aainstitutetext: Indian Institute of Science Education & Research Bhopal,
Bhopal Bypass Road, Bhauri, Bhopal 420 066,
Madhya Pradesh, India.bbinstitutetext: Harish-Chandra Research Institute,
Chhatnag Road, Jhusi, Prayagraj 211019,
Uttar Pradesh, India

# Soft Photon theorem in the small negative cosmological constant limit

Nabamita Banerjee b Karan Fernandes a and Arpita Mitra<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the effect of electromagnetic interactions on the classical soft
theorems on an asymptotically AdS background in 4 spacetime dimensions, in the
limit of a small cosmological constant or equivalently a large AdS radius $l$.
This identifies $1/l^{2}$ perturbative corrections to the known asymptotically
flat spacetime leading and subleading soft factors. Our analysis is only valid
to leading order in $1/l^{2}$. The leading soft factor can be expected to be
universal and holds beyond tree level. This allows us to derive a $1/l^{2}$
corrected Ward identity, following the known equivalence between large gauge
Ward identities and soft theorems in asymptotically flat spacetimes.

##  1 Introduction

Soft theorems are statements about quantum scattering amplitudes when one or
more of the external particles go soft, i.e. their momenta $k^{\mu}\rightarrow
0.$ These theorems state that a $(m+n)$ point scattering amplitude
${\cal{A}}_{m+n}$, where $m$ number of external particles go soft, is
proportional to ${\cal{A}}_{n}$, the $n$ point scattering amplitude involving
the other hard particles. The proportionality factor is universal at leading
order, irrespective of the details of the interactions Weinberg:1964ew ;
Weinberg:1965nx . The soft factor is also divergent at leading order in the
soft momenta expansion and depends on properties of the hard particles. The
theorems are valid for any gauge invariant quantum field theory in any
spacetime dimensions. In particular, the soft photon theorem follows from the
U(1) gauge invariance Kapec:2014zla ; He:2014cra  and the soft graviton
theorem follows from the diffeomorphism invariance of quantum field theories
Strominger:2013jfa ; Cachazo:2014fwa ; Campiglia:2015yka . Recent works
Lysov:2014csa ; Schwab:2014xua ; Campiglia:2014yka ; Casali:2014xpa ;
Broedel:2014fsa ; Conde:2016csj ; Chakrabarti:2017zmh ; Chakrabarti:2017ltl ;
Laddha:2017vfh ; AtulBhatkar:2018kfi ; Addazi:2019mjh ; Sahoo:2020ryf  have
extended soft theorems beyond leading orders, with results for subleading and
sub-subleading soft theorems in gravitational and U(1) gauge theories. In four
spacetime dimensions there exists an additional subtlety, in that the
subleading soft factor diverges as the logarithm of the frequency due to the
existence of asymptotically non-vanishing long range interactions. To be
precise, below we write the exact statement of the single soft photon theorem
to subleading order in four spacetime dimensions Sahoo:2018lxl :

$\displaystyle S_{\rm em}^{\text{flat}}$ $\displaystyle=S_{\rm
em;\text{leading}}^{\text{flat}}+S_{\rm
em;\text{subleading}}^{\text{flat}}\,,\qquad\text{with}$ $\displaystyle S_{\rm
em;\text{leading}}^{\text{flat}}$
$\displaystyle=\sum_{a=1}^{n}q_{(a)}\frac{\epsilon_{\mu}p^{\mu}_{(a)}}{p_{(a)}.k}\,,$
(1) $\displaystyle S_{\rm em;\text{subleading}}^{\text{flat}}$
$\displaystyle=i\sum_{a=1}^{n}q_{(a)}\frac{\epsilon_{\nu}k_{\rho}j^{\rho\nu}_{(a)}}{p_{(a)}.k}$
$\displaystyle=i\ln\omega^{-1}\sum_{a=1}^{n}q_{(a)}\frac{\epsilon_{\nu}k_{\rho}\bigg{(}c_{(a)}^{\rho}p_{(a)}^{\nu}-c_{(a)}^{\nu}p_{(a)}^{\rho}\bigg{)}}{p_{(a)}.k}+\cdots$
(2)

where $k^{\mu}$ and $\epsilon_{\mu}$ are respectively the momentum and
polarization of the soft photon, while $q_{(a)}$, ${p}^{\mu}_{(a)}$ and
${j}^{\mu\nu}_{(a)}$ are the charges, asymptotic momenta and angular momenta
of the $n$ hard particles. In going from the first to second equality of 2, we
have an expansion in terms of the classical trajectories of the hard
particles. The individual trajectories involve logarithmic contributions in
four dimensions due to the presence of long range interactions of the
electromagnetic fields. We can expand the classical trajectories of these
particles to find the following leading order contribution in proper time

$r^{\mu}_{(a)}(t)=\eta_{(a)}\frac{p_{(a)}^{\mu}}{m_{(a)}}t+c^{\mu}_{(a)}\ln|t|+\cdots,$

where $t$ is the proper time along the particle trajectories, $\eta_{(a)}$ is
$+1$ for incoming particles and $-1$ for outgoing particles, $m_{(a)}$ are the
particle masses and $c^{\mu}_{(a)}$ are coefficients which depend on the long
range electromagnetic force. Using this expression for $r^{\mu}_{(a)}(t)$ in
$j^{\mu\nu}_{(a)}=r^{\mu}_{(a)}(t)p_{(a)}^{\nu}-r^{\nu}(t)p_{(a)}^{\mu}$ then
provides the term in the second line of 2 on replacing $t$ with $\omega^{-1}$.
The additional terms not described in the second line of 2 are quantum
corrections. These terms can be ignored as long as they are much smaller than
the classical scattering contribution. This will be the case when the
wavelength of the soft particles are much larger than the impact parameter
involved in the scattering and when the total radiated energy is less than the
energy of the scatterer. In such cases, we can derive the universal
contributions entirely from the low frequency limit of the (gauge invariant)
classical radiative fields. This relation is provided by the classical soft
photon theorem Sahoo:2018lxl ; Laddha:2018rle ; Laddha:2018myi ,

$\displaystyle\lim_{\omega\to
0}\epsilon^{\mu}\tilde{a}_{\mu}\left(\omega\,,\vec{x}\right)$
$\displaystyle=e^{i\omega R}\left(\frac{\omega}{2\pi
iR}\right)^{\frac{D-2}{2}}\frac{1}{2\omega}S^{\text{flat}}_{\text{em}}$
$\displaystyle=-\frac{i}{4\pi R}e^{i\omega
R}S^{\text{flat}}_{\text{em}}\qquad\qquad\text{for}\;D=4\,,$ (3)

where $\tilde{a}_{\mu}$ is the radiative component of the electromagnetic
field in frequency space, $D$ is the spacetime dimension and $R$ denotes the
distance of the soft photon from the scatterer. A similar analysis holds for
the soft graviton theorem Sahoo:2018lxl ; Laddha:2018vbn ; Laddha:2019yaj ;
Saha:2019tub .

An interesting question to ask is: how does the above story change when we
study a quantum field theory in an asymptotically non-flat background? Since
the gauge invariance remains intact even for asymptotically non-flat theories,
we expect a version of soft theorems to be valid in this case as well. Non-
asymptotically flat backgrounds, particularly of the kind of asymptotically
Anti-de Sitter (AdS) or de Sitter (dS) types are of great importance in
physics. AdS arises as an interesting gravity background for some exact
computations in the context of String Theory and AdS/CFT conjectures. On the
other hand dS spacetime has its importance in cosmology. Thus, understanding
aspects of soft theorems in these spacetimes are important. In this paper we
shall look for classical soft photon theorems in asymptotically AdS spacetime,
where the radius of the AdS space is considered to be large (we shall make
this condition more precise in later sections). Our results, with slight
modifications, are also valid for asymptotically dS spacetimes in the large
radius limit.

AdS (dS) is a solution of Einstein’s gravity with a negative (positive)
cosmological constant. AdS spacetimes have an effective potential under which
particles behave like being confined in a box. The null rays bounce back from
the timelike boundary an infinite number of times. This creates the main
obstacle in defining the usual “in” and “out” states for a quantum field
theory in AdS backgrounds. Thus, unlike in asymptotically flat theories, the
definition of the usual scattering amplitudes Gary:2009mi ; Penedones:2010ue ;
Fitzpatrick:2011ia ; Rastelli:2016nze  and hence a soft theorem is not known
for quantum field theories defined in an asymptotically AdS spacetime111a
related work can be found in Hijano:2020szl . Instead we look for a possible
soft factorization on taking the classical limit of scattering amplitudes in
AdS backgrounds. This gives us the analogue of classical soft theorems known
from asymptotically flat spacetime classical scattering processes. While
computing the classical radiation profile, for technical simplification, we
consider the value of cosmological constant to be small , or equivalently the
radius $l$ of AdS large and treat it as a perturbation parameter in our
computations. Our results are exact up to order $1/l^{2}$ of the AdS radius.
Physically, we think of studying a scattering process in an asymptotically
flat theory modified by a small potential (inversely proportional to the
square of the AdS radius). Thus our results provide us perturbative
corrections to order $1/l^{2}$ of known results for classical photon and
graviton radiation profiles in the asymptotically flat Reissner-Nordström case
Fernandes:2020tsq . The details of the scattering process we consider will be
discussed in later sections.

Finally to study the “soft limit” of the classical radiation in asymptotically
AdS spacetime, we consider a double scaling limit Banerjee:2020dww : where the
frequency of radiation and cosmological constant simultaneously tend to zero,
keeping their ratio finite. This is due to the fact that a radiation mode in a
theory that asymptotes to AdS spacetime has a minimum frequency inversely
proportional to the size of the AdS and hence the frequency of the radiation
cannot limit to a zero value. In other words, there is a mass gap in AdS that
restricts the usual soft limit. Physically the double scaling limit implies
that we consider the radiation limits to a strictly soft one as the space is
limiting to an asymptotically flat spacetime. By taking this limit, we find
the classical soft photon theorem to leading and sub-leading order in an
asymptotically AdS theory, in the large AdS radius limit.

On asymptotically flat spacetimes, Weinberg’s soft theorems for scattering
amplitudes are known to be equivalent to Ward identities for large gauge
transformations He:2014cra , Lysov:2014csa ,He:2014laa ; Hamada:2018vrw ;
Campiglia:2019wxe ; AtulBhatkar:2019vcb . These identities represent the soft
charge conservation across null infinity $\mathscr{I}$. For asymptotically
flat spacetimes, it is well known by now that the classical soft factor
provide the same leading (quantum) soft factor in the classical limit up to
the usual gauge ambiguity. Hence we can as well recast the classical soft
theorem in terms of the large gauge Ward identity. In our present study, as
the classical soft photon theorem receives a $1/l^{2}$ correction (to its flat
space form) in taking the $l\to\infty$ limit of AdS spacetimes, we expect the
equivalence to imply a $1/l^{2}$ correction of the usual large gauge Ward
identity. This physically amounts to deriving $1/l^{2}$ perturbative
modifications in the Ward identity that correctly reproduces the modified
classical soft theorem.

The large gauge Ward identity on an asymptotically AdS spacetime involve two
subtleties related to their derivation and definition on an asymptotic
surface. A formal derivation of the large gauge Ward identity in AdS
spacetimes is complicated by the fact that there does not appear to be a
unique large $r$ saddle point corresponding to the low frequency result,
unlike in asymptotically flat spacetimes. As mentioned, the $1/l^{2}$
corrections to the soft factor results from a double scaling limit on the
cosmological constant and frequency, thus receiving contributions across
different length scales. We can nevertheless infer this Ward identity from the
soft photon theorem, following the procedure used in He:2014cra  to
demonstrate the equivalence. We find specific corrections of the soft photon
mode and gauge parameter that provide a Ward identity equivalent to the
classical soft photon theorem up to $1/l^{2}$ corrections. It has recently
been demonstrated that conformal Ward identities are equivalent to the
Weinberg’s soft theorems defined on a flat spacetime patch resulting from the
$l\to\infty$ limit on asymptotically AdS spacetimes Hijano:2020szl . Our Ward
identity result has a natural interpretation of being defined at “null
infinity” on this patch near the center of AdS Hijano:2020szl ;
Compere:2019bua , providing the leading $1/l^{2}$ corrections.

The current paper generalizes our previous work Banerjee:2020dww  on the
effect of the small AdS potential on the classical soft graviton theorem by
including an electromagnetic interaction. We also find the effect of the small
AdS potential on the classical soft photon theorem. The paper is organised as
follows: In section 2, we review basic properties of AdS Reissner-Nordström
spacetime and then study its perturbations by introducing a charged point
probe particle. In section 3, we have obtained the solution to the gauge and
gravity radiations. Next in section 4, we study the soft limit and extract the
classical soft photon factor from classical radiation profile. In taking a
similar limit, in section 5 we state the results for the classical soft
graviton factor. It turns out that the charge of the central black hole,
considered as the scatterer in the classical probe scattering process, has no
explicit effect on the soft graviton factor. Finally in section 6, we find the
Ward identity of large gauge transformations, perturbatively modified to
$1/l^{2}$ order using our classical soft photon factors that we derived in
section 4. We end the paper with a conclusion and some interesting open
questions in section 7. Appendix A contains the computation details for the
classical soft graviton factor.

##  2 Perturbations of AdS Reissner-Nordström spacetime

In this paper, we are interested in studying the classical soft photon theorem
in asymptotically AdS backgrounds. To achieve this, we study the classical
scattering of a charged and massive probe particle by a Reissner-Nordström
black hole placed in an asymptotically AdS spacetime in 4 spacetime
dimensions. The equations of motions result from the action, which consists of
the Einstein-Hilbert term and the Maxwell term,

$S=\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}(R-2\Lambda)-\frac{1}{16\pi}\int
d^{4}x\sqrt{-g}~{}F_{\mu\nu}F^{\mu\nu}.$ (4)

In 4 $R$ is the Ricci scalar for metric $g_{\mu\nu}$, $\Lambda$ is the
cosmological constant, $G$ is Newton’s constant and
$F_{\mu\nu}=A_{\nu,\mu}-A_{\mu,\nu}$ is the field strength tensor of the
electromagnetic field $A_{\mu}$. 222The Maxwell action is written using
Heaviside units and further details can be found in the Appendix E of
Wald:1984rg . We use the standard convention of denoting partial derivatives
by subscripted commas and covariant derivatives with semi-colons. Varying the
action in 4 with respect to the metric tensor one gets the Einstein equations,

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi GT_{\mu\nu}^{EM},$
(5)

where

$\displaystyle
T_{\mu\nu}^{EM}=\frac{1}{4\pi}\left(F_{\mu\alpha}F_{\nu\beta}g^{\alpha\beta}-\frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F_{\gamma\delta}g^{\alpha\gamma}g^{\beta\delta}\right)\,.$

Similarly for the gauge field $A_{\mu}$ we get the source-free Maxwell
equations

$\frac{\sqrt{-g}}{4\pi}F^{\mu\nu}{}_{;\nu}=0\,.$ (6)

The solutions of equations 5 and 6 for a static spherically symmetric
spacetime with mass M, charge Q and a negative cosmological constant
$\Lambda=-3/l^{2}$, provide the metric

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (7)

with a gauge potential

$A_{0}=\frac{Q}{r}\,.$ (8)

The lapse function $f(r)$ in global coordinates takes the form

$f(r)=1-\frac{2GM}{r}+\frac{GQ^{2}}{r^{2}}-\Lambda\frac{r^{2}}{3}=1-\frac{2GM}{r}+\frac{GQ^{2}}{r^{2}}+\frac{r^{2}}{l^{2}}.$
(9)

Since we are interested in studying the radiation emitted by the scattering of
a probe particle moving in an unbounded trajectory on the spacetime (from the
point of view of an asymptotic observer) we introduce isotropic coordinates.
We refer the reader to Banerjee:2020dww  for further justification on choosing
this particular coordinate system. In these coordinates, the resulting
radiation will be isotropic in all spatial directions. We assume that the
probe particle with mass $m\,(\ll M)$ and charge $q\,(\ll Q)$ has a large
impact parameter from the black hole which implies $GM/r\ll 1$ and
$\sqrt{G}Q/r\ll 1$. In addition, we also truncate our metric up to $1/l^{2}$
terms, as we consider radiative solutions in the large cosmological constant
limit. Therefore our result will be valid in the regime $\sqrt{G}Q\leq GM\ll
r\ll l$.333The equality of $\sqrt{G}Q$ and $GM$ holds in the extremal limit

The metric 7 in isotropic coordinates, expanded up to quadratic order in
$\rho$, takes the form

$ds^{2}=-g_{00}dt^{2}+g_{ij}dx^{i}dx^{j},$ (10)

where

$\displaystyle
g_{00}=-\left(1-\frac{2GM}{\rho}+\frac{GQ^{2}}{\rho^{2}}+\frac{\rho^{2}}{l^{2}}\right)\,,\qquad
g_{0i}$ $\displaystyle=0\,,\qquad
g_{ij}=\delta_{ij}\left(1+\frac{2GM}{\rho}-\frac{GQ^{2}}{\rho^{2}}+\frac{\rho^{2}}{2l^{2}}\right)\,.$
(11)

Here $(i,j)=1,2,3$, run over spatial directions and $\rho=|\vec{x}|$. The
isotropic coordinate $\rho$ is related to the Schwarzschild coordinate ‘$r$’
by

$\rho=r\left(1-\frac{GM}{r}+\frac{GQ^{2}}{2r^{2}}-\frac{r^{2}}{4l^{2}}\right)\,.$
(12)

We now impose the assumptions discussed above to express the metric of 11 in a
form relevant for our calculations. We set $8\pi G=1$ in the following.
Therefore we will replace $G$ by $1/8\pi$ in the remainder of the paper. The
condition of a large impact parameter amounts to considering the leading order
contribution of the gravitational and electromagnetic potential, which we will
denote by $\phi\left(\vec{x}\right)$. The potential goes like $r^{-1}$ and can
be defined either with respect to the mass or the charge. The analysis in this
paper is independent of either choice. We define

$\phi\left(\vec{x}\right)=-\frac{M}{8\pi\rho}\,.$ (13)

The gauge potential can then be expressed as

$A_{0}(\vec{x})=\frac{Q}{\rho}=-\frac{8\pi Q}{M}\phi\left(\vec{x}\right),$
(14)

Retaining terms up to leading order in $\phi$ and $1/l^{2}$, we then find that
the metric components in 11 take the form

$\displaystyle g_{00}=-\left(1+2\phi+\frac{\rho^{2}}{l^{2}}\right)\,,\qquad
g_{0i}=0\,,\qquad
g_{ij}=\delta_{ij}\left(1-2\phi+\frac{\rho^{2}}{2l^{2}}\right)\,.$ (15)

Note that the spacetime metric in 15 provides the leading AdS correction about
an asymptotically flat spacetime and in isotropic coordinates it just behaves
like the AdS-Schwarzschild metric with a gauge potential. The metric is
however equivalent to the metric in 7 up to leading order in $\phi$ and
$1/l^{2}$. In particular, the timelike boundary of the full AdS spacetime is
not part of the spacetime we have considered. In the remaining sections, we
shall investigate the scattering of a charged probe particle in this
background given by the metric in 15 and gauge field in 14.

###  2.1 Perturbations of Einstein-Maxwell equations

We now linearly perturb the spacetime by introducing a point probe particle
with mass ‘$m$’ and charge ‘$q$’ moving along a worldline trajectory
$r(\sigma)$, DeWitt:1960fc ; Peters:1966 ; Peters:1970mx ; Kovacs:1977uw ;
Poisson:2011nh  whose action is

$S_{P}=-m\int
d\sigma\sqrt{-g_{\mu\nu}\frac{dr^{\mu}}{d\sigma}\frac{dr^{\nu}}{d\sigma}}+\frac{q}{4\pi}\int
d\sigma A_{\mu}\frac{dr^{\mu}}{d\sigma}\,.$ (16)

where $\frac{dr^{\mu}}{d\sigma}=u^{\mu}$ is the tangent to the worldline of
the probe and the metric is evaluated at $r$. Variation of 16 gives the
following stress tensor $T^{\mu\nu}_{(P)}$ 444We have chosen $\sigma$ to be
proper time as measured in this spacetime and therefore $u^{\mu}$ satisfies
the relation $g_{\mu\nu}u^{\mu}u^{\nu}=-1$ and current $J^{\mu}_{(P)}$

$\displaystyle T^{\mu\nu}_{(P)}$
$\displaystyle=\frac{2}{\sqrt{-g}}\frac{\delta S_{P}}{\delta
g_{\mu\nu}}=m\int\delta(x,r(\sigma))\frac{dr^{\mu}}{d\sigma}\frac{dr^{\nu}}{d\sigma}\,d\sigma\,,$
$\displaystyle J_{(P)}^{\mu}$ $\displaystyle=\frac{1}{\sqrt{-g}}\frac{\delta
S_{P}}{\delta
A_{\mu}}=\frac{q}{4\pi}\int\delta(x,r(\sigma))\frac{dr^{\mu}}{d\sigma}\,d\sigma\,,$
(17)

where $\delta(x,r(\sigma))$ is the covariant delta function. It is related to
the flat spacetime delta function $\delta^{4}\left(x-r(\sigma)\right)$ via

$\delta(x,r(\sigma))\sqrt{-g}=\delta^{4}\left(x-r(\sigma)\right)=\delta(t-r^{0}(\sigma))\delta^{(3)}(\vec{x}-\vec{r}(\sigma))\,,$
(18)

and normalized as

$\int\sqrt{-g}~{}\delta(x,r(\sigma))d\sigma=1\,.$ (19)

The stress-energy tensor and current of the point particle induces a
perturbation of the background metric and gauge potential

$\displaystyle g_{\mu\nu}$ $\displaystyle\to g_{\mu\nu}+\delta
g_{\mu\nu}=g_{\mu\nu}+2h_{\mu\nu}\,,$ $\displaystyle A_{\mu}$
$\displaystyle\to A_{\mu}+\delta A_{\mu}=A_{\mu}+a_{\mu}\,.$ (20)

The variations of 5 and 6 yield

$\displaystyle\delta\tilde{G}_{\mu\nu}-\delta T^{h}_{\mu\nu}-\delta
T^{a}_{\mu\nu}$ $\displaystyle=T_{\mu\nu}^{(P)}\,,$ (21)
$\displaystyle\delta(F^{\mu\nu};_{\nu})$ $\displaystyle=4\pi J^{\mu}_{(P)}\,,$
(22)

where

$\tilde{G}_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\Lambda g_{\mu\nu}\,.$
(23)

In 21 we have split the total perturbation of the stress-energy tensor into
two components, one part $\delta T^{h}_{\mu\nu}$ is due to the perturbation of
the metric and another part $\delta T^{a}_{\mu\nu}$ is due to the perturbation
of the gauge potential.

On simplifying $\delta\tilde{G}_{\mu\nu}\,,\delta T^{h}_{\mu\nu}\,,\delta
T^{a}_{\mu\nu}$ and $\delta(F^{\mu\nu};_{\nu})$, we find

$\displaystyle\delta\tilde{G}_{\mu\nu}$
$\displaystyle=-e_{\mu\nu;\alpha}{}^{\alpha}+e_{\mu\alpha;}{}^{\alpha}{}_{\nu}+e_{\nu\alpha;\phantom{\alpha}\mu}^{\phantom{\nu\alpha;}\alpha}+\left(R_{\nu}{}^{\delta}e_{\delta\mu}+R_{\mu}{}^{\delta}e_{\delta\nu}\right)+2R^{\alpha}{}_{\nu\mu}{}^{\delta}e_{\delta\alpha}$
$\displaystyle\quad-
g_{\mu\nu}e_{\alpha\beta;}{}^{\alpha\beta}+g_{\mu\nu}R^{\alpha\beta}e_{\alpha\beta}-Re_{\mu\nu}+2\Lambda
e_{\mu\nu}-\Lambda g_{\mu\nu}e\,,$ $\displaystyle\delta T^{h}_{\mu\nu}$
$\displaystyle=eT_{\mu\nu}^{EM}-\frac{1}{8\pi}e_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}-\frac{1}{2\pi}g^{\alpha\epsilon}g^{\beta\delta}e_{\epsilon\delta}\left(F_{\alpha\mu}F_{\beta\nu}-\frac{1}{4}g_{\mu\nu}F_{\alpha\gamma}F_{\beta}{}^{\gamma}\right)\,,$
$\displaystyle\delta T^{a}_{\mu\nu}$
$\displaystyle=\frac{1}{4\pi}g^{\alpha\beta}\left(f_{\alpha\mu}F_{\beta\nu}+f_{\alpha\nu}F_{\beta\mu}-\frac{1}{2}g_{\mu\nu}g^{\gamma\delta}f_{\alpha\gamma}F_{\beta\delta}\right)\,,$
$\displaystyle\delta(F^{\mu\nu};_{\nu})$
$\displaystyle=-g^{\alpha\rho}g^{\mu\nu}\left[2g^{\beta\sigma}\left(e_{\rho\sigma}F_{\nu\beta;\alpha}+e_{\sigma\nu;\alpha}F_{\beta\rho}\right)-e_{,\rho}F_{\nu\alpha}-f_{\nu\rho;\alpha}-2F_{\alpha\nu}e_{\rho\beta;}{}^{\beta}\right]\,,$
(24)

where we have denoted the perturbed electromagnetic field strength tensor by

$f_{\mu\nu}=a_{\nu,\mu}-a_{\mu,\nu}\,,$ (25)

and have introduced the trace-reversed metric perturbations $e_{\mu\nu}$
defined by

$e_{\mu\nu}=h_{\mu\nu}-\frac{1}{2}hg_{\mu\nu}\,;\qquad
h=g^{\mu\nu}h_{\mu\nu}=-e=-g^{\mu\nu}e_{\mu\nu}\,.$ (26)

Substituting the first three expressions of 24 in 21, we find the following
expression for the perturbed Einstein equation

$\displaystyle-T^{(P)}_{\mu\nu}$
$\displaystyle=e_{\mu\nu;\alpha}{}^{\alpha}-e_{\mu\alpha;}{}^{\alpha}{}_{\nu}-e_{\nu\alpha;\phantom{\alpha}\mu}^{\phantom{\nu\alpha;}\alpha}-\left(R_{\nu}{}^{\delta}e_{\delta\mu}+R_{\mu}{}^{\delta}e_{\delta\nu}\right)-2R^{\alpha}{}_{\nu\mu}{}^{\delta}e_{\delta\alpha}+Re_{\mu\nu}-2\Lambda
e_{\mu\nu}$ $\displaystyle\quad+\Lambda
g_{\mu\nu}e-g_{\mu\nu}R^{\alpha\beta}e_{\alpha\beta}+g_{\mu\nu}e_{\alpha\beta;}{}^{\alpha\beta}+\frac{1}{4\pi}g^{\alpha\beta}\left(f_{\alpha\mu}F_{\beta\nu}+f_{\alpha\nu}F_{\beta\mu}-\frac{1}{2}g_{\mu\nu}g^{\gamma\delta}f_{\alpha\gamma}F_{\beta\delta}\right)$
$\displaystyle\qquad+eT_{\mu\nu}^{EM}-\frac{1}{8\pi}e_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}-\frac{1}{2\pi}g^{\alpha\epsilon}g^{\beta\delta}e_{\epsilon\delta}\left(F_{\alpha\mu}F_{\beta\nu}-\frac{1}{4}g_{\mu\nu}F_{\alpha\gamma}F_{\beta}{}^{\gamma}\right)\,.$
(27)

Similarly plugging the expression of $\delta(F^{\mu\nu};_{\nu})$ from 24 in 22
gives the perturbed Maxwell equation

$-4\pi
g^{\mu\nu}J_{\nu}^{(P)}=g^{\alpha\rho}g^{\mu\nu}\left[2g^{\beta\sigma}\left(e_{\rho\sigma}F_{\nu\beta;\alpha}+e_{\sigma\nu;\alpha}F_{\beta\rho}\right)-e_{,\rho}F_{\nu\alpha}-f_{\nu\rho;\alpha}-2F_{\alpha\nu}e_{\rho\beta;}{}^{\beta}\right].$
(28)

We will now express 27 and 28 about the background with the metric 15 and
gauge potential 14. We will also rewrite parts of the equations in terms of
the following quantities,

$k_{\mu}=e_{\mu\nu;}{}^{\nu}\,,\qquad b=-a_{0,0}+a_{i,i}\,.$ (29)

The radiative components of gravitational and electromagnetic perturbations
are spatial in isotropic coordinates. The spatial components of 27 are

$\displaystyle-T^{(P)}_{ij}$
$\displaystyle=\Box\left(\left(1+2\phi-\frac{\rho^{2}}{2l^{2}}\right)e_{ij}\right)-k_{i,j}-k_{j,i}-2(\left(\phi_{,k}-\frac{\rho^{2}_{,k}}{4l^{2}}\right)e_{ki})_{,j}-2(\left(\phi_{,k}-\frac{\rho^{2}_{,k}}{4l^{2}}\right)e_{kj})_{,i}$
$\displaystyle\quad-\left(k_{0,0}-k_{l,l}\right)\delta_{ij}+4\left[\phi
e_{ij,00}+\phi_{,i}e_{0j,0}+\phi_{,j}e_{i0,0}+\frac{1}{2}\left(\phi_{,ij}-\frac{1}{2}\phi_{,kk}\delta_{ij}\right)\left(e_{00}+e_{ll}\right)\right]$
$\displaystyle\qquad+2\delta_{ij}\left(\phi_{,kl}e_{kl}-\phi_{,k}k_{k}+2\phi
k_{0,0}\right)+\frac{\delta_{ij}}{4l^{2}}\left(\rho^{2}_{,kl}e_{kl}+\rho^{2}_{,k}k_{k}+2\rho^{2}k_{0,0}\right)$
$\displaystyle\qquad\;+\frac{1}{2l^{2}}\left[\rho^{2}e_{ij,00}+\rho^{2}_{,i}e_{j0,0}+\rho^{2}_{,j}e_{i0,0}+\rho_{,ij}^{2}(2e_{00}-e_{ll})-\frac{1}{2}\delta_{ij}\rho^{2}_{,kk}e_{ll}\right.$
$\displaystyle\left.\qquad\quad+\frac{3}{2}(\rho^{2}_{,ki}e_{kj}+\rho^{2}_{,kj}e_{ki})-\rho^{2}_{,kk}e_{ij}+\frac{3}{2}\rho_{,k}^{2}e_{ij,k}\right]-\frac{2Q}{M}\left(f_{0i}\phi_{,j}+f_{0j}\phi_{,i}-\delta_{ij}f_{0l}\phi_{,l}\right),$
(30)

where $\Box=-\partial_{0}^{2}+\partial_{i}^{2}$. The spatial component of
perturbed Maxwell equation 28 in the AdS-Reissner-Nordström background gives

$\displaystyle-4\pi
J^{i}_{(P)}=-4\pi\left(1+2\phi-\frac{\rho^{2}}{2l^{2}}\right)J_{i}^{(P)}$
$\displaystyle\,=\Box
a_{i}-b_{,i}+\frac{3\rho^{2}}{2l^{2}}\left(a_{i,00}-a_{0,0i}\right)+\left(4\phi-\frac{\rho^{2}}{l^{2}}\right)\left(a_{i,kk}-a_{k,ki}\right)-2\left(\phi_{,k}+\frac{\rho^{2}_{,k}}{8l^{2}}\right)f_{ik}$
$\displaystyle\qquad+\frac{16\pi
Q}{M}\left(e_{j0}\phi_{,ij}+\left(e_{ij,0}-e_{i0,j}\right)\phi_{,j}+\frac{1}{2}\left(e_{00,0}-e_{kk,0}\right)\phi_{,i}+\phi_{,i}k_{0}\right).$
(31)

We now need to implement gauge choices. To this end, we adopt the following
choice for $k_{\mu}$ and $b$ to simplify equations 30 and 31

$\displaystyle k_{\mu}$
$\displaystyle=-2\left(\phi_{,k}-\frac{\rho_{,k}^{2}}{4l^{2}}\right)e_{k\mu}+\frac{2Q}{M}a_{0}\phi_{,\mu}\,,$
(32) $\displaystyle b$
$\displaystyle=-2\left(\phi_{,k}+\frac{\rho^{2}_{,k}}{8l^{2}}\right)a_{k}-4\left(\phi+\frac{\rho^{2}}{8l^{2}}\right)a_{k,k}.$
(33)

Using 32 one can simplify 30 to

$\displaystyle\Box\left(\left(1+2\phi-\frac{\rho^{2}}{2l^{2}}\right)e_{ij}\right)+4\left[\phi
e_{ij,00}+\phi_{,i}e_{0j,0}+\phi_{,j}e_{i0,0}+\frac{1}{2}\left(\phi_{,ij}-\frac{1}{2}\phi_{,kk}\delta_{ij}\right)\left(e_{00}+e_{ll}\right)\right]$
$\displaystyle+\frac{1}{2l^{2}}\left[\rho^{2}e_{ij,00}+\rho^{2}_{,i}e_{j0,0}+\rho^{2}_{,j}e_{i0,0}+\rho_{,ij}^{2}(2e_{00}-e_{ll})-\frac{1}{2}\delta_{ij}\rho^{2}_{,kk}e_{ll}+\frac{3}{2}(\rho^{2}_{,ki}e_{kj}+\rho^{2}_{,kj}e_{ki})+\frac{3}{2}\delta_{ij}\rho^{2}_{,kl}e_{kl}\right.$
$\displaystyle\left.-\rho^{2}_{,kk}e_{ij}+\frac{3}{2}\rho_{,k}^{2}e_{ij,k}\right]-\frac{2Q}{M}\left[a_{i,0}\phi_{,j}+a_{j,0}\phi_{,i}-\delta_{ij}a_{l,0}\phi_{,l}+2a_{0}\left(\phi_{,ij}-\frac{1}{2}\delta_{ij}\phi_{,kk}\right)\right]=-T^{(P)}_{ij}.$
(34)

Similarly for the perturbed Maxwell equation 31 we use both 32 and 33 to get

$\displaystyle-4\pi\left(1-\phi+\frac{5\rho^{2}}{8l^{2}}\right)J_{i}^{(P)}=\Box\left(\left(1+\phi+\frac{\rho^{2}}{8l^{2}}\right)a_{i}\right)$
$\displaystyle\quad+4\left[\left(\phi+\frac{\rho^{2}}{8l^{2}}\right)a_{i,00}+\left(\phi_{,i}+\frac{\rho^{2}_{,i}}{8l^{2}}\right)a_{0,0}\right]+2\left(\phi_{,ki}+\frac{\rho^{2}_{,ki}}{8l^{2}}\right)a_{k}-\left(\phi_{,kk}+\frac{\rho_{,kk}^{2}}{8l^{2}}\right)a_{i}$
$\displaystyle\qquad+\frac{16\pi
Q}{M}\left[e_{j0}\phi_{,ij}+\left(e_{ij,0}-e_{i0,j}\right)\phi_{,j}+\frac{1}{2}\left(e_{00,0}-e_{kk,0}\right)\phi_{,i}\right]\,.$
(35)

Note that for both the equations 34 and 35 we keep the terms up to leading
order in $\phi$ and $1/l^{2}$. In the next section we will solve 34 and 35 for
$e_{ij}$ and $a_{i}$ in frequency space using the worldline formalism. The
result will involve a Green’s function for the $1/l^{2}$ correction which was
previously derived in Banerjee:2020dww .

##  3 Solutions of the field equations

To solve 34 and 35 we first briefly review the solution for a perturbed scalar
field equation. For an arbitrary source $f(\sigma)$, the solution of the
following scalar box equation in a curved spacetime

$\psi_{;\alpha}{}^{\alpha}(x)=-\int\delta(x,r(\sigma))f(\sigma)d\sigma\,.$
(36)

can be written as

$\displaystyle\psi^{(1)}(x)$ $\displaystyle=\psi^{(0)}(x)+\delta\psi^{(0)}(x)$
$\displaystyle=\frac{1}{4\pi}\int\limits_{-\infty}^{\sigma_{0}}\delta\left(-\Omega\left(x,r(\sigma)\right)\right)f(\sigma)d\sigma$
$\displaystyle\;+\frac{1}{16\pi^{2}}\int\sqrt{-g(y)}\delta\left(-\Omega\left(x,y\right)\right)d^{4}y\int\limits_{-\infty}^{\sigma_{0}}\delta^{\prime}\left(-\Omega\left(y,r(\sigma)\right)\right)F\left(y,r(\sigma)\right)f(\sigma)d\sigma\,.$
(37)

where $\Omega\left(x,r(\sigma)\right)$ is the Synge world function and
$F\left(x,r(\sigma)\right)$ is the Ricci tensor dependent term which arises
from derivatives of the world function 555For a more detailed discussion on
the world function in the context of our derivation we refer the reader to
Banerjee:2020dww .

$\displaystyle\Omega(x,r)=\frac{1}{2}(u_{1}-u_{0})\int_{u_{0}}^{u_{1}}~{}g_{\alpha\beta}U^{\alpha}U^{\beta}~{}du,$
(38) $\displaystyle
F\left(x,r\right)=\frac{1}{u_{1}-u_{0}}\int\limits_{u_{0}}^{u_{1}}\left(u-u_{0}\right)^{2}R_{\mu\nu}U^{\mu}U^{\nu}du\,.$
(39)

In 38 and 39 we assume that the observer ($x$) and probe particle source ($r$)
are joined by a unique geodesic $\xi^{\alpha}$ with affine parameter ‘u’ and
$U^{\alpha}=\frac{d\xi^{\alpha}}{du}$ is the tangent vector to the geodesic.
Gravitational and electromagnetic radiation follow this path from the source
to the observer. $R_{\mu\nu}$ in 39 gets the contribution from the black hole,
which in our scattering approximation can be treated as a point particle with
mass $M$ and charge $Q$. The integration limit is chosen up to $\sigma_{0}$
instead of infinity $(\infty)$. This ensures that $r^{\mu}(\sigma_{0})$ lies
outside the light cone centred at $x^{\mu}$ and the contribution to the scalar
pertubation $\psi$ only comes from the retarded part of the Green’s function.

Expanding 36 in terms of the d’Alembertian operator we get

$\displaystyle\Box\psi^{(1)}+4\phi\partial_{t}^{2}\psi^{(0)}-\frac{\rho^{2}}{4l^{2}}\partial_{t}^{2}\psi^{(0)}$
$\displaystyle+\frac{3}{4l^{2}}\rho^{2}_{k}\partial_{k}\psi^{(0)}+\frac{3\rho^{2}}{4l^{2}}\partial_{k}^{2}\psi^{(0)}$
$\displaystyle=-\int\delta^{4}\left(x-z(\sigma)\right)f(\sigma)d\sigma+\mathcal{O}\left(R^{2}\right).$
(40)

We can solve 40 by substituting 37 and performing a Fourier transformation

$\tilde{\psi}^{(1)}\left(\omega,\vec{x}\right)=\int dte^{i\omega
t}\psi^{(1)}\left(t,\vec{x}\right).$ (41)

The transformed field $\tilde{\psi}$ can be perturbatively solved about flat
spacetime. The solution of $\tilde{\psi}$ that are leading order in $\phi$ and
$1/l^{2}$ provide tail contributions to the flat spacetime Green’s function,
which arises due to the black hole potential and AdS potential. Denoting these
tail terms as $G_{M}$ and $G_{l}$, they have the solutions Fernandes:2020tsq ;
Banerjee:2020dww

$\displaystyle G_{M}\left(\omega,\vec{x},\vec{r}\right)$
$\displaystyle=-\frac{iM}{16\pi\omega}\left(\frac{e^{i\omega
R_{0}}\Gamma\left(\vec{x},\vec{r}\right)}{R_{0}}-\int\limits_{0}^{\infty}dv\frac{e^{i\omega\left(v+|\vec{z}|+\rho(v)\right)}}{\left(v+|\vec{z}|\right)\rho(v)}\right)\,,$
(42) $\displaystyle G_{l}\left(\omega,\vec{x},\vec{r}\right)$
$\displaystyle=-~{}\frac{4i}{\omega}~{}e^{i\omega
R_{0}}\left(\vec{x}.\vec{r}\right)\,,$ (43)

where $G_{M}$ and $G_{l}$ correspond to the contribution due to the black hole
mass and AdS radius respectively, $\vec{R}_{0}=\vec{x}-\vec{r}(\sigma)$ with
magnitude $R_{0}=|\vec{R}_{0}|$, and the following expressions for
$\Gamma\left(\vec{x},\vec{r}\right)$ and $\rho(v)$

$\Gamma\left(\vec{x},\vec{r}\right)=\text{ln}\left(\frac{|\vec{x}|R_{0}+\vec{x}.\vec{R}_{0}}{|\vec{r}|R_{0}+\vec{r}.\vec{R}_{0}}\right)\,,\qquad\rho(v)=\sqrt{x^{2}+v^{2}+\frac{2v\vec{x}.\vec{r}}{|\vec{r}|}}\,.$
(44)

The equations for the perturbed fields $e_{ij}$ and $a_{i}$ have additional
contributions from the background apart from those in 40 due to their
respective tensor and vector nature. These additional contributions are
frequency space solutions in terms of the derivatives of 42 and 43

$-\nabla_{i}\tilde{\Box}G_{M}\left(\omega,\vec{x},\vec{r}\right)=\phi_{,i}\frac{e^{i\omega
R_{0}}}{R_{0}}\,,\quad-\nabla_{i}\nabla_{k}\tilde{\Box}G_{M}\left(\omega,\vec{x},\vec{r}\right)=\phi_{,ik}\frac{e^{i\omega
R_{0}}}{R_{0}}\,,$ (45)

and

$-\nabla_{i}\tilde{\Box}G_{l}\left(\omega,\vec{x},\vec{r}\right)=\rho^{2}_{,i}\frac{e^{i\omega
R_{0}}}{R_{0}}\,,\quad-\nabla_{i}\nabla_{k}\tilde{\Box}G_{l}\left(\omega,\vec{x},\vec{r}\right)=\rho^{2}_{,ik}\frac{e^{i\omega
R_{0}}}{R_{0}}\,,$ (46)

where $\nabla_{i}=\frac{\partial}{\partial x^{i}}+\frac{\partial}{\partial
r^{i}}$ is an operator that acts on the two spatial arguments in the Green’s
functions and $\tilde{\Box}=(\omega^{2}+\partial_{i}^{2})$.

We can now derive the frequency space solutions $\tilde{e}_{ij}$ and
$\tilde{a}_{i}$. This requires substituting $e_{\mu\nu}$ and $a_{\mu}$ in
terms of their Fourier transformed fields
$\tilde{e}_{\mu\nu}\left(\omega\,,\vec{x}\right)$ and
$\tilde{a}_{\mu}\left(\omega\,,\vec{x}\right)$ in 34 and 35. In the case of
34, we find

$\displaystyle-T^{(P)}_{ij}$ $\displaystyle=\int d\omega\,e^{-i\omega
t}\,\widetilde{\Box}\left(\left(1+2\phi-\frac{\rho^{2}}{2l^{2}}\right)\tilde{e}_{ij}\right)$
$\displaystyle-\int d\omega\,e^{-i\omega
t}\,4\left[\omega^{2}\phi\tilde{e}_{ij}+i\omega\left(\phi_{,i}\tilde{e}_{j0}+\phi_{,j}\tilde{e}_{i0}\right)-\frac{1}{2}\left(\phi_{,ij}-\frac{1}{2}\phi_{,kk}\delta_{ij}\right)\left(\tilde{e}_{00}+\tilde{e}_{ll}\right)\right]$
$\displaystyle-\frac{1}{2l^{2}}\left[\omega^{2}\rho^{2}\tilde{e}_{ij}+i\omega\rho^{2}_{,i}\tilde{e}_{j0}+i\omega\rho^{2}_{,j}\tilde{e}_{i0}-\rho_{,ij}^{2}(2\tilde{e}_{00}-\tilde{e}_{ll})-\frac{3}{2}\delta_{ij}\rho^{2}_{,kl}\tilde{e}_{kl}+\frac{1}{2}\delta_{ij}\rho^{2}_{,kk}\tilde{e}_{ll}\right.$
$\displaystyle\left.\qquad-\frac{3}{2}(\rho^{2}_{,ki}\tilde{e}_{kj}+\rho^{2}_{,kj}\tilde{e}_{ki})-\rho^{2}_{,kk}\tilde{e}_{ij}-\frac{3}{2}\rho_{,k}^{2}\tilde{e}_{ij,k}\right]$
$\displaystyle\qquad+\frac{2Q}{M}\int d\omega\,e^{-i\omega
t}\left[i\omega\left(\tilde{a}_{i}\phi_{,j}+\tilde{a}_{j}\phi_{,i}-\delta_{ij}\tilde{a}_{l}\phi_{,l}\right)-2\tilde{a}_{0}\left(\phi_{,ij}-\frac{1}{2}\delta_{ij}\phi_{,kk}\right)\right]\,,$
(47)

where

$\displaystyle
T^{(P)}_{ij}=m\int\delta_{ki}\delta_{lj}~{}\frac{\delta^{4}(x-r(\sigma))}{1+2\phi(\vec{r})+\frac{r^{2}}{4l^{2}}}\frac{dr^{k}}{d\sigma}\frac{dr^{l}}{d\sigma}\,d\sigma\,.$
(48)

Similar steps for the perturbed Maxwell equation 35 gives

$\displaystyle-\left(1-\phi+\frac{5\rho^{2}}{8l^{2}}\right)4\pi
J^{(P)}_{i}=-q\int\delta_{ki}\frac{\delta^{4}(x-r(\sigma))}{1+\phi\left(\vec{r}\right)+\frac{r^{2}}{8l^{2}}}\frac{dr^{k}}{d\sigma}\,d\sigma$
$\displaystyle\hskip 100.00015pt=\int d\omega\,e^{-i\omega
t}\,\widetilde{\Box}\left(\left(1+\phi+\frac{\rho^{2}}{8l^{2}}\right)\tilde{a}_{i}\right)$
$\displaystyle\hskip 50.00008pt-\int d\omega\,e^{-i\omega
t}\left[4\left(\omega^{2}\tilde{a}_{i}\phi+i\omega\tilde{a}_{0}\phi_{,i}\right)+\tilde{a}_{i}\phi_{,kk}-2\tilde{a}_{k}\phi_{,ik}\right]$
$\displaystyle\hskip 50.00008pt+\int d\omega\,e^{-i\omega t}\frac{16\pi
Q}{M}\left(\tilde{e}_{0j}\phi_{,ij}-\left(i\omega\tilde{e}_{ij}+\tilde{e}_{i0,j}\right)\phi_{,j}-\frac{1}{2}\left(i\omega\tilde{e}_{00}-i\omega\tilde{e}_{kk}\right)\phi_{,i}\right)$
$\displaystyle\hskip 100.00015pt-\int d\omega\,e^{-i\omega
t}\left[4\left(\omega^{2}\tilde{a}_{i}\frac{\rho^{2}}{8l^{2}}+i\omega\tilde{a}_{0}\frac{\rho^{2}_{,i}}{8l^{2}}\right)+\tilde{a}_{i}\frac{\rho^{2}_{,kk}}{8l^{2}}-\tilde{a}_{k}\frac{\rho^{2}_{,ik}}{4l^{2}}\right]\,.$
(49)

We get the solution for $\tilde{e}_{ij}$ from the scalar perturbation solution
comparing $\tilde{\psi}^{(0)}_{0}(\omega\,,\vec{x})$ with
$\left(1+2\phi\left(\vec{x}\right)-\frac{x^{2}}{2l^{2}}\right)\tilde{e}_{ij}^{(0)}\left(\omega\,,\vec{x}\right)$
and replacing $f(\sigma)$ with
$2m\delta_{ki}\delta_{lj}\left(1-2\phi\left(\vec{r}\right)-\frac{r^{2}}{4l^{2}}\right)\frac{dr^{k}}{d\sigma}\frac{dr^{l}}{d\sigma}$.
The zeroth order solution of $\tilde{e}_{ij}(\omega\,,\vec{x})$ in frequency
space is

$\tilde{e}^{(0)}_{ij}(\omega\,,\vec{x})=m\int\frac{e^{i\omega\left(r^{0}+R_{0}\right)}}{4\pi
R_{0}}v_{i}v_{j}\,\frac{dr^{0}}{d\sigma}dr^{0}\,,$ (50)

where we have denoted $\frac{dr^{k}}{dr^{0}}$ as $v^{k}$.

Likewise, comparing $\tilde{\psi}^{(0)}_{0}(\omega\,,\vec{x})$ to
$\left(1+\phi\left(\vec{x}\right)+\frac{x^{2}}{8l^{2}}\right)\tilde{a}_{i}\left(\omega\,,\vec{x}\right)$
and replacing $f(\sigma)$ with
$q\delta_{ki}\left(1-\phi\left(\vec{r}\right)-\frac{r^{2}}{8l^{2}}\right)\frac{dr^{k}}{d\sigma}$,
we find the following zeroth order solution
$\tilde{a}^{(0)}_{i}(\omega\,,\vec{x})$ in Fourier space

$\tilde{a}^{(0)}_{i}(\omega\,,\vec{x})=q\int\frac{e^{i\omega\left(r^{0}+R_{0}\right)}}{4\pi
R_{0}}v_{i}\,dr^{0}\,.$ (51)

We can further compute the other components of gravitational and
electromagnetic perturbations from $\tilde{e}_{ij}$ and $\tilde{a}_{i}$, as
they are related among themselves by the gauge fixing condition. It follows
from our choice in 32 and 33 that 29 on flat spacetime simplifies to

$\displaystyle e_{ij,j}-e_{i0,0}$ $\displaystyle=0\,,\qquad
e_{0i,i}-e_{00,0}=0\,,$ (52) $\displaystyle a_{0,0}-a_{i,i}=0.$ (53)

These are simply the de Donder and Lorenz gauges in flat spacetime. By Fourier
transforming 52 and using 50, we can now derive the following zeroth order
solutions of $\tilde{e}_{i0}$ and $\tilde{e}_{00}$

$\displaystyle\tilde{e}^{(0)}_{i0}(\omega\,,\vec{x})$
$\displaystyle=-m\int\frac{e^{i\omega\left(r^{0}+R_{0}\right)}}{4\pi
R_{0}}v_{i}\,\frac{dr^{0}}{d\sigma}dr^{0}+\mathcal{O}(\phi)\,,$
$\displaystyle\tilde{e}^{(0)}_{00}(\omega\,,\vec{x})$
$\displaystyle=m\int\frac{e^{i\omega\left(r^{0}+R_{0}\right)}}{4\pi
R_{0}}\,\frac{dr^{0}}{d\sigma}dr^{0}+\mathcal{O}(\phi)\,.$ (54)

Using the Fourier transform of 53, we can similarly use 51 to determine the
solution for electromagnetic perturbation $\tilde{a}^{(0)}_{0}$

$\displaystyle\tilde{a}^{(0)}_{0}(\omega\,,\vec{x})$
$\displaystyle=-q\int\frac{e^{i\omega\left(r^{0}+R_{0}\right)}}{4\pi
R_{0}}dr^{0}+\mathcal{O}(\phi)\,.$ (55)

Hence the gauge conditions determine all the lowest order expressions. To find
the complete solution, we first substitute all zeroth order solutions 50, 54,
51 and 55 in all terms that are coefficients of $\phi$ and $1/l^{2}$ in 47. We
then use the expressions in 46 and 45 to determine the following solution for
$\tilde{e}_{ij}\left(\omega,\vec{x}\right)$

$\displaystyle\tilde{e}_{ij}\left(\omega,\vec{x}\right)$
$\displaystyle=\frac{m}{1+2\phi(\vec{x})-\frac{x^{2}}{2l^{2}}}\int
dr^{0}\,\frac{dr^{0}}{d\sigma}\,\frac{e^{i\omega\left(r^{0}+R_{0}\right)}}{4\pi
R_{0}}\frac{v_{i}v_{j}}{1+2\phi(\vec{r})+\frac{r^{2}}{4l^{2}}}$
$\displaystyle\quad-\int dr^{0}\,e^{i\omega r^{0}}\int
d^{3}\vec{r}\,^{\prime}\,\delta^{(3)}\left(\vec{r}\,^{\prime}-\vec{r}\left(r^{0}\right)\right)\Bigg{\\{}\frac{dr^{0}}{d\sigma}\frac{m}{\pi}\left[\omega^{2}v^{i}v^{j}-i\omega\left(v^{i}\nabla_{j}+v^{j}\nabla_{i}\right)\phantom{\left(\frac{1}{2}\right)}\right.$
$\displaystyle\left.\qquad-\frac{\left(1+\vec{v}^{2}\right)}{2}\left(\nabla_{i}\nabla_{j}-\frac{1}{2}\delta_{ij}\nabla_{k}\nabla_{k}\right)\right]+\frac{qQ}{2\pi
M}\left[i\omega\left(v^{i}\nabla_{j}+v^{j}\nabla_{j}-\delta_{ij}v^{k}\nabla_{k}\right)\phantom{\left(\frac{1}{2}\right)}\right.$
$\displaystyle\left.\qquad\qquad\quad\,+2\left(\nabla_{i}\nabla_{j}-\frac{1}{2}\delta_{ij}\nabla_{k}\nabla_{k}\right)\right]\Bigg{\\}}G_{M}\left(\omega,\vec{x},\vec{r}\,^{\prime}\right)$
$\displaystyle\quad-\int dr^{0}\,e^{i\omega r^{0}}\int
d^{3}\vec{r}\,^{\prime}\,\delta^{(3)}\left(\vec{r}\,^{\prime}-\vec{r}\left(r^{0}\right)\right)\Bigg{\\{}\frac{dr^{0}}{d\sigma}\frac{m}{8\pi
l^{2}}\left[\omega^{2}v_{i}v_{j}-i\omega\left(v_{i}\nabla_{j}+v_{j}\nabla_{i}\right)\right.$
$\displaystyle\left.-\left(2-\vec{v}^{2}\right)\nabla_{i}\nabla_{j}\right.-\frac{3}{2}\delta_{ij}v_{k}v_{m}\nabla_{k}\nabla_{m}-\frac{1}{2}\delta_{ij}v^{2}\nabla_{k}\nabla_{k}$
$\displaystyle\left.-\frac{3}{2}\left(v_{k}v_{j}\nabla_{k}\nabla_{i}+v_{k}v_{i}\nabla_{k}\nabla_{j}\right)+v_{i}v_{j}\nabla_{k}\nabla_{k}+\frac{3}{8}i\omega(v_{i}\nabla_{j}+v_{j}\nabla_{i})\right]\Bigg{\\}}G_{l}\left(\omega,\vec{x},\vec{r}~{}^{\prime}\right)\,.$
(56)

Carrying out similar steps for $\tilde{a}_{i}\left(\omega,\vec{x}\right)$,
from 49 we get

$\displaystyle\tilde{a}_{i}\left(\omega,\vec{x}\right)=\frac{q}{1+\phi(\vec{x})+\frac{x^{2}}{8l^{2}}}\int
dr^{0}\,\frac{e^{i\omega\left(r^{0}+R_{0}\right)}}{4\pi
R_{0}}\frac{v_{i}}{1+\phi(\vec{r})+\frac{r^{2}}{8l^{2}}}$
$\displaystyle\quad-\int dr^{0}\,e^{i\omega r^{0}}\int
d^{3}\vec{r}\,^{\prime}\,\delta^{(3)}\left(\vec{r}\,^{\prime}-\vec{r}\left(r^{0}\right)\right)\Bigg{\\{}\frac{q}{\pi}\left[\omega^{2}v_{i}-i\omega\nabla_{i}+\frac{1}{4}v_{i}\nabla_{k}\nabla_{k}-\frac{1}{2}v^{k}\nabla_{k}\nabla_{i}\right]$
$\displaystyle\qquad\qquad\qquad+\frac{dr^{0}}{d\sigma}\frac{4Qm}{M}\left[v^{j}\nabla_{i}\nabla_{j}+i\omega\left(v^{i}v^{j}\nabla_{j}+\frac{1}{6}\nabla_{i}-\frac{1}{2}\vec{v}^{2}\nabla_{i}\right)\right]\Bigg{\\}}G_{M}\left(\omega,\vec{x},\vec{r}\,^{\prime}\right)$
$\displaystyle-\frac{q}{4\pi}\frac{1}{l^{2}}\int dr^{0}\,e^{i\omega r^{0}}\int
d^{3}\vec{r}\,^{\prime}\,\delta^{(3)}\left(\vec{r}\,^{\prime}-\vec{r}\left(r^{0}\right)\right)\Bigg{\\{}\frac{1}{2}\omega^{2}\
v_{i}-\frac{1}{2}i\omega\nabla_{i}+\frac{1}{8}v_{i}\nabla_{k}\nabla_{k}-\frac{1}{4}v_{k}\nabla_{k}\nabla_{i}\Bigg{\\}}$
$\displaystyle\hskip
100.00015ptG_{l}\left(\omega,\vec{x},\vec{r}\,^{\prime}\right)\,.$ (57)

In the next section we explicitly carry out the soft expansion of
$\tilde{a}_{i}$ following the prescription described in Fernandes:2020tsq . We
will see that the contribution to the soft photon factor due to the AdS radius
will only come from the first line of 57. The terms in the last line of 57
give finite contributions in considering the double scaling limit of a
vanishing frequency and infinite AdS radius.

##  4 Classical soft photon factor

Computing the quantum soft factor of a quantum scattering amplitude in
asymptotically AdS spacetime is tricky. Asymptotic states cannot be defined in
AdS as it has timelike boundary and particle geodesics are periodic. Therefore
we choose to calculate the soft factor from a classical prescription. One can
compute the soft factors for photons or gravitons in asymptotically flat
spacetimes by considering the classical limit of single/multiple soft theorems
arising from a quantum scattering process Laddha:2018myi . The same factors
can also be derived from the low frequency classical radiation produced in a
classical scattering process. The classical scattering is subject to the
condition that the total energy carried by the soft radiation is small
compared to the energy carried by the scatterer. Finally in the classical
limit, the soft factor is extracted from the power spectrum of the low
frequency classical radiation.

Considering that the observer is far away from the probe, i.e.
$\vec{x}\gg\vec{r}(\sigma)$ and taking the frequency $\omega\rightarrow 0$
limit in 4 spacetime dimensions on asymptotically flat backgrounds, the
classical radiative field for the photon can be written in terms of the soft
factors as Laddha:2018rle , Laddha:2018myi

$\epsilon^{\alpha}\,\tilde{a}_{\alpha}(\omega,\vec{x})={\cal{N}}^{\prime}\,S_{\rm
em}(\epsilon,k),\,$ (58)

where $\epsilon^{\alpha}$ is an arbitrary polarization vector of the photon,
$S_{\rm em}$ is the soft photon factor and ‘$k$’ denotes the momentum of soft
photon. Similarly for the soft graviton factor one can write

$\displaystyle\epsilon^{\alpha\beta}\,\tilde{e}_{\alpha\beta}(\omega,\vec{x})={\cal{N}}^{\prime}\,S_{\rm
gr}(\epsilon,k)\,.$ (59)

In 59, $\epsilon^{\alpha\beta}$ is an arbitrary rank two polarization tensor
of the graviton, $S_{\rm gr}$ is the soft graviton factor and ‘$k$’ denotes
the momentum of the soft graviton Laddha:2018myi . For both 58 and 59

$\displaystyle
R\equiv|\vec{x}|,\quad{\cal{N}}^{\prime}\equiv-\frac{i}{4\pi}{e^{i\omega
R}\over R},\quad k\equiv-\omega(1,\hat{n}),\quad\hat{n}={\vec{x}\over R}.$
(60)

The soft factors $S_{\rm em}$ and $S_{\rm gr}$ have a term proportional to
$\omega^{-1}$ at leading order in frequency and another term proportional to
$\ln\omega^{-1}$ at subleading order666A detailed description of how to derive
$S_{\rm gr}$ is given in Laddha:2018rle ; Laddha:2018myi ; Laddha:2019yaj ;
Saha:2019tub .. Equations 58 and 59 can be considered as an alternate
definition for the soft factor which can be easily computed from considering
the soft limit of classical electromagnetic and gravitational radiation
profiles.

In the case of asymptotically flat backgrounds, to calculate the soft factor
one needs to consider the large $|t|$ limit and a suitable parametrization of
$\vec{r}(t)$, where $\vec{r}(t)$ is the position of the scattered probe
particle at time $t$. As shown in Banerjee:2020dww , for computing the soft
factor up to $1/l^{2}$ in an asymptotically AdS spacetime, we can still
consider the particle to follow an approximately straight line geodesic for
large values of $t$. In particular due to long range interaction caused by
black hole potential and AdS potential, the trajectory receives following
contribution

$\vec{r}(t)\sim c_{1}\ln|t|-\frac{c_{2}}{l^{2}}t^{2}\,.$ (61)

The second term in the RHS of 61 will effectively contribute
$1/(\omega^{2}l^{2})$ in frequency space. This is a constant contribution in
the limit that we will be interested in, which will be further elaborated in
this section. As the contribution from the AdS potential to the soft factor is
beyond the $\ln\omega^{-1}$ subleading order in frequency, we do not require
its correction to the particle trajectory for computing the soft factor up to
this order in $1/l^{2}$.

We hence parametrize $\vec{r}(t)$ for asymptotic trajectories at large $|t|$
in four spacetime dimensions similar to the flat spacetime case as
Laddha:2018myi

$\vec{r}(t)=\vec{\beta}_{\pm}t-C_{\pm}\,\vec{\beta}_{\pm}\,\ln|t|+\,\text{finite
terms},\qquad\vec{v}=\vec{\beta}_{\pm}\left(1-\frac{C_{\pm}}{t}\right).$ (62)

where $C_{\pm}$ are constants and $t$ denotes the proper time. The $\ln|t|$
terms are the contributions from long range interaction forces which only
exist in 4 spacetime dimensions. 62 can be written in the following covariant
form

$r^{\mu}_{(a)}(t)=\eta_{(a)}\frac{1}{m_{(a)}}p_{(a)}^{\mu}t+c_{(a)}^{\mu}\ln|t|\,,$
(63)

where $\eta_{(a)}$ is positive (negative) for incoming (outgoing) particles
and $m_{(a)}$ is the mass of the a-th particle. We will consider the proper
time as negative for incoming particles and positive for outgoing particles.
Using the parametrization 62, we will retain terms up to $1/t$ as these are
relevant in the soft expansion of $\tilde{e}_{ij}$ and $\tilde{a}_{i}$. Using
suitable integrals given in Laddha:2018myi  one can easily find the soft
factors following the relation 58 and 59.

For asymptotic AdS spacetimes, soft limits cannot merely imply a vanishing
frequency limit. Since there is no notion of null infinity in asymptotic AdS
spacetimes and any massless radiation gets bounced off an infinite number of
times from spatial infinity, the frequency of a radiation can never strictly
go to zero. To define a “soft limit” in this case one needs to consider a
double scaling limit. In Banerjee:2020dww  the soft limit for AdS was defined
as simultaneously taking the frequency of the radiation in AdS space to zero
and the radius of AdS space to infinity i.e. $\omega\rightarrow 0$ and
$l\rightarrow\infty$, keeping $\omega l$ fixed. Another interesting feature of
the metric 15 is that the effect of long range interactions are the same as in
asymptotically flat spacetimes Banerjee:2020dww . This happens because the
perturbed fields already contain terms up to $1/l^{2}$ and we are interested
in results up to this order only. As will be further explained in Sec. 6, the
large ‘$l$’ limit of an asymptotically AdS spacetime defines an asymptotically
flat spacetime patch near the center of the spacetime. Our $\frac{1}{l^{2}}$
correction results concern scattering processes at short times, with
asymptotic states defined on the boundaries of this embedded patch.

To calculate the soft photon factor, we set $r^{0}=t$ and simplify the tail
terms in the Green’s function for $\vec{x}>>\vec{r}\left(\sigma\right)$

$\displaystyle\tilde{G}_{M}\left(\omega,\vec{x},\vec{r}\right)$
$\displaystyle=\lim_{\vec{x}\gg\vec{r}}G_{M}\left(\omega,\vec{x},\vec{r}\right)=\frac{iM}{16\pi\omega}\left[\ln\left(\frac{|\vec{r}|+\hat{n}.\vec{r}}{R}\right)+\int_{|\vec{r}|+\hat{n}.\vec{r}}^{\infty}\frac{du}{u}e^{i\omega
u}\right]\frac{e^{i\omega(R-\hat{n}.\vec{r})}}{R}\,,$ (64)
$\displaystyle\tilde{G}_{l}\left(\omega,\vec{x},\vec{r}\right)$
$\displaystyle=\lim_{\vec{x}\gg\vec{r}}G_{l}\left(\omega,\vec{x},\vec{r}\right)=-\frac{4i}{\omega}e^{i\omega(R-\hat{n}.\vec{r})}\hat{n}.\vec{r}\,.$
(65)

We can then rewrite $\tilde{a}_{i}(\omega,\vec{x})$ in the large $|t|$ limit
as

$\displaystyle\tilde{a}_{i}(\omega,\vec{x})$
$\displaystyle=\tilde{a}_{i}^{(1)}(\omega,\vec{x})+\tilde{a}_{i}^{(2)}(\omega,\vec{x})+\tilde{a}_{i}^{(3)}(\omega,\vec{x})+\tilde{a}_{i}^{(4)}(\omega,\vec{x})+\tilde{a}_{i}^{(5)}(\omega,\vec{x})+\tilde{a}_{i}^{(6)}(\omega,\vec{x})$
$\displaystyle\qquad\qquad+\tilde{a}_{i}^{(7)}(\omega,\vec{x})+\tilde{a}_{i}^{(8)}(\omega,\vec{x})+\tilde{a}_{i}^{(9)}(\omega,\vec{x})+\tilde{a}_{i}^{(10)}(\omega,\vec{x})\,,$
(66)

where the individual terms present in the above equation are defined as below:

$\displaystyle\tilde{a}_{i}^{(1)}\left(\omega,\vec{x}\right)=\frac{q}{1+\frac{x^{2}}{8l^{2}}}\frac{e^{i\omega
R}}{4\pi R}\int
dt~{}e^{i\omega(t-\hat{n}.\vec{r})}\frac{v_{i}}{1+\phi(\vec{r})+\frac{r^{2}}{8l^{2}}}\,,$
(67) $\displaystyle\tilde{a}_{i}^{(2)}(\omega\,,\vec{x})$
$\displaystyle=\frac{q}{2\pi}\int dt\,e^{i\omega
t}\,v_{k}\nabla_{k}\nabla_{i}\tilde{G}_{M}\left(\omega,\vec{x},\vec{r}\right)\,,$
(68) $\displaystyle\tilde{a}_{i}^{(3)}(\omega\,,\vec{x})$
$\displaystyle=\frac{iq}{\pi}\omega\int dt\,e^{i\omega
t}\,\nabla_{i}\tilde{G}_{M}\left(\omega,\vec{x},\vec{r}\right)\,,$ (69)
$\displaystyle\tilde{a}_{i}^{(4)}(\omega\,,\vec{x})$
$\displaystyle=-\frac{iMq}{16\pi^{2}}\frac{e^{i\omega R}}{R}\omega\int
dtv_{i}\bigg{\\{}\ln\frac{|\vec{r}\,^{\prime}|+\hat{n}.\vec{r}\,^{\prime}}{R}\,e^{i\omega(t-\hat{n}.\vec{r}\,^{\prime})}+\int_{|\vec{r}\,^{\prime}|+\hat{n}.\vec{r}\,^{\prime}}^{\infty}\frac{du}{u}e^{i\omega(t-\hat{n}.\vec{r}\,^{\prime}+u)}\bigg{\\}}\,,$
(70) $\displaystyle\tilde{a}_{i}^{(5)}(\omega\,,\vec{x})$
$\displaystyle=-\frac{q}{4\pi}\int dt\,{e^{i\omega
t}}\,v_{i}\nabla_{k}\nabla_{k}\tilde{G}_{M}\left(\omega,\vec{x},\vec{r}\right)\,,$
(71) $\displaystyle\tilde{a}_{i}^{(6)}(\omega\,,\vec{x})$
$\displaystyle=\frac{4Qm}{M}\int dt\,\frac{dt}{d\sigma}\,e^{i\omega
t}\,v_{k}\nabla_{k}\nabla_{i}\tilde{G}_{M}\left(\omega,\vec{x},\vec{r}\right)\,,$
(72) $\displaystyle\tilde{a}_{i}^{(7)}(\omega\,,\vec{x})$
$\displaystyle=-\frac{i4Qm}{M}\omega\int dt\,\frac{dt}{d\sigma}\,e^{i\omega
t}\,v_{i}v_{k}\nabla_{k}\tilde{G}_{M}\left(\omega,\vec{x},\vec{r}\right)\,,$
(73) $\displaystyle\tilde{a}_{i}^{(8)}(\omega\,,\vec{x})$
$\displaystyle=-\frac{i2Qm}{3M}\omega\int dt\,\frac{dt}{d\sigma}\,e^{i\omega
t}\,\nabla_{i}\tilde{G}_{M}\left(\omega,\vec{x},\vec{r}\right)\,,$ (74)
$\displaystyle\tilde{a}_{i}^{(9)}(\omega\,,\vec{x})$
$\displaystyle=\frac{i2Qm}{M}\omega\int dt\,\frac{dt}{d\sigma}\,e^{i\omega
t}\,\vec{v}^{2}\nabla_{i}\tilde{G}_{M}\left(\omega,\vec{x},\vec{r}\right)\,,$
(75) $\displaystyle\tilde{a}_{i}^{(10)}(\omega\,,\vec{x})$
$\displaystyle=-\frac{q}{8\pi l^{2}}\omega^{2}\int dt\,{e^{i\omega
t}}v_{i}\tilde{G}_{l}\left(\omega,\vec{x},\vec{r}\right)\,,$ (76)
$\displaystyle\tilde{a}_{i}^{(11)}(\omega\,,\vec{x})$
$\displaystyle=\frac{iq}{8\pi l^{2}}\omega\int dt\,{e^{i\omega
t}}\nabla_{i}\tilde{G}_{l}\left(\omega,\vec{x},\vec{r}\right)\,,$ (77)
$\displaystyle\tilde{a}_{i}^{(12)}(\omega\,,\vec{x})$
$\displaystyle=-\frac{q}{32\pi l^{2}}\int dt\,{e^{i\omega
t}}v_{i}\nabla_{k}\nabla_{k}\tilde{G}_{l}\left(\omega,\vec{x},\vec{r}\right)\,,$
(78) $\displaystyle\tilde{a}_{i}^{(13)}(\omega\,,\vec{x})$
$\displaystyle=\frac{q}{16\pi l^{2}}\int dt\,{e^{i\omega
t}}v_{k}\nabla_{k}\nabla_{i}\tilde{G}_{l}\left(\omega,\vec{x},\vec{r}\right)\,.$
(79)

We can think of $\tilde{a}_{i}^{(2)}$ to $\tilde{a}_{i}^{(5)}$ as
contributions due to the scatterer black hole’s mass, $\tilde{a}_{i}^{(6)}$ to
$\tilde{a}_{i}^{(9)}$ arising due to the black hole’s charge and
$\tilde{a}_{i}^{(10)}$ to $\tilde{a}_{i}^{(13)}$ due to the AdS potential. The
soft limit evaluation of $\tilde{a}_{i}^{(2)}$ to $\tilde{a}_{i}^{(9)}$ has
been previously derived in Fernandes:2020tsq . Here we will evaluate
$\tilde{a}_{i}^{(1)}$ in the soft limit. This term will give us the leading
$1/l^{2}$ correction to the soft photon factor of asymptotically flat
spacetime due to the AdS spacetime.

To evaluate the soft limit of $\tilde{a}_{i}^{(1)}$ we will be using the
following relation

$e^{i\omega(t-\hat{n}.\vec{r}(t))}=\frac{1}{i\omega}\frac{1}{\partial_{t}(t-\hat{n}.\vec{r}(t))}\frac{d}{dt}e^{i\omega(t-\hat{n}.\vec{r}(t))}=\frac{1}{i\omega}\frac{1}{(1-\hat{n}.\vec{v}(t))}\frac{d}{dt}e^{i\omega(t-\hat{n}.\vec{r}(t))}\,.$
(80)

Using 80 and carrying out an integration by parts, we have

$\displaystyle\tilde{a}_{i}^{(1)}\left(\omega,\vec{x}\right)=-\frac{q}{1+\frac{x^{2}}{8l^{2}}}~{}\frac{e^{i\omega
R}}{4\pi R}~{}\frac{1}{i\omega}\int
dt~{}e^{i\omega(t-\hat{n}.\vec{r})}\frac{d}{dt}\left[\frac{v_{i}}{(1-\hat{n}.\vec{v}(t))}\frac{1}{1+\phi(\vec{r})+\frac{r^{2}}{8l^{2}}}\right]\,.$
(81)

Since we assume $\frac{r^{2}}{l^{2}}\ll 1$ throughout our calculation, this
implies that we have the following approximation

$\frac{1}{1+\phi(\vec{r})+\frac{r^{2}}{8l^{2}}}=\frac{1}{1+\phi(\vec{r})}\left(1-\frac{r^{2}}{8l^{2}}\right)\,.$

Hence 81 simplifies to

$\displaystyle\tilde{a}_{i}^{(1)}\left(\omega,\vec{x}\right)$
$\displaystyle=-\frac{q}{1+\frac{x^{2}}{8l^{2}}}~{}\frac{e^{i\omega R}}{4\pi
R}~{}\frac{1}{i\omega}\int
dt~{}e^{i\omega(t-\hat{n}.\vec{r})}\frac{d}{dt}\left[\frac{v_{i}}{(1-\hat{n}.\vec{v}(t))}\frac{1}{1+\phi(\vec{r})}\right]\,$
$\displaystyle+\frac{q}{8l^{2}}~{}\frac{e^{i\omega R}}{4\pi
R}~{}\frac{1}{i\omega}\int
dt~{}e^{i\omega(t-\hat{n}.\vec{r})}\frac{d}{dt}\left[\frac{v_{i}}{(1-\hat{n}.\vec{v}(t))}r^{2}\right]$
$\displaystyle={\cal{A}}_{1}+{\cal{A}}_{2}+{\cal{A}}_{3}+\mathcal{O}(1/l^{3},\phi^{2})\,,$
(82)

where

$\displaystyle{\cal{A}}_{1}$ $\displaystyle=-q~{}\frac{e^{i\omega R}}{4\pi
R}~{}\frac{1}{i\omega}\int
dt~{}e^{i\omega(t-\hat{n}.\vec{r})}\frac{d}{dt}\left[\frac{v_{i}}{(1-\hat{n}.\vec{v}(t))}\frac{1}{1+\phi(\vec{r})}\right]\,,$
(83) $\displaystyle{\cal{A}}_{2}$
$\displaystyle=q\frac{x^{2}}{8l^{2}}~{}\frac{e^{i\omega R}}{4\pi
R}~{}\frac{1}{i\omega}\int
dt~{}e^{i\omega(t-\hat{n}.\vec{r})}\frac{d}{dt}\left[\frac{v_{i}}{(1-\hat{n}.\vec{v}(t))}\right]\,,$
(84) $\displaystyle{\cal{A}}_{3}$
$\displaystyle=\frac{q}{8l^{2}}~{}\frac{e^{i\omega R}}{4\pi
R}~{}\frac{1}{i\omega}\int
dt~{}e^{i\omega(t-\hat{n}.\vec{r})}\frac{d}{dt}\left[\frac{v_{i}}{(1-\hat{n}.\vec{v}(t))}r^{2}\right]\,.$
(85)

Note that we are keeping terms up to leading order in $\phi$ and $1/l^{2}$. In
the asymptotic expansion $\phi$ will take the form

$\phi(\vec{r}(t))=-\frac{M}{8\pi|\vec{r}(t)|}=\mp\frac{M}{8\pi|\vec{\beta}_{\pm}|t}+\mathcal{O}\left(t^{-2}\right)\,,$
(86)

where we have used the parametrization of $r$ from 62. ${\cal{A}}_{1}$ is the
contribution on asymptotically flat spacetimes and has been evaluated in
Fernandes:2020tsq . Therefore we will concentrate on the other two integrals.
To evaluate ${\cal{A}}_{2}$, we will substitute $v$ from 62 and perform the
following expansion

$v_{i}\left(1-\hat{n}.\vec{v}\right)^{-1}=\left(1-\hat{n}.\vec{\beta}_{\pm}\right)^{-1}\beta_{\pm
i}\left[1-\frac{C_{\pm}}{t}\frac{1}{1-\hat{n}.\vec{\beta}_{\pm}}\right]+\mathcal{O}\left(t^{-2}\right)\,.$
(87)

Therefore 84 reduces to

$\displaystyle{\cal{A}}_{2}$
$\displaystyle=q\frac{x^{2}}{8l^{2}}~{}\frac{e^{i\omega R}}{4\pi
R}~{}\frac{1}{i\omega}\int
dt~{}e^{-i\omega\tilde{g}(t)}\frac{d}{dt}\left[\frac{\beta_{\pm
i}}{\left(1-\hat{n}.\vec{\beta}_{\pm}\right)}\left(1-\frac{C_{\pm}}{t}\frac{1}{1-\hat{n}.\vec{\beta}_{\pm}}\right)\right]\,,$
(88)
$\displaystyle\text{with}\quad\tilde{g}(t)=-(1-\hat{n}.\vec{\beta}_{\pm})t-C_{\pm}\hat{n}.\vec{\beta}\ln
t\,.$ (89)

In the $\omega\to 0$ limit, we have the following integral Laddha:2018myi

$\displaystyle
I_{1}=\frac{1}{\omega}\int\limits_{-\infty}^{+\infty}dte^{-i\omega
g(t)}\frac{d}{dt}f(t)=\frac{1}{\omega}\left(f_{+}-f_{-}\right)+i\left(a_{+}k_{+}-a_{-}k_{-}\right)\text{ln}\omega^{-1}+\text{finite}\,,$
(90)

where

$\displaystyle f(t)$ $\displaystyle\to f_{\pm}+\frac{k_{\pm}}{t}\,,\qquad
g(t)\to a_{\pm}t+b_{\pm}\ln|t|\,.$ (91)

Using 90, we find that the integral ${\cal{A}}_{2}$ in 88 evaluates to

$\displaystyle{\cal{A}}_{2}=$
$\displaystyle-\frac{i}{\omega}\frac{qx^{2}}{8l^{2}}\frac{e^{i\omega R}}{4\pi
R}\left[\frac{\beta_{+i}}{1-\hat{n}.\vec{\beta}_{+}}-\frac{\beta_{-i}}{1-\hat{n}.\vec{\beta}_{-}}\right]$
$\displaystyle\qquad+\frac{qx^{2}}{8l^{2}}\frac{e^{i\omega R}}{4\pi
R}\ln\omega^{-1}\left[\frac{\beta_{+i}}{1-\hat{n}.\vec{\beta}_{+}}C_{+}-\frac{\beta_{-i}}{1-\hat{n}.\vec{\beta}_{-}}C_{-}\right]\,.$
(92)

To consider the soft limit, we will need to take the simultaneous limits
$\omega\rightarrow 0$ and $l\rightarrow\infty$, while keeping $\omega l$
fixed. Therefore, by defining $\omega l=\gamma$, we find that 92 can be
written as

$\displaystyle{\cal{A}}_{2}=$
$\displaystyle-i\frac{qx^{2}}{8\gamma^{2}}\frac{e^{i\omega R}}{4\pi
R}\omega\left[\frac{\beta_{+i}}{1-\hat{n}.\vec{\beta}_{+}}-\frac{\beta_{-i}}{1-\hat{n}.\vec{\beta}_{-}}\right]$
$\displaystyle\qquad+\frac{qx^{2}}{8\gamma^{2}}\frac{e^{i\omega R}}{4\pi
R}\omega^{2}\ln\omega^{-1}\left[\frac{\beta_{+i}}{1-\hat{n}.\vec{\beta}_{+}}C_{+}-\frac{\beta_{-i}}{1-\hat{n}.\vec{\beta}_{-}}C_{-}\right]\,.$
(93)

We thus note that in the $\omega\rightarrow 0$ limit, the ${\cal{A}}_{2}$ term
is a finite contribution and does not provide the divergent terms in the soft
factor. It follows that divergent contributions due to $\frac{1}{l^{2}}$
corrections in the soft factor will arise from those integrals which at least
fall off like $\omega^{-3}$.

The integrand in ${\cal{A}}_{3}$ involves the term
$\frac{v_{i}}{(1-\hat{n}.\vec{v}(t))}r^{2}$, which has the following expansion
in $t$

$\displaystyle\frac{v_{i}}{(1-\hat{n}.\vec{v}(t))}r^{2}$
$\displaystyle=\frac{\beta_{\pm
i}}{1-\hat{n}.\vec{\beta}_{\pm}}\vec{\beta}_{\pm}^{2}\left[t^{2}-2C_{\pm}~{}t\ln|t|-\frac{C_{\pm}}{1-\hat{n}.\vec{\beta}_{\pm}}t+\frac{2C^{2}_{\pm}}{1-\hat{n}.\vec{\beta}_{\pm}}\ln|t|\right]$
$\displaystyle\qquad\qquad\qquad+\frac{\beta_{\pm
i}}{\left(1-\hat{n}.\vec{\beta}_{\pm}\right)^{3}}C_{\pm}^{2}\vec{\beta}_{\pm}^{2}\hat{n}.\vec{\beta}_{\pm}\left[1-\frac{C_{\pm}\hat{n}.\vec{\beta}_{\pm}}{1-\hat{n}.\vec{\beta}_{\pm}}\frac{1}{t}\right]+\cdots\,,$
(94)

where $\cdots$ in 94 refer to subleading terms that are
$\mathcal{O}\left(\frac{1}{t^{2}}\right)$ and
$\mathcal{O}\left(\frac{\ln|t|}{t}\right)$, which will always lead to finite
terms in the soft factor and are hence ignored. The terms in the second line
of 94 can be directly substituted into 85 to provide the following
contribution, which we denote as ${\cal{A}}^{(0)}_{3}$

$\displaystyle{\cal{A}}_{3}^{(0)}$
$\displaystyle=-\frac{iq}{8\gamma^{2}}~{}\frac{e^{i\omega R}}{4\pi
R}~{}\omega\int
dt~{}e^{-i\omega\tilde{g}(t)}\frac{d}{dt}\left[\frac{C_{\pm}^{2}\vec{\beta}_{\pm}^{2}\hat{n}.\vec{\beta}_{\pm}\beta_{\pm
i}}{\left(1-\hat{n}.\vec{\beta}_{\pm}\right)^{3}}\left[1-\frac{C_{\pm}\hat{n}.\vec{\beta}_{\pm}}{1-\hat{n}.\vec{\beta}_{\pm}}\frac{1}{t}\right]\right]\,.$
(95)

In taking the simultaneous limits $\omega\to 0$ and $l\to\infty$ of the
${\cal{A}}_{3}^{(0)}$ integral in 95, the overall coefficient ensures that we
will have a non-divergent contribution in $\omega$ to the soft factor.

To determine the other possible contributions of ${\cal{A}}_{3}$, we need to
differentiate 94 with respect to $t$ and retain terms up to $1/t$. This gives
us

$\displaystyle\frac{d}{dt}\left[\frac{v_{i}}{(1-\hat{n}.\vec{v}(t))}r^{2}\right]=\frac{\beta_{\pm
i}}{1-\hat{n}.\vec{\beta}_{\pm}}\vec{\beta}_{\pm}^{2}\left[2t-2C_{\pm}~{}\ln|t|-C_{\pm}\frac{3-2\hat{n}.\vec{\beta}_{\pm}}{1-\hat{n}.\vec{\beta}_{\pm}}+\frac{2C^{2}_{\pm}}{1-\hat{n}.\vec{\beta}_{\pm}}\frac{1}{t}\right]\,.$
(96)

To apply the integral 90 we have to perform another integration by parts using
80 until we get the form of 91. By using the expression in 96 and subsequently
performing the integration by parts, we find

$\displaystyle{\cal{A}}_{3}$
$\displaystyle=-\frac{q}{8l^{2}}~{}\frac{e^{i\omega R}}{4\pi
R}~{}\left(\frac{1}{i\omega}\right)^{2}\int
dt~{}e^{-i\omega\tilde{g}(t)}\frac{d}{dt}\left[\frac{1}{(1-\hat{n}.\vec{v}(t))}\frac{\beta_{\pm
i}}{1-\hat{n}.\vec{\beta}_{\pm}}\vec{\beta}_{\pm}^{2}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.\left(2t-2C_{\pm}~{}\ln|t|-C_{\pm}\frac{3-2\hat{n}.\vec{\beta}_{\pm}}{1-\hat{n}.\vec{\beta}_{\pm}}+\frac{2C^{2}_{\pm}}{1-\hat{n}.\vec{\beta}_{\pm}}\frac{1}{t}\right)\right]$
$\displaystyle=-\frac{q}{8l^{2}}~{}\frac{e^{i\omega R}}{4\pi
R}~{}\left(\frac{1}{i\omega}\right)^{2}\int
dt~{}e^{-i\omega\tilde{g}(t)}\frac{d}{dt}\left[\frac{\beta_{\pm
i}}{(1-\hat{n}.\vec{\beta}_{\pm})^{2}}\vec{\beta}_{\pm}^{2}\left(2t-2C_{\pm}~{}\ln|t|\right.\right.$
$\displaystyle\left.\left.\quad\qquad\qquad\qquad\qquad\qquad\qquad+\frac{1}{t}\left(\frac{C_{\pm}}{1-\hat{n}.\vec{\beta}_{\pm}}\right)^{2}\left(2+\hat{n}.\vec{\beta}_{\pm}\right)-3\frac{C_{\pm}}{1-\hat{n}.\vec{\beta}_{\pm}}\right)\right]\,.$
(97)

We will now separate the ${\cal{A}}_{3}$ integral above into terms which can
be evaluated using 90 and those which require an additional integration by
parts. The terms in the parenthesis of 97 which are constant and proportional
to $\frac{1}{t}$ provide the following contribution which we denote as
${\cal{A}}_{3}^{(1)}$

$\displaystyle{\cal{A}}_{3}^{(1)}$
$\displaystyle=-\frac{q}{8\gamma^{2}}~{}\frac{e^{i\omega R}}{4\pi R}\int
dt~{}e^{-i\omega\tilde{g}(t)}\frac{\beta_{\pm
i}}{(1-\hat{n}.\vec{\beta}_{\pm})^{2}}\frac{d}{dt}\left[3\frac{C_{\pm}}{1-\hat{n}.\vec{\beta}_{\pm}}-\frac{1}{t}\left(\frac{C_{\pm}}{1-\hat{n}.\vec{\beta}_{\pm}}\right)^{2}\left(2+\hat{n}.\vec{\beta}_{\pm}\right)\right]\,.$
(98)

Evaluating this integral in the soft limit using 90, we find a finite
contribution that is irrelevant to the soft photon factor. The remaining terms
in the integrand of 97 require another differentiation followed by an
integration by parts. This contribution, which we denote by
${\cal{A}}_{3}^{(2)}$, takes the form

$\displaystyle{\cal{A}}_{3}^{(2)}$
$\displaystyle=-\frac{q}{8l^{2}}~{}\frac{e^{i\omega R}}{4\pi
R}~{}\left(\frac{1}{i\omega}\right)^{2}\int
dt~{}e^{-i\omega\tilde{g}(t)}\frac{\beta_{\pm
i}}{(1-\hat{n}.\vec{\beta}_{\pm})^{2}}\left(2\vec{\beta}_{\pm}^{2}-2\vec{\beta}_{\pm}^{2}\frac{C_{\pm}}{t}\right)$
$\displaystyle=\frac{iq}{8\gamma^{2}}~{}\frac{e^{i\omega R}}{4\pi
R}~{}\frac{1}{\omega}\int dt~{}e^{-i\omega\tilde{g}(t)}\frac{\beta_{\pm
i}}{(1-\hat{n}.\vec{\beta}_{\pm})^{3}}\frac{d}{dt}\left(2\vec{\beta}_{\pm}^{2}-2\vec{\beta}_{\pm}^{2}\frac{C_{\pm}}{t}\frac{1}{1-\hat{n}.\vec{\beta}_{\pm}}\right)\,.$
(99)

This integral in the soft limit evaluates to

$\displaystyle{\cal{A}}_{3}^{(2)}$
$\displaystyle=\frac{1}{\gamma^{2}}\left[\frac{iq}{4\omega}~{}\frac{e^{i\omega
R}}{4\pi
R}~{}\Bigg{\\{}\frac{\beta_{+i}}{(1-\hat{n}.\vec{\beta}_{+})^{3}}\vec{\beta}_{+}^{2}-\frac{\beta_{-i}}{(1-\hat{n}.\vec{\beta}_{-})^{3}}\vec{\beta}_{-}^{2}\Bigg{\\}}\right.$
$\displaystyle\left.-\frac{q}{4}\ln\omega^{-1}~{}\frac{e^{i\omega R}}{4\pi
R}~{}\Bigg{\\{}\frac{\beta_{+i}}{(1-\hat{n}.\vec{\beta}_{+})^{3}}C_{+}\vec{\beta}_{+}^{2}-\frac{\beta_{-i}}{(1-\hat{n}.\vec{\beta}_{-})^{3}}C_{-}\vec{\beta}_{-}^{2}\Bigg{\\}}\right]\,.$
(100)

The divergent terms in this result contribute to the soft photon factor. From
100 we can write the soft factor following 58.

The other radiative field terms, those from $\tilde{a}_{i}^{(10)}$ to
$\tilde{a}_{i}^{(13)}$, are either finite in the soft limit or are
proportional to $k_{i}$. Gauge invariance can be used to show that terms which
go like $k_{i}$ do not contribute to the soft factor. The invariance of
$S_{\text{em}}$ in 58 under $\epsilon^{\mu}\to\epsilon^{\mu}+k^{\mu}$ imposes
the constraint $k^{\mu}\tilde{a}_{\mu}=0$. This implies that $\tilde{a}_{0}$
can be determined from $\tilde{a}_{i}$ and allows us to set $\epsilon^{0}=0$.
In addition, the expressions for $\tilde{a}_{\mu}$ are determined only up to a
choice in gauge. Denoting the arbitrary gauge parameters by $\lambda$, we have
the following gauge transformations

$\delta\tilde{a}_{\mu}=\lambda k_{\mu}\,.$ (101)

Using 101 in 58, we thus have $k_{\mu}\epsilon^{\mu}=0$. This provides the
condition on the polarization vector as $k_{i}\epsilon^{i}=0$.

Therefore the $1/l^{2}$ contribution to the soft photon factor arises only
from 100. On substituting in 58 we find

$\displaystyle S_{\rm em}^{l}=$
$\displaystyle\frac{q}{4\gamma^{2}}\epsilon^{i}\left[-\frac{1}{\omega}~{}~{}\Bigg{\\{}\frac{\beta_{+i}}{(1-\hat{n}.\vec{\beta}_{+})^{3}}\vec{\beta}^{2}_{+}-\frac{\beta_{-i}}{(1-\hat{n}.\vec{\beta}_{-})^{3}}\vec{\beta}^{2}_{-}\Bigg{\\}}\right.$
$\displaystyle\left.-i\ln\omega^{-1}~{}\Bigg{\\{}\frac{C_{+}\beta_{+i}}{(1-\hat{n}.\vec{\beta}_{+})^{3}}\vec{\beta}^{2}_{+}-\frac{C_{-}\beta_{-i}}{(1-\hat{n}.\vec{\beta}_{-})^{3}}\vec{\beta}^{2}_{-}\Bigg{\\}}\right]\,.$
(102)

We can describe the above result in terms of the momentum and angular momentum
of the probe particle, and the momentum of the soft photon. From 62 we find
the asymptotic momenta (as $t\to\infty$) to have the expressions

$p_{(1)}=\frac{m}{\sqrt{1-\vec{\beta}_{-}^{2}}}\left(1,\vec{\beta}_{-}\right)\,,\qquad
p_{(2)}=-\frac{m}{\sqrt{1-\vec{\beta}_{+}^{2}}}\left(1,\vec{\beta}_{+}\right)\,.$
(103)

The overall negative sign in $p_{(2)}$ reflects the convention for outgoing
momenta. Using 62 we also have for the position four vector

$r_{(1)}=\left(t,\vec{\beta}_{-}t-C_{-}\vec{\beta}_{-}\ln|t|\right)\,,\qquad
r_{(2)}=\left(t,\vec{\beta}_{+}t-C_{+}\vec{\beta}_{+}\ln|t|\right)\,,$ (104)

with the coefficients of the logarithmic terms in the position vector,
$c^{\mu}_{(a)}$, taking the form

$c_{(1)}=\left(1,-C_{-}\vec{\beta}_{-}\right)\,,\qquad
c_{(2)}=\left(1,-C_{+}\vec{\beta}_{+}\right)\,.$ (105)

Here for simplicity we are ignoring an overall constant term in the position
vector, which will not contribute to the soft factor result (as it is a finite
contribution in the $t\to\infty$ limit).

The angular momentum of the probe particle can be written as

$j^{\mu\nu}_{(a)}=r^{\mu}_{(a)}p^{\nu}_{(a)}-r^{\nu}_{(a)}p^{\mu}_{(a)}+\text{spin}=(c_{a}^{\mu}p_{a}^{\nu}-c_{a}^{\nu}p_{a}^{\mu})\ln|t|+\cdots\,,$
(106)

and thus has the following non-vanishing components

$j_{(1)}^{0i}=\frac{m}{\sqrt{1-\vec{\beta}_{-}^{2}}}C_{-}\vec{\beta}_{-}\ln|t|\,,\quad
j_{(2)}^{0i}=-\frac{m}{\sqrt{1-\vec{\beta}_{+}^{2}}}C_{+}\vec{\beta}_{+}\ln|t|\,.$
(107)

Lastly, the outgoing soft photon has the momentum $k=-\omega(1,\hat{n})$.
Using this along with the expressions in 103 and 107, and replacing $\ln|t|$
by $\ln\omega^{-1}$, we determine that 102 can be written as the following sum

$\displaystyle S_{\rm em}^{l}$ $\displaystyle=S_{\rm em}^{l\,(0)}+S_{\rm
em}^{l\,(1)}\,,$ (108)

where $S_{\rm em}^{l\,(0)}$ and $S_{\rm em}^{l\,(1)}$ are the universal
leading and subleading contributions respectively on AdS spacetimes, with the
expressions

$\displaystyle S_{\rm em}^{l\,(0)}$
$\displaystyle=\frac{q}{4l^{2}}\sum_{a=1}^{2}\left(-1\right)^{a-1}\frac{\epsilon_{\mu}p^{\mu}_{(a)}}{p_{(a)}.k}\frac{\vec{p}^{2}_{(a)}}{\left(p_{(a)}.k\right)^{2}}\,,$
(109) $\displaystyle S_{\rm em}^{l\,(1)}$
$\displaystyle=i\frac{q}{4l^{2}}\sum_{a=1}^{2}\left(-1\right)^{a-1}\frac{\epsilon_{\nu}k_{\rho}j^{\rho\nu}_{(a)}}{p_{(a)}.k}\frac{\vec{p}^{2}_{(a)}}{\left(p_{(a)}.k\right)^{2}}$
$\displaystyle=i\frac{q}{4l^{2}}\ln\omega^{-1}\sum_{a=1}^{2}\left(-1\right)^{a-1}\frac{\epsilon_{\nu}k_{\rho}(c_{a}^{\rho}p_{a}^{\nu}-c_{a}^{\nu}p_{a}^{\rho})}{p_{(a)}.k}\frac{\vec{p}^{2}_{(a)}}{\left(p_{(a)}.k\right)^{2}}\,.$
(110)

Apart from the factor $\frac{\vec{p}^{2}_{(a)}}{\left(p_{(a)}.k\right)^{2}}$
appearing in each of the above sums, the expressions are those for the
universal leading and subleading soft factors on asymptotically flat
spacetimes, in the case of the scattering of a single probe particle. The
summation index values $a=1$ and $a=2$ correspond to the outgoing and incoming
configuration of the probe, respectively.

It can be noted that the momentum soft factor expression is not covariant, due
to the involvement of $\vec{p}_{(a)}^{2}$. This is a consequence of working in
the ‘static gauge’ choice for the polarization vector of the electromagnetic
field. We would always be required to adopt some gauge in the probe scattering
calculations on a fixed curved spacetime, which is needed to derive the AdS
result. Furthermore, we see from the dependence of additional momenta factors
that the above result on AdS spacetimes cannot be represented as an overall
phase of the asymptotically flat spacetime result.

We can also note that the gauge transformation
$\epsilon^{\mu}\to\epsilon^{\mu}+k^{\mu}$ in 109 does not lead to the manifest
vanishing of these factors unlike on asymptotically flat spacetimes. Rather,
the requirement of the resulting sums over the momenta becomes a statement of
momentum conservation for scattering processes up to $1/l^{2}$ corrections on
asymptotically AdS spacetimes. Hence the $1/l^{2}$ corrected soft factor
results are gauge invariant for all viable scattering processes on the
spacetime.

For completeness, we note the soft photon factor that can be derived on
asymptotically flat backgrounds arising from contributions in ${\cal{A}}_{1}$,
$\tilde{a}_{i}^{(3)}$, $\tilde{a}_{i}^{(4)}$ and $\tilde{a}_{i}^{(5)}$
Fernandes:2020tsq

$\displaystyle
S_{\text{em}}^{\text{flat}}=-\frac{q}{\omega}\left(\frac{\vec{\epsilon}.\vec{\beta}_{+}}{1-\hat{n}.\vec{\beta}_{+}}-\frac{\vec{\epsilon}.\vec{\beta}_{-}}{1-\hat{n}.\vec{\beta}_{-}}\right)-iq\ln\omega^{-1}\left(\frac{C_{+}}{1-\hat{n}.\vec{\beta}_{+}}\vec{\epsilon}.\vec{\beta}_{+}-\frac{C_{-}}{1-\hat{n}.\vec{\beta}_{-}}\vec{\epsilon}.\vec{\beta}_{-}\right)$
$\displaystyle\qquad\qquad-iq\frac{M}{4\pi}\ln\left(\omega
R\right)\left(\frac{\vec{\epsilon}.\vec{\beta}_{+}}{1-\hat{n}.\vec{\beta}_{+}}-\frac{\vec{\epsilon}.\vec{\beta}_{-}}{1-\hat{n}.\vec{\beta}_{-}}\right)\,.$
(111)

In this case, the soft factor can be expressed in covariant form as given in
Laddha:2018myi .

##  5 Classical soft graviton factor

In this section, we will simply present the results on the contribution of the
charge and mass of the scatterer black hole to the soft graviton factor. For a
detailed calculation we refer the reader to Appendix A. The soft graviton
factor can be split in two parts as,

$S_{\text{gr}}(\epsilon,k)=S_{\rm gr}^{\rm flat}(\epsilon,k)+S_{\rm
gr}^{l}(\epsilon,k)\,,$ (112)

where $S_{\rm gr}^{\text{flat}}(\epsilon,k)$ is the contribution due to the
black hole mass and charge, which was derived in Fernandes:2020tsq . The
result in its covariant form can be expressed as Laddha:2018myi

$\displaystyle S_{\rm gr}^{\rm flat}(\epsilon,k)=$
$\displaystyle\sum_{a=1}^{2}{\epsilon_{\mu\nu}p_{(a)}^{\mu}p_{(a)}^{\nu}\over
p_{(a)}.k}+i\,\sum_{a=1}^{2}{\epsilon_{\mu\nu}p_{(a)}^{\mu}k_{\rho}\over
p_{(a)}.k}j^{\rho\nu}_{(a)}\,.$ (113)

In 113, $p_{(1)}$ and $p_{(2)}$ are the momenta of the probe particle before
and after the scattering, while $j_{(1)}$ and $j_{(2)}$ are the angular
momenta of the probe before and after the scattering. As in the previous
section, we choose the convention that all momenta and angular momenta are
positive for ingoing and negative for outgoing particles. The indices
$\mu,\nu,\cdots$ run over all spacetime coordinates in 113. The covariant
expression has an implicit dependence on the masses and charges in the
scattering process. The explicit form of the above equation on replacing the
momenta and angular momenta has been provided in A.

Further in Banerjee:2020dww , we derived the leading order contribution of the
cosmological constant to the soft graviton factor on the AdS Schwarzschild
black hole spacetime. Since to our order of approximation, the metric in the
isotropic coordinates does not receive any correction from the charge of the
black hole, the results for the soft factor remains the same. Here we present
this result in covariant form, in terms of the incoming and outgoing momenta
of the probe particle

$\displaystyle S_{\rm gr}^{l}$ $\displaystyle=S_{\rm gr}^{l\,(0)}+S_{\rm
gr}^{l\,(1)}\,,\qquad\text{with}$ $\displaystyle S_{\rm gr}^{l\,(0)}$
$\displaystyle=\frac{1}{2l^{2}}\sum_{a=1}^{2}\frac{\epsilon_{\mu\nu}p^{\mu}_{(a)}p^{\nu}_{(a)}}{p_{(a)}.k}\frac{\vec{p}^{2}_{(a)}}{\left(p_{(a)}.k\right)^{2}}\left(3+\frac{\vec{p}^{2}_{(a)}}{p^{2}_{(a)}}\right)\,,$
$\displaystyle S_{\rm gr}^{l\,(1)}$
$\displaystyle=i\frac{1}{2l^{2}}\sum_{a=1}^{2}\frac{\epsilon_{\mu\nu}p^{\mu}_{(a)}k_{\rho}j^{\rho\nu}_{(a)}}{p_{(a)}.k}\frac{\vec{p}^{2}_{(a)}}{\left(p_{(a)}.k\right)^{2}}\left(3+\frac{\vec{p}^{2}_{(a)}}{p^{2}_{(a)}}\right)\,,$
(114)

where $S_{\rm gr}^{l\,(0)}$ and $S_{\rm gr}^{l\,(1)}$ are respectively the
universal leading and subleading contributions to the soft graviton factor on
AdS spacetimes. Note that for the four momentum we have $p^{2}_{(a)}=-m^{2}$.
The leading and subleading soft factor sums are, as in the soft photon results
in 109 and 110, expressed in terms of a product of the flat spacetime result
and other momentum dependent terms. The gauge invariance followes for all
physical scattering process, as discussed in the last section.

##  6 Ward identities from soft photon factors

In the previous sections we derived classical soft photon and graviton factors
in an asymptotically AdS theory, to leading order $1/l^{2}$ in the large AdS
radius limit. Considering the large AdS radius limit is important for
computational simplifications as well as for defining a proper soft limit in
AdS space. Our analysis can be interpreted as $1/l^{2}$ corrections of
classical soft theorems in asymptotically flat spacetime in large AdS radius
limit. Thus we expect these AdS corrected soft factors derived in 109 and 114
to satisfy similar relations as the usual flat spacetime classical soft
factors. By now, it is a known fact that classical soft factors partially
reproduce the quantum soft factors. These results hold exactly at the level of
tree and all loop scattering amplitudes for the leading soft factor 777for
subleading and sub-subleading parts, the quantum soft factor reduces to the
classical one in the classical limit Sahoo:2020ryf . On the other hand, it is
also known that soft theorems with the leading quantum soft factor
contribution manifest as certain Ward identities arising from large gauge
transformations on asymptotically flat spacetimes. We can thus anticipate that
soft theorems involving leading soft factor terms, with AdS corrections,
should have a corresponding realization in a Ward identity.

In this section, we concern ourselves with the soft photon case. We will more
specifically be interested in the equivalence of large gauge Ward identity
with the soft photon theorem in the case of a massless scattering process.
While the soft factor expressions derived in previous sections resulted from a
massive probe particle scattering process, the universality of the tree level
soft photon factor enables us to consider its expression in any scattering
process involving soft particles. The analysis in particular involves the
derivation of an expression for the soft photon number mode from a saddle
point approximation. To maintain a flow of the paper and to establish
notations and definitions, we first review the correspondence between the
large gauge Ward identity and leading soft factor for a theory on
asymptotically flat spacetimes He:2014cra ; He:2014laa ; Strominger:2017zoo
to the leading order and then derive the same for our case 888Experts may skip
this section and directly go to section 6.1.

For a residual gauge parameter $\varepsilon(z\,,\bar{z})$ at null infinity
which satisfies the antipodal boundary condition

$\varepsilon(z\,,\bar{z})|_{\mathscr{I}^{+}_{-}}=\varepsilon(z\,,\bar{z})|_{\mathscr{I}^{-}_{+}}\,,$
(115)

we can define the total charge for the Maxwell field on $\mathscr{I}^{+}$ and
$\mathscr{I}^{-}$ in terms of boundary contributions on asymptotically flat
spacetimes

$Q_{\varepsilon}^{+}=\int_{\mathscr{I}^{+}_{-}}\varepsilon*F\,,\qquad
Q_{\varepsilon}^{-}=\int_{\mathscr{I}^{-}_{+}}\varepsilon*F\,.$ (116)

here $(z,\bar{z})$ are coordinates on the Celestial 2-sphere. The above
charges satisfy the following conservation equation

$Q_{\varepsilon}^{+}-Q_{\varepsilon}^{-}=0\,.$ (117)

We can now consider Maxwell’s equations in the presence of a source to re-
express the charges in 116 as the following integrals over $\mathscr{I}^{+}$
and $\mathscr{I}^{-}$

$\displaystyle
Q_{\varepsilon}^{+}=-\int_{\mathscr{I}^{+}}dud^{2}z\left(\partial_{z}\varepsilon
F^{(0)}_{u\bar{z}}+\partial_{\bar{z}}\varepsilon
F^{(0)}_{uz}\right)+\int_{\mathscr{I}^{+}}dud^{2}z\varepsilon\gamma_{z\bar{z}}j_{u}^{(2)}\,,$
(118) $\displaystyle
Q_{\varepsilon}^{-}=-\int_{\mathscr{I}^{-}}dvd^{2}z\left(\partial_{z}\varepsilon
F^{(0)}_{v\bar{z}}+\partial_{\bar{z}}\varepsilon
F^{(0)}_{vz}\right)+\int_{\mathscr{I}^{-}}dvd^{2}z\varepsilon\gamma_{z\bar{z}}j_{v}^{(2)}\,.$
(119)

In the above expressions $\gamma_{z\bar{z}}$ is the metric on the $2$-sphere
and the superscripts on the field strength tensors and currents represent the
order of the coefficient in their corresponding $r^{-1}$ expansions in
asymptotically flat spacetimes. The first integrals in $Q_{\varepsilon}^{+}$
and $Q_{\varepsilon}^{-}$ are known as the soft charges. The field strength
tensors involved in the soft charges can be expressed in terms of their soft
modes. Further in the absence of asymptotic magnetic fields and magnetic
monopoles, we have the following condition

$F_{z\bar{z}}|_{\mathscr{I}^{\pm}_{\pm}}=0\,.$ (120)

In this case, it can be shown that we can define the soft modes of the field
strength tensors in terms of the following real scalars $\mathcal{N}^{\pm}$
that satisfy

$\displaystyle\partial_{z}\mathcal{N}^{+}(z\,,\bar{z})=\int_{\mathscr{I}^{+}}duF^{(0)}_{uz}\,,\qquad$
$\displaystyle\partial_{\bar{z}}\mathcal{N}^{+}(z\,,\bar{z})=\int_{\mathscr{I}^{+}}duF^{(0)}_{u\bar{z}}\,,$
$\displaystyle\partial_{z}\mathcal{N}^{-}(z\,,\bar{z})=\int_{\mathscr{I}^{-}}dvF^{(0)}_{vz}\,,\qquad$
$\displaystyle\partial_{\bar{z}}\mathcal{N}^{-}(z\,,\bar{z})=\int_{\mathscr{I}^{-}}dvF^{(0)}_{v\bar{z}}.$
(121)

Using 121 in 118 and 119 then gives us the total charge expressions

$\displaystyle Q_{\varepsilon}^{+}=2\int
d^{2}z\mathcal{N}^{+}\partial_{z}\partial_{\bar{z}}\varepsilon+\int_{\mathscr{I}^{+}}dud^{2}z\varepsilon\gamma_{z\bar{z}}j_{u}^{(2)}\,,$
(122) $\displaystyle Q_{\varepsilon}^{-}=2\int
d^{2}z\mathcal{N}^{-}\partial_{z}\partial_{\bar{z}}\varepsilon+\int_{\mathscr{I}^{-}}dvd^{2}z\varepsilon\gamma_{z\bar{z}}j_{v}^{(2)}\,.$
(123)

With these classical results, we can now describe the Ward identity. Given a
scattering process going from an incoming state $|\text{in}\rangle$ on
$\mathscr{I}^{-}$ to a state $|\text{out}\rangle$ on $\mathscr{I}^{+}$
governed by an $S$-matrix $\mathcal{S}$, the charge conservation in 117 now
takes the form

$\langle\text{out}|\hat{Q}^{+}_{\varepsilon}\mathcal{S}-\mathcal{S}\hat{Q}^{-}_{\varepsilon}|\text{in}\rangle=0\,,$
(124)

where $\hat{Q}^{\pm}_{\varepsilon}$ are operator versions of the expressions
in 122 and 123. The integrated hard charges can be expressed as the following
sum over hard charges in the incoming and outgoing states

$\displaystyle\int_{\mathscr{I}^{+}}dud^{2}z\varepsilon\gamma_{z\bar{z}}j_{u}^{(2)}=\sum_{k=\text{out}}Q_{k}\varepsilon\left(z_{k}\,,\bar{z}_{k}\right)\,,$
(125)
$\displaystyle\int_{\mathscr{I}^{-}}dvd^{2}z\varepsilon\gamma_{z\bar{z}}j_{v}^{(2)}=\sum_{k=\text{in}}Q_{k}\varepsilon\left(z_{k}\,,\bar{z}_{k}\right)\,.$
(126)

The index ‘$k$’ run over incoming and outgoing hard particles in the
respective sums, while the coordinates $\\{z_{k}\,,\bar{z}_{k}\\}$ denote
their asymptotic positions on the Celestial $2$-sphere. Using the above
expressions in 122 and 123, we find that the operators
$\hat{Q}^{\pm}_{\varepsilon}$ take the form

$\displaystyle\langle\text{out}|\hat{Q}_{\varepsilon}^{+}=2\int
d^{2}z\partial_{z}\partial_{\bar{z}}\varepsilon\langle\text{out}|\mathcal{N}^{+}+\sum_{k=\text{out}}Q_{k}\varepsilon\left(z_{k}\,,\bar{z}_{k}\right)\langle\text{out}|\,,$
(127) $\displaystyle\hat{Q}_{\varepsilon}^{-}|\text{in}\rangle=2\int
d^{2}z\partial_{z}\partial_{\bar{z}}\varepsilon\mathcal{N}^{-}|\text{in}\rangle+\sum_{k=\text{in}}Q_{k}\varepsilon\left(z_{k}\,,\bar{z}_{k}\right)|\text{in}\rangle\,.$
(128)

Substituting 127 and 128 in 124 then gives us the following Ward identity
He:2014cra

$\displaystyle 2\int
d^{2}z\partial_{z}\partial_{\bar{z}}\varepsilon\left(z\,,\bar{z}\right)\langle\text{out}|\mathcal{N}^{+}\left(z\,,\bar{z}\right)\mathcal{S}-\mathcal{S}\mathcal{N}^{-}\left(z\,,\bar{z}\right)|\text{in}\rangle$
$\displaystyle\quad=\left[\sum_{k=\text{in}}Q_{k}\varepsilon\left(z_{k}\,,\bar{z}_{k}\right)-\sum_{k=\text{out}}Q_{k}\varepsilon\left(z_{k}\,,\bar{z}_{k}\right)\right]\langle\text{out}|\mathcal{S}|\text{in}\rangle\,.$
(129)

We can express the left hand side of 129 entirely in terms of either the soft
photon mode $\cal{N}^{+}$ or $\cal{N}^{-}$. This follows from the CPT
invariance of matrix elements involving the in and out soft photons. This
implies in particular that

$\langle\text{out}|\mathcal{N}^{+}\left(z\,,\bar{z}\right)\mathcal{S}|\text{in}\rangle=-\langle\text{out}|\mathcal{S}\mathcal{N}^{-}\left(z\,,\bar{z}\right)|\text{in}\rangle\,.$

Hence 129 can be expressed as

$\displaystyle 4\int
d^{2}z\partial_{z}\partial_{\bar{z}}\varepsilon\left(z\,,\bar{z}\right)\langle\text{out}|\mathcal{N}^{+}\left(z\,,\bar{z}\right)\mathcal{S}|\text{in}\rangle$
$\displaystyle\quad=\left[\sum_{k=\text{in}}Q_{k}\varepsilon\left(z_{k}\,,\bar{z}_{k}\right)-\sum_{k=\text{out}}Q_{k}\varepsilon\left(z_{k}\,,\bar{z}_{k}\right)\right]\langle\text{out}|\mathcal{S}|\text{in}\rangle\,.$
(130)

To simplify the notation going forward, we will refer to the soft photon mode
$\mathcal{N}^{+}$ as $\mathcal{N}$ and its associated outgoing operators
$a_{+}^{\text{out}}(\omega\hat{x})$ and
$a_{-}^{\text{out}}(\omega\hat{x})^{\dagger}$ as
$a_{+}^{\text{flat}}(\omega\hat{x})$ and
$a_{-}^{\text{flat}}(\omega\hat{x})^{\dagger}$ respectively. An expression for
$\partial_{z}\mathcal{N}$ can be formally derived from a mode expansion of the
gauge fields and a saddle point approximation Strominger:2017zoo

$\partial_{z}\mathcal{N}=-\frac{1}{8\pi}\frac{\sqrt{2}}{1+z\bar{z}}\lim_{\omega\to
0}\left[\omega a_{+}^{\text{flat}}(\omega\hat{x})+\omega
a_{-}^{\text{flat}}(\omega\hat{x})^{\dagger}\right]\,.$ (131)

We next consider the gauge parameter. A particularly convenient choice as
considered in Strominger:2017zoo  for $\varepsilon\left(z\,,\bar{z}\right)$ is
the following

$\varepsilon\left(z\,,\bar{z}\right)=\frac{1}{z-z_{k}}\,.$ (132)

In particular, the derivative of 132 satisfies

$\partial_{\bar{w}}\varepsilon(z\,,\bar{z})=2\pi\delta^{(2)}\left(z-w\right)\,.$
(133)

By substituting 133 and 131 in 130, we find

$\lim_{\omega\to 0}\langle\text{out}|\omega
a^{\text{flat}}_{+}(\omega\hat{x})\mathcal{S}|\text{in}\rangle=\frac{(1+z\bar{z})}{\sqrt{2}}\left[\sum_{k=\text{out}}\frac{Q_{k}}{z-z_{k}}-\sum_{k=\text{in}}\frac{Q_{k}}{z-z_{k}}\right]\langle\text{out}|\mathcal{S}|\text{in}\rangle\,.$
(134)

The large gauge Ward identity is formally derived as a property satisfied by
quantized fields of a given theory. However, we may also derive the Ward
identity using the soft photon theorem. We recall that the soft photon theorem
relates a scattering process with a soft photon insertion to the scattering
process without the soft photon by a soft factor. Assuming for simplicity the
insertion of the soft photon in the ‘out’ state, the theorem provides the
relation

$\displaystyle\langle\text{out}|a^{\text{flat}}_{+}(\omega\hat{x})\mathcal{S}|\text{in}\rangle=S^{\text{flat}}_{\text{em}}\langle\text{out}|\mathcal{S}|\text{in}\rangle\,,$
(135)

where $a^{\text{flat}}_{+}$ denotes the creation operator in the outgoing
state with helicity ‘$+$’ and $S^{\text{flat}}_{\text{em}}$ is taken to be the
leading contribution to the soft photon factor

$S^{\text{flat}\;(0)}_{\text{em}}=\left[\sum_{k=\text{out}}\frac{p_{(k)}.\epsilon_{+}}{p_{(k)}.q}Q_{k}-\sum_{k=\text{in}}\frac{p_{(k)}.\epsilon_{+}}{p_{(k)}.q}Q_{k}\right]\,,$
(136)

and $\mathcal{S}$ is a generic scattering process in flat spacetime. Our
analysis in previous sections considered a single probe particle with charge
$q$. Thus in this case, we set $Q_{k}=q$ in 136.

We can now readily derive the Ward identity in 130 from 135. For this
derivation, we only need to use the expressions for the momenta of all the
particles (hard and soft) and the polarization vector for the soft particle.
For the soft particle and its polarization, we have the following conditions

$q_{\mu}q^{\mu}=0\,,\quad
q_{\mu}\epsilon^{\pm\mu}(\vec{q})=0\,,\quad\epsilon^{\mu}_{\alpha}\epsilon^{*}_{\beta\mu}=\delta_{\alpha\beta}\,,$
(137)

where the indices $\alpha\,,\beta$ represent the polarization directions
$\pm$. A natural choice for the null vector $q^{\mu}$ is the following

$q^{\mu}=\omega\left(1\,,{\vec{x}\over
r}\right)=\frac{\omega}{1+z\bar{z}}\left(1+z\bar{z}\,,z+\bar{z}\,,-i(z-\bar{z})\,,1-z\bar{z}\right)\,,$
(138)

with $\omega$ the frequency of the soft particle. From this choice of
$q^{\mu}$ one can determine the polarization vectors

$\epsilon^{+\mu}=\frac{1}{\sqrt{2}}\left(\bar{z}\,,1\,,-i\,,-\bar{z}\right)\,,\quad\epsilon^{-\mu}=\frac{1}{\sqrt{2}}\left(z\,,1\,,i\,,-z\right)\,.$
(139)

Since the hard particles are also massless, but of finite energy, using 138 we
can choose

$p_{k}^{\mu}=\frac{E_{\vec{p}}}{1+z\bar{z}}\left(1+z\bar{z}\,,z+\bar{z}\,,-i(z-\bar{z})\,,1-z\bar{z}\right)\,.$
(140)

If we now substitute 138, 139 and 140 in 136, we find

$\displaystyle S^{\text{flat}\,(0)}_{\text{em}}$
$\displaystyle=\left[\sum_{k=\text{out}}\frac{p_{(k)}.\epsilon_{+}}{p_{(k)}.q}Q_{k}-\sum_{k=\text{in}}\frac{p_{(k)}.\epsilon_{+}}{p_{(k)}.q}Q_{k}\right]$
$\displaystyle=\frac{1+z\bar{z}}{\sqrt{2}\omega}\left[\sum_{k=\text{out}}\frac{1}{z-z_{k}}Q_{k}-\sum_{k=\text{in}}\frac{1}{z-z_{k}}Q_{k}\right]\,.$
(141)

Thus on substituting 141 in 135, we get

$\lim_{\omega\to 0}\frac{\sqrt{2}}{1+z\bar{z}}\langle\text{out}|\omega
a^{\text{flat}}_{+}(\omega\hat{x})\mathcal{S}|\text{in}\rangle=\left[\sum_{k=\text{out}}\frac{1}{z-z_{k}}Q_{k}-\sum_{k=\text{in}}\frac{1}{z-z_{k}}Q_{k}\right]\langle\text{out}|\mathcal{S}|\text{in}\rangle\,.$
(142)

This result agrees with the Ward identity in 134.

In the presence of AdS corrections of asymptotically flat spacetimes, we can
determine the tree level soft theorem which involves the soft factor derived
from our classical scattering analysis. However, a first principles derivation
of the Ward identity using quantized fields is not particularly clear. As
previously described, the reason for this is that the $\gamma^{-2}$ correction
to the asymptotically flat spacetime soft factor arises from a double scaling
limit. This implies that unlike asymptotically flat spacetimes, the
$\gamma^{-2}$ correction cannot result from a unique saddle point
approximation of the gauge fields. Rather, there can exist several
contributions to $\gamma=l\omega$ as we simultaneously take $l\to\infty$ and
$\omega\to 0$. Thus, while the saddle point approximation determines the soft
mode $\mathcal{N}$ in terms of creation and annihilation operators on
asymptotically flat spacetimes, this does not extend to possible $1/l^{2}$
corrections arising from the nearly flat limit of asymptotically AdS
spacetimes. These corrections could be determined by assuming that the
equivalence of tree level soft theorems and large gauge Ward identities on
asymptotically flat spacetimes continues to hold under small corrections. As
we will show in the following, this is highly constraining on the expressions
for the $1/l^{2}$ corrections of the soft photon mode and residual gauge
parameter.

###  6.1 $1/l^{2}$ correction of the large gauge Ward identity

We derived the leading soft photon factor correction $S^{l\,(0)}_{\text{em}}$
from a classical scattering analysis on an AdS spacetime in 109. To interpret
the corresponding soft theorem for a massless scattering process in terms of
an associated Ward identity, we must define the asymptotic surface on which it
is defined. Our scattering process with the assumptions on a large impact
parameter is well approximated by the large $l$ limit of a global AdS
spacetime, with the black hole as a point particle source located at the
origin. The global AdS4 metric is described by the metric

$ds^{2}=\frac{l^{2}}{\cos^{2}\rho}\left[-d\tau^{2}+d\rho^{2}+\sin^{2}\rho
d\Omega_{2}^{2}\right]\,.$ (143)

In the limit of $l\to\infty$, we can identify an asymptotically flat patch
close to the origin of the AdS spacetime Hijano:2019qmi  that has recently
been used to demonstrate the equivalence of the asymptotically flat spacetime
large gauge Ward identity with a CFT Ward identity Hijano:2020szl . In our
case, the effect of $1/l^{2}$ corrections will be shown to affect the soft
photon mode. Hence the effect of $1/l^{2}$ corrections in the soft photon case
has a natural interpretation as a perturbation of the asymptotically flat
spacetime large gauge Ward identity, evaluated across null infinity of the
patch embedded in the larger AdS spacetime. This is represented in Fig.1,

Figure 1: The Ward identity is defined across the null surfaces of the
asymptotically flat spacetime patch embedded within a larger AdS spacetime.
The black dot at the center represents the black hole treated as a point
particle in our scattering approximation. The effect of $1/l^{2}$ corrections
is to perturb the asymptotically flat spacetime Ward identity defined on the
surface of the patch.

As noted through the expression in 61, the $1/l^{2}$ corrections do not affect
the flat spacetime asymptotic trajectories of particles in any scattering
process that provide the soft factor expressions up to subleading order. Hence
the parametrizations for the momenta of the hard (and soft) massless particles
and the polarization can be taken to be the same as those on asymptotically
flat spacetime given in 138, 139 and 140.

Using these expressions in 109, we have the the following result for the soft
factor correction in terms of the asymptotic coordinates of the particles
$(z\,,\bar{z})$, $(z_{k}\,,\bar{z}_{k})$ and the soft photon frequency

$\displaystyle S^{l\,(0)}_{\text{em}}$
$\displaystyle=\frac{1}{4l^{2}}\left[\sum_{k=\text{out}}\frac{p_{(k)}.\epsilon_{+}}{p_{(k)}.q}\frac{\vec{p}^{2}_{(k)}}{\left(p_{(k)}.q\right)^{2}}Q_{k}-\sum_{k=\text{in}}\frac{p_{(k)}.\epsilon_{+}}{p_{(k)}.q}\frac{\vec{p}^{2}_{(k)}}{\left(p_{(k)}.q\right)^{2}}Q_{k}\right]$
$\displaystyle=\frac{\left(1+z\bar{z}\right)^{3}}{16\sqrt{2}\gamma^{2}\omega}\left[\sum_{k=\text{out}}\frac{\left(1+z_{k}\bar{z}_{k}\right)^{2}}{(\bar{z}-\bar{z}_{k})^{2}}\frac{1}{(z-z_{k})^{3}}Q_{k}-\sum_{k=\text{in}}\frac{\left(1+z_{k}\bar{z}_{k}\right)^{2}}{(\bar{z}-\bar{z}_{k})^{2}}\frac{1}{(z-z_{k})^{3}}Q_{k}\right]\,,$
(144)

where we have denoted $l^{2}\omega^{2}$ as $\gamma^{2}$.

A consistent deformation of the soft photon theorem given in 135 on
asymptotically flat spacetimes requires $1/l^{2}$ corrections on either side
of the equation. This leads to the following identity for scattering processes

$\displaystyle\langle\text{out}|a^{\text{flat}}_{+}(\omega\hat{x})\mathcal{S}|\text{in}\rangle+\langle\text{out}|\tilde{a}_{+}(\omega\hat{x})\mathcal{S}|\text{in}\rangle=\left(S^{\text{flat}\,(0)}_{\text{em}}+S^{l\,(0)}_{\text{em}}\right)\langle\text{out}|\mathcal{S}|\text{in}\rangle\,,$
(145)

with a separate theorem satisfied by the corrected components

$\displaystyle\langle\text{out}|\tilde{a}_{+}(\omega\hat{x})\mathcal{S}|\text{in}\rangle=S^{l\,(0)}_{\text{em}}\langle\text{out}|\mathcal{S}|\text{in}\rangle\,.$
(146)

In 145, $S^{\text{flat}\,(0)}_{\text{em}}$ and $S^{l\,(0)}_{\text{em}}$ are
the universal leading contributions without and with $1/l^{2}$ corrections
respectively, and $\tilde{a}_{+}$ denotes a creation operator in the out state
corresponding to the $1/l^{2}$ correction. Note that the $\tilde{a}_{+}$
operator must be distinct from the flat space soft photon operator and in
particular has a different dimension. We will make this point clear by
defining $\tilde{a}_{+}=\frac{1}{l^{2}}a_{+}^{\text{AdS}}$ later on.

The way to interpret 145 is that there exists a scattering process in flat
spacetime, and we insert a soft photon mode which involves a $1/l^{2}$
contribution. This leads to a corresponding $1/l^{2}$ correction of the soft
factor. This result informs us that there must be $1/l^{2}$ corrections of
both $\partial_{w}\mathcal{N}$ and the gauge parameter $\varepsilon$ appearing
in the usual flat spacetime Ward identity given in 130. The reason for
$1/l^{2}$ corrections in $\partial_{w}\mathcal{N}$ also follows from $1/l^{2}$
corrections of the soft photon operator.

On the other hand, the soft factor informs us on the $1/l^{2}$ correction of
the gauge parameter $\varepsilon(z_{k}\,,\bar{z}_{k})$. This must involve the
terms appearing in the parenthesis of 144. However, we desire that the gauge
parameter have some of the properties of the asymptotically flat spacetime
gauge parameter given in 132. We accordingly define the total gauge parameter
to be

$\displaystyle\varepsilon(w\,,\bar{w})$
$\displaystyle=\varepsilon^{\text{flat}}(w\,,\bar{w})+\frac{1}{l^{2}}\varepsilon^{\text{AdS}}(w\,,\bar{w})\,;$
$\displaystyle\varepsilon^{\text{flat}}(w\,,\bar{w})$
$\displaystyle=\frac{1}{\omega-z}\,,$
$\displaystyle\varepsilon^{\text{AdS}}(w\,,\bar{w})$
$\displaystyle=\frac{(1+z\bar{z})^{2}}{\left(\bar{w}-\bar{z}\right)^{2}}\frac{(1+w\bar{w})^{2}}{\left(w-z\right)^{3}}$
$\displaystyle=\varepsilon^{\text{flat}}(w\,,\bar{w})\frac{(1+z\bar{z})^{2}}{\left(\bar{w}-\bar{z}\right)^{2}}\frac{(1+w\bar{w})^{2}}{\left(w-z\right)^{2}}\,.$
(147)

We thus have defined
$\varepsilon^{\text{AdS}}=\Omega~{}\varepsilon^{\text{flat}}$, where
$\Omega=\frac{(1+z\bar{z})^{2}}{\left(\bar{w}-\bar{z}\right)^{2}}\frac{(1+w\bar{w})^{2}}{\left(w-z\right)^{2}}$
is a factor invariant under the interchange of $w\leftrightarrow z$ and
$\bar{w}\leftrightarrow\bar{z}$. We note that this property ensures that
$\varepsilon^{\text{AdS}}$ does change sign under the interchange of $w$ with
$z$, as in the case of $\varepsilon^{\text{flat}}$. This requirement led us to
the include the factor $(1+w\bar{w})^{2}$ in the definition of
$\varepsilon^{\text{AdS}}$ apart from the term appearing in the parenthesis of
144.

As noted above, we also have $1/l^{2}$ corrections of the soft photon creation
and annihilation operators. We accordingly have a correction of
$\partial_{z}\mathcal{N}$

$\displaystyle\partial_{w}\mathcal{N}$
$\displaystyle=\partial_{w}\mathcal{N}^{\text{flat}}+\frac{1}{l^{2}}\partial_{w}\mathcal{N}^{\text{AdS}}\,;$
(148) $\displaystyle\partial_{w}\mathcal{N}^{\text{flat}}$
$\displaystyle=-\frac{\sqrt{2}}{8\pi}\frac{1}{1+w\bar{w}}\lim_{\omega\to
0}\left[\omega a_{+}^{\text{flat}}(\omega\hat{x})+\omega
a_{-}^{\text{flat}}(\omega\hat{x})^{\dagger}\right]\,.$ (149)

The goal is to determine the form of $\partial_{z}\mathcal{N}^{\text{AdS}}$
such that $1/l^{2}$ corrections of the Ward identity for large gauge
transformations in 130 gives the soft photon theorem in 145, with the factor
given in 144. By using the expressions from 147 and 148 in 130, and collecting
the $1/l^{2}$ coefficient, we find

$\displaystyle 4$ $\displaystyle\int
d^{2}w\left[\partial_{\bar{w}}\varepsilon^{\text{flat}}(w\,,\bar{w})\langle\text{out}|\partial_{w}\mathcal{N}^{\text{AdS}}(w\,,\bar{w})\mathcal{S}|\text{in}\rangle+\partial_{\bar{w}}\varepsilon^{\text{AdS}}(w\,,\bar{w})\langle\text{out}|\partial_{w}\mathcal{N}^{\text{flat}}(w\,,\bar{w})\mathcal{S}|\text{in}\rangle\right]$
$\displaystyle=\left[\sum_{k=\text{in}}\frac{\left(1+z_{k}\bar{z}_{k}\right)^{2}}{(\bar{z}-\bar{z}_{k})^{2}}\frac{\left(1+z\bar{z}\right)^{2}}{(z-z_{k})^{3}}Q_{k}-\sum_{k=\text{out}}\frac{\left(1+z_{k}\bar{z}_{k}\right)^{2}}{(\bar{z}-\bar{z}_{k})^{2}}\frac{\left(1+z\bar{z}\right)^{2}}{(z-z_{k})^{3}}Q_{k}\right]\langle\text{out}|\mathcal{S}|\text{in}\rangle\,.$
(150)

150 will agree with the correction to the soft photon theorem given in 146 if

$\displaystyle\frac{4}{l^{2}}$ $\displaystyle\int
d^{2}w\left[\partial_{\bar{w}}\varepsilon^{\text{flat}}(w\,,\bar{w})\langle\text{out}|\partial_{w}\mathcal{N}^{\text{AdS}}(w\,,\bar{w})\mathcal{S}|\text{in}\rangle+\partial_{\bar{w}}\varepsilon^{\text{AdS}}(w\,,\bar{w})\langle\text{out}|\partial_{w}\mathcal{N}^{\text{flat}}(w\,,\bar{w})\mathcal{S}|\text{in}\rangle\right]$
$\displaystyle=-\frac{16\sqrt{2}}{(1+z\bar{z})}\lim_{\omega\to
0}\langle\text{out}|\omega^{3}\tilde{a}_{+}(\omega\hat{x})\mathcal{S}|\text{in}\rangle\,.$
(151)

This can only be true if $\partial_{w}\mathcal{N}^{\text{AdS}}$ involves a sum
of two parts – a term $\partial_{w}\mathcal{N}^{\text{AdS}}_{1}$ which
provides the $\tilde{a}_{+}$ contribution and another term
$\partial_{w}\mathcal{N}^{\text{AdS}}_{2}$ which only involves
$a^{\text{flat}}_{+}$. We hence assume

$\partial_{w}\mathcal{N}^{\text{AdS}}=\partial_{w}\mathcal{N}^{\text{AdS}}_{1}+\partial_{w}\mathcal{N}^{\text{AdS}}_{2}\,,$
(152)

with

$\displaystyle\partial_{w}\mathcal{N}^{\text{AdS}}_{1}$
$\displaystyle=h(w\,,\bar{w})\lim_{\omega\to
0}\left[\omega^{3}a^{\text{AdS}}_{+}(\omega\hat{x})+\omega^{3}a^{\text{AdS}}_{-}(\omega\hat{x})^{\dagger}\right]\,,$
(153) $\displaystyle\partial_{w}\mathcal{N}^{\text{AdS}}_{2}$
$\displaystyle=g(w\,,\bar{w})\lim_{\omega\to 0}\left[\omega
a_{+}^{\text{flat}}(\omega\hat{x})+\omega
a_{-}^{\text{flat}}(\omega\hat{x})^{\dagger}\right]\,,$ (154)

where

$\tilde{a}_{\pm}(\omega\hat{x})=\frac{1}{l^{2}}a_{\pm}^{\text{AdS}}(\omega\hat{x})\,,\quad\tilde{a}_{\pm}(\omega\hat{x})^{\dagger}=\frac{1}{l^{2}}a_{\pm}^{\text{AdS}}(\omega\hat{x})^{\dagger}\,.$
(155)

We can now determine the expressions for $h(w\,,\bar{w})$ and $g(w\,,\bar{w})$
from the LHS of 150, which can be expressed as

$\displaystyle 4$ $\displaystyle\int
d^{2}w\left[\partial_{\bar{w}}\varepsilon^{\text{flat}}\langle\text{out}|\partial_{w}\mathcal{N}^{\text{AdS}}_{1}\mathcal{S}|\text{in}\rangle+\partial_{\bar{w}}\varepsilon^{\text{flat}}\langle\text{out}|\partial_{w}\mathcal{N}^{\text{AdS}}_{2}\mathcal{S}|\text{in}\rangle+\partial_{\bar{w}}\varepsilon^{\text{AdS}}\langle\text{out}|\partial_{w}\mathcal{N}^{\text{flat}}\mathcal{S}|\text{in}\rangle\right]$
$\displaystyle=4\int
d^{2}w\partial_{\bar{w}}\varepsilon^{\text{flat}}\langle\text{out}|\partial_{w}\mathcal{N}^{\text{AdS}}_{1}\mathcal{S}|\text{in}\rangle$
$\displaystyle\qquad\qquad\qquad-4\int
d^{2}w\left[\varepsilon^{\text{flat}}\langle\text{out}|\partial_{\bar{w}}\partial_{w}\mathcal{N}^{\text{AdS}}_{2}\mathcal{S}|\text{in}\rangle+\varepsilon^{\text{AdS}}\langle\text{out}|\partial_{\bar{w}}\partial_{w}\mathcal{N}^{\text{flat}}\mathcal{S}|\text{in}\rangle\right]\,.$
(156)

Hence the first term in the RHS of 156 gives us

$4\int
d^{2}w\partial_{\bar{w}}\varepsilon^{\text{flat}}\langle\text{out}|\partial_{w}\mathcal{N}^{\text{AdS}}_{1}\mathcal{S}|\text{in}\rangle=8\pi\langle\text{out}|\partial_{z}\mathcal{N}^{\text{AdS}}_{1}(z\,,\bar{z})\mathcal{S}|\text{in}\rangle\,.$
(157)

Using the expression from 153, we then find

$\displaystyle h(z\,,\bar{z})=-\frac{16\sqrt{2}}{8\pi(1+z\bar{z})}\,,$
$\displaystyle\Rightarrow\partial_{z}\mathcal{N}^{\text{AdS}}_{1}(z\,,\bar{z})=-\frac{16\sqrt{2}}{8\pi(1+z\bar{z})}\lim_{\omega\to
0}\left[\omega^{3}a^{\text{AdS}}_{+}(\omega\hat{x})+\omega^{3}a^{\text{AdS}}_{-}(\omega\hat{x})^{\dagger}\right]\,.$
(158)

We now also require that the last two terms in the RHS of 156 cancel, which
determines the $\partial_{z}\mathcal{N}^{\text{AdS}}_{2}$ contribution. From
taking the $\bar{w}$ derivative of $\partial_{w}\mathcal{N}^{\text{flat}}$ in
149 we have

$\partial_{\bar{w}}\partial_{w}\mathcal{N}^{\text{flat}}=\frac{\sqrt{2}}{8\pi}\frac{w}{(1+w\bar{w})^{2}}\lim_{\omega\to
0}\left[\omega a_{+}^{\text{flat}}(\omega\hat{x})+\omega
a_{-}^{\text{flat}}(\omega\hat{x})^{\dagger}\right]\,.$ (159)

Using 159 and the expressions for $\varepsilon^{\text{AdS}}(w\,,\bar{w})$ and
$\partial_{w}\mathcal{N}^{\text{AdS}}_{2}$ from 147 and 154 respectively, we
then find that the terms in the last line of 156 to be

$\displaystyle\int d^{2}w$
$\displaystyle\left[\varepsilon^{\text{flat}}\langle\text{out}|\partial_{\bar{w}}\partial_{w}\mathcal{N}^{\text{AdS}}_{2}\mathcal{S}|\text{in}\rangle+\varepsilon^{\text{AdS}}\langle\text{out}|\partial_{\bar{w}}\partial_{w}\mathcal{N}^{\text{flat}}\mathcal{S}|\text{in}\rangle\right]$
$\displaystyle=\int
d^{2}w\varepsilon^{\text{flat}}\left[\partial_{\bar{w}}g(w\,,\bar{w})+\frac{\sqrt{2}}{8\pi}\frac{(1+z\bar{z})^{2}}{\left(\bar{w}-\bar{z}\right)^{2}}\frac{w}{\left(w-z\right)^{2}}\right]\langle\text{out}|\omega
a_{+}^{\text{flat}}(\omega\hat{x})\mathcal{S}|\text{in}\rangle\,.$ (160)

Thus 160 vanishes if

$g(w\,,\bar{w})=-\int
d\bar{w}\frac{\sqrt{2}}{8\pi}\frac{(1+z\bar{z})^{2}}{\left(\bar{w}-\bar{z}\right)^{2}}\frac{w}{\left(w-z\right)^{2}}=\frac{\sqrt{2}}{8\pi}\frac{(1+z\bar{z})^{2}}{\left(\bar{w}-\bar{z}\right)}\frac{w}{\left(w-z\right)^{2}}\,.$
(161)

Hence from 154 the result for $\partial_{w}\mathcal{N}^{\text{AdS}}_{2}$ takes
the form

$\partial_{w}\mathcal{N}^{\text{AdS}}_{2}=\frac{\sqrt{2}}{8\pi}\frac{(1+z\bar{z})^{2}}{\left(\bar{w}-\bar{z}\right)}\frac{w}{\left(w-z\right)^{2}}\lim_{\omega\to
0}\left[\omega a_{+}^{\text{flat}}(\omega\hat{x})+\omega
a_{-}^{\text{flat}}(\omega\hat{x})^{\dagger}\right]\,.$ (162)

Thus the equation for the $1/l^{2}$ corrected Ward identity on asymptotically
flat spacetimes is

$\displaystyle 4\int d^{2}w$
$\displaystyle\left[\partial_{\bar{w}}\varepsilon^{\text{flat}}(w\,,\bar{w})\langle\text{out}|\partial_{w}\mathcal{N}^{\text{AdS}}(w\,,\bar{w})\mathcal{S}|\text{in}\rangle+\partial_{\bar{w}}\varepsilon^{\text{AdS}}(w\,,\bar{w})\langle\text{out}|\partial_{w}\mathcal{N}^{\text{flat}}(w\,,\bar{w})\mathcal{S}|\text{in}\rangle\right]$
$\displaystyle\qquad\qquad=\left[\sum_{k=\text{in}}Q_{k}\varepsilon^{\text{AdS}}(z_{k}\,,\bar{z}_{k})-\sum_{k=\text{out}}Q_{k}\varepsilon^{\text{AdS}}(z_{k}\,,\bar{z}_{k})\right]\langle\text{out}|\mathcal{S}|\text{in}\rangle\,,$
(163)

which agrees with the corrected soft photon theorem in 146 by choosing
$\varepsilon(w\,,\bar{w})$ as in 147, and
$\partial_{w}\mathcal{N}(w\,,\bar{w})$ as in 152.

To summarize our main result in this section, we argued that $1/l^{2}$
corrections in the soft photon factor arise due to a perturbation of the soft
photon theorem on asymptotically flat spacetimes. The perturbed soft theorem
was given in 146. Using the known equivalence between soft theorems and large
gauge Ward identities on asymptotically flat spacetimes, we could then derive
a perturbed Ward identity which is given in 163. In the process of
demonstrating this equivalence up to $1/l^{2}$, we determined the corrections
of the gauge parameter in 147 and the soft photon mode in 158 and 162. Our
results are defined across null infinity on the asymptotically flat patch that
arises in the large $l$ limit of asymptotically AdS spacetimes, as indicated
in Fig. 1.

##  7 Conclusion and open questions

Defining a quantum soft theorem in asymptotically AdS spaces is not only a
technically involved problem but also is an unclear issue as the notion of
asymptotic in and out states are not well defined in AdS spacetime. Thus an
alternative way to look for a possible soft factorization is required. The
analysis of Laddha:2018rle ; Laddha:2018myi  for asymptotically flat theories
showed that the soft factorization is also evident in classical radiation
profiles. For asymptotically flat spacetimes, the radiative parts of the
electromagnetic and gravitational fields produced in a classical scattering
process provide the same leading quantum soft factor obtained from $S$-matrix,
up to the usual gauge ambiguity. Therefore, we looked for a similar behaviour
in asymptotically AdS systems with a small cosmological constant and found a
similar factorization by assuming the cosmological constant as a perturbation
parameter over asymptotically flat gravity. By considering the large impact
parameter scattering of a probe particle with a black hole in AdS spacetime,
we derived the $1/l^{2}$ corrections due to the AdS potential to the known
leading $\omega^{-1}$ and subleading $\ln\omega^{-1}$ soft photon and soft
graviton factors of four dimensional asymptotically flat spacetimes.

In Hijano:2020szl  and Hijano:2019qmi , asymptotically flat spacetime
scattering amplitudes were shown to result from the $l\to\infty$ limit of AdS
boundary correlation functions. The scattering amplitudes resulting from this
limit are defined on the boundary of an asymptotically flat spacetime patch
around the center of AdS. A description of bulk fields in terms of boundary
operators, determined by using the HKLL formalism Hamilton:2006az , can
identify photon operators in the flat spacetime patch in terms of a $U(1)$
current. Additionally, the large gauge Ward identity for the soft photon
theorem on asymptotically flat spacetimes was shown to be equivalent to a
conformal Ward identity in taking the limit of $l\to\infty$ and $\omega\to 0$
simultaneously while keeping $\omega l$ fixed Hijano:2020szl . This precisely
corresponds to the double scaling limit considered in this paper, and in a
previous derivation of the soft graviton factor from classical scattering
processes on asymptotically AdS spacetimes Banerjee:2020dww .

Our analysis reveals that up to $1/l^{2}$ corrections, the double scaling
limit does not affect hard particle trajectories from their flat spacetime
expressions, while soft factors do involve corrections due to the
consideration of an asymptotically AdS spacetime. This suggests that
asymptotically flat spacetime large gauge Ward identities should be perturbed
from $1/l^{2}$ corrections of the soft particles, while being defined on the
same asymptotic boundaries for hard particles in the $l\to\infty$ of
asymptotically AdS spacetimes, i.e the flat spacetime patch around the center.
In Section 6, we made use of the universality of the leading soft photon
factor to investigate $1/l^{2}$ corrections to the flat spacetime large gauge
Ward identity in a massless scattering process. We determined that the
perturbed identity results from $1/l^{2}$ corrections of the asymptotically
flat spacetime soft photon mode and gauge parameter, which we derived up to an
overall factor. This derivation only assumed that large gauge Ward identities
and soft theorems on asymptotically flat spacetimes continue to remain
equivalent under small perturbations of the soft particles.

It will be interesting to relate the $1/l^{2}$ corrected soft photon Ward
identity with a CFT Ward identity on asymptotically AdS spacetimes. These
results can be derived using the formalism in Hijano:2020szl  and
Hijano:2019qmi , while now retaining $1/l^{2}$ terms in the large AdS radius
limit. The derivation lies outside the scope of this paper and will be
presented in future work.
Acknowledgements
We would like to thank Sayali Bhatkar and Arindam Bhattacharjee for useful
discussions. Our work is partially supported by a SERB ECR grant, GOVT of
India. Finally, we thank the people of India for their generous support to the
basic sciences.

##  Appendix A Classical soft graviton theorem

In this section we will consider the contribution of the scatterer black hole
charge to the gravitational soft factor. For a detailed calculation of the
gravitation soft factor as a contribution of potential due to the black hole
mass and AdS radius we refer the reader to Fernandes:2020tsq  and
Banerjee:2020dww  respectively.

$S_{\text{gr}}$ in $D=4$ dimensions can be computed using

$S_{\text{gr}}=i\frac{4\pi R}{e^{i\omega
R}}\epsilon^{ij}\tilde{e}_{ij}(\omega,\vec{x})\,.$ (164)

Similar to the the electromagnetic case, we let $r^{0}=t$, assume $x>>r(t)$
and we will write 56 as

$\displaystyle\tilde{e}_{ij}(\omega,\vec{x})$
$\displaystyle=\tilde{e}^{(1)}_{ij}(\omega,\vec{x})+\tilde{e}^{(2)}_{ij}(\omega,\vec{x})+\tilde{e}^{(3)}_{ij}(\omega,\vec{x})+\tilde{e}^{(4)}_{ij}(\omega,\vec{x})+\tilde{e}^{(5)}_{ij}(\omega,\vec{x})+\tilde{e}^{(6)}_{ij}(\omega,\vec{x})+\tilde{e}^{(7)}_{ij}(\omega,\vec{x})$
$\displaystyle\qquad\qquad+\tilde{e}^{(8)}_{ij}(\omega,\vec{x})+\tilde{e}^{(9)}_{ij}(\omega,\vec{x})+\tilde{e}^{(10)}_{ij}(\omega,\vec{x})\,,$
(165)

where

$\displaystyle\tilde{e}^{(1)}_{ij}(\omega,\vec{x})$
$\displaystyle=\frac{m~{}e^{i\omega
R}}{4\,\pi\,R}\frac{1}{1-\frac{x^{2}}{2l^{2}}}\int\frac{dt}{\left(1+2\Phi(\vec{r}(t))+\frac{r^{2}}{4l^{2}}\right)}\frac{dt}{d\sigma}\,v_{i}v_{j}\,e^{i\omega(t-\hat{n}.\vec{r}(t))}+\text{boundary
terms}\,,$ (166) $\displaystyle\tilde{e}^{(2)}_{ij}(\omega,\vec{x})$
$\displaystyle=\frac{m}{2\pi}\int dt\frac{dt}{d\sigma}\,e^{i\omega
t}(1+\vec{v}^{2})\,\left(\nabla_{i}\nabla_{j}-\frac{1}{2}\delta_{ij}\,\nabla_{k}\nabla_{k}\right)\tilde{G}_{M}\left(\omega,\vec{x},\vec{r}\right)\,,$
(167) $\displaystyle\tilde{e}^{(3)}_{ij}(\omega,\vec{x})$
$\displaystyle=-i\,\frac{Mm}{16\,\pi^{2}}\,\omega\,\frac{e^{i\omega R}}{R}\int
dt\frac{dt}{d\sigma}\,v_{i}\,v_{j}\,$ $\displaystyle\hskip
50.00008pt\bigg{\\{}\ln\frac{|\vec{r}\,^{\prime}|+\hat{n}.\vec{r}\,^{\prime}}{R}\,e^{i\omega(t-\hat{n}.\vec{r}\,^{\prime})}+\int_{|\vec{r}\,^{\prime}|+\hat{n}.\vec{r}\,^{\prime}}^{\infty}\frac{du}{u}e^{i\omega(t-\hat{n}.\vec{r}\,^{\prime}+u)}\bigg{\\}}\,,$
(168) $\displaystyle\tilde{e}^{(4)}_{ij}(\omega,\vec{x})$
$\displaystyle=\frac{i\omega m}{\pi}\int dt\frac{dt}{d\sigma}\,e^{i\omega
t}\left(v_{i}\nabla_{j}+v_{j}\nabla_{i}\right)\,\tilde{G}_{M}\left(\omega,\vec{x},\vec{r}\right)\,,$
(169) $\displaystyle\tilde{e}^{(5)}_{ij}(\omega,\vec{x})$
$\displaystyle=-i\omega\frac{qQ}{2\pi M}\int dte^{i\omega
t}\delta_{ij}v^{k}\nabla_{k}\tilde{G}_{M}\left(\omega,\vec{x},\vec{r}\right)\,,$
(170) $\displaystyle\tilde{e}^{(6)}_{ij}(\omega,\vec{x})$
$\displaystyle=i\frac{qQ}{\pi M}\int dte^{i\omega
t}\left(\nabla_{i}\nabla_{j}-\frac{1}{2}\delta_{ij}\,\nabla_{k}\nabla_{k}\right)\tilde{G}_{M}\left(\omega,\vec{x},\vec{r}\right)\,,$
(171) $\displaystyle\tilde{e}^{(7)}_{ij}(\omega,\vec{x})$
$\displaystyle=i\omega\frac{qQ}{2\pi M}\int dte^{i\omega
t}\left(v_{i}\nabla_{j}+v_{j}\nabla_{i}\right)\tilde{G}_{M}\left(\omega,\vec{x},\vec{r}\right)\,,$
(172) $\displaystyle\tilde{e}^{(8)}_{ij}(\omega,\vec{x})=$
$\displaystyle-\frac{m}{8\pi l^{2}}\int
dt\frac{dt}{d\sigma}\left[-~{}\left(2-\vec{v}^{2}\right)\nabla_{i}\nabla_{j}-\frac{3}{2}\delta_{ij}v_{k}v_{m}\nabla_{k}\nabla_{m}-\frac{1}{2}\delta_{ij}v^{2}\nabla_{k}\nabla_{k}\right.$
$\displaystyle\left.-~{}\frac{3}{2}\left(v_{k}v_{j}\nabla_{k}\nabla_{i}+v_{k}v_{i}\nabla_{k}\nabla_{j}\right)+v_{i}v_{j}\nabla_{k}\nabla_{k}\right]\tilde{G}_{l}\left(\omega,\vec{x},\vec{r}\right)\,,$
(173) $\displaystyle\tilde{e}^{(9)}_{ij}(\omega,\vec{x})=$
$\displaystyle-\frac{m}{8\pi l^{2}}\omega^{2}\int
dt\frac{dt}{d\sigma}e^{i\omega
t}v_{i}v_{j}\tilde{G}_{l}\left(\omega,\vec{x},\vec{r}\right)\,,$ (174)
$\displaystyle\tilde{e}^{(10)}_{ij}(\omega,\vec{x})=$
$\displaystyle\frac{m}{8\pi l^{2}}\int
dt\frac{dt}{d\sigma}\frac{5}{8}e^{i\omega
t}(v_{i}\nabla_{j}+v_{j}\nabla_{i})\tilde{G}_{l}\left(\omega,\vec{x},\vec{r}\right)\,,$
(175)

$\tilde{e}^{(1)}_{ij}(\omega,\vec{x})$ to
$\tilde{e}^{(4)}_{ij}(\omega,\vec{x})$ are contributions of the black hole
mass. $\tilde{e}^{(5)}_{ij}(\omega,\vec{x})$ to
$\tilde{e}^{(7)}_{ij}(\omega,\vec{x})$ arises due to the charge of the black
hole. $\tilde{e}^{(8)}_{ij}(\omega,\vec{x})$ to
$\tilde{e}^{(10)}_{ij}(\omega,\vec{x})$ can be treated as the contribution
from the AdS potential.

The contribution of black hole mass and charge to the soft factor for
gravitation was explicitly derived in Fernandes:2020tsq

$\displaystyle S_{\text{gr}}^{\text{flat}}$ $\displaystyle=i\frac{4\pi
R}{e^{i\omega R}}\epsilon^{ij}\tilde{e}_{ij}(\omega,\vec{x})$
$\displaystyle=-\frac{m}{\omega}\epsilon^{ij}\left[\frac{1}{1-\hat{n}.\vec{\beta}_{+}}\frac{1}{\sqrt{1-\vec{\beta}^{2}_{+}}}\beta_{+i}\beta_{+j}-\frac{1}{1-\hat{n}.\vec{\beta}_{-}}\frac{1}{\sqrt{1-\vec{\beta}^{2}_{-}}}\beta_{-i}\beta_{-j}\right]$
$\displaystyle\quad-
im\ln\omega^{-1}\epsilon^{ij}\,\left[{1\over\sqrt{1-\vec{\beta}_{+}^{2}}}\,\left\\{C_{+}\left({1\over
1-\hat{n}.\vec{\beta}_{+}}+{1\over 1-\vec{\beta}_{+}^{2}}\right)-{M\over
8\,\pi\,|\vec{\beta}_{+}|^{3}}\,{3\vec{\beta}_{+}^{2}-1\over
1-\vec{\beta}_{+}^{2}}\right\\}\beta_{+i}\beta_{+j}\right.$
$\displaystyle\left.-{1\over\sqrt{1-\vec{\beta}_{-}^{2}}}\,\left\\{C_{-}\left({1\over
1-\hat{n}.\vec{\beta}_{-}}+{1\over 1-\vec{\beta}_{-}^{2}}\right)+{M\over
8\,\pi\,|\vec{\beta}_{-}|^{3}}{3\vec{\beta}_{-}^{2}-1\over
1-\vec{\beta}_{-}^{2}}\right\\}\beta_{-i}\beta_{-j}\right]$
$\displaystyle\qquad-
im\frac{M}{4\pi}\ln\left(R\omega\right)\epsilon^{ij}\left[\frac{1}{1-\hat{n}.\vec{\beta}_{+}}\frac{1}{\sqrt{1-\vec{\beta}^{2}_{+}}}\beta_{+i}\beta_{+j}-\frac{1}{1-\hat{n}.\vec{\beta}_{-}}\frac{1}{\sqrt{1-\vec{\beta}^{2}_{-}}}\beta_{-i}\beta_{-j}\right]$
$\displaystyle\quad\qquad-i\frac{qQ}{4\pi}\ln\omega^{-1}\epsilon^{ij}\,\left[\frac{\beta_{+i}\beta_{+j}}{|\vec{\beta}_{+}|^{3}}+\frac{\beta_{-i}\beta_{-j}}{|\vec{\beta}_{-}|^{3}}\right]+\text{finite}\,.$
(176)

To complete the analysis we need the expression for $C_{\pm}$, which can be
determined from considering the energy conservation equation. The energy of
the probe particle can be written from the point particle action in 16 as

$\displaystyle E=m|g_{00}|\frac{dt}{d\sigma}-\frac{q}{4\pi}A_{0}\,.$ (177)

Expanding this expression in powers of $t$, we then find from the
$\frac{1}{t}$ coefficient the following relation of $C_{\pm}$ with $M$ and $Q$
Fernandes:2020tsq

$\displaystyle C_{\pm}=\mp{M\over
8\,\pi\,|\vec{\beta}_{\pm}|^{3}}(1-3\vec{\beta}_{\pm}^{2})\mp\frac{qQ}{4\pi
m|\vec{\beta}_{\pm}|^{3}}(1-\vec{\beta}_{\pm}^{2})^{3/2}\,.$ (178)

Substituting for $M$ and $Q$ in 176 using 178 gives

$\displaystyle S_{\text{gr}}^{\text{flat}}=$
$\displaystyle-\frac{m}{\omega}\epsilon^{ij}\left[\frac{1}{1-\hat{n}.\vec{\beta}_{+}}\frac{1}{\sqrt{1-\vec{\beta}^{2}_{+}}}\beta_{+i}\beta_{+j}-\frac{1}{1-\hat{n}.\vec{\beta}_{-}}\frac{1}{\sqrt{1-\vec{\beta}^{2}_{-}}}\beta_{-i}\beta_{-j}\right]$
$\displaystyle-
im\ln\omega^{-1}\epsilon^{ij}\left[C_{+}\frac{1}{1-\hat{n}.\vec{\beta}_{+}}\frac{1}{\sqrt{1-\vec{\beta}^{2}_{+}}}\beta_{+i}\beta_{+j}-C_{-}\frac{1}{1-\hat{n}.\vec{\beta}_{-}}\frac{1}{\sqrt{1-\vec{\beta}^{2}_{-}}}\beta_{-i}\beta_{-j}\right]$
$\displaystyle-
im\frac{M}{4\pi}\ln\left(R\omega\right)\epsilon^{ij}\left[\frac{1}{1-\hat{n}.\vec{\beta}_{+}}\frac{1}{\sqrt{1-\vec{\beta}^{2}_{+}}}\beta_{+i}\beta_{+j}-\frac{1}{1-\hat{n}.\vec{\beta}_{-}}\frac{1}{\sqrt{1-\vec{\beta}^{2}_{-}}}\beta_{-i}\beta_{-j}\right]\,.$
(179)

This result agrees with the classical limit of the soft graviton factor, up to
the subleading logarithmic contribution, in asymptotically flat spacetimes. In
Banerjee:2020dww , the leading order contribution of cosmological constant to
the soft factor was derived with the following expression

$\displaystyle S_{\rm gr}^{l}=$
$\displaystyle-\frac{m}{2\gamma^{2}}\,\omega^{-1}\,\epsilon^{ij}\left\\{\frac{1}{(1-\hat{n}.\vec{\beta})^{3}}\beta_{+i}\beta_{+j}\frac{\vec{\beta}_{+}^{2}(3-2\vec{\beta}_{+}^{2})}{(1-\vec{\beta}_{+}^{2})^{\frac{3}{2}}}-\frac{1}{(1-\hat{n}.\vec{\beta})^{3}}\beta_{-i}\beta_{-j}\frac{\vec{\beta}_{-}^{2}(3-2\vec{\beta}_{-}^{2})}{(1-\vec{\beta}_{-}^{\frac{3}{2}})}\right\\}$
$\displaystyle-i\,\frac{m}{2\gamma^{2}}\,\ln\omega^{-1}\,\epsilon^{ij}\,\left[\frac{\beta_{+i}\beta_{+j}}{(1-\hat{n}.\vec{\beta}_{+})^{3}}\frac{C_{+}\vec{\beta}_{+}^{2}(3-2\vec{\beta}_{+}^{2})}{\left(1-\vec{\beta}_{+}^{2}\right)^{\frac{3}{2}}}-\frac{\beta_{-i}\beta_{-j}}{(1-\hat{n}.\vec{\beta}_{-})^{3}}\frac{C_{-}\vec{\beta}_{-}^{2}(3-2\vec{\beta}_{-}^{2})}{\left(1-\vec{\beta}_{-}^{2}\right)^{\frac{3}{2}}}\right]\,.$
(180)

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