$\eta\to 0^{-}$ corresponds to $|\sigma|\to\infty$ (c.f. eq. (F.6)). Assuming
the spectral densities satisfy the properties which we have listed at the
beginning of this section, we can consider the two sides of (4.50) at some
fixed $0<|\sigma|^{-1}\ll 1$ and close the contour of integration in the lower
side of the complex $\lambda$ plane and the contribution from the arc at
infinity will vanish212121Even if the spectral densities grow exponentially
with $\lambda$, we will always be able to pick a $\sigma$ that is large enough
such that the contribution of the arc at infinity vanishes.. Spurious poles
will give contributions that cancel with each other as discussed in section
4.2.1. The poles at $\lambda=-in$ in the gamma function appearing in (4.50)
are canceled by the zeroes in the spectral density. We are thus left with the
contributions of the non-spurious (let us call them physical) poles of the
spectral densities

$G_{\mathcal{O}^{(J)}}=2\sum_{\ell=0}^{J}\sum_{\lambda_{*}}\underset{\lambda=\lambda_{*}}{\text{Res}}\left[\rho^{\mathcal{P},\ell}_{\mathcal{O}^{(J)}}(\lambda)\right]\lambda_{*}\Gamma(\pm
i\lambda_{*})((W_{1}\cdot\nabla_{1})(W_{2}\cdot\nabla_{2}))^{J-\ell}\Pi_{\lambda_{*},\ell}(Y_{1},Y_{2};W_{1},W_{2})\,.$
(4.53)

Now, we take $Y_{2}$ to a point $P$ on the late time boundary and $W$ to a
null vector $Z$ such that $P\cdot Z=0$, as in (4.51). On the right hand side
of (4.53), we obtain a sum of bulk-to-boundary propagators and their
derivatives. By comparison with the left hand side, this suggests that a spin
$J$ operator $\mathcal{O}^{(J)}(Y,W)$ in de Sitter satisfies the following
late time expansion in terms of boundary operators

$\mathcal{O}^{(J)}(-P/\eta,-Z/\eta)\underset{\eta\to
0^{-}}{\approx}\sum_{\ell=0}^{J}\sum_{\Delta_{*}}c_{\mathcal{O}^{(J)}O^{(\ell)}_{\Delta_{*}}}(-\eta)^{\Delta_{*}-\ell}(Z\cdot\partial_{P})^{J-\ell}O^{(\ell)}_{\Delta_{*}}(P,Z)+\cdots\,,$
(4.54)

where $O^{(\ell)}_{\Delta_{*}}(P,Z)$ are boundary CFT primaries of spin $\ell$
and we call $c_{\mathcal{O}^{(J)}O^{(\ell)}_{\Delta_{*}}}$ the Boundary
Operator Expansion (BOE) coefficients. The dots stand for descendants, and
$\Delta_{*}\equiv\frac{d}{2}+i\lambda_{*}$ with $\lambda_{*}$ being the
position of the physical poles of the spectral density
$\rho_{\mathcal{O}^{(J)}}^{\mathcal{P},\ell}(\lambda)$. By comparing with
(4.53), we relate the BOE coefficients to the residues of the spectral density

$c_{\mathcal{O}^{(J)}O^{(\ell)}_{\Delta_{*}}}=2\underset{\lambda=\lambda^{*}}{\text{Res}}\left[\rho^{\mathcal{P},\ell}_{\mathcal{O}^{(J)}}(\lambda)\right]\lambda^{*}\Gamma(\pm
i\lambda^{*})\,.$ (4.55)

Let us stress that the existence of this BOE is dependant on the assumptions
we stated at the beginning of this section. It would be interesting to
understand what are its convergence properties and whether any of our
assumptions can be relaxed while maintaining its validity. We leave these for
future work. In the examples in section 5, where these assumptions are
verified, we will draw precise connections between the poles of the spectral
densities we will be studying and the associated boundary operators. When
extra representations other than the principal series appear in our examples,
their contributions are canceled when closing the contour of integration and
landing on the sum in (4.53). In practice this means that the BOE, once
derived by closing the contour of integration over the principal series, can
be trusted even if we continue the two-point function beyond the regime in
which it decomposes in principal series representations only. If more
representations than the principal series appeared in the Källén-Lehmann
decomposition, then we expect to find boundary operators with
$\text{Re}(\Delta)<\frac{d}{2}$.

Let us also note that, if this BOE exists, then the bulk two-point function of
$\mathcal{O}^{(J)}$ has to have a power law decay at late times, justifying
the discussion in section 4.3. Moreover, given (4.54), the power of this decay
corresponds to the conformal dimension of the lowest lying primary in the BOE
of $\mathcal{O}^{(J)}$.

Finally, an important open question is whether the same BOE (4.54) of a bulk
local operator can be used inside different correlation functions.

###  4.5 Inversion formula in dS2

In this section, we will derive an inversion formula to extract the spectral
densities in the dS2 Källén-Lehmann decomposition (3.2.2). For simplicity, we
first assume that the spectral density associated with the complementary
series is vanishing, and we will later discuss under what conditions such an
assumption is valid.

In general dimensions, the tensor structure of $G_{{\cal
O}^{(J)}}(Y_{1},Y_{2};W_{1},W_{2})$ has two building blocks, namely
$W_{1}\cdot W_{2}$ and $(Y_{1}\cdot W_{2})(Y_{2}\cdot W_{1})$. In dS2, because
of the relations in eq.(2.2.2), $G_{{\cal O}^{(J)}}$ is actually a scalar
function of $\sigma=Y_{1}\cdot Y_{2}$, multiplied by $(W_{1}\cdot W_{2})^{J}$,
and the scalar function depends on whether $W_{1}$ and $W_{2}$ have the same
chirality. Without loss of generality, fixing $W_{1}=W_{1}^{+}$, $G_{{\cal
O}^{(J)}}$ is encoded in two scalar functions $G^{\pm}_{{\cal
O}^{(J)}}(\sigma)$, defined by

$\displaystyle G_{{\cal
O}^{(J)}}(Y_{1},Y_{2};W_{1}^{+},W_{2}^{\pm})=(W_{1}^{+}\cdot
W_{2}^{\pm})^{J}\,G^{\pm}_{{\cal O}^{(J)}}(\sigma)~{}.$ (4.56)

Plugging it into (3.2.2), we should have

$\displaystyle(W_{1}^{+}\cdot W_{2}^{\pm})^{J}\,G^{\pm}_{{\cal
O}^{(J)}}(\sigma)$
$\displaystyle=\int_{\mathbb{R}}d\lambda\,\rho^{\mathcal{P},0}_{{\cal
O}^{(J)}}(\lambda)(W^{+}_{1}\cdot\nabla_{1})^{J}(W^{\pm}_{2}\cdot\nabla_{2})^{J}G_{\lambda,0}(Y_{1},Y_{2})$
$\displaystyle+\int_{\mathbb{R}}d\lambda\,\rho^{\mathcal{P},1}_{{\cal
O}^{(J)}}(\lambda)(W^{+}_{1}\cdot\nabla_{1})^{J-1}(W^{\pm}_{2}\cdot\nabla_{2})^{J-1}G_{\lambda,1}(Y_{1},Y_{2};W^{+}_{1},W^{\pm}_{2})$
$\displaystyle+\sum_{p=0}^{J}\rho^{\mathcal{D}_{p}}_{{\cal
O}^{(J)}}\left(W^{+}_{1}\cdot\nabla_{1}\right)^{J}\left(W^{\pm}_{2}\cdot\nabla_{2}\right)^{J}G_{-i(p-\frac{1}{2})}(Y_{1},Y_{2})~{}.$
(4.57)

The next task is to reduce the tensor structure on the R.H.S. For the first
line, it is actually solved in appendix A, c.f. eq. (A.3) and eq. (A.3)

$\displaystyle(W^{+}_{1}\cdot\nabla_{1})^{J}(W^{\pm}_{2}\cdot\nabla_{2})^{J}G_{\lambda,0}(Y_{1},Y_{2})=(W^{+}_{1}\cdot
W_{2}^{\pm})^{J}\,\phi^{\pm}_{\lambda,J}(\sigma)~{},$ (4.58)

where

$\displaystyle\phi^{\pm}_{\lambda,J}(\sigma)\equiv\partial^{J}_{\sigma}((\sigma\pm
1)^{J}\partial_{\sigma}^{J})G_{\lambda,0}(\sigma),\,\,\,\,\,G_{\lambda,0}(\sigma)=\frac{\Gamma(\frac{1}{2}\pm
i\lambda)}{4\pi}F\left(\frac{1}{2}+i\lambda,\frac{1}{2}-i\lambda,1,\frac{1+\sigma}{2}\right)~{}.$
(4.59)

For the second line, using the definition of $G_{\lambda,1}$ given by eq.
(A.2), we have

$\displaystyle\left(\frac{1}{4}+\lambda^{2}\right)G_{\lambda,1}(Y_{1},Y_{2};W^{+}_{1},W^{\pm}_{2})=\pm(W^{+}_{1}\cdot\nabla_{1})(W^{\pm}_{2}\cdot\nabla_{2})G_{\lambda,0}(\sigma)~{}.$
(4.60)

So it is equivalent to the first line. The reduction of the third line is
given by eq. (A.3) and eq. (A.24). Altogether, the spin $J$ Källén-Lehmann
decomposition (3.2.2) is equivalent to the following two scalar equations:

$\displaystyle G^{+}_{{\cal
O}^{(J)}}(\sigma)=\int_{\mathbb{R}}\,d\lambda\,\rho^{\mathcal{P},+}_{{\cal
O}^{(J)}}(\lambda)\phi^{+}_{\lambda,J}(\sigma)+\sum_{p=0}^{J}\rho^{\mathcal{D}_{p}}_{{\cal
O}^{(J)}}\psi_{p,J}(\sigma),\,\,\,\,\,\rho^{\mathcal{P},+}_{{\cal
O}^{(J)}}=\rho^{\mathcal{P},0}_{{\cal
O}^{(J)}}+\frac{1}{\frac{1}{4}+\lambda^{2}}\rho^{\mathcal{P},1}_{{\cal
O}^{(J)}}~{},$ (4.61)

and

$\displaystyle G^{-}_{{\cal
O}^{(J)}}(\sigma)=\int_{\mathbb{R}}\,d\lambda\,\rho^{\mathcal{P},-}_{{\cal
O}^{(J)}}(\lambda)\phi^{-}_{\lambda,J}(\sigma),\,\,\,\,\,\,\,\,\,\,\,\rho^{\mathcal{P},-}_{{\cal
O}^{(J)}}=\rho^{\mathcal{P},0}_{{\cal
O}^{(J)}}-\frac{1}{\frac{1}{4}+\lambda^{2}}\rho^{\mathcal{P},1}_{{\cal
O}^{(J)}}~{},$ (4.62)

where $\psi_{p,J}(\sigma)$ is defined in eq. (A.24). To invert these two
equations, we introduce $J$-dependent inner products for real functions
defined on $(-\infty,-1)$:

$\displaystyle(f,g)_{J}^{\pm}=\int_{-\infty}^{-1}d\sigma(\sigma\mp
1)^{2J}f(\sigma)g(\sigma)~{}.$ (4.63)

In appendix D, we show that $\\{\phi_{\lambda,J}^{+}\\}\cup\\{\psi_{p,J}\\}$
is an orthogonal basis with respect to $(\,,\,)_{J}^{+}$, and
$\\{\phi^{-}_{\lambda,J}\\}$ is an orthogonal basis with respect to
$(\,,\,)_{J}^{-}$. Using the orthogonality relations, c.f. eq. (D.6), (D.13)
and (D.15), we obtain the following inversion formulae for dS2

$\displaystyle\rho^{\mathcal{P},\pm}_{{\cal
O}^{(J)}}(\lambda)=\frac{4\lambda\sinh(2\pi\lambda)}{(\frac{1}{2}+i\lambda)_{J}^{2}(\frac{1}{2}-i\lambda)_{J}^{2}}\int_{-\infty}^{-1}d\sigma(\sigma\mp
1)^{2J}G^{\pm}_{{\cal O}^{(J)}}(\sigma)\phi_{\lambda,J}^{\pm}(\sigma)$
$\displaystyle\rho^{\mathcal{D}_{p}}_{{\cal
O}^{(J)}}=\frac{8\pi^{2}\,(2p-1)}{\Gamma(J+p)^{2}\Gamma(1+J-p)^{2}}\int_{-\infty}^{-1}d\sigma(\sigma-1)^{2J}G^{+}_{{\cal
O}^{(J)}}(\sigma)\psi_{p,J}(\sigma)$ (4.64)

and $\rho^{\mathcal{P},0}_{{\cal
O}^{(J)}}(\lambda),\rho^{\mathcal{P},1}_{{\cal O}^{(J)}}(\lambda)$ can be
recovered by taking linear combinations of $\rho^{\mathcal{P},\pm}_{{\cal
O}^{(J)}}(\lambda)$.

The expansions (4.61) and (4.62) are valid and unique when $G^{\pm}_{{\cal
O}^{(J)}}$ is integrable with respect to $(\,,\,)_{J}^{\pm}$. Alternatively,
it means that complementary series does not contribute to the two-point
function of ${\cal O}^{(J)}$ if $G^{\pm}_{{\cal O}^{(J)}}(\sigma)$ decays
faster than $(-\sigma)^{-J-\frac{1}{2}}$ at large $-\sigma$.

##  5 Applications

In this section, we apply the inversion formulae (4.21) and (4.5) to compute
the Källén-Lehmann decomposition of a variety of two-point functions. We study
two-point functions of composite operators in free theories, primary operators
in de Sitter Conformal Field Theories and composite operators in a weakly
coupled Quantum Field Theory. In the free theory case, studying which terms
appear in the Källén-Lehmann decomposition informs us on the decomposition of
tensor products of UIRs of the de Sitter group. In the CFT case, we study how
$SO(d+1,2)$ UIRs decompose into $SO(d+1,1)$ UIRs. In the weakly coupled case,
we use the Källén-Lehmann representation to compute anomalous dimensions of
boundary operators. All throughout, we compare the decomposition in $d>1$ with
the one in $d=1$, where the discrete series states contribute up to $\Delta=J$
for spin $J$ two-point functions. The two exceptional series never appear in
our examples.

###  5.1 Free QFTs

One of the uses of the Källén–Lehmann decomposition is to study the
decomposition of multi-particle states into single particle UIRs. By studying
the contents of the Källén-Lehmann decomposition of a two-point function of a
composite operator made of products of elementary fields, we infer the
complete set of UIRs that is generated by the action of that operator on the
Bunch-Davies vacuum. As shown in Penedones:2023uqc , the totality of UIRs that
appears in such a decomposition is almost exclusively composed of the
principal series, except for a few isolated complementary series states that
we recover by analytic continuation. In this section, to avoid clutter, we
will write

$\langle\mathcal{O}(Y_{1})\mathcal{O}(Y_{2})\rangle\equiv\langle\Omega|\mathcal{O}(Y_{1})\mathcal{O}(Y_{2})|\Omega\rangle\,.$
(5.1)

####  5.1.1 Spin 0 Examples

Let us start with the simplest possible case: the two-point function of a free
elementary massive scalar field $\phi$ with
$\Delta_{\phi}=\frac{d}{2}+i\lambda_{\phi}$ in the principal series

$\langle\phi(Y_{1})\phi(Y_{2})\rangle=G_{\lambda_{\phi},0}(Y_{1},Y_{2})\,.$
(5.2)

The Källén-Lehmann decomposition of this two-point function should read

$\langle\phi(Y_{1})\phi(Y_{2})\rangle=\int_{\mathbb{R}}d\lambda\
\rho^{\mathcal{P},0}_{\phi}(\lambda)G_{\lambda,0}(Y_{1},Y_{2})\,.$ (5.3)

It is then immediate to see that, necessarily,

$\displaystyle\rho_{\phi}^{\mathcal{P},0}(\lambda)$
$\displaystyle=\frac{1}{2}(\delta(\lambda+\lambda_{\phi})+\delta(\lambda-\lambda_{\phi}))$
(5.4) $\displaystyle=\lim_{\epsilon\to
0}\frac{\epsilon}{2\pi(\epsilon^{2}+(\lambda^{2}-\lambda_{\phi}^{2})^{2})}\,,$

which is a manifestly real and positive quantity. It has two poles in the
lower half of the complex $\lambda$ plane, signaling the presence of two
primary boundary operators in the BOE of $\phi$

$\phi(-P/\eta)\underset{\eta\to
0^{-}}{\approx}(-\eta)^{\Delta_{\phi}}\mathcal{O}(P)+(-\eta)^{\bar{\Delta}_{\phi}}\widetilde{\mathcal{O}}(P)+\ldots$
(5.5)

where $\mathcal{O}(P)$ and $\widetilde{\mathcal{O}}(P)$ are CFTd primaries
with scaling dimensions $\Delta_{\phi}$ and $d-\Delta_{\phi}$ respectively,
and the dots stand for descendants. The fact that the spectral density is a
delta function in this case makes sense, since a free field already falls into
a single particle UIR. To see more interesting features, like a decomposition
into a continuum of states, one has to instead consider two-point functions of
composite operators, such as $\phi_{1}\phi_{2}(Y)$ in a free theory of two
massive scalars

$\langle\phi_{1}\phi_{2}(Y_{1})\phi_{1}\phi_{2}(Y_{2})\rangle=\langle\phi_{1}(Y_{1})\phi_{1}(Y_{2})\rangle\langle\phi_{2}(Y_{1})\phi_{2}(Y_{2})\rangle=G_{\lambda_{1}}(Y_{1},Y_{2})G_{\lambda_{2}}(Y_{1},Y_{2})$
(5.6)

where we take the two fields to have scaling dimensions
$\Delta_{1}=\frac{d}{2}+i\lambda_{1}$ and
$\Delta_{2}=\frac{d}{2}+i\lambda_{2}$ in the principal series, so
$\lambda_{1},\lambda_{2}\in\mathbb{R}$. This two-point function is free of
antipodal singularities and decays at large distances as

$|G_{\phi_{1}\phi_{2}}(\sigma)|\underset{\sigma\to-\infty}{\sim}|\sigma|^{-d}\,,\qquad\sigma\equiv
Y_{1}\cdot Y_{2}$ (5.7)

Given the discussion in section 4.3, this means the Källén-Lehmann
decomposition of this two-point function will only include states in the
principal series, as long as $d>2$. We observe through numerical checks of
(5.12) that even in the limit case $d=2$ this two-point function decomposes
into principal series representations only. To apply the inversion formula
(4.21) to this two-point function, we analytically continue it to EAdS as
discussed in section 4.1. Under this continuation, (5.6) becomes a product of
two harmonic functions

$\langle\phi_{1}\phi_{2}(X_{1})\phi_{1}\phi_{2}(X_{2})\rangle=\Gamma(\pm
i\lambda_{1})\Gamma(\pm
i\lambda_{2})\Omega_{\lambda_{1},0}(X_{1},X_{2})\Omega_{\lambda_{2},0}(X_{1},X_{2})\,.$
(5.8)

Then, the inversion formula reads

$\rho^{\mathcal{P},0}_{\phi_{1}\phi_{2}}(\lambda)=\frac{\Gamma(\pm
i\lambda_{1})\Gamma(\pm
i\lambda_{2})}{\mathcal{N}_{0,0}}\int_{X_{1}}\Omega_{\lambda,0}(X_{1},X_{2})\Omega_{\lambda_{1},0}(X_{1},X_{2})\Omega_{\lambda_{2},0}(X_{1},X_{2})\,,$
(5.9)

where $\mathcal{N}_{J,\ell}$ is defined in (4.22). To make progress, we use
the split representation (F.14) on the three harmonic functions, following
what was first done in Bros:2009bz . Defining
$\Delta_{3}\equiv\frac{d}{2}+i\lambda$, we have

$\rho^{\mathcal{P},0}_{\phi_{1}\phi_{2}}(\lambda)=\frac{\lambda^{2}\lambda_{1}^{2}\lambda_{2}^{2}\Gamma(\pm
i\lambda_{1})\Gamma(\pm
i\lambda_{2})}{\pi^{3}\mathcal{N}_{0,0}}\int_{X_{1}}\prod_{k=1}^{3}\int_{P_{i}}\Pi_{\Delta_{k},0}(X_{1},P_{k})\Pi_{\bar{\Delta}_{k},0}(X_{2},P_{k})\,,$
(5.10)

where $\Pi_{\Delta,0}(X,P)$ is a EAdS scalar bulk-to-boundary propagator, of
which we report the definition in (F.15). The integral over $X_{1}$ leads to a
CFT three point function

$\rho^{\mathcal{P},0}_{\phi_{1}\phi_{2}}(\lambda)=\frac{\lambda^{2}\lambda_{1}^{2}\lambda_{2}^{2}\Gamma(\pm
i\lambda_{1})\Gamma(\pm
i\lambda_{2})b(\Delta_{1},\Delta_{2},\Delta,0)}{\pi^{3}\mathcal{N}_{0,0}}\int_{P_{1},P_{2},P_{3}}\frac{\prod_{k=1}^{3}\Pi_{\bar{\Delta}_{k},0}(X_{2},P_{k})}{{(P_{12})^{\Delta_{123}}(P_{13})^{\Delta_{132}}(P_{23})^{\Delta_{231}}}}\,,$
(5.11)

where the notation and all the coefficients are explicit in the Appendix H.1.
There, we also show how to solve the remaining integrals over the three
boundary points $P_{1}$, $P_{2}$ and $P_{3}$. Importantly, the spectral
density of every free QFT two-point function of composite operators made of
two fundamental fields with spin can be reduced to linear combinations of this
specific integral, so that in the spinning examples we will make extensive use
of it. In Appendix H.1 we show how to eventually obtain

$\rho^{\mathcal{P},0}_{\phi_{1}\phi_{2}}(\lambda)=\frac{\lambda\sinh(\pi\lambda)}{32\pi^{\frac{d}{2}+3}\Gamma(\frac{d}{2})\Gamma(\frac{d}{2}\pm
i\lambda)}\prod_{\pm,\pm,\pm}\Gamma\left(\frac{\frac{d}{2}\pm i\lambda\pm
i\lambda_{1}\pm i\lambda_{2}}{2}\right)\,.$ (5.12)

$\frac{d}{2}$$d$$\Delta$ Figure 5.1: The analytic structure of
$\rho^{\mathcal{P},0}_{\phi_{1}\phi_{2}}(\lambda)$, the spectral density for
$\langle\phi_{1}\phi_{2}(Y_{1})\phi_{1}\phi_{2}(Y_{2})\rangle$. Here we
represent the poles at $\lambda=\pm(\lambda_{1}+\lambda_{2})\pm
i(\frac{d}{2}+2n)$ for each combination of signs. As we continue $\lambda_{1}$
and $\lambda_{2}$ to the complementary series, we indicate with colored arrows
how the leading poles corresponding to $n=0$ move in the complex
$\Delta=\frac{d}{2}+i\lambda$ plane. Eventually, some of these poles can cross
the integration contour at $\text{Re}(\Delta)=\frac{d}{2}$ (we highlight their
path in orange). Their residues need to be summed, leading to the discrete sum
of complementary series states in (5.16).

It can be checked numerically that if $\lambda_{1},\lambda_{2}\in\mathbb{R}$,
the integral of (5.12) fully reproduces (5.6)

$\langle\phi_{1}\phi_{2}(Y_{1})\phi_{1}\phi_{2}(Y_{2})\rangle=\int_{\mathbb{R}}\mathrm{d}\lambda\
\rho^{\mathcal{P},0}_{\phi_{1}\phi_{2}}(\lambda)G_{\lambda,0}(Y_{1},Y_{2})\,,\qquad\text{if
}\lambda_{1},\lambda_{2}\in\mathbb{R}\,.$ (5.13)

Analytically continuing $\lambda_{1}$ and $\lambda_{2}$ to the complementary
series such that $i\lambda_{1}\in(0,\frac{d}{2})$ and
$i\lambda_{2}\in(0,\frac{d}{2})$ , poles of
$\rho_{\phi_{1}\phi_{2}}^{\mathcal{P},0}(\lambda)$ can cross the contour of
integration over the principal series, so that their residues need to be added
by hand, introducing some complementary series contributions to the Källén-
Lehmann decomposition of this two-point function. This is in agreement with
what is discussed in section 4.3. By studying the gamma functions in (5.12) we
see that poles cross the contour if there exists some $n\in\mathbb{N}$ such
that

$\frac{d}{2}+2n<i\lambda_{1}+i\lambda_{2}<d\,,$ (5.14)

where the second inequality comes from the fact that $\lambda_{1}$ and
$\lambda_{2}$ are constrained to be on the complementary series. Let us assume
more specifically that

$\frac{d}{2}+2N<i\lambda_{1}+i\lambda_{2}<\frac{d}{2}+2(N+1)\,,$ (5.15)

for some $N<\frac{d}{4}$. Then, the full decomposition reads

$\displaystyle\langle\phi_{1}\phi_{2}(Y_{1})\phi_{1}\phi_{2}(Y_{2})\rangle=$
$\displaystyle\int_{\mathbb{R}}\mathrm{d}\lambda\
\rho^{\mathcal{P},0}_{\phi_{1}\phi_{2}}(\lambda)G_{\lambda,0}(Y_{1},Y_{2})+\sum_{n=0}^{N}\rho^{\mathcal{C},0}_{\phi_{1}\phi_{2}}(n)G_{\lambda_{1}+\lambda_{2}+i\left(\frac{d}{2}+2n\right),0}(Y_{1},Y_{2})$
(5.16)

where

$\displaystyle\rho^{\mathcal{C},0}_{\phi_{1}\phi_{2}}(n)=$
$\displaystyle\frac{(-1)^{n}(\frac{d}{2})_{n}\Gamma(-n+i\lambda_{12})\Gamma(\frac{d}{2}+n-i\lambda_{12})\prod_{j=1,2}\Gamma(-n+i\lambda_{j})\Gamma(\frac{d}{2}+n-i\lambda_{j})}{4\pi^{1+\frac{d}{2}}n!\Gamma(-2n+i\lambda_{12})\Gamma(-\frac{d}{2}-2n+i\lambda_{12})\Gamma(d+2n-i\lambda_{12})\Gamma(\frac{d}{2}+2n-i\lambda_{12})}$
(5.17)

with $\lambda_{12}\equiv\lambda_{1}+\lambda_{2}$. The complementary series
densities are simply the residues on the poles of
$\rho_{\phi_{1}\phi_{2}}^{\mathcal{P},0}(\lambda)$

$\rho^{\mathcal{C},0}_{\phi_{1}\phi_{2}}(n)=4\pi
i\underset{\lambda=\lambda_{1}+\lambda_{2}+i\left(\frac{d}{2}+2n\right)}{\text{Res}}\rho^{\mathcal{P},0}_{\phi_{1}\phi_{2}}(\lambda)\,.$
(5.18)

As expected from the proof of the Källén-Lehmann decomposition, by studying
the sign of $\rho_{\phi_{1}\phi_{2}}^{\mathcal{C},0}(n)$, it can be verified
that these functions are positive as long as $\lambda_{1}$ and $\lambda_{2}$
are in the complementary series and lie in the interval (5.14). The appearance
of this discrete sum of complementary series UIRs is in agreement with
Penedones:2023uqc , see Table 1.3 there. Moreover, this sum was derived before
in Akhmedov:2017ooy  and we checked that our
$\rho_{\phi_{1}\phi_{2}}^{\mathcal{C},0}(n)$ matches their equation
(48)222222To verify the matching one needs to substitute
$\lambda_{1}\rightarrow-i\alpha$, $\lambda_{2}\rightarrow-i\beta$ and
$d\rightarrow D-1$.. The sum over $n$ runs up to $N<\lfloor\frac{d}{4}\rfloor$
because only in this range the interval $(\frac{d}{2}+2n,d)$ is non-vanishing.

##### Boundary Operator Expansion

The spectral density (5.12) satisfies all the assumptions of section 4.4. From
its poles in the lower half of the complex $\lambda$ plane we can thus read
off the primary operators which appear in the BOE of $\phi_{1}\phi_{2}$,
namely

$\displaystyle\phi_{1}\phi_{2}(-P/\eta)\underset{\eta\to
0^{-}}{\approx}\sum_{n=0}^{\infty}\Big{[}$ $\displaystyle
c_{\Delta_{1}\Delta_{2}}(-\eta)^{\Delta_{1}+\Delta_{2}+2n}[\mathcal{O}_{1}\mathcal{O}_{2}]_{n}(P)$
(5.19)
$\displaystyle+c_{\Delta_{1}\bar{\Delta}_{2}}(-\eta)^{\Delta_{1}+\bar{\Delta}_{2}+2n}[\mathcal{O}_{1}\widetilde{\mathcal{O}}_{2}]_{n}(P)+\cdots\Big{]}+\cdots$

where notation like $[\mathcal{O}_{1}\mathcal{O}_{2}]_{n}$ should be
understood to stand for all the boundary scalar primaries that can be
constructed with $\mathcal{O}_{1},\mathcal{O}_{2}$ and $2n$ contracted
derivatives while being symmetric under $1\leftrightarrow 2$. The dots in the
square brackets stand for contributions from primaries like
$[\widetilde{\mathcal{O}}_{1}\mathcal{O}_{2}]_{n}$ and
$[\widetilde{\mathcal{O}}_{1}\widetilde{\mathcal{O}}_{2}]_{n}$, while the dots
outside of the brackets stand for descendants. The operators
$\mathcal{O}_{i}(P)$ are defined as the leading late time behavior of the free
fields $\phi_{1}(Y)$ and $\phi_{2}(Y)$

$\displaystyle\phi_{1,2}(-P/\eta)\underset{\eta\to
0^{-}}{\sim}(-\eta)^{\Delta_{1,2}}\mathcal{O}_{1,2}(P)+(-\eta)^{\bar{\Delta}_{1,2}}\widetilde{\mathcal{O}}_{1,2}(P)\,.$
(5.20)

so that $\mathcal{O}_{1}(P)$ and $\widetilde{\mathcal{O}}_{1}(P)$ transform as
CFT scalar primaries with scaling dimensions
$\Delta_{1}=\frac{d}{2}+i\lambda_{1}$ and
$\bar{\Delta}_{1}=\frac{d}{2}-i\lambda_{1}$ respectively (and analogously
$\mathcal{O}_{2}(P)$ and $\widetilde{\mathcal{O}}_{2}(P)$). An extra comment:
when $\lambda_{1}$ and $\lambda_{2}$ satisfy (5.14), the poles of (5.12) at
$\lambda=\lambda_{1}+\lambda_{2}+i(\frac{d}{2}+2n)$ can be picked up when
closing the contour of integration to find the boundary operators in the late
time limit. One could thus expect there to be operators on the boundary with
$\Delta=d-\Delta_{1}-\Delta_{2}-2n$. But the residues on these poles are
precisely canceled by the complementary series sum in (5.16), and thus such
operators are actually not appearing in the bulk-boundary OPE of
$\phi_{1}\phi_{2}(Y)$.

####  5.1.2 Spin 1 Examples

Consider the correlator

$\langle V\phi(Y_{1};W_{1})V\phi(Y_{2};W_{2})\rangle=\langle
V(Y_{1};W_{1})V(Y_{2};W_{2})\rangle\langle\phi(Y_{1})\phi(Y_{2})\rangle$
(5.21)

in a free theory of a massive vector with
$\Delta_{V}=\frac{d}{2}+i\lambda_{V}$ and a massive scalar
$\Delta_{\phi}=\frac{d}{2}+i\lambda_{\phi}\,,$ both on the principal series.
This two-point function has two scalar components

$G_{V\phi}(Y_{1},Y_{2};W_{1},W_{2})=\mathcal{G}_{V\phi}^{(0)}(\sigma)(W_{1}\cdot
W_{2})+\mathcal{G}_{V\phi}^{(1)}(\sigma)(W_{1}\cdot Y_{2})(W_{2}\cdot
Y_{1})\,,$ (5.22)

which decay at large distances as

$|\mathcal{G}_{V\phi}^{(0)}(\sigma)|\underset{\sigma\to-\infty}{\sim}|\sigma|^{-d}\,,\qquad|\mathcal{G}_{V\phi}^{(1)}(\sigma)|\underset{\sigma\to-\infty}{\sim}|\sigma|^{-d-1}\,.$
(5.23)

Following the discussion in section 4.3, we can thus state that the Källén-
Lehmann decomposition of this two-point function will only include principal
series contributions, as long as $d>2$. We verified that this is true also in
the limit case $d=2$. These will be organized in two terms, related to
transverse and longitudinal degrees of freedom. In Appendix H.1 we show in
detail how to apply the inversion formula to this case and how to express the
two spectral densities as linear combinations of (5.9). Here we report the
result

$\displaystyle\rho^{\mathcal{P},0}_{V\phi}(\lambda)$
$\displaystyle=\frac{2^{-1}\pi^{-3-\frac{d}{2}}\lambda\sinh(\pi\lambda)}{(\Delta_{V}\\!-\\!1)(\bar{\Delta}_{V}\\!-\\!1)(d^{2}\\!+\\!4\lambda^{2})\Gamma(\frac{d}{2})\Gamma(\frac{d}{2}\pm
i\lambda+1)}\prod_{\pm,\pm,\pm}\Gamma\left(\frac{\frac{d}{2}+1\pm i\lambda\pm
i\lambda_{V}\pm i\lambda_{\phi}}{2}\right)$
$\displaystyle\rho^{\mathcal{P},1}_{V\phi}(\lambda)$
$\displaystyle=\frac{2^{-12}\pi^{-3-\frac{d}{2}}\lambda\sinh(\pi\lambda)f_{\lambda,\lambda_{V},\lambda_{\phi}}}{\Gamma(\frac{d+2}{2})(\Delta_{V}-1)(\bar{\Delta}_{V}-1)\Gamma(\frac{d}{2}\pm
i\lambda+1)}\prod_{\pm,\pm,\pm}\Gamma\left(\frac{\frac{d}{2}\pm i\lambda\pm
i\lambda_{\phi}\pm i\lambda_{V}}{2}\right)\,,$ (5.24)

with

$\displaystyle f_{\lambda,\lambda_{V},\lambda_{\phi}}=$ $\displaystyle
16\left(\lambda_{\phi}^{2}-(\lambda^{2}+\lambda_{V}^{2})\right)^{2}+64(d-1)\lambda^{2}\lambda_{V}^{2}$
$\displaystyle+8d(3d-4)\lambda_{\phi}^{2}+8d\left(2d^{2}-5d+4\right)\left(\lambda^{2}+\lambda_{V}^{2}\right)+d^{3}\left(4d^{2}-11d+8\right)\,,$
(5.25)

where we see the appearance of the spurious pole predicted in section 4.2.1.
When $\phi$ and $V$ have scaling dimensions in the principal series, the
integrals over the principal series reproduce the full two-point function. We
elaborate on how exactly to carry out numerical checks in section F.37.
Continuing their scaling dimensions to the complementary series,
$i\lambda_{V}\in\left(0,\frac{d}{2}-1\right)$ and
$i\lambda_{\phi}\in(0,\frac{d}{2})$, instead leads to different poles of
$\rho_{V\phi}^{\mathcal{P},0}(\lambda)$ and
$\rho_{V\phi}^{\mathcal{P},1}(\lambda)$ crossing the contour of integration.
This happens when the following conditions are satisfied for some integers
$N_{0}$ and $N_{1}$:

$\displaystyle\rho_{V\phi}^{\mathcal{P},0}:\qquad\frac{d}{2}+2N_{0}+1<i\lambda_{\phi}+i\lambda_{V}<\frac{d}{2}+2(N_{0}+1)+1\,,$
(5.26)
$\displaystyle\rho_{V\phi}^{\mathcal{P},1}:\qquad\frac{d}{2}+2N_{1}<i\lambda_{\phi}+i\lambda_{V}<\frac{d}{2}+2(N_{1}+1)\,,$

where the unitarity bounds for the complementary series impose
$N_{0}<\frac{d-4}{4}$ and $N_{1}<\frac{d-2}{4}\,.$ The complementary series
contributions to this two-point function when (5.26) are satisfied, then, are
given by the sum over the residues of $\rho_{V\phi}^{\mathcal{P},0}(\lambda)$
and $\rho_{V\phi}^{\mathcal{P},1}(\lambda)$ on those poles. Moreover, in $d=1$
the discrete series contributes as well. We can explicitly derive this extra
contribution by analytically continuing in the dimension until $d=1$, and
keeping track of any poles that cross the contour of integration over the
principal series. Specifically, what happens is that the poles at $\lambda=\pm
i\frac{d-2}{2}$ in the spin-1 free propagator (see section 4.2.1 for a
discussion about these poles) cross the contour of integration. At the precise
value $d=2$ these poles at $\lambda=\pm i\frac{d-2}{2}$ are canceled by zeroes
of the form $\lambda\sinh(\pi\lambda)$ which are present in the propagator,
but when continuing all the way to $d=1$, the poles need to be taken into
account. The complete decomposition thus reads

$\displaystyle\langle V\phi(Y_{1};W_{1})V\phi(Y_{2};W_{2})\rangle=$
$\displaystyle\sum_{\ell=0}^{1}\int_{\mathbb{R}}\mathrm{d}\lambda\
\rho^{\mathcal{P},\ell}_{V\phi}(\lambda)[(W_{1}\cdot\nabla_{1})(W_{2}\cdot\nabla_{2})]^{1-\ell}G_{\lambda,\ell}(Y_{1},Y_{2};W_{1},W_{2})$
$\displaystyle+\sum_{n=0}^{N_{0}}\rho^{\mathcal{C},0}_{V\phi,n}(W_{1}\cdot\nabla_{1})(W_{2}\cdot\nabla_{2})G_{\lambda_{\phi}+\lambda_{V}+i(\frac{d}{2}+2n+1),0}(Y_{1},Y_{2})$
$\displaystyle+\sum_{n=0}^{N_{1}}\rho^{\mathcal{C},1}_{V\phi,n}G_{\lambda_{\phi}+\lambda_{V}+i(\frac{d}{2}+2n),1}(Y_{1},Y_{2};W_{1},W_{2})$
(5.27)
$\displaystyle+\delta_{d,1}\rho^{\mathcal{D}_{1}}_{V\phi}\left(W_{1}\cdot\nabla_{1}\right)\left(W_{2}\cdot\nabla_{2}\right)G_{-\frac{i}{2},0}(Y_{1},Y_{2})\,,$

where $\delta_{d,1}$ is a Kronecker delta, because the discrete series term
only contributes in $d=1$. We stress that the complementary series
contributions appear only if (5.26) are satisfied for some $N_{0}$ and
$N_{1}$, and are absent otherwise. The spectral densities of the complementary
series contributions are, specifically,

$\displaystyle\rho^{\mathcal{C},0}_{V\phi,n}$ $\displaystyle=4\pi
i\underset{\lambda=\lambda_{\phi}+\lambda_{V}+i(\frac{d}{2}+2n+1)}{\text{Res}}\rho^{\mathcal{P}}_{V\phi,0}(\lambda)\,,$
(5.28) $\displaystyle\rho^{\mathcal{C},1}_{V\phi,n}$ $\displaystyle=4\pi
i\underset{\lambda=\lambda_{\phi}+\lambda_{V}+i(\frac{d}{2}+2n)}{\text{Res}}\rho^{\mathcal{P}}_{V\phi,1}(\lambda)\,,$

and we verify that they are positive functions for $\lambda_{V}$ and
$\lambda_{\phi}$ in (5.26). The discrete series density is instead given by

$\rho^{\mathcal{D}_{1}}_{V\phi}=\frac{\pi(\lambda_{V}^{2}-\lambda_{\phi}^{2})}{(1+4\lambda_{V}^{2})\sinh(\pi(\lambda_{V}-\lambda_{\phi}))\sinh(\pi(\lambda_{V}+\lambda_{\phi}))}\,.$
(5.29)

Now let us discuss the spectrum of boundary operators that we can infer from
this two-point function. As reviewed in Sun:2021thf  and discussed in section
2.2.3, bulk free vector fields have the following asymptotic behavior

$V_{i}(\eta,\mathbf{y})\underset{\eta\rightarrow
0^{-}}{\sim}(-\eta)^{\Delta_{V}-1}\mathcal{A}_{i}(\mathbf{y})+(-\eta)^{\bar{\Delta}_{V}-1}\widetilde{\mathcal{A}}_{i}(\mathbf{y})\,,$
(5.30)

with $\mathcal{A}_{i}(\mathbf{y})$ and
$\widetilde{\mathcal{A}}_{i}(\mathbf{y})$ transforming as CFT primaries with
scaling dimensions $\Delta_{V}$ and $\bar{\Delta}_{V}$. Using the fact that
$\nabla_{\mu}V^{\mu}=0$ we can fix the asymptotic behavior of
$V_{\eta}(\eta,\mathbf{y})$

$V_{\eta}(\eta,\mathbf{y})\underset{\eta\rightarrow
0^{-}}{\sim}\frac{1}{\bar{\Delta}_{V}-1}(-\eta)^{\Delta_{V}}\partial\cdot\mathcal{A}(\mathbf{y})+\frac{1}{\Delta_{V}-1}(-\eta)^{\bar{\Delta}_{V}}\partial\cdot\widetilde{\mathcal{A}}(\mathbf{y})\,.$
(5.31)

We recognize the appearance of these boundary operators in the poles of the
two spectral densities. Specifically, we can write the following BOE (c.f. eq.
(4.54))

$\displaystyle V\phi(-P/\eta,-Z/\eta)\underset{\eta\to
0^{-}}{\approx}\sum_{n=0}^{\infty}\Big{[}$ $\displaystyle
c^{(0)}_{\Delta_{\phi}\Delta_{V}}(-\eta)^{\Delta_{V}+\Delta_{\phi}+1+2n}(Z\cdot\partial_{P})[(D_{Z}\cdot\partial_{P})\mathcal{A}\mathcal{O}]_{n}(P,Z)$
$\displaystyle
c^{(1)}_{\Delta_{\phi}\Delta_{V}}(-\eta)^{\Delta_{V}+\Delta_{\phi}-1+2n}[\mathcal{A}\mathcal{O}]_{n}(P,Z)+\cdots\Big{]}+\cdots\,,$
(5.32)

where we see both spin $0$ and spin $1$ boundary operators appearing and the
dots stand for double trace primaries like
$[\widetilde{\mathcal{A}}\mathcal{O}]_{n}(P,Z)$ and descendants. The boundary
operator $\mathcal{O}$ is defined through the late time limit of the free
field

$\phi(-P/\eta)\underset{\eta\to
0^{-}}{\approx}(-\eta)^{\Delta_{\phi}}\mathcal{O}(P)+(-\eta)^{\bar{\Delta}_{\phi}}\widetilde{\mathcal{O}}(P)\,.$
(5.33)

Finally, we verify that the contributions from the spurious poles at
$\lambda=-i\frac{d}{2}$ and $\lambda=-i\frac{d-2}{2}$ in (5.27) exactly cancel
when closing the contour of integration due to the identities in 4.2.1. That
means they are not associated to any boundary operator. We also computed the
decomposition of the correlator

$\displaystyle\langle\phi_{1}\nabla\phi_{2}(Y_{1};W_{1})\phi_{1}\nabla\phi_{2}(Y_{2};W_{2})\rangle=\langle\phi_{1}(Y_{1})\phi_{1}(Y_{2})\rangle(W_{1}\cdot\nabla_{1})(W_{2}\cdot\nabla_{2})\langle\phi_{2}(Y_{1})\phi_{2}(Y_{2})\rangle\,.$
(5.34)

When $\lambda_{1},\lambda_{2}\in\mathbb{R},$ the following principal series
spectral densities account for the full Källén-Lehmann decomposition of this
two-point function (see appendix H.1 for more details)

$\displaystyle\rho^{\mathcal{P},0}_{\phi_{1}\nabla\phi_{2}}(\lambda)$
$\displaystyle=\frac{(d^{2}+4(\lambda^{2}-\lambda_{1}^{2}+\lambda_{2}^{2}))^{2}\Gamma(\frac{d+1}{2})\lambda\sinh(\pi\lambda)}{2^{10-d}\pi^{\frac{d+7}{2}}(d^{2}+4\lambda^{2})\Gamma(d)\Gamma(\frac{d}{2}+1\pm
i\lambda)}\prod_{\pm,\pm,\pm}\Gamma\left(\frac{\frac{d}{2}\pm i\lambda\pm
i\lambda_{1}\pm i\lambda_{2}}{2}\right)$
$\displaystyle\rho^{\mathcal{P},1}_{\phi_{1}\nabla\phi_{2}}(\lambda)$
$\displaystyle=\frac{\lambda\sinh(\pi\lambda)}{2^{4}\pi^{3+\frac{d}{2}}\Gamma(\frac{d+2}{2})\Gamma(\frac{d}{2}+1\pm
i\lambda)}\prod_{\pm,\pm,\pm}\Gamma\left(\frac{\frac{d}{2}+1\pm i\lambda\pm
i\lambda_{1}\pm i\lambda_{2}}{2}\right)$ (5.35)

When analytically continuing the conformal weights of $\phi_{1}$ and
$\phi_{2}$ to the complementary series $i\lambda_{1}\in(0,\frac{d}{2})$ and
$i\lambda_{2}\in(0,\frac{d}{2})$, poles of
$\rho_{\phi_{1}\nabla\phi_{2}}^{\mathcal{P},0}(\lambda)$ and
$\rho_{\phi_{1}\nabla\phi_{2}}^{\mathcal{P},1}(\lambda)$ cross the integration
contour when the following conditions are satisfied for some
$N_{0}<\frac{d-2}{4}$ and $N_{1}<\frac{d-4}{4}$

$\displaystyle\rho_{\phi_{1}\nabla\phi_{2}}^{\mathcal{P},0}:\qquad\frac{d}{2}+2N_{0}<i\lambda_{1}+i\lambda_{2}<\frac{d}{2}+2(N_{0}+1)\,,$
(5.36)
$\displaystyle\rho_{\phi_{1}\nabla\phi_{2}}^{\mathcal{P},1}:\qquad\frac{d}{2}+2N_{1}+1<i\lambda_{1}+i\lambda_{2}<\frac{d}{2}+2(N_{1}+1)+1\,.$

Notice that these are slightly different poles than (5.26). Moreover, in $d=1$
there is a discrete series state appearing corresponding to a massless scalar,
with $\Delta=1$. The full decomposition reads

$\displaystyle\langle\phi_{1}\nabla\phi_{2}(Y_{1};W_{1})\phi_{1}\nabla\phi_{2}(Y_{2};W_{2})\rangle=$
$\displaystyle\sum_{\ell=0}^{1}\int_{\mathbb{R}}\mathrm{d}\lambda\
\rho^{\mathcal{P},\ell}_{\phi_{1}\nabla\phi_{2}}(\lambda)[(W_{1}\cdot\nabla_{1})(W_{2}\cdot\nabla_{2})]^{1-\ell}G_{\lambda,\ell}(Y_{1},Y_{2};W_{1},W_{2})$
$\displaystyle+\sum_{n=0}^{N_{0}}\rho^{\mathcal{C},0}_{\phi_{1}\nabla\phi_{2},n}(W_{1}\cdot\nabla_{1})(W_{2}\cdot\nabla_{2})G_{\lambda_{\phi}+\lambda_{V}+i(\frac{d}{2}+2n),0}(Y_{1},Y_{2})$
$\displaystyle+\sum_{n=0}^{N_{1}}\rho^{\mathcal{C},1}_{\phi_{1}\nabla\phi_{2},n}G_{\lambda_{\phi}+\lambda_{V}+i(\frac{d}{2}+2n+1),1}(Y_{1},Y_{2};W_{1},W_{2})$
$\displaystyle+\delta_{d,1}\rho^{\mathcal{D}_{1}}_{\phi_{1}\nabla\phi_{2}}\left(W_{1}\cdot\nabla_{1}\right)\left(W_{2}\cdot\nabla_{2}\right)G_{-\frac{i}{2},0}(Y_{1},Y_{2})\,.$
(5.37)

The complementary series densities are once again positive functions, given by
the residues of the principal series densities on the poles that cross the
contour

$\displaystyle\rho^{\mathcal{C},0}_{\phi_{1}\nabla\phi_{2},n}$
$\displaystyle=4\pi
i\underset{\lambda=\lambda_{1}+\lambda_{2}+i(\frac{d}{2}+2n)}{\text{Res}}\rho^{\mathcal{P},0}_{\phi_{1}\nabla\phi_{2}}(\lambda)\,,$
(5.38) $\displaystyle\rho^{\mathcal{C},1}_{\phi_{1}\nabla\phi_{2},n}$
$\displaystyle=4\pi
i\underset{\lambda=\lambda_{1}+\lambda_{2}+i(\frac{d}{2}+2n+1)}{\text{Res}}\rho^{\mathcal{P},1}_{\phi_{1}\nabla\phi_{2}}(\lambda)\,,$

and their contribution is instead absent when $\lambda_{\phi}$ and
$\lambda_{V}$ are real or imaginary but outside of the intervals (5.36). The
discrete series density is again obtainable by analytically continuing in the
spacetime dimension and adding the residue on the pole that crosses the
contour of integration

$\rho^{\mathcal{D}_{1}}_{\phi_{1}\nabla\phi_{2}}=\frac{\pi(\lambda_{1}^{2}-\lambda_{2}^{2})}{4\sinh(\pi(\lambda_{1}-\lambda_{2}))\sinh(\pi(\lambda_{1}+\lambda_{2}))}\,.$
(5.39)

The difference in the pole structure of (5.35) compared to (5.24) is explained
when we consider the boundary operators appearing in the BOE of
$\phi_{1}\nabla\phi_{2}(Y)$

$\displaystyle\phi_{1}(Z\cdot\partial_{P})\phi_{2}(-P/\eta)\underset{\eta\to
0^{-}}{\approx}\sum_{n=0}^{\infty}\Big{[}$ $\displaystyle
c_{\Delta_{1}\Delta_{2}}^{(0)}(-\eta)^{\Delta_{1}+\Delta_{2}+2n}(Z\cdot\partial_{P})[\mathcal{O}_{1}\mathcal{O}_{2}]_{n}(P)$
(5.40) $\displaystyle
c_{\Delta_{1}\Delta_{2}}^{(1)}(-\eta)^{\Delta_{1}+\Delta_{2}+2n}[\mathcal{O}_{1}(Z\cdot\partial_{P})\mathcal{O}_{2}]_{n}(P)\Big{]}$

To form a boundary scalar, in fact, $V\phi$ needs the action of a derivative,
such that the scalar boundary operators with the lowest scaling dimension have
$\Delta=\Delta_{V}+\Delta_{\phi}+1$. The operator $\phi_{1}\nabla\phi_{2}$,
instead, can form a boundary scalar operator without the use of derivatives
and with scaling dimension $\Delta=\Delta_{1}+\Delta_{2}$. Vice versa for the
boundary vector operators. Finally, we verify that the contributions of the
spurious poles exactly cancel also in this case, because of the identities in
4.2.1.

###  5.2 Conformal Field Theories

We have shown some examples of Källén-Lehmann decompositions of two-point
functions of composite operators in free QFTs. In this section, we use the
Källén–Lehmann decomposition to study how states generated by the action of
bulk CFT primaries on the Euclidean vacuum decompose into UIRs of the de
Sitter group. That corresponds to decomposing irreps of $SO(d+1,2)$ into
irreps of $SO(d+1,1)\,.$ We test examples up to spin 2 and recover the fact
that for general $d>1$, CFT states decompose into principal series states and
complementary series states, while for $d=1$ there is the appearance of
discrete series states up to $\Delta=J,$ as in the free theory case. We verify
the validity of our results by comparing their flat space limit as described
in section 3.3 to the results presented in Karateev:2020axc .

##### Spinning CFT two-point functions in de Sitter

To start, let us review the form of CFT two-point functions of traceless
symmetric primary operators with spin in the bulk of de Sitter. The relevant
group of symmetries of these correlators is $SO(d+1,2).$ We thus embed the
$d+1$ dimensional de Sitter CFT in $\mathbb{R}^{d+1,2}$ with metric
$\eta=\text{diag}(-1,1,\ldots,1,-1)$. We denote points in this embedding space
$\mathcal{Y}\in\mathbb{R}^{d+1,2}$, and the invariance under $SO(d+1,2)$ is
enforced by $\mathcal{Y}^{2}=0$. Explicitly,

$\mathcal{Y}^{2}=Y^{2}-(\mathcal{Y}^{d+2})^{2}=0\,,$ (5.41)

where $Y\in\mathbb{R}^{d+1,1}$. The de Sitter hyperboloid constraint is
enforced by $(\mathcal{Y}^{d+2})^{2}=1$. That is the section of the lightcone
in $\mathbb{R}^{d+1,2}$ on which we will focus. We also embed the polarization
tensors as $\mathcal{Z}=(W,0),$ so that
$\mathcal{Y}_{1}\cdot\mathcal{Z}_{2}=Y_{1}\cdot W_{2}\,.$ In Costa:2011mg  it
is shown that CFT two-point functions of a spin $J$ primary operator of
conformal dimension $\mathbf{\Delta}$ is, in embedding space,

$\langle{\cal O}^{(J)}(\mathcal{Y}_{1},\mathcal{Z}_{1}){\cal
O}^{(J)}(\mathcal{Y}_{2},\mathcal{Z}_{2})\rangle=c_{{\cal
O}^{(J)}}\frac{(-2[(\mathcal{Z}_{1}\cdot\mathcal{Z}_{2})(\mathcal{Y}_{1}\cdot\mathcal{Y}_{2})-(\mathcal{Y}_{1}\cdot\mathcal{Z}_{2})(\mathcal{Y}_{2}\cdot\mathcal{Z}_{1})])^{J}}{(-2\mathcal{Y}_{1}\cdot\mathcal{Y}_{2})^{\mathbf{\Delta}+J}}\,.$
(5.42)

Projecting to de Sitter by using the lightcone constraint

$\mathcal{Y}_{1}\cdot\mathcal{Y}_{2}=Y_{1}\cdot Y_{2}-1\,,$ (5.43)

we obtain the general form of a spin $J$ CFT two-point function in de Sitter
expressed in the embedding space formalism

$\langle\mathcal{O}^{(J)}(Y_{1},W_{1})\mathcal{O}^{(J)}(Y_{2},W_{2})\rangle=c_{\mathcal{O}^{(J)}}\frac{[(W_{1}\cdot
W_{2})(1-Y_{1}\cdot Y_{2})+(Y_{1}\cdot W_{2})(Y_{2}\cdot
W_{1})]^{J}}{2^{\mathbf{\Delta}}(1-Y_{1}\cdot Y_{2})^{\mathbf{\Delta}+J}}\,,$
(5.44)

####  5.2.1 Spin 0 Example

Let us start by reviewing the Källén–Lehmann decomposition of the CFT two-
point function of a scalar primary operator ${\cal O}$ of conformal dimension
$\mathbf{\Delta}$ in de Sitter, which was computed before in
Hogervorst:2021uvp . The two-point function has the form

$\langle{\cal O}(Y_{1}){\cal O}(Y_{2})\rangle=\frac{c_{{\cal
O}}}{2^{\mathbf{\Delta}}(1-Y_{1}\cdot Y_{2})^{\mathbf{\Delta}}}\,.$ (5.45)

It has been argued in section 3 that only scalar principal series and
complementary series can contribute to such a two-point function. In addition,
using the criterion found in section 4.3, we know that complementary series is
also absent when $\mathbf{\Delta}>\frac{d}{2}$. In this case, the inversion
formula for the principal series contribution reads

$\rho^{\mathcal{P},0}_{\cal O}(\lambda)=\frac{c_{{\cal
O}}}{2^{\mathbf{\Delta}}\mathcal{N}_{0,0}}\int_{X_{1}}\Omega_{\lambda,0}(X_{1},X_{2})(1-X_{1}\cdot
X_{2})^{-\mathbf{\Delta}}\,.$ (5.46)

This integral can be solved by using the Mellin-Barnes representation of the
hypergeometric function and Barnes’ first lemma, as explained in Appendix H.2.
We obtain

$\rho^{\mathcal{P},0}_{\cal O}(\lambda)=c_{{\cal
O}}\frac{2^{1+d-2\mathbf{\Delta}}\pi^{\frac{d-1}{2}}\Gamma(-\frac{d}{2}+\mathbf{\Delta}\pm
i\lambda)}{\Gamma(\mathbf{\Delta})\Gamma(\frac{1-d}{2}+\mathbf{\Delta})}\lambda\sinh(\pi\lambda)\,.$
(5.47)

We numerically verified that the Källén–Lehmann integral over the principal
series of this spectral density fully reproduces the two-point function, if
$\mathbf{\Delta}>\frac{d}{2}$. When instead $\mathbf{\Delta}<\frac{d}{2}$, the
poles at

$\lambda=\pm i\left(\mathbf{\Delta}-\frac{d}{2}\right)$ (5.48)

cross the contour of integration and need to be added. This corresponds to a
complementary series state since $\lambda$ is imaginary. The contribution of
this state when $\mathbf{\Delta}<\frac{d}{2}$ is in agreement with the
decomposition of $SO(d+1,2)$ irreps into $SO(d+1,1)$ as shown in table 1.4 of
Penedones:2023uqc . Unitarity fixes $\rho_{\cal O}^{\mathcal{P}}(\lambda)$ to
be positive, and from this fact we can infer the usual unitarity bound on
$\mathbf{\Delta}$

$\mathbf{\Delta}>\frac{d-1}{2}\,.$ (5.49)

In general, we can write the full decomposition of a CFT scalar two-point
function as

$\langle{\cal O}(Y_{1}){\cal
O}(Y_{2})\rangle=\int_{\mathbb{R}}\mathrm{d}\lambda\ \rho_{\cal
O}^{\mathcal{P},0}(\lambda)G_{\lambda,0}(Y_{1},Y_{2})+\theta\left(\frac{d}{2}-\mathbf{\Delta}\right)\rho^{\mathcal{C},0}_{\cal
O}G_{i\left(\mathbf{\Delta}-\frac{d}{2}\right),0}(Y_{1},Y_{2})\,,$ (5.50)

where $\theta(x)$ is a Heaviside theta, and

$\rho_{\cal O}^{\mathcal{C},0}=c_{\cal
O}\frac{4\pi^{\frac{d}{2}}\Gamma(1-\frac{d}{2}+\mathbf{\Delta})\sin\left(\frac{\pi}{2}(d-2\mathbf{\Delta})\right)}{\Gamma(\mathbf{\Delta})}=4\pi
i\underset{\lambda=i(\mathbf{\Delta}-\frac{d}{2})}{\text{Res}}\rho_{\cal
O}^{\mathcal{P}}(\lambda)\,.$ (5.51)

Notice that the complementary series contribution is the two-point function of
a free field with $\Delta=d-\mathbf{\Delta}$. Furthermore, if
$\mathbf{\Delta}=\frac{d-1}{2}$, we have that $\rho_{\cal
O}^{\mathcal{P},0}(\lambda)=0$ and

$\langle{\cal O}(Y_{1}){\cal O}(Y_{2})\rangle=\rho_{\cal
O}^{\mathcal{C},0}G_{i\left(\mathbf{\Delta}-\frac{d}{2}\right),0}(Y_{1},Y_{2})\Big{|}_{\mathbf{\Delta}=\frac{d-1}{2}}\,.$
(5.52)

This is the two-point function of a conformally coupled scalar in dSd+1.
Finally, we comment on boundary operators. The poles in
$\rho_{\mathcal{O}}^{\mathcal{P},0}(\lambda)$ are at
$\Delta=\mathbf{\Delta}+\mathbb{N}$, signaling the appearance of boundary
operators with these weights in the Boundary Operator Expansion of the bulk
CFT primary $\mathcal{O}$. Let us explain their origin by considering a $d+1$
dimensional Lorentzian CFT on the Minkowski cylinder. The discussion is
entirely analogous in de Sitter because they are conformally equivalent
spacetimes. Consider a scalar primary $\mathcal{O}(t,\mathbf{x})$ in this CFT
with conformal dimension $\mathbf{\Delta}$ and let us focus on the timeslice
at $t=0$, where the primaries of the smaller $SO(d+1,1)$ group at that
timeslice are those operators $\mathcal{O}_{n}(\mathbf{x})$ which satisfy

$[D,\mathcal{O}_{n}(0)]=\Delta_{n}\mathcal{O}_{n}(0)\,,\qquad[K_{i},\mathcal{O}_{n}(0)]=0\qquad
i=1,\ldots,d\,,$ (5.53)

It can be checked that particular linear combinations of operators like
$(\bm{\partial}^{2})^{m}\partial_{t}^{n}\mathcal{O}(0,\mathbf{x})$ for
different $m$ and $n$ such that $\frac{m}{2}+n$ is constant, satisfy this
condition. For example, the operator
$\mathcal{O}_{0}(\mathbf{x})\equiv\mathcal{O}(0,\mathbf{x})$ trivially
satisfies (5.53) with $\Delta_{0}=\mathbf{\Delta}$. But also

$[K_{i},[P_{0},\mathcal{O}(0,0)]]=-2[M_{i0},\mathcal{O}(0,0)]=0$ (5.54)

so that
$\mathcal{O}_{1}(\mathbf{x})\equiv\partial_{t}\mathcal{O}(0,\mathbf{x})$ at
the timeslice $t=0$ is a boundary primary with $\Delta_{1}=\mathbf{\Delta}+1$.
If we go further, we find

$\displaystyle]=2[P_{i},\mathcal{O}(0,0)]\,,$ (5.55)

but also

$[K_{i},[P_{j}P^{j},\mathcal{O}(0,0)]]=2(2(\mathbf{\Delta}+1)-d)[P_{i},\mathcal{O}(0,0)]$
(5.56)

so that a good boundary primary is

$\mathcal{O}_{2}(\mathbf{x})\equiv\partial_{t}^{2}\mathcal{O}(0,\mathbf{x})-\frac{1}{2(\mathbf{\Delta}+1)-d}\bm{\partial}^{2}\mathcal{O}(0,\mathbf{x})\,,$
(5.57)

with $\Delta_{2}=\mathbf{\Delta}+2$. This can be iterated to find higher and
higher primaries with $\Delta=\mathbf{\Delta}+\mathbb{N}$. By Weyl
equivalence, the discussion in de Sitter space is entirely analogous and we
can thus explain the presence of the poles at
$\Delta=\mathbf{\Delta}+\mathbb{N}$ in the spectral density (5.47).

##### Flat space limit

To compute the flat space limit of the Källén-Lehmann decomposition of this
two-point function, we follow the discussion in section 3.3 by first restoring
the dimensions of the spectral densities by adding the appropriate factors of
the de Sitter radius $R$: $\rho_{{\cal O}}^{\mathcal{P},0}(\lambda)\to
R^{-2\mathbf{\Delta}+d-1}\rho_{{\cal O}}^{\mathcal{P},0}(\lambda)$. Then,
after changing variables to $\lambda=Rm$, we take the limit

$\lim_{R\to\infty}\frac{R}{m}\rho_{{\cal O}}^{\mathcal{P},0}(Rm)=c_{\cal
O}\frac{2^{d+1-2\mathbf{\Delta}}\pi^{\frac{d+1}{2}}}{\Gamma(\mathbf{\Delta})\Gamma(\frac{1-d}{2}+\mathbf{\Delta})}m^{2\mathbf{\Delta}-d-1}\,,$
(5.58)

which precisely matches the flat space CFT scalar spectral density (eq. (4.3)
in Karateev:2020axc ).232323To match conventions
$d_{\text{here}}=(d-1)_{\text{there}}$ and
$\mathbf{\Delta}_{\text{here}}=\Delta_{\mathcal{{\cal O}}\text{there}}$. Once
adjusted for dimensions, the complementary series spectral density instead
reads

$\rho^{\mathcal{C},0}_{\cal O}=c_{\cal
O}\frac{4\pi^{\frac{d}{2}}\Gamma(1-\frac{d}{2}+\mathbf{\Delta})\sin\left(\frac{\pi}{2}(d-2\mathbf{\Delta})\right)}{R^{2\mathbf{\Delta}-d+1}\Gamma(\mathbf{\Delta})}\,.$
(5.59)

In the large $R$ limit it survives only if $\mathbf{\Delta}=\frac{d-1}{2}$.
That is in agreement with the fact that in flat space, a free massless scalar
is a CFT primary operator. We can thus write that in flat space we have

$\rho^{\mathbb{M},0}_{\cal O}(m^{2})=c_{\cal
O}\frac{2^{d+1-2\mathbf{\Delta}}\pi^{\frac{d+1}{2}}}{\Gamma(\mathbf{\Delta})\Gamma(\frac{1-d}{2}+\mathbf{\Delta})}m^{2\mathbf{\Delta}-d-1}\,,$
(5.60)

and instead, if $\mathbf{\Delta}=\frac{d-1}{2}$,

$\rho^{\mathbb{M},0}_{\cal O}(m^{2})=c_{\cal
O}\delta(m^{2})\frac{4\pi^{\frac{d+1}{2}}}{\Gamma(\frac{d-1}{2})}\,.$ (5.61)

####  5.2.2 Spin 1 Example

For a spin-1 primary operator $J$ of conformal weight $\mathbf{\Delta}$, the
two-point function is

$\langle
J(Y_{1};W_{1})J(Y_{2};W_{2})\rangle=\frac{c_{J}}{2^{\mathbf{\Delta}}}\left[\frac{W_{1}\cdot
W_{2}}{(1-Y_{1}\cdot Y_{2})^{\mathbf{\Delta}}}+\frac{(Y_{1}\cdot
W_{2})(Y_{2}\cdot W_{1})}{(1-Y_{1}\cdot Y_{2})^{\mathbf{\Delta}+1}}\right]\,.$
(5.62)

In Appendix H.2 we show explicitly how to invert this two-point function and
find the principal series spectral densities:

$\displaystyle\rho_{J}^{\mathcal{P},0}(\lambda)$
$\displaystyle=c_{J}\frac{2^{3+d-2\mathbf{\Delta}}\pi^{\frac{d-1}{2}}(\mathbf{\Delta}-d)\Gamma\left(-\frac{d}{2}+\mathbf{\Delta}\pm
i\lambda\right)}{(d^{2}+4\lambda^{2})\Gamma(\mathbf{\Delta}+1)\Gamma(\frac{1-d}{2}+\mathbf{\Delta})}\lambda\sinh(\pi\lambda)\,,$
(5.63) $\displaystyle\rho_{J}^{\mathcal{P},1}(\lambda)$
$\displaystyle=c_{J}\frac{2^{1+d-2\mathbf{\Delta}}\pi^{\frac{d-1}{2}}(\mathbf{\Delta}-1)\Gamma(-\frac{d}{2}+\mathbf{\Delta}\pm
i\lambda)}{\Gamma(\mathbf{\Delta}+1)\Gamma(\frac{1-d}{2}+\mathbf{\Delta})}\lambda\sinh(\pi\lambda)\,.$

For $d\geq 2$, this is the complete Källén–Lehmann decomposition of this two-
point function, as can be verified numerically by performing the integral over
the principal series. In $d=1$, instead, we see the appearance of discrete
series states. We can either compute those directly from the inversion formula
in $d=1$, or just derive them by analytically continuing the higher
dimensional result all the way to $d=1$ while keeping track of any poles that
cross the principal series integration contour. The results one obtains in
these two ways agree. The full Källén-Lehmann decomposition for the two-point
function of a spin 1 primary CFT operator is thus

$\displaystyle\langle J(Y_{1};W_{1})J(Y_{2};W_{2})\rangle=$
$\displaystyle\sum_{\ell=0}^{1}\int_{\mathbb{R}}\mathrm{d}\lambda\
\rho_{J}^{\mathcal{P},\ell}(\lambda)[(W_{1}\cdot\nabla_{1})(W_{2}\cdot\nabla_{2})]^{1-\ell}G_{\lambda,\ell}(Y_{1},Y_{2};W_{1},W_{2})$
$\displaystyle+\delta_{d,1}\rho_{J}^{\mathcal{D}_{1}}(W_{1}\cdot\nabla_{1})(W_{2}\cdot\nabla_{2})G_{-i/2,0}(Y_{1},Y_{2})$
(5.64)

where

$\rho_{J}^{\mathcal{D}_{1}}=c_{J}\frac{2^{3-2\mathbf{\Delta}}\pi}{\mathbf{\Delta}}\,.$
(5.65)

When $d=1$ and $\mathbf{\Delta}=1$, we have that
$\rho^{\mathcal{P},0}_{J}=\rho^{\mathcal{P},1}_{J}=0$ and the left hand side
matches exactly the extra contribution from the discrete series. The discrete
series term is in fact the two-point function of the operator
$W\cdot\nabla\varphi$ with $\varphi$ being the massless scalar, corresponding
to the $\Delta=1$ operator in the discrete series

$(W_{1}\cdot\nabla_{1})(W_{2}\cdot\nabla_{2})\langle\varphi(Y_{1})\varphi(Y_{2})\rangle\,.$
(5.66)

In two-dimensional Minkowski space, the operator $\partial_{\mu}\varphi$ with
$\varphi$ being a free massless scalar is a CFT primary. It should not
surprise us then that the same is true in de Sitter, by Weyl equivalence. One
last observation is that from (5.63) we can recover unitarity bounds for spin
$1$ CFT primaries

$\mathbf{\Delta}\geq d\,.$ (5.67)

Moreover, we observe the expected feature that for a conserved current
$\mathbf{\Delta}=d$, only transverse states propagate, since
$\rho^{\mathcal{P},0}_{J}(\lambda)$ vanishes. Finally, we study the boundary
operators appearing in the Boundary Operator Expansion of the bulk CFT primary
$J(Y;W).$ The pole structure of the two spectral densities (5.63), after
verifying that the contributions of the spurious poles cancel as expected,
signals the appearance of scalar and vector boundary operators with conformal
dimensions $\mathbf{\Delta}+\mathbb{N}.$ Analogously to the scalar case, these
come from the late time expansion of the bulk operator and of linear
combinations of
$\partial_{t}^{n}(\bm{\partial}^{2})^{m}J^{\mu}(0,\mathbf{x})$. Scalar
boundary primary operators then come from particular linear combinations of
$\partial_{t}^{n}(\bm{\partial}^{2})^{m}J^{0}(0,\mathbf{x})$ and
$\partial_{t}^{n}(\bm{\partial}^{2})^{m}\bm{\partial}\cdot\mathbf{J}(0,\mathbf{x})$.
Vector operators come from
$\partial_{t}^{n}(\bm{\partial}^{2})^{m}J^{i}(0,\mathbf{x})$.

##### Flat space limit

We start by restoring the dimensions $\rho_{J}^{\mathcal{P},0}(\lambda)\to
R^{2\mathbf{\Delta}-d-1}\rho_{J}^{\mathcal{P},0}(\lambda)$ and
$\rho_{J}^{\mathcal{P},1}(\lambda)\to
R^{2\mathbf{\Delta}-d+1}\rho_{J}^{\mathcal{P},1}(\lambda)$. Then, we have

$\displaystyle\lim_{R\to\infty}\frac{R}{m}\rho^{\mathcal{P},0}_{J}(Rm)$
$\displaystyle=c_{J}\frac{2^{d+1-2\mathbf{\Delta}}\pi^{\frac{d+1}{2}}(\mathbf{\Delta}-d)}{\Gamma(1+\mathbf{\Delta})\Gamma(\frac{1-d}{2}+\mathbf{\Delta})}m^{2\mathbf{\Delta}-d-3}\,,$
(5.68)
$\displaystyle\lim_{R\to\infty}\frac{R}{m^{3}}\rho^{\mathcal{P},1}_{J}(Rm)$
$\displaystyle=c_{J}\frac{2^{d+1-2\mathbf{\Delta}}\pi^{\frac{d+1}{2}}(\mathbf{\Delta}-1)}{\Gamma(1+\mathbf{\Delta})\Gamma(\frac{1-d}{2}+\mathbf{\Delta})}m^{2\mathbf{\Delta}-d-3}\,.$

which match equations (4.10) in Karateev:2020axc . The discrete series
contribution with dimensions restored reads

$\rho^{\mathcal{D}_{1}}_{J}=c_{J}\frac{2^{3-2\mathbf{\Delta}}\pi}{R^{2\mathbf{\Delta}-2}\mathbf{\Delta}}\,.$
(5.69)

In the flat space limit, this survives only if $\mathbf{\Delta}=1$,
corresponding to the case in which we are decomposing a two-point function of
a conserved current in $d=1$. We can thus write, for $\mathbf{\Delta}>1$

$\displaystyle\rho^{\mathbb{M},0}_{J}(m^{2})$
$\displaystyle=c_{J}\frac{2^{d+1-2\mathbf{\Delta}}\pi^{\frac{d+1}{2}}(\mathbf{\Delta}-d)}{\Gamma(1+\mathbf{\Delta})\Gamma(\frac{1-d}{2}+\mathbf{\Delta})}m^{2\mathbf{\Delta}-d-3}\,,$
(5.70) $\displaystyle\rho^{\mathbb{M},1}_{J}(m^{2})$
$\displaystyle=c_{J}\frac{2^{d+1-2\mathbf{\Delta}}\pi^{\frac{d+1}{2}}(\mathbf{\Delta}-1)}{\Gamma(1+\mathbf{\Delta})\Gamma(\frac{1-d}{2}+\mathbf{\Delta})}m^{2\mathbf{\Delta}-d-3}\,,$

which precisely match eq. (4.10) in Karateev:2020axc . In the $d=1$ and
$\mathbf{\Delta}=1$ case, instead, the Källén-Lehmann decomposition is given
by a massless state

$\displaystyle\rho^{\mathbb{M},0}_{J}(m^{2})=2\pi
c_{J}\delta(m^{2})\,,\qquad\rho^{\mathbb{M},1}_{J}(m^{2})=0\,.$ (5.71)

####  5.2.3 Spin 2 Example

The two-point function of a spin-2 traceless and symmetric CFT primary with
conformal weight $\mathbf{\Delta}$ is

$\langle
T(Y_{1};W_{1})T(Y_{2};W_{2})\rangle=\frac{c_{T}}{2^{\mathbf{\Delta}}}\Big{[}\frac{(W_{1}\cdot
W_{2})^{2}}{(1-Y_{1}\cdot Y_{2})^{\mathbf{\Delta}}}+2\frac{(W_{1}\cdot
W_{2})(Y_{1}\cdot W_{2})(Y_{2}\cdot W_{1})}{(1-Y_{1}\cdot
Y_{2})^{\mathbf{\Delta}+1}}\\!+\\!\frac{[(Y_{1}\cdot W_{2})(Y_{2}\cdot
W_{1})]^{2}}{(1-Y_{1}\cdot Y_{2})^{\mathbf{\Delta}+2}}\Big{]}$ (5.72)

In Appendix H.2 we show how to apply the inversion formula to this case. The
resulting principal series contributions to the Källén-Lehmann decomposition
are

$\displaystyle\rho_{T}^{\mathcal{P},0}(\lambda)$
$\displaystyle=c_{T}\frac{2^{5+d-2\mathbf{\Delta}}(d+1)\pi^{\frac{d-1}{2}}(d-\mathbf{\Delta})(d+1-\mathbf{\Delta})\Gamma(-\frac{d}{2}+\mathbf{\Delta}\pm
i\lambda)}{d(d^{2}+4\lambda^{2})((d+2)^{2}+4\lambda^{2})\Gamma(\mathbf{\Delta}+2)\Gamma(\frac{1-d}{2}+\mathbf{\Delta})}\lambda\sinh(\pi\lambda)\,,$
(5.73) $\displaystyle\rho_{T}^{\mathcal{P},1}(\lambda)$
$\displaystyle=c_{T}\frac{2^{4+d-2\mathbf{\Delta}}\pi^{\frac{d-1}{2}}(1-\mathbf{\Delta})(d+1-\mathbf{\Delta})\Gamma(-\frac{d}{2}+\mathbf{\Delta}\pm
i\lambda)}{((d+2)^{2}+4\lambda^{2})\Gamma(\mathbf{\Delta}+2)\Gamma(\frac{1-d}{2}+\mathbf{\Delta})}\lambda\sinh(\pi\lambda)\,,$
$\displaystyle\rho_{T}^{\mathcal{P},2}(\lambda)$
$\displaystyle=c_{T}\frac{2^{1+d-2\mathbf{\Delta}}\pi^{\frac{d-1}{2}}(\mathbf{\Delta}-1)\mathbf{\Delta}\Gamma(-\frac{d}{2}+\mathbf{\Delta}\pm
i\lambda)}{\Gamma(\mathbf{\Delta}+2)\Gamma(\frac{1-d}{2}+\mathbf{\Delta})}\lambda\sinh(\pi\lambda)\,.$

Once again, for $d>1$ the Källén–Lehmann integral over the principal series
fully reproduces the two-point function, and the structure of spurious poles
matches the predictions made in section 4.2.1. In $d=1$, through the inversion
formula for the discrete series we find contributions from $\Delta=1$ and
$\Delta=2$ states.

$\displaystyle\langle T(Y_{1};W_{1})T(Y_{2};W_{2})\rangle=$
$\displaystyle\sum_{\ell=0}^{2}\int_{\mathbb{R}}\mathrm{d}\lambda\
\rho_{T}^{\mathcal{P},\ell}(\lambda)[(W_{1}\cdot\nabla_{1})(W_{2}\cdot\nabla_{2})]^{2-\ell}G_{\lambda,\ell}(Y_{1},Y_{2};W_{1},W_{2})$
$\displaystyle+\delta_{d,1}\sum_{p=1}^{2}\rho_{T}^{\mathcal{D}_{p}}(W_{1}\cdot\nabla_{1})^{2}(W_{2}\cdot\nabla_{2})^{2}G_{i(\frac{1}{2}-p),0}(Y_{1},Y_{2})$
(5.74)

with

$\rho_{T}^{\mathcal{D}_{2}}=c_{T}\frac{2^{3-2\mathbf{\Delta}}\pi}{\mathbf{\Delta}+1}\,,\qquad\rho_{T}^{\mathcal{D}_{1}}=c_{T}\frac{2^{3-2\mathbf{\Delta}}\pi(\mathbf{\Delta}-2)}{\mathbf{\Delta}(\mathbf{\Delta}+1)}\,.$
(5.75)

The positivity of (5.73) implies the expected unitarity bound for a spin 2 CFT
primary

$\mathbf{\Delta}\geq d+1\,,$ (5.76)

and when $T$ is a conserved stress tensor, $\mathbf{\Delta}=d+1$, both
$\rho^{\mathcal{P},0}_{T}$ and $\rho^{\mathcal{P},1}_{T}$ vanish. When $d=1$,
$G_{\lambda,2}$ vanishes and the decomposition matches (3.2.2) with vanishing
complementary series contributions. The two discrete series contributions
correspond to the two-point functions

$(W_{1}\cdot\nabla_{1})^{2}(W_{2}\cdot\nabla_{2})^{2}\langle\varphi(Y_{1})\varphi(Y_{2})\rangle\,,\qquad(W_{1}\cdot\nabla_{1})^{2}(W_{2}\cdot\nabla_{2})^{2}\langle\varphi^{\prime}(Y_{1})\varphi^{\prime}(Y_{2})\rangle\,,$
(5.77)

with $\varphi$ being a massless scalar, corresponding to $\Delta=1$, and
$\varphi^{\prime}$ being the first tachyon in the discrete series, the
$\Delta=2$ representation. When $d=1$ and $\mathbf{\Delta}=2$, all spectral
densities vanish except for $\rho_{T}^{\mathcal{D}_{2}}$, and then

$\langle
T(Y_{1};W_{1})T(Y_{2};W_{2})\rangle\Big{|}_{d=1,\mathbf{\Delta}=2}=\rho_{T}^{\mathcal{D}_{2}}\Big{|}_{\mathbf{\Delta}=2}(W_{1}\cdot\nabla_{1})^{2}(W_{2}\cdot\nabla_{2})^{2}G_{-\frac{3i}{2},0}(Y_{1},Y_{2})\,,$
(5.78)

meaning that in $d=1$ the operator $(W\cdot\nabla)^{2}\varphi^{\prime}$ with
$\varphi^{\prime}$ being a tachyonic free field with
$\Delta_{\varphi^{\prime}}=2$ is a CFT primary. Now let us comment on the
boundary operators. First of all, we verify that the contributions from the
spurious poles cancel. Then, we observe that the physical poles imply the
appearance of boundary operators with spins $\ell=0,1,2$ and conformal weight
$\mathbf{\Delta}+\mathbb{N}$ in the BOE of $T(Y,W)$. These again come from the
late time limit of linear combinations of
$\partial_{t}^{n}(\bm{\partial}^{2})^{m}T^{\mu\nu}(0,\mathbf{x})$. The spin 2
boundary operators are of the form
$\partial_{t}^{n}(\bm{\partial}^{2})^{m}T^{ij}(0,\mathbf{x})$, the spin 1
operators are combinations of derivatives and $\mu,\nu=0,i$ and the scalars
are combinations of the trace and of divergences.

##### Flat space limit

As done for the previous examples, we restore the factors of $R$ in the
spectral densities and then we take the large $R$ limit

$\displaystyle\lim_{R\to\infty}\frac{d}{d+1}\frac{R}{m}\rho^{\mathcal{P},0}_{T}(Rm)$
$\displaystyle=c_{T}\frac{2^{d+1-2\mathbf{\Delta}}\pi^{\frac{d+1}{2}}(\mathbf{\Delta}-d)(\mathbf{\Delta}-d-1)}{\Gamma(2+\mathbf{\Delta})\Gamma(\frac{1-d}{2}+\mathbf{\Delta})}m^{2\mathbf{\Delta}-d-5}\,,$
(5.79)
$\displaystyle\lim_{R\to\infty}\frac{1}{2}\frac{R}{m^{3}}\rho^{\mathcal{P},1}_{T}(Rm)$
$\displaystyle=c_{T}\frac{2^{d+1-2\mathbf{\Delta}}\pi^{\frac{d+1}{2}}(\mathbf{\Delta}-1)(\mathbf{\Delta}-d-1)}{\Gamma(2+\mathbf{\Delta})\Gamma(\frac{1-d}{2}+\mathbf{\Delta})}m^{2\mathbf{\Delta}-d-5}\,,$
$\displaystyle\lim_{R\to\infty}\frac{R}{m^{5}}\rho^{\mathcal{P},2}_{T}(Rm)$
$\displaystyle=c_{T}\frac{2^{d+1-2\mathbf{\Delta}}\pi^{\frac{d+1}{2}}(\mathbf{\Delta}-1)\mathbf{\Delta}}{\Gamma(2+\mathbf{\Delta})\Gamma(\frac{1-d}{2}+\mathbf{\Delta})}m^{2\mathbf{\Delta}-d-5}\,,$

where we have used $\beta_{2,0}=\frac{d}{d+1},\beta_{2,1}=\frac{1}{2}$ and
$\beta_{2,2}=1$. These results match equations (4.13) in Karateev:2020axc .
Restoring the dimensions in the discrete series densities, instead, gives

$\rho^{\mathcal{D}_{2}}_{T}=c_{T}\frac{2^{3-2\mathbf{\Delta}}\pi}{R^{2\mathbf{\Delta}-4}(\mathbf{\Delta}+1)}\,,\qquad\rho^{\mathcal{D}_{1}}_{T}=c_{T}\frac{2^{3-2\mathbf{\Delta}}\pi(\mathbf{\Delta}-2)}{R^{2\mathbf{\Delta}-4}\mathbf{\Delta}(\mathbf{\Delta}+1)}\,.$
(5.80)

Under the flat space limit, only when $\mathbf{\Delta}=2$ they have a chance
of surviving. The $p=1$ density vanishes even in this case, because of the
factor of $\mathbf{\Delta}-2$ in the numerator, and so we are left with the
following flat space densities for $\mathbf{\Delta}>2$

$\displaystyle\rho^{\mathbb{M},0}_{T}(m^{2})$
$\displaystyle=c_{T}\frac{2^{d+1-2\mathbf{\Delta}}\pi^{\frac{d+1}{2}}(\mathbf{\Delta}-d)(\mathbf{\Delta}-d-1)}{\Gamma(2+\mathbf{\Delta})\Gamma(\frac{1-d}{2}+\mathbf{\Delta})}\,m^{2\mathbf{\Delta}-d-5}\,,$
(5.81) $\displaystyle\rho^{\mathbb{M},1}_{T}(m^{2})$
$\displaystyle=c_{T}\frac{2^{d+1-2\mathbf{\Delta}}\pi^{\frac{d+1}{2}}(\mathbf{\Delta}-1)(\mathbf{\Delta}-d-1)}{\Gamma(1+\mathbf{\Delta})\Gamma(\frac{1-d}{2}+\mathbf{\Delta})}\,m^{2\mathbf{\Delta}-d-5}\,,$
$\displaystyle\rho^{\mathbb{M},2}_{T}(m^{2})$
$\displaystyle=c_{T}\frac{2^{d+1-2\mathbf{\Delta}}\pi^{\frac{d+1}{2}}(\mathbf{\Delta}-1)\mathbf{\Delta}}{\Gamma(2+\mathbf{\Delta})\Gamma(\frac{1-d}{2}+\mathbf{\Delta})}\,m^{2\mathbf{\Delta}-d-5}\,,$

and the special case $\mathbf{\Delta}=2$ and $d=1$ instead being

$\displaystyle\rho^{\mathbb{M},0}_{T}(m^{2})=\frac{\pi}{12}c_{T}\delta(m^{2})\,,\qquad\rho^{\mathbb{M},1}_{T}(m^{2})=0\,.$
(5.82)

Equations (5.81) and (5.82) match (4.13) in Karateev:2020axc .242424The
placement of the massless state in $\rho^{\mathbb{M},0}_{T}$ or
$\rho^{\mathbb{M},1}_{T}$ is arbitrary since the Wightman propagators
$\Delta_{0,1}^{(2)}$ and $\Delta_{0,0}^{(2)}$ are the same in $d=1$ in the
massless limit.

####  5.2.4 Higher spin examples in dS2

So far, we have derived the spectral densities for spin $J\in\\{0,1,2\\}$ CFT
operators ${\cal O}^{(J)}$ in dS2 by an analytical continuation from higher
$d$. This can also be done systematically using the dS2 inversion formula
developed in section 4.5. In general, the two-point function of a spin $J$
primary is given by eq. (5.44), and its corresponding chiral components as
defined in eq. (4.56) are

$\displaystyle G_{{\cal
O}^{(J)}}^{+}(\sigma)=\frac{2^{J-\mathbf{\Delta}}c_{\mathcal{O}^{(J)}}}{(1-\sigma)^{\mathbf{\Delta}+J}},\,\,\,\,\,\,\,G_{{\cal
O}^{(J)}}^{-}(\sigma)=0~{}.$ (5.83)

Assuming the unitarity bound $\mathbf{\Delta}\geq J$, the asymptotic behavior
$G_{{\cal O}^{(J)}}^{\pm}(\sigma)\sim(-\sigma)^{-(\mathbf{\Delta}+J)}$ ensures
the absence of complementary series in the Källén-Lehmann decomposition of
${\cal O}^{(J)}$. In addition, as a direct result of eq. (4.62), the vanishing
of $G_{{\cal O}^{(J)}}^{-}(\sigma)$ implies

$\displaystyle\rho^{\mathcal{P},1}_{{\cal
O}^{(J)}}(\lambda)=\left(\frac{1}{4}+\lambda^{2}\right)\rho^{\mathcal{P},0}_{{\cal
O}^{(J)}}(\lambda)~{}.$ (5.84)

Then we apply the inversion formula (4.5) to $G^{(+)}_{{\cal O}^{(J)}}$ given
by eq. (5.83)

$\displaystyle\rho^{\mathcal{P},0}_{{\cal
O}^{(J)}}(\lambda)=\frac{2^{J-\mathbf{\Delta}+1}c_{\mathcal{O}^{(J)}}\lambda\sinh(2\pi\lambda)}{(\frac{1}{2}+i\lambda)_{J}^{2}(\frac{1}{2}-i\lambda)_{J}^{2}}\int_{-\infty}^{-1}d\sigma\,\frac{\phi_{\lambda,J}^{+}(\sigma)}{(1-\sigma)^{\mathbf{\Delta}-J}}$
$\displaystyle\rho^{\mathcal{D}_{p}}_{{\cal
O}^{(J)}}=\frac{2^{J-\mathbf{\Delta}+3}c_{\mathcal{O}^{(J)}}\pi^{2}\,(2p-1)}{\Gamma(J+p)^{2}\Gamma(1+J-p)^{2}}\int_{-\infty}^{-1}d\sigma\,\frac{\psi_{p,J}(\sigma)}{(1-\sigma)^{\mathbf{\Delta}-J}}~{}.$
(5.85)

The evaluation of this type of integral is extensively discussed in appendix
H.2. Here we just report the final results

$\displaystyle\rho^{\mathcal{P},0}_{{\cal
O}^{(J)}}(\lambda)=\left(\frac{1}{4}+\lambda^{2}\right)^{-1}\rho^{\mathcal{P},1}_{{\cal
O}^{(J)}}(\lambda)=\frac{c_{\mathcal{O}^{(J)}}\Gamma(\pm
i\lambda+\mathbf{\Delta}-\frac{1}{2})}{2^{2\mathbf{\Delta}-J-1}(\frac{1}{2}\pm
i\lambda)_{J}\Gamma(\mathbf{\Delta}+J)\Gamma(\mathbf{\Delta}-J)}\lambda\sinh(\pi\lambda)$
$\displaystyle\rho^{\mathcal{D}_{p}}_{{\cal
O}^{(J)}}=\frac{c_{\mathcal{O}^{(J)}}\pi(2p-1)\Gamma(\mathbf{\Delta}-p)\Gamma(p+\mathbf{\Delta}-1)}{2^{2\mathbf{\Delta}-J-2}\Gamma(\mathbf{\Delta}-J)\Gamma(\mathbf{\Delta}+J)\Gamma(J-p+1)\Gamma(J+p)}~{}.$
(5.86)

When $\mathbf{\Delta}=J$, i.e. saturation of unitarity bound, only discrete
series with $p=J$ contributes.

##### Flat space limit

The spectral densities we have just derived allow us to make a prediction
about flat space two dimensional CFTs. After restoring the correct dimensions,
under the flat space limit we get

$\displaystyle\lim_{R\to\infty}\frac{R\,\beta_{J,0}}{m}\rho_{\mathcal{O}^{(J)}}^{\mathcal{P},0}(Rm)$
$\displaystyle=c_{\mathcal{O}^{(J)}}\frac{2^{2-2\mathbf{\Delta}}\pi}{\Gamma(\mathbf{\Delta}-J)\Gamma(\mathbf{\Delta}+J)}m^{2(\mathbf{\Delta}-J-1)}\,,$
(5.87)
$\displaystyle\lim_{R\to\infty}\frac{R\,\beta_{J,1}}{m^{3}}\rho_{\mathcal{O}^{(J)}}^{\mathcal{P},1}(Rm)$
$\displaystyle=c_{\mathcal{O}^{(J)}}\frac{2^{2-2\mathbf{\Delta}}\pi}{\Gamma(\mathbf{\Delta}-J)\Gamma(\mathbf{\Delta}+J)}m^{2(\mathbf{\Delta}-J-1)}\,,$

where we have used $\beta_{J,0}=\beta_{J,1}=2^{1-J}$, which can be easily read
off from eq. (A.4). For the discrete series contribution, since
$R^{2(J-\mathbf{\Delta})}$ has to be inserted in
$\rho^{\mathcal{D}_{p}}_{{\cal O}^{(J)}}$ before taking $R\to\infty$, it can
survive in the flat space limit only if $\mathbf{\Delta}=J$, which itself
forces $p=J$.

In total we thus have that in 2 spacetime dimensions, the flat space spectral
densities of a CFT primary of spin $J$ and conformal dimension
$\mathbf{\Delta}$ are, for $\mathbf{\Delta}>J$,

$\displaystyle\rho^{\mathbb{M},0}_{\mathcal{O}^{(J)}}(m^{2})=\rho^{\mathbb{M},1}_{\mathcal{O}^{(J)}}(m^{2})=c_{\mathcal{O}^{(J)}}\frac{2^{2-2\mathbf{\Delta}}\pi}{\Gamma(\mathbf{\Delta}-J)\Gamma(\mathbf{\Delta}+J)}m^{2(\mathbf{\Delta}-J-1)}\,.$
(5.88)

and for the special case $\mathbf{\Delta}=J$,

$\rho^{\mathbb{M},0}_{\mathcal{O}^{(J)}}(m^{2})=\delta(m^{2})c_{\mathcal{O}^{(J)}}\frac{2^{3-2J}\pi}{\Gamma(2J)}\,,\qquad\rho^{\mathbb{M},1}_{\mathcal{O}^{(J)}}(m^{2})=0\,.$
(5.89)

###  5.3 Weakly coupled QFT

The Källén–Lehmann decomposition is a non-perturbative representation of two-
point functions in de Sitter. At the same time, it can be used to decompose
two-point functions order by order in a perturbative expansion, when the QFT
is weakly coupled. Since poles in the spectral densities can be related to the
conformal dimensions of boundary operators Hogervorst:2021uvp  (see discussion
in section 4.4), we will observe their position shifting as we turn on
interactions in the bulk. From this shift, we can read off anomalous
dimensions for the boundary operators. At the same time, the fact that we will
stop to a definite order in the coupling expansion, means we will lose the
positivity of the spectral densities as a side effect. This is just an
artifact of perturbation theory and does not mean that the theory is not
unitary. We will now review how to derive anomalous dimensions of boundary
operators from the spectral densities and then show a practical example in a
scalar weakly interacting theory.

Consider the Källén–Lehmann decomposition of a two-point function of a scalar
operator $\Phi$ (which for simplicity we take to include only principal series
contributions) in an interacting theory governed by a coupling $g$. It will
present simple poles on the nonperturbative value of the conformal dimension
of each boundary operator that appears in the bulk-boundary OPE of $\Phi$
Hogervorst:2021uvp . Near a pole at $\Delta_{*}(g)$, it will behave as

$\rho(\Delta,g)\approx\frac{\underset{\Delta=\Delta_{*}(g)}{\text{Res}}\left[\rho(\Delta,g)\right]}{\Delta-\Delta_{*}(g)}\,.$
(5.90)

For a weakly coupled theory, we can expand this formula in $g$ and obtain to
first order

$\displaystyle\rho(\Delta,g)$
$\displaystyle\approx\frac{\underset{\Delta=\Delta_{*}(0)}{\text{Res}}\left[\rho(\Delta,0)\right]}{\Delta-\Delta_{*}(0)}+g\left(\frac{\partial_{g}\underset{\Delta=\Delta_{*}(0)}{\text{Res}}\left[\rho(\Delta,0)\right]}{\Delta-\Delta_{*}(0)}+\frac{\underset{\Delta=\Delta_{*}(0)}{\text{Res}}\left[\rho(\Delta,0)\right]\partial_{g}\Delta_{*}(0)}{(\Delta-\Delta_{*}(0))^{2}}\right)+O(g^{2})$
$\displaystyle\equiv\frac{c_{0}}{\Delta-\Delta_{*}(0)}+g\left(\frac{c_{1}}{\Delta-\Delta_{*}(0)}+\frac{c_{2}}{(\Delta-\Delta_{*}(0))^{2}}\right)+O(g^{2})\,,$
(5.91)

where in the second line we simply set up notation. If we consider the series
expansion of $\Delta_{*}(g)$,

$\Delta_{*}(g)=\Delta_{*}(0)+g\partial_{g}\Delta_{*}(0)+O(g^{2})\,,$ (5.92)

we recognize that the anomalous dimension of the boundary operator with
$\Delta=\Delta_{*}(g)$ at order $g$ is given by

$\gamma_{*}=g\frac{c_{2}}{c_{0}}\,.$ (5.93)

Simply put, the anomalous shift in the dimension $\Delta_{*}(0)$ at first
order in $g$ is given by the ratio between the coefficient of the double pole
in $\rho(\Delta,g)$ at the position $\Delta=\Delta_{*}(0)$ appearing at order
$g$ and the coefficient of the simple pole at the same position but in the
free theory. Let us now consider a concrete example.

####  5.3.1 Anomalous dimensions from quartic interactions

Consider the following weakly coupled theory for a massive real scalar in de
Sitter

$\mathcal{L}=-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}m^{2}\phi^{2}-\frac{g}{4!}\phi^{4}\,,$
(5.94)

with $\Delta=\frac{d}{2}+i\lambda_{\phi}$ and $\lambda_{\phi}\in\mathbb{R}.$
We are going to be interested in this theory when the interaction is relevant
or marginal, so $d=1,2,3$. We compute the correction to the free two-point
function of the composite operator $\phi^{2}$ by using the in-in formalism,
which we review in Appendix I. We choose to consider the Wightman function

$\langle\Omega|\phi^{2}(Y_{1})\phi^{2}(Y_{2})|\Omega\rangle\,,$ (5.95)

with $Y_{1}$ and $Y_{2}$ in the Expanding Poincaré Patch, $|\Omega\rangle$ is
the interacting Bunch-Davies vacuum and, as we discussed in 4.1, we are
avoiding the branch cut by taking $\eta_{1}\rightarrow e^{i\epsilon}\eta_{1}$
and $\eta_{2}\rightarrow e^{-i\epsilon}\eta_{2}\,.$ In the notation from
DiPietro:2021sjt , which we are going to adopt for the rest of this
subsection, it means we are selecting $Y_{2}\in r$ and $Y_{1}\in l$ (see
Appendix I for more details). When the coupling $g$ is turned off, the two-
point function is given by the free theory contribution, which has the
Källén–Lehmann representation shown in 5.1.1, with
$\lambda_{1}=\lambda_{2}=\lambda_{\phi}$ and a factor of 2 accounting for
symmetry

$\rho^{\mathcal{P},0}_{\phi^{2},\text{free}}(\lambda)=\frac{\lambda\sinh(\pi\lambda)}{16\pi^{3+\frac{d}{2}}\Gamma(\frac{d}{2})\Gamma(\frac{d}{2}\pm
i\lambda)}\Gamma\left(\frac{\frac{d}{2}\pm
i\lambda}{2}\right)^{2}\prod_{\pm,\pm}\Gamma\left(\frac{\frac{d}{2}\pm
i\lambda\pm 2i\lambda_{\phi}}{2}\right)\,.$ (5.96)

Importantly, this spectral density has simple poles at

$\displaystyle\lambda$
$\displaystyle=2\lambda_{\phi}-i\left(\frac{d}{2}+2n\right)\longrightarrow\Delta=2\Delta_{\phi}+2n\,,$
(5.97) $\displaystyle\lambda$
$\displaystyle=-2\lambda_{\phi}-i\left(\frac{d}{2}+2n\right)\longrightarrow\Delta=2\bar{\Delta}_{\phi}+2n\,,$

due to the fact that boundary operators of the form
$[\mathcal{O}\mathcal{O}]_{n}$ and
$[\widetilde{\mathcal{O}}\widetilde{\mathcal{O}}]_{n}$252525We remind the
reader that $[\mathcal{O}_{1}\mathcal{O}_{2}]_{n}$ is a schematic notation to
indicate all the scalar double trace operators one can form with
$\mathcal{O}_{1}$, $\mathcal{O}_{2}$ and $2n$ derivatives, and
$\phi(\eta,\mathbf{x})\underset{\eta\to
0^{-}}{\sim}(-\eta)^{\Delta_{\phi}}\mathcal{O}(\mathbf{x})+(-\eta)^{\bar{\Delta}_{\phi}}\widetilde{\mathcal{O}}(\mathbf{x})$.
appear in the bulk-boundary OPE of $\phi^{2}$. We expect these operators to
inherit anomalous dimensions once we turn on interactions. At leading order in
the coupling, the two-point function is corrected by the diagram shown in
Figure 5.2, which following the in-in formalism, corresponds to the following
integrals

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free theory and order $g$ contributions to the Wightman two-point function
$\langle\phi^{2}\phi^{2}\rangle$.

$\langle\phi^{2}(Y_{1})\phi^{2}(Y_{2})\rangle^{lr}_{(g)}=ig\left[\int_{Y^{l}}(G^{ll}_{\lambda_{\phi}}(Y_{1},Y)\,G^{lr}_{\lambda_{\phi}}(Y,Y_{2}))^{2}-\int_{Y^{r}}(G^{lr}_{\lambda_{\phi}}(Y_{1},Y)\,G^{rr}_{\lambda_{\phi}}(Y,Y_{2}))^{2}\right]$
(5.98)

In Appendix I we analytically continue these integrals to EAdS and solve them.
We obtain that the order $g$ contribution to the two-point function has the
following spectral density

$\displaystyle\rho^{\mathcal{P},0}_{\phi^{2},g}(\lambda)=$ $\displaystyle
g\frac{\rho^{\mathcal{P},0}_{\phi^{2},\rm
free}(\lambda)}{4\sinh^{2}(\pi\lambda_{\phi})}\Bigg{[}\sin\left(\pi\left(\frac{d}{2}+2i\lambda_{\phi}\right)\right)B_{\Delta_{\phi},\Delta_{\phi}}(\lambda)$
(5.99)
$\displaystyle+\sin\left(\pi\left(\frac{d}{2}-2i\lambda_{\phi}\right)\right)B_{\bar{\Delta}_{\phi},\bar{\Delta}_{\phi}}(\lambda)-2\sin\left(\frac{d\pi}{2}\right)B_{\Delta_{\phi},\bar{\Delta}_{\phi}}(\lambda)\Bigg{]}\,,$

with $B_{\Delta_{1}\Delta_{2}}(\lambda)$ defined as an infinite series in
(I.21). This function is well-defined when $d<3$ and suffers from a UV
divergence when $d\geq 3$. In $d=3$, as discussed in appendix I.3, we can make
sense of $B_{\Delta_{1}\Delta_{2}}(\lambda)$ by dimensional regularization,
i.e. $d=3-\epsilon$, and absorbing the divergence into the wavefunction
renormalization of $\phi^{2}$. In the same appendix, we also show how to
extract the anomalous dimensions of $[\mathcal{O}\mathcal{O}]_{n}$ and
$[\widetilde{\mathcal{O}}\widetilde{\mathcal{O}}]_{n}$ from eq. (5.99),
following the prescription outlined above. More precisely, we did that for
$\lambda_{\phi}\in\mathbb{R}$ with the final expressions given by eq. (I.3):

$\displaystyle\gamma_{[\mathcal{O}\mathcal{O}]_{n}}$
$\displaystyle=-g\frac{(\frac{d}{2})_{n}\Gamma(\frac{1}{2}+n+i\lambda_{\phi})\Gamma(\frac{d}{2}+n+i\lambda_{\phi})\Gamma(\frac{d}{2}+n+2i\lambda_{\phi})\sin(\frac{\pi}{2}(d+4i\lambda_{\phi}))}{2^{d+3}\pi^{\frac{d}{2}}n!\sinh^{2}(\pi\lambda_{\phi})\Gamma(1+n+i\lambda_{\phi})\Gamma(\frac{d+1}{2}+n+i\lambda_{\phi})\Gamma(1+n+2i\lambda_{\phi})}\,,$
(5.100)
$\displaystyle\gamma_{[\widetilde{\mathcal{O}}\widetilde{\mathcal{O}}]_{n}}$
$\displaystyle=-g\frac{(\frac{d}{2})_{n}\Gamma(\frac{1}{2}+n-i\lambda_{\phi})\Gamma(\frac{d}{2}+n-i\lambda_{\phi})\Gamma(\frac{d}{2}+n-2i\lambda_{\phi})\sin(\frac{\pi}{2}(d-4i\lambda_{\phi}))}{2^{d+3}\pi^{\frac{d}{2}}n!\sinh^{2}(\pi\lambda_{\phi})\Gamma(1+n-i\lambda_{\phi})\Gamma(\frac{d+1}{2}+n-i\lambda_{\phi})\Gamma(1+n-2i\lambda_{\phi})}\,.$

For an elementary field that in the absence of interactions is in the
principal series ($\lambda_{\phi}\in\mathbb{R}$), these anomalous dimensions
are complex, satisfying
$(\gamma_{[\mathcal{O}\mathcal{O}]_{n}})^{*}=\gamma_{[\widetilde{\mathcal{O}}\widetilde{\mathcal{O}}]_{n}}$
and with a positive real part. The late time boundary operators associated to
$\phi^{2}$ will thus decay faster once interactions are turned on (assuming
that $g>0$, or in other words that the Hamiltonian is bounded from below). In
Arkani-Hamed:2015bza ; Marolf:2010zp ; Krotov:2010ma ; Bros:2009bz , a similar
phenomenon was observed for the boundary operators $\mathcal{O}$ and
$\widetilde{\mathcal{O}}$ themselves. In figure 5.3 we plot
$\gamma_{[\mathcal{O}\mathcal{O}]_{n}}$ and
$\gamma_{[\widetilde{\mathcal{O}}\widetilde{\mathcal{O}}]_{n}}$ for
$\lambda_{\phi}\in(0,10)$ in $d=3$ and with $n=0$.

Figure 5.3: Anomalous dimensions of the boundary operators
$\widetilde{\mathcal{O}}^{2}$ and $\mathcal{O}^{2}$ in $\frac{g}{4!}\phi^{4}$
theory in dS4 for a field $\phi$ which, in the absence of interactions, has
$\Delta_{\phi}=\frac{d}{2}+i\lambda_{\phi}$ in the principal series.
$\mathcal{O}$ and $\widetilde{\mathcal{O}}$ are the leading boundary operators
in the BOE of $\phi$, and in the absence of interactions they transform as CFT
scalar primaries with conformal dimensions $\Delta_{\phi}$ and
$d-\Delta_{\phi}$ respectively. Notice that
$(\gamma_{\mathcal{O}^{2}})^{*}=\gamma_{\widetilde{\mathcal{O}}^{2}}$. The
divergence at $\lambda_{\phi}=0$ is due to the mixing of $\mathcal{O}^{2}$ and
$\widetilde{\mathcal{O}}^{2}$ in that degenerate case.

If we naively continue (5.100) to imaginary values of $\lambda_{\phi}$ to
study the case in which $\phi$ is in the complementary series, we can match
with known results in the literature Premkumar:2021mlz  on the anomalous
dimension of $[\widetilde{{\cal O}}\widetilde{{\cal O}}]_{0}$ in dS4, as shown
in Figure 5.4. We believe this analytic continuation should be done with care,
since many of the steps in Appendix I.3 do not trivially generalize to the
complementary series, but simply making $\lambda_{\phi}$ imaginary seems to
work, at least in this case.

Figure 5.4: Anomalous dimensions of $\widetilde{\mathcal{O}}^{2}$ in
$\frac{g}{4!}\phi^{4}$ theory in dS4, where $\widetilde{\mathcal{O}}$ is the
slowest decaying boundary operator associated to $\phi$ and $\phi$ is a field
that, in the absence of interactions, has
$\Delta_{\phi}=\frac{3}{2}+i\lambda_{\phi}$ in the complementary series. The
dots are from Figure 8 of Premkumar:2021mlz .

One can further compare (5.100) with the anomalous dimensions of the
corresponding boundary operators in AdS in a quartic theory Heemskerk:2009pn ;
Fitzpatrick:2010zm ; gravina ; Levine:2023ywq :

$\gamma^{\text{dS}}_{[\mathcal{O}\mathcal{O}]_{n}}=-\frac{1}{2}\text{csch}^{2}(\pi\lambda_{\phi})\sin\left(\pi\left(\frac{d}{2}+2i\lambda_{\phi}\right)\right)\gamma^{\text{AdS}}_{[\mathcal{O}\mathcal{O}]_{n}}\,.$
(5.101)

The trigonometric factors appearing in front have two separate origins. The
hyperbolic cosecant factor comes from the different normalizations for bulk-
to-boundary propagators in dS and AdS, while the sine factor originates from
the interference of the two branches of the in-in contour. Interestingly, its
role is to cancel unphysical singularities that are otherwise present in the
AdS result when analytically continued to the complementary series (which is
partly outside of the unitarity bounds in AdS). These facts were already
pointed out in Sleight:2021plv ; Sleight_2021Exch .

The anomalous dimensions of $\widetilde{\mathcal{O}}^{2}$ in dS diverge as one
approaches the two endpoints of the complementary series. The divergence as
$i\lambda_{\phi}\rightarrow\frac{d}{2}$ is a symptom of the breaking down of
perturbation theory due to the IR divergences associated to massless fields.
The divergence as $i\lambda_{\phi}\rightarrow 0$ instead corresponds to the
degeneracy between the boundary operators $\mathcal{O}^{2}$ and
$\widetilde{\mathcal{O}}^{2}$ when $\Delta_{\phi}=\frac{d}{2}\,.$

##  6 Outlook

In this paper we have derived the Källén-Lehmann decomposition for spinning
traceless symmetric bulk operators in dS (see section 3), and applied it to
many examples in section 5. Here we outline potential future applications that
we can imagine for this technology.

  * •

In the context of bootstrapping QFT in de Sitter baumann2022snowmass , it may
be useful to study a mixed system involving boundary and bulk correlation
functions analogous to what was recently done in flat space Karateev:2019ymz ;
Correia:2022dyp  and AdS meineri2023renormalization . In particular, it should
be possible to generalize Cardy:1988tj ; Zamolodchikov:1986gt  and derive a
sum rule for the spectral density of the trace of the stress tensor that gives
the central charge of the UV CFT in a two dimensional de Sitter background.

  * •

In this work, we mostly focused on the contributions of principal and
complementary series UIRs (except in dS2 where we also considered the discrete
series systematically). It would be interesting to study the effect of the
type ii@ exceptional series UIRs, which include photons and gravitons in
$d>2$. For example, in upcoming work loparcocong  we show that computing the
leading order correction to the two-point function of the conserved current in
scalar QED in de Sitter leads to the appearance of a photon in the Källén-
Lehmann decomposition.

Understanding the Ward identities of the boundary operators associated to
photons and gravitons is the natural next step towards understanding quantum
gravity in de Sitter.

  * •

It would be interesting to establish the convergence properties of the
Boundary Operator Expansion (BOE) non-perturbatively for QFT in dS. It is
likely that the connection between boundary operators and quasi-normal modes
of the static patch of de Sitter Ng:2012xp ; Jafferis:2013qia ; Sun:2020sgn ;
Anninos_2010 ; Anninos_2012 ; Anninos_2022  will be useful in this context.

## Acknowledgements

We are grateful to Tarek Anous, Miguel Correia, Frederik Denef, Victor
Gorbenko, Aditya Hebbar, Matthijs Hogervorst, Austin Joyce, Shota Komatsu,
Fedor Popov, Akhil Premkumar and Jiaxin Qiao for useful discussions.

JP and ML are supported by the Simons Foundation grant 488649 (Simons
Collaboration on the Nonperturbative Bootstrap) and the Swiss National Science
Foundation through the project 200020_197160 and through the National Centre
of Competence in Research SwissMAP. ZS is supported by the US National Science
Foundation under Grant No. PHY-2209997 and the Gravity Initiative at Princeton
University.

##  Appendix A Various properties of Green’s functions in de Sitter

###  A.1 Canonical quantization of a free scalar

Let $\Phi$ be a free massive scalar of mass $m\geq\frac{d}{2}$ in dSd+1. We
parametrize the mass by $m^{2}=\frac{d^{2}}{4}+\lambda^{2}$, with
$\lambda\in\mathbb{R}$ (principal series) or
$-\frac{d}{2}<i\lambda<\frac{d}{2}$ (complementary series). The standard bulk
mode expansion of $\phi$ in planar coordinates is

$\displaystyle\Phi(\eta,\mathbf{y}\,)=\int\frac{d^{d}\mathbf{y}}{(2\pi)^{\frac{d}{2}}}\left(a_{\mathbf{k}}\,\phi_{\mathbf{k}}(\eta)e^{i\mathbf{k}\cdot\mathbf{y}}+a^{\dagger}_{\mathbf{k}}\,\phi_{\mathbf{k}}(\eta)^{*}e^{-i\mathbf{k}\cdot\mathbf{y}}\right)~{},$
(A.1)

where $\phi_{\mathbf{k}}(\eta)$ satisfies the equations of motion

$\displaystyle\left((-\eta)^{d+1}\partial_{\eta}(-\eta)^{1-d}\partial_{\eta}+|\mathbf{k}|^{2}\eta^{2}+m^{2}\right)\phi_{\mathbf{k}}(\eta)=0~{}.$
(A.2)

and the Klein-Gordon normalization condition (needed to ensure the canonical
commutation relation between $\Phi$ and
$\Pi_{\Phi}=(-\eta)^{1-d}\partial_{\eta}\Phi$)

$\displaystyle
i(-\eta)^{1-d}\left(\phi^{*}_{\mathbf{k}}\partial_{\eta}\phi_{\mathbf{k}}-\partial_{\eta}\phi^{*}_{\mathbf{k}}\phi_{\mathbf{k}}\right)=1$
(A.3)

This is solved by

$\displaystyle\phi_{\mathbf{k}}(\eta)=(-\eta)^{\frac{d}{2}}\bar{h}_{i\lambda}(|\mathbf{k}|\eta),\,\,\,\,\,\phi_{\mathbf{k}}(\eta)^{*}=(-\eta)^{\frac{d}{2}}h_{i\lambda}(|\mathbf{k}|\eta)~{},$
(A.4)

where

$\displaystyle
h_{i\lambda}(\xi)=\frac{\sqrt{\pi}}{2}e^{\frac{\pi\lambda}{2}}H^{(2)}_{i\lambda}(-\xi),\,\,\,\,\,\bar{h}_{i\lambda}(\xi)=\frac{\sqrt{\pi}}{2}e^{-\frac{\pi\lambda}{2}}H^{(1)}_{i\lambda}(-\xi)~{}.$
(A.5)

The functions $h_{i\lambda}$ and $\bar{h}_{i\lambda}$ are invariant under
$\lambda\leftrightarrow-\lambda$. This is consistent with the fact that $m$ is
independent of the sign of $\lambda$. For light scalars, i.e.
$0<m<\frac{d}{2}$, $\lambda$ is purely imaginary, and
$e^{\pm\frac{\pi\lambda}{2}}$ becomes a phase.

The solution of eq. (A.2) and eq. (A.3) is not unique. Different linear
combinations of the Hankel functions also satisfy the equation of motion and
the Klein-Gordon normalization condition. This is related to the usual
ambiguity of choosing the vacuum in a curved spacetime. What singles out the
above choice is the early time $\eta\to-\infty$ asymptotic behavior

$\displaystyle\phi_{\mathbf{k}}(\eta)\approx
e^{-i\frac{\pi}{4}}(-\eta)^{\frac{d-1}{2}}\frac{e^{-i|\mathbf{k}|\eta}}{\sqrt{2|\mathbf{k}|}}~{}.$
(A.6)

It means that at early time, the corresponding choice of vacuum looks like the
canonical Minkowski vacuum. The vacuum selected in this way is called the
Bunch-Davies vacuum.

Given the mode functions (A.4), the Wightman two-point function of $\Phi$ in
the Bunch-Davies vacuum can be expressed as

$\displaystyle G_{\lambda,0}(\eta_{1},y_{1};\eta_{2},\mathbf{y}_{2}\,)$
$\displaystyle\equiv\langle\Omega|\Phi(\eta_{1},\mathbf{y}_{1}\,)\Phi(\eta_{2},\mathbf{y}_{2}\,)|\Omega\rangle$
$\displaystyle=(\eta_{1}\eta_{2})^{\frac{d}{2}}\int\frac{d^{d}\mathbf{k}}{(2\pi)^{d}}e^{-i\mathbf{k}\cdot(\mathbf{y}_{1}-\mathbf{y}_{2}\,)}\bar{h}_{i\lambda}(|\mathbf{k}|\eta_{1})h_{i\lambda}(|\mathbf{k}|\eta_{2})~{},$
(A.7)

Evaluating the Fourier transformation in eq. (A.1) yields the hypergeometric
representation of $G_{\lambda,0}$:

$G_{\lambda,0}(\eta_{1},\mathbf{y}_{1};\eta_{2},\mathbf{y}_{2}\,)=\frac{\Gamma(\Delta)\Gamma(\bar{\Delta})}{(4\pi)^{\frac{d+1}{2}}}\mathbf{F}\left(\Delta,\bar{\Delta},\frac{d+1}{2},\frac{1+\sigma}{2}\right)~{}.$
(A.8)

where $\Delta=\frac{1}{2}+i\lambda$, $\sigma$ is the chordal distance between
$(\eta_{1},\mathbf{y}_{1})$ and $(\eta_{2},\mathbf{y}_{2})$

$\sigma=Y_{1}\cdot
Y_{2}=\frac{\eta_{1}^{2}+\eta_{2}^{2}-\mathbf{y}_{12}^{\,2}}{2\eta_{1}\eta_{2}},\,\,\,\,\,\mathbf{y}_{12}=\mathbf{y}_{1}-\mathbf{y}_{2}~{},$
(A.9)

and $\mathbf{F}(a,b,c,z)\equiv\frac{1}{\Gamma(c)}F(a,b,c,z)$ is the
regularized hypergeometric function.

###  A.2 Proca fields in dS2

In dS2, the mode expansion of a Proca field $A_{\mu}$ is closely related to
that of $\Phi$. More precisely, consider the Proca Lagrangian

$\displaystyle\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}m^{2}A_{\mu}A^{\mu}~{},$
(A.10)

where $m^{2}=\frac{1}{4}+\lambda^{2}$. In planar coordinates, the equation of
motion of $A_{\mu}$ is satisfied by

$\displaystyle
A_{\eta}=\alpha\partial_{y}\Phi,\,\,\,\,\,A_{y}=\alpha\partial_{\eta}\Phi~{},$
(A.11)

where $\alpha$ is a normalization constant and $\Phi$ is a canonically
normalized scalar field of the same mass $m$. We fix $\alpha$ by requiring
$A_{y}$ and its conjugate momentum $\Pi_{y}$ satisfy the standard commutation
relation. The canonical momentum $\Pi_{y}$ is given by

$\displaystyle\Pi_{y}=\eta^{2}F_{\eta
y}=\alpha\,\eta^{2}(\partial^{2}_{\eta}-\partial_{y}^{2})\Phi=-\alpha
m^{2}\Phi~{},$ (A.12)

where in the last step we have used the equation of motion of $\Phi$. Using
the fact that $\partial_{\eta}\Phi=\Pi_{\Phi}$ is the canonical momentum of
$\Phi$, we get

$\displaystyle[\Pi_{y},A_{y}]=-\alpha^{2}m^{2}[\Phi,\Pi_{\Phi}]=-i\alpha^{2}m^{2}~{},$
(A.13)

which implies $\alpha^{2}=\frac{1}{m^{2}}$ for canonical quantization. Thus
the Green’s function of $A_{\mu}$ can be summarized as

$\displaystyle
d=1:\,\,\,\langle\Omega|A_{\mu}(\eta_{1},y_{1})A_{\nu}(\eta_{2},y_{2})|\Omega\rangle=\frac{1}{m^{2}}\,\epsilon_{\mu}^{\,\,\,\,\alpha}\partial_{y_{1}^{\alpha}}\epsilon_{\nu}^{\,\,\,\,\beta}\partial_{y_{2}^{\beta}}G_{\lambda,0}(\sigma)~{},$
(A.14)

where $\epsilon_{\mu\alpha}$ and $\epsilon_{\nu\beta}$ are totally
antisymmetric tensors at $(\eta_{1},y_{1})$ and $(\eta_{2},y_{2})$
respectively. To write the two-point function
$\langle\Omega|A_{\mu}(\eta_{1},y_{1})A_{\nu}(\eta_{2},y_{2})|\Omega\rangle$
in terms of embedding space coordinates, we need the following relation

$\displaystyle\epsilon_{ABC}\frac{\partial Y^{A}}{\partial
y^{\mu}}\frac{\partial Y^{B}}{\partial y^{\nu}}Y^{C}=-\epsilon_{\mu\nu}~{},$
(A.15)

which can be directly checked in any local coordinates $y^{\mu}$. It implies
that the embedding space counterpart of $\epsilon_{\mu\nu}$ is
$-\epsilon_{ABC}Y^{C}$. Then the uplift of
$\epsilon_{\mu}^{\,\,\,\,\alpha}\partial_{y^{\alpha}}$ to embedding space
should be

$\displaystyle\epsilon_{\mu}^{\,\,\,\,\alpha}\partial_{y^{\alpha}}\Longrightarrow-\epsilon_{ABC}Y^{C}(\partial_{Y^{B}}-Y_{B}\,Y\cdot\partial_{Y})=\epsilon_{ABC}Y^{B}\partial_{Y^{C}}~{}.$
(A.16)

Altogether, the two-point function $G_{\lambda,1}(Y_{1},Y_{2};W_{1},W_{2})$ of
$A(Y,W)$ is

$\displaystyle G_{\lambda,1}(Y_{1},Y_{2};W_{1},W_{2})$
$\displaystyle\equiv\langle\Omega|A(Y_{1},W_{1})A(Y_{2},W_{2})|\Omega\rangle$
$\displaystyle=\left(\frac{1}{4}+\lambda^{2}\right)^{-1}\epsilon\left(W_{1},Y_{1},\partial_{Y_{1}}\right)\epsilon\left(W_{2},Y_{2},\partial_{Y_{2}}\right)G_{\lambda,0}(Y_{1},Y_{2})~{},$
(A.17)

where
$\epsilon(U_{1},U_{2},U_{3})\equiv\epsilon_{ABC}U_{1}^{A}U_{2}^{B}U_{3}^{C}$.

###  A.3 Analytical continuation of $G_{\lambda,0}$ in dS2

In dS2, the Green’s function $G_{\lambda,0}$ in eq. (A.8) becomes divergent
when $\Delta\equiv\frac{1}{2}+i\lambda=p\in\mathbb{Z}_{+}$. Such divergences
can be removed by acting with derivatives. More precisely, let’s first apply
the series expansion of hypergeometric functions to $G_{\lambda,0}$

$\displaystyle G_{\lambda,0}(\sigma)=\sum_{n\geq
0}\frac{\Gamma(\Delta+n)\Gamma(\bar{\Delta}+n)}{(n!)^{2}}\left(\frac{1+\sigma}{2}\right)^{n}~{}.$
(A.18)

Then we take the limit $\Delta\to p$. The problematic terms in this limit
correspond to $n\leq p-1$, which means that the divergent part is a polynomial
in $\sigma=Y_{1}\cdot Y_{2}$ of degree $p-1$. This polynomial is obviously
annihilated by the differential operator
$(W_{1}\cdot\nabla_{1})^{p}(W_{2}\cdot\nabla_{2})^{p}$. Therefore, the
following function has a well-defined $\Delta\to p$ limit

$\displaystyle\Psi_{p,\Delta}(Y_{1},Y_{2};W_{1},W_{2})\equiv(W_{1}\cdot\nabla_{1})^{p}(W_{2}\cdot\nabla_{2})^{p}\,G_{-i(\Delta-\frac{1}{2}),0}(\sigma)~{}.$
(A.19)

In the remaining part of the section, we are going to compute
$\Psi_{p,q}\equiv\lim_{\Delta\to q}\Psi_{p,\Delta}$ for any $1\leq q\leq p$.
As discussed in section 2.2.2, it amounts to computing the matrix

$\displaystyle\begin{pmatrix}\Psi_{p,q}(Y_{1},Y_{2};W^{+}_{1},W^{+}_{2})&\Psi_{p,q}(Y_{1},Y_{2};W^{+}_{1},W^{-}_{2})\\\
\Psi_{p,q}(Y_{1},Y_{2};W^{-}_{1},W^{+}_{2})&\Psi_{p,q}(Y_{1},Y_{2};W^{-}_{1},W^{-}_{2})\end{pmatrix}~{},$
(A.20)

where $W^{\pm}$ are defined by eq. (2.31) to encode the two chiral components
of a symmetric and traceless tensor in dS2. Let’s start with the diagonal
entries $\Psi_{p,\Delta}(Y_{1},Y_{2};W^{\pm}_{1},W^{\pm}_{2})$

$\displaystyle\Psi_{p,\Delta}(Y_{1},Y_{2};W^{\pm}_{1},W^{\pm}_{2})$
$\displaystyle=\sum_{n=0}^{p}\binom{p}{n}(W^{\pm}_{1}\cdot\nabla_{1})^{p-n}(W^{\pm}_{2}\cdot
Y_{1})^{p}(W^{\pm}_{1}\cdot\nabla_{1})^{n}\partial_{\sigma}^{p}G_{-i(\Delta-\frac{1}{2}),0}(\sigma)$
$\displaystyle=\sum_{n=0}^{p}\binom{p}{n}\frac{p!}{n!}(W_{1}^{\pm}\\!\cdot\\!W_{2}^{\pm})^{p-n}(W_{1}^{\pm}\\!\cdot\\!Y_{2})^{n}(W_{2}^{\pm}\\!\cdot\\!Y_{1})^{n}\partial_{\sigma}^{p+n}G_{-i(\Delta-\frac{1}{2}),0}(\sigma)~{}.$
(A.21)

Using the relation $(W_{1}^{\pm}\cdot Y_{2})(W_{2}^{\pm}\cdot
Y_{1})=(\sigma+1)W_{1}^{\pm}\cdot W_{2}^{\pm}$ established in eq. (2.2.2), we
get

$\displaystyle\Psi_{p,\Delta}(Y_{1},Y_{2};W^{\pm}_{1},W^{\pm}_{2})$
$\displaystyle=(W_{1}^{\pm}\cdot
W_{2}^{\pm})^{p}\sum_{n=0}^{p}\binom{p}{n}\frac{p!}{n!}(\sigma+1)^{n}\partial_{\sigma}^{J+n}G_{-i(\Delta-\frac{1}{2}),0}(\sigma)$
$\displaystyle=\frac{\Gamma(\Delta)\Gamma(\bar{\Delta})}{4\pi}(W_{1}^{\pm}\cdot
W_{2}^{\pm})^{p}\partial_{\sigma}^{p}((\sigma+1)^{p}\partial_{\sigma}^{p})F\left(\Delta,\bar{\Delta},1,\frac{1+\sigma}{2}\right)~{},$
(A.22)

where the $2p$-th order differential operator
$\partial_{\sigma}^{p}((\sigma+1)^{p}\partial_{\sigma}^{p})$ acting on
$F\left(\Delta,\bar{\Delta},1,\frac{1+\sigma}{2}\right)$ yields another
hypergeometric function:

$\displaystyle\Psi_{p,\Delta}(Y_{1},Y_{2};W^{\pm}_{1},W^{\pm}_{2})=\frac{\Gamma(\Delta+p)\Gamma(\bar{\Delta}+p)}{2^{p+2}\pi}(W_{1}^{\pm}\cdot
W_{2}^{\pm})^{p}F\left(\Delta+p,\bar{\Delta}+p,1,\frac{1+\sigma}{2}\right)~{}.$
(A.23)

It clearly has a finite limit when $\Delta$ hits any positive integer $q$ that
is not larger than $p$:

$\displaystyle\Psi_{p,q}(Y_{1},Y_{2};W^{\pm}_{1},W^{\pm}_{2})=(W_{1}^{\pm}\cdot
W_{2}^{\pm})^{p}\,\psi_{p,q}(Y_{1},Y_{2})~{},$ (A.24)

where

$\displaystyle\psi_{p,q}(Y_{1},Y_{2})=\frac{\Gamma(p+q)\Gamma(p+1-q)}{2^{p+2}\pi}F\left(p+q,p+1-q,1,\frac{1+Y_{1}\cdot
Y_{2}}{2}\right)~{}.$ (A.25)

The off-diagonal matrix elements in (A.20) can be computed similarly

$\displaystyle\Psi_{p,\Delta}(Y_{1},Y_{2};W^{\pm}_{1},W^{\mp}_{2})$
$\displaystyle=\frac{\Gamma(\Delta)\Gamma(\bar{\Delta})}{4\pi}(W_{1}^{\pm}\cdot
W_{2}^{\mp})^{p}\partial_{\sigma}^{p}((\sigma-1)^{p}\partial_{\sigma}^{p})F\left(\Delta,\bar{\Delta},1,\frac{1+\sigma}{2}\right)$
$\displaystyle=\frac{\Gamma(\Delta\\!+\\!p)^{2}\Gamma(\bar{\Delta}\\!+\\!p)^{2}(W_{1}^{\pm}\\!\cdot\\!W_{2}^{\mp})^{p}}{(-2)^{p+2}(2p)!\pi\Gamma(\Delta)\Gamma(\bar{\Delta})}F\left(\Delta\\!+\\!p,\bar{\Delta}\\!+\\!p,2p\\!+\\!1,\frac{1\\!+\\!\sigma}{2}\right)~{}.$
(A.26)

They vanish when $\Delta$ approaches at positive integer
$q\in\\{1,2,\cdots,p\\}$, because $\lim_{\Delta\to q}(\bar{\Delta})_{p}=0$.
Altogether, we have

$\displaystyle(W^{\pm}_{1}\cdot\nabla_{1})^{p}(W^{\pm}_{2}\cdot\nabla_{2})^{p}\,G_{-i(q-\frac{1}{2}),0}(Y_{1},Y_{2})=(W_{1}^{\pm}\cdot
W_{2}^{\pm})^{p}\,\psi_{p,q}(Y_{1},Y_{2})~{},$
$\displaystyle(W^{\pm}_{1}\cdot\nabla_{1})^{p}(W^{\mp}_{2}\cdot\nabla_{2})^{p}\,G_{-i(q-\frac{1}{2}),0}(Y_{1},Y_{2})=0~{}.$
(A.27)

When $q=p$, $\psi_{p,p}(Y_{1},Y_{2})$ becomes particularly simple

$\displaystyle\psi_{p,p}(Y_{1},Y_{2})=\frac{2^{p-2}\Gamma(2p)}{\pi(1-Y_{1}\cdot
Y_{2})^{2p}}~{}.$ (A.28)

To end this section, we show the pull-back of
$\Psi_{p,p}(Y_{1},Y_{2};W_{1},W_{2})$ to conformal global coordinates.
According to the discussion in section 2.2.2, it amounts to replacing
$W^{A}_{\pm}$ by $\partial_{\pm}Y^{A}$ (where $\partial_{\pm}$ denotes the
ordinary derivative with respect to the local lightcone coordinates
$y^{\pm}=\tau\pm\varphi$), for example,

$\displaystyle\left(\nabla_{+}^{(1)}\right)^{p}\left(\nabla_{+}^{(2)}\right)^{p}G_{-i(p-\frac{1}{2})}(Y_{1},Y_{2})=\Psi_{p,p}(Y_{1},Y_{2};\partial_{+}Y_{1},\partial_{+}Y_{2})~{},$
$\displaystyle=\frac{2^{p-2}\Gamma(2p)}{\pi}\left(\frac{\partial_{y_{1}^{+}}\partial_{y_{2}^{+}}\sigma}{(1-\sigma)^{2}}\right)^{p}=\frac{\Gamma(2p)}{4\pi\left(-4\sin^{2}\frac{y^{+}_{12}}{2}\right)^{p}}~{},$
(A.29)

where $\nabla_{\pm}$ denotes covariant derivative along $y^{\pm}$. Similarly,
the other rest components are

$\displaystyle\left(\nabla_{-}^{(1)}\right)^{p}\left(\nabla_{-}^{(2)}\right)^{p}G_{-i(p-\frac{1}{2})}(Y_{1},Y_{2})=\frac{\Gamma(2p)}{4\pi\left(-4\sin^{2}\frac{y^{-}_{12}}{2}\right)^{p}}~{},$
$\displaystyle\left(\nabla_{+}^{(1)}\right)^{p}\left(\nabla_{-}^{(2)}\right)^{p}G_{-i(p-\frac{1}{2})}(Y_{1},Y_{2})=\left(\nabla_{-}^{(1)}\right)^{p}\left(\nabla_{+}^{(2)}\right)^{p}G_{-i(p+\frac{1}{2})}(Y_{1},Y_{2})=0~{}.$
(A.30)

###  A.4 Flat space limit of $G_{\lambda,\ell}$

In this subsection, we elaborate on the discussion in section 3.3 and show
that the canonically normalized de Sitter Green’s functions
$[(W_{1}\cdot\nabla_{1})(W_{2}\cdot\nabla_{2})]^{J-\ell}G_{\lambda,\ell}(Y_{1},Y_{2};W_{1},W_{2})$
reduce to the flat space Wightman functions
$m^{-2\ell}\Delta_{m^{2},\ell}^{(J)}(x_{1},x_{2};w_{1},w_{2})$ (which are
introduced in Karateev:2020axc  for $0\leq\ell\leq J\leq 2$), up to numerical
normalization factors. For example, in the $J=0$ case, we expect the flat
space limit of the scalar Green’s function $G_{\lambda,0}$ in dS to reproduce
$\Delta_{m^{2},0}^{(0)}$, which is given by Karateev:2020axc

$\Delta_{m^{2},0}^{(0)}(x_{1},x_{2})=\frac{1}{(2\pi)^{\frac{d+1}{2}}}m^{d-1}\frac{K_{\frac{d-1}{2}}\left(m\sqrt{x_{12}^{2}}\right)}{\left(m\sqrt{x_{12}^{2}}\right)^{\frac{d-1}{2}}}\,,$
(A.31)

where $x_{1}^{\mu},x_{2}^{\mu}$ are flat space coordinates of
$\mathbb{R}^{d,1}$, and $i\epsilon$ is suppressed in $\sqrt{x_{12}^{2}}$.

Let’s start with discussing the flat space limit of the coordinates and the
metric of dS. The dS metric in planar coordinate is
$ds^{2}=R^{2}\frac{-d\eta^{2}+d\mathbf{y}^{\,2}}{\eta^{2}}$, where the dS
radius $R$ is restored. We consider the large $R$ limit, with $R(\eta+1)\equiv
x^{0}$ and $R\,y^{i}\equiv x^{i}$ being fixed. Then the dS metric reduces to
the flat space metric $ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}$. It is also
useful to show how $\sigma=Y_{1}\cdot Y_{2}$ is related to the flat space
distance $x_{12}^{2}$ in this limit. For this purpose, we define a new
variable

$\displaystyle\rho\equiv\frac{1}{2}\left(1-\frac{\sigma}{R^{2}}\right)=\frac{-\eta_{12}^{2}+\mathbf{y}_{12}^{\,2}}{4\eta_{1}\eta_{2}}=\frac{-(x^{0}_{12})^{2}+\mathbf{x}_{12}^{\,2}}{4\eta_{1}\eta_{2}R^{2}}~{}.$
(A.32)

In the flat space limit, we can simply replace $4\eta_{1}\eta_{2}$ by $4$, and
hence obtain $\rho\approx\frac{x_{12}^{2}}{4R^{2}}$. In other words, $\rho\to
0$ in the flat space limit, but $4\rho R^{2}$ is finite and equal to the flat
space distance $x_{12}^{2}$. Using the large $R$ relation $\lambda=mR$, we can
also write $4\rho\lambda^{2}\approx m^{2}x_{12}^{2}$.

Next, we consider the de Sitter scalar propagator

$G_{\lambda,0}(\sigma)=\frac{\Gamma(\frac{d}{2}\pm
i\lambda)}{2^{d+1}\pi^{\frac{d+1}{2}}R^{d-1}}\mathbf{F}\left(\frac{d}{2}-i\lambda,\frac{d}{2}+i\lambda,\frac{d+1}{2},1-\rho\right)\,,$
(A.33)

which is expressed in terms of the new variable $\rho$. To retrieve the flat
space propagator we will thus have to take $\lambda^{2}\to\infty$ and $\rho\to
0$ while keeping the product fixed. This limit can be easily implemented if we
rewrite $G_{\lambda,0}$ by using the following Mellin representation of the
hypergeometric function

$\displaystyle\mathbf{F}(a,b,c,z)=\frac{\int_{\mathbb{R}+i\epsilon}\,dt\,\Gamma(a+it)\Gamma(b+it)\Gamma(c-a-
b-it)\Gamma(-it)(1-z)^{it}}{2\pi\Gamma(a)\Gamma(b)\Gamma(c-a)\Gamma(c-b)}~{}.$
(A.34)

which yields

$\displaystyle
G_{\lambda,0}(\sigma)=\frac{\int_{\mathbb{R}+i\epsilon}\,dt\,\Gamma(\frac{d}{2}+it\pm
i\lambda)\Gamma(\frac{1-d}{2}-it)\Gamma(-it)\rho^{it}}{2^{d+2}\pi^{\frac{d+3}{2}}\Gamma(\frac{1}{2}\pm
i\lambda)R^{d-1}}~{}.$ (A.35)

Taking the large $\lambda$ limit in eq. (A.35) :

$\displaystyle\frac{1}{\Gamma(\frac{1}{2}\pm
i\lambda)}\approx\frac{1}{2\pi}\,e^{\pi\lambda},\,\,\,\,\,\Gamma\left(\frac{d}{2}+it\pm
i\lambda\right)\approx 2\pi e^{-\pi\lambda}\lambda^{d-1+2it}~{},$ (A.36)

it is then easy to see that $G_{\lambda,0}(\sigma)$ becomes

$\displaystyle
G_{\lambda,0}(\sigma)\approx\frac{m^{d-1}}{2^{d+2}\pi^{\frac{d+3}{2}}}\int_{\mathbb{R}+i\epsilon}\,dt\,\Gamma\left(\frac{1-d}{2}-it\right)\Gamma\left(-it\right)(\lambda^{2}\rho)^{it}~{},$
(A.37)

where we have used $\lambda/R=m$. The remaining integral can be evaluated by
using the integral representation of $\Gamma$ functions

$\displaystyle
G_{\lambda,0}(\sigma)\approx\frac{m^{d-1}}{2^{d+2}\pi^{\frac{d+3}{2}}}\int_{0}^{\infty}\frac{du}{u}\,u^{\frac{1-d}{2}}\,e^{-u}\int_{0}^{\infty}\frac{dv}{v}\,e^{-v}\int_{\mathbb{R}}dt\,\left(\frac{\lambda^{2}\rho}{uv}\right)^{it}~{}.$
(A.38)

Performing the $t$ integral gives a delta function supported on
$uv=\lambda^{2}\rho$, so (A.38) reduces to a familiar form

$\displaystyle
G_{\lambda,0}(\sigma)\approx\frac{m^{d-1}}{2^{d+1}\pi^{\frac{d+1}{2}}}\int_{0}^{\infty}\frac{du}{u}\,u^{\frac{1-d}{2}}\,e^{-u-\frac{\lambda^{2}\rho}{u}}=\frac{m^{d-1}}{(2\pi)^{\frac{d+1}{2}}}\frac{K_{\frac{d-1}{2}}\left(2\lambda\sqrt{\rho}\right)}{(2\lambda\sqrt{\rho})^{\frac{d-1}{2}}}~{}.$
(A.39)

This is exactly (A.31) after using the relation $4\rho\lambda^{2}\approx
m^{2}x^{2}_{12}$. We want to mention that a EAdS version of this story was
discussed in Carmi:2018qzm .

The second example is $J=\ell=1$. On the flat space side, we have
Karateev:2020axc

$m^{-2}\Delta_{m^{2},1}^{(1)}(x_{1},x_{2};w_{1},w_{2})=(w_{1}\cdot
w_{2}+m^{-2}(w_{1}\cdot\partial_{1})(w_{2}\cdot\partial_{2}))\Delta_{m^{2},0}^{(0)}(x_{1},x_{2})$
(A.40)

where $w_{1}^{\mu},w_{2}^{\mu}$ are auxiliary null vectors. On the dS side,
the spin 1 Green’s function $G_{\lambda,1}$ can be obtained by evaluating the
split representation (3.26) for $\ell=1$ 262626One can check that when $d=1$,
eq. (A.41) is actually equivalent to eq.(A.2) by using the equation of motion
$(1-\sigma^{2})\partial_{\sigma}^{2}G_{\lambda,0}-2\sigma\partial_{\sigma}G_{\lambda,0}=(\frac{1}{4}+\lambda^{2})G_{\lambda,0}$.:

$\displaystyle
G_{\lambda,1}(Y_{1},Y_{2};W_{1},W_{2})=\frac{\left[\left(\frac{d^{2}}{4}\\!+\\!\lambda^{2}\right)(W_{1}\cdot
W_{2})\\!+\\!\sigma(W_{1}\cdot\partial_{Y_{1}})(W_{2}\cdot\partial_{Y_{2}})\\!+\\!d(W_{1}\cdot
Y_{2})(W_{2}\cdot\partial_{Y_{2}})\right]G_{\lambda,0}(\sigma)}{\frac{(d-2)^{2}}{4}+\lambda^{2}}$
(A.41)

where the $R$ dependence is implicitly restored in $G_{\lambda,0}(\sigma)$. To
take the flat space limit of $G_{\lambda,1}$, we should pull $W^{A}$ back to
local coordinates, which is realized by the relation
$W^{A}=\frac{w^{\mu}}{R}\frac{\partial Y^{A}}{\partial y^{\mu}}$ 272727In
general, the null vector $w^{\mu}$ here are different from the one used in the
flat space two-point function (A.40), because $w^{\mu}$ is null respect to the
dS metric $g_{\mu\nu}$. In this case, as we choose the planar coordinates,
$g_{\mu\nu}$ is conformally equivalent to $\eta_{\mu\nu}$ and hence $w^{\mu}$
is also null with repsect to the flat metric.. Here
$y^{\mu}=(\eta,\mathbf{y})$ denotes the planar coordinates. Applying this
pull-back rule to $W_{1}\cdot W_{2}$ yields $W_{1}\cdot
W_{2}=\frac{w_{1}^{\mu}w_{2}^{\nu}}{R^{2}}\partial_{y^{\mu}_{1}}\partial_{y^{\mu}_{2}}\sigma$.
Because of the identification $x^{\mu}=(R(\eta+1),Ry^{i})$, we can replace
$R^{-1}\partial_{y^{\mu}}$ by $\partial_{x^{\mu}}$ and then in the flat space
limit we get

$\displaystyle W_{1}\cdot
W_{2}=(w_{1}\cdot\partial_{1})(w_{2}\cdot\partial_{2})(-2\rho R^{2})\approx
w_{1}\cdot w_{2}$ (A.42)

where the substitution $\sigma=R^{2}-2\rho R^{2}$ is made in the first step,
and the relation $\rho R^{2}=\frac{1}{4}x_{12}^{2}$ is used in the second
step. Similarly, the term
$\sigma(W_{1}\cdot\partial_{Y_{1}})(W_{2}\cdot\partial_{Y_{2}})$ reduces to
$\sigma(w_{1}\cdot\partial_{1})(w_{2}\cdot\partial_{2})$. Although $\sigma$
itself diverges in the flat space limit, the combination $\sigma/\lambda^{2}$
is actually finite and equal to $m^{-2}$ because of the large $R$ relations
$\frac{\sigma}{R^{2}}=1$ and $mR=\lambda$. It is also easy to check that the
remaining term in $G_{\lambda,1}(Y_{1},Y_{2};W_{1},W_{2})$ does not survive in
the flat space limit. Altogether, the flat space limit of $G_{\lambda,1}$
should be

$\displaystyle G_{\lambda,1}\approx\left(w_{1}\cdot
w_{2}+m^{-2}(w_{1}\cdot\partial_{1})(w_{2}\cdot\partial_{2})\right)\Delta^{(0)}_{m^{2},0}(x_{1},x_{2})=m^{-2}\Delta_{m^{2},1}^{(1)}(x_{1},x_{2};w_{1},w_{2}).$
(A.43)

Let’s also consider the case $J=1$, $\ell=0$. In flat space,
$\Delta_{m^{2},0}^{(1)}$ is given by

$\Delta_{m^{2},0}^{(1)}(x_{1},x_{2};w_{1},w_{2})=(w_{1}\cdot\partial_{1})(w_{2}\cdot\partial_{2})\Delta_{m^{2},0}^{(0)}(x_{1},x_{2})~{},$
(A.44)

and its de Sitter counterpart is
$(W_{1}\cdot\nabla_{1})(W_{2}\cdot\nabla_{2})G_{\lambda,0}(\sigma)$. Since
$W\cdot\nabla=W\cdot\partial_{Y}$ reduces to $w\cdot\partial_{x}$ in the flat
space limit, we have

$(W_{1}\cdot\nabla_{1})(W_{2}\cdot\nabla_{2})G_{\lambda,0}(\sigma)\approx\Delta_{m^{2},0}^{(1)}(x_{1},x_{2};w_{1},w_{2})~{}.$
(A.45)

We report the flat space limit in the $J=2$ case without showing details. On
the flat space side, all $\Delta^{(2)}_{m^{2},\ell}$ with $0\leq\ell\leq 2$
are given by Karateev:2020axc

$\displaystyle\Delta^{(2)}_{m^{2},0}(x_{1},x_{2};w_{1},w_{2})\\!=\\!\frac{d+1}{d}(w_{1}\cdot\partial_{1})^{2}(w_{2}\cdot\partial_{2})^{2}\Delta^{(0)}_{m^{2},0}(x_{1},x_{2})\,$
$\displaystyle\Delta^{(2)}_{m^{2},1}(x_{1},x_{2};w_{1},w_{2})\\!=\\!2((w_{1}\cdot\partial_{1})^{2}(w_{2}\cdot\partial_{2})^{2}+m^{2}(w_{1}\cdot
w_{2})(w_{1}\cdot\partial_{1})(w_{2}\cdot\partial_{2}))\Delta^{(0)}_{m^{2},0}(x_{1},x_{2})$
(A.46)
$\displaystyle\Delta^{(2)}_{m^{2},2}(x_{1},x_{2};w_{1},w_{2})\\!=\\!\left((m^{2}\,w_{1}\cdot
w_{2}+w_{1}\cdot\partial_{1}\,w_{2}\cdot\partial_{2})^{2}-\frac{1}{d}(w_{1}\\!\cdot\\!\partial_{1})^{2}(w_{2}\\!\cdot\\!\partial_{2})^{2}\right)\Delta^{(0)}_{m^{2},0}(x_{1},x_{2})~{}.$

and on the de Sitter side we find

$\displaystyle(W_{1}\cdot\nabla_{1})^{2}(W_{2}\cdot\nabla_{2})^{2}G_{\lambda,0}(Y_{1},Y_{2})\approx\frac{d}{d+1}\Delta^{(2)}_{m^{2},0}(x_{1},x_{2};w_{1},w_{2})$
$\displaystyle(W_{1}\cdot\nabla_{1})(W_{2}\cdot\nabla_{2})G_{\lambda,1}(Y_{1},Y_{2};W_{1},W_{2})\approx\frac{1}{2\,m^{2}}\Delta^{(2)}_{m^{2},1}(x_{1},x_{2};w_{1},w_{2})$
$\displaystyle
G_{\lambda,2}(Y_{1},Y_{2};W_{1},W_{2})\approx\frac{1}{m^{4}}\Delta^{(2)}_{m^{2},2}(x_{1},x_{2};w_{1},w_{2})~{}.$
(A.47)

For generic $J$, the construction of $\Delta^{(J)}_{m^{2},\ell}$ is much more
involved since it requires $J+1$ projections operators $\Pi_{J\ell}$ that
effectively implement the branching rule from the spin $J$ representation of
$SO(d,1)$ to spin $\ell$ representations of $SO(d)$. The explicit expressions
of $\Pi_{J\ell}$ have been solved in Dobrev:1977qv  but they are very
complicated. We will not give these expressions. Instead, we will discuss
$\Delta^{(J)}_{m^{2},\ell}$ when $d=1$. This is a very degenerate case,
because all $\Pi_{J\ell}$ with $\ell\geq 2$ vanish. To find $\Pi_{J0}$ and
$\Pi_{J1}$, let’s pick a vector $p^{\mu}$ in $\mathbb{R}^{1,1}$. $\Pi_{J0}$
should only have longitudinal components along $p^{\mu}$, which in the index
free formalism means $\Pi_{J0}\propto p^{-2J}(p\cdot w_{1})^{J}(p\cdot
w_{2})^{J}$. The proportional constant can be fixed by using the fact that
$\Pi_{J0}$ is a projector. Indeed, we find

$\displaystyle\Pi_{J0}(p,w_{1},w_{2})=2^{J-1}\frac{(p\cdot w_{1})^{J}(p\cdot
w_{2})^{J}}{p^{2J}}~{}.$ (A.48)

By the completeness of $\Pi_{J0}$ and $\Pi_{J1}$, we know immediately that
$\Pi_{J1}=(w_{1}\cdot w_{2})^{J}-\Pi_{J0}$, with $\Pi_{J0}$ given by eq.
(A.48). In $\mathbb{R}^{1,1}$, there exists an interesting relation
$(w_{1}\cdot w_{2})(p^{2}\,w_{1}\cdot w_{2}-2p\cdot w_{1}\,p\cdot w_{2})=0$,
which allows us to rewrite $\Pi_{J1}$ as

$\displaystyle\Pi_{J1}(p,w_{1},w_{2})=2^{J-1}\frac{(p\cdot w_{1})^{J-1}(p\cdot
w_{2})^{J-1}}{p^{2J}}\left(p^{2}\,w_{1}\cdot w_{2}-p\cdot w_{1}\,p\cdot
w_{2}\right)~{}.$ (A.49)

With these projectors known, we can define $\Delta^{(J)}_{m^{2},\ell}$
following the general recipe given in Karateev:2020axc :

$\displaystyle\Delta^{(J)}_{m^{2},0}$
$\displaystyle=(-)^{J}m^{2J}\Pi_{J0}\left(p^{2}\to-m^{2},p_{\alpha}\to
i\partial_{x_{1}^{\alpha}}\right)\Delta^{(0)}_{m^{2},0}(x_{1},x_{2})$
$\displaystyle=2^{J-1}(w_{1}\cdot\partial_{1})^{J}(w_{2}\cdot\partial_{2})^{J}\Delta^{(0)}_{m^{2},0}(x_{1},x_{2})~{},$
(A.50)

and

$\displaystyle\Delta^{(J)}_{m^{2},1}$
$\displaystyle=(-)^{J-1}m^{2J}\Pi_{J1}\left(p^{2}\to-m^{2},p_{\alpha}\to
i\partial_{x_{1}^{\alpha}}\right)\Delta^{(0)}_{m^{2},0}(x_{1},x_{2})$
$\displaystyle=2^{J-1}(w_{1}\cdot\partial_{1})^{J-1}(w_{2}\cdot\partial_{2})^{J-1}\left(m^{2}\,w_{1}\cdot
w_{2}+w_{1}\cdot\partial_{1}\,w_{2}\cdot\partial_{2}\right)\Delta^{(0)}_{m^{2},0}(x_{1},x_{2})~{},$
(A.51)

where the extra signs $(-)^{J}$ and $(-)^{J-1}$ have been explained in section
3.3. On the de Sitter side, we thus find

$\displaystyle(W_{1}\cdot\nabla_{1})^{J}(W_{2}\cdot\nabla_{2})^{J}G_{\lambda,0}(Y_{1},Y_{2})\approx
2^{1-J}\Delta^{(J)}_{m^{2},0}(x_{1},x_{2};w_{1},w_{2})~{},$
$\displaystyle(W_{1}\cdot\nabla_{1})^{J-1}(W_{2}\cdot\nabla_{2})^{J-1}G_{\lambda,1}(Y_{1},Y_{2};W_{1},W_{2})\approx\frac{2^{1-J}}{m^{2}}\Delta^{(J)}_{m^{2},1}(x_{1},x_{2};w_{1},w_{2})~{}.$
(A.52)

For $d=1$ and $J=2$, eq. (A.4) and eq. (A.4) are consistent.

##  Appendix B Complementary series in the Källén-Lehmann decompositions

In this appendix, we show how complementary series contributes to the two-
point function of a scalar operator ${\cal O}(Y)$. We will only focus on
$\mathcal{C}_{\Delta,0}$ since $\mathcal{C}_{\Delta,\ell}$ with $\ell>0$
cannot contribute to $\langle\Omega|{\cal O}(Y_{1}){\cal
O}(Y_{2})|\Omega\rangle$.

Compared to the principal series, the main difference is the resolution of the
identity. For a principal series representation $\mathcal{P}_{\Delta,0}$, the
identity operator in its Hilbert space can be expressed as

$\displaystyle\mathbb{1}_{\mathcal{P}_{\Delta,0}}=\int
d^{d}\mathbf{y}\,|\Delta,\mathbf{y}\,\rangle\langle\Delta,\mathbf{y}\,|=\int_{P}|\Delta,P\rangle\langle\Delta,P|~{}.$
(B.1)

which follows from the inner product
$\langle\Delta,\mathbf{y}_{1}|\Delta,\mathbf{y}_{2}\,\rangle=\delta^{d}(\mathbf{y}_{1}-\mathbf{y}_{2}\,)$.
However, for complementary series, we are not allowed to choose such an inner
product since it does not respect the reality condition of
${\mathfrak{so}}(d+1,1)$ generators Sun:2021thf . Instead,
$\langle\Delta,\mathbf{y}_{1}|\Delta,\mathbf{y}_{2}\,\rangle$ has to be
proportional to the CFT two-point function of a scalar primary of scaling
dimension $\Delta$, i.e.
$\langle\Delta,\mathbf{y}_{1}|\Delta,\mathbf{y}_{2}\,\rangle\propto|\mathbf{y}_{1}-\mathbf{y}_{2}\,|^{-2\Delta}$.
From the embedding space point of view, it is obvious that
$\langle\Delta,P_{1}|\Delta,P_{2}\rangle\propto(-2P_{1}\cdot P_{2})^{-\Delta}$
is the only choice that is compatible with the $SO(d+1,1)$ invariance and the
scaling property imposed on $|\Delta,P\rangle$, when $\Delta$ is real. Also
because of $-2P_{1}\cdot P_{2}=|\mathbf{y}_{1}-\mathbf{y}_{2}|^{2}$, it is
consistent with the expression in local coordinates.

We fix the normalization of inner products in $\mathcal{C}_{\Delta,0}$ by
choosing

$\displaystyle\langle\Delta,P_{1}|\Delta,P_{2}\rangle=\frac{N_{\Delta}}{(-2P_{1}\cdot
P_{2})^{\Delta}},\,\,\,\,\,N_{\Delta}=\frac{\Gamma(\Delta)}{\pi^{\frac{d}{2}}\,\Gamma(\frac{d}{2}-\Delta)}~{},$
(B.2)

It fully determines the resolution of identity

$\displaystyle\mathbb{1}_{\mathcal{C}_{\Delta,0}}=\int_{P_{1},P_{2}}\,|\Delta,P_{1}\rangle\,\frac{N_{\bar{\Delta}}}{(-2P_{1}\cdot
P_{2})^{\bar{\Delta}}}\,\langle\Delta,P_{2}|~{}.$ (B.3)

For example, it is straightforward to check
$\langle\Delta,P_{1}|\mathbb{1}_{\mathcal{C}_{\Delta,0}}|\Delta,P_{2}\rangle=\langle\Delta,P_{1}|\Delta,P_{2}\rangle$
by using

$\displaystyle\int_{P_{1}}\frac{1}{(-2P_{0}\cdot
P_{1})^{\Delta}}\frac{1}{(-2P_{1}\cdot
P_{2})^{\bar{\Delta}}}=\int\,d^{d}\mathbf{y}_{1}\frac{1}{|\mathbf{y}_{0}-\mathbf{y}_{1}|^{2\Delta}}\frac{1}{|\mathbf{y}_{1}-\mathbf{y}_{2}|^{2\bar{\Delta}}}=\frac{\delta^{d}(\mathbf{y}_{0},\mathbf{y}_{2}\,)}{N_{\Delta}N_{\bar{\Delta}}}~{}.$
(B.4)

We insert $\mathbb{1}_{\mathcal{C}_{\Delta,0}}$ into the two-point function of
${\cal O}(Y)$

$\displaystyle\langle\Omega|{\cal
O}(Y_{1})|\mathbb{1}_{\mathcal{C}_{\Delta,0}}|{\cal
O}(Y_{2})|\Omega\rangle=\int_{P_{3},P_{4}}\,\langle\Omega|{\cal
O}(Y_{1})|\Delta,P_{3}\rangle\,\frac{N_{\bar{\Delta}}}{(-2P_{3}\cdot
P_{4})^{\bar{\Delta}}}\,\langle\Delta,P_{4}|{\cal O}(Y_{2})|\Omega\rangle~{},$
(B.5)

where $\langle 0|{\cal O}(Y)|\Delta,P\rangle=c_{\cal O}(\Delta)\,{\cal
K}_{\Delta}(Y,P)$. Next, we write the remaining double integral in local
coordinates and use the Fourier transformation

$\displaystyle\frac{N_{\bar{\Delta}}}{x^{2\bar{\Delta}}}=\int\frac{d^{d}\mathbf{k}}{(2\pi)^{d}}\left(\frac{k}{2}\right)^{d-2\Delta}e^{i\mathbf{k}\cdot\mathbf{x}}~{},$
(B.6)

which leads to

$\displaystyle\langle\Omega|{\cal
O}(\eta_{1},\mathbf{y}_{1})|\mathbb{1}_{\mathcal{C}_{\Delta,0}}|{\cal
O}(\eta_{2},\mathbf{y}_{2})|\Omega\rangle$ $\displaystyle=|c_{\cal
O}(\Delta)|^{2}\int\frac{d^{d}\mathbf{k}}{(2\pi)^{d}}\left(\frac{k}{2}\right)^{d-2\Delta}$
$\displaystyle\times\int\,d^{d}\mathbf{y}_{3}\,{\cal
K}_{\Delta}(\eta_{1},\mathbf{y}_{1};\mathbf{y}_{3})e^{i\mathbf{k}\cdot\mathbf{y}_{3}}\int\,d^{d}\mathbf{y}_{4}\,{\cal
K}_{\Delta}(\eta_{2},\mathbf{y}_{2};\mathbf{y}_{4})e^{-i\mathbf{k}\cdot\mathbf{y}_{4}}~{}.$
(B.7)

For the integral over $\mathbf{y}_{3}$ and $\mathbf{y}_{4}$, we use eq. (3.10)
and eq. (3.11) respectively. In the end, we obtain the mode expansion of the
free Green’s function $G_{-i(\Delta-\frac{1}{2})}$, c.f. eq. (A.1):

$\displaystyle\langle\Omega|{\cal
O}(\eta_{1},\mathbf{y}_{1})|\mathbb{1}_{\mathcal{C}_{\Delta,0}}|{\cal
O}(\eta_{2},\mathbf{y}_{2})|\Omega\rangle$ $\displaystyle=|c_{\cal
O}(\Delta)|^{2}G_{-i(\Delta-\frac{1}{2})}(\eta_{1},\mathbf{y}_{1};\eta_{2},\mathbf{y}_{2})~{}.$
(B.8)

Therefore the complementary series part of the ${\cal O}$ Källén-Lehmann
decomposition takes the form

$\displaystyle\int_{-\frac{d}{2}}^{\frac{d}{2}}\,d\lambda\,\rho_{\cal
O}^{\mathcal{C},0}(\lambda)\,G_{i\lambda}(Y_{1},Y_{2}),$ (B.9)

where $\rho_{\cal O}^{\mathcal{C},0}(\lambda)$ is a nonnegative function by
construction. For spinning operators, there is a similar result.

##  Appendix C Discrete series in free scalar theory in dS2

Let $\Phi(Y)$ be a free scalar field of scaling dimension
$\Delta=\frac{1}{2}+i\nu$. We have argued that discrete series representations
cannot contribute to the two-point function of $\Phi^{2}(Y)$ because it would
always lead to antipodal singularity while
$\langle\Omega|\Phi^{2}(Y_{1})\Phi^{2}(Y_{2})|\Omega\rangle$ is clearly free
of such a singularity given that $|\Omega\rangle$ is the BD vacuum. On the
other hand, according to the group theoretical analysis in Penedones:2023uqc ,
the two-particle Hilbert space ${\cal H}_{2}$ of $\Phi$ should contain all
${\cal D}^{\pm}_{k}$ for $k=2,4,6,\cdots$. To reconcile this contradiction, we
will explicitly compute the matrix elements of $\Phi^{2}$ between the vacuum
$|\Omega\rangle$ and discrete series states in ${\cal H}_{2}$. More precisely,
we will focus on the lowest-weight state $|k\rangle_{k}$ in each ${\cal
D}_{k}^{+}$, because all $\langle\Omega|\Phi^{2}(Y)|\ell\rangle_{k}$ should
vanish once $\langle\Omega|\Phi^{2}(Y)|k\rangle_{k}$ vanishes as a simple
result of the $SO(2,1)$ symmetry. For ${\cal D}_{k}^{-}$, the argument is
exactly the same.

Let’s first describe the single-particle states of $\Phi$ in conformally
global coordinates, using the following mode expansion in BD vacuum
Sun:2021thf :

$\displaystyle\Phi=\sum_{n\in\mathbb{Z}}\phi_{n}a_{n}+\phi_{n}^{*}a_{n}^{\dagger},\,\,\,\,\,\phi_{n}=g_{n}(\tau)\frac{e^{-in\varphi}}{\sqrt{2\pi}}~{},$
(C.1)

where282828We remind the reader that $\mathbf{F}$ is our notation for the
regularized hypergeometric function.

$\displaystyle
g_{n}(\tau)=\frac{\Gamma(n+\bar{\Delta})}{\sqrt{2}}e^{-in\tau}\mathbf{F}\left(\Delta,\bar{\Delta},n+1,\frac{1}{1+e^{2i\tau}}\right)~{}.$
(C.2)

The canonically normalized single-particle states are
$|n\rangle_{\Delta}\equiv a_{n}^{\dagger}|\Omega\rangle$, and the action of
${\mathfrak{so}}(2,1)$ on these states is computed in Sun:2021thf

$\displaystyle L_{\pm}|n\rangle_{\Delta}=(n\pm\Delta)|n\pm
1\rangle_{\Delta},\,\,\,\,\,L_{0}|n\rangle_{\Delta}=n|n\rangle_{\Delta}~{}.$
(C.3)

The two-particle Hilbert space ${\cal H}_{2}$ is spanned by
$|n,m\rangle_{\Delta}\equiv a_{n}^{\dagger}a_{m}^{\dagger}|\Omega\rangle$, and
the lowest-weight state $|k\rangle_{k}$ of ${\cal D}^{+}_{k}$ in ${\cal
H}_{2}$ can be written as Penedones:2023uqc

$\displaystyle|k\rangle_{k}=c\sum_{\ell\in\mathbb{Z}}\frac{\Gamma(\Delta+\ell-k)}{\Gamma(\bar{\Delta}+\ell)}|\ell,k-\ell\rangle_{\Delta}~{}.$
(C.4)

where $c$ is some unimportant normalization constant. With all these
ingredients, we can now compute the matrix element of $\Phi^{2}$ between
$\langle\Omega|$ and $|k\rangle_{k}$:

$\displaystyle\langle 0|\Phi^{2}(\tau,\varphi)|k\rangle_{k}$
$\displaystyle=2c\sum_{\ell}\frac{\Gamma(\Delta+\ell-k)}{\Gamma(\bar{\Delta}+\ell)}\phi_{\ell}(\tau,\varphi)\phi_{k-\ell}(\tau,\varphi)$
$\displaystyle=\frac{c\,e^{-ik(\tau+\varphi)}}{2\cosh(\pi\nu)}\sum_{\ell}(-)^{\ell}\mathbf{F}\left(\Delta,\bar{\Delta},\ell+1,\frac{1}{1+e^{2i\tau}}\right)\mathbf{F}\left(\Delta,\bar{\Delta},k-\ell+1,\frac{1}{1+e^{2i\tau}}\right)~{}.$
(C.5)

To evaluate the sum over $\ell$ in eq. (C), we use the series expansion of the
regularized hypergeometric function

$\displaystyle\langle 0|\Phi^{2}(\tau,\varphi)|k\rangle_{k}$
$\displaystyle=\frac{c\,e^{-ik(\tau+\varphi)}}{2\cosh(\pi\nu)}\sum_{n,m\geq
0}\frac{(\frac{1}{2}\pm i\nu)_{n}(\frac{1}{2}\pm
i\nu)_{m}}{n!\,m!}\,\mathcal{T}(n\\!+\\!1,m\\!+\\!k\\!+\\!1)\left(1\\!+\\!e^{2i\tau}\right)^{-n-m}~{},$
(C.6)

where

$\displaystyle\mathcal{T}(a,b)\equiv\sum_{\ell\in\mathbb{Z}}\frac{(-)^{\ell}}{\Gamma(a+\ell)\Gamma(b-\ell)},\,\,\,\,\,a,b\in\mathbb{Z}~{}.$
(C.7)

Since $a,b$ are integers, $\mathcal{T}(a,b)$ is actually a finite sum
supported on $1-a\leq\ell\leq b-1$, which implies that $\mathcal{T}$ vanishes