# AdS correction to the Faddeev-Kulish state: Migrating from the Flat
Peninsula

Sarthak Duary

###### Abstract

The IR finiteness of $\mathcal{S}$-matrix in flat spacetime is tied to the
Faddeev-Kulish dressed state, which suggests dressing the Fock space
scattering state with the soft photon modes. We instigate a construction for
the Faddeev-Kulish dressed state in the language of AdS/CFT. A salient feature
of AdS spacetime is that it acquits itself as a quintessential IR regulator.
The IR divergences will take shape after taking the zoomed in limit. We
explore the Faddeev-Kulish dressed state to account for the AdS radius
corrections. The Wilson line dressing stands as a guiding principle in the
study of AdS radius-corrected Faddeev-Kulish dressing. We construct the modes
of the Wilson line dressed massive scalar field implementing “vanilla HKLL
reconstruction” since the field is simply free field. This simplification is
owing to the use of soft photon modes in the Wilson line dressing. We map the
AdS radius-corrected soft photon modes in terms of CFT current operators. We
invert this mapping, use the mapping in the Wilson line dressing, and express
the AdS radius-corrected Faddeev-Kulish dressed state.

##  1 Introduction

Scattering amplitudes in QED vanish in four spacetime dimensions in flat space
because of IR divergences. The soft photon interchange between the external
legs due to long-range interactions is what causes these IR divergences. In
perturbation theory, to all loop orders, the scattering process suffers from
IR divergences. The soft contribution of each of the diagrams exponentiates
after resumming the series[1]. Therefore, the non-perturbative amplitude,
$\mathcal{A}$ vanishes after taking the infrared regulator,
$\Lambda^{\text{reg}}_{\text{IR}}$ to zero

$\mathcal{A}\to 0~{}~{},~{}~{}~{}\Lambda^{\text{reg}}_{\text{IR}}\to
0~{}~{}~{}.$ (1.1)

This means the probability of scattering is essentially nil. This is a quantum
mechanical statement but this is really just a reflection of a classical fact
that the power or energy radiated does not vanish for soft photons, basically
soft photons are produced in order to match the classical answer [2, 3]. The
typical textbook solution to this IR divergence problem is to consider
inclusive cross sections [3, 4] by taking a trace over the soft modes of the
photons in the scattering states. This trace shifts the zero and yields a
finite quantity. Since the trace is determined by the detector resolution,
this is typically fine for phenomenological scenarios. Numerous soft photon
modes evade detection and are thus regarded as unobservable. Nevertheless, in
order to explore fine-grained issues regarding the unitarity of
$\mathcal{S}$-matrix while taking soft modes of photon into consideration,
$\mathcal{S}$-matrix must be defined appropriately[11].

An upshot of the resolution of the IR divergence alternative to employing
inclusive cross sections is to use “dressed states” as physical scattering
states. While computing the $\mathcal{S}$-matrix, the “in” and “out”
scattering states we choose reside in a Fock space. This choice of
approximation is a nice one since while looking at timelike infinity (for
massive particles) or at null infinity (for massless particles), particles are
so far apart from one other that they barely interact and therefore are free.
Nevertheless, in this scenario, the states corresponding to the Fock space
basis is the “sick basis”, leading the $\mathcal{S}$-matrix to become IR
divergent. In order to construct an IR finite $\mathcal{S}$-matrix, the basis
of scattering states has to be modified to incorporate the soft modes of the
photons. The intuition is that because electro-magnetic interactions are long-
range interactions, soft modes of photons in the “in” and “out” scattering
states are always present. These dressed states by soft modes of the photons
is referred as Faddeev-Kulish dressed state [5]. The Faddeev-Kulish state is
such that it precisely cancels the IR divergences in the $\mathcal{S}$-matrix,
resulting in an IR finite $\mathcal{S}$-matrix [6, 7, 8, 9, 10]. Recently, it
was observed that the Faddeev-Kulish state arises as a consequence of
asymptotic symmetries [56, 57, 58, 59, 60], which indicates the existence of
selection sectors [61, 62, 63, 64]. For discussion of Faddeev-Kulish dressing
in celestial space which is achieved with dressing by edge modes, see ref.[65,
66, 67]. For a more recent discussion of Faddeev-Kulish state, see ref. [68].

In this paper, our goal is to explore the AdS radius correction to the
Faddeev-Kulish state. We construct the Faddeev-Kulish dressing in AdS/CFT from
the standpoint of the Wilson line dressing. The equivalence of the Faddeev-
Kulish dressing and the Wilson line dressing involving the soft modes of
photons or gravitons has been studied in [48, 49, 50, 51].111In flat
spacetime, the Wilson line path is time-like geodesic for massive scattering
states and for this geodesic, the Wilson line dressing describes the Faddeev-
Kulish dressing[48, 49, 50, 51].

We now talk about observables connected to the scattering process in the
AdS/CFT and how it relates to observables in flat spacetime. The boundary
observables in the AdS, thanks to the AdS/CFT correspondence are CFT
correlation functions. The CFT correlation functions can be computed using the
Witten diagrams in the bulk of AdS spacetime. Due to the absence of
boundaries, the flat spacetime observables is defined asymptotically. The
particular observable for scattering amplitude in flat spacetime is the
$\mathcal{S}$-matrix. Zooming in around the center of the AdS spacetime, AdS
spacetime manifests itself into flat spacetime. In this zommed in limit, the
$\mathcal{S}$-matrix can be obtained from the CFT correlation functions using
(i) Position space, (ii) Mellin space, and (iii) Momentum space
representations of the CFT correlation functions, see ref. [12, 13, 14, 15,
16, 17, 18, 19, 20] for earlier developments, see ref.[21, 22, 23, 24, 25, 26,
27, 28] for recent developments. Different CFT representations like Position
space [25, 26, 21, 22], Mellin space [17, 29, 31], and Momentum space [23, 28]
elucidate different roadmap to reach to the flat space $\mathcal{S}$-matrix.
In recent years, the flat limit of Position space CFT correlation function
using HKLL bulk reconstruction has been studied in [54, 55] to realize IR
sector physics in flat spacetime from techniques in AdS/CFT. This is
essentially a statement of deciphering the physics of flat spacetime that is
already stored in AdS/CFT. In AdS spacetime, causality is attributed to
analyticity and unitarity of CFT correlation functions [38, 39, 40, 41, 42,
43, 44, 45, 46, 47]. These conditions in the flat space $\mathcal{S}$-matrix
have spawned intriguing implementations in the “$\mathcal{S}$-matrix Bootstrap
program” from tools in “CFT Bootstrap program” [29, 30, 31, 32, 33, 34, 35,
36]. In a very recent paper [37], focusing bound from ANEC for the CFT
correlation function and it’s connection to the flat space
$\mathcal{S}$-matrix is studied.

Now, we turn to the discussion on Faddeev-Kulish dressing. Recently, we study
the Faddeev-Kulish dressing in the earlier paper [78] from the zoomed in limit
of AdS/CFT. In this paper, we explore the AdS radius corrections to the
Faddeev-Kulish dressed state which captures the outcome of cosmological
constant on the flat spacetime state. We know the fact of life that the AdS
radius acts as an IR regulator. In the zoomed in limit, the scattering
amplitudes will have IR divergences. If we consider the scattering states
dressed by the soft modes of the photons (Faddeev-Kulish dressed state), then
we can get rid of the IR divergences. With these dressed states, we need to
understand how the $\mathcal{S}$-matrix becomes the IR finite one from
AdS/CFT. To understand this, first we need to understand how IR divergences
manifest themselves after taking the zoomed in limit from the CFT correlation
function using the Fock-space scattering states. The AdS radius corrections to
this Faddeev-Kulish dressed state will provide new insight into an IR finite
$\mathcal{S}$-matrix.

Now, we discuss the strategy to construct the AdS radius correction to the
Faddeev-Kulish dressed state. As discussed before, we choose the Wilson line
dressing as our guiding principle to arrive at the Faddeev-Kulish dressing in
AdS/CFT. We are interested in the kinematic regime in which the scalar field
is dressed by the soft modes of the photons. The dressed scalar field are free
fields and the modes of the field can be reconstructed simply implementing the
vanilla HKLL reconstruction. Upon taking the zoomed in limit, the
creation/annihilation modes of the dressed field can be expressed in terms of
the CFT operator corresponding to the undressed field which is dressed by the
boundary-to-boundary Wilson line, which is the CFT representation. We can
study the flat space representation taking into account AdS radius corrections
as well by reexpressing the CFT operator corresponding to the undressed field
in terms of the undressed mode of the scalar field. In order to have a fully
fledged flat space representation, we must also express the Wilson line
operator having CFT current operator in terms of photon creation/annihilation
modes. In this paper, we explore AdS radius correction to the Faddeev-Kulish
dressed state. To accomplish so, we have to invert the map between the soft
modes of the photon as a smearing of the CFT current operators which shows up
in the Wilson line operator. The $1/L^{2}$ corrected modes of the photon is
derived in the paper [79]. The annihilation operator of an outgoing photon of
negative helicity is given by

$\begin{split}&\sqrt{2\omega_{\vec{q}}}~{}\mathbf{a}^{\text{AdS}(-)}_{\vec{q}}\\\
&=\frac{1}{32\pi\omega_{\vec{q}}^{2}L^{2}}\frac{1+z_{q}\bar{z}_{q}}{\sqrt{2}\omega_{\vec{q}}}\int
d\tau^{\prime}~{}e^{-i\omega_{\vec{q}}L\left(\frac{\pi}{2}-\tau^{\prime}\right)}\int
d^{2}z^{\prime}\int
d^{2}z_{w}\Bigg{[}\frac{(1+z^{\prime}\bar{z}^{\prime})^{2}(1+z_{w}\bar{z}_{w})^{2}}{(\bar{z}_{w}-\bar{z}^{\prime})^{2}(z_{q}-z_{w})^{3}}\Bigg{]}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}\partial_{z^{\prime}}j^{-}_{\bar{z}^{\prime}}(\tau^{\prime},z^{\prime},\bar{z}^{\prime})~{},\end{split}$
(1.2)

which is expressed in terms of a CFT current operator smeared over the
boundary $S^{2}$. The $1/L^{2}$ corrected mode denoted by
$\mathbf{a}_{\vec{q}}^{\text{AdS}(-)}$ is physically distinct from that of
flat space mode of the photon. Here, there is an extra $z_{w}$ integral
correponding to integration over intermediate angles $(z_{w},\bar{z}_{w})$
which is absent in the flat space mode. This result is used to study
properties of the $1/L^{2}$ corrected $\mathcal{S}$-matrix from CFT physics,
like soft photon theorem is derived from a CFT Ward identity [79], which takes
into consideration scattering in an asymptotically $AdS_{4}$ spacetimes where
the $\mathcal{S}$-matrix is defined in the zoomed in region where the flat
spacetime is located.

Now, in this paper we use these AdS corrected modes to express the Faddeev-
Kulish dressed state. The creation mode of the soft Wilson line dressed
massive scalar field is constructed implementing vanilla HKLL reconstruction
since the dressed field is simply free field. The result we highlight in
eq.(1.3) is the expression for the dressed creation mode expressed in terms of
the undressed creation mode with smearing over frequency and global time
coordinate. The dressed mode dressed by the soft modes of photon acting on the
vacuum state $\ket{0}$ is the Faddeev-Kulish state. The expression is the
following

$\begin{split}\sqrt{2\omega_{\vec{p}}}~{}\widetilde{a}^{\dagger}_{\omega_{\vec{p}}}&=\widetilde{\mathfrak{C}}~{}L\int
d\Delta_{\vec{p}}~{}e^{-i\Delta_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big(\frac{\Delta_{\vec{p}}+m}{\Delta_{\vec{p}}-m}\Big{missing})\Big{]}}~{}e^{i\omega_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{missing})\Big{]}}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}\int
d\tau~{}e^{-iL\tau(\omega_{\vec{p}}-\Delta_{\vec{p}})}~{}e^{iq\int_{\Gamma(\tau,\hat{p})}j_{a}dx^{a}}~{}\sqrt{2\Delta_{\vec{p}}}~{}\frac{a^{\dagger}_{\Delta_{\vec{p}}}}{\mathcal{C}}~{}~{}~{}.\end{split}$
(1.3)

Now, this expression of eq.(1.3) is written in terms of creation mode of the
scalar field but the Wilson line which is expressed in terms of the CFT
current operator. Therefore, this is in the “mixed representation” (mixed
between the “CFT representation” and the “flat space representation”). To
write this dressed creation mode in the full fledged flat space
representation, we have to express the CFT current operators in the Wilson
line in terms of creation/annihilation operators of photon. We express the
Wilson line for a particular path where global time coordinate varies from $0$
to $\tau$ keeping the angular direction fixed

$\begin{split}e^{iq\int_{\Gamma(\tau,\hat{p})}j_{a}dx^{a}}=e^{iq\int_{0}^{\tau}\left(j^{+}_{\tau^{\prime}}+j^{-}_{\tau^{\prime}}\right)d\tau^{\prime}}~{}~{}~{},\end{split}$
(1.4)

where, $j^{+}_{\tau^{\prime}}$ and $j^{-}_{\tau^{\prime}}$ corresponds to the
positive and negative frequency modes of the photon when mapped to the photon
modes. We evaluate the CFT current operators in terms of AdS corrected photon
creation/annihilation modes. After that, we find the inverse mapping. That
means we express the CFT current operators in terms of the AdS corrected modes
of the photon. Now, choosing a particular path we evaluate the Wilson line and
for that we express the global time component of the CFT current operator in
terms of the photon creation/annihilation modes. Finally, we express the
dressed creation mode of the scalar field in terms of the AdS corrected
creation/annihilation modes of the photon. The dressed creation operator
acting on the vacuum state gives the AdS radius-corrected Faddeev-Kulish
dressed state. The roadmap is summerized in the flowchart fig.1.

Figure 1: Flowchart: Roadmap of the AdS correction to the Faddeev-Kulish(FK)
state

##### Organization of the paper.

The paper is organized as follows. In the section 2 we review various things
to make the paper self-contained. First, in the section 2.1, we discuss the
zoomed in limit. We discuss how to extract the creation and annihilation
operators for the free massive scalar field in terms of the CFT operators in
this zoomed in limit. We use the equivalence between the Faddeev-Kulish
dressing and the Wilson line dressing as our main strategy to construct the
AdS correction to the Faddeev-Kulish dressed state. In the section 2.2, we
study the soft Wilson line dressed field in AdS and evaluate the CFT operator
corresponding to this dressed field. The soft Wilson line dressed scalar field
turns itself into the free field and thus can be reconstructed implementing
the vanilla HKLL reconstruction. In the section 3, we study the CFT
representation as well as the mixed representation of the Faddeev-Kulish
dressed state. In the section 3.1.1, we talk about some speculations to get an
IR finite $\mathcal{S}$-matrix from AdS/CFT. In the section 4, we calculate
the AdS corrected Faddeev-Kulish dressed state. To do this, first we express
the CFT current operators in terms of AdS radius-corrected photon
creation/annihilation operators in the section 4.1. Next, in the section 4.2,
we evaluate the inverse mapping, which implies we express the CFT current
operators in terms of the AdS radius-corrected modes of the photon. Next, in
the section 4.2.1, we express the global time component of the CFT current
operator in terms of the photon creation/annihilation modes. The global time
component is needed since we choose a particular path to evaluate the Wilson
line. Then, We express the dressed creation operator in terms of the AdS
radius-corrected creation/annihilation modes of the photon. The dressed
creation operator acting on the vacuum gives the AdS radius-corrected Faddeev-
Kulish dressed state. Finally, we save the section 5 for summarizing our
conclusions and discussing future directions.

##  2 Preliminaries

In this section, we revisit some known things in order to make the paper self-
contained.

###  2.1 Flat Peninsula inside AdS Lake: The zoomed in limit

In this section, we review the zoomed in limit of AdS/CFT and how to extract
the creation and annihilation operators for the massive scalar scalar field in
terms of the CFT operator in this limit using the vanilla HKLL bulk
reconstruction described in [55].

At the level of geometry, we take the large AdS radius limit such that the
global $AdS_{4}$ metric becomes that of flat spacetime. The $AdS_{4}$
lorentzian metric in global coordinates is given by

$ds^{2}=\frac{L^{2}}{\cos^{2}\rho}(-d\tau^{2}+d\rho^{2}+\sin^{2}\rho
d\Omega_{2}^{2})~{}~{}~{}.$ (2.1)

The coordinate replacement

$\tau=\frac{t}{L}~{}~{}\text{and}~{}~{}\rho=\frac{r}{L}~{}~{}~{},$

develops the $AdS_{4}$ metric into that of flat spacetime metric upon taking
the limit $L\to\infty$

$ds^{2}\xrightarrow[L\to\infty]{~{}}-dt^{2}+dr^{2}+r^{2}d\Omega_{2}^{2}~{}~{}~{}.$
(2.2)

Essentially, the flat peninsula can be thought of as being inside the AdS
lake, and this operation can be thought of as zoomed in limit.

### Massive scalar field modes in the zoomed in limit of AdS/CFT

Having discussed the zoomed in limit at the level of the geometry, we now turn
to a discussion on fields. The creation and annihilation modes for a free
massive scalar field in flat spacetime can be constructed in terms of the CFT
operator in the zoomed in limit of AdS/CFT. For normalizable modes of the bulk
AdS scalar field, the bulk AdS scalar field $\phi(\rho,x)$ is related to the
dual boundary CFT operator $\mathcal{O}(x)$ through the fall-off condition

$\phi(\rho,x)\xrightarrow[\rho\to\frac{\pi}{2}]{~{}}(\cos\rho)^{\Delta}\mathcal{O}(x)~{}~{}~{}.$
(2.3)

We have

$\begin{split}m^{2}L^{2}&=\Delta(\Delta-3)\\\
\implies\Delta&=\frac{3}{2}+mL+\mathcal{O}(L)^{-1}~{}~{}~{}.\end{split}$ (2.4)

To extract the creation and annihilation modes in flat spacetime in terms of
CFT operators, first we reconstruct bulk operators as operators in the CFT
using the vanilla HKLL bulk reconstruction prescription. The AdS scalar
operator has been sent to the scattering region of the flat spacetime

$\tau=\frac{t}{L}~{}~{}~{}\text{and}~{}~{}~{}\rho=\frac{r}{L}~{}~{}~{}.$ (2.5)

Free scalar fields in flat spacetime can be expanded in creation/annhilation
modes. The scalar field can be mode expanded as

$\phi(x)=\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}\frac{1}{\sqrt{2\omega_{\vec{p}}}}(a_{\omega_{\vec{p}}}e^{ip.x}+a^{\dagger}_{\omega_{\vec{p}}}e^{-ip.x})~{}~{}~{}.$
(2.6)

We can extract the creation/annhilation modes of the scalar field from the
position space field operator of the scalar field. The approach we follow is
the following. First, we construct the free local bulk operators in the CFT
using the free HKLL bulk reconstruction. Next, we extract the
creation/annhilation modes using fourier transform and then we take a large
AdS radius limit which is the zoomed in limit. The flat space creation mode
for the free massive scalar field $\phi$ is given by [55]

$\sqrt{2\omega_{\vec{p}}}~{}a^{\dagger}_{\omega_{\vec{p}}}=\mathcal{C}\int
d\tau^{\prime}e^{-i\omega_{\vec{p}}L\left[\tau^{\prime}-\frac{\pi}{2}-\frac{i}{2}\log\Big(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{missing})\right]}\mathcal{O}\left(\tau^{\prime},\hat{p}\right)~{}~{}~{},$
(2.7)

where

$\begin{split}\mathcal{C}&=\frac{1}{2\pi}\Bigg{(}\frac{mL}{\pi^{3}}\Bigg{)}^{\frac{1}{4}}\Bigg{(}\frac{2m}{i|\vec{p}|}\Bigg{)}^{mL+\frac{1}{2}}~{}L~{}~{}~{}.\\\
\end{split}$ (2.8)

In the formula of eq.(2.7), the exponential part in the integrand is highly
oscillatory as we take the zoomed in limit, $L\to\infty$, therefore the
insertion points of the operators are in windows of size $\mathcal{O}(1/L)$ at
the complex points

$\text{Re}(\tau^{\prime})=\frac{\pi}{2}~{}~{},~{}\text{and}~{}~{}~{}\text{Im}(\tau^{\prime})=\frac{1}{2}\text{log}\Bigg{(}\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Bigg{)}~{}~{}~{}.$
(2.9)

###  2.2 Soft Wilson line dressed scalar field in AdS and dual CFT operator

In this section, we review the set up of the Faddeev-Kulish dressed state in
the zoomed in limit of AdS/CFT from the earlier paper [78].

First, we explain the soft Wilson line dressed scalar field in AdS and
evaluate the CFT operator corresponding to this soft Wilson line dressed
field. The bulk massive scalar field dressed by the bulk-to-boundary Wilson
line $U_{\mathfrak{B}\partial}(y,x)$ is given by

$\widetilde{\phi}(y)=U_{\mathfrak{B}\partial}(y,x)\phi(y)~{}~{}~{},$ (2.10)

where the bulk-to-boundary Wilson line is

$\begin{split}U_{\mathfrak{B}\partial}(y,x)&=\mathcal{P}\Bigg{\\{}e^{iq\int_{\Gamma}dx^{M}A_{M}}\Bigg{\\}}\\\
&=\mathcal{P}\Bigg{\\{}e^{iq\int_{y}^{x}dx^{M}A_{M}}\Bigg{\\}}~{}~{}~{}.\end{split}$
(2.11)

Here, $\Gamma$ is the path in AdS that joins bulk point point $y$ to boundary
point $x$, $y\to x$. Now, considering scalar electrodynamics with action

$S=\int
d^{4}x\sqrt{-g}\Big{(}-D_{M}\phi^{\dagger}D^{M}\phi-\frac{1}{4}F_{MN}F^{MN}-m^{2}\phi^{\dagger}\phi\Big{)}~{},$
(2.12)

where $D_{M}=\partial_{M}-iqA_{M}$, the equation of motion of the scalar field
is given by

$\left(\square-m^{2}\right)\phi=iq(\phi\nabla_{M}A^{M}+2A^{M}\partial_{M}\phi)+q^{2}A^{2}\phi~{}~{}~{}.$
(2.13)

Now, the Wilson line dressed field $\widetilde{\phi}$ satisfies [69]

$\small\begin{split}\left(\square-m^{2}\right)\widetilde{\phi}=-iq\widetilde{\phi}\nabla^{M}\int_{\Gamma}F_{MP}dy^{P}-2iq\nabla^{M}\widetilde{\phi}\int_{\Gamma}F_{MP}~{}dy^{P}+q^{2}\widetilde{\phi}~{}g^{MN}\int_{\Gamma}F_{MP}dy^{P}\int_{\Gamma}F_{NQ}dy^{Q}~{}~{}~{}.\end{split}$
(2.14)

We choose the Wilson line dressing in such a way that the field strength
$F_{MN}$ becomes $\mathcal{O}\left(\frac{1}{L}\right)$. We dress the scalar
field with soft modes of the photon in the Wilson line and in AdS, the minimum
frequency of photon is of $\mathcal{O}\left(\frac{1}{L}\right)$. This dressed
scalar field we refer as soft Wilson line dressed scalar field. As a
consequence of this dressing, the field $\widetilde{\phi}$ is free field in
the zoomed in limit.222There is an alternative way to get the Faddeev-Kulish
dressed state by considering a ‘Soft-collinear effective theory(SCET)’
lagrangian [53] to construct an asymptotic Hamiltonian. While constructing
Faddeev-Kulish states we consider the soft part only, and put a hard cutoff on
the photon energy. See ref. [52] for the construction of Faddeev-Kulish states
within the framework of SCET. Using this simplification, $\widetilde{\phi}$
can ve obtained using vanilla HKLL. The boundary CFT operator dual to the
Wilson line dressed field $\widetilde{\phi}$ we denote by
$\widetilde{{\mathcal{O}}}$ and the operator $\widetilde{{\mathcal{O}}}$ is
non-local since it involves boundary-to-boundary Wilson line. The Wilson line
dressed scalar field $\widetilde{\phi}$ is given by

$\widetilde{\phi}(y)=\mathcal{P}\Bigg{\\{}e^{iq\int_{y}^{x}A_{M}dx^{M}}\Bigg{\\}}\phi(y)~{}~{}~{}.$
(2.15)

The fall-off conditions of the photon field and the scalar bulk fields are

$\begin{split}&A_{a}\left(\rho,x\right)~{}\xrightarrow[\rho\to\frac{\pi}{2}]{~{}}j_{a}(x)~{}\cos\rho~{}~{}~{}\\\
&\phi\left(\rho,x\right)~{}\xrightarrow[\rho\to\frac{\pi}{2}]{~{}}(\cos\rho)^{\Delta}\mathcal{O}(x)~{}~{}~{}\\\
&\widetilde{\phi}\left(\rho,x\right)~{}\xrightarrow[\rho\to\frac{\pi}{2}]{~{}}(\cos\rho)^{\Delta}\mathcal{\widetilde{O}}(x)~{}~{}~{}.\end{split}$
(2.16)

Therefore, the CFT operator at the boundary corresponding to the soft Wilson
line dressed field is given by

$\widetilde{\mathcal{O}}(x^{\prime})=U_{\partial\partial}(x^{\prime},x)\mathcal{O}(x^{\prime})~{}~{}~{}.$
(2.17)

The boundary-to-boundary Wilson line $U_{\partial\partial}$ is given by

$U_{\partial\partial}(x^{\prime},x)=\mathcal{P}\Bigg{\\{}e^{iq\int_{x^{\prime}}^{x}dx^{a}j_{a}}\Bigg{\\}}~{}~{}~{}.$
(2.18)

The eq.(2.17) can be straightway derived using the fall-off behaviors of the
fields. To derive this equation, there is a cute cancellation between the
fall-off of the photon field and the line element factor $(\cos\rho)^{-1}$
going from bulk to boundary line element.

##  3 CFT and Mixed representations of the Faddeev-Kulish dressed state:
Vanilla HKLL

###  3.1 CFT representation

In this section, we express the Faddeev-Kulish dressed state in CFT
representation. As discussed in the previous section, considering the field
strength $F_{MN}=\mathcal{O}\left(\frac{1}{L}\right)$, the Wilson line dressed
scalar field $\widetilde{\phi}$ is free field in the zoomed in limit and can
be obtained using vanilla HKLL. The creation mode for the free massive scalar
field $\widetilde{\phi}$ was constructed in [55] and it is given by

$\sqrt{2\omega_{\vec{p}}}~{}\widetilde{a}^{\dagger}_{\omega_{\vec{p}}}=\widetilde{c}\int
d\tau
e^{-i\omega_{\vec{p}}L\left[\tau-\frac{\pi}{2}-\frac{i}{2}\log\Big(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{missing})\right]}\widetilde{\mathcal{O}}\left(\tau,\hat{p}\right)~{}~{}~{}.$
(3.1)

where

$\begin{split}\widetilde{c}&=\frac{1}{2\pi}\Bigg{(}\frac{mL}{\pi^{3}}\Bigg{)}^{\frac{1}{4}}\Bigg{(}\frac{2m}{i|\vec{p}|}\Bigg{)}^{mL+\frac{1}{2}}~{}L~{}~{}~{}.\\\
\end{split}$ (3.2)

Now, the CFT operator of the dressed scalar field
$\widetilde{\mathcal{O}}\left(\tau,\hat{p}\right)$ is related to the CFT
operator $\mathcal{O}\left(\tau,\hat{p}\right)$ as

$\widetilde{\mathcal{O}}\left(\tau,\hat{p}\right)=e^{iq\int_{\Gamma(\tau,\hat{p})}j_{a}dx^{a}}\mathcal{O}\left(\tau,\hat{p}\right)~{}~{}~{}.$
(3.3)

Using eq.(3.3), we get the creation mode of the dressed massive scalar field
in terms of $\mathcal{O}\left(\tau,\hat{p}\right)$

$\sqrt{2\omega_{\vec{p}}}~{}\widetilde{a}^{\dagger}_{\omega_{\vec{p}}}=\widetilde{c}\int
d\tau
e^{-i\omega_{\vec{p}}L\left[\tau-\frac{\pi}{2}-\frac{i}{2}\log\Big(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{missing})\right]}e^{iq\int_{\Gamma(\tau,\hat{p})}j_{a}dx^{a}}\mathcal{O}\left(\tau,\hat{p}\right)~{}~{}~{}.\\\
$ (3.4)

The Wilson line dressed creation mode
$\widetilde{a}^{\dagger}_{\omega_{\vec{p}}}$ acting on the CFT vacuum
$\ket{0}$ is the CFT representation of the Faddeev-Kulish dressed state.

Now, in the expression of the CFT current operator in the Wilson line we
implement $F_{MN}\to 0$ limit and the Wilson line does not depend on path as a
consequence of this limit. Therefore, this enables us to choose a particular
path for the Wilson line $\Gamma(\tau,\hat{p})$ which connectes the two points
in the boundary to evaluate $e^{iq\int_{\Gamma(\tau,\hat{p})}j_{a}dx^{a}}$. We
choose the path to be global time coordinate varies from $0$ to $\tau$ while
keeping the coordinates of the angular direction fixed. Now, we evaluate the
global time integral. Denoting the global time component of the CFT current
operator as

$j_{\tau}(\tau^{\prime},\hat{p})=\sum_{m}e^{i\omega_{m}\tau^{\prime}}\hat{\chi}_{m}(\hat{p})~{}~{}~{},$
(3.5)

where $\hat{\chi}_{m}$ involves spherical harmonics $Y_{l}^{m}(\hat{\Omega})$.
Here, we denote the sum over modes by $m$. Integrating over global time yields

$\int_{0}^{\tau}j_{\tau}(\tau^{\prime},\hat{p})d\tau^{\prime}=\sum_{m}\frac{1}{i\omega_{m}}(e^{i\omega_{m}\tau}-1)\hat{\chi}_{m}(\hat{p})~{}~{}~{}.$
(3.6)

We expand the Wilson line operator
$e^{iq\int_{0}^{\tau}j_{\tau}(\tau^{\prime},~{}\hat{p})~{}d\tau^{\prime}}$ in
series to $\mathcal{O}(q)$

$\begin{split}e^{iq\int_{0}^{\tau}j_{\tau}(\tau^{\prime},~{}\hat{p})~{}d\tau^{\prime}}&=1+q\sum_{m}\frac{1}{\omega_{m}}(e^{i\omega_{m}\tau}-1)\hat{\chi}_{m}(\hat{p})+\mathcal{O}(q^{2})~{}~{}~{},\end{split}$
(3.7)

where, we use the expansion

$\begin{split}e^{iq\int_{0}^{\tau}j_{\tau}(\tau^{\prime},~{}\hat{p})~{}d\tau^{\prime}}&=1+iq\int_{0}^{\tau}j_{\tau}(\tau^{\prime},\hat{p})d\tau^{\prime}+\mathcal{O}(q^{2})~{}~{}~{}.\end{split}$
(3.8)

Scalar/vector soft modes have frequency $\omega_{m}=\begin{Bmatrix}1\\\
2\end{Bmatrix}+l+2\kappa$, $\kappa\in\mathbb{Z}^{+}$.333We refer the reader to
the paper [55] (the section “Reconstruction of $U(1)$ gauge fields in global
$AdS_{4}$”) for details regarding the scalar/vector modes. Now, there are two
different alternatives of the soft modes in the zoomed in limit. One is that
$\kappa\sim{\cal O}(1)$, and the frequency $\omega_{m}/L\rightarrow 0$. The
mode is given by

$\omega_{m}\sim\mathcal{O}(1)~{}~{}~{},~{}~{}~{}\frac{\omega_{m}}{L}\to
0~{}~{}~{}.$ (3.9)

Another alternative is to denote $\omega_{m}=kL$ and take the zoomed in limit,
$L\to\infty$, and then to take soft limit with $k\rightarrow 0$. The mode is
given by

$\omega_{m}=Lk~{}~{}~{},~{}~{}~{}\frac{\omega_{m}}{L}=k\to 0~{}~{}~{}.$ (3.10)

In fig.2, we draw a frequency scale to denote two different alternatives of
the soft modes in the zoomed in limit.

Figure 2: Frequency scale $\omega_{m}$: $\omega_{m}\sim\mathcal{O}(1)$ and
$\omega_{m}=Lk$.

In the zoomed in limit, $\omega_{m}\sim 2\kappa=Lk$ dominates the sum over
modes, and therefore

$\sum_{\kappa}\to\frac{L}{2}\int dk~{}~{}~{}.$ (3.11)

The Wilson line dressed creation mode of the massive scalar field at
$\mathcal{O}(q)$ is given by

$\begin{split}\sqrt{2\omega_{\vec{p}}}~{}\widetilde{a}^{\dagger}_{\omega_{\vec{p}}}=\frac{q}{2}\widetilde{c}\int
d\tau\int\frac{dk}{k}e^{-i\omega_{\vec{p}}L\left[\tau-\frac{\pi}{2}-\frac{i}{2}\log\left(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\right)\right]}\left(e^{iLk\tau}-1\right)\hat{\chi}_{k}(\hat{p})~{}{\mathcal{O}}\left(\tau,\hat{p}\right)~{}~{}~{}.\end{split}$
(3.12)

This expression for the dressed creation mode acting on CFT vacuum $\ket{0}$
state gives the “CFT representation” of the Faddeev-Kulish dressed state. We
expect the $\mathcal{S}$-matrix constructed with the dressed scattering state
will be IR-finite. In the next section 3.1.1, we discuss some speculations on
this issue.

####  3.1.1 Some speculation to get an IR finite $\mathcal{S}$-matrix from
AdS/CFT

In this section, we discuss some speculations to get an IR finite
$\mathcal{S}$-matrix from AdS/CFT. The CFT operator at the boundary
corresponding to the soft Wilson line dressed field is given by

$\widetilde{\mathcal{O}}(x^{\prime})=\mathcal{P}\Bigg{\\{}e^{iq\int_{x^{\prime}}^{x}dx^{a}j_{a}}\Bigg{\\}}\mathcal{O}(x^{\prime})~{}~{}~{}.$
(3.13)

The Wilson line doesn’t depend on the path and only depends on the endpoints
of the path is precisely when the field strength vanishes. When we dress the
scalar field $\phi$ with soft modes of photon, then this is exactly what
happens. The boundary-to-boundary Wilson line given by

$U_{\partial\partial}(x^{\prime},x)=\mathcal{P}\Bigg{\\{}e^{iq\int_{x^{\prime}}^{x}dx^{a}j_{a}}\Bigg{\\}}~{}~{}~{},$
(3.14)

is dependent only on the endpoints.

Now, The dressed $4$-point amplitude in terms of the undressed $4$-point
amplitude is expressed as

$\expectationvalue{\widetilde{O}\widetilde{\mathcal{O}}\widetilde{\mathcal{O}}\widetilde{\mathcal{O}}}=\sum_{p,q,r,s}\frac{1}{p!q!r!s!}\expectationvalue{\mathfrak{J}^{p}~{}\mathcal{O}~{}\mathfrak{J}^{q}~{}\mathcal{O}~{}\mathfrak{J}^{r}~{}\mathcal{O}~{}\mathfrak{J}^{s}~{}\mathcal{O}}~{}~{}~{},$
(3.15)

where, we use the shorthand notation

$\mathfrak{J}^{p}\equiv\Bigg{[}iq\int_{x^{\prime}}^{x}dx^{a}j_{a}\Bigg{]}^{p}~{}~{}~{}.$
(3.16)

The line integral in the Wilson line runs from the insertion point of the
boundary CFT operator to another point at the boundary. Now, the $4$-point
amplitude will involve OPE between the operator $\mathcal{O}$ with CFT current
operator $j_{a}$ since the original CFT operator of the undressed field
$\mathcal{O}(x^{\prime})$ will have the insertion point $x^{\prime}$. The
Wilson line depends on the end points therefore there is a CFT current
operator $j_{a}(x^{\prime})$ at the insertion point of the CFT operator
$\mathcal{O}$, dual to the undressed field.444See, for instance the OPE shows
up naturally for constructing IR-finite celestial amplitude. (Ref. [45]
section 3.3.3. “IR-finite celestial amplitude”.) We expect that the undressed
state which is the typical Fock-space state will bring IR divergence which
will be neatly cancelled by the piece of the dressing in the “CFT
representation” of the Faddeev-Kulish dressed state.

###  3.2 Mixed representation

We express the creation mode for the Wilson line dressed massive scalar field
with the soft modes of the photon in terms of undressed mode. To accomplish
so, we insert a delta function $\delta(\tau-\tau^{\prime})$ and express the
delta function in terms of integral over frequency, $\Delta_{\vec{p}}$ which
is the corresponding frequency of the undressed mode of the massive scalar
field

$\begin{split}\int d\tau^{\prime}\delta(\tau^{\prime}-\tau)=\int
d\tau^{\prime}\int
d\Delta_{\vec{p}}~{}\frac{L}{2\pi}e^{-i\Delta_{\vec{p}}L(\tau^{\prime}-\tau)}~{}~{}~{}.\end{split}$
(3.17)

Therefore, the creation mode of the soft Wilson line dressed massive scalar
field is expressed as

$\begin{split}\sqrt{2\omega_{\vec{p}}}~{}\widetilde{a}^{\dagger}_{\omega_{\vec{p}}}&=\widetilde{c}\int
d\tau
e^{-i\omega_{\vec{p}}L\left[\tau-\frac{\pi}{2}-\frac{i}{2}\log\Big(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{missing})\right]}e^{iq\int_{\Gamma(\tau,\hat{p})}j_{a}dx^{a}}\mathcal{O}\left(\tau,\hat{p}\right)\\\
&=\widetilde{c}\int d\tau~{}\Bigg{[}\int
d\tau^{\prime}{\delta(\tau^{\prime}-\tau)}\Bigg{]}~{}e^{-i\omega_{\vec{p}}L\left[\tau-\frac{\pi}{2}-\frac{i}{2}\log\Big(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{missing})\right]}e^{iq\int_{\Gamma(\tau,\hat{p})}j_{a}dx^{a}}\mathcal{O}\left(\tau,\hat{p}\right)\\\
&=\widetilde{c}\int d\tau\int d\tau^{\prime}{\int
d\Delta_{\vec{p}}~{}\frac{L}{2\pi}e^{-i\Delta_{\vec{p}}L(\tau^{\prime}-\tau)}}~{}e^{-i\omega_{\vec{p}}L\left[\tau-\frac{\pi}{2}-\frac{i}{2}\log\Big(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{missing})\right]}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}e^{iq\int_{\Gamma(\tau,\hat{p})}j_{a}dx^{a}}\mathcal{O}\left(\tau^{\prime},\hat{p}\right)\\\
&=\widetilde{\mathfrak{C}}L\int
d\Delta_{\vec{p}}~{}e^{-i\Delta_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big(\frac{\Delta_{\vec{p}}+m}{\Delta_{\vec{p}}-m}\Big{missing})\Big{]}}~{}\int
d\tau
e^{-i\omega_{\vec{p}}L\Big{[}\tau-\frac{\pi}{2}-\frac{i}{2}\log\Big(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{missing})\Big{]}}e^{i\Delta_{\vec{p}}L\tau}\\\
&~{}~{}~{}~{}~{}~{}e^{iq\int_{\Gamma(\tau,\hat{p})}j_{a}dx^{a}}\int
d\tau^{\prime}~{}e^{-i\Delta_{\vec{p}}L\Big{[}\tau^{\prime}-\frac{\pi}{2}-\frac{i}{2}\log\Big(\frac{\Delta_{\vec{p}}+m}{\Delta_{\vec{p}}-m}\Big{missing})\Big{]}}~{}\mathcal{O}\left(\tau^{\prime},\hat{p}\right)~{}~{}~{},\end{split}$
(3.18)

where, in the last step we use

$e^{-i\Delta_{\vec{p}}L(\tau^{\prime}-\tau)}=e^{-i\Delta_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big(\frac{\Delta_{\vec{p}}+m}{\Delta_{\vec{p}}-m}\Big{missing})\Big{]}}~{}e^{i\Delta_{\vec{p}}L\tau}~{}e^{-i\Delta_{\vec{p}}L\Big{[}\tau^{\prime}-\frac{\pi}{2}-\frac{i}{2}\log\Big(\frac{\Delta_{\vec{p}}+m}{\Delta_{\vec{p}}-m}\Big{missing})\Big{]}}~{}~{}~{}.$
(3.19)

where,

$\begin{split}\widetilde{c}&=\frac{1}{2\pi}\Bigg{(}\frac{mL}{\pi^{3}}\Bigg{)}^{\frac{1}{4}}\Bigg{(}\frac{2m}{i|\vec{p}|}\Bigg{)}^{mL+\frac{1}{2}}~{}L\\\
\widetilde{\mathfrak{C}}&=\frac{\widetilde{c}}{2\pi}~{}~{}~{}.\end{split}$
(3.20)

Now, using the expression for the creation mode
$a^{\dagger}_{\Delta_{\vec{p}}}$ corresponing to frequency of the undressed
mode, $\Delta_{\vec{p}}$ in terms of the boundary CFT operator
$\mathcal{O}\left(\tau^{\prime},\hat{p}\right)$

$\sqrt{2\Delta_{\vec{p}}}~{}a^{\dagger}_{\Delta_{\vec{p}}}=\mathcal{C}\int
d\tau^{\prime}~{}e^{-i\Delta_{\vec{p}}L\Big{[}\tau^{\prime}-\frac{\pi}{2}-\frac{i}{2}\log\Big(\frac{\Delta_{\vec{p}}+m}{\Delta_{\vec{p}}-m}\Big{missing})\Big{]}}\mathcal{O}\left(\tau^{\prime},\hat{p}\right)~{}~{}~{},$
(3.21)

eq.(3.18) is expressed as

$\begin{split}\sqrt{2\omega_{\vec{p}}}~{}\widetilde{a}^{\dagger}_{\omega_{\vec{p}}}&=\widetilde{\mathfrak{C}}~{}L\int
d\Delta_{\vec{p}}~{}e^{-i\Delta_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big(\frac{\Delta_{\vec{p}}+m}{\Delta_{\vec{p}}-m}\Big{missing})\Big{]}}~{}e^{i\omega_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{missing})\Big{]}}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}\int
d\tau~{}e^{-iL\tau(\omega_{\vec{p}}-\Delta_{\vec{p}})}~{}e^{iq\int_{\Gamma(\tau,\hat{p})}j_{a}dx^{a}}~{}\sqrt{2\Delta_{\vec{p}}}~{}\frac{a^{\dagger}_{\Delta_{\vec{p}}}}{\mathcal{C}}~{}~{}~{}.\end{split}$
(3.22)

In eq.(3.22), we express the creation mode of the soft Wilson line dressed
massive scalar field in terms of the undressed creation mode. We choose a
particular path for the Wilson line $\Gamma(\tau,\hat{p})$ as in the previous
section 3.1. We mode expand the $\tau$ component of the CFT current operator
as

$j_{\tau}(\tau^{\prime},\hat{p})=\sum_{m}e^{i\omega_{m}\tau^{\prime}}\hat{\chi}_{m}(\hat{p})~{}~{}~{}.$
(3.23)

Now, we perform the $\tau$ integral and get

$\begin{split}&\int
d\tau~{}e^{-iL\tau(\omega_{\vec{p}}-\Delta_{\vec{p}})}~{}e^{iq\int_{0}^{\tau}j_{\tau}(\tau^{\prime},~{}\hat{p})~{}d\tau^{\prime}}\\\
&=\int
d\tau~{}e^{-iL\tau(\omega_{\vec{p}}-\Delta_{\vec{p}})}~{}\Bigg{[}1+q\sum_{m}\frac{1}{\omega_{m}}(e^{i\omega_{m}\tau}-1)\hat{\chi}_{m}(\hat{p})+\mathcal{O}(q^{2})\Bigg{]}\\\
&=\frac{2\pi}{L}\delta(\omega_{\vec{p}}-\Delta_{\vec{p}})+\frac{2\pi}{L}\sum_{m}\frac{~{}q~{}\hat{\chi}_{m}}{\omega_{m}}\Bigg{(}\delta\Big{[}(\omega_{\vec{p}}-\Delta_{\vec{p}})+\frac{\omega_{m}}{L}\Big{]}-\delta(\omega_{\vec{p}}-\Delta_{\vec{p}})\Bigg{)}+\mathcal{O}(q^{2})~{}~{}~{}.\end{split}$
(3.24)

Scalar/Vector soft modes have frequency $\omega_{m}=\begin{Bmatrix}1\\\
2\end{Bmatrix}+l+2\kappa$, $\kappa\in\mathbb{Z}^{+}$. In the zoomed in limit,
$\omega_{m}\sim 2\kappa=Lk$ dominates the sum over modes, and therefore

$\sum_{\kappa}\to\frac{L}{2}\int dk~{}~{}~{}.$ (3.25)

We write eq.(3.22) by substituting $\omega_{m}=kL$ and replacing sum over
modes by integration over $k$ at $\mathcal{O}(q)$

$\begin{split}\sqrt{2\omega_{\vec{p}}}~{}\widetilde{a}^{\dagger}_{\omega_{\vec{p}}}&=\widetilde{\mathfrak{C}}~{}L\int
d\Delta_{\vec{p}}~{}e^{-i\Delta_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big(\frac{\Delta_{\vec{p}}+m}{\Delta_{\vec{p}}-m}\Big{missing})\Big{]}}~{}e^{i\omega_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{missing})\Bigg{]}}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}\frac{2\pi}{L}\sum_{m}\frac{~{}q~{}\hat{\chi}_{m}}{\omega_{m}}\Bigg{(}\delta\Big{[}(\omega_{\vec{p}}-\Delta_{\vec{p}})+\frac{\omega_{m}}{L}\Big{]}-\delta(\omega_{\vec{p}}-\Delta_{\vec{p}})\Bigg{)}\sqrt{2\Delta_{\vec{p}}}~{}\frac{a^{\dagger}_{\Delta_{\vec{p}}}}{\mathcal{C}}\\\
&=\widetilde{\mathfrak{C}}~{}L\frac{2\pi
q}{L}~{}\frac{L}{2}\int_{0}^{\infty}dk\frac{1}{L~{}k}\int_{-\infty}^{\infty}d\Delta_{\vec{p}}~{}~{}e^{-i\Delta_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big(\frac{\Delta_{\vec{p}}+m}{\Delta_{\vec{p}}-m}\Big{missing})\Big{]}}~{}e^{i\omega_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{missing})\Big{]}}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Bigg{(}\delta\Big{[}(\omega_{\vec{p}}-\Delta_{\vec{p}})+k\Big{]}-\delta(\omega_{\vec{p}}-\Delta_{\vec{p}})\Bigg{)}\hat{\chi}_{k}\sqrt{2\Delta_{\vec{p}}}~{}\frac{a^{\dagger}_{\Delta_{\vec{p}}}}{\mathcal{C}}~{}~{}~{}.\end{split}$
(3.26)

Now, we can perform the $\Delta_{\vec{p}}$ integral to simplify the expression
and finally the dreesed creation mode is given by

$\begin{split}\sqrt{2\omega_{\vec{p}}}~{}\widetilde{a}^{\dagger}_{\omega_{\vec{p}}}&=\widetilde{\mathfrak{C}}~{}\pi~{}q~{}\int_{0}^{\infty}dk\frac{1}{k}\Bigg{(}e^{-i(\omega_{\vec{p}}+k)L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big(\frac{\omega_{\vec{p}}+k+m}{\omega_{\vec{p}}+k-m}\Big{missing})\Big{]}}~{}e^{i\omega_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{missing})\Big{]}}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}\hat{\chi}_{k}\sqrt{2(\omega_{\vec{p}}+k)}\frac{a^{\dagger}_{\omega_{\vec{p}}+k}}{\mathcal{C}}-\sqrt{2\omega_{\vec{p}}}~{}\hat{\chi}_{k}~{}\frac{a^{\dagger}_{\omega_{\vec{p}}}}{\mathcal{C}}\Bigg{)}~{}~{}~{}.\end{split}$
(3.27)

The dressed creation mode acting on the vacuum $\ket{0}$ is the Faddeev-Kulish
dressed state. In this way of writing the expression, the frequency of the
creation mode of the scalar field will get shifted from $\omega_{\vec{p}}$ to
$\omega_{\vec{p}}+k$. The external leg of the scalar field will get dressed
due to the soft Wilson line and this has a nice pictorial representation in
terms of the frequency of the scalar field mode, see fig.3. After further
reexpressing the CFT operator in terms of the scalar field mode and perform
the frequency integral $\Delta_{\vec{p}}$, we get this representation. We
refer eq.(3.27) as “mixed representation” (mixed between the “CFT
representation” and the “flat space representation”) since the Wilson line
operator still contain the CFT current operator.555For understanding the soft
modes of photon in a pictorial way, it’s better to express the expression in
the mixed representation like in eq.(3.27). In terms of CFT operators, the
soft modes will correspond to the CFT current operator dual to the gauge
field.

Figure 3: Dressing with soft modes of Wilson line: The frequency of the
creation mode of the scalar field will get shifted from $\omega_{\vec{p}}$ to
$\omega_{\vec{p}}+k$ as a consequence of the dressing.

##  4 AdS correction to the Faddeev-Kulish dressed state

In this section, we will explore AdS radius correction to the Faddeev-Kulish
dressed state.

From the previous subsection 3.2 (ref.eq.(3.22)), we note the expression for
the dressed creation mode written in terms of the undressed mode

$\begin{split}\sqrt{2\omega_{\vec{p}}}~{}\widetilde{a}^{\dagger}_{\omega_{\vec{p}}}&=\widetilde{\mathfrak{C}}~{}L\int
d\Delta_{\vec{p}}~{}e^{-i\Delta_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big(\frac{\Delta_{\vec{p}}+m}{\Delta_{\vec{p}}-m}\Big{missing})\Big{]}}~{}e^{i\omega_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{missing})\Big{]}}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}\int
d\tau~{}e^{-iL\tau(\omega_{\vec{p}}-\Delta_{\vec{p}})}~{}e^{iq\int_{\Gamma(\tau,\hat{p})}j_{a}dx^{a}}~{}\sqrt{2\Delta_{\vec{p}}}~{}\frac{a^{\dagger}_{\Delta_{\vec{p}}}}{\mathcal{C}}~{}~{}~{}.\end{split}$
(4.1)

Now, in this expression the CFT current operator appears in the Wilson line
which makes the formula to be in the mixed representation. We map the CFT
current operators to creation/annhilation modes of the photon to express our
AdS correction to the Faddeev-Kulish dressed state. Now, we have to write the
CFT current operator appeared in the expression for the Wilson line

$e^{iq\int_{0}^{\tau}j_{\tau^{\prime}}d\tau^{\prime}}=e^{iq\int_{0}^{\tau}\left(j^{+}_{\tau^{\prime}}+j^{-}_{\tau^{\prime}}\right)d\tau^{\prime}}$

in terms of the photon creation/annihilation operatos. In the expression of
$j_{\tau^{\prime}}$, we use $j^{+}_{\tau^{\prime}}$ and
$j^{-}_{\tau^{\prime}}$ to denote the positive and negative frequency modes of
the photon.

Here, we choose a particular path for the Wilson line. The path is such that
the global time coordinate varies from $0$ to $\tau$ and the angular direction
remains unchanged. To emphasize, we can do this simplification since we are
interested in the soft Wilson line dressing to dress the field and in that
soft limit the Wilson line is independent of the path because the field
strength term can be ignored. Now, in next section 4.1, we explore the AdS
corrected modes of the photon in terms of CFT current operators. Then, we
invert this mapping in the next to next section 4.2, to express the CFT
current operators to AdS corrected modes of the photon.

###  4.1 AdS corrected photon modes in terms of CFT current operators

Free photon fields in flat spacetime can be mode expanded as

${A}_{\mu}(x)=\int{d^{3}\vec{q}\over(2\pi)^{3}}{1\over{\sqrt{2\omega_{\vec{q}}}}}\sum_{\lambda=\pm}\left(\varepsilon^{(\lambda)}_{\mu}\hat{a}^{(\lambda)}_{\vec{q}}\,e^{iq\cdot
x}+\varepsilon^{(\lambda)*}_{\mu}\hat{a}^{(\lambda)\dagger}_{\vec{q}}\,e^{-iq\cdot
x}\right)~{}~{}~{},$ (4.2)

where, $\varepsilon^{(\lambda)}_{\mu}$ are polarization vectors. The
creation/annihilation modes of the photon are given by

$\begin{split}\sqrt{2\omega_{\vec{q}}}~{}\mathbf{a}^{(\lambda)\dagger}_{\vec{q}}=&-i\int
d^{3}\vec{x}\,\varepsilon^{(\lambda),\mu}e^{iq\cdot
x}\overleftrightarrow{\partial_{0}}{{A}}_{\mu}(x)~{}~{}~{}\\\
\sqrt{2\omega_{\vec{q}}}~{}\mathbf{a}_{\vec{q}}^{(\lambda)}=&~{}i\int
d^{3}\vec{x}\,(\varepsilon^{(\lambda),\mu})^{*}e^{-iq\cdot
x}\overleftrightarrow{\partial_{0}}{{A}}_{\mu}(x)~{}~{}~{}.\\\ \end{split}$
(4.3)

Now, we express the creation/annihilation modes of the photon in terms of the
CFT current operators. We consider the magnetic boundary condition which
implies the the gauge field is related to a the bundary CFT current as

${A}_{\mu}(\rho,x)\xrightarrow[\rho\to\frac{\pi}{2}]{}\cos\rho\,j_{\mu}(x)~{}~{}~{}.$
(4.4)

Now, implementing HKLL bulk reconstruction, we have the free photon felds in
AdS expressed in terms of the CFT current operator as

${A}_{\mu}(\rho,x)=\frac{1}{\pi}\int_{\mathcal{T}}d\tau^{\prime}~{}~{}\int
d^{2}\Omega^{\prime}~{}\left[K^{V}_{\mu}(\rho,x;x^{\prime})\epsilon_{\tau^{\prime}ab}\partial^{a}j^{b}(x^{\prime})+K^{S}_{\mu}(\rho,x;x^{\prime})\partial_{a}j^{a}(x^{\prime})\right]+\text{h.c.}~{}~{}~{}.$
(4.5)

Here, the $a,b$ indices denotes the $S^{2}$ coordinates. The photon field
kernels $K^{V}$ and $K^{S}$ are obtained by vector harmonic decomposition of
the photon field using the treatment of Wald and Ishibashi[80]. The photon
field kernels are worked out in great detail in [55, 79].

In [79], the authors have studied the $1/L^{2}$ AdS correction to the photon
field kernels. Using that, the AdS correction to the photon field modes is
derived. The result is given by

$\small\begin{split}\sqrt{2\omega_{\vec{q}}}~{}\mathbf{a}_{\vec{q}}^{\text{AdS}\,(-)}&=i\int
r^{2}dr\,\int d\Omega\,\int\limits_{0}^{\pi}d\tau^{\prime}\int
d\Omega^{\prime}\int
d\omega\frac{1+z\bar{z}}{\sqrt{2}}\sum_{l,m,l^{\prime},m^{\prime}}j_{l^{\prime}}(r\omega_{\vec{q}})j_{l}(r\omega)\left(\frac{l(l+1)}{4\omega^{2}L^{2}}\right)\\\
&~{}\times\Bigg{\\{}i(\omega+\omega_{\vec{q}})\frac{Y_{lm}\left(\Omega^{\prime}\right)}{l(l+1)}Y_{l^{\prime}m^{\prime}}(\Omega)Y^{*}_{l^{\prime}m^{\prime}}(\Omega_{q})\partial_{\bar{z}}Y^{*}_{lm}\left(\Omega\right)(-i)^{-l+l^{\prime}}e^{-i(\omega-\omega_{\vec{q}})t}e^{i\omega
L\left(\tau^{\prime}-\frac{\pi}{2}\right)}D^{\bar{z}^{\prime}}j^{-}_{\bar{z}^{\prime}}\\\
&-i(\omega-\omega_{\vec{q}})\frac{Y^{*}_{lm}\left(\Omega^{\prime}\right)}{l(l+1)}Y_{l^{\prime}m^{\prime}}(\Omega)Y^{*}_{l^{\prime}m^{\prime}}(\Omega_{q})\partial_{\bar{z}}Y_{lm}\left(\Omega\right)(-i)^{l^{\prime}}(i)^{-l}e^{i(\omega+\omega_{\vec{q}})t}e^{-i\omega
L\left(\tau^{\prime}-\frac{\pi}{2}\right)}\,D^{\bar{z}^{\prime}}j^{+}_{\bar{z}^{\prime}}\Bigg{\\}}~{}~{}~{}.\end{split}$
(4.6)

We can further simplify the expression in eq.(4.6) by performing the
derivatives of the spherical harmonics on the $S^{2}$. We make use of the
orthonormality and completeness of the spherical harmonics and orthonormality
of the spherical bessel function to do further simplification

$\begin{split}\int d\Omega
Y_{lm}\left(\Omega\right)Y^{*}_{l^{\prime}m^{\prime}}\left(\Omega\right)&=\delta_{ll^{\prime}}\delta_{mm^{\prime}}\\\
\sum_{l,m}Y_{lm}\left(\Omega_{q}\right)Y^{*}_{lm}\left(\Omega\right)&=\delta^{(2)}\left(\Omega_{q},\Omega\right)\,\\\
\int\limits_{0}^{\infty}r^{2}drj_{l}(r\omega)j_{l}(r\omega_{\vec{q}})&=\frac{\pi}{2\omega^{2}_{\vec{q}}}\delta(\omega-\omega_{\vec{q}})~{}~{}~{}.\end{split}$
(4.7)

After simplification, we have the expression of the annihilation operator of
an outgoing photon of negative helicity given by

$\begin{split}&\sqrt{2\omega_{\vec{q}}}~{}\mathbf{a}^{\text{AdS}(-)}_{\vec{q}}\\\
&=\frac{1}{32\pi\omega_{\vec{q}}^{2}L^{2}}\frac{1+z_{q}\bar{z}_{q}}{\sqrt{2}\omega_{\vec{q}}}\int
d\tau^{\prime}~{}e^{-i\omega_{\vec{q}}L\left(\frac{\pi}{2}-\tau^{\prime}\right)}\int
d^{2}z^{\prime}\int
d^{2}z_{w}\Bigg{[}\frac{(1+z^{\prime}\bar{z}^{\prime})^{2}(1+z_{w}\bar{z}_{w})^{2}}{(\bar{z}_{w}-\bar{z}^{\prime})^{2}(z_{q}-z_{w})^{3}}\Bigg{]}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}\partial_{z^{\prime}}j^{-}_{\bar{z}^{\prime}}(\tau^{\prime},z^{\prime},\bar{z}^{\prime})~{}.\end{split}$
(4.8)

The $1/L^{2}$ corrected mode is expressed in terms of a CFT current operator
smeared over the boundary $S^{2}$ and we denote by
$\mathbf{a}_{\vec{q}}^{\text{AdS}(-)}$. This mode is physically a bit
different from the flat space mode of the photon obtained in the zoomed in
limit. The difference is that here an extra $z_{w}$ integral correponding to
integration over intermediate angles $(z_{w},\bar{z}_{w})$.

###  4.2 Inverse mapping: CFT current operators mapped to AdS corrected photon
modes

In this section, we evaluate the inverse mapping. That means, we express the
CFT current operators in terms of the AdS corrected creation/annihilation
modes of the photon.

From the previous section, we note that the annihilation operator of an
outgoing photon of negative helicity is given by

$\begin{split}&\sqrt{2\omega_{\vec{q}}}~{}\mathbf{a}^{\text{AdS}(-)}_{\vec{q}}\\\
&=\frac{1}{32\pi\omega_{\vec{q}}^{2}L^{2}}\frac{1+z_{q}\bar{z}_{q}}{\sqrt{2}\omega_{\vec{q}}}\int
d\tau^{\prime}~{}e^{-i\omega_{\vec{q}}L\left(\frac{\pi}{2}-\tau^{\prime}\right)}\int
d^{2}z^{\prime}\int
d^{2}z_{w}\Bigg{[}\frac{(1+z^{\prime}\bar{z}^{\prime})^{2}(1+z_{w}\bar{z}_{w})^{2}}{(\bar{z}_{w}-\bar{z}^{\prime})^{2}(z_{q}-z_{w})^{3}}\Bigg{]}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}\partial_{z^{\prime}}j^{-}_{\bar{z}^{\prime}}(\tau^{\prime},z^{\prime},\bar{z}^{\prime})~{}.\end{split}$
(4.9)

Acting with a $\partial_{\bar{z}_{q}}$ on both sides of the eq.(4.9) we have

$\begin{split}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\pi\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\text{AdS}(-)}_{\vec{q}}\Bigg{)}&=\int
d\tau^{\prime}~{}e^{-i\omega_{\vec{q}}L\left(\frac{\pi}{2}-\tau^{\prime}\right)}\int
d^{2}z^{\prime}~{}\int
d^{2}z_{w}~{}\partial_{\bar{z}_{q}}\Bigg{[}\frac{(1+z^{\prime}\bar{z}^{\prime})^{2}(1+z_{w}\bar{z}_{w})^{2}}{(\bar{z}_{w}-\bar{z}^{\prime})^{2}(z_{q}-z_{w})^{3}}\Bigg{]}\\\
&~{}~{}~{}~{}\partial_{z^{\prime}}j^{-}_{\bar{z}^{\prime}}\left(\tau^{\prime},z^{\prime},\bar{z}^{\prime}\right)~{}~{}~{}.\end{split}$
(4.10)

Now, using the identity

$\begin{split}\partial_{\bar{z}_{q}}\frac{1}{(z_{q}-z_{w})^{3}}&=(2\pi)\frac{(-1)^{2}}{2!}\partial_{z_{q}}^{2}\delta^{(2)}(z_{q},z_{w})\\\
&=(2\pi)\frac{(-1)^{2}}{2!}\partial_{z_{w}}^{2}\delta^{(2)}(z_{q},z_{w})~{}~{}~{},\end{split}$
(4.11)

we have

$\begin{split}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\pi\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\text{AdS}(-)}_{\vec{q}}\Bigg{)}&=\int
d\tau^{\prime}~{}e^{-i\omega_{\vec{q}}L\left(\frac{\pi}{2}-\tau^{\prime}\right)}\int
d^{2}z^{\prime}~{}\int
d^{2}z_{w}~{}\pi~{}\partial_{z_{w}}^{2}\delta^{(2)}(z_{q},z_{w})\\\
&~{}~{}~{}~{}~{}\times\Bigg{[}\frac{(1+z^{\prime}\bar{z}^{\prime})^{2}(1+z_{w}\bar{z}_{w})^{2}}{(\bar{z}_{w}-\bar{z}^{\prime})^{2}}\Bigg{]}\partial_{z^{\prime}}j^{-}_{\bar{z}^{\prime}}\left(\tau^{\prime},z^{\prime},\bar{z}^{\prime}\right)~{}~{}~{}.\end{split}$
(4.12)

After performing the $d^{2}z_{w}$ integral using the delta function
$\delta^{(2)}(z_{q},z_{w})$ to simplify the expression a bit. The steps of the
detailed calculation we save in the appendix A, section A.1. After
simplification the expression we get is

$\begin{split}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\text{AdS}(-)}_{\vec{q}}\Bigg{)}&=\int
d\tau^{\prime}~{}e^{-i\omega_{\vec{q}}L\left(\frac{\pi}{2}-\tau^{\prime}\right)}\int
d^{2}z^{\prime}~{}(1+z^{\prime}\bar{z}^{\prime})^{2}\frac{2\bar{z}_{q}^{2}}{(\bar{z}_{q}-\bar{z}^{\prime})^{2}}~{}~{}\partial_{z^{\prime}}j^{-}_{\bar{z}^{\prime}}\left(\tau^{\prime},z^{\prime},\bar{z}^{\prime}\right)~{}~{}~{}.\end{split}$
(4.13)

Now, further acting $\partial_{z_{q}}$ on the simplified expression eq.(4.13)
we finally get

$\begin{split}\partial_{z_{q}}\Bigg{[}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\text{AdS}(-)}_{\vec{q}}\Bigg{)}\Bigg{]}&=\int
d\tau^{\prime}~{}e^{-i\omega_{\vec{q}}L\left(\frac{\pi}{2}-\tau^{\prime}\right)}\Big{(}-8\pi
z_{q}(1+z_{q}\bar{z}_{q})^{2}\Big{)}\partial_{z_{q}}j^{-}_{\bar{z}_{q}}\left(\tau^{\prime},z_{q},\bar{z}_{q}\right)~{}~{}~{}.\end{split}$
(4.14)

The details is saved in the appendix A, section A.1.

The inverse mapping of the CFT current operator for the negative frequency in
terms of the AdS corrected photon annihilation mode is given by

$\small\partial_{z_{q}}j^{-}_{\bar{z}_{q}}\left(\tau^{\prime},z_{q},\bar{z}_{q}\right)=\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}e^{-i\omega_{\vec{q}}L\left(\tau^{\prime}-\frac{\pi}{2}\right)}~{}~{}\frac{1}{-4z_{q}(1+z_{q}\bar{z}_{q})^{2}}\partial_{z_{q}}\Bigg{[}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\text{AdS}(-)}_{\vec{q}}\Bigg{)}\Bigg{]}~{}~{}~{}.$
(4.15)

Now, we multiply both sides by $\frac{1}{\bar{z}_{q}-\bar{z}^{\prime}}$ and
integrate with respect to $d^{2}z_{q}$ and obtain

$\small
j^{-}_{\bar{z}^{\prime}}(\tau^{\prime},z^{\prime},\bar{z}^{\prime})=\frac{-L}{(2\pi)^{3}}\int
d\omega_{\vec{q}}~{}e^{-i\omega_{\vec{q}}L\left(\tau^{\prime}-\frac{\pi}{2}\right)}\int
d^{2}z_{q}\frac{1}{\bar{z}_{q}-\bar{z}^{\prime}}\frac{1}{-4z_{q}(1+z_{q}\bar{z}_{q})^{2}}\partial_{z_{q}}\Bigg{[}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\text{AdS}(-)}_{\vec{q}}\Bigg{)}\Bigg{]}~{}~{}~{}.$
(4.16)

Now, we can express eq.(4.15) in terms of the coordinates $z^{\prime}$ and
$\bar{z}^{\prime}$

$\small\partial_{z^{\prime}}j^{-}_{\bar{z}^{\prime}}\left(\tau^{\prime},z^{\prime},\bar{z}^{\prime}\right)=\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}e^{-i\omega_{\vec{q}}L\left(\tau^{\prime}-\frac{\pi}{2}\right)}~{}~{}\frac{1}{-4z^{\prime}(1+z^{\prime}\bar{z}^{\prime})^{2}}\partial_{z_{q}}\Bigg{[}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\text{AdS}(-)}_{\vec{q}}\Bigg{)}\Bigg{]}\Bigg{|}_{(z_{q},\bar{z}_{q})=(z^{\prime},\bar{z}^{\prime})}~{}~{}~{}.$
(4.17)

Similarly, we have

$\small\begin{split}\partial_{\bar{z}^{\prime}}j^{-}_{z^{\prime}}\left(\tau^{\prime},z^{\prime},\bar{z}^{\prime}\right)&=\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}e^{-i\omega_{\vec{q}}L\left(\tau^{\prime}-\frac{\pi}{2}\right)}~{}~{}\frac{1}{-4\bar{z}^{\prime}(1+z^{\prime}\bar{z}^{\prime})^{2}}\partial_{\bar{z}_{q}}\Bigg{[}\partial_{z_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\text{AdS}(+)}_{\vec{q}}\Bigg{)}\Bigg{]}\Bigg{|}_{(z_{q},\bar{z}_{q})=(z^{\prime},\bar{z}^{\prime})}\\\
j^{-}_{z^{\prime}}(\tau^{\prime},z^{\prime},\bar{z}^{\prime}&)=\frac{-L}{(2\pi)^{3}}\int
d\omega_{\vec{q}}~{}e^{-i\omega_{\vec{q}}L\left(\tau^{\prime}-\frac{\pi}{2}\right)}\int
d^{2}z_{q}\frac{1}{z_{q}-z^{\prime}}\frac{1}{-4\bar{z}_{q}(1+z_{q}\bar{z}_{q})^{2}}\partial_{\bar{z}_{q}}\Bigg{[}\partial_{z_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\text{AdS}(+)}_{\vec{q}}\Bigg{)}\Bigg{]}~{}~{}~{}.\end{split}$
(4.18)

For the creation mode of the photon we have CFT current operators
$j^{+}_{\bar{z}^{\prime}}$ and $j^{+}_{z^{\prime}}$. Here, the $+$ in the
superscript of $j$’s denotes the positive frequency mode of the photon. We
have the following expressions for the CFT current operators

$\small\begin{split}\partial_{z^{\prime}}j^{+}_{\bar{z}^{\prime}}\left(\tau^{\prime},z^{\prime},\bar{z}^{\prime}\right)&=\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}e^{i\omega_{\vec{q}}L\left(\tau^{\prime}-\frac{\pi}{2}\right)}~{}~{}\frac{1}{-4z^{\prime}(1+z^{\prime}\bar{z}^{\prime})^{2}}\partial_{z_{q}}\Bigg{[}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\dagger\text{AdS}(-)}_{\vec{q}}\Bigg{)}\Bigg{]}\Bigg{|}_{(z_{q},\bar{z}_{q})=(z^{\prime},\bar{z}^{\prime})}\\\
j^{+}_{\bar{z}^{\prime}}(\tau^{\prime},z^{\prime},\bar{z}^{\prime})&=\frac{-L}{(2\pi)^{3}}\int
d\omega_{\vec{q}}~{}e^{i\omega_{\vec{q}}L\left(\tau^{\prime}-\frac{\pi}{2}\right)}\int
d^{2}z_{q}\frac{1}{\bar{z}_{q}-\bar{z}^{\prime}}\frac{1}{-4z_{q}(1+z_{q}\bar{z}_{q})^{2}}\partial_{z_{q}}\Bigg{[}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\dagger\text{AdS}(-)}_{\vec{q}}\Bigg{)}\Bigg{]}\\\
\partial_{\bar{z}^{\prime}}j^{+}_{z^{\prime}}\left(\tau^{\prime},z^{\prime},\bar{z}^{\prime}\right)&=\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}e^{i\omega_{\vec{q}}L\left(\tau^{\prime}-\frac{\pi}{2}\right)}~{}~{}\frac{1}{-4\bar{z}^{\prime}(1+z^{\prime}\bar{z}^{\prime})^{2}}\partial_{\bar{z}_{q}}\Bigg{[}\partial_{z_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\dagger\text{AdS}(+)}_{\vec{q}}\Bigg{)}\Bigg{]}\Bigg{|}_{(z_{q},\bar{z}_{q})=(z^{\prime},\bar{z}^{\prime})}\\\
j^{+}_{z^{\prime}}(\tau^{\prime},z^{\prime},\bar{z}^{\prime})&=\frac{-L}{(2\pi)^{3}}\int
d\omega_{\vec{q}}~{}e^{i\omega_{\vec{q}}L\left(\tau^{\prime}-\frac{\pi}{2}\right)}\int
d^{2}z_{q}\frac{1}{z_{q}-z^{\prime}}\frac{1}{-4\bar{z}_{q}(1+z_{q}\bar{z}_{q})^{2}}\partial_{\bar{z}_{q}}\Bigg{[}\partial_{z_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\dagger\text{AdS}(+)}_{\vec{q}}\Bigg{)}\Bigg{]}~{}~{}~{}.\end{split}$
(4.19)

####  4.2.1 Global time component of the CFT current operator mapped to photon
modes

In this section, we evaluate the global time component of the CFT current
operator in terms of the photon creation/annihilation modes. Since, in the
regime where $F_{MN}=0$, the Wilson line doesn’t depend on path. The path for
Wilson line we choose in such a way that the global time coordinate varies
from $0$ to $\tau$ and the angular direction remains unchanged. The CFT
current operator in the Wilson line operator

$e^{iq\int_{0}^{\tau}j_{\tau^{\prime}}d\tau^{\prime}}=e^{iq\int_{0}^{\tau}\left(j^{+}_{\tau^{\prime}}+j^{-}_{\tau^{\prime}}\right)d\tau^{\prime}}~{}~{}~{},$
(4.20)

can be expressed in terms of the photon creation/annihilation modes.

We use the current conservation equation in the boundary CFT
$\partial_{a}j^{a}=0$ to relate the global time component of the CFT current
operator $j_{\tau}$ in terms of $z$ and $\bar{z}$ components, $j_{z}$ and
$j_{\bar{z}}$.

Using the current conservation equation we get,

$\begin{split}&\partial_{\tau}j^{\tau}+\partial_{z}j^{z}+\partial_{\bar{z}}j^{\bar{z}}=0\\\
\implies&-\partial_{\tau}j_{\tau}+\partial_{z}(g^{z\bar{z}}j_{\bar{z}})+\partial_{\bar{z}}(g^{\bar{z}z}j_{z})=0\\\
\implies&\partial_{\tau}j_{\tau}=\frac{1}{2}(1+z\bar{z})^{2}(\partial_{z}j_{\bar{z}}+\partial_{\bar{z}}j_{z})+\bar{z}(1+z\bar{z})j_{\bar{z}}+z(1+z\bar{z})j_{z}~{}~{}~{}.\end{split}$
(4.21)

Here, the $S^{2}$ part of the boundary metric is
$ds^{2}=\frac{4}{(1+z\bar{z})^{2}}dzd\bar{z}$ which gives
$g_{z\bar{z}}=g_{\bar{z}z}=\frac{2}{(1+z\bar{z})^{2}}$ $\&$
$g^{z\bar{z}}=g^{\bar{z}z}=\frac{1}{2}(1+z\bar{z})^{2}$. Here, in eq.(4.21) we
drop $+$ and $-$ superscript in the $z$ component, $\bar{z}$ component and the
global time component of the the CFT current operators. Using the current
conservation equation, global time component of the CFT current operators
corresponding to positive and negative frequency modes
$j_{\tau}^{+}~{}\text{and}~{}j_{\tau}^{-}$ are given by

$\small\begin{split}j_{\tau}^{+}&=\int_{0}^{\tau}d\tau^{\prime}\Bigg{[}\frac{1}{2}(1+z^{\prime}\bar{z}^{\prime})^{2}(\partial_{z^{\prime}}j^{+}_{\bar{z}^{\prime}}+\partial_{\bar{z}^{\prime}}j^{+}_{z^{\prime}})+\bar{z}^{\prime}(1+z^{\prime}\bar{z}^{\prime})j^{+}_{\bar{z}^{\prime}}+z^{\prime}(1+z^{\prime}\bar{z}^{\prime})j^{+}_{z^{\prime}}\Bigg{]}\\\
&=\frac{1}{2}(1+z^{\prime}\bar{z}^{\prime})^{2}\\\
&~{}~{}~{}~{}~{}\times\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)\frac{1}{-4z^{\prime}(1+z^{\prime}\bar{z}^{\prime})^{2}}\partial_{z_{q}}\Bigg{[}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\dagger\text{AdS}(-)}_{\vec{q}}\Bigg{)}\Bigg{]}\Bigg{|}_{(z_{q},\bar{z}_{q})=(z^{\prime},\bar{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)\frac{1}{-4\bar{z}^{\prime}(1+z^{\prime}\bar{z}^{\prime})^{2}}\partial_{\bar{z}_{q}}\Bigg{[}\partial_{z_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\dagger\text{AdS}(+)}_{\vec{q}}\Bigg{)}\Bigg{]}\Bigg{|}_{(z_{q},\bar{z}_{q})=(z^{\prime},\bar{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}}\frac{1}{-4z_{q}(1+z_{q}\bar{z}_{q})^{2}}\partial_{z_{q}}\Bigg{[}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\dagger\text{AdS}(-)}_{\vec{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}}\frac{1}{-4z_{q}(1+z_{q}\bar{z}_{q})^{2}}\partial_{z_{q}}\Bigg{[}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\dagger\text{AdS}(+)}_{\vec{q}}\Bigg{)}~{}\Bigg{]}~{},\end{split}$
(4.22)

$\small\begin{split}j_{\tau}^{-}&=\int_{0}^{\tau}d\tau^{\prime}\Bigg{[}\frac{1}{2}(1+z^{\prime}\bar{z}^{\prime})^{2}(\partial_{z^{\prime}}j^{-}_{\bar{z}^{\prime}}+\partial_{\bar{z}^{\prime}}j^{-}_{z^{\prime}})+\bar{z}^{\prime}(1+z^{\prime}\bar{z}^{\prime})j^{-}_{\bar{z}^{\prime}}+z^{\prime}(1+z^{\prime}\bar{z}^{\prime})j^{-}_{z^{\prime}}\Bigg{]}\\\
&=\frac{1}{2}(1+z^{\prime}\bar{z}^{\prime})^{2}\\\
&\times\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)\frac{1}{-4z^{\prime}(1+z^{\prime}\bar{z}^{\prime})^{2}}\partial_{z_{q}}\Bigg{[}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\text{AdS}(-)}_{\vec{q}}\Bigg{)}\Bigg{]}\Bigg{|}_{(z_{q},\bar{z}_{q})=(z^{\prime},\bar{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)\frac{1}{-4\bar{z}^{\prime}(1+z^{\prime}\bar{z}^{\prime})^{2}}\partial_{\bar{z}_{q}}\Bigg{[}\partial_{z_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\text{AdS}(+)}_{\vec{q}}\Bigg{)}\Bigg{]}\Bigg{|}_{(z_{q},\bar{z}_{q})=(z^{\prime},\bar{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}}\frac{1}{-4z_{q}(1+z_{q}\bar{z}_{q})^{2}}\partial_{z_{q}}\Bigg{[}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\text{AdS}(-)}_{\vec{q}}\Bigg{)}\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}}\frac{1}{-4z_{q}(1+z_{q}\bar{z}_{q})^{2}}\partial_{z_{q}}\Bigg{[}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\text{AdS}(+)}_{\vec{q}}\Bigg{)}\Bigg{]}\Bigg{]}~{}.\end{split}$
(4.23)

In deriving the expression for global time component of the CFT current
operators, $j_{\tau}^{+}$ and $j_{\tau}^{-}$, we perform the integration over
global time coordinates.

The results are given by

$\begin{split}\int_{0}^{\tau}e^{i\omega_{\vec{q}}L(\tau^{\prime}-\frac{\pi}{2})}d\tau^{\prime}&=\frac{1}{iL\omega_{\vec{q}}}e^{-\frac{1}{2}i\pi
L\omega_{\vec{q}}}(e^{iL\tau\omega_{\vec{q}}}-1)\\\
\int_{0}^{\tau}e^{-i\omega_{\vec{q}}L(\tau^{\prime}-\frac{\pi}{2})}d\tau^{\prime}&=\frac{1}{-iL\omega_{\vec{q}}}e^{\frac{1}{2}i\pi
L\omega_{\vec{q}}}(e^{-iL\tau\omega_{\vec{q}}}-1)~{}~{}~{}.\end{split}$ (4.24)

###  4.3 Faddeev-Kulish dressed state

Now, we express the dressed creation mode for the massive scalar field in
terms of the photon creation/annihilation modes and creation mode of the
undressed scalar operator. We substitute the expression for CFT current
operators, $j^{\pm}_{\tau}$ as

$e^{iq\int_{\Gamma(\tau,\hat{p})}j_{a}dx^{a}}=e^{iq\int_{0}^{\tau}\left(j^{+}_{\tau^{\prime}}+j^{-}_{\tau^{\prime}}\right)d\tau^{\prime}}$

in the expression of the dressed creation mode of the massive scalar field

$\begin{split}\sqrt{2\omega_{\vec{p}}}~{}\widetilde{a}^{\dagger}_{\omega_{\vec{p}}}&=\widetilde{\mathfrak{C}}~{}L\int
d\Delta_{\vec{p}}~{}e^{-i\Delta_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big(\frac{\Delta_{\vec{p}}+m}{\Delta_{\vec{p}}-m}\Big{missing})\Big{]}}~{}e^{i\omega_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{missing})\Big{]}}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}\int
d\tau~{}e^{-iL\tau(\omega_{\vec{p}}-\Delta_{\vec{p}})}~{}e^{iq\int_{\Gamma(\tau,\hat{p})}j_{a}dx^{a}}~{}\sqrt{2\Delta_{\vec{p}}}~{}\frac{a^{\dagger}_{\Delta_{\vec{p}}}}{\mathcal{C}}~{}~{}~{},\end{split}$
(4.25)

and express the AdS radius-corrected dressed operator at $\mathcal{O}(q)$. The
AdS corrected dressed creation operator at $\mathcal{O}(q)$ is given by

$\small\begin{split}\sqrt{2\omega_{\vec{p}}}~{}\widetilde{a}^{\dagger}_{\omega_{\vec{p}}}&=\widetilde{\mathfrak{C}}~{}L\int
d\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{missing})\Big{]}}~{}e^{i\omega_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{missing})\Big{]}}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}\int
d\tau~{}e^{-iL\tau(\omega_{\vec{p}}-\Delta_{\vec{p}})}~{}iq~{}\Bigg{(}\int_{0}^{\tau}j^{+}_{\tau}(\tau^{\prime},\hat{p})d\tau^{\prime}+\int_{0}^{\tau}j^{-}_{\tau}(\tau^{\prime},\hat{p})d\tau^{\prime}\Bigg{)}\sqrt{2\Delta_{\vec{p}}}~{}\frac{a^{\dagger}_{\Delta_{\vec{p}}}}{\mathcal{C}}\\\
&=\widetilde{\mathfrak{C}}~{}L\int
d\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{missing})\Big{]}}~{}e^{i\omega_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{missing})\Big{]}}\int
d\tau~{}e^{-iL\tau(\omega_{\vec{p}}-\Delta_{\vec{p}})}\\\
&\Bigg{[}iq\times\Bigg{\\{}\frac{1}{2}(1+z^{\prime}\bar{z}^{\prime})^{2}\\\
&\times\Bigg{[}\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}\mathcal{I}(\tau,\omega_{q})\frac{1}{-4z^{\prime}(1+z^{\prime}\bar{z}^{\prime})^{2}}\partial_{z_{q}}\Bigg{[}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\dagger\text{AdS}(-)}_{\vec{q}}\Bigg{)}\Bigg{]}\Bigg{|}_{(z_{q},\bar{z}_{q})=(z^{\prime},\bar{z}^{\prime})}\\\
&~{}~{}+\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}\mathcal{I}(\tau,\omega_{q})\frac{1}{-4\bar{z}^{\prime}(1+z^{\prime}\bar{z}^{\prime})^{2}}\partial_{\bar{z}_{q}}\Bigg{[}\partial_{z_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\dagger\text{AdS}(+)}_{\vec{q}}\Bigg{)}\Bigg{]}\Bigg{|}_{(z_{q},\bar{z}_{q})=(z^{\prime},\bar{z}^{\prime})}\Bigg{]}\Bigg{\\}}\\\
&+\bar{z}^{\prime}(1+z^{\prime}\bar{z}^{\prime})\\\
&\Bigg{[}-\frac{L}{(2\pi)^{3}}\int
d\omega_{\vec{q}}~{}\mathcal{I}(\tau,\omega_{q})\int
d^{2}z_{q}\frac{1}{\bar{z}_{q}-\bar{z}^{\prime}}\frac{1}{-4z_{q}(1+z_{q}\bar{z}_{q})^{2}}\partial_{z_{q}}\Bigg{[}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\dagger\text{AdS}(-)}_{\vec{q}}\Bigg{)}\Bigg{]}\Bigg{]}\\\
&+z^{\prime}(1+z^{\prime}\bar{z}^{\prime})\\\
&\Bigg{[}-\frac{L}{(2\pi)^{3}}\int
d\omega_{\vec{q}}~{}\mathcal{I}(\tau,\omega_{q})\int
d^{2}z_{q}\frac{1}{z_{q}-z^{\prime}}\frac{1}{-4\bar{z}_{q}(1+z_{q}\bar{z}_{q})^{2}}\partial_{\bar{z}_{q}}\Bigg{[}\partial_{z_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\dagger\text{AdS}(+)}_{\vec{q}}\Bigg{)}\Bigg{]}\Bigg{]}\\\
+&\Bigg{(}\text{terms for the negative frequency mode involving annihilation
modes of photon}~{}\mathbf{a}_{\vec{q}}^{\text{AdS}(\pm)}\Bigg{)}\Bigg{]}\\\
&~{}~{}~{}~{}\sqrt{2\Delta_{\vec{p}}}~{}~{}\frac{a^{\dagger}_{\Delta_{\vec{p}}}}{\mathcal{C}}~{}~{}~{},\\\
\end{split}$ (4.26)

where, $\mathcal{I}(\tau,\omega_{q})$ is given by

$\mathcal{I}(\tau,\omega_{q})=\frac{1}{iL\omega_{\vec{q}}}e^{-\frac{1}{2}i\pi
L\omega_{\vec{q}}}\Big{(}\frac{1}{iL\omega_{\vec{q}}}(e^{iL\tau\omega_{\vec{q}}}-1)-\tau\Big{)}~{}~{}~{}.$
(4.27)

Now, we note that, when acting on the vacuum state, the set of modes with the
negative frequency involves annihilation modes and thus kills the state. In
the expression for dressed state, these terms will not contribute. The dressed
creation operator $\widetilde{a}^{\dagger}_{\omega_{\vec{p}}}$ acting on the
vacuum $\ket{0}$ gives the AdS corrected Faddeev-Kulish dressed state.

In appendix A, in eq.(A.1) we have written the expression by dropping the
terms for the negative frequency mode involving annihilation modes of photon
$\mathbf{a}_{\vec{q}}^{\text{AdS}(\pm)}$. We can further perform the global
time, $\tau$ integral to simplify the expression. After performing the global
time, $\tau$ integral the integral evaluates to delta function, using which we
can further perform frequency, $\Delta_{\vec{p}}$ integral. We have shifted
the final expression to appendix A. After performing the frequency,
$\Delta_{\vec{p}}$ integral we have the AdS corrected dressed creation
operator in eq.(A.4) of appendix A.

##  5 Conclusions and future directions

In this paper, we have constructed AdS radius correction to the Faddeev-Kulish
dressed state. We follow the philosophy of Wilson line dressing in the context
of AdS/CFT to arrive at the Faddeev-Kulish dressed state. We study the CFT
representation and the mixed representation of the Faddeev-Kulish dressed
state. We comment on some speculations to get an IR finite
$\mathcal{S}$-matrix from AdS/CFT. We use HKLL bulk reconstruction
prescription to construct the soft photon modes in terms of the CFT current
operators. Then, after expressing the $1/L^{2}$ correction to the soft photon
modes, we implement AdS radius correction to the Wilson line dressing. We
invert the mapping of the AdS radius-corrected soft photon modes in terms of
CFT current operators, that means we evaluate the CFT current operators in
terms of the AdS radius-corrected modes of the photon. In the Wilson line
dressing, we use this inverse mapping between the CFT current operators and
soft photon modes to construct the AdS radius-corrected creation mode of the
Wilson line dressed scalar field. The dressed mode acting on the vaccuum is
the desired Faddeev-Kulish dressed state, which takes into account the AdS
radius correction.

### Future directions.

In this section, we list some excellent open questions that would be worth
exploring in the future.

### CFT representation of the Faddeev-Kulish dressed state outside the zoomed
in limit.

It would be interesting to obtain a formula for the Faddeev-Kulish dressed
states in the CFT outside the zoomed in limit. For that, we first have to
construct the AdS radius-corrected modes of the Wilson line dressed massive
scalar field in terms of the dual CFT operator. We will report results of this
analysis in an upcoming work.

### Asymptotic charges and states from AdS/CFT.

The asymptotic symmetries inevitably lead to the Faddeev-Kulish dressed state
since the amplitudes that preserve the asymtotic charge are IR finite, the
explicit construction was done in [64]. The study of generalizations of
asymptotic symmetries in AdS was done in [70, 71, 72] using different boundary
conditions. An excellent open problem is to apply them to create the physical
asymptotic states. It would be fascinating to have the states from AdS/CFT and
AdS corrections to it.

### Faddeev-Kulish state involving soft gravitons from AdS/CFT.

It would be interesting to construct the Faddeev-Kulish asymptotic state
involving soft gravitons in the flat limit of AdS/CFT and AdS corrections to
it. To do this, first we have to construct the graviton scattering states in
terms of the CFT stress tensor operators, then following the Wilson line
dressing as studied in [51] we can construct the Faddeev-Kulish state. Also,
using the mapping between flat space scattering states and CFT Stress tensor
operators, we can derive the soft graviton theorem and AdS correction to soft
graviton theorem from the $CFT_{3}$ Stress tensor Ward identities.

### Principal series decomposition in AdS.

In celestial amplitude, generally a principal series decomposition of
scattering states are used to translate the IR effects in flat spacetime into
celestial amplitudes [73, 74, 75, 76, 77]. It would be interesting to study
the principal series decomposition in AdS. For that, we have to perform
principal series decomposition of the CFT correlation function in terms of the
bulk Witten diagram in AdS before taking the flat limit. This would open up a
new avenue for investigation into the IR effects in AdS. Moving away from the
flat limit, this full fledged AdS principal series decomposition will pave the
way for the AdS correction to the Faddeev-Kulish dressing.

##### Acknowledgement

I would especially like to thank Eliot Hijano for useful discussions and
collaboration in the earlier work. I also thank Joydeep Chakravarty, Sangmin
Choi, Nava Gaddam, Hofie Sigridar Hannesdottir, Chethan Krishnan, Priyadarshi
Paul, Suvrat Raju, Iain Stewart, Becher Thomas, and definately Daniel Kabat,
Alok Laddha, R. Loganayagam, Pronobesh Maity, Jyotirmoy Mukherjee, and Pabitra
Ray for correspondence and useful discussions. I gratefully acknowledge
support from the grant of the Department of Atomic Energy, Government of
India, under project no. RTI4001.

##  Appendix A AdS corrected Faddeev-Kulish dressed state: some
simplifications

In this appendix A, we express the AdS corrected Faddeev-Kulish dressed state.

The AdS corrected dressed creation operator at $\mathcal{O}(q)$ is given by

$\begin{split}\sqrt{2\omega_{\vec{p}}}~{}\widetilde{a}^{\dagger}_{\omega_{\vec{p}}}&=\widetilde{\mathfrak{C}}~{}L\int
d\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{missing})\Big{]}}~{}e^{i\omega_{\vec{p}}L\Big{[}\frac{\pi}{2}+\frac{i}{2}\log\Big(\frac{\omega_{\vec{p}}+m}{\omega_{\vec{p}}-m}\Big{missing})\Big{]}}\int
d\tau~{}e^{-iL\tau(\omega_{\vec{p}}-\Delta_{\vec{p}})}\\\
&\Bigg{[}iq\times\Bigg{\\{}\frac{1}{2}(1+z^{\prime}\bar{z}^{\prime})^{2}\\\
&\times\Bigg{[}\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}\mathcal{I}(\tau,\omega_{q})\frac{1}{-4z^{\prime}(1+z^{\prime}\bar{z}^{\prime})^{2}}\partial_{z_{q}}\Bigg{[}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\dagger\text{AdS}(-)}_{\vec{q}}\Bigg{)}\Bigg{]}\Bigg{|}_{(z_{q},\bar{z}_{q})=(z^{\prime},\bar{z}^{\prime})}\\\
&~{}~{}+\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}\mathcal{I}(\tau,\omega_{q})\frac{1}{-4\bar{z}^{\prime}(1+z^{\prime}\bar{z}^{\prime})^{2}}\partial_{\bar{z}_{q}}\Bigg{[}\partial_{z_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\dagger\text{AdS}(+)}_{\vec{q}}\Bigg{)}\Bigg{]}\Bigg{|}_{(z_{q},\bar{z}_{q})=(z^{\prime},\bar{z}^{\prime})}\Bigg{]}\Bigg{\\}}\\\
&+\bar{z}^{\prime}(1+z^{\prime}\bar{z}^{\prime})\\\
&\Bigg{[}-\frac{L}{(2\pi)^{3}}\int
d\omega_{\vec{q}}~{}\mathcal{I}(\tau,\omega_{q})\int
d^{2}z_{q}\frac{1}{\bar{z}_{q}-\bar{z}^{\prime}}\frac{1}{-4z_{q}(1+z_{q}\bar{z}_{q})^{2}}\partial_{z_{q}}\Bigg{[}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\dagger\text{AdS}(-)}_{\vec{q}}\Bigg{)}\Bigg{]}\Bigg{]}\\\
&+z^{\prime}(1+z^{\prime}\bar{z}^{\prime})\\\
&\Bigg{[}-\frac{L}{(2\pi)^{3}}\int
d\omega_{\vec{q}}~{}\mathcal{I}(\tau,\omega_{q})\int
d^{2}z_{q}\frac{1}{z_{q}-z^{\prime}}\frac{1}{-4\bar{z}_{q}(1+z_{q}\bar{z}_{q})^{2}}\partial_{\bar{z}_{q}}\Bigg{[}\partial_{z_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\dagger\text{AdS}(+)}_{\vec{q}}\Bigg{)}\Bigg{]}\Bigg{]}\Bigg{]}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sqrt{2\Delta_{\vec{p}}}~{}~{}\frac{a^{\dagger}_{\Delta_{\vec{p}}}}{\mathcal{C}}~{}~{}~{},\\\
\end{split}$ (A.1)

where, $\mathcal{I}(\tau,\omega_{q})$ is given by

$\mathcal{I}(\tau,\omega_{q})=\frac{1}{iL\omega_{\vec{q}}}e^{-\frac{1}{2}i\pi
L\omega_{\vec{q}}}\Big{(}\frac{1}{iL\omega_{\vec{q}}}(e^{iL\tau\omega_{\vec{q}}}-1)-\tau\Big{)}~{}~{}~{}.$
(A.2)

Now, after performing the global time, $\tau$ integral we have

$\begin{split}&\int
d\tau~{}e^{-iL\tau(\omega_{\vec{p}}-\Delta_{\vec{p}})}\Big{(}\frac{1}{iL\omega_{\vec{q}}}(e^{iL\tau\omega_{\vec{q}}}-1)-\tau\Big{)}\\\
&=\frac{1}{iL\omega_{\vec{q}}}\frac{2\pi}{L}\Bigg{(}\delta(\omega_{\vec{p}}+\omega_{\vec{q}}-\Delta_{\vec{p}})-\delta(\omega_{\vec{p}}-\Delta_{\vec{p}})\Bigg{)}-i\frac{2\pi}{L}\frac{\partial}{\partial\Delta_{\vec{p}}}\delta(\omega_{\vec{p}}-\Delta_{\vec{p}})~{}~{}~{}.\end{split}$
(A.3)

After performing the frequency, $\Delta_{\vec{p}}$ integral we have the AdS
corrected dressed creation operator

$\small\begin{split}\sqrt{2\omega_{\vec{p}}}~{}\widetilde{a}^{\dagger}_{\omega_{\vec{p}}}&=\frac{2\pi\widetilde{\mathfrak{C}}}{\mathcal{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{missing})+(\omega_{\vec{p}}+\omega_{\vec{q}})\log\Big(\frac{\omega_{\vec{p}}+\omega_{\vec{q}}+m}{\omega_{\vec{p}}+\omega_{\vec{q}}-m}\Big{missing})\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{missing})\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{missing})\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}}}~{}\frac{1}{-4z^{\prime}(1+z^{\prime}\bar{z}^{\prime})^{2}}\partial_{z_{q}}\Bigg{[}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\dagger\text{AdS}(-)}_{\vec{q}}\Bigg{)}\Bigg{]}\Bigg{|}_{(z_{q},\bar{z}_{q})=(z^{\prime},\bar{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{missing})+(\omega_{\vec{p}}+\omega_{\vec{q}})\log\Big(\frac{\omega_{\vec{p}}+\omega_{\vec{q}}+m}{\omega_{\vec{p}}+\omega_{\vec{q}}-m}\Big{missing})\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{missing})\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{missing})\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}}}\frac{1}{-4\bar{z}^{\prime}(1+z^{\prime}\bar{z}^{\prime})^{2}}\partial_{\bar{z}_{q}}\Bigg{[}\partial_{z_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\dagger\text{AdS}(+)}_{\vec{q}}\Bigg{)}\Bigg{]}\Bigg{|}_{(z_{q},\bar{z}_{q})=(z^{\prime},\bar{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{missing})+(\omega_{\vec{p}}+\omega_{\vec{q}})\log\Big(\frac{\omega_{\vec{p}}+\omega_{\vec{q}}+m}{\omega_{\vec{p}}+\omega_{\vec{q}}-m}\Big{missing})\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{missing})\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{missing})\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}}\frac{1}{-4z_{q}(1+z_{q}\bar{z}_{q})^{2}}\partial_{z_{q}}\Bigg{[}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\dagger\text{AdS}(+)}_{\vec{q}}\Bigg{)}\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{missing})+(\omega_{\vec{p}}+\omega_{\vec{q}})\log\Big(\frac{\omega_{\vec{p}}+\omega_{\vec{q}}+m}{\omega_{\vec{p}}+\omega_{\vec{q}}-m}\Big{missing})\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{missing})\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{missing})\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}}\frac{1}{-4\bar{z}_{q}(1+z_{q}\bar{z}_{q})^{2}}\partial_{\bar{z}_{q}}\Bigg{[}\partial_{z_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\dagger\text{AdS}(+)}_{\vec{q}}\Bigg{)}~{}\Bigg{]}~{}\Bigg{]}\Bigg{\\}}\Bigg{]}\left|0\right>~{}.\\\
\end{split}$ (A.4)

###  A.1 More detailed steps of the derivation of the inverse mapping

In this section A.1, we provide the detailed steps to derive the inverse
mapping between the CFT current operators and the photon modes.

The annihilation operator of an outgoing photon of negative helicity is
expressed in terms of an integrated expression of the boundary CFT current
operator

$\begin{split}&\sqrt{2\omega_{\vec{q}}}~{}\mathbf{a}^{\text{AdS}(-)}_{\vec{q}}\\\
&=\frac{1}{32\pi\omega_{\vec{q}}^{2}L^{2}}\frac{1+z_{q}\bar{z}_{q}}{\sqrt{2}\omega_{\vec{q}}}\int
d\tau^{\prime}~{}e^{-i\omega_{\vec{q}}L\left(\frac{\pi}{2}-\tau^{\prime}\right)}\int
d^{2}z^{\prime}\int
d^{2}z_{w}\Bigg{[}\frac{(1+z^{\prime}\bar{z}^{\prime})^{2}(1+z_{w}\bar{z}_{w})^{2}}{(\bar{z}_{w}-\bar{z}^{\prime})^{2}(z_{q}-z_{w})^{3}}\Bigg{]}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}\partial_{z^{\prime}}j^{-}_{\bar{z}^{\prime}}(\tau^{\prime},z^{\prime},\bar{z}^{\prime})~{}.\end{split}$
(A.5)

Now, we act with a $\partial_{\bar{z}_{q}}$ on both sides of the eq.(A.5) and
get

$\begin{split}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\pi\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\text{AdS}(-)}_{\vec{q}}\Bigg{)}&=\int
d\tau^{\prime}~{}e^{-i\omega_{\vec{q}}L\left(\frac{\pi}{2}-\tau^{\prime}\right)}\int
d^{2}z^{\prime}~{}\int
d^{2}z_{w}~{}\partial_{\bar{z}_{q}}\Bigg{[}\frac{(1+z^{\prime}\bar{z}^{\prime})^{2}(1+z_{w}\bar{z}_{w})^{2}}{(\bar{z}_{w}-\bar{z}^{\prime})^{2}(z_{q}-z_{w})^{3}}\Bigg{]}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\partial_{z^{\prime}}j^{-}_{\bar{z}^{\prime}}\left(\tau^{\prime},z^{\prime},\bar{z}^{\prime}\right)~{}~{}~{}.\end{split}$
(A.6)

Now, using the identity

$\begin{split}\partial_{\bar{z}_{q}}\frac{1}{(z_{q}-z_{w})^{3}}&=(2\pi)\frac{(-1)^{2}}{2!}\partial_{z_{q}}^{2}\delta^{(2)}(z_{q},z_{w})\\\
&=(2\pi)\frac{(-1)^{2}}{2!}\partial_{z_{w}}^{2}\delta^{(2)}(z_{q},z_{w})~{}~{}~{},\end{split}$
(A.7)

we simplify the expression and get

$\begin{split}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\pi\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\text{AdS}(-)}_{\vec{q}}\Bigg{)}&=\int
d\tau^{\prime}~{}e^{-i\omega_{\vec{q}}L\left(\frac{\pi}{2}-\tau^{\prime}\right)}\int
d^{2}z^{\prime}~{}\int
d^{2}z_{w}~{}\pi~{}\partial_{z_{w}}^{2}\delta^{(2)}(z_{q},z_{w})\\\
&~{}~{}~{}~{}~{}~{}~{}\times\Bigg{[}\frac{(1+z^{\prime}\bar{z}^{\prime})^{2}(1+z_{w}\bar{z}_{w})^{2}}{(\bar{z}_{w}-\bar{z}^{\prime})^{2}}\Bigg{]}\partial_{z^{\prime}}j^{-}_{\bar{z}^{\prime}}\left(\tau^{\prime},z^{\prime},\bar{z}^{\prime}\right)~{}~{}~{}.\end{split}$
(A.8)

Now, we perform the $d^{2}z_{w}$ integral using the delta function
$\delta^{(2)}(z_{q},z_{w})$ to simplify the expression a bit

$\begin{split}&\int
d^{2}z_{w}~{}\partial^{2}_{z_{w}}\delta^{(2)}(z_{q},z_{w})\Bigg{[}\frac{(1+z_{w}\bar{z}_{w})^{2}}{(\bar{z}_{w}-\bar{z}^{\prime})^{2}}\Bigg{]}\\\
&=\frac{2\bar{z}_{q}^{2}}{(\bar{z}_{q}-\bar{z}^{\prime})^{2}}~{}~{}~{},\end{split}$
(A.9)

where, in the intermediate steps of eq.(A.9) we use the product rule for
double differentiation and expand each term by the following way

$\begin{split}\partial_{z_{w}}^{2}\frac{(1+z_{w}\bar{z}_{w})^{2}}{(\bar{z}_{w}-\bar{z}^{\prime})^{2}}&=\frac{1}{(\bar{z}_{w}-\bar{z}^{\prime})^{2}}\partial_{z_{w}}^{2}\Big{[}(1+z_{w}\bar{z}_{w})^{2}\Big{]}+(1+z_{w}\bar{z}_{w})^{2}~{}\partial_{z_{w}}^{2}\frac{1}{(\bar{z}_{w}-\bar{z}^{\prime})^{2}}\\\
&~{}~{}~{}~{}~{}+2\partial_{z_{w}}[(1+z_{w}\bar{z}_{w})^{2}]~{}\partial_{z_{w}}\frac{1}{(\bar{z}_{w}-\bar{z}^{\prime})^{2}}\\\
&=\frac{2\bar{z}_{w}^{2}}{(\bar{z}_{w}-\bar{z}^{\prime})^{2}}-2\pi(1+z_{w}\bar{z}_{w})^{2}~{}\partial_{z_{w}}\partial_{\bar{z}_{w}}\delta^{(2)}(z_{w},z^{\prime})\\\
&~{}~{}~{}~{}~{}~{}~{}~{}-4\pi\bar{z}_{w}(1+z_{w}\bar{z}_{w})~{}\partial_{\bar{z}_{w}}\delta^{(2)}(z_{w},z^{\prime})~{}~{}~{}.\end{split}$
(A.10)

After simplification the expression becomes

$\begin{split}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\text{AdS}(-)}_{\vec{q}}\Bigg{)}&=\int
d\tau^{\prime}~{}e^{-i\omega_{\vec{q}}L\left(\frac{\pi}{2}-\tau^{\prime}\right)}\int
d^{2}z^{\prime}~{}(1+z^{\prime}\bar{z}^{\prime})^{2}\frac{2\bar{z}_{q}^{2}}{(\bar{z}_{q}-\bar{z}^{\prime})^{2}}~{}~{}\partial_{z^{\prime}}j^{-}_{\bar{z}^{\prime}}\left(\tau^{\prime},z^{\prime},\bar{z}^{\prime}\right)~{}~{}~{}.\end{split}$
(A.11)

Now, further acting $\partial_{z_{q}}$ on the simplified expression eq.(A.11)
and using the identity

$\partial_{z_{q}}\frac{1}{(\bar{z}_{q}-\bar{z}^{\prime})^{2}}=2\pi(-1)\partial_{\bar{z}_{q}}\delta^{(2)}(z_{q},z^{\prime})~{}~{}~{},$
(A.12)

we get

$\begin{split}\partial_{z_{q}}\Bigg{[}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\text{AdS}(-)}_{\vec{q}}\Bigg{)}\Bigg{]}&=\int
d\tau^{\prime}~{}e^{-i\omega_{\vec{q}}L\left(\frac{\pi}{2}-\tau^{\prime}\right)}\Big{(}-8\pi
z_{q}(1+z_{q}\bar{z}_{q})^{2}\Big{)}\partial_{z_{q}}j^{-}_{\bar{z}_{q}}\left(\tau^{\prime},z_{q},\bar{z}_{q}\right)~{}~{}~{}.\end{split}$
(A.13)

Now, we get the inverse mapping of the CFT current operator for the negative
frequency in terms of the AdS corrected photon annihilation mode

$\partial_{z_{q}}j^{-}_{\bar{z}_{q}}\left(\tau^{\prime},z_{q},\bar{z}_{q}\right)=\frac{L}{(2\pi)^{2}}\int
d\omega_{\vec{q}}~{}e^{-i\omega_{\vec{q}}L\left(\tau^{\prime}-\frac{\pi}{2}\right)}~{}~{}\frac{1}{-4z_{q}(1+z_{q}\bar{z}_{q})^{2}}\partial_{z_{q}}\Bigg{[}\partial_{\bar{z}_{q}}\Bigg{(}\frac{64\omega_{\vec{q}}^{7/2}L^{2}}{1+z_{q}\bar{z}_{q}}\mathbf{a}^{\text{AdS}(-)}_{\vec{q}}\Bigg{)}\Bigg{]}~{}~{}~{}.$
(A.14)

The useful identities used to derive the inverse mapping, we list in appendix,
section A.1.1.

####  A.1.1 Useful identities used to derive the inverse mapping

Here, we list the useful identities used to derive the inverse mapping

$\begin{split}\partial_{z}^{n}\delta(z)&=\frac{(-1)^{n}n!}{z^{n}}\delta(z)\\\
\partial_{z}\delta(z,w)&=-\partial_{w}\delta(z,w)\\\
\int\partial_{z}^{n}\delta(z,w)f(z)dz&=(-1)^{n}\partial_{z}^{n}f(z)\Big{|}_{z=w}\\\
\partial_{z}\delta(z,w)f(w,\bar{w})&=\partial_{z}\Big{[}\delta^{(2)}(z,w)f(w,\bar{w})\Big{]}\\\
&=\partial_{z}\Big{[}\delta(z,w)f(z,\bar{z})\Big{]}\\\
&=\partial_{z}\delta(z,w)f(z,\bar{z})+\delta(z,w)\partial_{z}f(z,\bar{z})\\\
\partial^{2}_{z}\delta(z,w)f(w,\bar{w})&=\partial^{2}_{z}\Big{[}\delta(z,w)f(w,\bar{w})\Big{]}\\\
&=\partial^{2}_{z}\Big{[}\delta(z,w)f(z,\bar{z})\Big{]}\\\
&=\partial_{z}^{2}\delta(z,w)f(z,\bar{z})+\delta(z,w)\partial_{z}^{2}f(z,\bar{z})+2\partial_{z}\delta(z,w)\partial_{z}f(z,\bar{z})~{}~{}~{}~{}\\\
\partial_{\bar{z}}\frac{1}{(z-w)^{n+1}}&=(2\pi)\frac{(-1)^{n}}{n!}\partial_{z}^{n}\delta^{(2)}(z,w)~{}~{}~{},\end{split}$
(A.15)

where, the delta function $\delta^{(2)}(z,w)$ in the last equation is a
complex plane delta function.

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