# Notes on Resonances and Unitarity from Celestial Amplitudes

Diego García-Sepúlveda 2,3,4 Alfredo Guevara 5,6 Justin Kulp 7 and Jingxiang
Wu<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Kadanoff Center for Theoretical
Physics & Enrico Fermi Institute, University of ChicagoCenter for the
Fundamental Laws of Nature, Harvard University, Cambridge, MA 02138, USABlack
Hole Initiative, Harvard University, Cambridge, MA 02138, USASociety of
Fellows, Harvard University, Cambridge, MA 02138, USAPerimeter Institute for
Theoretical Physics, Waterloo, ON N2L 2Y5, CanadaDepartment of Physics &
Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, CanadaMathematical
Institute, University of Oxford, Andrew-Wiles Building, Woodstock Road,
Oxford, OX2 6GG, UK

###### Abstract

We study the celestial description of the $O(N)$ sigma model in the large $N$
limit as introduced by Coleman, Jackiw and Politzer. Focusing on three-
dimensions, we analyze the implications of a UV complete, all-loop order
4-point amplitude of pions in terms of correlation functions defined on the
celestial circle. We find these retain many key features from the previously
studied tree-level case, such as their relation to Generalized Free Field
theories and crossing-symmetry, but also incorporate new properties such as
IR/UV softness and S-matrix metastable states. In particular, to understand
unitarity, we propose a form of the optical theorem that controls the
imaginary part of the correlator based solely on the presence of these
resonances. We also explicitly analyze the conformal block expansions and
factorization of four-point functions into three-point functions. We find that
summing over resonances is key for these factorization properties to hold. We
end with some topics for future study.

##  1 Introduction

The celestial holography program aims to characterize flat-space S-matrices of
diverse theories in terms of boundary conformal correlators
stromingerCelestial ; pasterskiShao ; fullVirasoro:3 , constituting a
promising approach to the long-standing problem of flat-space holography. The
framework has been succesful in unveiling new structures of the S-matrix, in
particular regarding its infrared structure Strominger:2017zoo , yet many
questions arise regarding the nature of the so-defined celestial conformal
field theories (CCFTs), see Raclariu:2021zjz ; Pasterski:2021rjz ;
Pasterski:2021raf  for recent reviews. In particular, a precise statement of
unitarity and a classification of the spectrum are missing in CCFTs, although
recent progress has been made for both massive and massless S-matrices in
lamShao ; Nandan:2019jas ; Chang:2021wvv ; Melton:2021kkz ; Fan:2021isc ;
Fan:2021pbp ; Atanasov:2021cje ; Guevara:2021tvr .

Along such lines, it has been observed that well-understood analytic
properties of scattering amplitudes have a novel realization in celestial
correlation functions, yielding surprising features from the viewpoint of
standard CFTs celestialSymmetries ; celestialConstraints ; Pate:2019lpp ;
Fotopoulos:2019tpe ; Fotopoulos:2019vac ; Fotopoulos:2020bqj ; Fan:2019emx ;
Arkani-Hamed:2020gyp . In particular, references lamShao ; Nandan:2019jas ;
Chang:2021wvv ; Atanasov:2021cje ; Guevara:2021tvr ; Melton:2021kkz ;
ShuHengTalk  have considered the celestial realization of the tree-level
S-matrix with massive particle exchange at both three and four points. It was
observed that the three-point correlation functions are regular and non-
distributional (as opposed to the all-massless ones) and the four-point
function has a simple structure closely related to a Generalized Free Field
Theory (GFFT) (see also ShuHengTalk ). Indeed, using a scalar theory as an
example, lamShao  pointed out that such four-point correlation functions in
CCFT satisfy a novel realization of the optical theorem. More precisely, the
four-point function has an imaginary piece controlled by three-point
correlation functions through a conformal partial-wave expansion lamShao ;
Nandan:2019jas ; Atanasov:2021cje ; Melton:2021kkz . This follows directly
from the analytic factorization of the S-matrix and provides a first approach
to the underlying unitarity of CCFTs.

A number of pressing questions arise from the previous discussion:

  * •

The optical theorem applies to the imaginary piece of the 4-point function.
How does unitarity extend to the full correlation function?

  * •

What is the precise relation of the partial wave expansion to the standard
conformal block decomposition from the CCFT point of view? How does the CCFT
spectrum emerge from the principal continuous series?

  * •

Do any of the above conclusions hold beyond tree level? Do they hold at finite
coupling?

The last point is important because, as observed in Arkani-Hamed:2020gyp ,
celestial amplitudes are “anti-Wilsonian.” This means that unlike their bulk
analogs, they are highly sensitive to UV physics, because the scattering of
boost eigenstates covers the complete range of energies.

In this work we will introduce a novel realization of unitarity that is
particular to celestial CFTs as opposed to their underlying QFT. In QFT,
unitarity (and generalized unitarity) requires one to sum over a stable
spectrum of states VELTMAN  (see also Denner:2014zga ; Donoghue:2019fcb ;
Menezes:2021tsj ; Chang:2021wvv . In turn, the analysis of singularities
associated to bound states and metastable particles (resonances) requires a
particular prescription for analytic continuation of the S-matrix, i.e
Hannesdottir:2022bmo . Here, we will show that in the CCFT context the
presence of these singularities is manifest in four-point correlation
functions. The choice of S-matrix prescription turns into a definition of the
Mellin integral transform, which connects the momentum basis to the conformal
basis. In this setup, one finds that in the conformal basis resonances
completely control the imaginary part of the correlation function and resum
the contribution from bound states.

The object under study is a UV/IR finite scattering amplitude valid to all
loop-orders in the large $N$ limit, corresponding to the three-dimensional
$O(N)$ model first studied by Coleman, Jackiw, and Politzer. It exhibits
spontaneous symmetry breaking, and by introducing auxiliary fields the
effective action can be summed to all orders. This allows us to explicitly
derive and check the unitarity properties of the corresponding 1d CCFT non-
perturbatively. This includes partial wave decomposition for the full
correlator and factorization of the imaginary part. Interestingly, in this
particular case the four-point amplitude retains much of the GGFT structure
previously observed at tree-level by Lam and Shao lamShao , which passes the
usual unitarity constraints such as positivity, but also incorporates new
relevant features of CCFT such as color structure. We will also detail the
general extent to which our arguments are expected to hold, in a more diverse
class of theories.

This paper is organized as follows: In Section 2 we recount the famous
solution to the $O(N)$ model described by Coleman, Jackiw, and Politzer, and
use it to obtain the $\pi^{a}\pi^{b}\to\pi^{c}\pi^{d}$ and
$\pi^{a}\pi^{b}\sigma$ scattering amplitudes in Section 2.1, with focus on
(2+1)d.

In Section 3 we introduce the celestial amplitude that we will study for the
rest of the paper. In Section 3.1 we recap the structure of a celestial CFT
correlator and set our notation, contrasting 1d CCFTs to 2d CCFTs. We compute
the actual $O(N)$ CCFT amplitude in 3.2 and consider the IR and UV limits in
Section 3.3. In Section 3.4 we look at the permutation relations coming from
scattering four bosons, and the implications for the (stripped) CCFT four-
point function.

In Section 4 we study how resonances manifest in celestial amplitudes via non-
zero contributions to the imaginary part of the celestial amplitude, extending
a result of Chang:2021wvv  to the case of resonances. We illustrate our
observations in the case of the $O(N)$ model.

In Section 5 we look at various CFT data implied by the CCFT correlator. We
lay out the amplitudes in Section 5.1 then study the conformal block
decompositions in Section 5.2. We verify that the same result is obtained for
conformal partial waves using the Euclidean inversion formula in Section 5.3.
In Section 5.4 we analyze the factorization of four-point function
coefficients into three-point function coefficients and explain the
modification to the celestial optical theorem proposed in lamShao .

We conclude in Section 6 and end with a broad list of open problems and
conjectures for future study inspired by our work in Section 6.1.

We set our conventions for CFT in Appendix A and list various formula for
four-point functions and conformal blocks and partial waves in Appendix A.1
and Appendix A.2 respectively.

##  2 Effective Description of the $O(N)$ Model

In this section we review the relevant pieces in the large $N$ analysis of the
$O(N)$ model as described in Coleman:1974jh  (see also the review
mosheQuantumFieldTheory2003  and more recently Carmi:2018qzm ), which will
also serve to establish some notation. We comment on the analytic structure of
the correlation functions, and compute the three and four-point functions for
use in celestial $\pi\pi\to\pi\pi$ scattering. For further details we refer
the reader to the original paper Coleman:1974jh  which we follow closely. We
use Euclidean signature throughout this section.

The $O(N)$ model is given by the following $O(N)$ symmetric Lagrangian of $N$
real scalar fields $\varphi^{a}$

$\mathcal{L}=\frac{1}{2}\partial_{\mu}\varphi^{a}\partial^{\mu}\varphi^{a}-\frac{1}{2}\mu_{0}^{2}\varphi^{a}\varphi^{a}-\frac{\lambda_{0}}{8N}\left(\varphi^{a}\varphi^{a}\right)^{2}\,,$
(1)

where $\mu_{0}^{2}$ and $\lambda_{0}$ are the bare squared mass and bare
coupling constant. The corresponding renormalized quantities will be denoted
$\mu^{2}$ and $\lambda$ respectively. It is known that this model can be
solved exactly in the large $N$ limit Coleman:1974jh . We will not review that
computation here, but instead we will simply list the key results needed.

The first point to consider is that when $\lambda>0$ and $\mu^{2}<0$,
$\varphi$ obtains a vacuum expectation value (vev)

$\expectationvalue{\varphi}^{2}=-2\mu^{2}N/\lambda\,,$ (2)

which spontaneously breaks the $O(N)$ symmetry to $O(N-1)$. Without loss of
generality, we choose to preserve the $O(N-1)$ symmetry in the first $N-1$
components, which are our massless Goldstone bosons, and take $\varphi^{N}$ to
have the vev. We thus introduce more suitable variables

$\sigma\coloneqq\varphi^{N}-\langle\varphi\rangle,\quad\pi^{a}\coloneqq\varphi^{a}\quad(1\leq
a<N)\,.$ (3)

Before switching to these new variables, it is helpful for the large $N$ part
of the analysis to introduce a Hubbard-Stratonovich (HS) field $\chi$,
rewriting the Lagrangian as

$\mathcal{L}=\frac{1}{2}\partial_{\mu}\varphi^{a}\partial^{\mu}\varphi^{a}+\frac{1}{2}\frac{N}{\lambda_{0}}\chi^{2}-\frac{1}{2}\chi\varphi^{a}\varphi^{a}-\frac{N\mu_{0}^{2}}{\lambda_{0}}\chi\,.$
(4)

The introduction of $\chi$ is a combinatorial trick which greatly simplifies
the large-$N$ analysis: there is now only one three-point vertex
$\chi\varphi^{2}$, all factors of $\frac{1}{N}$ come from the $\chi$
propagator, and all factors of $N$ come from $\varphi$ loops. Hence, to
leading order in large $N$, the only diagrams that contribute to the effective
action are those 1PI diagrams with a single loop of $\varphi$ and an arbitrary
number of insertions of the field $\chi$, as shown in Figure 1.

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Figure 1: Solid lines denote pions $\pi^{a}$ and dashed lines denote the
Hubbard-Stratonovich field $\chi$. Left: 1PI diagrams that contribute to the
effective action at leading order in $1/N$. Right: Bubble diagrams that
contribute to the $\pi\pi\to\pi\pi$ amplitude at leading order.

Putting both of the pieces together, the effective action is found to be
Coleman:1974jh

$\displaystyle\Gamma=\int d^{n}\chi$
$\displaystyle\left[\frac{1}{2}\pi^{a}\Box\pi^{a}+\frac{1}{2}\sigma\Box\sigma+\frac{1}{2}\frac{N}{\lambda_{0}}\chi^{2}-\chi\sigma\langle\varphi\rangle-\frac{1}{2}\chi\sigma^{2}\right.$
$\displaystyle\left.-\frac{1}{2}\chi\pi^{a}\pi^{a}-\left(\frac{N\mu_{0}^{2}}{\lambda}+\frac{1}{2}\langle\varphi\rangle^{2}\right)\chi\right]-\frac{1}{2}N\operatorname{tr}\ln\left(-\Box+\chi\right).$
(5)

The HS field $\chi$ is taken to have zero expection value
$\langle\chi\rangle=0$. This is guaranteed by requiring the sum of all tadpole
diagrams to vanish, i.e. terms linear in $\chi$ in the expansion, which gives
rise exactly to Equation (2).

Following Coleman:1974jh , we compute the propagators from the effective
action in Equation (5). The propagators for the pions take the expected form
$D_{\pi^{a}\pi^{b}}=\delta^{ab}/p^{2}$. On the other hand, there is a mixing
between $\sigma$ and $\chi$, i.e.

$-\frac{1}{2}\begin{pmatrix}\sigma&\chi\end{pmatrix}D^{-1}(p^{2})\begin{pmatrix}\sigma\\\
\chi\end{pmatrix},$ (6)

with the non-diagonal matrix

$D^{-1}(p^{2})=\begin{pmatrix}p^{2}&\langle\varphi\rangle\\\
\langle\varphi\rangle\,\,&-\frac{N}{\lambda}-\frac{N}{16p}\end{pmatrix}.$ (7)

Inverting this matrix, we find the propagators are double-valued functions of
$p^{2}$

$\displaystyle D_{\sigma\sigma}$
$\displaystyle=\frac{p+\frac{\lambda}{16}}{p\left(p^{2}+\frac{\lambda}{16}p-2\mu^{2}\right)}$
(8) $\displaystyle D_{\chi\chi}$
$\displaystyle=-\frac{\lambda}{N}\frac{p^{2}}{p^{2}+\frac{\lambda}{16}p-2\mu^{2}}$
(9) $\displaystyle D_{\sigma\chi}=D_{\chi\sigma}$
$\displaystyle=\sqrt{\frac{-2\mu^{2}\lambda}{N}}\frac{1}{p^{2}+\frac{\lambda}{16}p-2\mu^{2}}\,,$
(10)

where $p$ is the positive square root of $p^{2}$. Importantly,
$D_{\sigma\sigma}$ diverges at $p=0$ because of a pole in $p$, and not because
of a pole in $p^{2}$ (in $p^{2}$ this is a branch point from the cut
associated with two-$\pi$ intermediate states). Both propagators have poles on
the second sheet of the $p^{2}$-plane, given by the zeros of $\det
D^{-1}(p^{2})$

$p_{\pm}=-\frac{\lambda}{32}\pm
i\sqrt{-2\mu^{2}-\left(\frac{\lambda}{32}\right)^{2}}\,.$ (11)

Physically, the two poles are interpreted as two resonances. All of the above
appears in the analysis of Coleman:1974jh .

For simplicity, we will assume that $p_{\pm}$ have both nonzero real and
imaginary components, which motivates the definition

$\cos(\theta)\coloneqq\frac{\lambda}{16\sqrt{-8\mu^{2}}}<1\,.$ (12)

However, it will become clear that this is not a restrictive assumption in any
way, since the resonance exists as long as $\lambda\neq 0$ and no further
bound is required for our derivation.111If $\lambda$ is large enough $p_{\pm}$
will become purely real. This is however not an issue since one can check that
the two poles are always on the unphysical sheet. Indeed, all of the
discussion in this article holds for $\lambda>0$. Moreover, recall that
boundedness of the potential requires $\lambda>0$.

Now let us briefly discuss the continuation to Lorentzian signature, which
will be necessary later. In order to see the correct continuation in the
energy variable, consider the Riemann surface for the Lorentzian222From here
out, when we use the variable $p$ in scattering amplitudes, we are working in
Euclidean space; when we use variables like $s$, $t$, $u$, and $\omega$, we
are working in Lorentzian. momentum variable $s:=\omega^{2}$. To identify the
correct Wick rotation, we locate the branch cut of the effective propagator
along the positive real $s$ axis; this starts at $s=0$. From here, only the
choice $p\to-i\omega$ is physical, sending the resonances in Equation (11) to
the second-sheet in the $s$-variable.

In the small coupling limit, the zeroes of the Wick rotated propagator are at

$\omega\to\pm\sqrt{-2\mu^{2}}-i\frac{\lambda}{32}\,.$ (13)

From this, one can see how the resonances approach the tree level expression
$\pm\sqrt{-2\mu^{2}}$ as $\lambda\to 0$, and how they emerge at higher-loop
order because of the non-zero Higgs width $\lambda$. Now, at leading order in
$\lambda$ and close to $\omega=+\sqrt{-2\mu^{2}}$

$\frac{1}{\omega^{2}+i\frac{\lambda}{16}\omega+2\mu^{2}}\to\frac{1}{\omega^{2}+i\frac{\lambda}{16}\sqrt{-2\mu^{2}}+2\mu^{2}}\,,$
(14)

which is of the standard Breit-Wigner form $\frac{1}{s-m^{2}+im\Gamma}$.
Clearly since this is valid only close to $\omega=+\sqrt{-2\mu^{2}}$, only the
root $\omega_{+}=+\sqrt{-2\mu^{2}}-i\epsilon$ of the denominator is
meaningful. This also has the correct sign for the $i\epsilon$-prescription
for tree level propagators where the identification
$\epsilon=\sqrt{-2\mu^{2}}\frac{\lambda}{16}$ has been made (the analogous
analysis holds close to $\omega=-\sqrt{-2\mu^{2}}$). Notice that if the wrong
continuation $p\to i\omega$ had been used, then the tree level result for the
propagators would have appeared with the wrong sign for the
$i\epsilon$-prescription.

###  2.1 Three and Four-point Momentum Amplitudes

We are now in position to discuss scattering amplitudes. For example, it is
easy to write down the $\pi^{a}\pi^{b}\rightarrow\pi^{c}\pi^{d}$ amplitude
using the full propagator $D_{\chi\chi}$ in (9) repeatedly

$\displaystyle
T_{4}^{ab,cd}(p_{1},p_{2},p_{3},p_{4})=-\frac{\lambda}{N}\bigg{[}$
$\displaystyle\frac{\,(p_{12})^{2}\,\delta^{ab}\delta^{cd}}{(p_{12})^{2}+\sqrt{(p_{12})^{2}}\lambda/16-2\mu^{2}}+\frac{\,(p_{13})^{2}\,\delta^{ac}\delta^{bd}}{(p_{13})^{2}+\sqrt{(p_{13})^{2}}\lambda/16-2\mu^{2}}$
$\displaystyle\hskip
99.58464pt+\frac{\,(p_{14})^{2}\,\delta^{ad}\delta^{bc}}{(p_{14})^{2}+\sqrt{(p_{14})^{2}}\lambda/16-2\mu^{2}}\bigg{]}\,,$
(15)

where $(p_{ij})^{2}=(p_{i}+p_{j})^{2}$ and we have used the convention that
all the momenta are incoming.

The tree-level result, obtained as $\lambda\to 0$, corresponds to the scalar
tree-level amplitude studied by Lam and Shao in lamShao . Indeed, as $\lambda$
stays positive but becomes small, the two poles approach the point
$p^{2}=2\mu^{2}$ in the physical region with the physical $i\epsilon$
deformation. However, an important difference between (15) and the amplitude
of lamShao  is the presence of additional powers of $(p_{ij})^{2}$ in the
numerators of (15). This reflects the Adler’s zero phenomenon: the fact that
the scattering of Goldstone bosons must vanish as $(p_{ij})^{2}\rightarrow 0$
Adler:1964um . As we will see in Section 3.2, this soft behaviour will be
reflected in the analyticity domain for the celestial amplitude as a function
of the boost weight $\Delta$. Note further that the full large $N$ amplitude
is finite in the high-energy limit at fixed scattering angle. Such UV softness
will be also essential for the existence of a convergence domain in $\Delta$.

The $\pi^{a}\pi^{b}\rightarrow\sigma$ scattering amplitude can be found by
applying the standard Lehmann Symanzik Zimmermann (LSZ) reduction formula to
the three-point function
$\expectationvalue{\pi^{a}\left(p_{1}\right)\pi^{b}\left(p_{2}\right)\sigma\left(p_{3}\right)}$.
Recall that the LSZ reduction formula works by bringing the field operator to
the mass shell and studying the poles of the correlation functions. The $S$
matrix elements are simply given by the corresponding residues after we take
care of the normalization coming from wavefunction renormalization.

More explicitly, by Wick contraction, we find the three point function to be

$\displaystyle\left\langle\pi^{a}\left(p_{1}\right)\pi^{b}\left(p_{2}\right)\sigma\left(p_{3}\right)\right\rangle$
$\displaystyle=D_{\pi^{a}\pi^{c}}(p_{1})D_{\pi^{b}\pi^{c}}(p_{2})D_{\sigma\chi}(p_{3})\,,$
$\displaystyle=\delta^{ab}\frac{1}{p_{1}^{2}}\frac{1}{p_{2}^{2}}\sqrt{\frac{-2\mu^{2}\lambda}{N}}\frac{1}{p_{3}^{2}+\frac{\lambda}{16}p_{3}-2\mu^{2}}\,.$
(16)

If we bring all three external particles on shell, we can read off S matrix
elements as

$\displaystyle\left\langle\pi^{a}\left(p_{1}\right)\pi^{b}\left(p_{2}\right)\sigma\left(p_{3}\right)\right\rangle\sim\frac{1}{p_{1}^{2}}\frac{1}{p_{2}^{2}}\frac{\sqrt{Z}}{p_{3}^{2}-p_{\pm}^{2}}\langle\sigma(p_{\pm})|\pi^{a}(p_{1})\pi^{b}(p_{2})\rangle\,,$
(17)

where $Z_{\pm}$ denotes the wavefunction renormalization appearing in the
residue at the poles $D_{\sigma\sigma}\sim Z_{\pm}/(p^{2}-p^{2}_{\pm})$.
Altogether we have

$T_{3}^{ab}(p_{\pm})\coloneqq\langle\sigma(p_{\pm})|\pi^{a}(p_{1})\pi^{b}(p_{2})\rangle=\sqrt{\frac{-\mu^{2}\lambda}{N}}\frac{-2p_{\pm}}{\sqrt{p_{\mp}(p_{\mp}-p_{\pm})}}\delta^{ab}\,.$
(18)

Finally, as a consistency check, notice that

$\underset{p_{12}^{2}=p^{2}_{\pm}}{\mathbf{Res}}\langle\pi^{c}(p_{3})\pi^{d}(p_{4})|\pi^{a}(p_{1})\pi^{b}(p_{2})\rangle=\langle\pi^{c}(p_{3})\pi^{d}(p_{4})|\sigma(p_{\pm})\rangle\langle\sigma(p_{\pm})|\pi^{a}(p_{1})\pi^{b}(p_{2})\rangle,$
(19)

where only the first term in (15) contributes.

##  3 The Celestial $O(N)$ Amplitude

In this section, we introduce the celestial amplitude for massless external
particles, as defined in stromingerCelestial ; pasterskiShao , and then apply
it to $2\to 2$ pion scattering in the symmetry broken phase of the 3d $O(N)$
model. We see how the all-loop resummation affects convergence of the Mellin
integral, study the EFT expansion, and examine the constraints of scattering
bosons for the celestial four-point function.

###  3.1 Massless Scattering and the Celestial Circle

The celestial sphere formalism recasts the S-matrix elements of a Poincaré-
invariant theory in $\operatorname{\mathbb{R}}^{1,d+1}$ as “correlation
functions” on the $d$-dimensional celestial sphere by re-writing the
scattering amplitude in a basis of $SO(1,d+1)$ covariant wavefunctions, rather
than the usual plane-wave basis given by the eigenfunctions of translation
operators stromingerCelestial ; pasterskiShao  (see also deBoer:2003vf ;
fullVirasoro:3  for related work). Since we will only be studying massless
$2\to 2$ scattering, we will focus on just the details we need for this case.

Given a massless particle in $\operatorname{\mathbb{R}}^{1,d+1}$ its momentum
can be parametrized as

$p^{\mu}=\epsilon\omega(1+\absolutevalue{\vec{x}}^{2},2\vec{x},1-\absolutevalue{\vec{x}}^{2})\,,$
(20)

where $\epsilon=\pm 1$ designates whether it is outgoing/incoming, $\omega>0$
is a positive energy scale, and $\vec{x}\in\operatorname{\mathbb{R}}^{d}$
gives a position on the (past or future) celestial sphere: the projective
null-cone in momentum space. Following Strominger:2017zoo , we identify past
($\epsilon=-1$) and future ($\epsilon=+1$) spheres via antipodal matching.

For $n$ massless external particles in $\operatorname{\mathbb{R}}^{1,d+1}$,
the celestial amplitude is obtained from the flat space scattering amplitude
$T(s,t)$ by a Mellin transform in each of the scale variables

$\widetilde{\mathcal{A}}(\Delta_{i},x_{i})=\prod_{i=1}^{n}\int_{0}^{\infty}d\omega_{i}\,\omega_{i}^{\Delta_{i}-1}\,T(s,t)\,\delta^{(d+2)}\left(\textstyle{\sum}\,p_{i}\right)\,.$
(21)

The celestial amplitude’s construction makes its $SO(1,d+1)$ covariance
manifest and ensures that it is at least as constrained as an $n$-point
correlator in a “typical” $d$-dimensional CFT stromingerCelestial ;
pasterskiShao ; lamShao . Readers familiar with the embedding space formalism
used in, for example, AdS/CFT or the conformal bootstrap, may find this
procedure familiar Dirac:1936fq ; Mack:1969rr ; Weinberg:2010fx ; Costa:2011mg
; Costa:2011dw ; simmonsDuffin:shadows .

However, the original amplitude transformed under the full Poincaré group,
which leads to additional constraints on the celestial amplitudes beyond the
$SO(1,d+1)$ covariance of a generic CFT correlator. As highlighted in
celestialConstraints ; Law:2020tsg ; Law:2020xcf ; Arkani-Hamed:2020gyp ;
Atanasov:2021cje , the invariance under translations leads to phenomena like
distribution valued conformal correlators if $n\leq d+2$, and special
restrictions for the celestial amplitude on the external scaling dimensions
$\Delta_{i}$ (see celestialConstraints  for specifics).

For massless $2\to 2$ amplitudes in, say, $d=1$ and $d=2$, a straightforward
evaluation of equation (21) demonstrates a factorization into a universal
“kinematic part” $X(\Delta_{i},x_{i})$, and a “dynamical part”
$\operatorname{\mathcal{A}}(\Delta_{T},z)$ which depends on the scattering
amplitude $T(s,t)$, i.e.

$\displaystyle\widetilde{\mathcal{A}}_{d=1}(\Delta_{i},x_{i})$
$\displaystyle=X_{d=1}(\Delta_{i},x_{i})\operatorname{\mathcal{A}}(\Delta_{T},z)\,,$
(22) $\displaystyle\widetilde{\mathcal{A}}_{d=2}(\Delta_{i},x_{i})$
$\displaystyle=X_{d=2}(\Delta_{i},x_{i})\operatorname{\mathcal{A}}(\Delta_{T},z)\,.$
(23)

Here $\Delta_{T}=\sum\Delta_{i}$ and $z$ is the conformal cross-ratio

$z\coloneqq\frac{x_{12}x_{34}}{x_{13}x_{24}}\,.$ (24)

To see this, note that on the support of the momentum conserving delta
function in Equation (21) the Mandelstam variables can be written as

$s=-4\frac{x_{12}^{2}x_{13}x_{14}}{x_{23}x_{24}}\omega_{1}^{2}\,,\quad
t=-\frac{s}{z}\,,\quad u=\frac{1-z}{z}s\,.$ (25)

Moreover, for each of the three physical scattering channels, we have a
different admissible range of cross-ratio $z$

$u$-channel: $\displaystyle z\in(-\infty,0)\,,$ (26) $t$-channel:
$\displaystyle z\in(0,1)\,,$ (27) $s$-channel: $\displaystyle
z\in(1,\infty)\,.$ (28)

See lamShao ; Chang:2021wvv ; Mizera:2022sln  for in-depth discussions.

Evaluating three of the delta functions, the kinematic parts can be written
explicitly as

$\displaystyle X_{d=1}(\Delta_{i},x_{i})$
$\displaystyle=\left(\prod_{i<j}x_{ij}^{\Delta_{T}/3-\Delta_{i}-\Delta_{j}}\right)2^{-\Delta_{T}+1}\left(\frac{z}{\sqrt{1-z}}\right)^{1-\Delta_{T}/3}\,,$
(29) $\displaystyle X_{d=2}(\Delta_{i},x_{i})$
$\displaystyle=\left(\prod_{i<j}x_{ij}^{h/3-h_{i}-h_{j}}\bar{x}_{ij}^{\bar{h}/3-\bar{h}_{i}-\bar{h}_{j}}\right)\delta(z-\bar{z})2^{-\Delta_{T}+2}\absolutevalue{\frac{z}{\sqrt{1-z}}}^{-\frac{\Delta_{T}}{3}}\,.$
(30)

Note that the 1d kinematic piece does not possess the infamous momentum
conservation $\delta(z-\bar{z})$ which shows up for massless $2\to 2$
scattering in 2d.

Meanwhile, the dynamical data is contained in
$\operatorname{\mathcal{A}}(\Delta_{T},z)$, which boils down to a single
Mellin transform333For convenience, we will write $T(\omega,z)$ whenever we
refer to a general amplitude in the $(\omega,z)$ variables i.e. before Mellin-
transforming, as no confusions should arise. of $T(\omega,z)$ in center of
mass energy $\omega\coloneqq\sqrt{\epsilon_{s}s}$ where
$\epsilon_{s}\coloneqq\epsilon_{1}\epsilon_{2}$

$\operatorname{\mathcal{A}}(\Delta_{T},z)=\int_{0}^{\infty}d\omega\,\omega^{\Delta_{T}-4}\,T\Big{(}\epsilon_{s}\omega^{2},-\epsilon_{s}\frac{\omega^{2}}{z}\Big{)}\,.$
(31)

Altogether, any celestial amplitude
$\widetilde{\mathcal{A}}_{d=1}(\Delta_{i},x_{i})$ takes the form of a 1d CFT
correlator

$\widetilde{\mathcal{A}}_{d=1}(\Delta_{i},x_{i})=\frac{\left(\frac{x_{14}^{2}}{x_{24}^{2}}\right)^{-\frac{1}{2}\Delta_{12}}\left(\frac{x_{14}^{2}}{x_{13}^{2}}\right)^{\frac{1}{2}\Delta_{34}}}{\left(x_{12}^{2}\right)^{\frac{1}{2}(\Delta_{1}+\Delta_{2})}\left(x_{34}^{2}\right)^{\frac{1}{2}(\Delta_{3}+\Delta_{4})}}f(z)$
(32)

with the conformally invariant part (aka “stripped four-point function”)

$f(z)=2^{-\Delta_{T}+1}z(1-z)^{-\frac{1}{2}(1-\Delta_{12}+\Delta_{34})}\operatorname{\mathcal{A}}(\Delta_{T},z)\,.$
(33)

For four identical particles with $\Delta_{i}=\Delta$, this reduces to

$\displaystyle\widetilde{\mathcal{A}}\left(\Delta,x_{i}\right)$
$\displaystyle=\frac{1}{\left|x_{12}\right|^{2\Delta}\left|x_{34}\right|^{2\Delta}}f(z)\,,$
(34) $\displaystyle f(z)$
$\displaystyle=2^{-\Delta_{T}+1}\frac{|z|}{\sqrt{|z-1|}}\operatorname{\mathcal{A}}(\Delta_{T},z)\,.$
(35)

###  3.2 The $O(N)$ Amplitude

We may now compute and analyze the $O(N)$ celestial amplitude by directly
applying the tools from the previous section. We focus on the physical
$s$-channel $12\to 34$ so that $z\in(1,\infty)$ for concreteness and compare
each of our results to the (2+1)d massless scalar scattering studied in
lamShao . As clear from the previous subsection, to perform the Mellin
transform (31) we need to recast the amplitude (15) in Lorentzian signature.
The physical analytic continuation consistent with the
$i\epsilon$-prescription at small $\lambda$ leads to

$\displaystyle
T_{4}^{ab,cd}\Big{(}\omega^{2},-\frac{\omega^{2}}{z}\Big{)}=-\frac{\lambda}{N}\bigg{[}\frac{\omega^{2}\delta^{ab}\delta^{cd}}{\omega^{2}+i\frac{\lambda}{16}\omega+2\mu^{2}}$
$\displaystyle+\frac{\omega^{2}\delta^{ac}\delta^{bd}}{\omega^{2}+\frac{\lambda}{16}\sqrt{z}\omega-2\mu^{2}z}$
$\displaystyle+\frac{\omega^{2}\delta^{ad}\delta^{bc}}{\omega^{2}+\frac{\lambda}{16}\sqrt{\frac{z}{z-1}}\omega-2\mu^{2}\frac{z}{z-1}}\bigg{]}\,.$
(36)

For future reference, let us spell out the position of the different poles in
this expression:

$\displaystyle\omega^{\mp}_{ab,cd}$
$\displaystyle=\frac{1}{2}\Bigg{(}-i\frac{\lambda}{16}\pm\sqrt{-\Big{(}\frac{\lambda}{16}\Big{)}^{2}-8\mu^{2}}\Bigg{)},$
(37) $\displaystyle\omega^{\mp}_{ac,bd}$
$\displaystyle=\frac{1}{2}\sqrt{z}\Bigg{(}-\frac{\lambda}{16}\pm
i\sqrt{-\Big{(}\frac{\lambda}{16}\Big{)}^{2}-8\mu^{2}}\Bigg{)},$ (38)
$\displaystyle\omega^{\mp}_{ad,bc}$
$\displaystyle=\frac{1}{2}\sqrt{\frac{z}{z-1}}\Bigg{(}-\frac{\lambda}{16}\pm
i\sqrt{-\Big{(}\frac{\lambda}{16}\Big{)}^{2}-8\mu^{2}}\Bigg{)},$ (39)

where $\omega^{\mp}_{ab,cd}$ stands for the poles associated to the term
proportional to $\delta^{ab}\delta^{cd}$ in (36),444The “mismatch” between the
signs in the left and right sides of these expressions come from the fact that
under Wick rotation, $p_{\pm}$ in (11) becomes $\omega_{\pm}$. and similarly
for $\omega^{\mp}_{ac,bd}$ and $\omega^{\mp}_{ad,bc}$.

All three of the contributions in (36) require Mellin transforms of the
form555See entry (12) of Section 6.2 of bateman1954tables .

$\int_{0}^{\infty}\frac{\omega^{\Delta_{T}-2}}{\omega^{2}+2a\cos\theta\,\omega+a^{2}}d\omega=\pi
a^{\Delta_{T}-3}\frac{\sin{[(\Delta_{T}-2)\theta]}}{\sin{\theta}\sin{(\pi\Delta_{T})}}\,,$
(40)

where, to avoid IR and UV divergences, the integral requires
$-1<\real(\Delta_{T})-2<1$. We also note that none of the three terms have
poles on the positive real axis (recall $\mu^{2}<0$ and $z\in(1,\infty)$),
hence the function is integrable and admits a Mellin transform in the usual
sense. This is in contrast with the tree-level case $\lambda\to 0$ where the
poles pinch the integration region and need to be regulated lamShao . In our
case the finite coupling acts as a natural regulator.

Putting this together, the Mellin-transformed $s$-channel amplitude takes the
following form:

$\operatorname{\mathcal{A}}^{ab,cd}(\alpha,z)=2^{2+2\alpha}\operatorname{\mathcal{N}}\operatorname{\mathcal{N}}_{\alpha}\left[\delta^{ab}\delta^{cd}e^{i\pi\alpha}+\delta^{ac}\delta^{bd}z^{\alpha}+\delta^{ad}\delta^{bc}\left(\frac{z}{z-1}\right)^{\alpha}\right],\quad
z\in(1,\infty),$ (41)

where we have defined

$\operatorname{\mathcal{N}}\coloneqq\frac{\pi\lambda}{N\sin\theta}\,,\qquad\operatorname{\mathcal{N}}_{\alpha}\coloneqq\frac{(-2\mu^{2})^{\alpha}}{2^{2+2\alpha}}\frac{\sin[(2\alpha+1)\theta]}{\sin(2\pi\alpha)}\,,\qquad\alpha\coloneqq(\Delta_{T}-3)/2\,.$
(42)

For the case of four identical external weights $\Delta_{i}=\Delta$ the
(conformally invariant piece of the) celestial amplitude is

$\displaystyle f^{ab,cd}_{s}(z)$
$\displaystyle=\frac{1}{2^{2+2\alpha}}\frac{z}{\sqrt{z-1}}\operatorname{\mathcal{A}}(\alpha,z),$
(43)
$\displaystyle=\operatorname{\mathcal{N}}\operatorname{\mathcal{N}}_{\alpha}\frac{z}{\sqrt{z-1}}\left[\delta^{ab}\delta^{cd}e^{i\pi\alpha}+\delta^{ac}\delta^{bd}z^{\alpha}+\delta^{ad}\delta^{bc}\left(\frac{z}{z-1}\right)^{\alpha}\right],\quad
z\in(1,\infty).$ (44)

We find that (up to color structures) this result has the same $z$-dependence
as the tree-level example originally studied by Lam and Shao in lamShao ,
which is itself closely related to a Generalized Free Field theory, with an
overall kinematic pre-factor and an imaginary piece which marks the physical
channel.

It is instructive to see how the result of lamShao  appears in the tree-level
approximation. We approach this limit by taking $\lambda\to 0$ while keeping
$-\mu^{2}$ fixed, which amounts to take $\theta\to\pi/2$. The result is

$f^{ab,cd}_{s}(z)\xrightarrow[\text{$\lambda\to
0$}]{\text{}}\frac{\pi\lambda\big{(}-2\mu^{2}\big{)}^{\alpha}}{2^{4\Delta}N\cos{(2\pi\Delta)}}\frac{z}{\sqrt{z-1}}\left[\delta^{ab}\delta^{cd}e^{i\pi\alpha}+\delta^{ac}\delta^{bd}z^{\alpha}+\delta^{ad}\delta^{bc}\left(\frac{z}{z-1}\right)^{\alpha}\right],$
(45)

which is precisely the result of lamShao  up to an extra power of
$-2\mu^{2}=M^{2}$. This is nothing but the effect of the extra power of
$\omega^{2}$ dictated by Adler zero.

One distinct effect of the all-loop resummation in comparison with the tree-
level amplitude, besides the modification of the prefactors
$\operatorname{\mathcal{N}}$ and $\operatorname{\mathcal{N}}_{\alpha}$, is the
modification in the convergence region for the Mellin integral. This
modification occurs because of an overall extra power of $\omega^{2}$ in the
numerator of the resummed $T_{4}^{ab,cd}$ relative to the tree-level result.
Indeed, we have that:

Tree-Level: $\displaystyle 3<\real(\Delta_{T})<5,$ (46) Resummed:
$\displaystyle 1<\real(\Delta_{T})<3.$ (47)

Interestingly, this convergence range for the resummed result allows all of
the scattering $\Delta_{i}$ to be on the principal continuous series of
$SO(1,2)$, i.e. $\Delta_{i}=\frac{1}{2}+i\ell$,
$\ell\in\operatorname{\mathbb{R}}$. Famously, conformal primary wavefunctions
with $\Delta$ on the principal continuous series form a complete set of
normalizable solutions to the massless Klein-Gordon equation, with respect to
the Klein-Gordon inner product pasterskiShao .

###  3.3 Analyticity in $\Delta$ and EFT Expansion

Let us close this subsection making explicit the analytic structure of the
celestial amplitude in the complex $\Delta_{T}$-plane. As expected from a
field theory result, we observe that there are infinitely many simple poles at
both positive and negative values of $\Delta_{T}$ (in contrast to a quantum
gravity/string theory result, as discussed in Arkani-Hamed:2020gyp ). Clearly
there are simple poles at integers $\Delta_{T}=m$ for $m\leq 1$ and $m\geq 3$.
At $m=2$, the celestial amplitude is finite in accordance with our
aforementioned comment on the amplitude being well-defined on the principal
continuous series $\Delta_{i}=\frac{1}{2}+i\ell$,
$\ell\in\operatorname{\mathbb{R}}$.

As pointed out in Arkani-Hamed:2020gyp , the residues of the celestial
amplitude in $\Delta_{T}$ are not arbitrary but are in direct correspondence
to EFT operators at four points. They correspond to the coefficients of the
low energy expansion of the corresponding momentum space amplitude
$T(\omega,z)$ if the poles in $\Delta_{T}$ organize in the left side of the
$\Delta_{T}$-plane, and to the coefficients of the high energy expansion if
the poles organize in the right side of the $\Delta_{T}$-plane. In our
context, these are the $m\leq 1$ and $m\geq 3$ families of poles respectively.

Let us then illustrate a specific all-loop resummed instance of this
observation in both the UV and IR. This means we perform the expansion of the
field theory result as $\omega\to 0$:666This expansion can be performed with
the aid of the series
$\sum_{k=1}^{\infty}p^{k}\sin{(kx)}=\frac{p\sin{x}}{1-2p\cos{x}+p^{2}}\,,\qquad|p|<1\,.$

$\displaystyle
T(\omega,z)=-\frac{\lambda}{N}\sum_{k=1}^{\infty}\Bigg{[}\delta^{ab}\delta^{cd}\Bigg{(}\frac{e^{i\frac{\pi}{2}}\omega}{\sqrt{-2\mu^{2}}}\Bigg{)}^{k+1}$
$\displaystyle+\delta^{ac}\delta^{bd}\Bigg{(}\frac{e^{i\pi}\omega}{\sqrt{-2\mu^{2}}\sqrt{z}}\Bigg{)}^{k+1}$
$\displaystyle+\delta^{ad}\delta^{bc}\Bigg{(}\frac{e^{i\pi}\omega\sqrt{\frac{z-1}{z}}}{\sqrt{-2\mu^{2}}}\Bigg{)}^{k+1}\Bigg{]}\frac{\sin{(k\theta)}}{\sin{\theta}}\,,$
(48)

whose coefficients indeed reproduce the residues of (41) as $\Delta_{T}\to m$
for $m\leq 1$. Similarly, the high energy expansion

$\displaystyle
T(\omega,z)=-\frac{\lambda}{N}\sum_{k=1}^{\infty}\Bigg{[}\delta^{ab}\delta^{cd}\Bigg{(}e^{-i\frac{\pi}{2}}\frac{\sqrt{-2\mu^{2}}}{\omega}\Bigg{)}^{k-1}$
$\displaystyle+\delta^{ac}\delta^{bd}\Bigg{(}\frac{e^{i\pi}\sqrt{-2\mu^{2}}\sqrt{z}}{\omega}\Bigg{)}^{k-1}$
$\displaystyle+\delta^{ad}\delta^{bc}\Bigg{(}\frac{e^{i\pi}\sqrt{-2\mu^{2}}\sqrt{\frac{z}{z-1}}}{\omega}\Bigg{)}^{k-1}\Bigg{]}\frac{\sin{(k\theta)}}{\sin{\theta}}\,,$
(49)

is analogously related to the poles of (41) as $\Delta_{T}\to m$ for $m\geq
3$.

###  3.4 S-Matrix Permutation Symmetry

Repeating the results of the previous section for different physical regions,
we can write the three celestial amplitudes as:

$\displaystyle f_{u}^{abcd}(z)$
$\displaystyle=\operatorname{\mathcal{N}}\operatorname{\mathcal{N}}_{\alpha}\frac{-z}{\sqrt{1-z}}\left(\delta^{ab}\delta^{cd}+\delta^{ac}\delta^{bd}(-z)^{\alpha}+\delta^{ad}\delta^{bc}e^{i\pi\alpha}\left(\frac{-z}{1-z}\right)^{\alpha}\right)\,,$
$\displaystyle\quad z\in(-\infty,0)\,,$ (50) $\displaystyle f_{t}^{abcd}(z)$
$\displaystyle=\operatorname{\mathcal{N}}\operatorname{\mathcal{N}}_{\alpha}\frac{z}{\sqrt{1-z}}\left(\delta^{ab}\delta^{cd}+\delta^{ac}\delta^{bd}e^{i\pi\alpha}z^{\alpha}+\delta^{ad}\delta^{bc}\left(\frac{z}{1-z}\right)^{\alpha}\right)\,,$
$\displaystyle\quad z\in(0,1)\,,$ (51) $\displaystyle f_{s}^{abcd}(z)$
$\displaystyle=\operatorname{\mathcal{N}}\operatorname{\mathcal{N}}_{\alpha}\frac{z}{\sqrt{z-1}}\left(\delta^{ab}\delta^{cd}e^{i\pi\alpha}+\delta^{ac}\delta^{bd}z^{\alpha}+\delta^{ad}\delta^{bc}\left(\frac{z}{z-1}\right)^{\alpha}\right)\,,$
$\displaystyle\quad z\in(1,\infty)\,.$ (52)

The functions are not analytic continuations of one another in $z$, but
“permutation symmetry” (i.e. relabelling the external scattering states) of
the scattering amplitude implies:

$\displaystyle f_{t}^{abcd}\left(\frac{z}{z-1}\right)$
$\displaystyle=f_{u}^{bacd}(z)=f_{u}^{abdc}(z)\,,$ $\displaystyle\quad
z\in(-\infty,0)\,,$ (53) $\displaystyle
z^{2\Delta}f_{t}^{abcd}\left(\frac{1}{z}\right)$
$\displaystyle=f_{s}^{acbd}(z)=f_{s}^{dbca}(z)\,,$ $\displaystyle\quad
z\in(1,\infty)\,,$ (54)

as well as

$z^{-2\Delta}f_{t}^{abcd}\left(\frac{1}{z}\right)=(1-z)^{-2\Delta}f_{t}^{adcb}(z)=(1-z)^{-2\Delta}f_{t}^{cbad}(z)\,,\quad
z\in(0,1)\,.\\\ $ (55)

These relations are the color analogs of those given for the 1d CCFT coming
from massless scattering in lamShao .

These equations are all familiar from 1d CFT as we explain in Appendix A.1
(see also Section 2 of Mazac:2018qmi ). In short, a 1d CFT correlator is
defined piecewise on the three disconnected regions $(-\infty,0)$, $(0,1)$,
and $(1,\infty)$. Similar to our 1d celestial amplitudes, the pieces cannot
generally be related by analytic continuation in $z$, but if it is a
correlator of bosons, then the pieces are related by Bose symmetry. Those
relations implied on the 1d boson correlators are precisely those we have
found above.

In other words, the original Bose symmetry of the S-matrix implies
relationships between the celestial amplitudes in their different physical
regions, and those relations are necessary for the celestial amplitude to
literally “be a CFT four point function of bosons” on the celestial circle.
This is consistent with the extrapolate dictionary Pasterski:2021dqe ;
Donnay:2022sdg .

Note: this is not crossing symmetry, which is a statement about obtaining
scattering amplitudes in different physical channels by analytic continuation
of the complexified S-matrix. First steps to understanding crossing symmetry
in celestial holography have appeared in Mizera:2022sln  (see also
Hannesdottir:2022bmo  for related discussion).

##  4 The Imaginary Part of the Celestial Amplitude and Resonances

In this section we study the analytic properties of scattering amplitudes in
celestial variables, extending the work of Chang:2021wvv  to the non-
perturbative regime by allowing for unstable resonances (like those in the
$O(N)$ model). In particular, we see how poles associated with unstable
particles control the imaginary part of the correlator relevant to the optical
theorem. We will check all of our results explicitly in the $O(N)$ model case,
whose resonances we have reviewed in Section 2.

$\omega$-plane Figure 2: A generic amplitude $T(s,t)$ when translated to the
$\omega$ plane will NOT satisfy $T(\omega,z)^{*}=T(\omega^{*},z)$. Note that
the $t$-channel branch cuts of the amplitude unfold (green) along the
imaginary $\omega$-axis.

Generally we will be interested in quantities of the form

$\int_{0}^{\infty}d\omega\ \omega^{\Delta_{T}-4}\,T(\omega,z)\,,$ (56)

where in general $T(\omega,z)$ could have branch cuts in addition to that of
$\omega^{\Delta_{T}-4}$ (see Figure 2). Commonly, one relates the imaginary
part of the scattering amplitude in the Mandelstam variables $(s,t,u)$

$\operatorname{Im}T\left(s,t\right)=\frac{1}{2i}\left(T\left(s,t\right)-{T\left(s,t\right)^{*}}\right)$
(57)

to the corresponding discontinuity across the s-channel cut

$\operatorname{Disc}T\left(s,t\right)=\lim_{\varepsilon\rightarrow
0^{+}}\frac{1}{2i}\left({T}\left(s+i\varepsilon,t\right)-{T}\left(s-i\varepsilon,t\right)\right)$
(58)

However, notice that this is based on the Schwartz reflection principle which
requires the amplitude $T(s,t)$ in the $s$-plane to be real in an interval on
the negative real $s$-axis.777We can illustrate this with the following
prototypical example: Take $f(s)=\ln{(e^{-i\pi}s)}$ with
$0\leq\arg{(s)}<2\pi$. If we evaluate $s$ on the negative real axis, i.e.
$s=e^{i\pi}\bar{s}$ for $\bar{s}$ some positive real number, then
$f(e^{i\pi}\bar{s})=\ln{(\bar{s})}$ is real so the discontinuity and imaginary
parts should be related. Indeed, $\imaginary(f(s))\big{|}_{s=\bar{s}}=-\pi$,
while
$\mathrm{Disc}(f(s))\big{|}_{s=\bar{s}}=\ln{(e^{-i\pi}\bar{s})}-\ln{(e^{-i\pi}e^{2\pi
i}\bar{s})}=-2\pi i$ for $\bar{s}$ real, so that
$\imaginary(f(s))=\frac{1}{2i}\mathrm{Disc}(f(s))$ as expected. Notice that
had we defined $f(s)=\ln{(e^{-in\pi}s)}$ for some other odd integer $n$, the
function would not have been real at $s=\bar{s}$ real; and indeed, then the
imaginary parts and discontinuity would not have matched.

In contrast, in $(\omega,z)$ variables the amplitude $T(\omega,z)$ is
generically not real along the negative $\omega$-axis, and as a consequence
one cannot simply equate the imaginary part with the discontinuity across some
branch cut. One instance where this phenomenon arises is the following: take
particles $1$ and $2$ to be incoming and consider possible resonances which
manifest themselves as poles of $T(s,t)$ in $s=(p_{1}+p_{2})^{2}$ in the
second-sheet of $s$. In the $\omega$-plane this translates to the fact that
resonances will appear in the lower-half $\omega$-plane with no conjugate pole
in the upper-half $\omega$-plane, thus making $T(\omega,z)$ a complex function
of $\omega$, i.e. such that $T(\omega,z)^{*}\neq T(\omega^{*},z)$ (see Figure
2). Indeed, the $O(N)$ model amplitude (36) manifestly provides an example
where $T(\omega,z)$ is such that it is never real in an interval in the real
$\omega$-axis, and the amplitude has a non-trivial imaginary part on the
positive $\omega$-axis despite having no discontinuity.

In general, we will assume $T(\omega,z)$ can be written as a sum of two
pieces:

$T(\omega,z)=t_{0}(\omega,z)+t_{\log}(\omega,z)\,.$ (59)

Here $t_{0}(\omega,z)$ is defined as a rational piece with no branch cut and
being possibly complex through the real axis. The remaining
$t_{\log}(\omega,z)$ is defined as the piece which contains the logarithmic
branch cuts; we assume it to be a real function of $\omega$ up to the
discontinuities arising from said branch cuts.888This is reminiscent of the
expansion of $1$-loop amplitudes in terms of $m$-gon $1$-loop scalar integrals
in $D$ dimensions
$A_{n}^{1-\mathrm{loop}}=\sum_{i}C_{D}^{(i)}I^{(i)}_{D;n}+\sum_{j}C_{D-1}^{(j)}I^{(j)}_{D-1;n}+\cdots+\sum_{k}C_{2}^{(k)}I^{(k)}_{2;n}+\mathrm{Rational\
Terms},$ (60) with the integrals implying the logarithmic pieces, and the rest
contained in the rational terms (see e.g. Elvang:2013cua  and references
within).

To properly define the expression (56) we need to specify the integration
contour by taking the different branch cuts into account. Generically, the
amplitude $T(s,t)$ has logarithmic branch cuts in the $s$-channel,
$t$-channel, or $u$-channel. Since we will be working with massless theories,
let us consider the case where the $s$-channel branch point sits at the
origin. Notice that the $i\epsilon$-prescription then instructs us to consider
a small gap with the $u$-channel cut as we approach the forward limit, i.e. as
$t\to 0\iff z\to\infty$. Since we will be interested in highlighting the
effect of resonances let us take the poles to lie away from the positive real
$\omega$-axis; although it will not be difficult to include such poles at
positive real $\omega$ and massive branch cuts later as done in Chang:2021wvv
. Altogether, the generic cases we consider have a branch cut lying on (or
rather, slightly below) the positive real axis, loosely referred to as the
$s$-channel cut.999See Hannesdottir:2022bmo  for an extended discussion of the
$i\epsilon$-prescription for the S-matrix, including the effects of
resonances.

In order to see the effect of resonances on the imaginary part of the
celestial amplitude let us focus on the case where $t_{0}(\omega,z)$ fails to
be a real function of $\omega$ because of these resonances. For instance,
taking particles 1 and 2 to be incoming, there are poles of $t_{0}(\omega,z)$
in the lower-half $\omega$-plane which appear with no complex conjugate. These
are the $s$-channel resonances of $T(\omega,z)$, which we label as
$\omega_{\mathrm{res}}$. All other poles come then in conjugate pairs with
conjugate residues. An illustration of this analytic structure is depicted in
Figure 3.

To compute the Mellin-transform of $t_{0}(\omega,z)$, we can proceed by
extending the integral to the whole real line, and then deforming the contour
to pick the residues in the upper $\omega$-plane. The integral has a branch
cut beginning at $\omega=0$, for which we choose the branch cut as depicted in
Figure 3. The extension of the integral to the whole real line is then,
explicitly:

$\displaystyle\int_{0}^{\infty}d\omega\,\omega^{\Delta_{T}-4}\,t_{0}(\omega,z)$
$\displaystyle=\frac{1}{(1+e^{i\pi\Delta_{T}})}\int_{-\infty}^{\infty}d\omega\,\omega^{\Delta_{T}-4}\,\frac{t_{0}(\omega,z)+t_{0}(-\omega,z)}{2}$
$\displaystyle+\frac{1}{(1-e^{i\pi\Delta_{T}})}\int_{-\infty}^{\infty}d\omega\,\omega^{\Delta_{T}-4}\,\frac{t_{0}(\omega,z)-t_{0}(-\omega,z)}{2}$
$\displaystyle=\frac{ie^{-i\pi\Delta_{T}}}{4\sin{(\frac{\pi\Delta_{T}}{2})}\cos{(\frac{\pi\Delta_{T}}{2})}}\int_{-\infty}^{\infty}d\omega\,\omega^{\Delta_{T}-4}\,t_{0}(\omega,z)$
$\displaystyle-\frac{i}{4\sin{(\frac{\pi\Delta_{T}}{2})}\cos{(\frac{\pi\Delta_{T}}{2})}}\int_{-\infty}^{\infty}d\omega\,\omega^{\Delta_{T}-4}\,t_{0}(-\omega,z)\,.$
(61)

Specializing to the imaginary part
$\mathrm{Im}(t_{0})(\omega,z)=(t_{0}(\omega,z)-t_{0}^{*}(\omega,z))/2i$ we
find poles in both the lower-half and upper-half $\omega$-plane, at
$\omega_{\mathrm{res}}$ and $\omega^{*}_{\mathrm{res}}$, arising from the
$t_{0}(\omega,z)$ and $t_{0}^{*}(\omega,z)$ terms respectively. Any other
poles that could have appeared conjugated cancel when taking the imaginary
part (see Figure 3). In other words, the poles corresponding to resonances
from the second-sheet appear in the lower-half $\omega$-plane and without
complex conjugate partners. When looking at the imaginary part of the
amplitude these contributions do not disappear, the way poles not in the
scattering channel would, and so provide residues for the contour integral in
(61).101010It is interesting to note that if we had poles on the first-sheet
of the S-matrix (which generally indicates issues with causality of the
scattering amplitude), they would turn into non-complex conjugate partners in
the upper half $\omega$-plane, and so would contribute to the contour
integrals with the wrong-sign residues.

$\omega$-plane$\times$$\times$$\times$$\times$$\times$$\times$

$\omega$-plane$\times$$\times$$\times$$\times$

Figure 3: The analytic structure for $s$-channel resonances. Left: Example of
the $\omega$-poles of the momentum space amplitude $T$ in $(\omega,z)$
coordinates. The red poles appear without complex conjugate partners, and
denote the resonances of the amplitude. Right: $\omega$-pole structure for the
imaginary part $(T(\omega,z)-T^{*}(\omega,zd))/2i$. We have used the $O(N)$
model to exemplify the analytic structure.

Since any poles appearing with their conjugate in $t_{0}(\omega,z)$ have been
cancelled in $\mathrm{Im}(t_{0})(\omega,z)$, we can restrict to only those
poles that are associated with resonances. We do this in the following.
Applying (61) to $\mathrm{Im}(t_{0})(\omega,z)$ instead of $t_{0}(\omega,z)$,
we see that in the first term of the last expression only
$-t_{0}^{*}(\omega,z)$ contributes since $t_{0}(\omega,z)$ has no poles in the
upper $\omega$-plane, and similarly in the second term
$\mathrm{Im}(t_{0})(-\omega,z)$ where only $t_{0}(-\omega,z)$ contributes (the
poles of $t_{0}^{*}(-\omega,z)$ now appearing in the lower $\omega$-plane).
Altogether, we obtain the contribution of $t_{0}(\omega,z)$ to the imaginary
part of the celestial amplitude as

$\frac{-i\pi}{4\sin{(\frac{\pi\Delta_{T}}{2})}\cos{(\frac{\pi\Delta_{T}}{2})}}\sum_{\omega_{\mathrm{res}}}\bigg{[}e^{-i\pi\Delta_{T}}\,\underset{\omega=\omega^{*}_{\mathrm{res}}}{\mathrm{\mathbf{Res}}}\Big{(}\omega^{\Delta_{T}-4}t_{0}^{*}(\omega,z)\Big{)}+\underset{\omega=-\omega_{\mathrm{res}}}{\mathrm{\mathbf{Res}}}\Big{(}\omega^{\Delta_{T}-4}t_{0}(-\omega,z)\Big{)}\bigg{]}\,,$
(62)

which is solely determined by the residues at the $s$-channel resonances. In
particular, when $t_{\log}(\omega,z)=0$, the imaginary part of the celestial
amplitude is fully determined by the $s$-channel resonances, and is equal to
the above expression.

Interestingly, the previous expression allows us to consider the question of
how to identify resonances directly from the celestial perspective, with no
reference to the more standard momentum space. Namely, expand the imaginary
part of the celestial amplitude in the form (62) and read-off the $\Delta_{T}$
dependence. This should be given by contributions that schematically depend as
$(\omega_{\mathrm{res}}^{*})^{\Delta_{T}-4}$ or
$(-\omega_{\mathrm{res}})^{\Delta_{T}-4}$ with $\omega_{\mathrm{res}}^{*}$ or
$-\omega_{\mathrm{res}}$ having a positive imaginary part, recalling that
$\omega_{\mathrm{res}}$ corresponds to a resonance that in the $\omega$-plane
lied in the lower-half plane. The resonances can then be identified as the
conjugates of those contributions providing the non-trivial $\Delta_{T}$
dependence in (62). At the end of this section we will illustrate the previous
point using the $O(N)$ model amplitude as an example.

If we also consider the logarithmic pieces, we can write the imaginary part of
the celestial amplitude as:

$\displaystyle\imaginary(\mathcal{A}\big{(}\Delta_{T},z)\big{)}=$
$\displaystyle\frac{-i\pi}{4\sin{(\frac{\pi\Delta_{T}}{2})}\cos{(\frac{\pi\Delta_{T}}{2})}}$
$\displaystyle\times\sum_{\omega_{\mathrm{res}}}\bigg{[}e^{-i\pi\Delta_{T}}\,\underset{\omega=\omega^{*}_{\mathrm{res}}}{\mathrm{\mathbf{Res}}}\Big{(}\omega^{\Delta_{T}-4}t_{0}^{*}(\omega,z)\Big{)}+\underset{\omega=-\omega_{\mathrm{res}}}{\mathrm{\mathbf{Res}}}\Big{(}\omega^{\Delta_{T}-4}t_{0}(-\omega,z)\Big{)}\bigg{]}$
$\displaystyle+\int_{0}^{\infty}d\omega\,\omega^{\Delta_{T}-4}\big{[}\mathrm{Disc}(t_{\log})\big{]}(\omega,z)\,.$
(63)

If instead of a branch cut at the origin we have a discrete series of simple
poles at positive masses $\omega=m_{i}$ and a massive branch cut starting at
$\omega=M$ described by a real function of $\omega$ (up to the discontinuities
implied by such branch cut as in Chang:2021wvv ), then we just replace the
last integral in (63) by a discontinuity integral starting at $M$ instead and
a series of residues at $\omega=m_{i}$. That is, the result of Chang:2021wvv
is recovered, but with contributions coming from resonances added (indeed, as
we will see next, the imaginary part of the $O(N)$ model is fully explained by
the latter, with no contributions arising either from discontinuities or poles
at positive masses $\omega=m_{i}$).

We see that the imaginary part of a celestial amplitude captures both “the
data at the positive real $\omega$-axis” such as discontinuities or residues
of poles at positive masses (had we considered massive theories), as well as
the data from resonances at the lower-half $\omega$-plane through non-trivial
contributions.

Let us illustrate the previous discussion for the case of the $O(N)$ model.
Since we are considering $12\rightarrow 34$ scattering, the contributions
proportional to $\delta^{ac}\delta^{bd}$ and $\delta^{ad}\delta^{bc}$ have
complex conjugate poles (depicted blue in Figure 3) and by the previous
discussion we know they will not play a role in what follows.111111Similarly,
if we were considering physical $13\rightarrow 24$ ($14\rightarrow 23$)
scattering, then $t>0$ ($u>0$) and the $\delta^{ac}\delta^{bd}$
($\delta^{ad}\delta^{bc}$) term would have the resonance poles in the lower-
half $\omega$-plane. The remaining terms would have conjugate poles. The
$O(N)$ model is also a case where $t_{\log}(\omega,z)=0$ and where no massive
poles appear in the real positive axis for finite values of the coupling
$\lambda$, so, we can cleanly concentrate on the non-trivial aspects of
resonances only.

The previous paragraph implies we only need to consider the contribution
proportional to $\delta^{ab}\delta^{cd}$:

$\frac{\omega^{2}}{(\omega^{2}+i\frac{\lambda}{16}\omega+2\mu^{2})}\delta^{ab}\delta^{cd}=\frac{\omega^{2}}{{(\omega-\omega_{+}})(\omega-\omega_{-})}\delta^{ab}\delta^{cd}\,,$
(64)

where the $s$-channel resonances of $T^{ab,cd}_{4}(\omega,z)$ –which lie in
the lower-half $\omega$-plane– are the $\omega^{\pm}_{ab,cd}$ poles in (37),
which we have written here merely as
$\omega_{\pm}\coloneqq\omega^{\pm}_{ab,cd}$ for simplicity. These appear
depicted in red in Figure 3.

We could now directly apply (62), but let us notice that the $O(N)$ model is a
case where (64) satisfies $T^{*}(\omega,z)=T(-\omega,z)$. In general, when
this condition is satisfied for some $T(\omega,z)$ we have that

$\sum_{\omega_{\mathrm{res}}}\underset{\
\omega=\omega^{*}_{\mathrm{res}}}{\mathrm{\mathbf{Res}}}\Big{(}\omega^{\Delta_{T}-4}T^{*}(\omega,z)\Big{)}=\sum_{\omega_{\mathrm{res}}}\underset{\
\omega=-\omega_{\mathrm{res}}}{\mathrm{\mathbf{Res}}}\Big{(}\omega^{\Delta_{T}-4}T(-\omega,z)\Big{)}\,,$
(65)

in which case Equation (62) simplifies to

$\boxed{\mathrm{Im}\big{(}\mathcal{A}(\Delta_{T},z)\big{)}=-\frac{\pi}{(1-e^{\pi
i\Delta_{T}})}\sum_{\omega_{\mathrm{res}}}\underset{\
\omega=\omega^{*}_{\mathrm{res}}}{\mathrm{\mathbf{Res}}}\Big{(}\omega^{\Delta_{T}-4}T^{*}(\omega,z)\Big{)}\,.}$
(66)

In particular, this expression can be used to more directly evaluate
$\mathrm{Im}(\mathcal{A})$ in the case of the $O(N)$ model. The relevant
residues of $T^{*}(\omega,z)$ for the $O(N)$ model are located at

$\omega_{\pm}^{*}=\sqrt{-2\mu^{2}}e^{i\pi/2}e^{\pm i\theta}\,,$ (67)

and are straightforwardly evaluated to verify that:

$\underset{\
\omega=\omega_{\pm}^{*}}{\mathrm{\mathbf{Res}}}\big{(}\omega^{\Delta_{T}-4}T^{*}(\omega)\big{)}=-\frac{(e^{-i\pi}\omega_{\mp})^{\Delta_{T}-4}}{2\omega_{\mp}}\underset{\
\omega^{2}=\omega_{\mp}^{2}}{\mathrm{\mathbf{Res}}}\big{(}T(\omega)\big{)}=\frac{(e^{-i\pi}\omega_{\mp})^{\Delta_{T}-4}}{2\omega_{\mp}}T^{ab}_{3}(\omega_{\mp})T^{cd}_{3}(\omega_{\mp})\,.$
(68)

In this expression the first equality stresses the point that the imaginary
part of the celestial amplitudes arises from resonances going on-shell. This
is also recognized in the last equality, which has been written as a product
of the three-point function (18) satisfying (19).121212Up to a Wick-rotation
taking us from Euclidean to Lorentzian signature. Obviously, this leads to the
imaginary part of the celestial amplitude:

$\displaystyle\mathrm{Im}\big{(}\mathcal{A}(\Delta_{T},z)\big{)}$
$\displaystyle=-\frac{\pi e^{-i\pi(\Delta_{T}-4)}}{2(1-e^{\pi
i\Delta_{T}})}\bigg{[}\omega_{+}^{\Delta_{T}-5}T^{ab}_{3}(\omega_{+})T^{cd}_{3}(\omega_{+})+\omega_{-}^{\Delta_{T}-5}T^{ab}_{3}(\omega_{-})T^{cd}_{3}(\omega_{-})\bigg{]}$
(69)
$\displaystyle=-\Big{(}\frac{\pi\lambda}{2N}\Big{)}(-2\mu^{2})^{(\Delta_{T}-3)/2}\frac{\sin{[(\Delta_{T}-2)\theta]}}{\sin{\theta}\sin{(\frac{\pi\Delta_{T}}{2})}}\,\delta^{ab}\delta^{cd},$
(70)

fully determined by the resonances of the theory going on-shell. Note: To
obtain the correct pre-factors for the imaginary part of the celestial
amplitude it was crucial that both resonances appeared in the sum. This fact
will play a role for us again when we study the four-point functions
decomposition into conformal blocks and factorization into three-point
coefficients in Section 5.4.

Let us note that when considering the limit $\lambda\rightarrow 0$, formula
(66) leads to two contributions at
$\omega=\pm\sqrt{-2\mu^{2}}+\mathcal{O}(\lambda)$. These mix together to
cancel the monodromy factor in (66), obtaining:

$\lim_{\lambda\rightarrow
0}\mathrm{Im}\Big{(}\mathcal{A}(\Delta_{T},z)\Big{)}=\pi\underset{\
\omega=-2\mu^{2}}{\mathrm{\mathbf{Res}}}\bigg{(}\omega^{\Delta_{T}-4}T(\omega,z)\Big{|}_{\lambda=0}\bigg{)},$
(71)

in accordance with the non-trivial imaginary part at tree-level as considered
in Chang:2021wvv .

In the converse direction, let us illustrate how to locate the resonances of
the $O(N)$ model directly from the celestial amplitude following the
discussion in the paragraph right below Equation (62). That is, suppose we are
handed the celestial amplitude (41) and we are asked to check for the presence
of resonances. According to the previous discussion, we can take the imaginary
part given by Equation (70), and expand it in the form (62). A quick bit of
algebra shows that:

$\displaystyle\mathrm{Im}\big{(}\mathcal{A}(\Delta_{T},z)\big{)}=\frac{-i\pi\lambda\delta^{ab}\delta^{cd}}{8N\sin{(\frac{\pi\Delta_{T}}{2})}\cos{(\frac{\pi\Delta_{T}}{2})}}\frac{(-2\mu^{2})^{1/2}}{\sin{\theta}}\bigg{\\{}e^{-i\pi\Delta_{T}}\Big{[}(\sqrt{-2\mu^{2}}e^{i(\frac{\pi}{2}-\theta)})^{\Delta_{T}-4}e^{-2i\theta}$
$\displaystyle-(\sqrt{-2\mu^{2}}e^{i(\frac{\pi}{2}+\theta)})^{\Delta_{T}-4}e^{2i\theta}\Big{]}+\Big{[}(\sqrt{-2\mu^{2}}e^{i(\frac{\pi}{2}-\theta)})^{\Delta_{T}-4}e^{-2i\theta}-(\sqrt{-2\mu^{2}}e^{i(\frac{\pi}{2}+\theta)})^{\Delta_{T}-4}e^{2i\theta}\Big{]}\bigg{\\}}.$
(72)

Clearly, the different summands carry factors of the form
$\omega_{*}^{\Delta_{T}-4}$ with $\omega_{*}$ having a positive imaginary part
(e.g. $\omega_{*}=(\sqrt{-2\mu^{2}}e^{i(\frac{\pi}{2}-\theta)}$). Following
our previous discussion the position of the resonances can then be read off as
the conjugates, and as obviously expected they correspond to the resonances
(37), although here we have obtained them from the celestial amplitude instead
of the momentum space amplitude.

##  5 Conformal Decompositions

Studying the conformal partial wave decompositions of 2d celestial amplitudes
has been of great interest in elucidating the optical theorem, soft limits,
light ray operators, and more (see e.g. lamShao ; celestialAmplitudes ;
Fan:2021isc ; Atanasov:2021cje ; Chang:2021wvv ). In this section, we start by
looking at the conformal block decompositions of our 1d celestial amplitudes
(assuming all external scaling dimensions are the same for simplicity). We
find that there are two infinite towers of operators: “single-trace” at
exchange dimension $\Delta_{\operatorname{\mathcal{O}}}=2n+1$, coming
(roughly) from the kinematic pre-factor of the celestial amplitude; and
“double-trace” at exchange dimension
$\Delta_{\operatorname{\mathcal{O}}}=\alpha+n+1$, where
$n\in\operatorname{\mathbb{N}}^{0}$.

Focusing on the imaginary part of the amplitude, we verify that we can
reproduce the coefficients of the conformal block decomposition by the related
conformal partial wave decomposition. The $J=1$ series of antisymmetric
conformal partial waves appear because we consider four-point functions with
colour. We see the cancellations that occur between the discrete series
conformal partial waves and spurious poles obtained via the continuous series.

An important question is to what extent the conformal block coefficients in
the expansions can be viewed as “three-point function coefficients squared.”
We argue that for the single-trace operators, the fact that there are two
poles on the second sheet of the $p^{2}$-plane in the $O(N)$ model is directly
relevant for this to hold. We briefly comment on the extent to which the
double-trace operators can be factorized.

Background on correlation functions in 1d CFT, conformal blocks, and conformal
partial waves appear in Appendix A.

###  5.1 Color Structure of Celestial Amplitudes

The celestial amplitudes in Equations (50) - (52) sew together piecewise into
a reasonable 1d CFT correlator of bosons (see Appendix A.1 for properties of
1d CFT correlators), given by

$f^{abcd}(z)\coloneqq\begin{cases}f_{u}^{abcd}(z)&\text{if
$z\in(-\infty,0)$}\\\ f_{t}^{abcd}(z)&\text{if $z\in(0,1)$}\\\
f_{s}^{abcd}(z)&\text{if $z\in(1,\infty)$}\end{cases}\,.$ (73)

We may further decompose $f^{abcd}(z)$ into its $O(N)$ representations. We
define $O(N)$ color tensors

$T_{S}^{abcd}=\delta^{ab}\delta^{cd},\quad
T_{A}^{abcd}=\delta^{ac}\delta^{bd}-\delta^{ad}\delta^{bc},\quad
T_{T}^{abcd}=\delta^{ac}\delta^{bd}+\delta^{ad}\delta^{bc}-\frac{2}{N}\delta^{ab}\delta^{cd},$
(74)

labelling symmetric, antisymmetric, and traceless components respectively, and
corresponding components $f_{I}(z)$ for $f^{abcd}(z)$, i.e.

$f^{abcd}(z)=f_{S}(z)T_{S}^{abcd}+f_{A}(z)T_{A}^{abcd}+f_{T}(z)T_{T}^{abcd}\,.$
(75)

We might call the choice of definitions in Equation (74) the “$s$-channel
color tensors” since, for example, $T_{S}^{abcd}$ really only looks like the
singlet when scattering $12\to 34$. In decomposing $t$-channel amplitudes, we
instead use the basis given by $T_{S}^{acbd}$, $T_{A}^{acbd}$, $T_{T}^{acbd}$,
and similarly for $u$. i.e.

$\displaystyle f_{u}^{abcd}(z)$ $\displaystyle=f_{u,S}(z)T_{S}^{adcb}$
$\displaystyle+f_{u,A}(z)T_{A}^{adcb}$
$\displaystyle+f_{u,T}(z)T_{T}^{adcb}\,,$ (76) $\displaystyle f_{t}^{abcd}(z)$
$\displaystyle=f_{t,S}(z)T_{S}^{acbd}$ $\displaystyle+f_{t,A}(z)T_{A}^{acbd}$
$\displaystyle+f_{t,T}(z)T_{T}^{acbd}\,,$ (77) $\displaystyle f_{s}^{abcd}(z)$
$\displaystyle=f_{s,S}(z)T_{S}^{abcd}$ $\displaystyle+f_{s,A}(z)T_{A}^{abcd}$
$\displaystyle+f_{s,T}(z)T_{T}^{abcd}\,.$ (78)

If we write out the celestial amplitudes in components we see that the
“imaginary pieces” that show up in the optical theorem, distinguished by the
$e^{i\pi\alpha}$ factors, find themselves entirely in the color-singlet
channel:

$\displaystyle f_{S}(z)$
$\displaystyle=\frac{\operatorname{\mathcal{N}}\operatorname{\mathcal{N}}_{\alpha}}{N}\frac{\absolutevalue{z}}{\sqrt{\absolutevalue{1-z}}}\times\begin{cases}1+(-z)^{\alpha}+Ne^{i\pi\alpha}\left(\frac{-z}{1-z}\right)^{\alpha}&\text{if
$z\in(-\infty,0)$}\\\
1+Ne^{i\pi\alpha}z^{\alpha}+\left(\frac{z}{1-z}\right)^{\alpha}&\text{if
$z\in(0,1)$}\\\
Ne^{i\pi\alpha}+z^{\alpha}+\left(\frac{z}{z-1}\right)^{\alpha}&\text{if
$z\in(1,\infty)$}\end{cases}\,,$ (79) $\displaystyle f_{A}(z)$
$\displaystyle=\frac{\operatorname{\mathcal{N}}\operatorname{\mathcal{N}}_{\alpha}}{2}\frac{\absolutevalue{z}}{\sqrt{\absolutevalue{1-z}}}\times\begin{cases}-1+(-z)^{\alpha}&\text{if
$z\in(-\infty,0)$}\\\ 1-\left(\frac{z}{1-z}\right)^{\alpha}&\text{if
$z\in(0,1)$}\\\ z^{\alpha}-\left(\frac{z}{z-1}\right)^{\alpha}&\text{if
$z\in(1,\infty)$}\end{cases}\,,$ (80) $\displaystyle f_{T}(z)$
$\displaystyle=\frac{\operatorname{\mathcal{N}}\operatorname{\mathcal{N}}_{\alpha}}{2}\frac{\absolutevalue{z}}{\sqrt{\absolutevalue{1-z}}}\times\begin{cases}1+(-z)^{\alpha}&\text{if
$z\in(-\infty,0)$}\\\ 1+\left(\frac{z}{1-z}\right)^{\alpha}&\text{if
$z\in(0,1)$}\\\ z^{\alpha}+\left(\frac{z}{z-1}\right)^{\alpha}&\text{if
$z\in(1,\infty)$}\end{cases}\,.$ (81)

We’ve assumed all external scaling dimensions $\Delta_{i}$ are the same above.
For comparison, the four-point function of the $O(N)$ generalized free boson
with scaling dimension $\Delta_{\mathrm{GFB}}$ is obtained by removing the
kinematic prefactors, removing the exponential factors $e^{i\pi\alpha}$, and
replacing $\alpha\mapsto 2\Delta_{\mathrm{GFB}}$. It can be decomposed in the
same way.

###  5.2 Conformal Block Decomposition

We start by considering the conformal block decomposition of $f_{t}^{abcd}(z)$
where $z\in(0,1)$. The components in the singlet, anti-symmetric, and
symmetric traceless reps respectively are

$\displaystyle f_{t,S}(z)$
$\displaystyle=\frac{\operatorname{\mathcal{N}}\operatorname{\mathcal{N}}_{\alpha}}{N}\frac{z}{\sqrt{1-z}}$
$\displaystyle\left(1+Ne^{i\pi\alpha}z^{\alpha}+\left(\frac{z}{1-z}\right)^{\alpha}\right)\,,$
(82) $\displaystyle f_{t,A}(z)$
$\displaystyle=\frac{\operatorname{\mathcal{N}}\operatorname{\mathcal{N}}_{\alpha}}{2}\frac{z}{\sqrt{1-z}}$
$\displaystyle\left(1-\left(\frac{z}{1-z}\right)^{\alpha}\right)\,,$ (83)
$\displaystyle f_{t,T}(z)$
$\displaystyle=\frac{\operatorname{\mathcal{N}}\operatorname{\mathcal{N}}_{\alpha}}{2}\frac{z}{\sqrt{1-z}}$
$\displaystyle\left(1+\left(\frac{z}{1-z}\right)^{\alpha}\right)\,.$ (84)

To proceed with a conformal block decomposition, we just need the conformal
block decomposition of some common $z$-dependent pieces. In Section 2 of
Hogervorst:2017sfd  the authors give the conformal block decompositions we
need as

$z^{p}(1-z)^{-q}=\sum_{n=0}^{\infty}c_{p,q}(n)k_{p+n}(z)\,,\qquad 0<z<1\,,$
(85)

where $k_{\Delta}$ is the $SL(2,\mathbb{R})$ conformal block with all external
$\Delta_{i}$ the same,131313In a CCFT, where scaling dimensions are
continuous, taking all external dimensions to be the same could be a large
loss of information. However, we make this specialization so that we can
obtain closed form expressions in terms of hypergeometrics and not just series
expansions. Using external scaling dimensions with $\Delta_{12}=\Delta_{34}=0$
isn’t much harder (see footnote 15). i.e.

$k_{\Delta}(z)=z^{\Delta}{}_{2}F_{1}(\Delta,\Delta;2\Delta;z)$ (86)

(see also Appendix A.2.1). The function $c_{p,q}(n)$ is defined as

$c_{p,q}(n)\coloneqq\frac{(p)^{2}_{n}}{n!(2p-1+n)_{n}}{}_{3}F_{2}\biggl{[}\genfrac{.}{.}{0.0pt}{}{p-q\mathchar
44\relax\mkern 6.0mu2p-1+n\mathchar 44\relax\mkern 6.0mu-n}{p\mathchar
44\relax\mkern 6.0mup};1\biggr{]}\,.$ (87)

We note that the three quantities relevant to us satisfy141414To see the last
relation, one can manipulate gamma functions, or use the following identity
for $n$ a positive integer (see Sections 3.5 and 3.9 of bailey1935generalized
)  ${}_{3}\tilde{F}_{2}\biggl{[}\genfrac{.}{.}{0.0pt}{}{a\mathchar
44\relax\mkern 6.0mub\mathchar 44\relax\mkern 6.0mu-n}{e\mathchar
44\relax\mkern
6.0muf};1\biggr{]}=(-1)^{n}\frac{\Gamma(1-f+b)}{\Gamma(f+n)}{}_{3}\tilde{F}_{2}\biggl{[}\genfrac{.}{.}{0.0pt}{}{e-a\mathchar
44\relax\mkern 6.0mub\mathchar 44\relax\mkern 6.0mu-n}{e\mathchar
44\relax\mkern 6.0mu1-f+b-n};1\biggr{]}\,.$ (88)

$\displaystyle c_{1,\frac{1}{2}}(2k+1)$ $\displaystyle=0\,,$ (89)
$\displaystyle(-1)^{n}c_{\alpha+1,\frac{1}{2}}(n)$
$\displaystyle=c_{\alpha+1,\alpha+\frac{1}{2}}(n)\,.$ (90)

Putting all the pieces together, we can write $f_{t,I}$ as a sum over two
towers of exchange operators

$\frac{1}{\operatorname{\mathcal{N}}\operatorname{\mathcal{N}}_{\alpha}}f_{t,I}(z)=\sum_{n=0}^{\infty}B_{I}(n)k_{2n+1}(z)+\sum_{n=0}^{\infty}C_{I}(n)k_{n+\alpha+1}(z)$
(91)

with

$NB_{S}(n)=2B_{A}(n)=2B_{T}(n)=c_{1,\frac{1}{2}}(2n)=\frac{16^{n}\,\Gamma(\frac{1}{2}+n)^{2}}{\Gamma(4n+1)\,\Gamma(\frac{1}{2}-n)^{2}}\,,$
(92)

and

$C_{I}(n)=c_{\alpha+1,\alpha+\frac{1}{2}}(n)\times\begin{cases}\frac{1}{N}(1+(-1)^{n}Ne^{i\pi\alpha})&\text{if
$I=S$}\\\ -\frac{1}{2}&\text{if $I=A$}\\\ \phantom{-}\frac{1}{2}&\text{if
$I=T$}\\\ \end{cases}\,.$ (93)

All-in-all, we see that there are “single-trace operators” exchanged in the
$\phi\times\phi$ OPE with odd-integer dimension,151515When all $\Delta_{i}$
are not the same, the exchanges at even integer dimension don’t vanish. In
other words, there is no special mechanism, besides our specificity of
external scaling dimension, that makes the
$\Delta_{\operatorname{\mathcal{O}}}=2n$ contributions disappear.  i.e. no
dependence on the scaling dimensions of the external particles; and also
operators that look more like familiar “double-trace” operators

$\Delta_{\operatorname{\mathcal{O}}}=2n+1\,,\qquad\Delta_{\operatorname{\mathcal{O}}}=\alpha+n+1=2\Delta_{\phi}+n-\frac{1}{2}\,,\qquad
n\in\operatorname{\mathbb{N}}^{0}\,.$ (94)

Recall that for $z\in(0,1)$ there are actually two convergent OPE channels,
the $z\to 0$ channel and the $z\to 1$ channel. Our decomposition in Equation
(91) is for the $z\to 0$ channel, but it’s not hard to see that this
decomposition holds for the crossed channel coming from the $z\to 1$ OPE. In
other words, we have that

$\frac{1}{\operatorname{\mathcal{N}}\operatorname{\mathcal{N}}_{\alpha}}\left(\frac{1-z}{z}\right)^{\alpha+3/2}f_{t,I}(z)=\sum_{n=0}^{\infty}B_{I}(n)k_{2n+1}(1-z)+\sum_{n=0}^{\infty}C_{I}(n)k_{\alpha+n+1}(1-z)$
(95)

for $I=S,T$, and the same with an overall minus sign for $I=A$ from swapping
$1$ and $3$ in the OPE. Of course, this is trivially true because
$f_{S,T}(z)=f_{S,T}(1-z)$ and $f_{A}(z)=-f_{A}(1-z)$, but it is nice to check
the decompositions.

####  5.2.1 The Imaginary Pieces

We might also consider just the “imaginary piece” of the celestial amplitude,
$\imaginary(f^{abcd}(z))$, which is also a well-defined correlator of bosons
(see Equations (139)-(141)). It is given by

$\imaginary(f^{abcd}(z))=\sin({\pi\alpha})\,\operatorname{\mathcal{N}}\operatorname{\mathcal{N}}_{\alpha}\frac{\absolutevalue{z}}{\sqrt{\absolutevalue{1-z}}}\times\begin{cases}\delta^{ad}\delta^{bc}\left(\frac{-z}{1-z}\right)^{\alpha}&\text{if
$z\in(-\infty,0)$}\\\ \delta^{ac}\delta^{bd}z^{\alpha}&\text{if
$z\in(0,1)$}\\\ \delta^{ab}\delta^{cd}&\text{if
$z\in(1,\infty)$}\end{cases}\,.$ (96)

Re-purposing the computations from above, we have the CFT crossing-symmetric
results

$\displaystyle\frac{\csc({\pi\alpha})}{\operatorname{\mathcal{N}}\operatorname{\mathcal{N}}_{\alpha}}\imaginary(f_{u}^{abcd}(z))$
$\displaystyle=\delta^{ad}\delta^{bc}\sum_{n=0}^{\infty}c_{\alpha+1,\frac{1}{2}}(n)k_{\alpha+n+1}\left(\frac{z}{z-1}\right)\,,$
(97)
$\displaystyle\frac{\csc({\pi\alpha})}{\operatorname{\mathcal{N}}\operatorname{\mathcal{N}}_{\alpha}}\imaginary(f_{t}^{abcd}(z))$
$\displaystyle=\delta^{ac}\delta^{bd}\sum_{n=0}^{\infty}c_{\alpha+1,\frac{1}{2}}(n)k_{\alpha+n+1}(z)\,,$
(98) $\displaystyle
z^{-(\alpha+\frac{3}{2})}\frac{\csc({\pi\alpha})}{\operatorname{\mathcal{N}}\operatorname{\mathcal{N}}_{\alpha}}\imaginary(f_{s}^{abcd}(z))$
$\displaystyle=\delta^{ab}\delta^{cd}\sum_{n=0}^{\infty}c_{\alpha+1,\frac{1}{2}}(n)k_{\alpha+n+1}\left(\frac{1}{z}\right)\,.$
(99)

We see that the imaginary pieces, which play a role in the optical theorem,
contribute to the tower of double-exchange operators that we saw earlier.

###  5.3 Conformal Partial Waves: Euclidean Inversion of $\imaginary(f)$

Here we explicitly apply the Euclidean inversion formula to the imaginary
piece of the celestial amplitude $\imaginary(f^{abcd}(z))$ in an attempt to
reproduce the conformal block decomposition answer. We also see the nice
cancellations that occur with the discrete series. Our conventions,
definitions, and identities for 1d conformal partial waves are in Appendix
A.2.2.

In the case of the Euclidean inversion formula, we just naively apply the
inner product to Equation (96). We will temporarily put aside the
contributions from the discrete series $\tilde{I}_{m}^{abcd}$ and focus on the
integrals to do in computing

$I_{\Delta,J}^{abcd}\coloneqq(\Psi_{\Delta,J},\imaginary
f^{abcd})=\int_{-\infty}^{\infty}\frac{dz}{z^{2}}\imaginary(f^{abcd})(z)\Psi_{\Delta,J}(z)$
(100)

along the principal continuous series conformal partial waves,
$\Psi_{\Delta,J}(z)$.

It is easiest to start with the integral over the $z\in(0,1)$ region. For
this, we compute

$\displaystyle I^{(0)}_{\Delta,J}$
$\displaystyle\coloneqq\int_{0}^{1}\frac{dz}{z^{2}}\frac{z^{\alpha+1}}{\sqrt{1-z}}\Psi^{(0)}_{\Delta,J}(z)$
(101)
$\displaystyle=\sqrt{\pi}\left((-1)^{J}+\sec(\pi\Delta)\right)\Gamma(\Delta)^{2}\Gamma(\alpha+\Delta)\,{}_{3}\tilde{F}_{2}\biggl{[}\genfrac{.}{.}{0.0pt}{}{\Delta\mathchar
44\relax\mkern 6.0mu\Delta\mathchar 44\relax\mkern
6.0mu\alpha+\Delta}{2\Delta\mathchar 44\relax\mkern
6.0mu\alpha+\Delta+\frac{1}{2}};1\biggr{]}$ (102)
$\displaystyle+(\Delta\leftrightarrow 1-\Delta)\,.$

The integral over the $z\in(-\infty,0)$ region can be completed by using the
identity in Equation (159)

$I^{(-)}_{\Delta,J}\coloneqq\int_{-\infty}^{0}\frac{dz}{z^{2}}\frac{(-z)^{\alpha+1}}{(1-z)^{\alpha+\frac{1}{2}}}\Psi^{(-)}_{\Delta,J}(z)=\int_{0}^{1}\frac{dw}{w^{2}}\frac{w^{\alpha+1}}{\sqrt{1-w}}(-1)^{J}\Psi^{(0)}_{\Delta,J}(w)\,,$
(103)

so that $I^{(-)}_{\Delta,J}=(-1)^{J}I^{(0)}_{\Delta,J}$.

For the integral over $z\in(1,\infty)$, one must use the integral form of the
partial wave and break up the absolute values over different regions. Luckily,
we can instead reuse the result from Equation (4.20) of lamShao , which tells
us that

$I_{\Delta,0}^{(+)}\coloneqq\int_{1}^{\infty}\frac{dz}{z^{2}}\frac{z}{\sqrt{z-1}}\Psi^{(+)}_{\Delta,0}(z)\\\
=\frac{4\pi^{3}}{\sin(\pi\Delta)\Gamma\left(\frac{2-\Delta}{2}\right)^{2}\Gamma\left(\frac{\Delta+1}{2}\right)^{2}}\,.$
(104)

Because $\imaginary(f_{s}^{abcd}(z))$ is symmetric under
$z\mapsto\frac{z}{z-1}$, the $J=1$ integral vanishes and
$I^{(+)}_{\Delta,1}=0$.

We can now decompose the four-point function over the conformal partial waves
using the Euclidean inner product (see Equation (154) and surrounding
discussion), and shadow symmetry of the various components, to write

$\imaginary(f^{abcd})(z)=\sum_{J=0,1}\int_{\frac{1}{2}-i\infty}^{\frac{1}{2}+i\infty}\frac{d\Delta}{2\pi
i}\frac{I^{abcd}_{\Delta,J}}{2n_{\Delta}}\Psi_{\Delta,J}(z)+\sum_{m=0}^{\infty}\frac{2m-1}{4\pi^{2}}\tilde{I}^{abcd}_{m}\Psi_{m,J}(z)\,,$
(105)

where the combined coefficient is

$I^{abcd}_{\Delta,J}=\sin(\pi\alpha)\,{\operatorname{\mathcal{N}}}\,{\operatorname{\mathcal{N}}}_{\alpha}\left((-1)^{J}\delta^{ad}\delta^{bc}I_{\Delta,J}^{(0)}+\delta^{ac}\delta^{bd}I_{\Delta,J}^{(0)}+\delta^{ab}\delta^{cd}I_{\Delta,J}^{(+)}\right)\,.$
(106)

To recover the conformal block decomposition is easy; for instance, take
$z\in(0,1)$ and use the form of $\Psi_{\Delta,J}^{(0)}(z)$ given in Equation
(157). Then

$\imaginary(f^{abcd})(z)=\sum_{J=0,1}\int_{\frac{1}{2}-i\infty}^{\frac{1}{2}+i\infty}\frac{d\Delta}{2\pi
i}\frac{(-1)^{J}I^{abcd}_{\Delta,J}}{2K_{\Delta,J}}k_{\Delta}(z)+\sum_{m=0}^{\infty}\frac{(-1)^{m}\Gamma(m)^{2}}{2\pi^{2}\Gamma(2m-1)}\tilde{I}^{abcd}_{m}k_{m}(z)\,,$
(107)

and we can close the contour of the integral to the right of $\frac{1}{2}$ in
the $\Delta$ plane to produce a sum over conformal blocks if we know the
locations of poles of $\sum_{J}(-1)^{J}I_{\Delta,J}^{abcd}/2K_{\Delta,J}$. The
location of a pole at $\Delta=\Delta_{\operatorname{\mathcal{O}}}$ denotes the
dimension of an exchange operator $\operatorname{\mathcal{O}}$ in the OPE, and
the residue is minus the coefficient in front of the conformal block (and, in
principle, these are also three-point function coefficients squared,
normalized by the two-point function coefficient, i.e.
$-c_{\phi\phi\operatorname{\mathcal{O}}}^{2}/d_{\operatorname{\mathcal{O}}\operatorname{\mathcal{O}}}$).

We proceed with the analysis in pieces. $I^{(+)}_{\Delta,J}/2K_{\Delta,J}$ has
no poles to the right of $\Delta=\frac{1}{2}$ so contributes nothing, hence we
turn to $I^{(0)}_{\Delta,J}/2K_{\Delta,J}$. When
$\real(\alpha)\geq-\frac{1}{2}$ there are two infinite towers of poles which
contribute, at $\Delta_{\operatorname{\mathcal{O}}}=\alpha+n+1$ and
$\Delta_{\operatorname{\mathcal{O}}}=2n+J$; the former come from poles of
$I_{\Delta,J}^{(0)}$ and the latter from zeroes of $K_{\Delta,J}$. The
residues are161616To get from Equation (109) to Equation (110) we need to use
another identity from Sections 3.5 and 3.9 of bailey1935generalized . In
particular,  ${}_{3}\tilde{F}_{2}\biggl{[}\genfrac{.}{.}{0.0pt}{}{a\mathchar
44\relax\mkern 6.0mub\mathchar 44\relax\mkern 6.0mu-n}{e\mathchar
44\relax\mkern
6.0muf};1\biggr{]}=(-1)^{n}\frac{\Gamma(1-f+b)\Gamma(1-f+a)}{\Gamma(e-c)\Gamma(f-c)}{}_{3}\tilde{F}_{2}\biggl{[}\genfrac{.}{.}{0.0pt}{}{1-s\mathchar
44\relax\mkern 6.0mu1-f+c\mathchar 44\relax\mkern 6.0mu-n}{1+b+c-f\mathchar
44\relax\mkern 6.0mu1+a+c-f};1\biggr{]}\,$ (108) where $s=e+f-a-b-c$. In
passing from Equation (110) to Equation (111) we use our old identity from
Equation (90).

$\displaystyle\underset{\Delta=\alpha+n+1}{\mathbf{Res}}\frac{I^{(0)}_{\Delta,J}}{2K_{\Delta,J}}$
$\displaystyle=(-1)^{n+J+1}\frac{\sqrt{\pi}\Gamma(-2\alpha-2n)}{2\Gamma(n+1)}{}_{3}\tilde{F}_{2}\biggl{[}\genfrac{.}{.}{0.0pt}{}{-\alpha-n\mathchar
44\relax\mkern 6.0mu-\alpha-n\mathchar 44\relax\mkern
6.0mu-n}{\frac{1}{2}-n\mathchar 44\relax\mkern
6.0mu-2\alpha-2n};1\biggr{]}\,,$ (109)
$\displaystyle=(-1)^{J+1}\frac{\Gamma(-2\alpha-2n)\Gamma(\alpha+n+1)^{2}}{2\Gamma(-2\alpha-n)\Gamma(n+1)}{}_{3}\tilde{F}_{2}\biggl{[}\genfrac{.}{.}{0.0pt}{}{\frac{1}{2}\mathchar
44\relax\mkern 6.0mu2\alpha+n+1\mathchar 44\relax\mkern
6.0mu-n}{\alpha+1\mathchar 44\relax\mkern 6.0mu\alpha+1};1\biggr{]}\,,$ (110)
$\displaystyle=(-1)^{J+1}\frac{1}{2}c_{\alpha+1,\frac{1}{2}}(n)\,;$ (111)
$\displaystyle\underset{\Delta=2n+J}{\mathbf{Res}}\frac{I^{(0)}_{\Delta,J}}{2K_{\Delta,J}}$
$\displaystyle=(-1)^{J}\frac{4^{1-2n-J}\Gamma(2n+J)^{3}\Gamma(\alpha+2n+J)}{\pi\Gamma(2n+J-\frac{1}{2})}$
(112)
$\displaystyle\qquad\times\,{}_{3}\tilde{F}_{2}\biggl{[}\genfrac{.}{.}{0.0pt}{}{2n+J\mathchar
44\relax\mkern 6.0mu2n+J\mathchar 44\relax\mkern
6.0mu\alpha+2n+J}{4n+2J\mathchar 44\relax\mkern
6.0mu\alpha+2n+J+\frac{1}{2}};1\biggr{]}\,.$

We can combine all of these results to obtain the residues of
$\sum_{J}(-1)^{J}I_{\Delta,J}^{abcd}/2K_{\Delta,J}$ at
$\Delta_{\operatorname{\mathcal{O}}}=\alpha+n+1$. We have

$\displaystyle\frac{\csc({\pi\alpha})}{\operatorname{\mathcal{N}}\operatorname{\mathcal{N}}_{\alpha}}\underset{\Delta=\alpha+n+1}{\mathbf{Res}}\sum_{J=0,1}\frac{(-1)^{J}I^{abcd}_{\Delta,J}}{2K_{\Delta,J}}$
$\displaystyle=-\sum_{J=0,1}\frac{1}{2}c_{\alpha+1,\frac{1}{2}}(n)\left((-1)^{J}\delta^{ad}\delta^{bc}+\delta^{ac}\delta^{bd}\right)\,,$
(113) $\displaystyle=-\delta^{ac}\delta^{bd}c_{\alpha+1,\frac{1}{2}}(n)\,,$
(114)

which successfully reproduces the conformal block expansion. If instead we’d
considered $z\in(-\infty,0)$ and the $\Psi^{(-)}(z)$ conformal partial waves,
the calculations would proceed identically except Equation (113) would collect
an extra factor of $(-1)^{J}$, projecting us onto the $\delta^{ad}\delta^{bc}$
sector.

Of course, we expect no physical operators at exchange dimensions
$\Delta_{\operatorname{\mathcal{O}}}=2n$ and
$\Delta_{\operatorname{\mathcal{O}}}=2n+1$ (atleast for
$\imaginary(f^{abcd})$). As explained in Maldacena_2016 ; Mazac:2018qmi , if
we evaluate the contributions to the discrete series $\tilde{I}^{abcd}_{m}$ in
Equation (107), we get a cancellation with the spurrious residues at
$\Delta_{\operatorname{\mathcal{O}}}=2n$ and
$\Delta_{\operatorname{\mathcal{O}}}=2n+1$. To illustrate the point, consider

$\displaystyle\tilde{I}^{(0)}_{m}$
$\displaystyle\coloneqq\int_{0}^{1}\frac{dz}{z^{2}}\frac{z^{\alpha+1}}{\sqrt{1-z}}\Psi^{(0)}_{m,J}(z)$
(115)
$\displaystyle=2\sqrt{\pi}(-1)^{m}\Gamma(m)^{2}\Gamma(\alpha+m)\,{}_{3}\tilde{F}_{2}\biggl{[}\genfrac{.}{.}{0.0pt}{}{m\mathchar
44\relax\mkern 6.0mum\mathchar 44\relax\mkern 6.0mu\alpha+m}{2m\mathchar
44\relax\mkern 6.0mu\alpha+m+\frac{1}{2}};1\biggr{]}\,,$ (116)

then quantities such as

$\displaystyle\frac{(-1)^{m}\Gamma(m)^{2}}{2\pi^{2}\Gamma(2m-1)}\tilde{I}^{(0)}_{m}$
$\displaystyle=\frac{\Gamma(m)^{4}\Gamma(\alpha+m)}{\pi^{3/2}\Gamma(2m-1)}{}_{3}\tilde{F}_{2}\biggl{[}\genfrac{.}{.}{0.0pt}{}{m\mathchar
44\relax\mkern 6.0mum\mathchar 44\relax\mkern 6.0mu\alpha+m}{2m\mathchar
44\relax\mkern 6.0mu\alpha+m+\frac{1}{2}};1\biggr{]}\,.$ (117)

appear in the decomposition in Equation (107). For $m=2n$, this is Equation
(112) with $J=0$; for $m=2n+1$, this is negative Equation (112) with $J=1$. By
following the colour-index bookkeeping, we see that the discrete series
contributions and the spurious poles cancel, as we anticipated!

Note the important role played by the $J=1$ eigenfunctions
$\Psi_{\Delta,1}(z)$: if we had not included them, we would not have obtained
the right result in Equation (114). Instead, we would only have the colour-
symmetric contribution, which is $\frac{1}{2}$ as large. This is familiar from
the GFB case when we recall that $J=1$ is our $1d$ analogue of spin: the four-
point function of four identical scalars involves the exchange of double trace
operators with dimension
$\Delta_{\operatorname{\mathcal{O}}}=2\Delta_{\phi}+2n$ (or
$\Delta_{\operatorname{\mathcal{O}}}=2\Delta_{\phi}+\ell+2m$ with spin
$\ell\in 2\operatorname{\mathbb{N}}^{0}$ in $d>1$). However, when colour is
added, the colour-antisymmetric channel instead involves the exchange of
operators with dimension
$\Delta_{\operatorname{\mathcal{O}}}=2\Delta_{\phi}+2n+1$.

Although there are a lot of similarities with the GFB and more conventional
CFT, there are some differences. Consider the GFB, whose stripped four-point
function is

$f_{\mathrm{GFB}}(z)=1+\absolutevalue{z}^{2\Delta_{\phi}}+\absolutevalue{\frac{z}{z-1}}^{2\Delta_{\phi}}\,.$
(118)

Before studying conformal partial waves, one must first remove the identity
piece, $1$, as it’s not normalizable with respect to the inner-product.
However, because of the overall kinematic pre-factor $z/{\sqrt{1-z}}$, we seem
to avoid this altogether, as all of the three pieces then have sensible
conformal partial wave decompositions.

The best analogue to a Generalized Free Theory, without the kinematic
prefactor, actually comes from doing a celestial transform of the identity
piece of a massive scattering amplitude ShuHengTalk , i.e. the $\mathds{1}$ in
$S=\mathds{1}+iT$. However, for massless scalars, the celestial transform of
the identity piece only leads to contact terms unless some of the operators
are at shadow weights of one another.

###  5.4 Factorization

As explained in Section 2, in the symmetry breaking phase of the $O(N)$ model,
we are effectively scattering massless pions mediated by exchange of a massive
$\sigma$-particle with some mass $M$ (we expand on this below). The CCFT
three-point function coefficient for the scattering of two massless scalars
with one massive scalar was computed in lamShao , it is

$C_{g}(\Delta_{1},\Delta_{2};\Delta_{M})=g\frac{M^{\Delta_{1}+\Delta_{2}-3}}{2^{\Delta_{1}+\Delta_{2}}}\frac{\Gamma\left(\frac{\Delta_{1}+\Delta_{M}-\Delta_{2}}{2}\right)\Gamma\left(\frac{\Delta_{2}+\Delta_{M}-\Delta_{1}}{2}\right)}{\Gamma(\Delta_{M})}\,,$
(119)

where $g$ is the coupling for the three-point interaction. Note: we have
suppressed the colour indices in the following calculations, but it is
straightforward to add them back in to the three-point functions by appending
them with the appropriate $\delta^{ab}$ factors.

Meanwhile, the two point-function normalization is not “1,” because (in our
conventions) the two point function is defined by the Mellin transform of a 2
particle amplitude in the original theory. We obtain the two-point function
normalization for two massive particles $\sigma$ with mass $M$ by the same
method as Appendix B of Atanasov:2021cje . For a $d$-dimensional CCFT
amplitude, the result is

$\expectationvalue{\sigma_{\Delta_{1}}(\vec{x}_{1})\,\sigma_{\Delta_{2}}(\vec{x}_{2})}=\frac{2\pi^{d/2}}{M^{d}}\frac{\Gamma(\Delta_{1}-\frac{d}{2})}{\Gamma(\Delta_{1})}\delta(\Delta_{1}-\Delta_{2})\frac{1}{\absolutevalue{\vec{x}_{1}-\vec{x}_{2}}^{2\Delta_{1}}}\,.$
(120)

We will call the overall normalization $D_{\Delta_{1}\Delta_{2}}^{M}$.

Now, when the external particles are the same, the three-point function should
square to the four point function (taking the two-point normalization into
account). Recall that there were actually two poles on the second sheet of the
$p^{2}$-plane, at $p_{\pm}$, associated with $\sigma$. We thus have two masses
$M_{\pm}^{2}:=p_{\pm}^{2}$ in the $D_{\sigma\sigma}$ propagator and two
couplings $g_{\pm}$ from Equation (18). Using that

$p_{\pm}=-\sqrt{-2\mu^{2}}\,e^{\mp i\theta}\,,$ (121)

we have

$\displaystyle M_{\pm}^{2}$ $\displaystyle=(-2\mu^{2})\,e^{\mp 2i\theta}\,,$
(122) $\displaystyle g_{\pm}^{2}$
$\displaystyle=M_{\pm}^{2}\left(\frac{\lambda}{N}\csc\theta\right)(\mp ie^{\mp
i\theta})\,.$ (123)

Note: in Lorentzian signature these poles become
$\omega_{\pm}=\sqrt{-2\mu^{2}}e^{-i\pi/2}e^{\mp i\theta}$.

And we can verify, at least for the ($\alpha$-independent) single trace tower
in Equation (91) with $\Delta_{M}=2n+1$, that

$\displaystyle\sum_{s=\pm}\frac{C_{g_{s}}(\Delta_{\phi},\Delta_{\phi};\Delta_{M_{s}})^{2}}{D_{\Delta_{M}\Delta_{M}}^{M_{s}}}$
$\displaystyle=\sum_{s=\pm}\pi
g_{s}^{2}\frac{M_{s}^{4\Delta_{\phi}-5}}{2^{4\Delta_{\phi}+1}}c_{1,\frac{1}{2}}(2n)$
(124)
$\displaystyle=2\pi\left(\frac{\lambda}{N}\csc\theta\right)(-2\mu^{2})^{2\Delta_{\phi}-\frac{3}{2}}\sin\Big(\left(2-4\Delta_{\phi}\right)\theta\Big{missing})c_{1,\frac{1}{2}}(2n)$
(125)
$\displaystyle=\frac{\sin(4\pi\Delta_{\phi})}{2}\operatorname{\mathcal{N}}\operatorname{\mathcal{N}}_{\alpha}c_{1,\frac{1}{2}}(2n)\,.$
(126)

where $c_{p,q}(n)$ is defined in (87). Two comments are in order. First, we
see that the three-point functions squared, normalized, and summed reproduces
the coefficients in the conformal block decomposition up to an overall
$\Delta_{\phi}$ dependent factor. This is similar to the findings of
Atanasov:2021cje  on integer mode exchanges in the conformal block
decomposition of 4d massless scalars mediated by a massive exchange, and still
requires explanation.

We also note the importance of summing over both resonances. If we had not
summed over them we would not have been able to reproduce terms like
$\sin((2-4\Delta_{\phi})\theta)$ living inside the prefactor
$\operatorname{\mathcal{N}}_{\alpha}$.

The double trace operators at $\Delta_{\operatorname{\mathcal{O}}}=\alpha+n+1$
aren’t as straightforward. If we focus on just the $n$-dependent pieces of the
four-point function coefficients, i.e. terms like

$\displaystyle c_{\alpha+1,\alpha+\frac{1}{2}}(n)$
$\displaystyle\coloneqq\frac{(\alpha+1)^{2}_{n}}{n!(2\alpha+n+1)_{n}}{}_{3}F_{2}\biggl{[}\genfrac{.}{.}{0.0pt}{}{\frac{1}{2}\mathchar
44\relax\mkern 6.0mu2\alpha+n+1\mathchar 44\relax\mkern
6.0mu-n}{\alpha+1\mathchar 44\relax\mkern 6.0mu\alpha+1};1\biggr{]}\,,$ (127)
$\displaystyle=2^{-2(\alpha+n)}\frac{\Gamma(\alpha+n+1)}{\Gamma(\alpha+n+\frac{1}{2})}\sum_{k=0}^{n}\frac{(-1)^{k}\Gamma(k+\frac{1}{2})}{\Gamma(k+1)\Gamma(\alpha+k+1)^{2}}\frac{\Gamma(2\alpha+n+k+1)}{\Gamma(n-k+1)}$
(128)

then it’s easy to see, by comparing to the $n$-dependence of our previous
guess, that something of the form

$\frac{C_{g}(\Delta_{\phi},\Delta_{\phi};\Delta_{\operatorname{\mathcal{O}}})^{2}}{D_{\Delta_{\operatorname{\mathcal{O}}}\Delta_{\operatorname{\mathcal{O}}}}^{M}}$
(129)

will not match, although the expansion of $c_{\alpha+1,\alpha+\frac{1}{2}}(n)$
has a very suggestive form.

There are a couple possible explanations for this non-matching of the three-
point functions to the four-point function coefficients. Some of them include:

  1. 1.

There are multiple operators with the same scaling dimension
$\Delta_{\operatorname{\mathcal{O}}}=\alpha+n+1$, so one cannot just naively
identify the coefficients of a four-point function with one “three-point
function squared.”

  2. 2.

There are other three-point functions in the CCFT that do not come from the
celestial transform of a three-particle amplitude. A straightforward
conjecture is that the “double trace operators” come from scattering two-
particle states.

  3. 3.

CCFTs are different from standard CFTs.

Finally, in lamShao , the authors show that the optical theorem for the
scattering amplitude translates into a decomposition of $\imaginary(f(z))$
along the principal continuous series partial waves. In the case of the $O(N)$
amplitude (or more broadly, any amplitude with resonances), this is modified
in the same way as our previous discussions, in that one must sum over
resonances to obtain the analogous formula, i.e.

$\imaginary
f(z)=\pi\sum_{s=\pm}M_{\pm}\int_{-\infty}^{\infty}d\nu\,\mu(\nu)C_{g_{s}}\left(\Delta_{1},\Delta_{2};\frac{1}{2}+i\nu\right)C_{g_{s}}\left(\Delta_{3},\Delta_{4};\frac{1}{2}-i\nu\right)\Psi_{\frac{1}{2}+i\nu}(z)\,.$
(130)

This follows directly from applying the same manipulations in lamShao  to
Equation (69), since Equation (69) shows exactly how the four-point celestial
amplitude is writable as a (sum of) three-point amplitude(s) squared, and
(aside from the fact that there are two terms) takes the exact same form as
the amplitude in the reference.171717Another more physical way to see this
follows from the results in Section 6 of Melton:2021kkz . In particular, the
author shows how an amplitude which can be factorized into two pieces,
connected by the exchange of an off-shell massive particle, can be written as
an integral over conformal partial waves with an additional integral over
internal momentum of the exchange particle weighted by a Feynman propagator.
Using the usual Cutkosy rules on this line (replacing the Feynman propagator
with a delta function) so that the exchange particle is on-shell, one obtains
the celestial optical theorem from lamShao . This will be explained in more
detail in forthcoming work.

##  6 Conclusion

In this paper we studied the effects of unstable particles on unitarity of
scattering amplitudes (via the optical theorem) when phrased in variables on
the celestial circle (or the celestial sphere more generally), and the effects
on factorization of celestial amplitudes and the celestial optical theorem. We
also discussed the 3d $O(N)$ model as an interesting example for celestial
holography in it’s own right.

In particular, we studied how a $2\to 2$ scattering amplitude written in terms
of Mandelstam variables could be re-written in terms of center of mass energy
$\omega$ and a coordinate $z$ on the celestial circle. In this basis, the
poles of the S-matrix “unfold” into poles in the complex $\omega$-plane, where
their effects on the optical theorem manifest geometrically as whether or not
they are caught in simple contour integrals. For example, an unstable particle
(a pole on the second sheet in Mandelstam variables) becomes a pole in the
lower-half $\omega$ plane, which is then turned into a pole in the upper-half
plane in the imaginary amplitude and caught by a contour integral when doing a
Mellin transform. Our discussion also points at inherently celestial CFT ways
to identify unstable particles in scattering amplitudes. We leave development
of a complex-analytic description of celestial amplitudes (in the spirit of
the S-matrix bootstrap program) to future work.

As a motivating example, we applied our results to the 3d $O(N)$ model and
worked with the fully resummed $\pi\pi\to\pi\pi$ scattering amplitude in the
large $N$ limit. The 3d $O(N)$ model is interesting as it is both UV and IR
finite, satisfies positivity constraints in the EFT expansion, and contains
unstable states in the spectrum. Moreover, taking the large $N$ limit allowed
us to study the theory at all loop orders and finite coupling. From the CCFT
point-of-view, this gives an interesting example of an interacting non-
supersymmetric celestial amplitude beyond tree-level, which could be a useful
example for understanding both renormalization and spontaneous symmetry
breaking in CCFT; we also leave these studies to future investigations. When
converted to a celestial amplitude, this model showcased a number of
interesting features:

  1. 1.

The resummed amplitude shifted the convergence strip for the celestial scaling
dimension $\Delta_{T}$ (also called $\alpha$ or $\beta$ in the literature) in
the Mellin transform to include the principal continuous series.

  2. 2.

A CCFT four-point function analogous to a Generalized Free Field Theory.

  3. 3.

Confirmed that scattering of bulk pions with “Bose statistics” behave like a
correlation function of bosons in the CCFT.

  4. 4.

The effects of unstable particles and/or resonances on CCFT amplitudes.

To elaborate on the last point. In the final section of the paper we provided
independent conformal block and conformal partial wave decompositions of the
$\pi\pi\to\pi\pi$ amplitude; and verified they were equivalent after a
standard cancellation of spurious poles along the principal discrete series.
In particular, we found that there were two towers of exchanged particles in
these conformal decompositions, one exchanging “single trace” states, and one
exchanging “double trace” states. On one hand, we found that the single trace
tower was able to be given an interpretation as the exchange of massive
particles with increasing conformal dimension. However, we also found that it
was essential to interpret the mass of the particles to be coming from both of
the massive unstable resonances when doing so, or else one would not be able
to produce the correct prefactors (with the correct dependence on external
scaling dimensions and couplings). On the other hand, the double trace towers
did not factorize, and we conjectured some reasons for this; we leave the
resolution of this to future work. Finally, we comment on the CCFT realization
of the optical theorem, and explain how it must also be modified to include a
sum over resonances in the case of the $O(N)$ model, for the exact same
reasoning as in the conformal block decomposition; we also leave a more
detailed description of cutting rules in CCFT to future work.

###  6.1 Open Problems

In this section, we enumerate a few more speculative and conjectural
directions spurred on by our investigations, but not necessarily with an
immediate connection to the main text.

  1. 1.

Other large $N$ theories. In general, the large $N$ limit allows one to go
beyond tree level and compute observables exactly as functions of the coupling
constants, providing a standard approach to non-perturbative physics. In our
case, this led to an interesting shift of the convergence region for the
Mellin integrals, amongst other properties. There are many other interesting
theories which might be elucidated by a celestial analysis, or conversely,
provide interesting case studies to probe the CCFT formalism. One example
might be the Gross-Neveu model Gross:1974jv ; for (2+1)d Gross-Neveu it would
be satisfying to see relationships to Generalized Free Fermions or SYK emerge
(see also Pasterski:2022joy ; Kar:2022vqy ).

  2. 2.

Reversing the celestial transform. In practice, we typically start with a
scattering amplitude in $(d+2)$-dimensions then turn it into the correlator of
a $d$-dimensional CCFT. It could be interesting to see if there are any
structures about $(d+2)$-dimensional amplitudes that are revealed by reversing
this procedure. For example, the CCFT correlation function obtained from the
$O(N)$ model roughly looks like a modification of a generalized free boson by
a “kinematic pre-factor” and an exponential related to the scattering channel.
Can one start with a (possibly modified) CFT correlation function, e.g. for a
1d generalized free fermion or the SYK model, and see what (2+1)d amplitudes
it produces? Since we are currently missing an intrinsic definition of a CCFT,
it may be hard to justify any starting correlation function a priori.

In a similar vein, are there any “manipulations” that can be performed on a
CCFT that become new (or recover known) dualities of scattering amplitudes
when both the starting and ending CCFT are reverse-transformed? If a duality
is apparent at the level of S-matrix elements (e.g. as in Chern-Simons Matter
theories Jain:2014nza ), then this should have some manifestation in CCFT
correlation functions. It would be curious to see if whatever relationship
appears could be abstracted though to find new dualities.

More generally we are asking to what extent we can commute around diagrams
like

${{\mathcal{A}_{1}}}$${{\mathcal{A}_{2}}}$${{\mathrm{CCFT}_{1}}}$${{\mathrm{CCFT}_{2}}}$
(131)

and track interesting properties of QFTs like dualities, RG flows, etc., with
the goal of applying CCFT formalism to understanding QFT more broadly.

  3. 3.

Theories with Anyons. In Section 3.4 we pointed out that a naive permutation
symmetry of a (2+1)d scattering amplitude led to the condition that
correlation functions of the CCFT behaved “like bosons.” One might conjecture
that given an S-matrix in a theory such as those in Jain:2014nza , where
particles are dressed with Wilson lines with a Chern-Simons part of the
action, that some non-trivial features of the S-matrix could be revealed.

  4. 4.

CCFTs and Discrete Symmetries. Discrete actions on $(d+2)$-dimensional
Minkowski spacetime induce discrete actions on the $d$-dimensional celestial
circle. For example, in (2+1)d a generic scattering amplitude can depend on
the orientation of the scattering momenta (see Jain:2014nza  for example). The
non-invariance of such a scattering amplitude under reflections should imply
non-invariance of the 1d CCFT under orientation-reversal.

  5. 5.

Lorentzian inversion formula and dispersion relations. Another way to obtain
the coefficients $I_{\Delta,J}$ is given by the Lorentzian inversion formula
Caron_Huot_2017 ; Simmons_Duffin_2018  (see also Maldacena_2016 ). It is
easier in that one only needs to integrate against the
$\operatorname{\mathrm{dDisc}}$, defined by

$\operatorname{\mathrm{dDisc}}[f(z)]\coloneqq
f_{t}(z)-\frac{1}{2}\left(f_{s}^{\operatorname{\rotatebox[origin={c}]{0.0}{$\curvearrowleft$}}}(z)+f_{s}^{\operatorname{\rotatebox[origin={c}]{180.0}{$\curvearrowright$}}}(z)\right)\,,\qquad
z\in(0,1)\,,$ (132)

where
$f_{s}^{\operatorname{\rotatebox[origin={c}]{0.0}{$\curvearrowleft$}}}(z)$ and
$f_{s}^{\operatorname{\rotatebox[origin={c}]{180.0}{$\curvearrowright$}}}(z)$
are the continuations of $f(z)$ from the $z\in(1,\infty)$ region to the
$z\in(0,1)$ region above or below the real axis.

As an example, for our $O(N)$ model the negative average is

$-\frac{1}{2}\left(f_{s}^{\operatorname{\rotatebox[origin={c}]{0.0}{$\curvearrowleft$}}}(z)+f_{s}^{\operatorname{\rotatebox[origin={c}]{180.0}{$\curvearrowright$}}}(z)\right)=\delta^{ad}\delta^{bc}\sin(\pi\alpha)\frac{z}{\sqrt{1-z}}\left(\frac{z}{1-z}\right)^{\alpha}\,,$
(133)

so that $\operatorname{\mathrm{dDisc}}[f(z)]$ in $z\in(0,1)$ is just

$\operatorname{\mathrm{dDisc}}[f(z)]=\operatorname{\mathcal{N}}\operatorname{\mathcal{N}}_{\alpha}\frac{z}{\sqrt{1-z}}\left(\delta^{ab}\delta^{cd}+\delta^{ac}\delta^{bd}e^{i\pi\alpha}z^{\alpha}+\delta^{ad}\delta^{bc}(1+\sin(\pi\alpha))\left(\frac{z}{1-z}\right)^{\alpha}\right)\,.$
(134)

Part of the elegance of the Lorentzian inversion formula is that it gives
$I_{\Delta,J}$ in a way analytic in spin, in addition to various analytic
bounds on OPE data and diagnostics for chaos (see Maldacena_2016 ;
Mazac:2018qmi  and references within). In $d=1$ there is no analyticity in
spin, but analyticity in the discrete series $I_{m,J}$ Simmons_Duffin_2018 .

Since analytic continuation in $z$ relates one scattering channel to another
lamShao ; Chang:2021wvv , it could be interesting to understand which data of
scattering amplitudes can be extracted from understanding
$\operatorname{\mathrm{dDisc}}$’s in the CCFT.

  6. 6.

2d Bulks. In this paper we studied (2+1)d amplitudes which became correlation
functions on the celestial circle. We might try to study (1+1)d bulks so that
the boundary theory is supported entirely on two points, leaving almost all
physics to the $\Delta_{T}$-plane (or $\alpha$ or $\beta$ plane in other
conventions). It could be interesting to transform the Sine-Gordon theory to
these “celestial points,” or other integrable theories, to see how interesting
non-perturbative phenomena manifest in the celestial presentation.

###### Acknowledgements.

We would like to thank Davide Gaiotto for his comments throughout the research
process. We would like to thank Davide Gaiotto and Sabrina Pasterski for
feedback on the draft. AG thanks Shu-Heng Shao and Andy Strominger for
discussions, as well as the organizers and participants of the "S-Matrix
Bootstrap 2022" conference for hospitality and valuable exchange of ideas. The
work of DGS is supported by a Plotnick Fellowship in the Enrico Fermi
Institute at the University of Chicago and in part by the US Department of
Energy DE-SC0021432. JK is funded through the NSERC CGS-D program. JW is
supported in part by a grant from the Krembil foundation by the Perimeter
Institute for Theoretical Physics. AG was partially supported by DOE grant de-
sc/000787, by the Society of Fellows and by the Black Hole Initiative at
Harvard University, which is funded by grants from the John Templeton
Foundation and the Gordon and Betty Moore Foundation. Research at Perimeter
Institute is supported in part by the Government of Canada through the
Department of Innovation, Science and Economic Development Canada and by the
Province of Ontario through the Ministry of Colleges and Universities. JW is
also supported by the European Union’s Horizon 2020 Framework: ERC grant
682608 and the "Simons Collaboration on Special Holonomy in Geometry, Analysis
and Physics’

##  Appendix A CFT Background and Conventions

Here we record background and references on CFT, general reviews of CFT are
Rychkov:2016iqz ; simmonsduffin2016tasi . We often specialize to 1d where
things are slightly different from other dimensions. An axiomatic approach to
unitary 1d CFT appears in Qiao:2017xif , but some lessons are transmutable;
helpful reviews of 1d conformal partial waves appear in gadde2017search ;
Mazac:2018qmi ; Hogervorst:2021uvp .

###  A.1 Four-Point Functions in 1d CFT

In a 1d CFT, as in higher dimensions, a four-point correlation function is
fixed by conformal invariance to take the form

$\expectationvalue{\phi_{1}^{a}(x_{1})\phi_{2}^{b}(x_{2})\phi_{3}^{c}(x_{3})\phi_{4}^{d}(x_{4})}=\frac{\left(\frac{x_{14}^{2}}{x_{24}^{2}}\right)^{-\frac{1}{2}\Delta_{12}}\left(\frac{x_{14}^{2}}{x_{13}^{2}}\right)^{\frac{1}{2}\Delta_{34}}}{\left(x_{12}^{2}\right)^{\frac{1}{2}(\Delta_{1}+\Delta_{2})}\left(x_{34}^{2}\right)^{\frac{1}{2}(\Delta_{3}+\Delta_{4})}}f^{abcd}(z)\,,$
(135)

where $\Delta_{ij}\coloneqq\Delta_{i}-\Delta_{j}$ and the conformal cross-
ratio is

$z\coloneqq\frac{x_{12}x_{34}}{x_{13}x_{24}}\,.$ (136)

If we take all $\phi_{i}$ to have the same conformal weight $\Delta_{\phi}$,
then this becomes

$\expectationvalue{\phi^{a}(x_{1})\phi^{b}(x_{2})\phi^{c}(x_{3})\phi^{d}(x_{4})}=\frac{1}{x_{12}^{2\Delta_{\phi}}x_{34}^{2\Delta_{\phi}}}f^{abcd}(z)\,.$
(137)

We will also call (stripped) four-point functions, like $f^{abcd}(z)$, “four-
point functions.”

It is convenient for intuition to spend the $SL(2,\operatorname{\mathbb{R}})$
symmetry and set $(x_{1},x_{2},x_{3},x_{4})=(0,z,1,\infty)$. Since the four
point function is singular at $z=0$, $1$, $\infty$, the 1d CFT four-point
function is really only defined piecewise. We suggestively label the three
disconnected regions with $u$, $t$, and $s$, anticipating applications to
CCFT, so that

$f^{abcd}(z)\coloneqq\begin{cases}f_{u}^{abcd}(z)&\text{if
$z\in(-\infty,0)$}\\\ f_{t}^{abcd}(z)&\text{if $z\in(0,1)$}\\\
f_{s}^{abcd}(z)&\text{if $z\in(1,\infty)$}\end{cases}\,.$ (138)

As explained in Section 2 of Mazac:2018qmi , these $f^{abcd}_{u,t,s}(z)$ can
be analytically continued to other regions, but they are not necessarily the
analytic continuations of one another.181818Although in the specific case of
1d CCFT correlation functions, they are actually related in a slightly non-
trivial way, see e.g. lamShao ; Chang:2021wvv .

However, if we started with a correlation function of four bosons, then
various $f^{abcd}(z)$ are related by relabelling. For instance, invariance
under relabelling $1\leftrightarrow 2$ (equivalently $3\leftrightarrow 4$) and
$2\leftrightarrow 3$ (or $1\leftrightarrow 4$) imply

$\displaystyle f_{t}^{abcd}\left(\frac{z}{z-1}\right)$
$\displaystyle=f_{u}^{bacd}(z)=f_{u}^{abdc}(z)\,,$ $\displaystyle\quad
z\in(-\infty,0)\,,$ (139) $\displaystyle
z^{2\Delta_{\phi}}f_{t}^{abcd}\left(\frac{1}{z}\right)$
$\displaystyle=f_{s}^{acbd}(z)=f_{s}^{dbca}(z)\,,$ $\displaystyle\quad
z\in(1,\infty)\,,$ (140)

respectively. Moreover, swapping $1\leftrightarrow 3$ (or $2\leftrightarrow
4$) gives CFT crossing symmetry

$z^{-2\Delta_{\phi}}f_{t}^{abcd}(z)=(1-z)^{-2\Delta_{\phi}}f_{t}^{adcb}(1-z)=(1-z)^{-2\Delta_{\phi}}f_{t}^{cbad}(1-z)\,,\quad
z\in(0,1)\,.\\\ $ (141)

Similar relations exist for fermionic statistics Mazac:2018qmi . Moreover,
care should be taken about some transformations inducing time-reversal when it
is a concern Bulycheva_2017 . These CFT crossing symmetry constraints
(including the additional 1d relations) are familiar and heavily used in, for
example, the conformal bootstrap program (see e.g. Rattazzi:2008pe ;
Mazac:2018qmi ; El-Showk:2012cjh ; Kos:2013tga ; Poland:2018epd ;
Ferrero:2019luz ; He:2020azu  for various references including recent results
in 1d). We expand a bit more on this in the next section.

###  A.2 Decomposing CFT Four-Point Functions

####  A.2.1 Conformal Blocks

There are 3 different channel decompositions for $f(z)$ that can be performed
using the OPE, but the convergence of the various OPEs depends on the value of
$z$ (which translates to physical region for the CCFT scattering amplitude).
In particular, when $z\in(0,1)$ the expansion where $z\to 0$ (contracting
$\phi_{1}\phi_{2}$ and $\phi_{3}\phi_{4}$) converges, telling us191919This
$z\to 0$ contraction is sometimes called the $s$-channel OPE, while the $z\to
1$ contraction is called $t$-channel OPE, and the $z\to\infty$ contraction is
called $u$-channel OPE. We will avoid overloading the terms $s$, $t$, and $u$
any further.

$f(z)=\sum_{\operatorname{\mathcal{O}}}\frac{c_{\phi\phi\operatorname{\mathcal{O}}}^{2}}{d_{\operatorname{\mathcal{O}}\operatorname{\mathcal{O}}}}k_{\Delta_{\operatorname{\mathcal{O}}}}(z)\,.$
(142)

Here $\operatorname{\mathcal{O}}$ denotes an exchanged conformal primary in
the $\phi\times\phi$ OPE with scaling dimension
$\Delta_{\operatorname{\mathcal{O}}}$, and
$k_{\Delta_{\operatorname{\mathcal{O}}}}(z)$ are the
$SL(2,\operatorname{\mathbb{R}})$ conformal blocks, proto-correlators that
span the vector space of solutions to the Ward identities. The coefficients
$c_{\phi\phi\operatorname{\mathcal{O}}}$ are the usual three point function
coefficients, and $d_{\operatorname{\mathcal{O}}\operatorname{\mathcal{O}}}$
is the normalization of the two-point function. In a unitary theory, the
$d_{\operatorname{\mathcal{O}}\operatorname{\mathcal{O}}}$ are typically
chosen to be $1$ (see Rychkov:2016iqz ), but in CCFT we do not use this
freedom, opting to define the two-point function from the celestial transform.

In less obscure terms, the conformal blocks are functions which re-sum over
all descendants of the primary $\operatorname{\mathcal{O}}$, and hence
describe the contribution to
$\expectationvalue{\phi_{1}\phi_{2}\phi_{3}\phi_{4}}$ by exchange of
$\operatorname{\mathcal{O}}$ between $\phi_{1}\phi_{2}$ and $\phi_{3}\phi_{4}$
Dolan:2003hv ; Dolan:2011dv . They are known to have a nice closed form in
terms of hypergeometric functions in $d=1$,$2$,$4$. Focusing on $d=1$, the
$SL(2,\operatorname{\mathbb{R}})$ conformal blocks are

$k^{\Delta_{i}}_{\Delta}(z)=z^{\Delta}{}_{2}F_{1}(\Delta-\Delta_{12},\Delta+\Delta_{34};2\Delta;z)\,.$
(143)

Expanding in other channels, and enforcing associativity of OPE, is what gives
crossing symmetry in the CFT.

####  A.2.2 Conformal Partial Waves

Another related expansion is the “conformal partial wave expansion,” which is
an expansion in a complete basis of single-valued functions
$\Psi_{\Delta,J}(z,\bar{z})$ called conformal partial waves, which are analogs
of harmonic functions for the conformal group gadde2017search . They are
typically chosen to be the eigenfunctions of the $z\to 0$ channel Casimir.

For general $d>1$, the so-called “principal series representations” consisting
of those eigenfunctions having $\Delta=\frac{d}{2}+ir$ with
$r\in\operatorname{\mathbb{R}}_{\geq 0}$, and integral spin $J$, form a
complete and orthogonal set of $\Psi$ with respect to an inner product
$(\,\,\,,\,\,\,)$.202020For brevity we suppress the details of the integral
defining the inner product in arbitrary dimensions, see Simmons_Duffin_2018
for details. The moral of the story is that this allows one to write

$f(z,\bar{z})=\sum_{J=0}^{\infty}\int_{\frac{d}{2}}^{\frac{d}{2}+i\infty}\frac{d\Delta}{2\pi
i}\frac{I_{\Delta,J}}{n_{\Delta,J}}\Psi_{\Delta,J}(z,\bar{z})+(\textrm{non-
norm.}),$ (144)

where $I_{\Delta,J}$ contains the dynamical data of the CFT (i.e. the data in
three-point functions) and can be naively extracted using the aforementioned
inner product; this is called the Euclidean inversion formula. Or, famously,
extracted in a way analytic in $J$ using the Lorentzian inversion formula
Caron_Huot_2017 ; Simmons_Duffin_2018 .

Specializing to $d=1$, we also need to include the “principal discrete series”
to form a complete set of partial waves. Altogether we have a basis consisting
of:212121We will also use the notation $\Psi_{m,J}$ for the discrete series
wavefunctions, keeping in mind that $\Psi_{m,J}$ is only normalizable (and
thus relevant for us) when $J\equiv m\ (\mathrm{mod}\ 2)$.

$\displaystyle\Psi_{\Delta,J}\,,$
$\displaystyle\quad\Delta\in\frac{1}{2}+i\operatorname{\mathbb{R}}_{\geq
0}\,,$ (145) $\displaystyle\Psi_{m,0}\,,$ $\displaystyle\quad m\in
2\operatorname{\mathbb{N}}\,,$ (146) $\displaystyle\Psi_{m,1}\,,$
$\displaystyle\quad m\in 2\operatorname{\mathbb{N}}-1\,,$ (147)

where $J=0,1$ is a 1d analog of spin, and denotes if $\Psi_{\Delta,J}$ is
symmetric or anti-symmetric respectively under $z\mapsto\frac{z}{z-1}$. The
inner product is given by

$(g,f)=\int_{-\infty}^{\infty}\frac{dz}{z^{2}}\bar{g}(z)f(z)\,,$ (148)

and the eigenfunctions satisfy

$\displaystyle\left(\Psi_{\frac{1}{2}+ir,J},\Psi_{\frac{1}{2}+ir^{\prime},J^{\prime}}\right)$
$\displaystyle=2\pi
n_{\frac{1}{2}+ir}\delta(r-r^{\prime})\delta_{J,J^{\prime}}\,,$ (149)
$\displaystyle\left(\Psi_{m,J},\Psi_{n,J^{\prime}}\right)$
$\displaystyle=\frac{4\pi^{2}}{2m-1}\delta_{m,n}\delta_{J,J^{\prime}}\,,$
(150) $\displaystyle\left(\Psi_{\frac{1}{2}+ir,J},\Psi_{n,J^{\prime}}\right)$
$\displaystyle=0\,.$ (151)

Here we define

$K_{\Delta,J}\coloneqq\frac{\sqrt{\pi}\Gamma(\Delta-\frac{1}{2})\Gamma(\Delta-1+J)\Gamma\left(\frac{1}{2}(1-\Delta+J)\right)^{2}}{\Gamma(\Delta-1)\Gamma(1-\Delta+J)\Gamma\left(\frac{\Delta+J}{2}\right)^{2}}\,,$
(152) $n_{\Delta}\coloneqq
2K_{\Delta,J}K_{1-\Delta,J}=\frac{4\pi\tan(\pi\Delta)}{2\Delta-1}\,.$ (153)

So that in total we can write

$f(z)=\sum_{J}\int_{\frac{1}{2}}^{\frac{1}{2}+i\infty}\frac{d\Delta}{2\pi
i}\frac{I_{\Delta,J}}{n_{\Delta}}\Psi_{\Delta,J}(z)+\sum_{m=1}^{\infty}\frac{2m-1}{4\pi^{2}}\tilde{I}_{m}\Psi_{m,J}(z)\,,$
(154)

recalling again that for the discrete series $J\equiv m\ (\mathrm{mod}\ 2)$.

The explicit formulas for the 1d conformal partial waves are

$\displaystyle\Psi_{\Delta,0}(z)$ $\displaystyle=\int
dx_{5}\,\left(\frac{\absolutevalue{x_{12}}}{\absolutevalue{x_{15}}\absolutevalue{x_{25}}}\right)^{\Delta}\left(\frac{\absolutevalue{x_{34}}}{\absolutevalue{x_{35}}\absolutevalue{x_{45}}}\right)^{1-\Delta}\,,$
(155) $\displaystyle\Psi_{\Delta,1}(z)$ $\displaystyle=\int
dx_{5}\,\left(\frac{\absolutevalue{x_{12}}}{\absolutevalue{x_{15}}\absolutevalue{x_{25}}}\right)^{\Delta}\left(\frac{\absolutevalue{x_{34}}}{\absolutevalue{x_{35}}\absolutevalue{x_{45}}}\right)^{1-\Delta}\mathrm{sgn}(x_{12}x_{15}x_{25}x_{34}x_{35}x_{45})\,.$
(156)

The integrals converge when $0<\real(\Delta)<1$, and other values can be
obtained by analytic continuation from these answers. One can also evaluate
these integrals explicitly in the three different regions for $z$. For the
principal continuous series we have222222We will use the labels $(-)$, $(0)$,
and $(+)$ for the conformal partial waves in the regions $(-\infty,0)$,
$(0,1)$, and $(1,\infty)$ respectively, to avoid deviations of notation from