Completeness from Laplacian. The spherical function (D.23) satisfies the
eigenequation of the radial Laplacian (D.12), even for non-unitary principal
series $\Delta\in\mathbb{C}$. Using the Sturm-Liouville theory, in Appendix
D.5.1 we show that the spherical functions of principal series provide a
complete basis for the radial Laplacian. The orthogonality and completeness
relations of are

$\displaystyle 2\pi
N_{\text{H}}(\Delta_{1})\delta(s_{1}-s_{2})=\int_{0}^{\infty}dt\sinh^{d}t\,\phi_{\Delta_{1}}^{*}(t)\phi_{\Delta_{2}}(t),$
(D.24)
$\displaystyle\sinh^{-d}t\,\delta(t_{1}-t_{2})=\int_{0}^{\infty}\frac{ds}{2\pi
N_{\text{H}}(\Delta)}\,\phi_{\Delta}^{*}(t_{1})\phi_{\Delta}(t_{2}),$ (D.25)

where $\Delta_{i}=\frac{d}{2}+is_{i},\,s_{i}>0$, and the density
$N_{\text{H}}(\Delta)$ will be derived later.

The integration over $s$ can be rewritten as $\frac{1}{2\pi
i}\int_{\Gamma}\frac{d\Delta}{N_{\text{H}}(\Delta)}\,$, where the contour is
half of the principal series from $\Delta=\frac{d}{2}$ to
$\Delta=\frac{d}{2}+i\infty$. Then for normalizable functions we have the
decomposition and inversion formulas

$\widetilde{f}(\Delta)=\int_{0}^{\infty}dt\sinh^{d}t\,\phi_{\Delta}^{*}(t)f(t),\,\text{
and }\,f(t)=\frac{1}{2\pi
i}\int_{\Gamma}\frac{d\Delta}{N_{\text{H}}(\Delta)}\,\widetilde{f}(\Delta)\phi_{\Delta}(t).$
(D.26)

They can be interpreted geometrically on the EAdS space
$\operatorname{H}_{d+1}$. The Hilbert space with respect to the radial
Laplacian contains normalizable $\operatorname{SO}(d+1)$-invariant functions
on $\operatorname{H}_{d+1}$. The spectral decomposition of the radial
Laplacian ensures that the $\operatorname{SO}(d+1)$-invariant functions can be
decomposed and inverted by the spherical functions, which are called the
spherical transforms in the theory of Gelfand pairs.

Plancherel measure. The density $N_{\text{H}}(\Delta)$ can be read off by the
following localization technique. The integrand $I(t,\Delta_{1},\Delta_{2})$
of the right side of (D.24) has the asymptotic form as $t\to\infty$,

$I(t,\Delta_{1},\Delta_{2})\sim
e^{-i(s_{1}-s_{2})t}c_{0}c^{*}(\Delta_{1})c(\Delta_{2})+e^{i(s_{1}-s_{2})t}c_{0}c^{*}(\widetilde{\Delta}_{1})c(\widetilde{\Delta}_{2})+\text{terms
with }e^{\pm i(s_{1}+s_{2})t}.$ (D.27)

where $c(\Delta)$ is the leading coefficient of the spherical function (D.70)
and the factor $c_{0}=2^{-d}$ is the leading coefficient of the measure
$\sinh^{d}t$.

Using the substitution $t\to\lambda t$ and taking the limit
$\lambda\to\infty$, the terms with $e^{\pm i(s_{1}+s_{2})t}$ do not contribute
due to $s_{1},s_{2}>0$, and the first two terms combine into
$\int_{\mathbb{R}}dt\,e^{-i(s_{1}-s_{2})t}=2\pi\delta(s_{1}-s_{2})$. Hence the
measure of the inversion formula (D.26) is

$N_{\text{H}}(\Delta)=c_{0}|c(\Delta)|^{2}=c_{0}c(\Delta)c(\widetilde{\Delta})=\frac{(2\pi)^{d}\Gamma(\Delta-\frac{d}{2})\Gamma(\widetilde{\Delta}-\frac{d}{2})}{\Gamma(\Delta)\Gamma(\widetilde{\Delta})}.$
(D.28)

In history [59, 60], Harish-Chandra was the first to establish the connection
between the Plancherel measure and the asymptotic behaviour of the spherical
functions, and for this reason the leading coefficient is called the Harish-
Chandra $c$-function. In the EAdS case, the Plancherel measure of
$\operatorname{EAdS}_{d+1}$ coincides with the prefactor of double shadow
transform. This fact is not accidental and holds for other semisimple groups,
see [61, 62].

Fourier transform and inversion formula. The Poisson transform (C.15), the
Fourier transform (C.16) and the projector (C.17) on $\operatorname{H}_{d+1}$
are

$\displaystyle\mathcal{F}^{\dagger}_{\Delta}:\,f(P)\,\mapsto\int_{\operatorname{LC}_{d+1}}d^{\prime}P\,(-X\cdot
P)^{-\widetilde{\Delta}}f(P)\in L^{2}(\operatorname{H}_{d+1}),$ (D.29)
$\displaystyle\mathcal{F}_{\Delta}:\,f(X)\,\mapsto\int_{\operatorname{H}_{d+1}}dX\,(-X\cdot
P)^{-\Delta}f(X)\in\mathcal{E}_{\Delta},$ (D.30)
$\displaystyle{\mathcal{P}}_{\Delta}:\,f(X)\,\mapsto\int_{\operatorname{H}_{d+1}}dX^{\prime}\,\phi_{\Delta}(X,X^{\prime})f(X^{\prime}),$
(D.31)

where the two-variable spherical function is

$\phi_{\Delta}(X,X^{\prime})=\int_{\operatorname{LC}_{d+1}}d^{\prime}P\,(-X\cdot
P)^{-\widetilde{\Delta}}(-X^{\prime}\cdot P)^{-\Delta}.$ (D.32)

From (C.22), the inversion formula reads as

$f(X)=\frac{1}{2\pi
i}\int_{\Gamma}\frac{d\Delta}{N_{\text{H}}(\Delta)}\,{\mathcal{P}}_{\Delta}[f(X)],$
(D.33)

where the contour $\Gamma$ is half of the principal series.

For the Dirac delta distribution we have the resolution of identity, i.e. the
split representation on the Euclidean AdS space

$\delta(X,X^{\prime})=\frac{1}{2\pi
i}\int_{\Gamma}\frac{d\Delta}{\mu(\Delta)}\,\,\int_{\operatorname{LC}_{d+1}}d^{\prime}P\,(-2X\cdot
P)^{-\widetilde{\Delta}}(-2X^{\prime}\cdot P)^{-\Delta},$ (D.34)

where for later convenience we have introduced the factor

$\mu(\Delta)=2^{-d}N_{\text{H}}(\Delta)=\frac{\pi^{d}\Gamma(\Delta-\frac{d}{2})\Gamma(\widetilde{\Delta}-\frac{d}{2})}{\Gamma(\Delta)\Gamma(\widetilde{\Delta})}.$
(D.35)

###  D.4 Harmonic analysis on dS

In this subsection we provide a derivation of the harmonic analysis on
$\operatorname{dS}_{d+1},\,d\geqslant 2$. The main difference from the EAdS
case is that the bulk-to-boundary covariant quantity $X\cdot P$ is indefinite
in the dS spacetime, and this allows the existence of two independent
spherical functions associated with the same unitary irrep
$\mathcal{E}_{\Delta},\,\Delta=\frac{d}{2}+is,\,s\in\mathbb{R}$. For $d=1$,
the discussion is slightly more complicated than that of $d\geqslant 2$ since
the exceptional series split into two discrete series, but the resulting
Fourier transform and inversion formula share the same forms with $d\geqslant
2$.

Spherical function. The derivation of spherical functions is almost the same
as that on EAdS space. We choose a reference point
$X_{0}=(0,\dots,0,1)\in\operatorname{dS}_{d+1}$ and
$H\simeq\operatorname{SO}(d,1)$ is the stabilizer subgroup of $X_{0}$. In the
EAdS case, the lightcone is foliated into orbits of the
$\operatorname{SO}(d+1)$-action, $P^{0}=\text{const.}$, and homogeneous
functions are completely determined by values on one of the orbits $P^{0}=1$.
While in the dS case, the $\operatorname{SO}(d,1)$-orbits are
$P^{d+1}=\text{const.}$, and the homogeneous functions are determined by
values on the two orbits $P^{d+1}=\pm 1$. Hence there are two $H$-fixed
homogeneous functions on $\operatorname{LC}_{d+1}$:

$f_{\Delta,\epsilon}(P)=|P^{d+1}|^{-\Delta,\epsilon}=|P^{d+1}|^{-\Delta}\operatorname{sgn}^{\epsilon}(P^{d+1}),\,\text{
for }\,P\in\operatorname{LC}_{d+1}.$ (D.36)

where the superscript notation $\epsilon=0,1$ follows from Appendix B and
labels the parity of $f_{\Delta,\epsilon}(P)$ under $P\to-P$. The matrix
element $\pi(X)f_{\Delta}(P),\,X\in\operatorname{dS}_{d+1}$ can be deduced
similarly: $\pi(X)f_{\Delta,\epsilon}(P)=|X\cdot P|^{-\Delta,\epsilon}$.

The functions $f_{\Delta,\epsilon}(P)$ have singularities along the
hypersurface $P^{d+1}=0$, and are not $H$-fixed vectors but $H$-fixed
distributions associated with $\mathcal{E}_{\Delta}$. For the principal series
$\mathcal{E}_{\frac{d}{2}+is},\,s\in\mathbb{R}$, The corresponding spherical
functions are

$\phi_{\Delta,\epsilon}(T)=\phi_{\Delta,\epsilon}(X_{1},X_{2})=\int_{\operatorname{LC}_{d+1}}d^{\prime}P\,|X_{1}\cdot
P|^{-\widetilde{\Delta},\epsilon}|X_{2}\cdot P|^{-\Delta,\epsilon},$ (D.37)

where $T=X_{1}\cdot X_{2}$, and the derivations can be found in Appendix
D.6.2. They respect the parity symmetry and the shadow symmetry,

$\phi_{\Delta,\epsilon}(T)=(-1)^{\epsilon}\phi_{\Delta,\epsilon}(-T)=\phi_{\widetilde{\Delta},\epsilon}(T).$
(D.38)

In the third hyperbolic chart $T\in(1,\infty)$, the spherical functions are

$\phi_{\Delta,\epsilon}(T)=2^{\frac{d+1}{2}}\pi^{\frac{d-1}{2}}\Gamma(1-\Delta)\left[\Gamma(-d+1+\Delta)\left((-1)^{\epsilon}\sin\frac{d\pi}{2}+\sin\frac{(d-2\Delta)\pi}{2}\right)\phi_{1}(T)+2\phi_{2}(T)\right],$
(D.39)

and in the second hyperbolic chart $T\in(-1,1)$, they are

$\phi_{\Delta,\epsilon}(T)=2^{\frac{d+1}{2}}\pi^{\frac{d-1}{2}}\Gamma(1-\Delta)\Gamma(-d+\Delta+1)\left[\phi_{1}(T)\left((-1)^{\epsilon}-\cos\pi(d-\Delta)\right)+\phi_{2}(T)\frac{2\sin\pi(d-\Delta)}{\pi}\right],$
(D.40)

where $\phi_{1,2}$-s are proportional to the Legendre functions, see (D.60),
(D.61), (D.62) and (D.63), and we have set
$a_{1}=1,a_{2}=\frac{e^{-\frac{1}{2}i\pi(d-1)}}{\Gamma(d-\Delta)},a^{\prime}_{1}=1,a^{\prime}_{2}=1$
therein.

As explained in Appendix B, the $H$-fixed distributions
$f_{\Delta,\epsilon}(P)$ are meromorphic functions of $\Delta$ with simple
poles at positive integers, and the regularized ones
$\frac{1}{\Gamma(\frac{-\Delta+\epsilon+1}{2})}f_{\Delta,\epsilon}(P)$ are
holomorphic with respect to $\Delta$. We introduce the regularized spherical
functions $\varphi_{\Delta,\epsilon}(T)$, which are well-defined for integer
$\Delta$-s,

$\varphi_{\Delta,\epsilon}(T)=A^{-1}(\Delta,\epsilon)\phi_{\Delta,\epsilon}(T),\,\text{
where
}\,A(\Delta,\epsilon)=\Gamma(\frac{-\Delta+\epsilon+1}{2})\Gamma(\frac{-\widetilde{\Delta}+\epsilon+1}{2}).$
(D.41)

Completeness and Plancherel measure. Similar to the EAdS case, in Appendix
D.5.2 we show that the (regularized) spherical functions of principal,
complementary and exceptional series representations provide a complete basis
for the radial Laplacian (D.13), including the continuous part
$\\{\varphi_{\frac{d}{2}+is,\epsilon}:s>0,\epsilon=0,1\\}$ and the discrete
part $\\{\varphi_{\Delta,\epsilon}:(\Delta,\epsilon)\in D^{-}_{\text{dS}}\\}$,
where

$D^{-}_{\text{dS}}=\\{(\Delta\in\mathbb{Z},\epsilon):\,\Delta<\frac{d}{2}\,\text{
and }\,\epsilon=1+\Delta\,\operatorname{mod}\mathbb{Z}_{2}\\}.$ (D.42)

By the Sturm-Liouville theory, the nonvanishing orthogonality relations are

$\displaystyle 2\pi
N^{c}_{\text{dS}}(\Delta_{1},\epsilon_{1})\delta(s_{1}-s_{2})\delta_{\epsilon_{1},\epsilon_{2}}=(\varphi_{\Delta_{1},\epsilon_{1}},\varphi_{\Delta_{2},\epsilon_{2}}),$
(D.43) $\displaystyle 2\pi
N^{d}_{\text{dS}}(\Delta_{1})\delta_{\Delta_{1},\Delta_{2}}=(\varphi_{\Delta_{1}},\varphi_{\Delta_{2}}),$
(D.44)

where the inner product in the right side is

$(f,g)=\int_{\mathbb{R}}dT|T^{2}-1|^{\frac{d-1}{2}}\,f^{*}(T)g(T).$ (D.45)

And the completeness relation is

$\displaystyle\begin{split}&|T^{2}-1|^{\frac{d-1}{2}}\delta(T_{1}-T_{2})\\\
&\phantom{=}=\frac{1}{2\pi
i}\sum_{\epsilon=0,1}\int_{\Gamma}\frac{d\Delta}{N^{c}_{\text{dS}}(\Delta,\epsilon)}\,\varphi_{\Delta,\epsilon}^{*}(T_{1})\varphi_{\Delta,\epsilon}(T_{2})+\frac{1}{2\pi}\sum_{(\Delta,\epsilon)\in
D^{-}_{\text{dS}}}\frac{1}{N^{d}_{\text{dS}}(\Delta)}\varphi_{\Delta}^{*}(T_{1})\varphi_{\Delta}(T_{2}).\end{split}$
(D.46)

Similarly, the density $N^{c}_{\text{dS}}(\Delta,\epsilon)$ is determined by
the asymptotic behaviour of the spherical functions

$N^{c}_{\text{dS}}(\Delta,\epsilon)=2c(\Delta,\epsilon)c(\widetilde{\Delta},\epsilon),$
(D.47)

where $c(\Delta,\epsilon)$ is the $c$-function of dS spacetime (D.72), and the
factor $2$ is due to the two boundary $T\to\pm\infty$. We find and check
numerically that the density $N^{d}_{\text{dS}}(\Delta)$ is related to
$N^{c}_{\text{dS}}(\Delta,\epsilon)$ by

$\frac{1}{N^{d}_{\text{dS}}(\Delta)}=-\alpha_{d}\operatorname{Res}_{\Delta\in
D^{-}_{\text{dS}}}\frac{1}{N^{c}_{\text{dS}}(\Delta,\epsilon)},$ (D.48)

where the constant $\alpha_{d}=1$ for odd $d$ and $\alpha_{d}=\frac{1}{2}$ for
even $d$.

Fourier transform and inversion formula. For later convenience, we turn back
to the unregularized spherical functions $\varphi_{\Delta,\epsilon}(T)$ by
recovering the regularization factor $A(\Delta,\epsilon)$. The Poisson
transform and the Fourier transform are

$\displaystyle\mathcal{F}^{\dagger}_{\Delta,\epsilon}:\,f(P)\,\mapsto\int_{\operatorname{LC}_{d+1}}d^{\prime}P\,|X\cdot
P|^{-\Delta,\epsilon}f(P)\in L^{2}(\operatorname{dS}_{d+1}),$ (D.49)
$\displaystyle\mathcal{F}_{\Delta,\epsilon}:\,f(X)\,\mapsto\int_{\operatorname{dS}_{d+1}}dX\,|X\cdot
P|^{-\widetilde{\Delta},\epsilon}f(X)\in\mathcal{E}_{\Delta},$ (D.50)
$\displaystyle{\mathcal{P}}_{\Delta,\epsilon}:\,f(X)\,\mapsto\int_{\operatorname{dS}_{d+1}}dX^{\prime}\,\phi_{\Delta,\epsilon}(X,X^{\prime})f(X^{\prime}).$
(D.51)

Similar to the spherical functions, the Fourier transform of $f(X)$ has poles
at $D^{-}_{\text{dS}}$, and the residue is proportional to the Fourier
transform with respect to $\varphi_{\Delta,\epsilon}(X)$. The inversion
formula is

$f(X)=\frac{1}{2\pi
i}\sum_{\epsilon=0,1}\int_{\Gamma}\frac{d\Delta}{2^{d+1}\mu(\Delta)}\,{\mathcal{P}}_{\Delta,\epsilon}[f(X)]-\alpha_{d}\underset{(\Delta,\epsilon)\in
D^{-}_{\text{dS}}}{\sum\operatorname{Res}}\frac{1}{2^{d+1}\mu(\Delta)}{\mathcal{P}}_{\Delta,\epsilon}[f(X)],$
(D.52)

where we have rewritten the density $N^{d}_{\text{dS}}(\Delta)$ by

$\mu(\Delta)=2^{-d}A^{-1}(\Delta,\epsilon)N^{d}_{\text{dS}}(\Delta)=\frac{\pi^{d}\Gamma(\Delta-\frac{d}{2})\Gamma(\widetilde{\Delta}-\frac{d}{2})}{\Gamma(\Delta)\Gamma(\widetilde{\Delta})}.$
(D.53)

For the Dirac delta distribution we have the split representation on the dS
spacetime

$\begin{split}\delta(X,X^{\prime})&=\frac{1}{2\pi
i}\sum_{\epsilon=0,1}\int_{\Gamma}\frac{d\Delta}{2\mu(\Delta)}\,\int_{\operatorname{LC}_{d+1}}d^{\prime}P\,|2X\cdot
P|^{-\widetilde{\Delta},\epsilon}|2X^{\prime}\cdot P|^{-\Delta,\epsilon}\\\
&\phantom{=}-\alpha_{d}\underset{(\Delta,\epsilon)\in
D^{-}_{\text{dS}}}{\sum\operatorname{Res}}\int_{\operatorname{LC}_{d+1}}d^{\prime}P\,\frac{1}{2\mu(\Delta)}|2X\cdot
P|^{-\widetilde{\Delta},\epsilon}|2X^{\prime}\cdot
P|^{-\Delta,\epsilon},\end{split}$ (D.54)

By the shadow symmetry of the spherical functions, the discrete part of (D.54)
flips sign and with $D^{-}_{\text{dS}}$ changing to $D_{\text{dS}}$,

$D_{\text{dS}}=\\{(\Delta\in\mathbb{Z},\epsilon):\,\Delta>\frac{d}{2}\,\text{
and }\,\epsilon=1+d+\Delta\,\operatorname{mod}\mathbb{Z}_{2}\\}.$ (D.55)

###  D.5 Sturm-Liouville problem and boundary conditions

In this appendix we analyse the Sturm-Liouville equation (D.12) with different
boundary conditions,

$\left(z^{2}-1\right)f^{\prime\prime}(z)+(d+1)zf^{\prime}(z)-\Delta(\Delta-d)f(z)=0.$
(D.56)

This is a singular151515The term “singular” means the interval is infinite and
the function $\left(z^{2}-1\right)^{-\frac{1+d}{2}}$ is singular at the
boundary $z=\pm 1$. In this case the spectrum may contain both continuous and
discrete parts, see e.g. [66]. Sturm-Liouville problem and can be solved by
the following steps:

  * •

find all solutions and determine the inner product;

  * •

choose suitable boundary conditions;

  * •

select the complete basis from the solutions.

The allowed boundary conditions are usually not unique. They lead to different
set of complete basis and correspond to inequivalent self-adjoint extensions
of the same differential operator. This phenomenon has already occurred in the
discussion of CFT inversion formulas [63, 64, 65, 42].

The Sturm-Liouville standard form of (D.56) is

${\mathcal{L}}[f](x)=-|z^{2}-1|^{\frac{1-d}{2}}\frac{d}{dz}\left[\operatorname{sgn}(z-1)|z^{2}-1|^{\frac{1+d}{2}}\frac{d}{dz}f(z)\right]=\Delta(\Delta-d)f(x),$
(D.57)

where we keep the weight function $w(z)=|z^{2}-1|^{\frac{1-d}{2}}$ positive
and absorb the sign into
$p(z)=\operatorname{sgn}(z-1)|z^{2}-1|^{\frac{1+d}{2}}$. Then the Sturm-
Liouville inner product agrees with that in the global coordinates of
$\operatorname{H}_{d+1}$ and in the hyperbolic coordinates of
$\operatorname{dS}_{d+1}$,

$(f,g)=\int dz|z^{2}-1|^{\frac{d-1}{2}}\,f^{*}(z)g(z),$ (D.58)

and the differential operator ${\mathcal{L}}$ is formally self-adjoint upto
the boundary terms

$(f,{\mathcal{L}}g)-({\mathcal{L}}f,g)=\operatorname{sgn}(z-1)|z^{2}-1|^{\frac{1+d}{2}}\left[\frac{d}{dz}f(z)g^{*}(z)-f(z)\frac{d}{dz}g^{*}(z)\right]_{z=-\infty}^{z=\infty}.$
(D.59)

The region $z\in(-\infty,-1)$ is related to $z\in(1,\infty)$ by $z\to-z$. For
$z\in(1,\infty)$ using the substitution $f(z)=(z^{2}-1)^{\frac{1-d}{4}}g(z)$,
$g(z)$ satisfies the standard Legendre equation, hence the two independent
solutions of (D.56) are

$\displaystyle\phi_{1}(z)$
$\displaystyle=a_{1}\left(z^{2}-1\right)^{\frac{1-d}{4}}P_{\frac{1}{2}(d-2\Delta-1)}^{\frac{d-1}{2}}(z),$
(D.60) $\displaystyle\phi_{2}(z)$
$\displaystyle=a_{2}\left(z^{2}-1\right)^{\frac{1-d}{4}}Q_{\frac{1}{2}(d-2\Delta-1)}^{\frac{d-1}{2}}(z),$
(D.61)

where the Legendre functions $P^{a}_{b}(z),Q^{a}_{b}(z)$ have a branch cut
along $z\in(-\infty,1)$, and in Mathematica they are `LegendreP[b,a,3,z]`,
`LegendreQ[b,a,3,z]` respectively. For example, if choosing
$a_{2}=\frac{e^{-\frac{1}{2}i\pi(d-1)}}{\Gamma(d-\Delta)}$, then $f_{2}(z)$ is
well-defined and real-valued for any $d,\Delta\in\mathbb{R}$ and
$z\in(1,\infty)$.

For $z\in(-1,1)$ we choose the following solutions

$\displaystyle\phi_{1}(z)$
$\displaystyle=a^{\prime}_{1}\left(1-z^{2}\right)^{\frac{1-d}{4}}\widetilde{P}_{\frac{1}{2}(d-2\Delta-1)}^{\frac{d-1}{2}}(z),$
(D.62) $\displaystyle\phi_{2}(z)$
$\displaystyle=a^{\prime}_{2}\left(1-z^{2}\right)^{\frac{1-d}{4}}\widetilde{Q}_{\frac{1}{2}(d-2\Delta-1)}^{\frac{d-1}{2}}(z),$
(D.63)

where $\widetilde{P},\widetilde{Q}$ have a branch cut along
$z\in(-\infty,-1)\cup(1,\infty)$, and in Mathematica they are
`LegendreP[b,a,2,z]`, `LegendreQ[b,a,2,z]` respectively.

####  D.5.1 EAdS case

The radial direction of the EAdS space corresponds to the region
$z\in(1,\infty)$. For convenience we work in the $t$-coordinate with $z=\cosh
t,\,t>0$. The Sturm–Liouville equation is

$\sinh^{-d}t\,\frac{d}{dt}\left[\sinh^{d}t\frac{df}{dt}\right]=\Delta(\Delta-d)f.$
(D.64)

Then the inner product is

$(f,g)=\int_{0}^{\infty}dt\sinh^{d}t\,f^{*}(t)g(t),$ (D.65)

and the boundary terms are

$\sinh^{d}t\left[\frac{d}{dt}f_{1}(t)f_{2}^{*}(t)-f_{1}(t)\frac{d}{dt}f_{2}^{*}(t)\right]_{t=0}^{t=\infty}.$
(D.66)

The spherical function is a linear combination of Legendre functions,

$\phi_{\Delta}(t)=\frac{(2\pi)^{\frac{d+1}{2}}\Gamma(1-\Delta)}{\Gamma(d-\Delta)}\left(\phi_{1}(\cosh
t)+\frac{2}{(\tan\frac{\pi d}{2}-i)\pi}\phi_{2}(\cosh t)\right),$ (D.67)

where we have set $a_{1}=a_{2}=1$ in (D.60) and (D.61). By the theory of ODE,
the other solution of (D.64) besides $\phi_{\Delta}(t)$ is singular at $t=0$,
since the index equation of (D.64) at $t=0$ is $x(x-1)+xd=0$ and the
difference of roots $x_{1}-x_{2}=d-1$ is an integer.

To establish the completeness of the spherical functions, we require that the
boundary conditions should eliminate the boundary terms (D.66) and exclude the
non-spherical singular solutions. The correct choice is

$\displaystyle f(0)<\infty,\,f^{\prime}(0)=0\,\text{ at }\,t=0,$ (D.68)
$\displaystyle f(t)\sim O(e^{-\frac{d}{2}t})\,\text{ as }\,t\to\infty,$ (D.69)

and this determines the Hilbert space $\mathcal{H}_{{\mathcal{L}}}$ of the
Sturm-Liouville problem.

The reality condition of the eigenvalues $\Delta(\Delta-d)$ implies: 1)
$\Delta=\frac{d}{2}+is,\,s\geqslant 0$; 2) $\Delta=\frac{d}{2}+s,\,s<0$, where
the redundancy from the shadow symmetry
$\phi_{\Delta}(t)=\phi_{\widetilde{\Delta}}(t)$ has been reduced. The
asymptotic behaviour of $\phi_{\Delta}(t)$ as $t\to\infty$ is

$\phi_{\Delta}(t)\sim e^{-\widetilde{\Delta}t}c(\Delta)+e^{-\Delta
t}c(\widetilde{\Delta}),\,\text{ where
}\,c(\Delta)=\frac{2^{d}\pi^{\frac{d}{2}}\Gamma(\Delta-\frac{d}{2})}{\Gamma(\Delta)}.$
(D.70)

For the first case, $\phi_{\Delta}(t)\sim
e^{-\frac{d}{2}t}(e^{ist}c(\Delta)+e^{-ist}c(\widetilde{\Delta}))$ saturates
the boundary condition and oscillates like plane-wave, hence the spherical
functions $\phi_{\frac{d}{2}+is}(t),\,s\geqslant 0$ of unitary principal
series belong to the continuous spectrum of the Laplacian. For the second
case, the term $\phi_{\Delta}(t)\sim
e^{-\frac{d}{2}t}e^{-st}c(\widetilde{\Delta})$ dominates the growth and
exceeds the boundary condition unless the coefficient vanishes
$c(\widetilde{\Delta})=0$. The equation $c(\widetilde{\Delta})=0$ does not
have solutions for $s<0$, hence all the eigenfunctions
$\phi_{\frac{d}{2}+s}(t),\,s<0$ are ruled out and there is no discrete
spectrum.

Hence the spherical functions $\phi_{\frac{d}{2}+is}(t),\,s\geqslant 0$
associated with the unitary principal series representations provide a
orthogonal complete basis of the radial Laplacian (D.64).

####  D.5.2 dS case

For the dS case, there are several differences from the EAdS case: 1) the
region is $z\in\mathbb{R}$ and the joint points $z=\pm 1$ are singular; 2) the
well-defined eigenfunctions are the regularized spherical functions
$\varphi_{\Delta,\epsilon}(z)$ and they exhaust all the solutions of the
Sturm-Liouville equation.

Since the dS spherical functions are even/odd functions of $z$, we can
restrict to $z>0$ and the boundary terms at $z=0$ get canceled by the parity
symmetry. As $z\to 1^{\pm}$ the asymptotic behaviours of
$\varphi_{\Delta,\epsilon}(z)$ take the same form, the boundary terms at $z=1$
cancel with each other due to the factor $\operatorname{sgn}(z-1)$. Similar to
the EAdS case, to cancel the boundary term at $z=\infty$ we choose the
following boundary condition

$f(z)\sim O(z^{-\frac{d}{2}})\,\text{ as }\,z\to\infty,$ (D.71)

and this determines the Hilbert space ${\mathcal{H}}_{{\mathcal{L}}}$.

The reality condition of eigenvalues is the same as before: 1)
$\Delta=\frac{d}{2}+is,\,s\geqslant 0$; 2) $\Delta=\frac{d}{2}+s,\,s<0$. The
asymptotic behaviours of the spherical functions as $z\to\infty$ are

$\varphi_{\Delta,\epsilon}(z)\sim
z^{-\widetilde{\Delta}}c(\Delta,\epsilon)+z^{-\Delta}c(\widetilde{\Delta},\epsilon),$
(D.72)

where the $c$-function is

$c(\Delta,\epsilon)=\frac{2\pi^{\frac{d-1}{2}}\Gamma(\Delta-\frac{d}{2})\cos\frac{\pi}{2}(d-\Delta+\epsilon)}{\Gamma(\frac{\Delta-d+\epsilon+1}{2})\Gamma(\frac{\Delta+\epsilon}{2})}.$
(D.73)

For the first case, all the spherical functions associated with the principal
series saturate the boundary condition, hence they are in the continuum
spectrum of the Laplacian. For the second case, the dominate term is
$z^{-\Delta}c(\widetilde{\Delta},\epsilon)$. But unlike the EAdS case, for
$s<0$ the $c$-function
$c(\widetilde{\Delta},\epsilon)\sim\cos\frac{\pi}{2}(\Delta+\epsilon)=0$ has
solutions at the set

$D^{-}_{\text{dS}}=\\{(\Delta\in\mathbb{Z},\epsilon):\,\Delta<\frac{d}{2}\,\text{
and }\,\epsilon=1+\Delta\,\operatorname{mod}\mathbb{Z}_{2}\\}.$ (D.74)

Namely, $\Delta$ is an integer at the left of the principal series and the
parity $\epsilon$ is determined by $\Delta$. For $(\Delta,\epsilon)\in
D^{-}_{\text{dS}}$, the spherical functions $\varphi_{\Delta,\epsilon}(z)$ are
square integrable due to the estimate,

$\int_{\Lambda}^{\infty}(z^{2}-1)^{\frac{d-1}{2}}dz\,|\phi_{\Delta,\epsilon}(t)|^{2}\sim\int_{\Lambda}^{\infty}dz\,z^{2s-1}<\infty,\,\text{
for }\,s<0.$ (D.75)

Hence the corresponding representations are in the discrete part of
$L^{2}(\operatorname{dS}_{d+1})$. By comparing the Casimir eigenvalues, they
are the exceptional series representations for $\Delta>d$ and the
complementary series representations for $\frac{d}{2}<\Delta\leqslant d$.

###  D.6 EAdS/dS spherical functions

In this section we derive the $H$-spherical functions on the EAdS and dS
spacetimes. We need the following conformal integrals: for timelike $X$,
$-X^{2}>0$,

$\int_{\operatorname{LC}_{d+1}}d^{\prime}P\,(-X\cdot
P)^{-d}=\operatorname{vol}{\operatorname{S}^{d}}(-X^{2})^{-\frac{d}{2}},$
(D.76)

and for spacelike $X$, $X^{2}>0$,

$\int_{\operatorname{LC}_{d+1}}d^{\prime}P\,|X\cdot
P|^{-d}=2\pi^{\frac{d-1}{2}}\Gamma(\frac{1-d}{2})(X^{2})^{-\frac{d}{2}}=\frac{1}{\cos\frac{d}{2}\pi}\operatorname{vol}{\operatorname{S}^{d}}(X^{2})^{-\frac{d}{2}}.$
(D.77)

They can be computed in the global coordinates, and the remaining integrals
are

$\displaystyle\operatorname{vol}{\operatorname{S}^{d-1}}\int_{0}^{\pi}d\theta\,\sin^{d-1}\theta=\operatorname{vol}\operatorname{S}^{d},$
(D.78)
$\displaystyle\operatorname{vol}{\operatorname{S}^{d-1}}\int_{0}^{\pi}d\theta\,\sin^{d-1}\theta|\cos^{-d}\theta|=2\pi^{\frac{d-1}{2}}\Gamma(\frac{1-d}{2}).$
(D.79)

The prefactor $\Gamma(\frac{1-d}{2})$ in (D.77) is exactly the regularization
factor of the distribution $|X\cdot P|^{-d}$. By the relations (B.4), the
conformal integral (D.77) is equivalent to

$\int_{\operatorname{LC}_{d+1}}d^{\prime}P\,(X\cdot P\pm
i\epsilon)^{-d}=e^{\mp\frac{d}{2}\pi
i}\operatorname{vol}{\operatorname{S}^{d}}(X^{2})^{-\frac{d}{2}}.$ (D.80)

The Feynman-Schwinger parameterization is

$\displaystyle\prod_{i=1}^{n}A_{i}^{-a_{i}}$
$\displaystyle=\frac{\Gamma(d)}{\prod_{i=1}^{n}\Gamma(a_{i})}\int_{0}^{\infty}\prod_{i=2}^{n}ds_{i}s_{i}^{a_{i}-1}\,(A_{1}+\sum_{i=2}^{n}s_{i}A_{i})^{-d}$
(D.81)
$\displaystyle=\frac{\Gamma(d)}{\prod_{i=1}^{n}\Gamma(a_{i})}\int_{0}^{1}\prod_{i=1}^{n}du_{i}u_{i}^{a_{i}-1}\,\delta(1-\sum_{i=1}^{n}u_{i})(\sum_{i=1}^{n}u_{i}A_{i})^{-d},$
(D.82)

where $\Re(a_{i})>0$ and $d=\sum_{i}a_{i}$.

The Gegenbauer integral is related to the hypergeometric function by

$\displaystyle I_{G}(z)$
$\displaystyle=C^{\frac{d}{2}}_{-\Delta}(z)\frac{\pi}{\sin\pi
z}=\int_{0}^{\infty}dss^{\Delta-1}\,(1+s^{2}+2sz)^{-\frac{d}{2}}$ (D.83)
$\displaystyle=\frac{\Gamma(\Delta)\Gamma(d-\Delta)}{\Gamma(d)}{}_{2}F_{1}(\Delta,d-\Delta,\frac{d+1}{2},\frac{1-z}{2}).$
(D.84)

where $C^{\frac{d}{2}}_{-\Delta}(z)$ is the Gegenbauer function, see section
3.15.2 of [67].

####  D.6.1 EAdS spherical functions

In the global coordinates $X=(\cosh t,\widehat{X}\sinh t)$,
$P=(1,\widehat{P})$ and $\cos\theta=\widehat{X}\cdot\widehat{P}$, the
spherical function (D.23) is

$\displaystyle\phi_{\Delta}(t)$
$\displaystyle=\int_{\operatorname{S}^{d}}d\widehat{P}\,(-X\cdot
P)^{-\widetilde{\Delta}}$ (D.85)
$\displaystyle=\operatorname{vol}{\operatorname{S}^{d-1}}\int_{0}^{\pi}d\theta\,\sin^{d-1}\theta(\cosh
t-\sinh t\cos\theta)^{-\widetilde{\Delta}}$ (D.86)
$\displaystyle=\operatorname{vol}{\operatorname{S}^{d-1}}2^{d-1}e^{-\Delta
t}\int_{0}^{1}dz\,z^{\frac{d}{2}-1}(1-z)^{\frac{d}{2}-1}(1-(1-e^{-2t})z)^{-\widetilde{\Delta}}$
(D.87)
$\displaystyle=\operatorname{vol}{\operatorname{S}^{d}}\,e^{-\widetilde{\Delta}t}\,_{2}F_{1}(\frac{d}{2},\widetilde{\Delta},d,1-e^{-2t}),$
(D.88)

where in the third line the $\theta$-integral is substituted by
$z=\frac{1}{2}(1+\cos\theta)$. As a crosscheck, we re-derive the spherical
function (D.23) by the Feynman-Schwinger parameterization: with $T=-X_{1}\cdot
X_{2}\geqslant 1$ the two-point spherical function is

$\displaystyle\phi_{\Delta}(X_{1},X_{2})$
$\displaystyle=\int_{\operatorname{LC}_{d+1}}d^{\prime}P\,(-X_{1}\cdot
P)^{-\widetilde{\Delta}}(-X_{2}\cdot P)^{-\Delta}$ (D.89)
$\displaystyle=\frac{\Gamma(d)}{\Gamma(\Delta)\Gamma(\widetilde{\Delta})}\int_{0}^{\infty}dss^{\Delta-1}\,\int_{\operatorname{LC}_{d+1}}d^{\prime}P\,(-(X_{1}+sX_{2})\cdot
P)^{-d}$ (D.90)
$\displaystyle=\operatorname{vol}{\operatorname{S}^{d}}\frac{\Gamma(d)}{\Gamma(\Delta)\Gamma(\widetilde{\Delta})}\int_{0}^{\infty}dss^{\Delta-1}\,(1+s^{2}+2sT)^{-\frac{d}{2}}$
(D.91)
$\displaystyle=\operatorname{vol}{\operatorname{S}^{d}}{}_{2}F_{1}(\Delta,\widetilde{\Delta},\frac{d+1}{2},\frac{1-T}{2}),$
(D.92)

where in the third line, the $P$-integral is done by (D.76), and in the last
line the $s$-integral is done by (D.83). The two results (D.88) and (D.92) are
related by the quadratic transformations of hypergeometric functions.

We show that the spherical functions of complementary series are analytic
continuation of $\Delta$ from that of principal series, although they do not
enter into the complete basis. In this case $\Delta$ is real and the inner
product follows from (D.19). The spherical function is

$\displaystyle\phantom{=}c_{\text{cp}}\int
d^{\prime}P_{1}d^{\prime}P_{2}\,(-X_{1}\cdot P_{1})^{-\Delta}(-P_{1}\cdot
P_{2})^{-\widetilde{\Delta}}(-X_{2}\cdot P_{2})^{-\Delta}$ (D.93)
$\displaystyle=c_{\text{cp}}\frac{\Gamma(d)}{\Gamma(\Delta)\Gamma(\widetilde{\Delta})}\int
d^{\prime}P_{1}d^{\prime}P_{2}\,\int_{0}^{\infty}d\alpha\alpha^{\widetilde{\Delta}-1}(-(X_{1}+\alpha
P_{2})\cdot P_{1})^{-d}(-X_{2}\cdot P_{2})^{-\Delta}$ (D.94)
$\displaystyle=c_{\text{cp}}\frac{\operatorname{vol}{\operatorname{S}^{d}}\Gamma(d)}{\Gamma(\Delta)\Gamma(\widetilde{\Delta})}\int
d^{\prime}P_{2}\,\int_{0}^{\infty}d\alpha\alpha^{\widetilde{\Delta}-1}(1-2\alpha
X_{1}\cdot P_{2})^{-\frac{d}{2}}(-X_{2}\cdot P_{2})^{-\Delta}$ (D.95)
$\displaystyle=c_{\text{cp}}\frac{2^{\Delta}\pi^{\frac{d}{2}}\Gamma(\Delta-\frac{d}{2})}{\Gamma(\Delta)}\int_{\operatorname{LC}_{d+1}}d^{\prime}P_{2}\,(-X_{1}\cdot
P_{2})^{-\widetilde{\Delta}}(-X_{2}\cdot P_{2})^{-\Delta}$ (D.96)
$\displaystyle\sim\phi_{\Delta,\text{principal}}(X_{1},X_{2}).$ (D.97)

In the second line the Feynman-Schwinger parameterization is used to merge the
factors, and in the third line the integral over $P_{1}$ is done by (D.76).

####  D.6.2 dS spherical functions

The dS two-point spherical funcions (D.37) take different forms in the three
hyperbolic charts $T=X_{1}\cdot X_{2}\in(-\infty,-1)$, $(-1,1)$ and
$(1,\infty)$,

$\phi_{\Delta,\epsilon}(T)=\int_{\operatorname{LC}_{d+1}}d^{\prime}P\,|X_{1}\cdot
P|^{-\widetilde{\Delta},\epsilon}|X_{2}\cdot P|^{-\Delta,\epsilon},$ (D.98)

and the result of $T\in(-\infty,-1)$ is related to that of $T\in(1,\infty)$ by
the parity symmetry

$\phi_{\Delta,\epsilon}(T)=(-1)^{\epsilon}\phi_{\Delta,\epsilon}(-T).$ (D.99)

As a crosscheck, we also evaluate the spherical functions by two different
methods.

First method. We fix $X_{2}$ to the reference point $X_{0}=(0,\dots,1)$ and
parametrize $X_{1}$ and $P$ by the hyperbolic coordinates (D.4),

$\displaystyle\phi_{\Delta,\epsilon}(T)$
$\displaystyle=\int_{\operatorname{LC}_{d+1}}d^{\prime}P\,|X_{1}\cdot
P|^{-\widetilde{\Delta},\epsilon}|P^{d+1}|^{-\Delta,\epsilon}$ (D.100)
$\displaystyle=\int_{\operatorname{H}_{d}}d\widecheck{P}\,|X_{1}\cdot(\widecheck{P},1)|^{-\widetilde{\Delta},\epsilon}+\int_{\operatorname{H}_{d}}d\widecheck{P}\,|X_{1}\cdot(\widecheck{P},-1)|^{-\widetilde{\Delta},\epsilon}(-1)^{\epsilon}.$
(D.101)

In the third chart of the hyperbolic coordinates, with
$X_{1}=(\widecheck{X}\sinh t,\cosh t)$, $T=\cosh t,\,t\geqslant 0$,
$P=(\widecheck{P},\pm 1)$ and $\cosh
s=-\widecheck{X}\cdot\widecheck{P},\,s\geqslant 0$, we have

$\phi_{\Delta,\epsilon}(t)=\phi_{\Delta,\epsilon}^{+}(t)+\phi_{\Delta,\epsilon}^{-}(t),$
(D.102)

where $\pm$ labels the contributions from the two terms in (D.101). For the
minus term, the factor $\cosh t+\sinh t\cosh s>0$ is positive, and

$\displaystyle\phi_{\Delta,\epsilon}^{-}(t)$
$\displaystyle=\operatorname{vol}\operatorname{S}^{d-1}(-1)^{\epsilon}\int_{0}^{\infty}ds\,\sinh^{d-1}s\,|-\cosh
t-\sinh t\cosh s|^{\Delta-d,\epsilon}$ (D.103)
$\displaystyle=\operatorname{vol}\operatorname{S}^{d-1}\int_{0}^{\infty}ds\,\sinh^{d-1}s\,(\cosh
t+\sinh t\cosh s)^{\Delta-d}$ (D.104)
$\displaystyle=2^{d-1}e^{(d-\Delta)t}\int_{0}^{1}dz\,z^{\frac{d}{2}-1}(1-z)^{-\Delta}(e^{2t}-z)^{\Delta-d}$
(D.105)
$\displaystyle=\frac{2^{d}\pi^{d/2}\Gamma(1-\Delta)}{\Gamma(\frac{d}{2}-\Delta+1)}e^{(\Delta-d)t}{}_{2}F_{1}(\frac{d}{2},d-\Delta,\frac{d}{2}-\Delta+1,e^{-2t}),$
(D.106)

where in the third line $s$ is substituted by $z=\tanh^{2}\frac{s}{2}$. The
plus term can be separated into two parts at $\cosh s=\coth t$ according to
the sign of $X_{1}\cdot P$, denoted as $I^{p/n}$,

$\phi_{\Delta,\epsilon}^{+}(t)=\operatorname{vol}\operatorname{S}^{d-1}\int_{0}^{\infty}ds\,\sinh^{d-1}s\,|\cosh
t-\sinh t\cosh s|^{\Delta-d,\epsilon}=I^{p}(t)+(-1)^{\epsilon}I^{n}(t).$
(D.107)

For the positive part $I^{p}(t)$ with $\cosh s\in(1,\coth t)$, we use the
substitution $z=\frac{\cosh s-1}{\coth t-1}$,

$\displaystyle I^{p}(t)$
$\displaystyle=\operatorname{vol}{\operatorname{S}^{d-1}}e^{(d-\Delta)t}(\coth
t-1)^{d-1}\int_{0}^{1}dz\,(e^{2t}+z-1)^{\frac{d-2}{2}}z^{\frac{d-2}{2}}(1-z)^{\Delta-d}$
(D.108)
$\displaystyle=\frac{2^{d}\pi^{d/2}\Gamma(-d+\Delta+1)}{\Gamma(-\frac{d}{2}+\Delta+1)}e^{(d-\Delta)t}(e^{2t}-1)^{-d/2}\,{}_{2}F_{1}(1-\frac{d}{2},\frac{d}{2},-\frac{d}{2}+\Delta+1,\frac{1}{1-e^{2t}}).$

And for the negative part $I^{n}(t)$ with $\cosh s\in(\coth t,\infty)$, we use
the substitution $z=\frac{\coth t-1}{\cosh s-1}$,

$\displaystyle I^{n}(t)$
$\displaystyle=\operatorname{vol}{\operatorname{S}^{d-1}}e^{(d-\Delta)t}(\coth
t-1)^{d-1}\int_{0}^{1}dz\,\left((e^{2t}-1)z+1\right)^{\frac{d-2}{2}}z^{-\Delta}(z-1)^{\Delta-d}$
(D.109)
$\displaystyle=\frac{2^{d}\pi^{d/2}\Gamma(1-\Delta)\Gamma(-d+\Delta+1)}{\Gamma(\frac{d}{2})\Gamma(2-d)}e^{(d-\Delta)t}(e^{2t}-1)^{1-d}{}_{2}F_{1}(1-\frac{d}{2},1-\Delta,2-d,1-e^{2t}).$

Then the total contributions to the spherical functions are

$\phi_{\Delta,\epsilon}(T)=2^{\frac{d+1}{2}}\pi^{\frac{d-1}{2}}\Gamma(1-\Delta)\left[\Gamma(-d+1+\Delta)\left((-1)^{\epsilon}\sin\frac{d\pi}{2}+\sin\frac{(d-2\Delta)\pi}{2}\right)\phi_{1}(T)+2\phi_{2}(T)\right],$
(D.110)

where we have set
$a_{1}=1,a_{2}=\frac{e^{-\frac{1}{2}i\pi(d-1)}}{\Gamma(d-\Delta)}$ in (D.60)
and (D.61).

For the second chart $T\in(-1,1)$, with
$X_{1}=(\widetilde{X}\sin\theta,\cos\theta)$, $P=(\widecheck{P},\pm 1)$ and
$T=\cos\theta$, the integral also contains two terms from (D.101),

$\phi_{\Delta,\epsilon}(\theta)=\phi_{\Delta,\epsilon}^{+}(\theta)+\phi_{\Delta,\epsilon}^{-}(\theta),$
(D.111)

and each term is of the form
$\phi^{\pm}_{\Delta,\epsilon}(\theta)=\int_{\operatorname{H}_{d}}d\widecheck{P}\,f^{\pm}(\widetilde{X}\cdot\widecheck{P},\theta)$.
For the plus term $\phi_{\Delta,\epsilon}^{+}(\theta)$, using the symmetry
$\operatorname{SO}(d,1)$ of $\operatorname{H}_{d}$ and
$\operatorname{dS}_{d}$, we can fix $\widetilde{X}=(0,\dots,1)$ and
parametrize $\widecheck{P}$ as $\widecheck{P}=(\cosh s,\dots,\sinh
s\cos\varphi)$, then

$\displaystyle\phi_{\Delta,\epsilon}^{+}(\theta)$
$\displaystyle=\operatorname{vol}\operatorname{S}^{d-2}\int_{0}^{\pi}d\varphi\,\int_{0}^{\infty}ds\,\sin^{d-2}\varphi\sinh^{d-1}s|\cos\theta+\sin\theta\sinh
s\cos\varphi|^{\Delta-d,\epsilon}$ (D.112)
$\displaystyle=\operatorname{vol}\operatorname{S}^{d-2}\int_{0}^{\infty}dy\,\int_{\mathbb{R}}dx\,\frac{y^{d-2}}{\sqrt{1+x^{2}+y^{2}}}|\cos\theta+x\sin\theta|^{\Delta-d,\epsilon}$
(D.113)
$\displaystyle=\pi^{\frac{d}{2}-1}\Gamma(1-\frac{d}{2})\int_{\mathbb{R}}dx\,(1+x^{2})^{\frac{d}{2}-1}|\cos\theta+x\sin\theta|^{\Delta-d,\epsilon}$
(D.114) $\displaystyle=J^{p}(\theta)+(-1)^{\epsilon}J^{n}(\theta),$ (D.115)

where we have substituted $x=\sinh s\cos\varphi,y=\sinh s\sin\varphi$ and
separated the integral into two parts according to the sign of
$\cos\theta+x\sin\theta$. The minus term $\phi_{\Delta,\epsilon}^{-}(\theta)$
is equal to the plus term by

$\phi_{\Delta,\epsilon}^{-}(\theta)=(-1)^{\epsilon}\left(J^{p}(\pi-\theta)+(-1)^{\epsilon}J^{n}(\pi-\theta)\right)=(-1)^{\epsilon}\left(J^{n}(\theta)+(-1)^{\epsilon}J^{p}(\theta)\right)=\phi_{\Delta,\epsilon}^{+}(\theta).$
(D.116)

For the postive part $J^{p}(\theta)$ with $x\in(-\cot\theta,\infty)$, by the
substitution $z=\cos\theta+x\sin\theta\in(0,\infty)$, we have

$\displaystyle J^{p}(\theta)$
$\displaystyle=\pi^{\frac{d}{2}-1}\Gamma(1-\frac{d}{2})\sin^{1-d}\theta\int_{0}^{\infty}dz\,z^{\Delta-d}(1+z^{2}-2z\cos\theta)^{\frac{d}{2}-1}$
(D.117)
$\displaystyle=\frac{2^{d-1}\pi^{\frac{d-1}{2}}\Gamma(1-\Delta)\Gamma(-d+\Delta+1)}{\Gamma(\frac{3}{2}-\frac{d}{2})}\sin^{1-d}\theta{}_{2}F_{1}(1-\Delta,1-d+\Delta,\frac{3-d}{2},\frac{1+\cos\theta}{2}),$
(D.118)

and for the minus part $J^{n}(\theta)$ with $x\in(-\infty,-\cot\theta)$, by
the substitution $z=-\cos\theta-x\sin\theta\in(0,\infty)$, we have

$\displaystyle J^{n}(\theta)$
$\displaystyle=\pi^{\frac{d}{2}-1}\Gamma(1-\frac{d}{2})\sin^{1-d}\theta\int_{0}^{\infty}dz\,z^{\Delta-d}(1+z^{2}+2z\cos\theta)^{\frac{d}{2}-1}$
(D.119)
$\displaystyle=\frac{2^{d-1}\pi^{\frac{d-1}{2}}\Gamma(1-\Delta)\Gamma(-d+\Delta+1)}{\Gamma(\frac{3}{2}-\frac{d}{2})}\sin^{1-d}\theta{}_{2}F_{1}(1-\Delta,1-d+\Delta,\frac{3-d}{2},\frac{1-\cos\theta}{2}).$
(D.120)

Then the spherical functions are

$\phi_{\Delta,\epsilon}(T)=2^{\frac{d+1}{2}}\pi^{\frac{d-1}{2}}\Gamma(1-\Delta)\Gamma(-d+\Delta+1)\left[\phi_{1}(T)\left((-1)^{\epsilon}-\cos\pi(d-\Delta)\right)+\phi_{2}(T)\frac{2\sin\pi(d-\Delta)}{\pi}\right],$
(D.121)

where we have set $a^{\prime}_{1}=1,a^{\prime}_{2}=1$ in (D.62) and (D.63).

Second method. We evaluate the two-point spherical functions by the Feynman-
Schwinger parameterization. By the relations (B.6), the integrand $|X_{1}\cdot
P|^{-\widetilde{\Delta},\epsilon}|X_{2}\cdot P|^{-\Delta,\epsilon}$ can be
rewritten as linear combinations of

$h_{\pm,\pm}=(X_{1}\cdot P\pm i\epsilon)^{-\widetilde{\Delta}}(X_{2}\cdot P\pm
i\epsilon)^{-\Delta}.$ (D.122)

Explicitly, they are

$\displaystyle\phi_{\Delta,0}(T)$
$\displaystyle\ni\frac{e^{2i\pi\Delta}h_{-,+}}{\left(1+e^{i\pi\Delta}\right)\left(e^{i\pi
d}+e^{i\pi\Delta}\right)}+\frac{h_{+,-}}{\left(1+e^{i\pi\Delta}\right)\left(1+e^{i\pi(\Delta-d)}\right)}$
(D.123)
$\displaystyle+\frac{e^{i\pi\Delta}h_{-,-}}{\left(1+e^{i\pi\Delta}\right)\left(e^{i\pi
d}+e^{i\pi\Delta}\right)}+\frac{h_{+,+}}{\left(1+e^{-i\pi\Delta}\right)\left(1+e^{i\pi(\Delta-d)}\right)},$
(D.124) $\displaystyle\phi_{\Delta,1}(T)$
$\displaystyle\ni\frac{e^{2i\pi\Delta}h_{-,+}}{\left(-1+e^{i\pi\Delta}\right)\left(e^{i\pi\Delta}-e^{i\pi
d}\right)}+\frac{h_{+,-}}{\left(-1+e^{i\pi\Delta}\right)\left(-1+e^{i\pi(\Delta-d)}\right)}$
(D.125)
$\displaystyle+\frac{e^{i\pi\Delta}h_{-,-}}{\left(-1+e^{i\pi\Delta}\right)\left(e^{i\pi
d}-e^{i\pi\Delta}\right)}+\frac{h_{+,+}}{\left(-1+e^{-i\pi\Delta}\right)\left(-1+e^{i\pi(\Delta-d)}\right)}.$
(D.126)

Then similar to (D.92) we combine the two factor $(X_{1}\cdot P\pm
i\epsilon)^{-\widetilde{\Delta}}$ and $(X_{2}\cdot P\pm i\epsilon)^{-\Delta}$
by the Feynman-Schwinger parameterization and do the $P$-integral by the
conformal integral (D.80).

For $T\in(-1,1),\,s\geqslant 0$, the factor $1+s^{2}\pm 2sT>0$ is positive,
and the $s$-integrand can be done by (D.83). The four terms contribute to

$\displaystyle h_{+,+}$ $\displaystyle\to e^{-\frac{d}{2}\pi
i}\operatorname{vol}{\operatorname{S}^{d}}I_{G,0}(T),$ (D.127) $\displaystyle
h_{-,-}$ $\displaystyle\to e^{d\pi i}e^{-\frac{d}{2}\pi
i}\operatorname{vol}{\operatorname{S}^{d}}I_{G,0}(T),$ (D.128) $\displaystyle
h_{+,-}$ $\displaystyle\to e^{\Delta\pi i}e^{-\frac{d}{2}\pi
i}\operatorname{vol}{\operatorname{S}^{d}}I_{G,0}(-T),$ (D.129) $\displaystyle
h_{-,+}$ $\displaystyle\to e^{(d-\Delta)\pi i}e^{-\frac{d}{2}\pi
i}\operatorname{vol}{\operatorname{S}^{d}}I_{G,0}(-T),$ (D.130)

where we have relabeled $I_{G}$ to $I_{G,0}$,

$I_{G,0}(T)=\frac{\Gamma(\Delta)\Gamma(\widetilde{\Delta})}{\Gamma(d)}{}_{2}F_{1}(\Delta,d-\Delta,\frac{d+1}{2},\frac{1-T}{2}).$
(D.131)

Then the spherical functions are

$\displaystyle\phi_{\Delta,0}(T)$
$\displaystyle=2\pi^{\frac{d}{2}-1}\Gamma(\frac{1}{2}-\frac{\Delta}{2})\Gamma(\frac{-d+\Delta+1}{2}){}_{2}F_{1}(\frac{d-\Delta}{2},\frac{\Delta}{2},\frac{1}{2},T^{2}),$
(D.132) $\displaystyle\phi_{\Delta,1}(T)$
$\displaystyle=4\pi^{\frac{d}{2}-1}T\Gamma(1-\frac{\Delta}{2})\Gamma(\frac{-d+\Delta+2}{2}){}_{2}F_{1}(\frac{d-\Delta+1}{2},\frac{\Delta+1}{2},\frac{3}{2},T^{2}),$
(D.133)

and they match with (D.121).

For $T\in(1,\infty),s\geqslant 0$, the factor $1+s^{2}\pm 2sT$ is not definite
and the integrals should be divided into three parts according to the signs.
The four terms contribute to

$\displaystyle h_{+,+}$ $\displaystyle\to e^{-\frac{d}{2}\pi
i}\operatorname{vol}{\operatorname{S}^{d}}I_{G,0}(T),$ (D.134) $\displaystyle
h_{-,-}$ $\displaystyle\to e^{d\pi i}e^{-\frac{d}{2}\pi
i}\operatorname{vol}{\operatorname{S}^{d}}I_{G,0}(T),$ (D.135) $\displaystyle
h_{+,-}$ $\displaystyle\to e^{\Delta\pi
i}\operatorname{vol}{\operatorname{S}^{d}}\left(e^{-\frac{d}{2}\pi
i}I_{G,1}(T)+e^{-\frac{d}{2}\pi i}I_{G,3}(T)+I_{G,2}(T)\right),$ (D.136)
$\displaystyle h_{-,+}$ $\displaystyle\to e^{(d-\Delta)\pi
i}\operatorname{vol}{\operatorname{S}^{d}}\left(e^{-\frac{d}{2}\pi
i}I_{G,1}(T)+e^{-\frac{d}{2}\pi i}I_{G,3}(T)+I_{G,2}(T)\right),$ (D.137)

where $T=\cosh t$ and

$\displaystyle I_{G,1}(T)$
$\displaystyle=\frac{\Gamma(1-\frac{d}{2})\Gamma(\Delta)}{\Gamma(-\frac{d}{2}+\Delta+1)}e^{-\Delta
t}{}_{2}F_{1}(\frac{d}{2},\Delta,-\frac{d}{2}+\Delta+1,e^{-2t}),$ (D.138)
$\displaystyle I_{G,2}(T)$
$\displaystyle=\frac{\Gamma(1-\frac{d}{2})^{2}}{\Gamma(2-d)}(e^{2t}-1)^{1-d}e^{dt-\Delta
t}\,{}_{2}F_{1}(1-\frac{d}{2},1-\Delta,2-d,1-e^{2t}),$ (D.139) $\displaystyle
I_{G,3}(T)$
$\displaystyle=\frac{\Gamma(1-\frac{d}{2})\Gamma(d-\Delta)}{\Gamma(\frac{d-2\Delta+2}{2})}e^{t(d-\Delta)}{}_{2}F_{1}(\frac{d}{2},d-\Delta,\frac{d-2\Delta+2}{2},e^{2t}).$
(D.140)

Then by the relations between hypergeometric functions and Legendre functions,
the spherical functions match with (D.110).

##  Appendix E Massless scalar-massless scalar-tachyonic scalar

In this appendix, we compute the three-point celestial amplitude
$\mathcal{A}^{\Delta_{i}}_{1^{0}_{0}\rightarrow
2^{0}_{0}+3^{0}_{iM,\epsilon}}$ which involves one incoming massless scalar,
one outgoing massless scalar and one outgoing tachyonic scalar with imaginary
mass $iM$. In terms of the conformal primary basis,
$\mathcal{A}^{\Delta_{i}}_{1^{0}_{0}\rightarrow
2^{0}_{0}+3^{0}_{iM,\epsilon}}$ takes the form as

$\displaystyle\mathcal{A}^{\Delta_{i}}_{1^{0}_{0}\rightarrow
2^{0}_{0}+3^{0}_{iM,\epsilon}}=\int_{0}^{\infty}d\omega_{1}d\omega_{2}\omega_{1}^{\Delta_{1}-1}\omega_{2}^{\Delta_{2}-1}\int\frac{d^{d+1}\widehat{k}}{|\widehat{k}^{+}|}\frac{1}{|\widehat{q}_{3}\cdot\widehat{k}|^{\Delta_{3}}}\text{sgn}^{\epsilon}(\widehat{q}_{3}\cdot\widehat{k})\delta^{(d+2)}(q_{1}-q_{2}-k)\;,$
(E.1)

where $k^{2}=M^{2}$. The region with $k^{2}=M^{2}$ can be divided into two
regions corresponding to expanding and contracting patches of the dS
hypersurfaces

$\displaystyle\begin{split}\mathcal{D}^{+}:\qquad
k^{2}=M^{2}\;,\qquad\widehat{k}^{+}\equiv\widehat{k}^{0}+\widehat{k}^{d+1}>0\;,\\\
\mathcal{D}^{-}:\qquad
k^{2}=M^{2}\;,\qquad\widehat{k}^{+}\equiv\widehat{k}^{0}+\widehat{k}^{d+1}<0\;,\end{split}$
(E.2)

respectively. We can thus define the celestial coordinates in
$\mathcal{D}^{+}$ as

$\displaystyle\begin{split}&k=M\widehat{k}=\frac{M}{2y}(1-y^{2}+|\vec{w}|^{2},2\vec{w},1+y^{2}-|\vec{w}|^{2})\;,\end{split}$
(E.3)

and the celestial coordinates in $\mathcal{D}^{-}$ as

$\displaystyle\begin{split}&k=M\widehat{k}=-\frac{M}{2y}(1-y^{2}+|\vec{w}|^{2},2\vec{w},1+y^{2}-|\vec{w}|^{2})\;,\end{split}$
(E.4)

with $y\in[0,\infty)$ in both regions. As a result,
$\mathcal{A}^{\Delta_{i}}_{1^{0}_{0}\rightarrow
2^{0}_{0}+3^{0}_{iM,\epsilon}}$ can be written as

$\displaystyle\mathcal{A}^{\Delta_{i}}_{1^{0}_{0}\rightarrow
2^{0}_{0}+3^{0}_{iM,\epsilon}}=\mathcal{A}^{+,\Delta_{i}}_{1^{0}_{0}\rightarrow
2^{0}_{0}+3^{0}_{iM,\epsilon}}+\mathcal{A}^{-,\Delta_{i}}_{1^{0}_{0}\rightarrow
2^{0}_{0}+3^{0}_{iM,\epsilon}}\;,$ (E.5)

where $\mathcal{A}^{\pm,\Delta_{i}}_{1^{0}_{0}\rightarrow
2^{0}_{0}+3^{0}_{iM,\epsilon}}$ are obtained by integral $\widehat{k}$ over
region $\mathcal{D}^{\pm}$. We first focus on
$\mathcal{A}^{+,\Delta_{i}}_{1^{0}_{0}\rightarrow
2^{0}_{0}+3^{0}_{iM,\epsilon}}$. Using the fact that

$\displaystyle\int_{\mathcal{D}^{+}}\frac{d^{d+1}\widehat{k}}{|k^{+}|}\frac{1}{|\widehat{q}_{3}\cdot\widehat{k}|^{\Delta_{3}}}=\int_{0}^{\infty}\frac{dy}{y^{d+1}}\int
d^{d}\vec{w}\bigg{|}\frac{y}{|x_{3}-w|^{2}-y^{2}}\bigg{|}^{\Delta_{3}}\;,$
(E.6)

we find that

$\displaystyle\mathcal{A}^{+,\Delta_{i}}_{1^{0}_{0}\rightarrow
2^{0}_{0}+3^{0}_{iM,\epsilon}}=\int_{0}^{\infty}d\omega_{1}d\omega_{2}\omega_{1}^{\Delta_{1}-1}\omega_{2}^{\Delta_{2}-1}\int_{0}^{\infty}\frac{dy}{y^{d+1}}\int
d^{d}\vec{w}\bigg{|}\frac{y}{|x_{3}-w|^{2}-y^{2}}\bigg{|}^{\Delta_{3}}\delta^{(d+2)}(q_{1}-q_{2}-k)\;.$
(E.7)

The support of the momentum-conserving delta-function is

$\displaystyle\begin{split}&\omega_{2}=\frac{M^{2}}{4|x_{12}|^{2}\omega_{1}}\;,\qquad
y=\frac{2M|x_{12}|^{2}\omega_{1}}{4|x_{12}|^{2}\omega_{1}^{2}-M^{2}}\;,\qquad\vec{w}=\frac{M^{2}\vec{x}_{2}-4\vec{x}_{1}|x_{12}|^{2}\omega_{1}^{2}}{M^{2}-4|x_{12}|^{2}\omega_{1}^{2}}\end{split}$
(E.8)

with the Jacobian

$\displaystyle|J|=\left|\frac{M^{d+1}(y^{2}-|x_{2}-w|^{2})}{y^{d+2}}\right|\;.$
(E.9)

We note that $\omega_{1}\geqslant M/(2|x_{12}|)$ since $y\geqslant 0$. The
integrand in (E.7) then becomes

$\displaystyle\begin{split}&\frac{\omega_{1}^{\Delta_{1}-1}\omega_{2}^{\Delta_{2}-1}}{y^{d+1}}\bigg{|}\frac{y}{|x_{3}-w|^{2}-y^{2}}\bigg{|}^{\Delta_{3}}\delta^{(d+2)}(q_{1}-q_{2}-k)\\\
&=\frac{2^{1-2\Delta_{2}+\Delta_{3}}M^{2\Delta_{2}+\Delta_{3}-d-2}\omega_{1}^{\Delta_{1}-\Delta_{2}+\Delta_{3}-1}}{|x_{12}|^{2\Delta_{2}-2\Delta_{3}}}\bigg{|}\frac{1}{-M^{2}|x_{23}|^{2}+4|x_{12}|^{2}|x_{13}|^{2}\omega_{1}^{2}}\bigg{|}^{\Delta_{3}}\\\
&\quad\times\delta\bigg{(}y-\frac{2M|x_{12}|^{2}\omega_{1}}{4|x_{12}|^{2}\omega_{1}^{2}-M^{2}}\bigg{)}\delta\bigg{(}\omega_{2}-\frac{M^{2}}{4|x_{12}|^{2}\omega_{1}}\bigg{)}\delta^{(d)}\bigg{(}\vec{w}-\frac{M^{2}\vec{x}_{2}-4\vec{x}_{1}|x_{12}|^{2}\omega_{1}^{2}}{M^{2}-4|x_{12}|^{2}\omega_{1}^{2}}\bigg{)}\;.\end{split}$
(E.10)

Performing the integral over $y$, $\omega_{2}$, and $\vec{w}$ leads to

$\displaystyle\begin{split}\mathcal{A}^{+,\Delta_{i}}_{1^{0}_{0}\rightarrow
2^{0}_{0}+3^{0}_{iM,\epsilon}}=&\frac{2^{1-2\Delta_{2}+\Delta_{3}}M^{2\Delta_{2}+\Delta_{3}-d-2}}{|x_{12}|^{2\Delta_{2}-2\Delta_{3}}}\int_{\frac{M}{2|x_{12}|}}^{\infty}d\omega_{1}\frac{\omega_{1}^{\Delta_{1}-\Delta_{2}+\Delta_{3}-1}\text{sgn}^{\epsilon}(-M^{2}|x_{23}|^{2}+4|x_{12}|^{2}|x_{13})}{\big{|}-M^{2}|x_{23}|^{2}+4|x_{12}|^{2}|x_{13}|^{2}\omega_{1}^{2}\big{|}^{\Delta_{3}}}\;.\end{split}$
(E.11)

Now, we turn to the computation of
$\mathcal{A}^{-,\Delta_{i}}_{1^{0}_{0}\rightarrow
2^{0}_{0}+3^{0}_{iM,\epsilon}}$. In the region $\mathcal{D}^{-}$, we have

$\displaystyle\int_{\mathcal{D}^{-}}\frac{d^{d+1}\widehat{k}}{|k^{+}|}\frac{1}{|\widehat{q}_{3}\cdot\widehat{k}|^{\Delta_{3}}}=\int_{0}^{\infty}\frac{dy}{y^{d+1}}\int
d^{d}\vec{w}\bigg{|}\frac{-y}{|x_{3}-w|^{2}-y^{2}}\bigg{|}^{\Delta_{3}}\;,$
(E.12)

leading to

$\displaystyle\mathcal{A}^{-,\Delta_{i}}_{1^{0}_{0}\rightarrow
2^{0}_{0}+3^{0}_{iM,\epsilon}}=\int_{0}^{\infty}d\omega_{1}d\omega_{2}\omega_{1}^{\Delta_{1}-1}\omega_{2}^{\Delta_{2}-1}\int_{0}^{\infty}\frac{dy}{y^{d+1}}\int
d^{d}\vec{w}\bigg{|}\frac{-y}{|x_{3}-w|^{2}-y^{2}}\bigg{|}^{\Delta_{3}}\delta^{(d)}(q_{1}-q_{2}-k)\;.$
(E.13)

The support of the momentum-conserving delta-function is

$\displaystyle\begin{split}&\omega_{2}=\frac{M^{2}}{4|x_{12}|^{2}\omega_{1}}\;,\qquad
y=\frac{2M|x_{12}|^{2}\omega_{1}}{M^{2}-4|x_{12}|^{2}\omega_{1}^{2}}\;,\qquad\vec{w}=\frac{M^{2}\vec{x}_{2}-4\vec{x}_{1}|x_{12}|^{2}\omega_{1}^{2}}{M^{2}-4|x_{12}|^{2}\omega_{1}^{2}}\end{split}$
(E.14)

with the Jacobian

$\displaystyle|J|=\left|\frac{M^{d+1}(y^{2}-|x_{2}-w|^{2})}{y^{d+2}}\right|\;.$
(E.15)

We note that in the region $\mathcal{D}^{-}$, $y\geqslant 0$ demands that
$0\leqslant\omega_{1}\leqslant M/(2|x_{12}|)$. The integrand in (E.13) then
becomes

$\displaystyle\begin{split}&\frac{\omega_{1}^{\Delta_{1}-1}\omega_{2}^{\Delta_{2}-1}}{y^{d+1}}\bigg{|}\frac{-y}{|x_{3}-w|^{2}-y^{2}}\bigg{|}^{\Delta_{3}}\delta^{(d+2)}(q_{1}-q_{2}-k)\\\
&=\frac{2^{1-2\Delta_{2}+\Delta_{3}}M^{2\Delta_{2}+\Delta_{3}-d-2}\omega_{1}^{\Delta_{1}-\Delta_{2}+\Delta_{3}-1}}{|x_{12}|^{2\Delta_{2}-2\Delta_{3}}}\bigg{|}\frac{1}{-M^{2}|x_{23}|^{2}+4|x_{12}|^{2}|x_{13}|^{2}\omega_{1}^{2}}\bigg{|}^{\Delta_{3}}\\\
&\quad\times\delta\bigg{(}y-\frac{2M|x_{12}|^{2}\omega_{1}}{M^{2}-4|x_{12}|^{2}\omega_{1}^{2}}\bigg{)}\delta\bigg{(}\omega_{2}-\frac{M^{2}}{4|x_{12}|^{2}\omega_{1}}\bigg{)}\delta^{(d)}\bigg{(}\vec{w}-\frac{M^{2}\vec{x}_{2}-4\vec{x}_{1}|x_{12}|^{2}\omega_{1}^{2}}{M^{2}-4|x_{12}|^{2}\omega_{1}^{2}}\bigg{)}\;.\end{split}$
(E.16)

Performing the integral over $y$, $\omega_{2}$, and $\vec{w}$ leads to

$\displaystyle\begin{split}\mathcal{A}^{-,\Delta_{i}}_{1^{0}_{0}\rightarrow
2^{0}_{0}+3^{0}_{iM,\epsilon}}=&\frac{2^{1-2\Delta_{2}+\Delta_{3}}M^{2\Delta_{2}+\Delta_{3}-d-2}}{|x_{12}|^{2\Delta_{2}-2\Delta_{3}}}\int_{0}^{\frac{M}{2|x_{12}|}}d\omega_{1}\frac{\omega_{1}^{\Delta_{1}-\Delta_{2}+\Delta_{3}-1}\text{sgn}^{\epsilon}(-M^{2}|x_{23}|^{2}+4|x_{12}|^{2}|x_{13}|^{2})}{\big{|}-M^{2}|x_{23}|^{2}+4|x_{12}|^{2}|x_{13}|^{2}\omega_{1}^{2}\big{|}^{\Delta_{3}}}\;.\end{split}$
(E.17)

Thus we have

$\displaystyle\begin{split}\mathcal{A}^{\Delta_{i}}_{1^{0}_{0}\rightarrow
2^{0}_{0}+3^{0}_{iM,\epsilon}}=&\frac{2^{1-2\Delta_{2}+\Delta_{3}}M^{2\Delta_{2}+\Delta_{3}-d-2}}{|x_{12}|^{2\Delta_{2}-2\Delta_{3}}}\int_{0}^{\infty}d\omega_{1}\frac{\omega_{1}^{\Delta_{1}-\Delta_{2}+\Delta_{3}-1}\text{sgn}^{\epsilon}(-M^{2}|x_{23}|^{2}+4|x_{12}|^{2}|x_{13})}{\big{|}-M^{2}|x_{23}|^{2}+4|x_{12}|^{2}|x_{13}|^{2}\omega_{1}^{2}\big{|}^{\Delta_{3}}}\\\
=&\frac{2^{1-2\Delta_{2}+\Delta_{3}}M^{2\Delta_{2}+\Delta_{3}-d-2}}{|x_{12}|^{2\Delta_{2}-2\Delta_{3}}}\bigg{(}\int_{0}^{\frac{M|x_{23}|}{2|x_{12}||x_{13}|}}d\omega_{1}\frac{\omega_{1}^{\Delta_{1}-\Delta_{2}+\Delta_{3}-1}(-1)^{\epsilon}}{(M^{2}|x_{23}|^{2}-4|x_{12}|^{2}|x_{13}|^{2}\omega_{1}^{2})^{\Delta_{3}}}\\\
&\qquad+\int_{\frac{M|x_{23}|}{2|x_{12}||x_{13}|}}^{\infty}d\omega_{1}\frac{\omega_{1}^{\Delta_{1}-\Delta_{2}+\Delta_{3}-1}}{(-M^{2}|x_{23}|^{2}+4|x_{12}|^{2}|x_{13}|^{2}\omega_{1}^{2})^{\Delta_{3}}}\bigg{)}\\\
=&\frac{C^{\Delta_{i}}_{1^{0}_{0}\rightarrow
2^{0}_{0}+3^{0}_{iM,\epsilon}}}{|x_{12}|^{\Delta_{1}+\Delta_{2}-\Delta_{3}}|x_{13}|^{\Delta_{1}-\Delta_{2}+\Delta_{3}}|x_{23}|^{\Delta_{2}+\Delta_{3}-\Delta_{1}}}\;,\end{split}$
(E.18)

where $C^{\Delta_{i}}_{1^{0}_{0}\rightarrow 2^{0}_{0}+3^{0}_{iM,\epsilon}}$ is
given by

$\displaystyle\begin{split}C^{\Delta_{i}}_{1^{0}_{0}\rightarrow
2^{0}_{0}+3^{0}_{iM,\epsilon}}=&\frac{M^{\Delta_{1}+\Delta_{2}-d-2}\Gamma[1-\Delta_{3}]}{2^{\Delta_{1}+\Delta_{2}}}\bigg{(}\frac{\Gamma[\frac{\Delta_{3}-\Delta_{1}+\Delta_{2}}{2}]}{\Gamma[\frac{2-\Delta_{1}+\Delta_{2}-\Delta_{3}}{2}]}+(-1)^{\epsilon}\frac{\Gamma[\frac{\Delta_{3}+\Delta_{1}-\Delta_{2}}{2}]}{\Gamma[\frac{2+\Delta_{1}-\Delta_{2}-\Delta_{3}}{2}]}\bigg{)}\;.\end{split}$
(E.19)

Using the identity

$\displaystyle\Gamma[x]\Gamma[1-x]=\frac{\pi}{\sin[\pi x]}\;,$ (E.20)

we get

$\displaystyle\begin{split}C^{\Delta_{i}}_{1^{0}_{0}\rightarrow
2^{0}_{0}+3^{0}_{iM,\epsilon}}=&\frac{M^{\Delta_{1}+\Delta_{2}-d-2}\Gamma[1-\Delta_{3}]\Gamma[\frac{\Delta_{3}-\Delta_{1}+\Delta_{2}}{2}]\Gamma[\frac{\Delta_{3}+\Delta_{1}-\Delta_{2}}{2}]}{2^{\Delta_{1}+\Delta_{2}}}\\\
&\times\bigg{(}2\cos[\frac{\Delta_{3}}{2}\pi]\sin[\frac{(\Delta_{1}-\Delta_{2})}{2}\pi]\bigg{)}^{\epsilon}\bigg{(}2\sin[\frac{\Delta_{3}}{2}\pi]\cos[\frac{(\Delta_{1}-\Delta_{2})}{2}\pi]\bigg{)}^{1-\epsilon}\;.\end{split}$
(E.21)

Moreover, using the symmetric property $\delta^{(4)}(x)=\delta^{(4)}(-x)$, we
find that $\mathcal{A}^{\Delta_{i}}_{2^{0}_{0}+3^{0}_{iM,\epsilon}\rightarrow
1^{0}_{0}}=\mathcal{A}^{\Delta_{i}}_{1^{0}_{0}\rightarrow
2^{0}_{0}+3^{0}_{iM,\epsilon}}$ .

##  Appendix F $t$-channel conformal block expansion from alpha space approach

In this section we derive the $t$-channel conformal block expansion of the
$1d$ four-point massless scalar $t$-channel celestial amplitudes by the alpha
space approach. The four-point massless scalar $t$-channel celestial
amplitudes in one dimensional CCFT is

${}_{t}\mathcal{A}^{\Delta_{i}}_{1^{0}_{0}+2^{0}_{0}\rightarrow
3^{0}_{0}+4^{0}_{0}}=\frac{\pi
m^{\beta-1}}{2^{\beta+4}}\sec[\frac{\pi\beta}{2}]I_{13-24}f(\chi)\;,$ (F.1)

where $f(\chi)=\chi(1-\chi)^{\frac{1}{2}(-1+\Delta_{13}-\Delta_{24})}$. In the
following, we will first consider the conformal block expansion of
$\mathcal{G}(\chi)=\chi^{a}(1-\chi)^{b}$ and then take the special value
$a=1,b=\frac{1}{2}(-1+\Delta_{13}-\Delta_{24})$.

Due to the different boundary condition of the conformal Casimir equation, in
the alpha space approach the integral region is $x\in(0,1)$ and there are no
contributions from the discrete series representations, in contrast to the
Euclidean inversion formula [63, 64], see also the application in celestial
CFT [7].

In the alpha space approach, the stripped conformal blocks and partial waves
are

$\displaystyle G^{\Delta_{13},\Delta_{24}}_{\Delta}(\chi)$
$\displaystyle=\chi^{\Delta}{}_{2}F_{1}(\Delta-\Delta_{13},\Delta+\Delta_{24},2\Delta,\chi),$
(F.2) $\displaystyle\psi_{\Delta}(\chi)$
$\displaystyle=\frac{1}{2}(Q(\Delta)G^{\Delta_{13},\Delta_{24}}_{\Delta}(\chi)+Q(1-\Delta)G^{\Delta_{13},\Delta_{24}}_{1-\Delta}(\chi))$
(F.3)
$\displaystyle=\chi^{\Delta_{13}}{}_{2}F_{1}(\Delta-\Delta_{13},1-\Delta-\Delta_{13},1-\Delta_{13}+\Delta_{24},1-\frac{1}{\chi}),$
(F.4)

with the boundary condition $\psi_{\Delta}(1)=1$. The inversion function and
formula are

$\displaystyle I(\Delta)$
$\displaystyle=\int_{0}^{1}d\chi\,\chi^{-2}(1-\chi)^{-\Delta_{13}+\Delta_{24}}\mathcal{G}(\chi)\psi^{*}_{\Delta}(\chi),$
(F.5) $\displaystyle\mathcal{G}(\chi)$ $\displaystyle=\frac{1}{2\pi
i}\int_{\Gamma}\frac{d\Delta}{N(\Delta)}\,I(\Delta)\psi_{\Delta}(\chi),$ (F.6)

where $\Gamma$ is the principal series, and

$Q(\Delta)=\frac{2\Gamma(1-2\Delta)\Gamma(1-\Delta_{13}+\Delta_{24})}{\Gamma(1-\Delta-\Delta_{13})\Gamma(1-\Delta+\Delta_{24})},\,\text{
and }\,N(\Delta)=\frac{1}{2}Q(\Delta)Q(1-\Delta).$ (F.7)

For the four-point function $\mathcal{G}(\chi)=\chi^{a}(1-\chi)^{b}$ using the
Mellin-Barnes integral and changing the order of integration, we have

$\displaystyle I(\Delta)$
$\displaystyle=\int_{0}^{1}d\chi\,\chi^{a-2}(1-\chi)^{b-\Delta_{13}+\Delta_{24}}\psi_{\Delta}(\chi)$
(F.8) $\displaystyle=\int_{-i\infty}^{+i\infty}\frac{ds}{2\pi
i}\,\frac{(\Delta-\Delta_{13})_{s}(1-\Delta-\Delta_{13})_{s}}{(1-\Delta_{13}+\Delta_{24})_{s}}\Gamma(-s)\cdot\int_{0}^{1}dx\,x^{a+\Delta_{13}-s-2}(1-x)^{b+s-\Delta_{13}+\Delta_{24}}$
$\displaystyle=\int_{-i\infty}^{+i\infty}\frac{ds}{2\pi
i}\,\frac{(\Delta-\Delta_{13})_{s}(1-\Delta-\Delta_{13})_{s}}{(1-\Delta_{13}+\Delta_{24})_{s}}\frac{\Gamma(b+s-\Delta_{13}+\Delta_{24}+1)}{\Gamma(a+b+\Delta_{24})}\Gamma(-s)\Gamma(a+\Delta_{13}-s-1).$

There are two series of poles in the right half-plane, $s=n$ and
$s=a+\Delta_{13}+n-1$ for $n\in\mathbb{N}$, and their contributions are

$\displaystyle I(\Delta)$
$\displaystyle=-a_{1}\,\,{}_{3}F_{2}\biggl{(}\genfrac{.}{.}{0.0pt}{}{-\Delta-\Delta_{13}+1,\Delta-\Delta_{13},b-\Delta_{13}+\Delta_{24}+1}{2-a-\Delta_{13},-\Delta_{13}+\Delta_{24}+1};1\biggr{)}$
(F.9) $\displaystyle-
a_{2}\,\,{}_{3}F_{2}\biggl{(}\genfrac{.}{.}{0.0pt}{}{a-\Delta,a+\Delta-1,a+b+\Delta_{24}}{a+\Delta_{13},a+\Delta_{24}};1\biggr{)}$
(F.10)

where the minus sign is due to the orientation of contour, and the
coefficients are

$\displaystyle a_{1}$
$\displaystyle=\frac{\Gamma\left(a+\Delta_{13}-1\right)\Gamma\left(b-\Delta_{13}+\Delta_{24}+1\right)}{\Gamma\left(a+b+\Delta_{24}\right)},$
(F.11) $\displaystyle a_{2}$
$\displaystyle=\frac{\Gamma\left(-\Delta_{13}+\Delta_{24}+1\right)\Gamma(a-\Delta)\Gamma(a+\Delta-1)\Gamma\left(-a-\Delta_{13}+1\right)}{\Gamma\left(-\Delta-\Delta_{13}+1\right)\Gamma\left(\Delta-\Delta_{13}\right)\Gamma\left(a+\Delta_{24}\right)}.$
(F.12)

As a first check, for four identical external operators
$\Delta_{13}=\Delta_{24}=0$, this result matches with (2.58) in [63] with the
conventions $(\Delta,a,b)=(\frac{1}{2}+\alpha,p,-q)$.

Now we can derive the conformal block expansion by deforming the
$\Delta$-contour to the right infinity. The first term is analytic in
$\Delta$, and the second term contains simple poles
$\Delta_{n}=a+n,\,n\in\mathbb{N}$ at the right side of principal series. Hence
the four-point function can be expanded as

$\displaystyle\mathcal{G}(\chi)=\chi^{a}(1-\chi)^{b}=\sum_{n}\mathcal{C}_{1+n}^{\Delta_{i}}G^{\Delta_{13},\Delta_{24}}_{1+n}(\chi)=-2\sum_{n}\frac{\operatorname{Res}_{\Delta=\Delta_{n}}I(\Delta)}{Q(1-\Delta_{n})}G^{\Delta_{13},\Delta_{24}}_{1+n}(\chi),$
(F.13)

where the orientation of contour contributes to factor $-1$, and the shadow
term contributes to factor $2$. The block coefficients are

$\mathcal{C}_{1+n}^{\Delta_{i}}=\frac{\left(a+\Delta_{13}\right)_{n}\left(a+\Delta_{24}\right)_{n}}{n!(2a+n-1)_{n}}\,\,{}_{3}F_{2}\biggl{(}\genfrac{.}{.}{0.0pt}{}{-n,2a+n-1,a+b+\Delta_{24}}{a+\Delta_{13},a+\Delta_{24}};1\biggr{)}$
(F.14)

and we can match the block expansion with the four-point function order by
order.

Back to the original four-point function (F.1), taking
$a=1,b=\frac{1}{2}(-1+\Delta_{13}-\Delta_{24})$ we have

$\mathcal{C}_{1+n}^{\Delta_{i}}=\frac{\pi
m^{\beta-1}}{2^{\beta+4}}\sec[\frac{\pi\beta}{2}]\frac{(-1)^{n}2^{2n}\Gamma\left(\frac{n-\Delta_{13}+1}{2}\right)\Gamma\left(\frac{n+\Delta_{24}+1}{2}\right)}{\Gamma(2n+1)\Gamma\left(\frac{-n-\Delta_{13}+1}{2}\right)\Gamma\left(\frac{-n+\Delta_{24}+1}{2}\right)}\;,$
(F.15)

which agrees with (5.38) computed by using the split representation.

##  Appendix G Expansion of conformal block around its poles

In this appendix, we will derive the expansion of $1D$ and $2D$ conformal
block around its poles. We start with the $1D$ conformal block
$G^{\Delta_{13},\Delta_{24}}_{\Delta}$ which takes the following form

$\displaystyle\begin{split}G^{\Delta_{13},\Delta_{24}}_{\Delta}(\chi)=&\chi^{\Delta}\sum_{k=0}^{\infty}\frac{(\Delta-\Delta_{13})_{k}(\Delta+\Delta_{24})_{k}}{\Gamma[k+1](2\Delta)_{k}}\chi^{k}=\Gamma[2\Delta]H^{\Delta_{13},\Delta_{24}}_{\Delta}(\chi)\;,\end{split}$
(G.1)

where we defined $H^{\Delta_{13},\Delta_{24}}_{\Delta}$ as

$\displaystyle\begin{split}H^{\Delta_{13},\Delta_{24}}_{\Delta}(\chi)=&\chi^{\Delta}\sum_{k=0}^{\infty}\frac{(\Delta-\Delta_{13})_{k}(\Delta+\Delta_{24})_{k}}{\Gamma[k+1]\Gamma[2\Delta+k]}\chi^{k}\;.\end{split}$
(G.2)

We note that $G^{\Delta_{13},\Delta_{24}}_{\Delta}$ has simple poles at
$2\Delta=-n$ with $n=0,1,2,\cdots$. This admits the following expansion of
${}_{t}F^{\Delta_{i}}_{\Delta}$ around $2\Delta=-n$

$\displaystyle\begin{split}G^{\Delta_{13},\Delta_{24}}_{\Delta}(\chi)=&\frac{a^{\Delta_{i},n}_{-1}(\chi)}{2\Delta+n}+a^{\Delta_{i},n}_{0}(\chi)+O^{1}(2\Delta+n)\;.\end{split}$
(G.3)

The coefficient $a^{\Delta_{i},n}_{-1}(\chi)$ can be computed from

$\displaystyle\begin{split}a^{\Delta_{i},n}_{-1}(\chi)=\bigg{(}(2\Delta+n)G^{\Delta_{13},\Delta_{24}}_{\Delta}(\chi)\bigg{)}\bigg{|}_{\Delta=-\frac{n}{2}}=\frac{(-1)^{n}}{n!}G^{\Delta_{13},\Delta_{24}}_{-\frac{n}{2}}(\chi)\;.\end{split}$
(G.4)

Using the definition (G.2) of $H^{\Delta_{13},\Delta_{24}}_{\Delta}(\chi)$, we
find that

$\displaystyle\begin{split}H^{\Delta_{13},\Delta_{24}}_{-\frac{n}{2}}(\chi)=&\sum_{k=0}^{\infty}\chi^{-\frac{n}{2}}\frac{(-\frac{n}{2}-\Delta_{13})_{k}(-\frac{n}{2}+\Delta_{24})_{k}}{\Gamma[k+1]\Gamma[k-n]}\chi^{k}\;.\end{split}$
(G.5)

Shifting $k$ by $k\rightarrow k+n+1$ leads to

$\displaystyle\begin{split}H^{\Delta_{13},\Delta_{24}}_{-\frac{n}{2}}(\chi)=&\sum_{k=0}^{\infty}\chi^{\frac{n+2}{2}}\frac{(-\frac{n}{2}-\Delta_{13})_{k+n+1}(-\frac{n}{2}+\Delta_{24})_{k+n+1}}{\Gamma[k+n+2]\Gamma[k+1]}\chi^{k}\\\
=&(-\frac{n}{2}-\Delta_{13})_{n+1}(-\frac{n}{2}+\Delta_{24})_{n+1}H^{\Delta_{13},\Delta_{24}}_{\frac{n+2}{2}}(\chi)\;.\end{split}$
(G.6)

This leads to

$\displaystyle\begin{split}a^{\Delta_{i},n}_{-1}(\chi)=\frac{(-1)^{n}(-\frac{n}{2}-\Delta_{13})_{n+1}(-\frac{n}{2}+\Delta_{24})_{n+1}}{\Gamma[n+1]\Gamma[n+2]}G^{\Delta_{13},\Delta_{24}}_{\frac{n+2}{2}}(\chi)\;.\end{split}$
(G.7)

The coefficient $a^{\Delta_{i},n}_{0}(\chi)$ can be computed from

$\displaystyle\begin{split}a^{\Delta_{i},n}_{0}(\chi)&=\frac{1}{2}\frac{\partial}{\partial\Delta}\bigg{(}(2\Delta+n)G^{\Delta_{13},\Delta_{24}}_{\Delta}(\chi)\bigg{)}\bigg{|}_{\Delta=-\frac{n}{2}}\\\
&=\frac{(-1)^{n}}{n!}\psi(n+1)H^{\Delta_{13},\Delta_{24}}_{-\frac{n}{2}}(\chi)+\frac{(-1)^{n}}{2n!}\frac{\partial
H^{\Delta_{13},\Delta_{24}}_{\Delta}(\chi)}{\partial\Delta}\bigg{|}_{\Delta=-\frac{n}{2}}\\\
&=\frac{(-1)^{n}(-\frac{n}{2}-\Delta_{13})_{n+1}(-\frac{n}{2}+\Delta_{24})_{n+1}}{\Gamma[n+1]\Gamma[n+2]}\psi(n+1)G^{\Delta_{13},\Delta_{24}}_{\frac{n+2}{2}}+\frac{(-1)^{n}}{2n!}\frac{\partial
H^{\Delta_{13},\Delta_{24}}_{\Delta}}{\partial\Delta}\bigg{|}_{\Delta=-\frac{n}{2}}\;,\end{split}$
(G.8)

where we used (G.6). To compute $\frac{1}{2}\frac{\partial
H^{\Delta_{13},\Delta_{24}}_{\Delta}(\chi)}{\partial\Delta}\bigg{|}_{\Delta=-\frac{n}{2}}$,
we note that

$\displaystyle\begin{split}\frac{1}{2}\frac{\partial
H^{\Delta_{13},\Delta_{24}}_{\Delta}(\chi)}{\partial\Delta}\bigg{|}_{\Delta=-\frac{n}{2}}={}_{1}S^{\Delta_{i},-\frac{n}{2}}_{13-24}(\chi)+{}_{2}S^{\Delta_{i},-\frac{n}{2}}_{13-24}(\chi)+{}_{3}S^{\Delta_{i},-\frac{n}{2}}_{13-24}(\chi)+{}_{4}S^{\Delta_{i},-\frac{n}{2}}_{13-24}(\chi)\;,\end{split}$
(G.9)

where ${}_{i}S^{\Delta_{i},-\frac{n}{2}}_{13-24}(z)$ are defined as

$\displaystyle\begin{split}{}_{1}S^{\Delta_{i},-\frac{n}{2}}_{13-24}(\chi)\equiv&\frac{1}{2}\frac{\partial\chi^{\Delta}}{\partial\Delta}\bigg{|}_{\Delta=-\frac{n}{2}}\chi^{\frac{n}{2}}H^{\Delta_{13},\Delta_{24}}_{-\frac{n}{2}}(\chi)\;,\end{split}$
(G.10)

and

$\displaystyle\begin{split}{}_{2}S^{\Delta_{i},-\frac{n}{2}}_{13-24}(\chi)\equiv&\frac{1}{2}\sum_{k=0}^{\infty}\chi^{\frac{-n}{2}}\frac{(-\frac{n}{2}+\Delta_{24})_{k}}{\Gamma[k+1]\Gamma[k-n]}\frac{\partial(\Delta-\Delta_{13})_{k}}{\partial\Delta}\bigg{|}_{\Delta=-\frac{n}{2}}\chi^{k}\\\
=&\sum_{k=0}^{\infty}\chi^{\frac{-n}{2}}\frac{(-\frac{n}{2}-\Delta_{13})_{k}(-\frac{n}{2}+\Delta_{24})_{k}}{\Gamma[k+1]\Gamma[k-n]}\frac{\psi(-\frac{n}{2}-\Delta_{13}+k)-\psi(-\frac{n}{2}-\Delta_{13})}{2}\chi^{k}\;,\end{split}$
(G.11)

and

$\displaystyle\begin{split}{}_{3}S^{\Delta_{i},-\frac{n}{2}}_{13-24}(\chi)\equiv&\frac{1}{2}\sum_{k=0}^{\infty}\chi^{\frac{-n}{2}}\frac{(-\frac{n}{2}-\Delta_{13})_{k}}{\Gamma[k+1]\Gamma[k-n]}\frac{\partial(\Delta+\Delta_{24})_{k}}{\partial\Delta}\bigg{|}_{\Delta=-\frac{n}{2}}\chi^{k}\\\
=&\sum_{k=0}^{\infty}\chi^{\frac{-n}{2}}\frac{(-\frac{n}{2}-\Delta_{13})_{k}(-\frac{n}{2}+\Delta_{24})_{k}}{\Gamma[k+1]\Gamma[k-n]}\frac{\psi(-\frac{n}{2}+\Delta_{24}+k)-\psi(-\frac{n}{2}+\Delta_{24})}{2}\chi^{k}\;,\end{split}$
(G.12)

and

$\displaystyle\begin{split}{}_{4}S^{\Delta_{i},-\frac{n}{2}}_{13-24}(\chi)\equiv&\frac{1}{2}\sum_{k=0}^{\infty}\chi^{\frac{-n}{2}}\frac{(-\frac{n}{2}-\Delta_{13})_{k}(-\frac{n}{2}+\Delta_{24})_{k}}{\Gamma[k+1]}\bigg{(}\frac{\partial}{\partial\Delta}\frac{1}{\Gamma[k+2\Delta]}\bigg{)}\bigg{|}_{\Delta=-\frac{n}{2}}\chi^{k}\\\
=&-\sum_{k=0}^{\infty}\chi^{\frac{-n}{2}}\frac{-(\frac{n}{2}-\Delta_{13})_{k}(-\frac{n}{2}+\Delta_{24})_{k}}{\Gamma[k+1]\Gamma[k-n]}\psi(k-n)\chi^{k}\;.\end{split}$
(G.13)

Using (G.6), we can re-write ${}_{1}S^{\Delta_{i},-\frac{n}{2}}_{13-24}(\chi)$
as

$\displaystyle\begin{split}{}_{1}S^{\Delta_{i},-\frac{n}{2}}_{13-24}(\chi)=&\frac{1}{2}\log(\chi)\;{}_{t}H^{\Delta_{i}}_{-\frac{n}{2}}(\chi)\\\
=&\frac{1}{2}\log(\chi)(-\frac{n}{2}-\Delta_{13})_{n+1}(-\frac{n}{2}+\Delta_{24})_{n+1}H^{\Delta_{13},\Delta_{24}}_{\frac{n+2}{2}}(\chi)\\\
=&(-\frac{n}{2}-\Delta_{13})_{n+1}(-\frac{n}{2}+\Delta_{24})_{n+1}\,{}_{1}S^{\Delta_{i},\frac{n+2}{2}}_{13-24}(\chi)\;.\end{split}$
(G.14)

Moreover, after shifting the dummy index $k$ by $k\rightarrow k+n+1$, we can
re-write ${}_{2}S^{\Delta_{i},-\frac{n}{2}}_{13-24}(z)$ ast

$\displaystyle\begin{split}{}_{2}S^{\Delta_{i},-\frac{n}{2}}_{13-24}=&\sum_{k=0}^{\infty}\chi^{\frac{n+2}{2}}\frac{(-\frac{n}{2}-\Delta_{13})_{k+n+1}(-\frac{n}{2}+\Delta_{24})_{k+n+1}}{\Gamma[k+n+2]\Gamma[k+1]}\frac{\psi(\frac{n+2}{2}-\Delta_{13}+k)-\psi(-\frac{n}{2}-\Delta_{13})}{2}\chi^{k}\\\
=&(-\frac{n}{2}-\Delta_{13})_{n+1}(-\frac{n}{2}+\Delta_{24})_{n+1}\bigg{[}{}_{2}S^{\Delta_{i},\frac{n+2}{2}}_{13-24}+\frac{\psi(\frac{2+n}{2}-\Delta_{13})-\psi(-\frac{n}{2}-\Delta_{13})}{2\Gamma[n+2]}G^{\Delta_{13},\Delta_{24}}_{\frac{n+2}{2}}\bigg{]}\;.\end{split}$
(G.15)

Similarly, we get

$\displaystyle\begin{split}{}_{3}S^{\Delta_{i},-\frac{n}{2}}_{13,24}=&(-\frac{n}{2}-\Delta_{13})_{n+1}(-\frac{n}{2}+\Delta_{24})_{n+1}\bigg{[}{}_{3}S^{\Delta_{i},\frac{n+2}{2}}_{13,24}+\frac{\psi(\frac{2+n}{2}+\Delta_{24})-\psi(-\frac{n}{2}+\Delta_{24})}{2\Gamma[n+2]}G^{\Delta_{13},\Delta_{24}}_{\frac{n+2}{2}}\bigg{]}\;.\end{split}$
(G.16)

To compute ${}_{4}S^{\Delta_{i},-\frac{n}{2}}_{13-24}$, we split the sum over
$k\geqslant 0$ into two sums, i.e.,

$\displaystyle\begin{split}{}_{4}S^{\Delta_{i},-\frac{n}{2}}_{13-24}=&-\bigg{(}\sum_{k=0}^{n}+\sum_{k=n+1}^{\infty}\bigg{)}\chi^{\frac{-n}{2}}\frac{(-\frac{n}{2}-\Delta_{13})_{k}(-\frac{n}{2}+\Delta_{24})_{k}}{\Gamma[k+1]\Gamma[k-n]}\psi(k-n)\chi^{k}\;.\end{split}$
(G.17)

The sum with $k\geqslant n+1$ can be computed by shifting $k$ to $k+n+1$,
giving

$\displaystyle\begin{split}-\sum_{k=0}^{\infty}\chi^{\frac{n+2}{2}}\frac{(-\frac{n}{2}-\Delta_{13})_{k+n+1}(-\frac{n}{2}+\Delta_{24})_{k+n+1}}{\Gamma[k+n+2]\Gamma[k+1]}\psi(k+1)\chi^{k}\;.\end{split}$
(G.18)

Armed with the identity

$\displaystyle\begin{split}\psi(k+n+2)=\psi(k+1)+\sum_{s=0}^{n}\frac{1}{k+1+s}\;,\end{split}$
(G.19)

It can be written as

$\displaystyle\begin{split}&-\sum_{k=0}^{\infty}\chi^{\frac{n+2}{2}}\frac{(-\frac{n}{2}-\Delta_{13})_{k+n+1}(-\frac{n}{2}+\Delta_{24})_{k+n+1}}{\Gamma[k+n+2]\Gamma[k+1]}\psi(k+1)\chi^{k}\\\
&\quad=(-\frac{n}{2}-\Delta_{13})_{n+1}(-\frac{n}{2}+\Delta_{24})_{n+1}\,{}_{4}S^{\Delta_{i},\frac{n+2}{2}}_{13-24}+\sum_{s=0}^{n}\sum_{k=0}^{\infty}\chi^{\frac{n+2}{2}}\frac{(-\frac{n}{2}-\Delta_{13})_{k+n+1}(-\frac{n}{2}+\Delta_{24})_{k+n+1}}{\Gamma[k+n+2]\Gamma[k+1](k+1+s)}\chi^{k}\;.\end{split}$
(G.20)

Using the fact that $\psi(-n)/\Gamma[-n]=(-1)^{n+1}n!$, we find that

$\displaystyle\begin{split}-\sum_{k=0}^{n}\frac{(-\frac{n}{2}-\Delta_{13})_{k}(-\frac{n}{2}+\Delta_{24})_{k}\psi(k-n)}{\Gamma[k+1]\Gamma[k-n]}\chi^{k-\frac{n}{2}}+\sum_{s=0}^{n}\sum_{k=0}^{\infty}\frac{(-\frac{n}{2}-\Delta_{13})_{k+n+1}(-\frac{n}{2}+\Delta_{24})_{k+n+1}}{\Gamma[k+n+2]\Gamma[k+1](k+1+s)}\chi^{k+\frac{n+2}{2}}\end{split}$
(G.21)

is equal to
$(-1)^{n}n!G^{\Delta_{13},\Delta_{24}}_{\text{sta},-\frac{n}{2}}(\chi)$ where
$G^{\Delta_{13},\Delta_{24}}_{\text{sta},-\frac{n}{2}}(\chi)$ is the conformal
block associated with the staggered module (A.3). Thus we find that

$\displaystyle\begin{split}{}_{4}S^{\Delta_{i},-\frac{n}{2}}_{13-24}=&(-\frac{n}{2}-\Delta_{13})_{n+1}(-\frac{n}{2}+\Delta_{24})_{n+1}\,{}_{4}S^{\Delta_{i},\frac{n+2}{2}}_{13-24}+(-1)^{n}n!G^{\Delta_{13},\Delta_{24}}_{\text{sta},-\frac{n}{2}}\;.\end{split}$
(G.22)

Combining (G.14), (G.15), (G.16), and (G.22), we get

$\displaystyle\begin{split}&\frac{1}{2}\frac{\partial
H^{\Delta_{13},\Delta_{24}}_{\Delta}(\chi)}{\partial\Delta}\bigg{|}_{\Delta=-\frac{n}{2}}\\\
&\quad=\frac{(-\frac{n}{2}-\Delta_{13})_{n+1}(-\frac{n}{2}+\Delta_{24})_{n+1}}{2\Gamma[n+2]}\bigg{(}b^{\Delta_{i}}_{n}G^{\Delta_{13},\Delta_{24}}_{\frac{n+2}{2}}+\frac{\partial
G^{\Delta_{13},\Delta_{24}}_{\Delta}}{\partial\Delta}\bigg{|}_{\Delta=\frac{n+2}{2}}\bigg{)}+(-1)^{n}n!G^{\Delta_{13,\Delta_{24}}}_{\text{sta},-\frac{n}{2}}\;,\end{split}$
(G.23)

where we defined $b^{\Delta_{i}}_{n}$ as

$\displaystyle\begin{split}b^{\Delta_{i}}_{n}\equiv&\psi(\frac{2+n}{2}-\Delta_{13})-\psi(\frac{-n}{2}-\Delta_{13})+\psi(\frac{2+n}{2}+\Delta_{24})-\psi(\frac{-n}{2}+\Delta_{24})-2\psi(n+2)\;.\end{split}$
(G.24)

We conclude that

$\displaystyle\begin{split}G^{\Delta_{13},\Delta_{24}}_{\Delta}(\chi)=&\frac{a^{\Delta_{i},n}_{-1}(\chi)}{2\Delta+n}+a^{\Delta_{i},n}_{0}(\chi)+O^{1}(2\Delta+n)\;,\end{split}$
(G.25)

where $a^{\Delta_{i},n}_{-1}(\chi)$ is given by

$\displaystyle\begin{split}a^{\Delta_{i},n}_{-1}(\chi)=\frac{(-1)^{n}(-\frac{n}{2}-\Delta_{13})_{n+1}(-\frac{n}{2}+\Delta_{24})_{n+1}}{\Gamma[n+1]\Gamma[n+2]}{}_{t}F^{\Delta_{i}}_{\frac{n+2}{2}}(\chi)\;,\end{split}$
(G.26)

and $a^{\Delta_{i},n}_{0}(\chi)$ is given by

$\displaystyle\begin{split}a^{\Delta_{i},n}_{0}=\frac{(-1)^{n}(-\frac{n}{2}-\Delta_{13})_{n+1}(-\frac{n}{2}+\Delta_{24})_{n+1}}{2\Gamma[n+1]\Gamma[n+2]}\bigg{(}B^{\Delta_{i}}_{n}\;{}_{t}F^{\Delta_{i}}_{\frac{n+2}{2}}+\frac{\partial{}_{t}F^{\Delta_{i}}_{\Delta}}{\partial\Delta}\bigg{|}_{\Delta=\frac{n+2}{2}}\bigg{)}+(-1)^{n}n!{}_{t}F^{\Delta_{i}}_{\text{sta},-\frac{n}{2}}\end{split}$
(G.27)

with

$\displaystyle\begin{split}B^{\Delta_{i}}_{n}\equiv&\psi(\frac{2+n}{2}-\Delta_{13})-\psi(\frac{-n}{2}-\Delta_{13})+\psi(\frac{2+n}{2}+\Delta_{24})-\psi(\frac{-n}{2}+\Delta_{24})-\frac{2}{n+1}\;.\end{split}$
(G.28)

Since the two dimensional global conformal block is a product of two one-
dimensional conformal blocks, i.e.,

$\displaystyle\begin{split}G^{h_{13},h_{24};\overline{h}_{13},\overline{h}_{24}}_{h,\overline{h}}(\chi,\overline{\chi})=G^{h_{13},h_{24}}_{h}(\chi)G^{\overline{h}_{13},\overline{h}_{24}}_{h,\overline{h}}(\overline{\chi})\;,\end{split}$
(G.29)

we find that
$G^{h_{13},h_{24};\overline{h}_{13},\overline{h}_{24}}_{h,\overline{h}}(\chi,\overline{\chi})$
admits the following expansion around the pole $2h=-n$ and
$2\overline{h}=-\overline{n}$ or $\Delta=(-n-\overline{n})/2$ and
$\ell=(-n+\overline{n})/2$

$\displaystyle\begin{split}G^{h_{13},h_{24};\overline{h}_{13},\overline{h}_{24}}_{h,\overline{h}}(\chi,\overline{\chi})=&\frac{a^{h_{i},n}_{-1}(\chi)a_{-1}^{\overline{h}_{i},\overline{n}}(\overline{\chi})}{(2h+n)(2\overline{h}+\overline{n})}+\frac{a^{h_{i},n}_{-1}(\chi)a^{\overline{h}_{i},\overline{n}}_{0}(\overline{\chi})}{2h+n}+\frac{a^{h_{i},n}_{0}(\chi)a^{\overline{h}_{i},\overline{n}}_{-1}(\overline{\chi})}{2\overline{h}+\overline{n}}+\cdots\;.\end{split}$
(G.30)

Especially, when $h_{i}=\overline{h}_{i}=\Delta_{i}/2$ and
$h=\overline{h}=\Delta/2$, we have

$\displaystyle\begin{split}&G^{\Delta_{13},\Delta_{24}}_{h,\overline{h}}(\chi,\overline{\chi})\\\
&=\frac{a^{h_{i},n}_{-1}(\chi)a_{-1}^{h_{i},n}(\overline{\chi})}{(2h+n)^{2}}+\frac{a^{h_{i},n}_{-1}(\chi)a^{h_{i},n}_{0}(\overline{\chi})+a^{h_{i},n}_{0}(\chi)a^{h_{i},n}_{-1}(\overline{\chi})}{2h+n}+\cdots\\\
&=\bigg{[}\frac{(-\frac{n}{2}-h_{13})_{n+1}(-\frac{n}{2}+h_{24})_{n+1}}{\Gamma[n+1]\Gamma[n+2]}\bigg{]}^{2}\bigg{[}\frac{G^{\Delta_{13},\Delta_{24}}_{\frac{n+2}{2},\frac{n+2}{2}}}{(\Delta+n)^{2}}+\frac{1}{\Delta+n}\bigg{(}B_{n}^{h_{i}}G^{\Delta_{13},\Delta_{24}}_{\frac{n+2}{2},\frac{n+2}{2}}\\\
&\quad+\frac{\partial
G^{\Delta_{13},\Delta_{24}}_{h,\overline{h}}}{\partial\Delta}\bigg{|}_{\Delta=n+2}\bigg{)}\bigg{]}+\frac{(-\frac{n}{2}-h_{13})_{n+1}(-\frac{n}{2}+h_{24})_{n+1}}{\Gamma[n+2](\Delta+n)}\bigg{(}G^{\Delta_{13},\Delta_{24}}_{\text{sta},-\frac{n}{2},\frac{n}{2}+1}+G^{\Delta_{13},\Delta_{24}}_{\text{sta},\frac{n}{2}+1,-\frac{n}{2}}\bigg{)}+\cdots\;,\end{split}$
(G.31)

where $G^{\Delta_{13},\Delta_{24}}_{\text{sta},-\frac{n}{2},\frac{n}{2}+1}$
and $G^{\Delta_{13},\Delta_{24}}_{\text{sta},\frac{n}{2}+1,-\frac{n}{2}}$ are
conformal blocks associated with chiral staggered module given in (A.6).

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