b}}=\frac{S_{\Delta_{a}}^{\Delta_{1},\Delta_{2}}}{S_{\Delta_{a}}^{\Delta_{3},\Delta_{b}}}I_{\Delta_{a},\Delta_{b}}\;,\quad
I_{\Delta_{a},\widetilde{\Delta}_{b}}=\frac{S_{\Delta_{b}}^{\widetilde{\Delta}_{a},\Delta_{3}}}{S_{\Delta_{b}}^{\Delta_{4},\Delta_{5}}}I_{\Delta_{a},\Delta_{b}}\;.$
(6.10)

The by the shadow symmetry of the conformal partial wave (3.9) and the
relation to the conformal blocks (3.15), we can rewrite (6.7) as a double
contour integral,

$\displaystyle\begin{split}\mathcal{A}_{1_{0}^{0}+2_{0}^{0}+3_{0}^{0}\to
4_{0}^{0}+5_{0}^{0}}^{\Delta_{i}}&=\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d\Delta_{a}}{2\pi
i}\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d\Delta_{b}}{2\pi
i}\rho_{\Delta_{a},\Delta_{b}}G^{\Delta_{i}}_{\Delta_{a},\Delta_{b}}\;,\end{split}$
(6.11)

where

$\displaystyle\begin{split}&\phantom{{}={}}\rho_{\Delta_{a},\Delta_{b}}=\int_{0}^{+\infty}\frac{dm_{a}}{(2\pi)^{d+2}}(-im_{a}^{d-1})\int_{m_{a}}^{+\infty}\frac{dm_{b}}{(2\pi)^{d+2}}(-im_{b}^{d-1})\mu^{-1}_{\Delta_{a}}\mu^{-1}_{\Delta_{b}}I_{\Delta_{a},\Delta_{b}}S^{\Delta_{3},\Delta_{b}}_{\widetilde{\Delta}_{a}}S^{\Delta_{4},\Delta_{5}}_{\widetilde{\Delta}_{b}}\;.\end{split}$
(6.12)

We first use the Mellin-Barnes representation of the hypergeometric function
(A.4) with the Mellin parameter $s$, then integrate out $\alpha$ by the change
of variable $\alpha=m_{b}^{2}-m_{a}^{2}$,

$\displaystyle\rho_{\Delta_{a},\Delta_{b}}$
$\displaystyle=(\cdots)\int_{0}^{+\infty}dm_{a}m_{a}^{\Delta_{12345}-d-7}\int_{-i\infty}^{+i\infty}\frac{ds}{2\pi
i}\,$ (6.13)
$\displaystyle\phantom{{}={}}\times\Gamma\left[{-s,-s-\Delta_{3}+\frac{-\Delta_{45b}+d+4}{2},s+\frac{\Delta_{3b,a}}{2},s+\frac{\Delta_{3ab}-d}{2}}\right]\;.$

The $m_{a}$-integral gives a $\delta$-function with complex argument (A.16),
and the $s$-integral can be done by the first Barnes lemma (A.2), leading to

$\displaystyle\rho_{\Delta_{a},\Delta_{b}}$
$\displaystyle=-\pi^{-\frac{5d}{2}-4}2^{-\Delta_{12345}-2d-5}2\pi\delta_{c}\left(i(\Delta_{12345}-d-6)\right)$
(6.14)
$\displaystyle\phantom{{}={}}\times\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\mathcolor{red}{\frac{-\Delta_{345a}+d+4}{2}},\frac{\Delta_{1a,2}}{2},\frac{\Delta_{2a,1}}{2},\frac{\Delta_{a,345}+4}{2},\mathcolor{blue}{\frac{\Delta_{3a,b}}{2}},\frac{\Delta_{4b,5}}{2},\frac{\Delta_{5b,4}}{2},\mathcolor{red}{\frac{\Delta_{3b,a}}{2}},\frac{\Delta_{ab,3}}{2},\frac{\Delta_{3ab}-d}{2}}{\Delta_{a},\Delta_{a}-\frac{d}{2},\mathcolor{blue}{\frac{-\Delta_{45b}+d+4}{2}},\Delta_{b},\Delta_{b}-\frac{d}{2},\frac{\Delta_{b,45}+4}{2}}\biggr{]}\;.$

If we first perform the $\Delta_{a}$-integral, there are two series of
$\Delta_{a}$-poles to the right of the principal series. But in any case we
cannot enclose the contour of $\Delta_{b}$, and the conformal block expansion
can be schematically written as

$\displaystyle\mathcal{A}_{1_{0}^{0}+2_{0}^{0}+3_{0}^{0}\to
4_{0}^{0}+5_{0}^{0}}^{\Delta_{i}}$
$\displaystyle=\sum_{N=0}^{+\infty}\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d\Delta_{b}}{2\pi
i}\bigl{(}\cdots G^{\Delta_{i}}_{\Delta_{12}-2+2N,\Delta_{b}}+\cdots
G^{\Delta_{i}}_{\Delta_{3b}+2N,\Delta_{b}}\bigr{)}\;.$ (6.15)

Similarly, if firstly performing the $\Delta_{b}$-integral, we cannot enclose
the contour of $\Delta_{a}$ to the right of the principal series, and the
conformal block expansion is schematically

$\displaystyle\mathcal{A}_{1_{0}^{0}+2_{0}^{0}+3_{0}^{0}\to
4_{0}^{0}+5_{0}^{0}}^{\Delta_{i}}$
$\displaystyle=\sum_{N=0}^{+\infty}\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d\Delta_{a}}{2\pi
i}\cdots G^{\Delta_{i}}_{\Delta_{a},\Delta_{3a}+2N}\;.$ (6.16)

The fact that the double contour integral (6.11) cannot be jointly enclosed
can be understood that the origin in the cross-ratio space is disallowed by
the momentum conservation. As discussed in Section 5.1, this has already
happened in the case of four-point celestial amplitudes of four massless
scalars: the momentum conservation contributes a non-analytic factor
$\theta(|\chi|-1)$ to $\mathcal{A}_{1_{0}^{0}+2_{0}^{0}\to
4_{0}^{0}+5_{0}^{0}}^{\Delta_{i}}$ that obstructs the $s$-channel conformal
block expansion.

Instead, we choose the shadow conformal basis for outgoing states. Comparing
with (6.11), the new double contour integral is

$\displaystyle\mathcal{A}_{1_{0}^{0}+2_{0}^{0}+3_{0}^{0}\to\mathcal{S}[4_{0}^{0}]+\mathcal{S}[5_{0}^{0}]}^{\Delta_{1},\Delta_{2},\Delta_{3},\widetilde{\Delta}_{4},\widetilde{\Delta}_{5}}$
$\displaystyle=\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d\Delta_{a}}{2\pi
i}\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d\Delta_{b}}{2\pi
i}\sigma_{\Delta_{a},\Delta_{b}}G_{\Delta_{a},\Delta_{b}}^{\Delta_{1},\Delta_{2},\Delta_{3},\widetilde{\Delta}_{4},\widetilde{\Delta}_{5}}\;,$
(6.17)

where

$\displaystyle\sigma_{\Delta_{a},\Delta_{b}}$
$\displaystyle=\rho_{\Delta_{a},\Delta_{b}}N_{\Delta_{4}}S_{\Delta_{4}}^{\Delta_{5},\widetilde{\Delta}_{b}}N_{\Delta_{5}}S_{\Delta_{5}}^{\widetilde{\Delta}_{4},\widetilde{\Delta}_{b}}S_{\widetilde{\Delta}_{b}}^{\widetilde{\Delta}_{4},\widetilde{\Delta}_{5}}/S_{\widetilde{\Delta}_{b}}^{\Delta_{4},\Delta_{5}}$
(6.18)
$\displaystyle=-\pi^{-\frac{5d}{2}-4}2^{-\Delta_{12345}-2d-5}2\pi\delta_{c}\left(i(\Delta_{12345}-d-6)\right)$
$\displaystyle\phantom{{}={}}\times\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\Delta_{4},\Delta_{5},\frac{\Delta_{1a,2}}{2},\frac{\Delta_{2a,1}}{2},\mathcolor{blue}{\frac{\Delta_{3a,b}}{2}},\frac{\Delta_{4b,5}}{2},\frac{\Delta_{5b,4}}{2},\mathcolor{red}{\frac{\Delta_{3b,a}}{2}},\frac{\Delta_{ab,3}}{2}}{d-\Delta_{4},d-\Delta_{5},\Delta_{a},\Delta_{a}-\frac{d}{2},\mathcolor{blue}{\frac{\Delta_{45,b}}{2}},\Delta_{b},\Delta_{b}-\frac{d}{2}}\biggr{]}$
$\displaystyle\phantom{{}={}}\times\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\mathcolor{red}{\frac{-\Delta_{345a}+d+4}{2}},\frac{\Delta_{a,345}+4}{2},\mathcolor{blue}{\frac{2d-\Delta_{45b}}{2}},\frac{\Delta_{b,45}+d}{2},\frac{\Delta_{3ab}-d}{2}}{\mathcolor{blue}{\frac{-\Delta_{45b}+d+4}{2}},\frac{\Delta_{b,45}+4}{2},\frac{\Delta_{45b}-d}{2}}\biggr{]}\;.$

We perform the integrals in the order $\Delta_{a}\to\Delta_{b}$. There are two
series of $\Delta_{a}$-poles (to the right of the principal series).

  1. 1.

The factor $\Gamma[\frac{-\Delta_{345a}+d+4}{2}]$ provides double-trace
$\Delta_{a}$-poles at

$\Delta_{a}=\Delta_{12}-2+2N\;,\quad N\in\mathbb{N}\;.$ (6.19)

After integrating out $\Delta_{a}$, the $\Delta_{b}$-poles are captured by

$\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\frac{\Delta_{123,b}+d-6}{2},\frac{\Delta_{123,b}+2N-2}{2}}{\frac{\Delta_{123,b}-2}{2}}\biggr{]}\;.$
(6.20)

  2. 2.

The factor $\Gamma[\frac{\Delta_{3b,a}}{2}]$ provides “running”
$\Delta_{a}$-poles at

$\Delta_{a}=\Delta_{3b}+2N\;,\quad N\in\mathbb{N}\;.$ (6.21)

After integrating out $\Delta_{a}$, the $\Delta_{b}$-poles are captured by

$\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\frac{\Delta_{123,b}+d-6}{2}}{\frac{\Delta_{123,b}-2}{2}}\biggr{]}\;.$
(6.22)

Due to the factors (6.20) and (6.22), the pole structure depends on the
dimension $d$, and we summarize the result in Table 1. The type-1
$\Delta_{a}$-poles (6.19) accompanied with the type-1 $\Delta_{b}$-poles
(6.20) are present in any dimension, but the type-2 $\Delta_{a}$-poles (6.21)
disappear in even dimensions except $d=2$.

Now we can compare with the results in [16]. When $d=2$, the leading type-1
$\Delta_{a}$-pole is $\Delta_{a}=\Delta_{12}-2$, matching with the conformal
dimension of the single-particle operator in [16, Equation 4.1]; the leading
type-2 $\Delta_{a}$-pole is $\Delta_{a}=\Delta_{12}+2\Delta_{3}-4$, matching
with the conformal dimension of the two-particle operator in [16, Equation
4.1].

At this stage, it is unclear whether the subleading $\Delta_{a}$-poles
correspond to single- or two particle exchanged operators, so we will retain
the current naming convention (6.5). We leave this question in a future work.
Another concern arises in even dimensions, the first bulk dimension $(d+2)$ at
which the type-2 poles disappear coincides with the critical dimension of the
$\phi^{3}$ theory. Is this merely a coincidence, or are there more
informations of the bulk theory hidden behind the tree-level celestial
amplitudes?

###  6.2 One massive and four massless scalars

To test the stability of the type-2 poles, in the tree-level diagram 1, we
replace the first incoming massless scalar with a massive one, and choose the
first propagator as massless or massive. The computation is similar to that in
Section 6.1, and in this section, we will only present the celestial
amplitudes in the shadow conformal basis for outgoing states. As before, the
celestial amplitudes are represented as

$\mathcal{A}_{1_{m}^{0}+2_{0}^{0}+3_{0}^{0}\to\mathcal{S}[4_{0}^{0}]+\mathcal{S}[5_{0}^{0}]}^{\Delta_{1},\Delta_{2},\Delta_{3},\widetilde{\Delta}_{4},\widetilde{\Delta}_{5}}=\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d\Delta_{a}}{2\pi
i}\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d\Delta_{b}}{2\pi
i}\sigma_{\Delta_{a},\Delta_{b}}G_{\Delta_{a},\Delta_{b}}^{\Delta_{1},\Delta_{2},\Delta_{3},\widetilde{\Delta}_{4},\widetilde{\Delta}_{5}}\;.$
(6.23)

####  6.2.1 Massless exchanges

The scattering amplitude is

$\displaystyle\mathcal{M}_{1_{m}^{0}+2_{0}^{0}+3_{0}^{0}\to
4_{0}^{0}+5_{0}^{0}}=i^{2}\frac{\delta^{(d+2)}(q_{1}+q_{2}+q_{3}-q_{4}-q_{5})}{(q_{1}+q_{2})^{2}(q_{1}+q_{2}+q_{3})^{2}}\;,$
(6.24)

and the corresponding celestial amplitude (6.23) is characterized by

$\displaystyle\sigma_{\Delta_{a},\Delta_{b}}$
$\displaystyle=-\pi^{-2(d+2)}2^{-\Delta_{2345}-2d-6}m^{\Delta_{2345}-d-6}$
(6.25)
$\displaystyle\phantom{{}={}}\times\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\Delta_{4},\Delta_{5},\mathcolor{red}{\frac{\Delta_{12,a}}{2}},\frac{\Delta_{1a,2}}{2},\frac{\Delta_{2a,1}}{2},\mathcolor{blue}{\frac{\Delta_{3a,b}}{2}},\frac{\Delta_{4b,5}}{2},\frac{\Delta_{5b,4}}{2},\mathcolor{red}{\frac{\Delta_{3b,a}}{2}},\frac{\Delta_{ab,3}}{2}}{\Delta_{1},d-\Delta_{4},d-\Delta_{5},\Delta_{a},\Delta_{a}-\frac{d}{2},\mathcolor{blue}{\frac{\Delta_{45,b}}{2}},\Delta_{b},\Delta_{b}-\frac{d}{2}}\biggr{]}$
$\displaystyle\phantom{{}={}}\times\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\frac{-\Delta_{12345}+d+6}{2},\frac{\Delta_{1,2345}+6}{2},\mathcolor{red}{\frac{-\Delta_{345a}+d+4}{2}},\frac{\Delta_{12a}-d}{2},\frac{\Delta_{a,345}+4}{2},\mathcolor{blue}{\frac{2d-\Delta_{45b}}{2}},\frac{\Delta_{b,45}+d}{2},\frac{\Delta_{3ab}-d}{2}}{\mathcolor{red}{\frac{-\Delta_{345a}+d+6}{2}},\frac{\Delta_{a,345}+6}{2},\mathcolor{blue}{\frac{-\Delta_{45b}+d+4}{2}},\frac{\Delta_{b,45}+4}{2},\frac{\Delta_{45b}-d}{2}}\biggr{]}\;.$

We first check the massless limit $m\to 0$. After taking into account the
prefactor in (2.23), the factor
$m^{\Delta_{12345}-d-6}\Gamma[\frac{-\Delta_{12345}+d+6}{2}]$ recovers
$2\pi\delta_{c}\left(i(\Delta_{12345}-d-6)\right)$ in (6.18) by the
approximation of the $\delta$-function with complex argument (A.18), leading
to

$\displaystyle\phantom{{}={}}\lim_{m\rightarrow
0}2^{-\Delta_{1}}\pi^{-\frac{d}{2}}m^{\Delta_{1}}\frac{\Gamma[\Delta_{1}]}{\Gamma[\Delta_{1}-\frac{d}{2}]}\mathcal{A}_{1_{m}^{0}+2_{0}^{0}+3_{0}^{0}\to\mathcal{S}[4_{0}^{0}]+\mathcal{S}[5_{0}^{0}]}^{\Delta_{1},\Delta_{2},\Delta_{3},\widetilde{\Delta}_{4},\widetilde{\Delta}_{5}}=\mathcal{A}_{1_{0}^{0}+2_{0}^{0}+3_{0}^{0}\to\mathcal{S}[4_{0}^{0}]+\mathcal{S}[5_{0}^{0}]}^{\Delta_{1},\Delta_{2},\Delta_{3},\widetilde{\Delta}_{4},\widetilde{\Delta}_{5}}\;.$
(6.26)

The pole structure of (6.25) is summarized in Table 2. The names of poles are
chosen by comparing with the massless limit $m\to 0$. For example, the
$\Delta_{a}$-poles at $\Delta_{a}=\Delta_{3,45}+4+2N$ in Table 2 correspond to
the type-2 poles in Table 1. We notice that after turning on the mass of the
first particle, the type-1 poles $\Delta_{a}=\Delta_{12}-2+2N$ in the massless
case split into two parts: $\Delta_{a}=\Delta_{12}+2N$ and
$\Delta_{a}=-\Delta_{345}+d+4$.

####  6.2.2 One massive exchange

The scattering amplitude is

$\displaystyle\mathcal{M}_{1_{m}^{0}+2_{0}^{0}+3_{0}^{0}\to
4_{0}^{0}+5_{0}^{0}}=i^{2}\frac{\delta^{(d+2)}(q_{1}+q_{2}+q_{3}-q_{4}-q_{5})}{\left((q_{1}+q_{2})^{2}+m^{2}\right)(q_{1}+q_{2}+q_{3})^{2}}\;.$
(6.27)

and the corresponding celestial amplitude (6.23) is characterized by

$\displaystyle\sigma_{\Delta_{a},\Delta_{b}}$
$\displaystyle=-\pi^{-2(d+2)}2^{-\Delta_{2345}-2d-6}m^{\Delta_{2345}-d-6}\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\frac{\Delta_{a,345}+4}{2},\mathcolor{blue}{\frac{2d-\Delta_{45b}}{2}},\frac{\Delta_{b,45}+d}{2},\frac{\Delta_{3ab}-d}{2}}{\mathcolor{blue}{\frac{-\Delta_{45b}+d+4}{2}},\frac{\Delta_{b,45}+4}{2},\frac{\Delta_{45b}-d}{2}}\biggr{]}$
(6.28)
$\displaystyle\phantom{{}={}}\times\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\Delta_{4},\Delta_{5},\mathcolor{red}{\frac{\Delta_{12,a}}{2}},\frac{\Delta_{1a,2}}{2},\mathcolor{blue}{\frac{\Delta_{3a,b}}{2}},\frac{\Delta_{4b,5}}{2},\frac{\Delta_{5b,4}}{2},\mathcolor{red}{\frac{\Delta_{3b,a}}{2}},\frac{\Delta_{ab,3}}{2}}{\Delta_{1},d-\Delta_{4},d-\Delta_{5},\Delta_{a},\Delta_{a}-\frac{d}{2},\mathcolor{blue}{\frac{\Delta_{45,b}}{2}},\Delta_{b},\Delta_{b}-\frac{d}{2}}\biggr{]}$
$\displaystyle\phantom{{}={}}\times\int_{-i\infty}^{+i\infty}\frac{ds}{2\pi
i}\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{-s,\Delta_{2}+s-1,\mathcolor{red}{\frac{-\Delta_{345a}+d-2\Delta_{2}+6}{2}-s},\frac{\Delta_{2a,1}}{2}+s,\frac{\Delta_{12a}-d}{2}+s}{\Delta_{2}+s}\biggr{]}\;.$

We first check the massless limit $m\to 0$. By the method of pole pinching,
during the $s$-integral the factors
$\Gamma[\frac{-\Delta_{345a}+d-2\Delta_{2}+6}{2}-s]$ and
$\Gamma[\frac{\Delta_{2a,1}}{2}+s]$ provide a simple pole at
$\Delta_{1}=d+6-\Delta_{2345}$. Then the factor
$\frac{1}{\Delta_{12345}-d-6}m^{\Delta_{12345}-d-6}$ recovers
$2\pi\delta_{c}\left(i(\Delta_{12345}-d-6)\right)$ in (6.18). After performing
the $s$-integral and reading off the residue at
$s=\frac{1}{2}(-\Delta_{345a}-2\Delta_{2}+d+6)$, we obtain the expected
massless limit,

$\displaystyle\phantom{{}={}}\lim_{m\rightarrow
0}2^{-\Delta_{1}}\pi^{-\frac{d}{2}}m^{\Delta_{1}}\frac{\Gamma[\Delta_{1}]}{\Gamma[\Delta_{1}-\frac{d}{2}]}\mathcal{A}_{1_{m}^{0}+2_{0}^{0}+3_{0}^{0}\to\mathcal{S}[4_{0}^{0}]+\mathcal{S}[5_{0}^{0}]}^{\Delta_{1},\Delta_{2},\Delta_{3},\widetilde{\Delta}_{4},\widetilde{\Delta}_{5}}=\mathcal{A}_{1_{0}^{0}+2_{0}^{0}+3_{0}^{0}\to\mathcal{S}[4_{0}^{0}]+\mathcal{S}[5_{0}^{0}]}^{\Delta_{1},\Delta_{2},\Delta_{3},\widetilde{\Delta}_{4},\widetilde{\Delta}_{5}}\;.$
(6.29)

The pole structure of (6.28) is summarized in Table 3. The names of poles are
chosen by comparing with the massless limit $m\to 0$. We notice that after
turning on the mass of the first propagator, the type-1 poles
$\Delta_{a}=-\Delta_{345}+d+4$ in the massive case extend to a series poles at
$\Delta_{a}=-\Delta_{345}+d+4+2N$. This can be understood by the small mass
expansion of the propagator,

$\frac{1}{p^{2}+m^{2}}=\sum_{n=0}^{+\infty}(-1)^{n}\frac{m^{2n}}{(p^{2})^{n+1}}\;,$
(6.30)

and the new $\Delta_{a}$-poles correspond to the higher order terms in this
expansion.

###  6.3 Six massless scalars with massless exchange

$1_{0}^{0}$$2_{0}^{0}$$3_{0}^{0}$$4_{0}^{0}$$5_{0}^{0}$$6_{0}^{0}$ Figure 2:
A tree-level diagram for $1_{0}^{0}+2_{0}^{0}+3_{0}^{0}+4_{0}^{0}\to
5_{0}^{0}+6_{0}^{0}$. The external and internal particles are all massless
scalars.

In this subsection we explore whether there are further new exchanged
operators in the tree-level celestial amplitudes in the $\phi^{3}$ theory. We
consider the scattering amplitude of the tree-level diagram 2,

$\displaystyle\mathcal{M}_{1_{0}^{0}+2_{0}^{0}+3_{0}^{0}\to
4_{0}^{0}+5_{0}^{0}}=i^{3}\frac{\delta^{(d+2)}(q_{1}+q_{2}+q_{3}+q_{4}-q_{5}-q_{6})}{(q_{1}+q_{2})^{2}(q_{1}+q_{2}+q_{3})^{2}(q_{1}+q_{2}+q_{3}+q_{4})^{2}}\;.$
(6.31)

Using the split representation (2.26) and the integral expression of the six-
point conformal partial wave (3.16), the celestial amplitude of (6.31) is

$\displaystyle\mathcal{A}_{1_{0}^{0}+2_{0}^{0}+3_{0}^{0}+4_{0}^{0}\to\mathcal{S}[5_{0}^{0}]+\mathcal{S}[6_{0}^{0}]}^{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4},\widetilde{\Delta}_{5},\widetilde{\Delta}_{6}}$
$\displaystyle=\int_{0}^{+\infty}dm_{a}\int_{m_{a}}^{+\infty}dm_{b}\int_{m_{b}}^{+\infty}dm_{c}\left(\prod_{n\in\\{a,b,c\\}}\frac{-im_{n}^{d-1}}{(2\pi)^{d+2}}\int_{\frac{d}{2}}^{\frac{d}{2}+i\infty}\frac{d\Delta_{n}}{2\pi
i\,\mu_{\Delta_{n}}}\right)$ $\displaystyle\times C_{1^{0}_{0}+2^{0}_{0}\to
a^{0}_{m_{a}}}^{\Delta_{1},\Delta_{2},\Delta_{a}}C_{a^{0}_{m_{a}}+3^{0}_{0}\to
b^{0}_{m_{b}}}^{\widetilde{\Delta}_{a},\Delta_{3},\Delta_{b}}C_{b^{0}_{m_{b}}+4^{0}_{0}\to
c^{0}_{m_{c}}}^{\widetilde{\Delta}_{b},\Delta_{4},\Delta_{c}}C_{c^{0}_{m_{c}}\to\mathcal{S}[5_{0}^{0}]+\mathcal{S}[6_{0}^{0}]}^{\widetilde{\Delta}_{c},\widetilde{\Delta}_{5},\widetilde{\Delta}_{6}}\Psi^{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4},\widetilde{\Delta}_{5},\widetilde{\Delta}_{6}}_{\Delta_{a},\Delta_{b},\Delta_{c}}\;.$
(6.32)

There are two hypergeometric functions in the integrand, and by (A.4) we
rewrite them as contour integrals of $s$ and $t$. Then the integrals of
$m_{b}$ and $m_{c}$ can be done by the change of variables
$\alpha=m_{b}^{2}-m_{a}^{2},\,\beta=m_{c}^{2}-m_{b}^{2}$, and the
$m_{a}$-integral gives a $\delta$-function with complex argument (A.16).
Finally we integrate out $s$ and $t$ by the first Barnes lemma (A.2). The
result is

$\displaystyle\mathcal{A}_{1_{0}^{0}+2_{0}^{0}+3_{0}^{0}+4_{0}^{0}\to\mathcal{S}[5_{0}^{0}]+\mathcal{S}[6_{0}^{0}]}^{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4},\widetilde{\Delta}_{5},\widetilde{\Delta}_{6}}$
$\displaystyle=\left(\prod_{n\in\\{a,b,c\\}}\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d\Delta_{n}}{2\pi
i}\right)\sigma_{\Delta_{a},\Delta_{b},\Delta_{c}}G_{\Delta_{a},\Delta_{b},\Delta_{c}}^{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4},\widetilde{\Delta}_{5},\widetilde{\Delta}_{6}}\;,$
(6.33)

where

$\displaystyle\sigma_{\Delta_{a},\Delta_{b},\Delta_{c}}$
$\displaystyle=-i\,\pi^{-\frac{7d}{2}-6}2^{-\Delta_{123456}-3d-8}2\pi\delta_{c}\left(i(\Delta_{123456}-d-8)\right)$
$\displaystyle\phantom{{}={}}\times\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\Delta_{5},\Delta_{6},\frac{\Delta_{1a,2}}{2},\frac{\Delta_{2a,1}}{2},\mathcolor{blue}{\frac{\Delta_{3a,b}}{2}},\mathcolor{red}{\frac{\Delta_{3b,a}}{2}},\frac{\Delta_{ab,3}}{2},\mathcolor{orange}{\frac{\Delta_{4b,c}}{2}},\frac{\Delta_{5c,6}}{2},\frac{\Delta_{6c,5}}{2},\mathcolor{blue}{\frac{\Delta_{4c,b}}{2}},\frac{\Delta_{bc,4}}{2}}{d-\Delta_{5},d-\Delta_{6},\Delta_{a},\Delta_{a}-\frac{d}{2},\Delta_{b},\Delta_{b}-\frac{d}{2},\mathcolor{orange}{\frac{\Delta_{56,c}}{2}},\Delta_{c},\Delta_{c}-\frac{d}{2}}\biggr{]}$
(6.34)
$\displaystyle\phantom{{}={}}\times\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\mathcolor{red}{\frac{-\Delta_{3456a}+d+6}{2}},\frac{\Delta_{a,3456}+6}{2},\mathcolor{blue}{\frac{-\Delta_{456b}+d+4}{2}},\frac{\Delta_{b,456}+4}{2},\frac{\Delta_{3ab}-d}{2},\mathcolor{orange}{\frac{2d-\Delta_{56c}}{2}},\frac{\Delta_{c,56}+d}{2},\frac{\Delta_{4bc}-d}{2}}{\mathcolor{blue}{\frac{-\Delta_{456b}+d+6}{2}},\frac{\Delta_{b,456}+6}{2},\mathcolor{orange}{\frac{-\Delta_{56c}+d+4}{2}},\frac{\Delta_{c,56}+4}{2},\frac{\Delta_{56c}-d}{2}}\biggr{]}\;.$

We perform the triple contour integrals in (6.33) in the order
$\Delta_{a}\to\Delta_{b}\to\Delta_{c}$. Firstly there are two series of
$\Delta_{a}$-poles (to the right of the principal series).

  1. 1.

The factor $\Gamma[\frac{-\Delta_{3456a}+d+6}{2}]$ provides the single-
particle $\Delta_{a}$-poles at

$\Delta_{a}=\Delta_{12}-2+2N\;,\quad N\in\mathbb{N}\;.$ (6.35)

After integrating out $\Delta_{a}$, there are three types of
$\Delta_{b}$-poles.

    1. (a)

The first type is an alone single-particle $\Delta_{b}$-pole at

$\Delta_{b}=\Delta_{123}-4\;.$ (6.36)

Then after integrating out $\Delta_{b}$, the $\Delta_{c}$-poles are captured
by

$\Gamma\left[\frac{\Delta_{1234,c}+d-8}{2}\right]\;.$ (6.37)

    2. (b)

During the $\Delta_{a}$-integral, the factors
$\Gamma[\frac{-\Delta_{3456a}+d+6}{2}]$ and $\Gamma[\frac{\Delta_{3a,b}}{2}]$
provide single-particle $\Delta_{b}$-poles at

$\Delta_{b}=\Delta_{123}-2+2N+2M\;,\quad M\in\mathbb{N}\;.$ (6.38)

Then after integrating out $\Delta_{b}$, the $\Delta_{c}$-poles are captured
by

$\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\frac{\Delta_{1234,c}+d-8}{2},\frac{\Delta_{1234,c}+2N+2M-2}{2}}{\frac{\Delta_{1234,c}-4}{2}}\biggr{]}\;.$
(6.39)

    3. (c)

The factor $\Gamma[\frac{\Delta_{4c,b}}{2}]$ provides a series of “running”
$\Delta_{b}$-poles at

$\Delta_{b}=\Delta_{4c}+2M\;,\quad M\in\mathbb{N}\;,$ (6.40)

Then after integrating out $\Delta_{b}$, the $\Delta_{c}$-poles are captured
by

$\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\frac{\Delta_{1234,c}+d-8}{2}}{\frac{\Delta_{1234,c}-4}{2}}\biggr{]}\;.$
(6.41)

  2. 2.

The factor $\Gamma[\frac{\Delta_{3b,a}}{2}]$ provides a series of “running”
$\Delta_{a}$-poles at

$\Delta_{a}=\Delta_{3b}+2N\;,\quad N\in\mathbb{N}\;.$ (6.42)

After integrating out $\Delta_{a}$, there are two types of $\Delta_{b}$-poles.

    1. (a)

The first type of poles of $\Delta_{b}$ and $\Delta_{c}$ are the same as
(6.36) and (6.37) respectively.

    2. (b)

The second type of poles of $\Delta_{b}$ and $\Delta_{c}$ are the same as
(6.40) and (6.41) respectively.

The pole structure is summarized in Table 4. We notice that there is a new
series type-3 poles appearing in the conformal partial wave expansion (6.33)
comparing to the five-point case.

##  7 Conclusion

In this paper, we computed scalar three-point celestial amplitudes involving
both two and three massive scalars. We found that the three-point coefficient
of celestial amplitudes with two massive scalars contains a hypergeometric
function (4.8), while the three-point coefficient of celestial amplitudes with
three massive scalars can be written as a triple Mellin-Barnes integral
(4.25).

Based on these results and using the split representation (2.26), we studied
the conformal block expansion of four-point celestial amplitudes involving two
massive scalars. By choosing the conformal basis and shadow conformal basis
for the incoming and outgoing particles respectively, we found that in the
conformal block expansion of four-point celestial amplitudes
$\mathcal{A}_{1^{0}_{m}+2^{0}_{0}\to\mathcal{S}[3^{0}_{m}]+\mathcal{S}[4^{0}_{0}]}^{\Delta_{1},\Delta_{2},\widetilde{\Delta}_{3},\widetilde{\Delta}_{4}}$,
there exist two series of double-trace poles (5.17), which is similar to the
spectrum of four-point contact diagrams in AdS.

In addition, we obtained the conformal block expansion of five- and six-point
massless celestial amplitudes. We found that when choosing the usual conformal
basis for all massless particles, it is impossible to enclose the contour in
the conformal partial wave expansion. To address this issue, we chose the
shadow conformal basis for outgoing particles, while still using the usual
conformal basis for the incoming particles, and the exchanged operators in the
conformal block expansion are summarized in Table 1 and 4. It is interesting
that the operator spectrum depends on the spacetime dimension $d$. For $d=2$,
we found that the $\mathcal{O}_{1}^{+}\mathcal{O}_{2}^{+}$ OPE contains three
series of exchanged operators:

$\displaystyle\mathcal{O}_{1}^{+}\mathcal{O}_{2}^{+}$
$\displaystyle\sim\mathcal{C}_{\Delta_{12}-2+2N}\mathcal{O}_{\Delta_{12}-2+2N}^{+}+\mathcal{C}_{\Delta_{12}+2\Delta_{3}-4+2N}\mathcal{O}_{\Delta_{12}+2\Delta_{3}-4+2N}^{+}$
(7.1)
$\displaystyle\phantom{{}\sim{}}+\mathcal{C}_{\Delta_{12}+2\Delta_{34}-6+2N}\mathcal{O}_{\Delta_{12}+2\Delta_{34}-6+2N}^{+}\;.$

The existence of the first two series of operators in the conformal block
expansion of both five- and six-point celestial amplitudes confirms the
analysis in [16, 17]. The leading terms $\mathcal{O}_{\Delta_{12}-2}^{+}$ and
$\mathcal{O}_{\Delta_{12}+2\Delta_{3}-4}^{+}$ in these two series of operators
have been interpreted as single- and two-particle operators, respectively. The
third series of operators is new and only appears in the conformal block
expansion of six-point celestial amplitudes. Following the same spirit of [16,
17], we propose that the leading term
$\mathcal{O}_{\Delta_{12}+2\Delta_{34}-6}^{+}$ in this series corresponds to a
three-particle operator. It is now natural to conjecture that there would be
$n$-particle operators $\mathcal{O}_{\Delta_{12}+2\Delta_{34\cdots
n+1}-2(n-1)}^{+}$ in the $\mathcal{O}_{1}^{+}\mathcal{O}_{2}^{+}$ OPE. This
conjecture can be confirmed by examining $(n+3)$-point celestial amplitudes.
Although we currently do not have the conformal block expansion of
$(n+3)$-point celestial amplitudes, the computation can be done without much
effort using the split representations.

We also extended our computations to five-point celestial amplitudes involving
one massive scalar. The results are summarized in Table 2 and 3. For $d=2$, we
found that the OPE between one massive scalar and one massless scalar contains
three types of operators:

$\displaystyle\mathcal{O}_{1,m}^{+}\mathcal{O}_{2}^{+}$
$\displaystyle\sim\mathcal{C}_{\Delta_{12}+2N}\mathcal{O}_{\Delta_{12}+2N}^{+}+\mathcal{C}_{-\Delta_{345}+6}\mathcal{O}_{-\Delta_{345}+6}^{+}$
(7.2)
$\displaystyle\phantom{{}\sim{}}+\mathcal{C}_{\Delta_{3,45}+4+2N}\mathcal{O}_{\Delta_{3,45}+4+2N}^{+}\;.$

The first series consists of double-trace operators, which also appear in the
conformal block expansion of four-point celestial amplitudes (5.17). The
second and the third types appear only in the conformal block expansion of
five-point celestial amplitudes. In the massless limit $m\rightarrow 0$, due
to the emergence of $2\pi\delta_{c}\left(i(\Delta_{12345}-8)\right)$,
$\mathcal{O}_{-\Delta_{345}+6}^{+}$ and $\mathcal{O}_{\Delta_{3,45}+4+2N}^{+}$
are reduced to $\mathcal{O}_{\Delta_{12}-2}^{+}$ and
$\mathcal{O}_{\Delta_{12}+2\Delta_{3}-4+2N}^{+}$ respectively. Hence we
conjecture that the leading operator $\mathcal{O}_{\Delta_{3,45}+4}^{+}$ can
be interpreted as a two-particle operator in the
$\mathcal{O}_{1,m}^{+}\mathcal{O}_{2}^{+}$ OPE. Furthermore, if turning on the
mass of the first propagator, the OPE is corrected as:

$\displaystyle\mathcal{O}_{1,m}^{+}\mathcal{O}_{2}^{+}$
$\displaystyle\sim\mathcal{C}_{\Delta_{12}+2N}\mathcal{O}_{\Delta_{12}+2N}^{+}+\mathcal{C}_{-\Delta_{345}+6+2N}\mathcal{O}_{-\Delta_{345}+6+2N}^{+}$
(7.3)
$\displaystyle\phantom{{}\sim{}}+\mathcal{C}_{\Delta_{3,45}+4+2N}\mathcal{O}_{\Delta_{3,45}+4+2N}^{+}\;.$

We explained that the extension of the series
$\mathcal{O}_{-\Delta_{345}+6+2N}^{+}$ can be understood from the small mass
expansion of the propagator.

Our work leads to many directions for future research. Firstly, it would be
intriguing to study the conformal block expansion of six-point celestial
amplitudes in other topologies and to deduce the corresponding operator
spectra. It is known that at six-point there are two topologies: the comb
topology and the snowflake topology [35]. We have shown that the comb channel
conformal block expansion of six-point celestial amplitudes contains three-
particle operators. To compute the one in the snowflake channel, it is
necessary to use the three-point coefficients of three massive scalars (4.25).
This computation is significantly more intricate and we leave it to the future
work.

Another avenue is to extend our methods to calculate three-point celestial
amplitudes of massive spinning particles. These three-point celestial
amplitudes play a crucial role in obtaining the conformal block expansion of
higher-point celestial MHV amplitudes. The authors of [16] and [17] have shown
that two-particle operators also appear in the OPEs of gluons. It would be
important to verify this at the level of conformal block expansion and to
investigate if there are additional operators appearing in the OPE of gluons.

Finally, it would be of great interest to examine the consistency conditions
of OPEs. Using our explicit results of conformal block expansions, it is
important to study whether the coefficients can be factorized into a product
of three-point coefficients, as expected from our understanding of CFTs.

## Acknowledgements

We thank Chi-Ming Chang and Yangrui Hu for the useful discussions.

##  Appendix A A Useful formulas

In this appendix we list some useful formulas. The Mellin-Barnes relation is

$(A+B)^{-\Delta}=\int_{-i\infty}^{+i\infty}\frac{ds}{2\pi
i}\,\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\Delta+s,-s}{\Delta}\biggr{]}A^{s}B^{-s-\Delta}\;.$
(A.1)

The first Barnes lemma is

$\displaystyle\int_{-i\infty}^{+i\infty}\frac{ds}{2\pi
i}\Gamma[a_{1}+s,a_{2}+s,b_{1}-s,b_{2}-s]=\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{a_{1}+b_{1},a_{1}+b_{2},a_{2}+b_{1},a_{2}+b_{2}}{a_{1}+a_{2}+b_{1}+b_{2}}\biggr{]}\;.$
(A.2)

For $\operatorname{Re}(\Delta_{i})>0$ and $d=\sum_{i=1}^{n}\Delta_{i}$, the
Feynman-Schwinger parameterization is

$\displaystyle\prod_{i=1}^{n}A_{i}^{-\Delta_{i}}$
$\displaystyle=\frac{\Gamma[d]}{\prod_{i=1}^{n}\Gamma[\Delta_{i}]}\int_{0}^{\infty}\prod_{i=2}^{n}ds_{i}s_{i}^{\Delta_{i}-1}\,(A_{1}+\sum_{i=2}^{n}s_{i}A_{i})^{-d}\;.$
(A.3)

The Mellin-Barnes representation of the hypergeometric function is

${}_{2}F_{1}(a,b,c,z)=\int_{-i\infty}^{+i\infty}ds\,(-z)^{s}\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{c,-s,a+s,b+s}{a,b,c+s}\biggr{]}\;.$
(A.4)

If $\arg(1-z)<\pi$, the hypergeometric function satisfies

${}_{2}F_{1}(a,b,c,z)=(1-z)^{c-a-b}{}_{2}F_{1}(c-a,c-b,c,z)\;.$ (A.5)

If $\arg(-z)<\pi$, the hypergeometric function near $z=0$ is related to that
near $z=\infty$ by

$\displaystyle{}_{2}F_{1}(a,b,c,z)$
$\displaystyle=(-z)^{-a}\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{b-a,c}{b,c-a}\biggr{]}{}_{2}F_{1}(a,a-c+1,a-b+1,\frac{1}{z})$
(A.6)
$\displaystyle\phantom{{}={}}+(-z)^{-b}\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{a-b,c}{a,c-b}\biggr{]}{}_{2}F_{1}(b,b-c+1,-a+b+1,\frac{1}{z})\;.$

If $\operatorname{Re}(c-a-b)>0$, the hypergeometric function with argument
unity is

${}_{2}F_{1}(a,b,c,1)=\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{c,c-a-b}{c-a,c-b}\biggr{]}\;.$
(A.7)

###  A.1 EAdS integrals

The tree-level three-point contact Witten diagram is

$D_{\Delta_{1},\Delta_{2},\Delta_{3}}=\int\frac{d^{d+1}\widehat{p}}{\widehat{p}^{0}}\,(-\widehat{q}_{1}\cdot\widehat{p})^{-\Delta_{1}}(-\widehat{q}_{2}\cdot\widehat{p})^{-\Delta_{2}}(-\widehat{q}_{3}\cdot\widehat{p})^{-\Delta_{3}}=d_{\Delta_{1},\Delta_{2},\Delta_{3}}\langle\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\rangle\;,$
(A.8)

where

$d_{\Delta_{1},\Delta_{2},\Delta_{3}}=\frac{1}{2}\pi^{d/2}\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\frac{\Delta_{12,3}}{2},\frac{\Delta_{13,2}}{2},\frac{\Delta_{23,1}}{2},\frac{\Delta_{123}-d}{2}}{\Delta_{1},\Delta_{2},\Delta_{3}}\biggr{]}\;.$
(A.9)

When evaluating the massive-massive-massive three-point function we need the
following EAdS integral,

$\displaystyle
I_{d,\Delta}(p)=\int\frac{d^{d+1}\widehat{p}_{1}}{\widehat{p}_{1}^{0}}\,(-\widehat{p}_{1}\cdot
p)^{-\Delta}\;,$ (A.10) $\displaystyle
I_{d,\Delta}(p,a)=\int\frac{d^{d+1}\widehat{p}_{1}}{\widehat{p}_{1}^{0}}\,(a-\widehat{p}_{1}\cdot
p)^{-\Delta}\;,$ (A.11)

where $a\leqslant 0$ and $p\in\mathbb{R}^{1,d+1}$ is a timelike vector. The
first integral can be computed as follows: using the Lorentz symmetry and the
scaling behavior of $p$, it takes the form as

$I_{d,\Delta}(p)=f(\Delta)(-p^{2})^{-\frac{\Delta}{2}}\;.$ (A.12)

We choose the rest frame of $p$, _i.e_. $p^{\mu}=(1,0,\cdots,0)$ to determine
the $p$-independent function $f(\Delta)$, and the above integral becomes

$\displaystyle f(\Delta)=\int
d^{d+1}\widehat{p}_{1}\,(\widehat{p}_{1}^{0})^{-\Delta-1}=\int_{0}^{+\infty}dr\int_{\operatorname{S}^{d}}d\Omega\,r^{d}(1+r^{2})^{-\frac{\Delta+1}{2}}=\pi^{\frac{d+1}{2}}\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\frac{\Delta-d}{2}}{\frac{\Delta+1}{2}}\biggr{]}\;.$
(A.13)

This implies

$I_{d,\Delta}(p)=\pi^{\frac{d+1}{2}}\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\frac{\Delta-d}{2}}{\frac{\Delta+1}{2}}\biggr{]}(-p^{2})^{-\frac{\Delta}{2}}\;.$
(A.14)

By the Mellin-Barnes relation (A.1) and the integral (A.10), we have

$\displaystyle I_{d,\Delta}(p,a)$
$\displaystyle=\int\frac{d^{d+1}\widehat{p}_{1}}{\widehat{p}_{1}^{0}}\,\int_{-i\infty}^{+i\infty}\frac{ds}{2\pi
i}\,\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\Delta+s,-s}{\Delta}\biggr{]}(-\widehat{p}_{1}\cdot
p)^{-s-\Delta}a^{s}$ (A.15)
$\displaystyle=\pi^{\frac{d+1}{2}}\int_{-i\infty}^{+i\infty}\frac{ds}{2\pi
i}\,\Gamma\biggl{[}\genfrac{.}{.}{0.0pt}{}{\Delta+s,-s,\frac{\Delta+s-d}{2}}{\Delta,\frac{\Delta+s+1}{2}}\biggr{]}(-p^{2})^{-\frac{\Delta+s}{2}}a^{s}\;.$

###  A.2 Analytic functional

The $\delta$-function with complex argument is defined by the following
properties,

$\displaystyle
2\pi\delta_{c}\left(i(z-w)\right)=\int_{0}^{+\infty}d\alpha\;\alpha^{z-w-1}\;,$
(A.16) $\displaystyle\int_{a-i\infty}^{a+i\infty}\frac{dz}{2\pi
i}\,2\pi\delta_{c}\left(i(z-w)\right)f(z)=f(w)\;,$ (A.17)

and when the argument locates on the integration contour, it reduces to the
usual $\delta$-function. This is an analytic functional belonging to the
Gelfand–Shilov space $Z^{\prime}$, see e.g. [49, 50] and [51, Section 4].

The $\delta$-function with complex argument admits the following
approximations in the sense of weak limit,

$\lim_{m\to
0}\frac{1}{2}\Gamma[-\frac{\Delta}{2}]m^{\Delta}=2\pi\delta_{c}\left(i(\Delta)\right)\;.$
(A.18)

To prove this, we integrate the left side with an analytic test function
$f(\Delta)$, then enclose the contour to the right half $\Delta$-plane and
pick up poles at $\Delta=2n,\,n\in\mathbb{N}$, leading to

$\displaystyle\phantom{{}={}}\lim_{m\to
0}\int_{a-i\infty}^{a+i\infty}\frac{d\Delta}{2\pi
i}\,\frac{1}{2}\Gamma[-\frac{\Delta}{2}]m^{\Delta}f(\Delta)=\lim_{m\to
0}\sum_{n=0}^{+\infty}\frac{(-1)^{n}}{n!}m^{2n}f(2n)=f(0)\;,$ (A.19)

which justifies the defining property (A.17).

##  Appendix B B Tables of operator spectra

$d$ | $\begin{array}[]{l}\text{type-1 $\Delta_{a}$}\\\ \text{type-1 $\Delta_{b}$}\end{array}$ | $\begin{array}[]{l}\text{type-2 $\Delta_{a}$}\\\ \text{type-1 $\Delta_{b}$}\end{array}$
---|---|---
$2$ | $\begin{array}[]{l}\Delta_{a}=\Delta_{12}-2+2N\\\ \Delta_{b}=\Delta_{123}-4\,\text{ or }\\\ \phantom{\Delta_{c}={}}\Delta_{123}-2+2N+2M\end{array}$ | $\begin{array}[]{l}\Delta_{a}=\Delta_{12}+2\Delta_{3}-4+2N\\\ \Delta_{b}=\Delta_{123}-4\\\ \phantom{\Delta_{c}={}}\end{array}$
$4,6,\dots$ | $\begin{array}[]{l}\Delta_{a}=\Delta_{12}-2+2N\\\ \Delta_{b}=\Delta_{123}-2+2N+2M\end{array}$ |  /
$1,3,\dots$ | $\begin{array}[]{l}\Delta_{a}=\Delta_{12}-2+2N\\\ \Delta_{b}=\Delta_{123}+d-6+2M\end{array}$ | $\begin{array}[]{l}\Delta_{a}=\Delta_{12}+2\Delta_{3}+d-6+2N+2M\\\ \Delta_{b}=\Delta_{123}+d-6+2M\end{array}$
Table 1:  The conformal dimensions of exchanged operators in the $s$-channel conformal block expansion of $\mathcal{A}_{1_{0}^{0}+2_{0}^{0}+3_{0}^{0}\to\mathcal{S}[4_{0}^{0}]+\mathcal{S}[5_{0}^{0}]}^{\Delta_{1},\Delta_{2},\Delta_{3},\widetilde{\Delta}_{4},\widetilde{\Delta}_{5}}$, where $N,M\in\mathbb{N}$. $d$ | $\begin{array}[]{l}\text{type-1 $\Delta_{a}$}\\\ \text{type-1 $\Delta_{b}$}\end{array}$ | $\begin{array}[]{l}\text{type-2 $\Delta_{a}$}\\\ \text{type-1 $\Delta_{b}$}\end{array}$
---|---|---
$2$ | $\begin{array}[]{l}\Delta_{a}=\Delta_{12}+2N\\\ \Delta_{b}=-\Delta_{45}+4\,\text{ or }\\\ \phantom{\Delta_{b}={}}\Delta_{123}+2N+2M\end{array}$ | $\begin{array}[]{l}\Delta_{a}=\Delta_{3,45}+4+2N\\\ \Delta_{b}=-\Delta_{45}+4\\\ \phantom{\Delta_{b}={}}\end{array}$
$4,6,\dots$ | $\begin{array}[]{l}\Delta_{a}=\Delta_{12}+2N\\\ \Delta_{b}=\Delta_{123}+2N+2M\end{array}$ |  /
$1,3,\dots$ | $\begin{array}[]{l}\Delta_{a}=\Delta_{12}+2N\\\ \Delta_{b}=-\Delta_{45}+2d+2M\,\text{ or }\\\ \phantom{\Delta_{b}={}}\Delta_{123}+2N+2M\end{array}$ | $\begin{array}[]{l}\Delta_{a}=\Delta_{3,45}+2d+2N+2M\\\ \Delta_{b}=-\Delta_{45}+2d+2M\\\ \phantom{\Delta_{b}={}}\end{array}$
|  |
$d$ | $\begin{array}[]{l}\text{type-1 $\Delta_{a}$}\\\ \text{type-1 $\Delta_{b}$}\end{array}$ |
$1,2,\dots$ | $\begin{array}[]{l}\Delta_{a}=-\Delta_{345}+d+4\\\ \Delta_{b}=-\Delta_{45}+2d+2M\end{array}$ |
Table 2:  The conformal dimensions of exchanged operators in the $s$-channel
conformal block expansion of
$\mathcal{A}_{1_{m}^{0}+2_{0}^{0}+3_{0}^{0}\to\mathcal{S}[4_{0}^{0}]+\mathcal{S}[5_{0}^{0}]}^{\Delta_{1},\Delta_{2},\Delta_{3},\widetilde{\Delta}_{4},\widetilde{\Delta}_{5}}$
with massless exchange, where $N,M\in\mathbb{N}$.

$d$ | $\begin{array}[]{l}\text{type-1 $\Delta_{a}$}\\\ \text{type-1 $\Delta_{b}$}\end{array}$ | $\begin{array}[]{l}\text{type-2 $\Delta_{a}$}\\\ \text{type-1 $\Delta_{b}$}\end{array}$
---|---|---
$2$ | $\begin{array}[]{l}\Delta_{a}=\Delta_{12}+2N\\\ \Delta_{b}=-\Delta_{45}+4\,\text{ or }\\\ \phantom{\Delta_{b}={}}\Delta_{123}+2N+2M\end{array}$ | $\begin{array}[]{l}\Delta_{a}=\Delta_{3,45}+4+2N\\\ \Delta_{b}=-\Delta_{45}+4\\\ \phantom{\Delta_{b}={}}\end{array}$
$4,6,\dots$ | $\begin{array}[]{l}\Delta_{a}=\Delta_{12}+2N\\\ \Delta_{b}=\Delta_{123}+2N+2M\end{array}$ |  /
$1,3,\dots$ | $\begin{array}[]{l}\Delta_{a}=\Delta_{12}+2N\\\ \Delta_{b}=-\Delta_{45}+2d+2M\,\text{ or }\\\ \phantom{\Delta_{b}={}}\Delta_{123}+2N+2M\end{array}$ | $\begin{array}[]{l}\Delta_{a}=\Delta_{3,45}+2d+2N+2M\\\ \Delta_{b}=-\Delta_{45}+2d+2M\\\ \phantom{\Delta_{b}={}}\end{array}$
|  |
$d$ | $\begin{array}[]{l}\text{type-1 $\Delta_{a}$}\\\ \text{type-1 $\Delta_{b}$}\end{array}$ |
$2$ | $\begin{array}[]{l}\Delta_{a}=-\Delta_{345}+6+2N\\\ \Delta_{b}=-\Delta_{45}+4\,\text{ or }\\\ \phantom{\Delta_{b}={}}-\Delta_{45}+6+2N+2M\end{array}$ |
$4,6,\dots$ | $\begin{array}[]{l}\Delta_{a}=-\Delta_{345}+d+4+2N\\\ \Delta_{b}=-\Delta_{45}+d+4+2N+2M\end{array}$ |
$1,3,\dots$ | $\begin{array}[]{l}\Delta_{a}=-\Delta_{345}+d+4+2N\\\ \Delta_{b}=-\Delta_{45}+2d+2M\end{array}$ |
Table 3:  The conformal dimensions of exchanged operators in the $s$-channel
conformal block expansion of
$\mathcal{A}_{1_{m}^{0}+2_{0}^{0}+3_{0}^{0}\to\mathcal{S}[4_{0}^{0}]+\mathcal{S}[5_{0}^{0}]}^{\Delta_{1},\Delta_{2},\Delta_{3},\widetilde{\Delta}_{4},\widetilde{\Delta}_{5}}$
with massive exchange, where $N,M\in\mathbb{N}$.

$d$ | $\begin{array}[]{l}\text{type-1 $\Delta_{a}$}\\\ \text{type-1 $\Delta_{b}$}\\\ \text{type-1 $\Delta_{c}$}\end{array}$ | $\begin{array}[]{l}\text{type-1,2 $\Delta_{a}$}\\\ \text{type-1 $\Delta_{b}$}\\\ \text{type-1 $\Delta_{c}$}\end{array}$
---|---|---
$2$ | $\begin{array}[]{l}\Delta_{a}=\Delta_{12}-2+2N\\\ \Delta_{b}=\Delta_{123}-2+2N+2M\\\ \Delta_{c}=\Delta_{1234}-6\,\text{ or }\\\ \phantom{\Delta_{c}={}}\Delta_{1234}-2+2N+2M+2K\end{array}$ |  $\\!\begin{array}[]{l}\Delta_{a}=\Delta_{12}-2+2N\,\text{ or }\\\ \phantom{\Delta_{a}={}}\Delta_{12}+2\Delta_{3}-4+2N\\\ \Delta_{b}=\Delta_{123}-4\\\ \Delta_{c}=\Delta_{1234}+d-8+2K\end{array}$
$4,6,\dots$ | $\begin{array}[]{l}\Delta_{a}=\Delta_{12}-2+2N\\\ \Delta_{b}=\Delta_{123}-2+2N+2M\\\ \Delta_{c}=\Delta_{1234}-2+2N+2M+2K\end{array}$ |
$1,3,\dots$ | $\begin{array}[]{l}\Delta_{a}=\Delta_{12}-2+2N\\\ \Delta_{b}=\Delta_{123}-2+2N+2M\\\ \Delta_{c}=\Delta_{1234}+d-8+2K\end{array}$ |
|  |
$d$ | $\begin{array}[]{l}\text{type-1,3 $\Delta_{a}$}\\\ \text{type-2 $\Delta_{b}$}\\\ \text{type-1 $\Delta_{c}$}\end{array}$ |
$2$ | $\begin{array}[]{l}\Delta_{a}=\Delta_{12}-2+2N\,\text{ or }\\\ \phantom{\Delta_{a}={}}\Delta_{12}+2\Delta_{34}-6+2N+2M\\\ \Delta_{b}=\Delta_{123}+2\Delta_{4}-6+2M\\\ \Delta_{c}=\Delta_{1234}-6\end{array}$ |
$4,6,\dots$ |  / |
$1,3,\dots$ | $\begin{array}[]{l}\Delta_{a}=\Delta_{12}-2+2N\,\text{ or }\\\ \phantom{\Delta_{a}={}}\Delta_{12}+2\Delta_{34}+d-8+2N+2M+2K\\\ \Delta_{b}=\Delta_{123}+2\Delta_{4}+d-8+2M+2K\\\ \Delta_{c}=\Delta_{1234}+d-8+2K\end{array}$ |
Table 4:  The conformal dimensions of exchanged operators in the $s$-channel
conformal block expansion of
$\mathcal{A}_{1_{0}^{0}+2_{0}^{0}+3_{0}^{0}+4_{0}^{0}\to\mathcal{S}[5_{0}^{0}]+\mathcal{S}[6_{0}^{0}]}^{\Delta_{1},\Delta_{2},\Delta_{3},\Delta_{4},\widetilde{\Delta}_{5},\widetilde{\Delta}_{6}}$,
where $N,M,K\in\mathbb{N}$.

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