aainstitutetext: International Centre for Theoretical Sciences-TIFR,
Shivakote, Hesaraghatta Hobli, Bengaluru North 560089, Indiabbinstitutetext:
Department of Physics, Princeton University, Princeton, NJ 08544,
USAccinstitutetext: School of Physical Sciences, National Institute of Science
Education and Research
An OCC of Homi Bhabha National Institute, Jatani-752050, India

# Towards an IR finite $\mathcal{S}$-matrix in the flat limit of AdS/CFT

Sarthak Duary b Eliot Hijano c and Milan Patra<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Asymptotic Fock spaces lead to IR divergences in S-matrices. The issue can be
traced back to the assumption of asymptotic decoupling, and its relaxation
leads to Faddeev-Kulish states and an IR-finite S-matrix. In this paper we
initiate the exploration of these states in the context of the flat limit of
AdS/CFT. We construct asymptotic states as flat limits of fields that interact
with the electro-magnetic field in AdS, and provide a reconstruction in terms
of the dual CFT operators.

##  1 Introduction

The appearance of IR divergences in flat spacetime is one of the features of
the IR structure of QED. Physically, this is due to the presence of long-range
interaction and was first studied by Weinberg Weinberg:1965nx . The
cancellation of the IR divergences was studied by restricting to sufficiently
inclusive cross sections Donoghue:1999qh . But, at the level of
$\mathcal{S}$-matrix element, the IR divergence is related to the assumption
of asymptotic decoupling which casts the asymptotic Hamiltonian as free. This
means the asymptotic states are Fock space states and the fields are
completely free in the asymptotic regime of flat spacetime, that is in the
region $r\to\infty$ with $u,v=t\pm r$ fixed. Relaxing the assumption of
asymptotic decoupling is a hard thing to do. It is certainly unreasonable to
consider the full fledged Hamiltonian asymptotically. We can however consider
a Hamiltonian that is a good approximation to the original one when the field
profiles are well separated. Any interaction involving short wavelengths can
thus be disregarded. However, long wavelengths or soft frequencies correspond
to long range interactions that should be taken into account. Indeed, it is
these frequencies that participate in the resolution of the IR divergences in
terms of Faddeev-Kulish (F-K) states Kulish:1970ut . The resulting asymptotic
Hamiltonian leads to the construction of asymptotic states consisting on Fock
space states with clouds of soft photons. The $\mathcal{S}$-matrix between
these dressed states are IR finite Chung:1965zza ; Kibble:1968oug ;
Kibble:1968npb ; Kibble:1968lka ; Ware:2013zja .

In this paper, our objective is to understand the IR finite
$\mathcal{S}$-matrix in the flat limit of AdS/CFT. The $\mathcal{S}$-matrix is
the only precise observable in flat spacetime. In the presence of gravity,
diffeomorphism invariance makes it difficult to define local correlation
functions. In AdS, due to the AdS/CFT correspondence, gauge invariant
correlation functions can be defined at the AdS boundary as conformal
correlators. In flat limit of the AdS/CFT correspondence, the CFT correlation
functions are related to the flat space $\mathcal{S}$-matrix, see
e.g.Polchinski:1999ry ; Susskind:1998vk ; Giddings:1999qu ; Giddings:1999jq ;
Gary:2009mi ; Penedones:2010ue ; Fitzpatrick:2011ia ; Fitzpatrick:2011jn ;
Fitzpatrick:2010zm ; Gary:2009ae ; Komatsu:2020sag ; vanRees:2022itk ;
Raju:2012zr ; Li:2021snj . Using the flat limit, the $2\to 2$ scattering
amplitude of massless string states in type IIB superstring theory on $10d$
flat space is evaluated from the $\mathcal{N}=4$ SYM four point
functionsOkuda:2010ym . The analyticity and unitarity of $\mathcal{S}$-matrix
from holographic CFT in the flat limit has been elaborated upon in
Fitzpatrick:2011hu ; Fitzpatrick:2011dm  and found applications in the
$\mathcal{S}$-matrix bootstrap programPaulos:2016fap ; Kruczenski:2022lot .
The flat limit of position space CFT correlation function introduced using
HKLL bulk reconstruction prescription introduced in Hijano:2019qmi ;
Hijano:2020szl  is another way to formulate holography in flat spacetime. The
flat limit relates the bulk scattering states in flat spacetime to boundary
CFT operators. Using the map between the scattering states and CFT operators
in Hijano:2020szl  the Weinberg soft theorems are derived from CFT Ward
identities. The scattering states for massive scalar particle form a Fock
space and in Hijano:2020szl  the construction of these states are described in
AdS/CFT.

Faddeev-Kulish dressings address the IR divergences of the
$\mathcal{S}$-matrix in flat spacetime. The dressing can be understood as a
Wilson line involving the soft modes of the gauge field Mandelstam:1962mi ;
Jakob:1990zi ; Choi:2018oel ; Choi:2019fuq . In this paper, we construct the
dressing in AdS/CFT in the language of Wilson lines. We then take the flat
limit as a large radius of AdS limit. It is common lore that the radius of AdS
acts as an IR regulator. As the large limit is taken, IR divergences are
expected to reappear unless the asymptotic dynamics of fields is studied
appropriately. By constructing dressed fields in AdS, the large radius of AdS
limit can be taken while avoiding IR divergences in the flat space
observables.

The dressing in a local bulk massive scalar field operator in AdS can be
understood as a Wilson line following a path from a bulk point to the
boundary. In the soft limit, the scalar field dynamics are coupled to the soft
modes of the gauge field. However, the dressed scalar field solves the free
Klein-Gordon equation. Therefore, we can reconstruct the dressed scalar field
using free HKLL reconstruction method. The interaction of the scalar field
with the soft modes of the photon provides the main simplification in our
analysis. Now, we can extract the creation mode for the massive scalar field
dressed by the Wilson line in terms of dressed CFT operator. The dual CFT
operator for the dressed field can be related to the dual CFT operator of the
undressed field where the Wilson line end-points are at the boundary. The
Wilson line itself can be constructed in terms of current operators in the
CFT. The main result of this draft is an expression for the Faddeev-Kulish
dressed state in terms of the CFT operators at the boundary. Such expression
can potentially be used to compute IR finite S-matrices from flat limits of
CFT correlation functions.

#### Outline of the paper:

The note is organized as follows. In section 2 we review very briefly the
equivalence between Wilson line dressing and Faddeev-Kulish dressing in flat
space and why they are equivalent. In section 3 we note down the expressions
for annihilation and creation operators in flat limit of AdS for scalar field
and photons, which will be crucial in the latter sections. In section 4 we
define the soft dressing in AdS. Then in section 5 we present the derivation
of dressed scattering state in terms of boundary CFT operator and creation and
annihilation operator of photon. Finally in section 6 we conclude.

##  2 Review: Faddeev-Kulish states as Wilson line dressings

In the gauge invariant formulation of QED conventional matter fields are
dressed by a path dependent Wilson line extending all the way to
infinity.111In this section, we will denote charge by $e$ to avoid notational
inconsistency. But in later sections we will again switch to $q$ to denote
charge.

$\begin{split}\hat{\Phi}\left(x,\Gamma\right)=\mathcal{P}~{}\exp\left(ie\int_{\Gamma}^{x}d\xi^{\mu}~{}A_{\mu}(\xi)\right)\Phi(x)~{}.\end{split}$
(1)

In flat space this path dependent dressing describes the Faddeev-Kulish
dressing for a particular time-like geodesic path for the asymptotic field
(Choi:2018oel, ). In this note our goal is to understand how to derive this as
a flat limit of AdS/CFT.

In flat space in Lorenz gauge, the Maxwell field equations can be written as

$\square A_{\mu}=J_{\mu}\,,$ (2)

where, $J_{\mu}$ is the source term. Asymptotically, the solution to this
equation consists of a homogeneous solution and a source term

$A_{\mu}(x)=A^{h}_{\mu}(x)+\int
d^{4}x^{\prime}G_{\text{ret}}(x,x^{\prime})J_{\mu}(x^{\prime})\,.$ (3)

where $G_{\text{ret}}(x,x^{\prime})$ solves $\Box
G_{\text{ret}}(x,x^{\prime})=\delta^{(4)}(x,x^{\prime})$.
While source term will contribute by a phase factor, the homogeneous solution
will have the following mode expansion

$\begin{split}A^{h}_{\mu}(x)=\int\frac{d^{3}q}{(2\pi)^{3}2\omega_{\vec{q}}}\sum_{\lambda=\pm}\left(\varepsilon^{(\lambda)}_{\mu}\mathbf{a}^{(\lambda)}_{\vec{q}}e^{iq\cdot
x}+{\varepsilon^{(\lambda)}_{\mu}}^{*}{\mathbf{a}^{(\lambda)}_{\vec{q}}}^{\dagger}e^{-iq\cdot
x}\right)\end{split}$ (4)

where, $\omega_{\vec{q}}=|\vec{q}|$, and $\varepsilon^{(\lambda)}_{\mu}$ are
the polarization tensor. Choosing the Wilson line path to be the time-like
geodesic followed by a semiclassical particle,
$\xi^{\mu}=x^{\mu}+\frac{p^{\mu}}{m}\tau$, we have

$\begin{split}\exp\left(ie\int_{\Gamma}^{x}d\xi^{\mu}~{}A_{\mu}(\xi)\right)&=\exp\left\\{ie\frac{p^{\mu}}{m}\int_{-\infty}^{0}d\tau~{}A_{\mu}^{h}\left(x^{\nu}+\frac{p^{\nu}}{m}\tau\right)\right\\}\\\
&=\exp\left\\{-e\int\frac{d^{3}q}{(2\pi)^{3}2\omega_{\vec{q}}}\frac{p^{\mu}}{p\cdot
q}\sum_{\lambda=\pm}\left({\varepsilon^{(\lambda)}_{\mu}}^{*}{\mathbf{a}^{(\lambda)}_{\vec{q}}}^{\dagger}e^{-iq\cdot
x}-\varepsilon^{(\lambda)}_{\mu}\mathbf{a}^{(\lambda)}_{\vec{q}}e^{iq\cdot
x}\right)\right\\}\end{split}$ (5)

which is the Wilson line dressing of an asymptotic particle with momentum
$p^{\mu}$. As we are interested in asymptotic field we will take
$x^{0}\to-\infty$ limit for an asymptotic incoming particle. In that limit the
integral in (5) will have contribution in $q\to 0$ limit and we can replace
$e^{\pm iq\cdot x}$ with a function $\phi(p,q)$ which has support in a small
neighbourhood of $q\to 0$ with $\phi\to 1$ as $q\to 0$. The resulting
expression is the well known formula for the F-K dressing.

$\begin{split}\mathcal{W}(p)=\exp\left\\{-e\int\frac{d^{3}q}{(2\pi)^{3}2\omega_{\vec{q}}}\frac{p^{\mu}}{p\cdot
q}\phi(p,q)\sum_{\lambda=\pm}\left({\varepsilon^{(\lambda)}_{\mu}}^{*}{\mathbf{a}^{(\lambda)}_{\vec{q}}}^{\dagger}-\varepsilon^{(\lambda)}_{\mu}\mathbf{a}^{(\lambda)}_{\vec{q}}\right)\right\\}~{}.\end{split}$
(6)

The expectation is that dressing fields with Wilson lines in the context of
AdS/CFT will yield a CFT operator formula for FK dressings upon a flat limit.

###  2.1 Why dressing by Wilson line is equivalent to the F-K dressing?

One may ask behind the motivation for considering a field dressed by a Wilson
line. To understand that let us study the equation of motion for the field
$\hat{\Phi}$. The scalar electrodynamics action is

$\begin{split}S=\int
d^{4}x~{}\sqrt{-g}\left(-D_{M}\Phi^{\dagger}D^{M}\phi-m^{2}\Phi^{\dagger}\Phi-\frac{1}{4}F_{MN}F^{MN}\right)\,,\end{split}$
(7)

where $D_{M}=\partial_{M}-ieA_{M}$. If we neglect the backreaction of the
scalar field $\Phi$ on the field strength $F_{AB}$ we have the equation of
motion for the dressed field $\hat{\Phi}$ as (Monica:2017, )

$\begin{split}\left(\square-m^{2}\right)\hat{\Phi}=-ie~{}\hat{\Phi}~{}\nabla^{M}\int_{\Gamma}F_{MA}~{}dy^{A}-2ie~{}\nabla^{M}\hat{\Phi}~{}\int_{\Gamma}F_{MA}~{}dy^{A}+e^{2}~{}\hat{\Phi}~{}g^{MN}\int_{\Gamma}F_{MA}~{}dy^{A}\int_{\Gamma}F_{NB}~{}dy^{B}\end{split}$
(8)

Now $F_{AB}~{}\approx\mathcal{O}(q)$ and if we consider the soft limit (i.e
$q\to 0$ limit) then we can neglect the right hand side of the equation (8)
and eventually we have

$\begin{split}\left(\square-m^{2}\right)\hat{\Phi}=\mathcal{O}(q)\end{split}$
(9)

Hence, in the soft limit the dressed field actually behaves like a free field.
IR divergences are a consequence of the non-trivial interactions among fields.
In particular interactions involving long range forces. We can expect states
corresponding to free fields will not have the problem of bringing IR
divergences.

##  3 Free scalar and gauge field in AdS and it’s flat space limit

In this note our main goal is to construct these dressed state as a flat limit
from AdS. For that let us first recap HKLL bulk field reconstruction in global
AdS.

We start with the $\text{AdS}_{4}$ global coordinate where the line element
reads

$\begin{split}ds^{2}=\frac{L^{2}}{\cos^{2}\rho}\left(-d\tau^{2}+d\rho^{2}+\sin^{2}\rho~{}d\Omega^{2}_{2}\right)\end{split}$
(10)

Where $\rho$ is the radial coordinate and the boundary of the space is located
at $\rho=\frac{\pi}{2}$. The coordinates parametrizes the CFT are
$x\equiv\left(\tau,\Omega_{2}\right)$ and $L$ is the AdS radius.

Now let us try to understand how to perform a flat limit in this space. We
need to take AdS radius $L$ to be large with $\tau=t/L$ and $\rho=r/L$ such
that

$\begin{split}ds^{2}~{}\xrightarrow[L\to\infty]{~{}}-dt^{2}+dr^{2}+r^{2}~{}d\Omega^{2}_{2}\end{split}$
(11)

In this global coordinate we can think of the the space-time a cylinder and
and taking a flat limit implies zooming into the centre of the cylinder where
geometry is locally flat.

###  3.1 Free scalar field in AdS

In this note we will use HKLL bulk operator reconstruction in global AdS and
discuss explicitly the flat limit to obtain the flat space creation and
annihilation operators in terms of CFT operator smeared along time-like
direction.

We consider the scattering region at $\tau=0$, at the centre of the AdS. Now
if we consider the assumption of asymptotic decoupling then before and after
the scattering region the scalar field $\Phi$ must obey free Klein-Gordon
equation on AdS

$\begin{split}\left(\square-m^{2}\right)\Phi=0,~{}~{}\text{with}~{}~{}m^{2}L^{2}=\Delta\left(\Delta-3\right)\end{split}$
(12)

with the following fall-off behaviour

$\begin{split}\Phi(\rho,x)~{}\xrightarrow[\rho\to\frac{\pi}{2}]{~{}}\left(\cos\rho\right)^{\Delta}\mathcal{O}(x)\end{split}$
(13)

After reconstructing these operators in HKLL bulk reconstruction method we
need to take a flat limit to extract the creation and annihilation operators.
For massive field this reads (Hijano:2020szl, )

$\begin{split}\sqrt{2\omega_{\vec{p}}}~{}{a}^{\dagger}_{\omega_{\vec{p}}}&={C}\left(L,m,|\vec{p}|\right)\int_{0}^{\pi}d\tau
e^{-i\omega_{\vec{p}}L\left[\tau-\frac{\pi}{2}-\frac{i}{2}\log\left(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\right)\right]}\mathcal{O}\left(\tau,\hat{p}\right)\end{split}$
(14)

where, $\hat{p}$ is the angle of the momentum $\vec{p}$ and
${C}=\frac{1}{2\pi}\Bigg{(}\frac{mL}{\pi^{3}}\Bigg{)}^{\frac{1}{4}}\Bigg{(}\frac{2m}{i|\vec{p}|}\Bigg{)}^{mL+\frac{1}{2}}~{}L\,.$
This integral will be dominated by a window of size of
$\mathcal{O}\left(\frac{1}{L}\right)$ at $\text{Re}(\tau)=\pm\frac{\pi}{2}$
and
$\text{Im}(\tau)=\frac{1}{2}\log\left(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\right)$.
This has been denoted by the caps in figure 1.

###  3.2 Free $U(1)$ gauge field in AdS

Following (Hijano:2020szl, ) in this section we review the construction of
photon scattering states in flat limit of AdS. Again, by virtue of asymptotic
decoupling, we will assume that photons obey the free Maxwell equation at
early and late times. There are two possibilities to quantize the bulk gauge
field, which are specified by either electric or magnetic boundary conditions.
In this note we will consider the magnetic boundary condition, which fixes the
magnetic field at the boundary to zero. In this case the gauge is dual to a
current operator in the CFT. We thus look for a field operator with boundary
condition

$\begin{split}{\mathcal{A}_{\mu}}(\rho,x)\xrightarrow[\rho\to\frac{\pi}{2}]{~{}}\cos\rho\,j_{\mu}(x)\,.\end{split}$
(15)

A free $U(1)$ gauge field operator in Minkowaski space can be written in the
following form.

$\begin{split}{\cal
A}_{\mu}(x)=\int\frac{d^{3}\vec{q}}{(2\pi)^{3}}\frac{1}{\sqrt{2\omega_{\vec{q}}}}\sum_{\lambda=\pm}\left(\varepsilon^{(\lambda)}_{\mu}\mathbf{a}^{(\lambda)}_{\vec{q}}e^{iq\cdot
x}+{\varepsilon^{(\lambda)}_{\mu}}^{*}{\mathbf{a}^{(\lambda)}_{\vec{q}}}^{\dagger}e^{-iq\cdot
x}\right),~{}~{}x\in\text{Mink}_{3+1}\,,\end{split}$ (16)

where, $\varepsilon^{(\lambda)}_{\mu}$ are the null polarization vector with
$\lambda=\pm$ denote the two different polarization and it is normalized such
that $\varepsilon^{(+)}_{\mu}\varepsilon^{(-),\mu}=1$. We will also work in
Lorenz gauge such that $\varepsilon_{\mu}q^{\mu}=0$.

In Lorenz gauge we can reconstruct the gauge field explicitly using HKLL and
then taking a flat limit and using (16) we can derive the
creation/annihilation modes explicitly. This has been done in (Hijano:2020szl,
). We have

$\begin{split}&\sqrt{2\omega_{\vec{q}}}~{}\mathbf{a}^{\dagger(-)}_{\vec{q}}=\frac{-1}{4\omega_{\vec{q}}}\frac{1+z_{q}\bar{z}_{q}}{\sqrt{2}}\int
d\tau^{\prime}~{}e^{i\omega_{\vec{q}}L\left(\frac{\pi}{2}-\tau^{\prime}\right)}\int
d^{2}z^{\prime}\frac{1}{\left(z_{q}-z^{\prime}\right)^{2}}j^{+}_{\bar{z}^{\prime}}(\tau^{\prime},z^{\prime},\bar{z}^{\prime})\\\
&\sqrt{2\omega_{\vec{q}}}~{}\mathbf{a}^{\dagger(+)}_{\vec{q}}=\frac{-1}{4\omega_{\vec{q}}}\frac{1+z_{q}\bar{z}_{q}}{\sqrt{2}}\int
d\tau^{\prime}~{}e^{i\omega_{\vec{q}}L\left(\frac{\pi}{2}-\tau^{\prime}\right)}\int
d^{2}z^{\prime}\frac{1}{\left(\bar{z}_{q}-\bar{z}^{\prime}\right)^{2}}j^{+}_{{z}^{\prime}}(\tau^{\prime},z^{\prime},\bar{z}^{\prime})\\\
&\sqrt{2\omega_{\vec{q}}}~{}\mathbf{a}_{\vec{q}}^{(-)}=-\frac{1}{4\omega_{\vec{q}}}\frac{1+z_{q}\bar{z}_{q}}{\sqrt{2}}\int
d\tau^{\prime}e^{-i\omega_{\vec{q}}L(\frac{\pi}{2}-\tau^{\prime})}\int
d^{2}z^{\prime}\frac{1}{(z_{q}-z^{\prime})^{2}}j^{-}_{\bar{z}^{\prime}}(\tau^{\prime},z^{\prime},\bar{z}^{\prime})~{}~{}~{}\\\
&\sqrt{2\omega_{\vec{q}}}~{}\mathbf{a}_{\vec{q}}^{(+)}=-\frac{1}{4\omega_{\vec{q}}}\frac{1+z_{q}\bar{z}_{q}}{\sqrt{2}}\int
d\tau^{\prime}e^{-i\omega_{\vec{q}}L(\frac{\pi}{2}-\tau^{\prime})}\int
d^{2}z^{\prime}\frac{1}{(\bar{z}_{q}-\bar{z}^{\prime})^{2}}j^{-}_{z^{\prime}}(\tau^{\prime},z^{\prime},\bar{z}^{\prime})\end{split}$
(17)

where $(z,\bar{z})$ are the complex coordinate on the sphere and
$\mathbf{a}^{\dagger(\pm)}_{\vec{q}},\mathbf{a}^{(\pm)}_{\vec{q}}$ are the
photon creation and annihilation operators respectively. These expression
tells us that the integral will be dominated by a window of size of
$\mathcal{O}\left(\frac{1}{L}\right)$ around the region $\tau=+\frac{\pi}{2}$
for outgoing particle and $\tau=-\frac{\pi}{2}$ for incoming particle. A
pictorial representation of these formulas can be found in figure 1.

$\partial{\cal M}_{-}$$\tilde{\cal I}^{-}$$\tilde{\cal
I}^{+}$$\tau={{\pi}\over 2}$$\tau=-{{\pi}\over 2}$$\partial{\cal M}_{+}$
Figure 1: Pictorial representation of the map between the asymptotic regions
of flat space and the boundary of AdS. The euclidean domes
$\partial\mathcal{M}_{\pm}$ are analytic continuation of the boundary CFT in
the imaginary global time direction playing the role of future/past infinity
$i^{\pm}$. $\widetilde{\mathcal{I}}^{\pm}$ are future/past null infinity. This
figure is taken from (Hijano:2020szl, ).

##  4 Soft dressing in AdS

In this section we will try to define the dressing in AdS that could
correspond to the F-K state after taking a flat limit.

In (Hijano:2020szl, ) the authors have studied fields that are completely free
in the asymptotic regime of Minkowaski space. That is the fields are free in
the region $r\to\infty$ with $u=t-r$ fixed. Now relaxing the assumption of
asymptotic decoupling involves including interaction in the definition of
scattering states. Then instead of studying free fields we could study fields
that interact with the long wavelength modes of the electro-magnetic field. We
could then think of reconstructing such interacting fields in AdS/CFT and
taking a flat limit described in previous section. Then one can expect that
the resulting scattering state correspond to F-K state that are usual Fock
space states dressed by clouds of photons.

###  4.1 Defining the soft dressing in AdS

Now the question is how to define the dressing in AdS. Taking the motivation
from flat space described in 2.1 we will define the dressing in the bulk of
AdS and then consider reconstruction of those fields in the bulk of AdS by
HKLL bulk reconstruction method.

In flat space, the soft limit is attained by the frequency of the photon being
zero. Soft frequencies are the ones participating in long range forces, and so
they are the only ones that should be considered when constructing asymptotic
fields in the flat limit. In flat space we can take minimum momentum of photon
to be zero, $q\to 0$. But in AdS this is not possible because minimum
frequency would be inversely proportional to the size of the space. Hence we
can think that in the soft limit the field strength $F_{AB}$ would be
$\mathcal{O}\left(\frac{1}{L}\right)$. Taking this as the input we can define
the dressing in AdS in the following way.

Let us consider a free scalar field $\Phi$ dual to an operator $\mathcal{O}$
and a gauge field $\mathcal{A}$ dual to an operator $j$. Now we will define
the dressing by the following non-local but gauge invariant bulk operator

$\tilde{\Phi}(y)=U(x,y)\Phi(y)$ (18)

along with $F_{AB}=\mathcal{O}\left(\frac{1}{L}\right)$.

The point $y$ is a point in the bulk of AdS and the point $x$ is a point in
the boundary of AdS. $U(x,y)$ is a Wilson line constructed from the gauge
field. Explicitly we have

$U(x,y)=\mathcal{P}\Bigg{\\{}e^{iq\int_{x}^{y}\mathcal{A}_{M}dx^{M}}\Bigg{\\}}$
(19)

Now the bulk field $\tilde{\Phi}$ is gauge invariant by definition but as a
trade-off we have introduced a nonlocality because the field now depend on a
boundary point $x$.

If we neglect the backreaction of the scalar field $\Phi$ on the field
strength $F_{AB}$ we have the equation of motion for the dressed field
$\tilde{\Phi}$ as (Monica:2017, )

$\begin{split}\left(\square-m^{2}\right)\tilde{\Phi}=-iq~{}\tilde{\Phi}~{}\nabla^{M}\int_{\Gamma}F_{MA}~{}dy^{A}-2iq~{}\nabla^{M}\tilde{\Phi}~{}\int_{\Gamma}F_{MA}~{}dy^{A}+q^{2}~{}\tilde{\Phi}~{}g^{MN}\int_{\Gamma}F_{MA}~{}dy^{A}\int_{\Gamma}F_{NB}~{}dy^{B}\end{split}$
(20)

In the soft limit as we have $F_{AB}=\mathcal{O}\left(\frac{1}{L}\right)$,
then the equation of motion for the field $\tilde{\Phi}$, becomes

$\begin{split}\left(\square-m^{2}\right)\tilde{\Phi}=\mathcal{O}\left(\frac{1}{L}\right)\end{split}$
(21)

Hence, in the kinematic regime we are interested the field $\tilde{\Phi}$
obeys free Klein-Gordon equation. This means we can actually reconstruct
$\tilde{\Phi}$ by using free HKLL, without worrying about interactions. Of
course, the operator at the boundary is now different. Instead of having usual
operator $\mathcal{O}$ dual to free bulk field $\Phi$, now we have a non-local
operator $\tilde{O}$.

###  4.2 The dual operator of the AdS dressing in CFT

Let us now determine the dual operator to the dressed field $\tilde{\Phi}$

$\tilde{\Phi}(y)=\mathcal{P}\Bigg{\\{}e^{iq\int_{x}^{y}\mathcal{A}_{M}dx^{M}}\Bigg{\\}}\Phi(y)~{}~{}~{}.$
(22)

Since,222where $a$ are coordinates at the boundary.

$\begin{split}\Phi(\rho,x)~{}&\xrightarrow[\rho\to\frac{\pi}{2}]{~{}}\left(\cos\rho\right)^{\Delta}\mathcal{O}(x)\\\
\mathcal{A}_{a}\left(\rho,x\right)~{}&\xrightarrow[\rho\to\frac{\pi}{2}]{~{}}j_{a}~{}\cos\rho\end{split}$
(23)

and the line element in the bulk varies with the boundary line element with a
factor of $\frac{1}{\cos\rho}$, the dual operator should be

$\tilde{O}(x^{\prime})=\mathcal{P}\Bigg{\\{}e^{iq\int_{x}^{x^{\prime}}dx^{a}j_{a}}\Bigg{\\}}\mathcal{O}(x^{\prime})~{}.$
(24)

As the end-points of this Wilson line are at the boundary, we can thought this
Wilson line as an expression in the boundary. In fig.2 we draw two Wilson
lines, one representing the Wilson line in AdS joining bulk point to boundary
point and the other is the Wilson line joining two boundary points.

Figure 2: Wilson line: the magenta line is the Wilson line connecting bulk
point $y$ to boundary point $x$. The cyan line is the Wilson line connecting
two boundary points $x^{\prime}$ and $x$.

##  5 The flat space scattering state for the dressed operator

###  5.1 The dressed scattering state in terms of CFT operators

In flat space the decomposition of a scalar field $\Phi$ can be done at early
and late times. Hence, there exists two sets of creation and annihilation
operators -one for early and the other for the late time. This means the “in”
and “out” scattering state can be built using the formulas

$\begin{split}&\left|\vec{k},in\right>=\sqrt{2\omega_{\vec{k}}}~{}a^{\dagger}_{in,\vec{k}}\left|0\right>\\\
&\left<out,\vec{p}\right|=\sqrt{2\omega_{\vec{p}}}~{}\left<0\right|a_{out,\vec{p}}\end{split}$
(25)

The $\mathcal{S}$ matrix is computed as an overlap between “in” and “out”
states

$\mathcal{S}_{fi}=\left<out|in\right>$ (26)

In this way the $1\rightarrow 1$ $\mathcal{S}$-matrix element is normalized as

$\left<k_{1}|\mathcal{S}|k_{2}\right>=2\omega_{\vec{k}}~{}\delta^{(3)}\left(\vec{k_{1}}-\vec{k_{2}}\right)$
(27)

Now let us discuss how to construct the dressed “in” and “out” state in flat
space. As we have reviewed in 3.1 for a free scalar field $\Phi$ in the bulk
of AdS with the dual operator $\mathcal{O}$, the creation operator in flat
space reads

$\begin{split}\sqrt{2\omega_{\vec{p}}}~{}{a}^{\dagger}_{\omega_{\vec{p}}}&=\tilde{C}\left(L,m,|\vec{p}|\right)\int_{0}^{\pi}d\tau
e^{-i\omega_{\vec{p}}L\left[\tau-\frac{\pi}{2}-\frac{i}{2}\log\left(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\right)\right]}\mathcal{O}\left(\tau,\hat{p}\right)\end{split}$
(28)

Now instead of a free scalar field $\Phi$ we have a free field $\tilde{\Phi}$
with dual operator $\tilde{\mathcal{O}}$ . Hence, the creation modes extracted
from the position space scalar field operator as

$\begin{split}\sqrt{2\omega_{\vec{p}}}~{}\tilde{a}^{\dagger}_{\omega_{\vec{p}}}&=\tilde{C}\left(L,m,|\vec{p}|\right)\int_{0}^{\pi}d\tau
e^{-i\omega_{\vec{p}}L\left[\tau-\frac{\pi}{2}-\frac{i}{2}\log\left(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\right)\right]}\tilde{\mathcal{O}}\left(\tau,\hat{p}\right)\end{split}$
(29)

This operator when act on the CFT vacuum will give the IR finite $\mathcal{S}$
matrix in CFT.

Now using the expression for $\tilde{\mathcal{O}}$ we can write the dressed
creation operator as

$\begin{split}\sqrt{2\omega_{\vec{p}}}~{}\tilde{a}^{\dagger}_{\omega_{\vec{p}}}&=\tilde{C}\left(L,m,|\vec{p}|\right)\int_{0}^{\pi}d\tau
e^{-i\omega_{\vec{p}}L\left[\tau-\frac{\pi}{2}-\frac{i}{2}\log\left(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\right)\right]}\exp\left(iq\int_{\Gamma(\tau,\hat{p})}j_{a}~{}dx^{a}\right){\mathcal{O}}\left(\tau,\hat{p}\right)\end{split}$
(30)

As in the kinematic regime we are interested we can neglect $F_{AB}$, which
implies that the Wilson line is path independent and we can choose any path.
We will choose

$\begin{split}&\Gamma\left(\tau,\hat{p}\right)~{}:~{}x\rightarrow x_{0}\\\
&x^{a}_{0}~{}:~{}x^{\tau}_{0}=0,\hat{\Omega}_{0}=\hat{p}\\\
&x^{a}~{}:~{}x^{\tau}=\tau,\hat{\Omega}_{0}=\hat{p}\\\ \end{split}$ (31)

Then,

$\begin{split}\exp\left(iq\int_{\Gamma(\tau,\hat{p})}j_{a}~{}dx^{a}\right)&=\exp\left(iq\int_{0}^{\tau}j_{\tau^{\prime}}d\tau^{\prime}\right)\\\
&=1+iq\int_{0}^{\tau}j_{\tau^{\prime}}d\tau^{\prime}+\mathcal{O}(q^{2})\\\
&=1+q\sum_{m}\frac{1}{\omega_{m}}\left[e^{i\omega_{m}\tau}-1\right]\hat{X}_{m}+\mathcal{O}(q^{2})\\\
\end{split}$ (32)

Where we have used the form of the CFT operator $j_{a}$, which is

$\begin{split}j_{\tau^{\prime}}\left(\tau^{\prime},\hat{p}\right)=\sum_{m}e^{i\omega_{m}\tau^{\prime}}\hat{X}_{m}(\hat{p})+h\cdot
c\end{split}$ (33)

where,

$\omega_{m}=1+l+2\kappa~{}~{}\text{or}~{}~{}2+l+2\kappa\,.$ (34)

Upon taking the flat limit $L\to\infty$, we need to specify what happens to
the integers $\kappa$. A possibility is to interpret any mode with
$\kappa\sim{\cal O}(1)$ as a soft mode, as their frequency tends to zero
$\omega_{m}/L\rightarrow 0$. This is however not what we usually consider a
soft field. We usually take the limit from finite momentum to zero, which
means we first write $\kappa=kL$, then take the large $L$ limit, and finally
we take the $k\rightarrow 0$ limit. This is the limit we will consider here.
It is however interesting to wonder what the physical meaning of modes with
finite $\kappa$ is. All in all, we will define the soft limit as
$\frac{\omega_{m}}{L}=k$ with $k\rightarrow 0$ after $L\to\infty$.
Now as $\omega_{m}=1+l+2\kappa=kL$, for such large value of $\kappa$ the sum
over integers turn into integral over $k$.

$\sum_{m}\rightarrow\frac{L}{2}\int dk$ (35)

So, we have

$q\sum_{m}\frac{1}{\omega_{m}}\left[e^{i\omega_{m}\tau}-1\right]\hat{X}_{m}(\hat{p})=\frac{q}{2}\int\frac{dk}{k}\left[e^{iLk\tau}-1\right]\hat{X}_{k}(\hat{p})$

So, up to $\mathcal{O}(q)$ the dressed creation operator can be written as

$\begin{split}\sqrt{2\omega_{\vec{p}}}~{}\tilde{a}^{\dagger}_{\omega_{\vec{p}}}=&\tilde{C}\left(L,m,|\vec{p}|\right)\int_{0}^{\pi}d\tau
e^{-i\omega_{\vec{p}}L\left[\tau-\frac{\pi}{2}-\frac{i}{2}\log\left(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\right)\right]}{\mathcal{O}}\left(\tau,\hat{p}\right)\\\
&+\frac{q}{2}\tilde{C}\left(L,m,|\vec{p}|\right)\int_{0}^{\pi}d\tau\int\frac{dk}{k}e^{-i\omega_{\vec{p}}L\left[\tau-\frac{\pi}{2}-\frac{i}{2}\log\left(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\right)\right]}\left[e^{iLk\tau}-1\right]\hat{X}_{k}(\hat{p})~{}{\mathcal{O}}\left(\tau,\hat{p}\right)+\mathcal{O}(q^{2})\end{split}$
(36)

This formula can potentially be used to compute IR finite S-matrices. Firstly,
one would have to evaluate the CFT correlator in an integral representation
like a Mellin amplitude, and then perform the smearing integrals in a saddle
point approximation involving the large $L$ limit, and finally, the soft $k$
limit.

###  5.2 The dressed state in terms of creation and annihilation operator of
scalar field and photon

From the previous section we have the dressed creation operator as

$\begin{split}\sqrt{2\omega_{\vec{p}}}\tilde{a}^{\dagger}_{\omega_{\vec{p}}}&=\tilde{C}\int
d\tau
e^{-i\omega_{\vec{p}}L\left[\tau-\frac{\pi}{2}-\frac{i}{2}\log\left(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\right)\right]}e^{iq\int_{\Gamma(\tau,\hat{p})}j_{a}dx^{a}}\mathcal{O}\left(\tau,\hat{p}\right)\\\
\end{split}$ (37)

Now we will analyze the expression in the following way. At first we will
introduce a delta function inside the integral and then use the following
expression of the Dirac delta function

$\begin{split}\delta(\tau-\tau^{\prime})=\frac{L}{2\pi}\int
d\Delta_{\vec{p}}~{}e^{i\Delta_{\vec{p}}L(\tau-\tau^{\prime})}\end{split}$
(38)

Then we can write the dressed operator as

$\begin{split}\sqrt{2\omega_{\vec{p}}}\tilde{a}^{\dagger}_{\omega_{\vec{p}}}&=\tilde{C}\int
d\tau\left[\int
d\tau^{\prime}\delta\left(\tau-\tau^{\prime}\right)\right]e^{-i\omega_{\vec{p}}L\left[\tau-\frac{\pi}{2}-\frac{i}{2}\log\left(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\right)\right]}e^{iq\int_{\Gamma(\tau,\hat{p})}j_{a}dx^{a}}\mathcal{O}\left(\tau,\hat{p}\right)\\\
&=\frac{\tilde{C}L}{2\pi}\int d\tau
d\tau^{\prime}d\Delta_{\vec{p}}~{}e^{i\Delta_{\vec{p}}L\left(\tau-\tau^{\prime}\right)}e^{-i\omega_{\vec{p}}L\left[\tau-\frac{\pi}{2}-\frac{i}{2}\log\left(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\right)\right]}e^{iq\int_{\Gamma(\tau,\hat{p})}j_{a}dx^{a}}\mathcal{O}\left(\tau^{\prime},\hat{p}\right)\\\
\end{split}$ (39)

Now after rearranging the expression a little bit we have the following
expression

$\begin{split}\sqrt{2\omega_{\vec{p}}}\tilde{a}^{\dagger}_{\omega_{\vec{p}}}=&\frac{\tilde{C}L}{2\pi}\int
d\Delta_{\vec{p}}~{}e^{-i\Delta_{\vec{p}}L\left[\frac{\pi}{2}+\frac{i}{2}\log\left(\frac{\Delta_{\vec{p}}+m}{\Delta_{\vec{p}}-m}\right)\right]}e^{i\omega_{\vec{p}}L\left[\frac{\pi}{2}+\frac{i}{2}\log\left(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\right)\right]}\int
d\tau
e^{iL\tau\left(\Delta_{\vec{p}}-\omega_{\vec{p}}\right)}e^{iq\int_{\Gamma(\tau,\hat{p})}j_{a}dx^{a}}\\\
&\frac{1}{C}~{}C~{}\int
d\tau^{\prime}e^{-i\Delta_{\vec{p}}L\left[\tau^{\prime}-\frac{\pi}{2}-\frac{i}{2}\log\left(\frac{\Delta_{\vec{p}}+m}{\Delta_{\vec{p}}-m}\right)\right]}\mathcal{O}\left(\tau^{\prime},\hat{p}\right)\\\
\end{split}$ (40)

Now we use the definition of the creation operator

$\sqrt{2\Delta_{\vec{p}}}~{}{a}^{\dagger}_{\Delta_{\vec{p}}}=C\int
d\tau^{\prime}e^{-i\Delta_{\vec{p}}L\left[\tau^{\prime}-\frac{\pi}{2}-\frac{i}{2}\log\left(\frac{\Delta_{\vec{p}}+m}{\Delta_{\vec{p}}-m}\right)\right]}\mathcal{O}\left(\tau^{\prime},\hat{p}\right)$
(41)

Then the dressed creation operator will have the following form

$\begin{split}\sqrt{2\omega_{\vec{p}}}\tilde{a}^{\dagger}_{\omega_{\vec{p}}}=\frac{\tilde{C}L}{2\pi}&\int
d\Delta_{\vec{p}}~{}e^{-i\Delta_{\vec{p}}L\left[\frac{\pi}{2}+\frac{i}{2}\log\left(\frac{\Delta_{\vec{p}}+m}{\Delta_{\vec{p}}-m}\right)\right]}e^{i\omega_{\vec{p}}L\left[\frac{\pi}{2}+\frac{i}{2}\log\left(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\right)\right]}\\\
&\int d\tau
e^{iL\tau\left(\Delta_{\vec{p}}-\omega_{\vec{p}}\right)}e^{iq\int_{\Gamma(\tau,\hat{p})}j_{a}dx^{a}}\frac{\sqrt{2\Delta_{\vec{p}}}}{C}~{}{a}^{\dagger}_{\Delta_{\vec{p}}}\\\
\end{split}$ (42)

Now the above expression has been written in terms of creation/annihilation
operator of the scalar field and an operator constructed from the CFT current
$j_{a}$, dual to the gauge field. So to write this dressed creation operator
totally in terms of creation/annihilation operator the remaining task is to
write the expression $e^{iq\int_{\Gamma(\tau,\hat{p})}j_{a}dx^{a}}$ in terms
of creation/annihilation of photon. To do that first note from the previous
section that for a particular path of (31) we have

$\begin{split}\exp\left(iq\int_{\Gamma(\tau,\hat{p})}j_{a}~{}dx^{a}\right)&=\exp\left(iq\int_{0}^{\tau}j_{\tau^{\prime}}d\tau^{\prime}\right)=\exp\left(iq\int_{0}^{\tau}\left(j^{+}_{\tau^{\prime}}+j^{-}_{\tau^{\prime}}\right)d\tau^{\prime}\right)\end{split}$
(43)

where, $+$ and $-$ symbol are due to the positive and negative frequency modes
of the gauge field.
Then we use the conservation equation $\partial_{\mu}j^{\mu}=0$ to write
$j_{\tau^{\prime}}$ in terms of creation/annihilation operators as follows

$\begin{split}&\partial_{\mu}j^{\mu}=0\\\
\implies&\partial_{\tilde{\tau}}j^{\tilde{\tau}}+\partial_{z}j^{z}+\partial_{\bar{z}}j^{\bar{z}}=0\\\
\implies&\partial_{\tilde{\tau}}j_{\tilde{\tau}}=\frac{1}{2}(1+z\bar{z})^{2}(\partial_{z}j_{\bar{z}}+\partial_{\bar{z}}j_{z})+(1+z\bar{z})\left(zj_{z}+\bar{z}j_{\bar{z}}\right)\end{split}$
(44)

,where we have used the fact that the metric on the sphere is
$ds^{2}=\frac{4}{(1+z\bar{z})^{2}}dz~{}d\bar{z}$. Then integrating out the
above expression we have the following expression for $j_{\tau^{\prime}}$.

$\begin{split}j_{\tau^{\prime}}^{\pm}=\int_{0}^{\tau^{\prime}}d{\tilde{\tau}}\Bigg{(}\frac{1}{2}(1+z\bar{z})^{2}(\partial_{z}j_{\bar{z}}^{\pm}+\partial_{\bar{z}}j_{z}^{\pm})+(1+z\bar{z})\left(zj_{z}^{\pm}+\bar{z}j_{\bar{z}}^{\pm}\right)\Bigg{)}\end{split}$
(45)

Now to write the above expression in terms of creation/annihilation operator
of photon we need to invert the expression of (17), which have the following
expressions333The derivation of the expressions has been written in A.

$\begin{split}\partial_{z_{q}}j^{+}_{\bar{z}_{q}}(\tau,z_{q},\bar{z}_{q})&=\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}e^{i\omega_{\vec{q}}L\left(\tau-\pi/2\right)}\partial_{\bar{z}_{q}}\left(\frac{8\omega_{\vec{q}}\sqrt{\omega_{\vec{q}}}~{}\mathbf{a}^{\dagger(-)}_{\vec{q}}}{1+z_{q}\bar{z}_{q}}\right)\\\
j^{+}_{\bar{w}}(\tau,w,\bar{w})&=\frac{-L}{(2\pi)^{3}}\int
d\omega_{\vec{q}}~{}e^{i\omega_{\vec{q}}L\left(\tau-\pi/2\right)}\int
d^{2}z_{q}\frac{1}{(\bar{z}_{q}-\bar{w})}\partial_{\bar{z}_{q}}\left(\frac{8\omega_{\vec{q}}\sqrt{\omega_{\vec{q}}}~{}\mathbf{a}^{\dagger(-)}_{\vec{q}}}{1+z_{q}\bar{z}_{q}}\right)\\\
\partial_{\bar{z}_{q}}j^{+}_{{z}_{q}}(\tau,z_{q},\bar{z}_{q})&=\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}e^{i\omega_{\vec{q}}L\left(\tau-\pi/2\right)}\partial_{{z}_{q}}\left(\frac{8\omega_{\vec{q}}\sqrt{\omega_{\vec{q}}}~{}\mathbf{a}^{\dagger(+)}_{\vec{q}}}{1+z_{q}\bar{z}_{q}}\right)\\\
j^{+}_{{w}}(\tau,w,\bar{w})&=\frac{-L}{(2\pi)^{3}}\int
d\omega_{\vec{q}}~{}e^{i\omega_{\vec{q}}L\left(\tau-\pi/2\right)}\int
d^{2}z_{q}\frac{1}{({z}_{q}-{w})}\partial_{{z}_{q}}\left(\frac{8\omega_{\vec{q}}\sqrt{\omega_{\vec{q}}}~{}\mathbf{a}^{\dagger(+)}_{\vec{q}}}{1+z_{q}\bar{z}_{q}}\right)\\\
\partial_{{z}_{q}}j^{-}_{\bar{z}_{q}}(\tau,z_{q},\bar{z}_{q})&=\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}e^{-i\omega_{\vec{q}}L\left(\tau-\pi/2\right)}\partial_{\bar{z}_{q}}\left(\frac{8\omega_{\vec{q}}\sqrt{\omega_{\vec{q}}}~{}\mathbf{a}^{(-)}_{\vec{q}}}{1+z_{q}\bar{z}_{q}}\right)\\\
j^{-}_{\bar{w}}(\tau,w,\bar{w})&=\frac{-L}{(2\pi)^{3}}\int
d\omega_{\vec{q}}~{}e^{-i\omega_{\vec{q}}L\left(\tau-\pi/2\right)}\int
d^{2}z_{q}\frac{1}{(\bar{z}_{q}-\bar{w})}\partial_{\bar{z}_{q}}\left(\frac{8\omega_{\vec{q}}\sqrt{\omega_{\vec{q}}}~{}\mathbf{a}^{(-)}_{\vec{q}}}{1+z_{q}\bar{z}_{q}}\right)\\\
\partial_{\bar{z}_{q}}j^{-}_{{z}_{q}}(\tau,z_{q},\bar{z}_{q})&=\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}e^{-i\omega_{\vec{q}}L\left(\tau-\pi/2\right)}\partial_{{z}_{q}}\left(\frac{8\omega_{\vec{q}}\sqrt{\omega_{\vec{q}}}~{}\mathbf{a}^{(+)}_{\vec{q}}}{1+z_{q}\bar{z}_{q}}\right)\\\
j^{-}_{{w}}(\tau,w,\bar{w})&=\frac{-L}{(2\pi)^{3}}\int
d\omega_{\vec{q}}~{}e^{-i\omega_{\vec{q}}L\left(\tau-\pi/2\right)}\int
d^{2}z_{q}\frac{1}{({z}_{q}-{w})}\partial_{{z}_{q}}\left(\frac{8\omega_{\vec{q}}\sqrt{\omega_{\vec{q}}}~{}\mathbf{a}^{(+)}_{\vec{q}}}{1+z_{q}\bar{z}_{q}}\right)\end{split}$
(46)

Using these expressions we can write (58) in terms of creation/annihilation
operator of photon. Then using the expression of (43) and putting that into
(42) we will have the dressed creation operator in terms of
creation/annihilation operator of scalar field and photon. The final
expression is a bit long to be presented here. We have written it in A.

##  6 Conclusions and future directions

We conclude by summarizing our findings and listing some future directions. In
this paper, we have constructed a version of the Faddeev-Kulish dressed state
in terms of CFT operators. This has been done by utilizing AdS dressed fields
and existing maps between the photon creation operators and the CFT current
operator.

### Future directions:

There are several future directions related to this work.

### Construction of BMS supertranslation charge in the flat limit of AdS/CFT:

The Faddeev-Kulish dressing can be realized in terms of the eigenstate of BMS
supertranslation charge Choi:2017ylo . It would be interesting to construct
the BMS supertranslation charge in the flat limit of AdS/CFT. This would give
an alternative way to arrive the Faddeev-Kulish dressing from AdS/CFT.

### Correction to the Faddeev-Kulish dressed state:

It would be interesting to analyze the
$\mathcal{O}\Big{(}\frac{1}{L^{2}}\Big{)}$ correction to the Faddeev-Kulish
dressed state. Recently, in a paper Banerjee:2022oll
$\mathcal{O}\Big{(}\frac{1}{L^{2}}\Big{)}$ correction to the soft photon
theorem was studied where they derive the
$\mathcal{O}\Big{(}\frac{1}{L^{2}}\Big{)}$ corrections of the gauge field HKLL
kernels. We can use this to calculate the
$\mathcal{O}\Big{(}\frac{1}{L^{2}}\Big{)}$ to the Faddeev-Kulish dressed
state. We can further study the $\mathcal{O}\Big{(}\frac{1}{L^{2}}\Big{)}$
correction to the free massive scalar field mode as well. We hope to address
this question in an upcoming workfuture2 .

### CCFT from averaging over global time of AdS/CFT:

Using the mapping between the creation/annihilation operators and CFT
operators Hijano:2020szl , we can construct $\mathcal{S}$-matrix from CFT
correlators. Now, we can further perform a Mellin transform/convolute with the
bulk-to-boundary propagators to get the celestial amplitude in CCFT. It would
be exciting to get the CCFT correlators from flat limit of $CFT_{3}$
correlatorsfuture3 . Also, we can study $\mathcal{O}\Big{(}\frac{1}{L}\Big{)}$
correction to the $\mathcal{S}$-matrix from $CFT_{3}$ correlators and relate
it to ‘moving away from CCFT’future3 . From this, we can translate the IR
divergence in $CFT_{3}$ into CCFT.

### Celestial Integrability and moving away from it:

For $2d$ $\mathcal{S}$-matrix, integrability can be formulated in terms of the
Yang-Baxter equation which expresses $3\to 3$ scattering processes in terms of
$2\to 2$. Recently, celestial amplitudes are explored for integrable
theoriesDuary:2022onm ; Kapec:2022xjw . It’s interesting to find the map
between scattering states and CFT operators in the flat limit of
$AdS_{2}/CFT_{1}$. We can also try to see the
$\mathcal{O}\Big{(}\frac{1}{L}\Big{)}$ correction to the flat space
$\mathcal{S}$-matrix and CFT correlators. After translating into celestial
space, we can try to see the Yang-Baxter equation in celestial space and try
to see the corrections to it and how far we can go to solve the
$\mathcal{S}$-matrix exactlyfuture4 .

## Acknowledgements

We would like to thank Shamik Banerjee, Sayantani Bhattacharyya, Sangmin Choi,
Hofie Sigridar Hannesdottir, Daniel Kabat, Chethan Krishnan, Alok Laddha,
Gilad Lifschytz, R.Loganayagam, Pronobesh Maity, Priyadarshi Paul, Suvrat
Raju, Pabitra Ray, Iain Stewart, Ashoke Sen, Becher Thomas for useful
discussions. S.D. would especially like to thank Alok Laddha and R.Loganayagam
for giving him mental support throughout this work. M.P would especially like
to thank Sayantani Bhattacharyya and Alok Laddha for encouragement throughout
the course of the work. The work of S.D. is supported by the Department of
Atomic Energy, Government of India, under project no. RTI4001.

##  Appendix A Detailed derivation of some equations in section 5.2

We will use complex coordinates on the two sphere, where the standard
coordinates $\theta\in\left[0,\pi\right]$ and $\phi\in(0,2\pi]$ is related to
the complex coordinates $z$ and $\bar{z}$ as follows.

$\begin{split}\cos\theta=\frac{1-z\bar{z}}{1+z\bar{z}},~{}~{}\sin\theta\cos\phi=\frac{z+\bar{z}}{1+z\bar{z}},~{}~{}\sin\theta\sin\phi=-i\frac{z-\bar{z}}{1+z\bar{z}}~{}.\end{split}$
(47)

Then the metric in these coordinates reads

$\begin{split}ds^{2}=\frac{4dzd\bar{z}}{\left(1+z\bar{z}\right)^{2}}.\end{split}$
(48)

We have the photon creation operator from (Hijano:2020szl, ) as

$\begin{split}&\sqrt{2\omega_{\vec{q}}}~{}\mathbf{a}^{\dagger(-)}_{\vec{q}}=\frac{-1}{4\omega_{\vec{q}}}\frac{1+z_{q}\bar{z}_{q}}{\sqrt{2}}\int
d\tau^{\prime}~{}e^{i\omega_{\vec{q}}L\left(\frac{\pi}{2}-\tau^{\prime}\right)}\int
d^{2}z^{\prime}\frac{1}{\left(z_{q}-z^{\prime}\right)^{2}}j^{+}_{\bar{z}^{\prime}}(\tau^{\prime},z^{\prime},\bar{z}^{\prime})\\\
&\sqrt{2\omega_{\vec{q}}}~{}\mathbf{a}^{\dagger(+)}_{\vec{q}}=\frac{-1}{4\omega_{\vec{q}}}\frac{1+z_{q}\bar{z}_{q}}{\sqrt{2}}\int
d\tau^{\prime}~{}e^{i\omega_{\vec{q}}L\left(\frac{\pi}{2}-\tau^{\prime}\right)}\int
d^{2}z^{\prime}\frac{1}{\left(\bar{z}_{q}-\bar{z}^{\prime}\right)^{2}}j^{+}_{{z}^{\prime}}(\tau^{\prime},z^{\prime},\bar{z}^{\prime})\\\
\end{split}$ (49)

Note that here the $+$ symbol in $j^{+}_{\bar{z}^{\prime}}$ stands for the
positive frequency mode of the gauge field. We first invert this formula to
write $j^{+}_{\bar{z}^{\prime}}$ in terms of the creation operator of photon.
Acting with a $\partial_{\bar{z}_{q}}$ on both sides of the first equation of
(49) we have

$\begin{split}\partial_{\bar{z}_{q}}\left(\frac{(-8)\omega_{\vec{q}}\sqrt{\omega_{\vec{q}}}~{}\mathbf{a}^{\dagger(-)}_{\vec{q}}}{1+z_{q}\bar{z}_{q}}\right)=\int
d\tau^{\prime}~{}e^{i\omega_{\vec{q}}L\left(\pi/2-\tau^{\prime}\right)}\int
d^{2}z^{\prime}~{}\partial_{\bar{z}_{q}}\frac{1}{(z_{q}-z^{\prime})^{2}}~{}j^{+}_{\bar{z}^{\prime}}\left(\tau^{\prime},z^{\prime},\bar{z}^{\prime}\right)\end{split}$
(50)

Now we use the following identity

$\begin{split}\partial_{\bar{z}}\frac{1}{(z-z_{q})^{n+1}}=(2\pi)\frac{(-1)^{n}}{n!}\partial_{z}^{n}\delta^{(2)}(z,z_{q})\end{split}$
(51)

Then we have

$\begin{split}\partial_{z_{q}}j^{+}_{\bar{z}_{q}}(\tau,z_{q},\bar{z}_{q})=\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}e^{i\omega_{\vec{q}}L\left(\tau-\pi/2\right)}\partial_{\bar{z}_{q}}\left(\frac{8\omega_{\vec{q}}\sqrt{\omega_{\vec{q}}}~{}\mathbf{a}^{\dagger(-)}_{\vec{q}}}{1+z_{q}\bar{z}_{q}}\right)\end{split}$
(52)

Now we multiply both sides of this equation by $\frac{1}{\bar{z}_{q}-\bar{w}}$
and then integrate with respect to $z_{q}$, then we have

$\begin{split}j^{+}_{\bar{w}}(\tau,w,\bar{w})=\frac{-L}{(2\pi)^{3}}\int
d\omega_{\vec{q}}~{}e^{i\omega_{\vec{q}}L\left(\tau-\pi/2\right)}\int
d^{2}z_{q}\frac{1}{(\bar{z}_{q}-\bar{w})}\partial_{\bar{z}_{q}}\left(\frac{8\omega_{\vec{q}}\sqrt{\omega_{\vec{q}}}~{}\mathbf{a}^{\dagger(-)}_{\vec{q}}}{1+z_{q}\bar{z}_{q}}\right)\end{split}$
(53)

where we have used the following identity

$\begin{split}\partial_{z}\frac{1}{(\bar{z}-\bar{z}_{q})}=(2\pi)\delta^{(2)}(z_{q},z).\end{split}$
(54)

Following similar steps we have

$\begin{split}\partial_{\bar{z}_{q}}j^{+}_{{z}_{q}}(\tau,z_{q},\bar{z}_{q})&=\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}e^{i\omega_{\vec{q}}L\left(\tau-\pi/2\right)}\partial_{{z}_{q}}\left(\frac{8\omega_{\vec{q}}\sqrt{\omega_{\vec{q}}}~{}\mathbf{a}^{\dagger(+)}_{\vec{q}}}{1+z_{q}\bar{z}_{q}}\right)\\\
j^{+}_{{w}}(\tau,w,\bar{w})&=\frac{-L}{(2\pi)^{3}}\int
d\omega_{\vec{q}}~{}e^{i\omega_{\vec{q}}L\left(\tau-\pi/2\right)}\int
d^{2}z_{q}\frac{1}{({z}_{q}-{w})}\partial_{{z}_{q}}\left(\frac{8\omega_{\vec{q}}\sqrt{\omega_{\vec{q}}}~{}\mathbf{a}^{\dagger(+)}_{\vec{q}}}{1+z_{q}\bar{z}_{q}}\right)\end{split}$
(55)

Similarly, we have the photon annihilation operator from (Hijano:2020szl, ) as

$\begin{split}&\sqrt{2\omega_{\vec{q}}}~{}\mathbf{a}^{(-)}_{\vec{q}}=\frac{-1}{4\omega_{\vec{q}}}\frac{1+z_{q}\bar{z}_{q}}{\sqrt{2}}\int
d\tau^{\prime}~{}e^{-i\omega_{\vec{q}}L\left(\frac{\pi}{2}-\tau^{\prime}\right)}\int
d^{2}z^{\prime}\frac{1}{\left(z^{\prime}-z_{q}\right)}\partial_{z^{\prime}}j^{-}_{\bar{z}^{\prime}}(\tau^{\prime},z^{\prime},\bar{z}^{\prime})\\\
&\sqrt{2\omega_{\vec{q}}}~{}\mathbf{a}^{(+)}_{\vec{q}}=\frac{-1}{4\omega_{\vec{q}}}\frac{1+z_{q}\bar{z}_{q}}{\sqrt{2}}\int
d\tau^{\prime}~{}e^{-i\omega_{\vec{q}}L\left(\frac{\pi}{2}-\tau^{\prime}\right)}\int
d^{2}z^{\prime}\frac{1}{\left(\bar{z}^{\prime}-\bar{z}_{q}\right)}\partial_{\bar{z}^{\prime}}j^{-}_{{z}^{\prime}}(\tau^{\prime},z^{\prime},\bar{z}^{\prime})\\\
\end{split}$ (56)

Inverting these equations following similar steps as before we have

$\begin{split}\partial_{{z}_{q}}j^{-}_{\bar{z}_{q}}(\tau,z_{q},\bar{z}_{q})&=\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}e^{-i\omega_{\vec{q}}L\left(\tau-\pi/2\right)}\partial_{\bar{z}_{q}}\left(\frac{8\omega_{\vec{q}}\sqrt{\omega_{\vec{q}}}~{}\mathbf{a}^{(-)}_{\vec{q}}}{1+z_{q}\bar{z}_{q}}\right)\\\
j^{-}_{\bar{w}}(\tau,w,\bar{w})&=\frac{-L}{(2\pi)^{3}}\int
d\omega_{\vec{q}}~{}e^{-i\omega_{\vec{q}}L\left(\tau-\pi/2\right)}\int
d^{2}z_{q}\frac{1}{(\bar{z}_{q}-\bar{w})}\partial_{\bar{z}_{q}}\left(\frac{8\omega_{\vec{q}}\sqrt{\omega_{\vec{q}}}~{}\mathbf{a}^{(-)}_{\vec{q}}}{1+z_{q}\bar{z}_{q}}\right)\\\
\partial_{\bar{z}_{q}}j^{-}_{{z}_{q}}(\tau,z_{q},\bar{z}_{q})&=\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}e^{-i\omega_{\vec{q}}L\left(\tau-\pi/2\right)}\partial_{{z}_{q}}\left(\frac{8\omega_{\vec{q}}\sqrt{\omega_{\vec{q}}}~{}\mathbf{a}^{(+)}_{\vec{q}}}{1+z_{q}\bar{z}_{q}}\right)\\\
j^{-}_{{w}}(\tau,w,\bar{w})&=\frac{-L}{(2\pi)^{3}}\int
d\omega_{\vec{q}}~{}e^{-i\omega_{\vec{q}}L\left(\tau-\pi/2\right)}\int
d^{2}z_{q}\frac{1}{({z}_{q}-{w})}\partial_{{z}_{q}}\left(\frac{8\omega_{\vec{q}}\sqrt{\omega_{\vec{q}}}~{}\mathbf{a}^{(+)}_{\vec{q}}}{1+z_{q}\bar{z}_{q}}\right)\end{split}$
(57)

The expression for the $\tau$ component of the boundary current for positive
and negative frequency modes are

$\begin{split}j_{\tau^{\prime}}^{\pm}=\int_{0}^{\tau^{\prime}}d{\tilde{\tau}}\Bigg{(}\frac{1}{2}(1+z\bar{z})^{2}(\partial_{z}j_{\bar{z}}^{\pm}+\partial_{\bar{z}}j_{z}^{\pm})+(1+z\bar{z})\left(zj_{z}^{\pm}+\bar{z}j_{\bar{z}}^{\pm}\right)\Bigg{)}~{}.\end{split}$
(58)

Putting the expressions for
$\partial_{z}j_{\bar{z}}^{\pm},\partial_{\bar{z}}j_{z}^{\pm},j_{z}^{\pm}$ and
$j_{\bar{z}}^{\pm}$ we have

$\small\begin{split}j_{\tau}^{+}&=\frac{1}{2}(1+z^{\prime}\bar{z}^{\prime})^{2}\Bigg{[}\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}\frac{1}{iL\omega_{\vec{q}}}e^{-\frac{1}{2}i\pi
L\omega_{\vec{q}}}(e^{iL\tau\omega_{\vec{q}}}-1)\partial_{\bar{z}_{q}}~{}\Bigg{(}\sqrt{2\omega_{\vec{q}}}~{}4\omega_{\vec{q}}~{}\mathbf{a}_{\vec{q}}^{\dagger(-)}\frac{\sqrt{2}}{1+z_{q}\bar{z}_{q}}\Bigg{)}\Bigg{|}_{z_{q}=z^{\prime}}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}\frac{1}{iL\omega_{\vec{q}}}e^{-\frac{1}{2}i\pi
L\omega_{\vec{q}}}(e^{iL\tau\omega_{\vec{q}}}-1)\partial_{z_{q}}~{}\Bigg{(}\sqrt{2\omega_{\vec{q}}}~{}4\omega_{\vec{q}}~{}\mathbf{a}_{\vec{q}}^{\dagger(+)}\frac{\sqrt{2}}{1+z_{q}\bar{z}_{q}}\Bigg{)}\Bigg{|}_{z_{q}=z^{\prime}}\Bigg{]}\\\
&+\bar{z}^{\prime}(1+z^{\prime}\bar{z}^{\prime})\Bigg{[}-\frac{L}{(2\pi)^{3}}\int
d\omega_{\vec{q}}~{}\frac{1}{iL\omega_{\vec{q}}}e^{-\frac{1}{2}i\pi
L\omega_{\vec{q}}}(e^{iL\tau\omega_{\vec{q}}}-1)\int
d^{2}z_{q}\frac{1}{\bar{z}_{q}-\bar{z}^{\prime}}\partial_{\bar{z}_{q}}~{}\Bigg{(}\sqrt{2\omega_{\vec{q}}}~{}4\omega_{\vec{q}}~{}\mathbf{a}_{\vec{q}}^{\dagger(-)}\frac{\sqrt{2}}{1+z_{q}\bar{z}_{q}}\Bigg{)}\Bigg{]}\\\
&+z^{\prime}(1+z^{\prime}\bar{z}^{\prime})\Bigg{[}-\frac{L}{(2\pi)^{3}}\int
d\omega_{\vec{q}}~{}\frac{1}{iL\omega_{\vec{q}}}e^{-\frac{1}{2}i\pi
L\omega_{\vec{q}}}(e^{iL\tau\omega_{\vec{q}}}-1)\int
d^{2}z_{q}\frac{1}{z_{q}-z^{\prime}}\partial_{z_{q}}~{}\Bigg{(}\sqrt{2\omega_{\vec{q}}}~{}4\omega_{\vec{q}}~{}\mathbf{a}_{\vec{q}}^{\dagger(+)}\frac{\sqrt{2}}{1+z_{q}\bar{z}_{q}}\Bigg{)}~{}\Bigg{]}~{}~{}~{},\end{split}$
(59)

$\small\begin{split}j_{\tau}^{-}&=\frac{1}{2}(1+z^{\prime}\bar{z}^{\prime})^{2}\Bigg{[}\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}\frac{1}{-iL\omega_{\vec{q}}}e^{\frac{1}{2}i\pi
L\omega_{\vec{q}}}(e^{-iL\tau\omega_{\vec{q}}}-1)\partial_{\bar{z}_{q}}~{}\Bigg{(}\sqrt{2\omega_{\vec{q}}}~{}4\omega_{\vec{q}}~{}\mathbf{a}_{\vec{q}}^{(-)}\frac{\sqrt{2}}{1+z_{q}\bar{z}_{q}}\Bigg{)}\Bigg{|}_{z_{q}=z^{\prime}}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}\frac{1}{-iL\omega_{\vec{q}}}e^{\frac{1}{2}i\pi
L\omega_{\vec{q}}}(e^{-iL\tau\omega_{\vec{q}}}-1)\partial_{z_{q}}~{}\Bigg{(}\sqrt{2\omega_{\vec{q}}}~{}4\omega_{\vec{q}}~{}\mathbf{a}_{\vec{q}}^{(+)}\frac{\sqrt{2}}{1+z_{q}\bar{z}_{q}}\Bigg{)}\Bigg{|}_{z_{q}=z^{\prime}}\Bigg{]}\\\
&+\bar{z}^{\prime}(1+z^{\prime}\bar{z}^{\prime})\Bigg{[}-\frac{L}{(2\pi)^{3}}\int
d\omega_{\vec{q}}~{}\frac{1}{-iL\omega_{\vec{q}}}e^{\frac{1}{2}i\pi
L\omega_{\vec{q}}}(e^{-iL\tau\omega_{\vec{q}}}-1)\int
d^{2}z_{q}\frac{1}{\bar{z}_{q}-\bar{z}^{\prime}}\partial_{\bar{z}_{q}}~{}\Bigg{(}\sqrt{2\omega_{\vec{q}}}~{}4\omega_{\vec{q}}~{}\mathbf{a}_{\vec{q}}^{(-)}\frac{\sqrt{2}}{1+z_{q}\bar{z}_{q}}\Bigg{)}\Bigg{]}\\\
&+z^{\prime}(1+z^{\prime}\bar{z}^{\prime})\Bigg{[}-\frac{L}{(2\pi)^{3}}\int
d\omega_{\vec{q}}~{}\frac{1}{-iL\omega_{\vec{q}}}e^{\frac{1}{2}i\pi
L\omega_{\vec{q}}}(e^{-iL\tau\omega_{\vec{q}}}-1)\int
d^{2}z_{q}\frac{1}{z_{q}-z^{\prime}}\partial_{z_{q}}~{}\Bigg{(}\sqrt{2\omega_{\vec{q}}}~{}4\omega_{\vec{q}}~{}\mathbf{a}_{\vec{q}}^{(+)}\frac{\sqrt{2}}{1+z_{q}\bar{z}_{q}}\Bigg{)}~{}\Bigg{]}~{}~{}~{}.\end{split}$
(60)

Putting these expressions in (42) and after some simplification, at
$\mathcal{O}(q)$ we have the expression for the dressed state as

$\small\begin{split}\sqrt{2\omega_{\vec{p}}}~{}\widetilde{a}^{\dagger}_{\omega_{\vec{p}}}\left|0\right>&=\frac{\tilde{{C}}}{{C}}~{}~{}\Bigg{[}-iq\Bigg{\\{}\frac{L}{2}(1+z^{\prime}\bar{z}^{\prime})^{2}\times\\\
&\Bigg{[}\int
d\omega_{\vec{q}}~{}\Bigg{\\{}\frac{1}{iL\omega_{\vec{q}}}\Big{(}e^{\frac{L}{2}\Big{[}-i\pi\omega_{\vec{q}}-\omega_{\vec{p}}\log\Big{(}\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{)}+(\omega_{\vec{p}}+\omega_{\vec{q}})\log\Big{(}\frac{\omega_{\vec{p}}+\omega_{\vec{q}}+m}{\omega_{\vec{p}}+\omega_{\vec{q}}-m}\Big{)}\Big{]}}\sqrt{2(\omega_{\vec{p}}+\omega_{\vec{q}})}a^{\dagger}_{\omega_{\vec{p}}+\omega_{\vec{q}}}-\sqrt{2\omega_{\vec{p}}}a^{\dagger}_{\omega_{\vec{p}}}\Big{)}\\\
&+ie^{i\omega_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big{(}\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{)}\Big{]}}\frac{\partial}{\partial\Delta_{\vec{p}}}\Big{(}\sqrt{2\Delta_{\vec{p}}}a^{\dagger}_{\Delta_{\vec{p}}}e^{-i\Delta_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big{(}\frac{\Delta_{\vec{p}}+m}{\Delta_{\vec{p}}-m}\Big{)}\Big{]}}\Big{)}\Bigg{|}_{\Delta_{\vec{p}}=\omega_{\vec{p}}}\Bigg{\\}}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}\times-\frac{1}{(2\pi)^{2}}\frac{1}{iL\omega_{\vec{q}}}e^{-\frac{1}{2}i\pi
L\omega_{\vec{q}}}\partial_{\bar{z}_{q}}~{}\Bigg{(}\sqrt{2\omega_{\vec{q}}}~{}4\omega_{\vec{q}}~{}\mathbf{a}_{\vec{q}}^{\dagger(-)}\frac{\sqrt{2}}{1+z_{q}\bar{z}_{q}}\Bigg{)}\Bigg{|}_{z_{q}=z^{\prime}}\\\
&-\int
d\omega_{\vec{q}}~{}\Bigg{\\{}\frac{1}{iL\omega_{\vec{q}}}\Big{(}e^{\frac{L}{2}\Big{[}-i\pi\omega_{\vec{q}}-\omega_{\vec{p}}\log\Big{(}\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{)}+(\omega_{\vec{p}}+\omega_{\vec{q}})\log\Big{(}\frac{\omega_{\vec{p}}+\omega_{\vec{q}}+m}{\omega_{\vec{p}}+\omega_{\vec{q}}-m}\Big{)}\Big{]}}\sqrt{2(\omega_{\vec{p}}+\omega_{\vec{q}})}a^{\dagger}_{\omega_{\vec{p}}+\omega_{\vec{q}}}-\sqrt{2\omega_{\vec{p}}}a^{\dagger}_{\omega_{\vec{p}}}\Big{)}\\\
&+ie^{i\omega_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big{(}\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{)}\Big{]}}\frac{\partial}{\partial\Delta_{\vec{p}}}\Big{(}\sqrt{2\Delta_{\vec{p}}}a^{\dagger}_{\Delta_{\vec{p}}}e^{-i\Delta_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big{(}\frac{\Delta_{\vec{p}}+m}{\Delta_{\vec{p}}-m}\Big{)}\Big{]}}\Big{)}\Bigg{|}_{\Delta_{\vec{p}}=\omega_{\vec{p}}}\Bigg{\\}}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}\times-\frac{1}{(2\pi)^{2}}\frac{1}{iL\omega_{\vec{q}}}e^{-\frac{1}{2}i\pi
L\omega_{\vec{q}}}\partial_{z_{q}}~{}\Bigg{(}\sqrt{2\omega_{\vec{q}}}~{}4\omega_{\vec{q}}~{}\mathbf{a}_{\vec{q}}^{\dagger(+)}\frac{\sqrt{2}}{1+z_{q}\bar{z}_{q}}\Bigg{)}\Bigg{|}_{z_{q}=z^{\prime}}\Bigg{]}\\\
&+\bar{z}^{\prime}(1+z^{\prime}\bar{z}^{\prime})L\\\ &\Bigg{[}\int
d\omega_{\vec{q}}~{}\Bigg{\\{}\frac{1}{iL\omega_{\vec{q}}}\Big{(}e^{\frac{L}{2}\Big{[}-i\pi\omega_{\vec{q}}-\omega_{\vec{p}}\log\Big{(}\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{)}+(\omega_{\vec{p}}+\omega_{\vec{q}})\log\Big{(}\frac{\omega_{\vec{p}}+\omega_{\vec{q}}+m}{\omega_{\vec{p}}+\omega_{\vec{q}}-m}\Big{)}\Big{]}}\sqrt{2(\omega_{\vec{p}}+\omega_{\vec{q}})}a^{\dagger}_{\omega_{\vec{p}}+\omega_{\vec{q}}}-\sqrt{2\omega_{\vec{p}}}a^{\dagger}_{\omega_{\vec{p}}}\Big{)}\\\
&+ie^{i\omega_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big{(}\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{)}\Big{]}}\frac{\partial}{\partial\Delta_{\vec{p}}}\Big{(}\sqrt{2\Delta_{\vec{p}}}a^{\dagger}_{\Delta_{\vec{p}}}e^{-i\Delta_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big{(}\frac{\Delta_{\vec{p}}+m}{\Delta_{\vec{p}}-m}\Big{)}\Big{]}}\Big{)}\Bigg{|}_{\Delta_{\vec{p}}=\omega_{\vec{p}}}\Bigg{\\}}\\\
&\times\frac{1}{(2\pi)^{3}}\frac{1}{iL\omega_{\vec{q}}}e^{-\frac{1}{2}i\pi
L\omega_{\vec{q}}}\int
d^{2}z_{q}\frac{1}{\bar{z}_{q}-\bar{z}^{\prime}}\partial_{\bar{z}_{q}}~{}\Bigg{(}\sqrt{2\omega_{\vec{q}}}~{}4\omega_{\vec{q}}~{}\mathbf{a}_{\vec{q}}^{\dagger(-)}\frac{\sqrt{2}}{1+z_{q}\bar{z}_{q}}\Bigg{)}\Bigg{]}\\\
&+z^{\prime}(1+z^{\prime}\bar{z}^{\prime})L\\\ &\Bigg{[}\int
d\omega_{\vec{q}}~{}\Bigg{\\{}\frac{1}{iL\omega_{\vec{q}}}\Big{(}e^{\frac{L}{2}\Big{[}-i\pi\omega_{\vec{q}}-\omega_{\vec{p}}\log\Big{(}\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{)}+(\omega_{\vec{p}}+\omega_{\vec{q}})\log\Big{(}\frac{\omega_{\vec{p}}+\omega_{\vec{q}}+m}{\omega_{\vec{p}}+\omega_{\vec{q}}-m}\Big{)}\Big{]}}\sqrt{2(\omega_{\vec{p}}+\omega_{\vec{q}})}a^{\dagger}_{\omega_{\vec{p}}+\omega_{\vec{q}}}-\sqrt{2\omega_{\vec{p}}}a^{\dagger}_{\omega_{\vec{p}}}\Big{)}\\\
&+ie^{i\omega_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big{(}\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{)}\Big{]}}\frac{\partial}{\partial\Delta_{\vec{p}}}\Big{(}\sqrt{2\Delta_{\vec{p}}}a^{\dagger}_{\Delta_{\vec{p}}}e^{-i\Delta_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big{(}\frac{\Delta_{\vec{p}}+m}{\Delta_{\vec{p}}-m}\Big{)}\Big{]}}\Big{)}\Bigg{|}_{\Delta_{\vec{p}}=\omega_{\vec{p}}}\Bigg{\\}}\\\
&\times\frac{1}{(2\pi)^{3}}\frac{1}{iL\omega_{\vec{q}}}e^{-\frac{1}{2}i\pi
L\omega_{\vec{q}}}\int
d^{2}z_{q}\frac{1}{z_{q}-z^{\prime}}\partial_{z_{q}}~{}\Bigg{(}\sqrt{2\omega_{\vec{q}}}~{}4\omega_{\vec{q}}~{}\mathbf{a}_{\vec{q}}^{\dagger(+)}\frac{\sqrt{2}}{1+z_{q}\bar{z}_{q}}\Bigg{)}~{}\Bigg{]}\Bigg{\\}}\Bigg{]}\left|0\right>~{}.\\\
\end{split}$ (61)

## References

  * (1) S. Weinberg, “Infrared photons and gravitons,” Phys. Rev. 140 (1965), B516-B524 doi:10.1103/PhysRev.140.B516.
  * (2) J. F. Donoghue and T. Torma, “Infrared behavior of graviton-graviton scattering,” Phys. Rev. D 60 (1999), 024003 doi:10.1103/PhysRevD.60.024003 [arXiv:hep-th/9901156 [hep-th]].
  * (3) P. P. Kulish and L. D. Faddeev, “Asymptotic conditions and infrared divergences in quantum electrodynamics,” Theor. Math. Phys. 4 (1970), 745 doi:10.1007/BF01066485.
  * (4) V. Chung, “Infrared Divergence in Quantum Electrodynamics,” Phys. Rev. 140 (1965), B1110-B1122 doi:10.1103/PhysRev.140.B1110.
  * (5) T. W. B. Kibble, “Coherent soft-photon states and infrared divergences. ii. mass-shell singularities of green’s functions,” Phys. Rev. 173 (1968), 1527-1535 doi:10.1103/PhysRev.173.1527.
  * (6) T. W. B. Kibble, “Coherent soft-photon states and infrared divergences. iii. asymptotic states and reduction formulas,” Phys. Rev. 174 (1968), 1882-1901 doi:10.1103/PhysRev.174.1882.
  * (7) T. W. B. Kibble, “Coherent soft-photon states and infrared divergences. iv. the scattering operator,” Phys. Rev. 175 (1968), 1624-1640 doi:10.1103/PhysRev.175.1624.
  * (8) J. Ware, R. Saotome and R. Akhoury, “Construction of an asymptotic S matrix for perturbative quantum gravity,” JHEP 10 (2013), 159 doi:10.1007/JHEP10(2013)159 [arXiv:1308.6285 [hep-th]].
  * (9) J. Polchinski, “S matrices from AdS space-time,” [arXiv:hep-th/9901076 [hep-th]].
  * (10) L. Susskind, “Holography in the flat space limit,” AIP Conf. Proc. 493 (1999) no.1, 98-112 doi:10.1063/1.1301570 [arXiv:hep-th/9901079 [hep-th]].
  * (11) S. B. Giddings, “The Boundary S matrix and the AdS to CFT dictionary,” Phys. Rev. Lett. 83 (1999), 2707-2710 doi:10.1103/PhysRevLett.83.2707 [arXiv:hep-th/9903048 [hep-th]].
  * (12) S. B. Giddings, “Flat space scattering and bulk locality in the AdS/CFT correspondence,” Phys. Rev. D 61 (2000), 106008 doi:10.1103/PhysRevD.61.106008 [arXiv:hep-th/9907129 [hep-th]].
  * (13) M. Gary and S. B. Giddings, “The Flat space S-matrix from the AdS/CFT correspondence?,” Phys. Rev. D 80 (2009), 046008 doi:10.1103/PhysRevD.80.046008 [arXiv:0904.3544 [hep-th]].
  * (14) J. Penedones, “Writing CFT correlation functions as AdS scattering amplitudes,” JHEP 03 (2011), 025 doi:10.1007/JHEP03(2011)025 [arXiv:1011.1485 [hep-th]].
  * (15) A. L. Fitzpatrick, J. Kaplan, J. Penedones, S. Raju and B. C. van Rees, “A Natural Language for AdS/CFT Correlators,” JHEP 11 (2011), 095 doi:10.1007/JHEP11(2011)095 [arXiv:1107.1499 [hep-th]].
  * (16) A. L. Fitzpatrick and J. Kaplan, “Scattering States in AdS/CFT,” [arXiv:1104.2597 [hep-th]].
  * (17) A. L. Fitzpatrick, E. Katz, D. Poland and D. Simmons-Duffin, “Effective Conformal Theory and the Flat-Space Limit of AdS,” JHEP 07 (2011), 023 doi:10.1007/JHEP07(2011)023 [arXiv:1007.2412 [hep-th]].
  * (18) M. Gary, S. B. Giddings and J. Penedones, “Local bulk S-matrix elements and CFT singularities,” Phys. Rev. D 80 (2009), 085005 doi:10.1103/PhysRevD.80.085005 [arXiv:0903.4437 [hep-th]].
  * (19) S. Komatsu, M. F. Paulos, B. C. Van Rees and X. Zhao, “Landau diagrams in AdS and S-matrices from conformal correlators,” JHEP 11 (2020), 046 doi:10.1007/JHEP11(2020)046 [arXiv:2007.13745 [hep-th]].
  * (20) B. C. van Rees and X. Zhao, “QFT in AdS instead of LSZ,” [arXiv:2210.15683 [hep-th]].
  * (21) S. Raju, “New Recursion Relations and a Flat Space Limit for AdS/CFT Correlators,” Phys. Rev. D 85 (2012), 126009 doi:10.1103/PhysRevD.85.126009 [arXiv:1201.6449 [hep-th]].
  * (22) Y. Z. Li, “Notes on flat-space limit of AdS/CFT,” JHEP 09 (2021), 027 doi:10.1007/JHEP09(2021)027 [arXiv:2106.04606 [hep-th]].
  * (23) T. Okuda and J. Penedones, “String scattering in flat space and a scaling limit of Yang-Mills correlators,” Phys. Rev. D 83 (2011), 086001 doi:10.1103/PhysRevD.83.086001 [arXiv:1002.2641 [hep-th]].
  * (24) A. L. Fitzpatrick and J. Kaplan, “Analyticity and the Holographic S-Matrix,” JHEP 10 (2012), 127 doi:10.1007/JHEP10(2012)127 [arXiv:1111.6972 [hep-th]].
  * (25) A. L. Fitzpatrick and J. Kaplan, “Unitarity and the Holographic S-Matrix,” JHEP 10 (2012), 032 doi:10.1007/JHEP10(2012)032 [arXiv:1112.4845 [hep-th]].
  * (26) M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees and P. Vieira, “The S-matrix bootstrap. Part I: QFT in AdS,” JHEP 11 (2017), 133 doi:10.1007/JHEP11(2017)133 [arXiv:1607.06109 [hep-th]].
  * (27) M. Kruczenski, J. Penedones and B. C. van Rees, “Snowmass White Paper: S-matrix Bootstrap,” [arXiv:2203.02421 [hep-th]].
  * (28) S. Mandelstam, “Quantum electrodynamics without potentials,” _Annals Phys._ 19 (1962) 1–24.
  * (29) R. Jakob and N. G. Stefanis, “Path dependent phase factors and the infrared problem in QED,” _Annals Phys._ 210 (1991) 112–136.
  * (30) S. Choi and R. Akhoury, “Soft Photon Hair on Schwarzschild Horizon from a Wilson Line Perspective,” JHEP 12 (2018), 074 doi:10.1007/JHEP12(2018)074 [arXiv:1809.03467 [hep-th]].
  * (31) S. Choi, S. Sandeep Pradhan and R. Akhoury, “Supertranslation Hair of Schwarzschild Black Hole: A Wilson Line Perspective,” JHEP 01 (2020), 013 doi:10.1007/JHEP01(2020)013 [arXiv:1910.05882 [hep-th]].
  * (32) E. Hijano, “Flat space physics from AdS/CFT,” JHEP 07 (2019), 132 doi:10.1007/JHEP07(2019)132 [arXiv:1905.02729 [hep-th]].
  * (33) E. Hijano and D. Neuenfeld, “Soft photon theorems from CFT Ward identites in the flat limit of AdS/CFT,” JHEP 11 (2020), 009 doi:10.1007/JHEP11(2020)009 [arXiv:2005.03667 [hep-th]].
  * (34) S. Choi and R. Akhoury, “BMS Supertranslation Symmetry Implies Faddeev-Kulish Amplitudes,” JHEP 02 (2018), 171 doi:10.1007/JHEP02(2018)171 [arXiv:1712.04551 [hep-th]].
  * (35) M. Guica and D. L. Jafferis, _On the construction of charged operators inside an eternal black hole_ , _SciPost_ 03 (2017) 03, [arXiv:1511.05627].
  * (36) S. Duary and E. Hijano, work in progress.
  * (37) S. Duary and P. Maity, work in progress.
  * (38) S. Duary, work in progress.
  * (39) S. Duary, “Celestial amplitude for 2d theory,” [arXiv:2209.02776 [hep-th]].
  * (40) D. Kapec and A. Tropper, “Integrable Field Theories and Their CCFT Duals,” [arXiv:2210.16861 [hep-th]].
  * (41) N. Banerjee, K. Fernandes and A. Mitra, “$1/L^{2}$ corrected soft photon theorem from a CFT3 Ward identity,” [arXiv:2209.06802 [hep-th]].