CALT-TH 2024-012

An On-Shell Derivation of the
Soft Effective Action in Abelian Gauge Theories

Temple He‡, Prahar Mitra◇, Allic Sivaramakrishnan‡, Kathryn M. Zurek‡

‡Walter Burke Institute for Theoretical Physics, California Institute of
Technology,
Pasadena, CA 91125 USA
◇Institute for Theoretical Physics, University of Amsterdam, Science Park 904,
Postbus 94485, 1090 GL Amsterdam, The Netherlands

We derive the soft effective action in $(d+2)$-dimensional abelian gauge
theories from the on-shell action obeying Neumann boundary conditions at
timelike and null infinity and Dirichlet boundary conditions at spatial
infinity. This allows us to identify the on-shell degrees of freedom on the
boundary with the soft modes living on the celestial sphere. Following the
work of Donnelly and Wall, this suggests that we can interpret soft modes as
entanglement edge modes on the celestial sphere and study entanglement
properties of soft modes in abelian gauge theories.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Notations and Conventions
    2. 2.2 Soft Factorization
      1. 2.2.1 Real Soft Photons
      2. 2.2.2 Virtual Soft Photons
    3. 2.3 Soft Effective Action
  3. 3 Soft On-Shell Action $\to$ Soft Effective Action
    1. 3.1 Solutions to Maxwell’s Equations
    2. 3.2 On-Shell Action
      1. 3.2.1 Bulk Term
      2. 3.2.2 Boundary Terms
  4. 4 Summary
  5. A Relating Current to its Shadow
  6. B Massive Particles

##  1 Introduction

The infrared (IR) sector of quantum field theories (QFTs) has recently enjoyed
much attention, primarily due to the seminal work of Strominger [1, 2], which
showed that Weinberg’s leading soft graviton theorem [3] is the Ward identity
for the BMS supertranslation symmetry [4, 5]. This relationship between soft
theorems and the so-called _asymptotic symmetries_ is now understood to be a
universal feature of all gauge and gravitational theories in asymptotically
flat spacetimes (see [6, 7, 8] for a review). For example, the leading soft
photon or soft gluon theorems in gauge theories [3, 9] are the Ward identity
for large gauge symmetries [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], and
the subleading soft graviton theorem in gravitational theories [21] is the
Ward identity for BMS superrotations [22, 23].

A particularly important consequence of these symmetries is that gauge and
gravitational theories do not have a unique vacuum state. Rather, they have
infinitely many vacua, all related via action by the asymptotic symmetry
charge. In other words, the asymptotic symmetry is spontaneously broken by the
choice of vacuum state.111The vacuum degeneracy being discussed here is _not_
the one associated with the ${\theta}$-angle in gauge theories, which is
related to gauge transformations that are constant on the boundary but are
topologically non-trivial (i.e., they have a non-zero winding number). We are
interested in degeneracy due to gauge transformations that are non-constant on
the boundary but are topologically trivial. The Goldstone mode ${\theta}(x)$
associated with this spontaneous breaking lives on the celestial sphere
${\mathbb{S}}^{d}$, the codimension-two boundary of the spacetime (the bulk
spacetime dimension is $d+2$). The symplectic conjugate of the Goldstone mode
is the so-called _soft photon operator_ $\phi(x)$, which inserts a soft (low
energy) photon in a scattering amplitude [18]. Together, the fields
${\theta},\phi$ constitute the low energy sector of the theory. They live on a
codimension-two boundary of the spacetime and, in this sense, are the boundary
or edge modes of abelian gauge theories.

The effective dynamics of the edge modes are described by a codimension-two
action which was constructed in [24] (similar actions in various other forms
have previously appeared in [25, 26, 27, 28, 29] as well, though they all
exclusively work in four spacetime dimensions). The so-called _soft effective
action_ reproduces all universal soft features of abelian gauge theories,
namely Weinberg’s leading soft photon theorem [3] and the IR factorization of
scattering amplitudes (IR divergences in four dimensions) [30]. More
precisely, given a gauge theory with IR cutoff $\mu$, and denoting the energy
scale separating the soft modes from the hard ones as ${\Lambda}$, scattering
amplitudes take the form

$\displaystyle\langle\,\phi(x_{1})\cdots\phi(x_{m}){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\mu}=\left({\mathcal{J}}(x_{1})\cdots{\mathcal{J}}(x_{m})e^{-{\Gamma}(\mu,{\Lambda})}\right)\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\Lambda},$
(1.1)

where $\langle\cdots\rangle_{E}$ denotes a scattering amplitude evaluated with
IR cutoff $E$, ${\mathcal{O}}_{k}$ are the hard insertions (energy above the
scale ${\Lambda}$), $\phi(x_{i})$ are the soft photon insertion (with energy
below ${\Lambda}$ but above $\mu$), ${\mathcal{J}}(x_{i})$ are related to the
leading Weinberg soft factor, and ${\Gamma}(\mu,{\Lambda})$ is the
contribution from virtual soft photons. Thus, we see the amplitude factorizes
into a “hard amplitude” and a soft factor that receives contributions from
virtual soft photons and external soft photons. When $\mu$ and ${\Lambda}$ are
small compared to all the other energy scales in the amplitude, the soft
factor is universal and can be reproduced by a path integral of the form

$\displaystyle\int[{{\mathsf{d}}}\phi][{{\mathsf{d}}}{\theta}]\,e^{-S_{\text{eff}}[\phi,{\theta}]}\phi(x_{1})\cdots\phi(x_{m})={\mathcal{J}}(x_{1})\cdots{\mathcal{J}}(x_{m})e^{-{\Gamma}[\mu,{\Lambda}]},$
(1.2)

where the effective action $S_{\text{eff}}[\phi,{\theta}]$ for the soft or
edge modes is given by (see Section 2.3 for details)

$\begin{split}S_{{\text{eff}}}[\phi,{\theta}]={\alpha}\int_{{\mathbb{S}}^{d}}\frac{{{\mathsf{d}}}^{d}x}{(2\pi)^{d}}\big{(}\partial_{a}\phi(x)\big{)}^{2}-\frac{i}{2c_{1,1}}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,\widetilde{\partial^{a}{\theta}}(x)\big{(}\partial_{a}\phi(x)-\partial_{a}{\mathcal{J}}(x)\big{)},\end{split}$
(1.3)

where ${\mathbb{S}}^{d}$ is the celestial sphere. Interestingly, this action
is neither real nor local (the tilde superscript denotes the shadow transform
(2.26), which is a non-local integral transform). It was constructed in [24]
using the asymptotic symmetries of the theory (in this case, large gauge
transformations) and relied heavily on effective field theory techniques.
Therefore, it is interesting to ask whether the action (1.3) involving soft
modes can be derived directly from the bulk action, whose on-shell degrees of
freedom are the edge modes.

Edge modes in gauge theories were introduced to the study of entanglement
entropy by Donnelly [31]. It is known from [32] that the entanglement entropy
of Maxwell theory in $d+2$ dimensions is equal to that of $d$ scalar fields
plus an additional contact term, whose physical significance was not clear. In
[33, 34], Donnelly and Wall showed that this contact term _is_ physical and
is, in fact, the contribution of edge modes to the entanglement entropy. They
further showed that the effective action for the edge modes could be obtained
by evaluating the Maxwell action on-shell with “magnetic conductor boundary
conditions.” These “entanglement edge modes” also live on a codimension-two
boundary of the spacetime (specifically, on the entangling surface), and given
their remarkable similarity to the “soft edge modes” on the celestial sphere
appearing in (1.3), it is reasonable to expect that they are in fact related
(for example, see [35]). In this paper, we prove that this is indeed the case
by showing that the soft contribution to the on-shell action of abelian gauge
theories is _exactly_ equal to the soft effective action (1.3). There are,
however, two crucial ways in which our setup differs from that of Donnelly and
Wall in [33, 34].

Firstly, as was remarked previously, Donnelly and Wall imposed magnetic
conductor boundary conditions, which fixes $B_{\|}=0$ and $E_{\perp}$, so that
the effective action for the entanglement edge modes is a function of
$E_{\perp}$.222The entanglement entropy of the entanglement edge modes is then
determined by evaluating the path integral over the modes $E_{\perp}$. On the
other hand, the soft degrees of freedom that we are interested in live on a
cut of asymptotic null infinity ${\mathcal{I}}^{\pm}$, so we will instead
impose Neumann boundary conditions, which allows non-trivial radiation flux
through the boundary. This requires us to add extra boundary terms similar to
Gibbons-Hawking-York (GHY) terms in general relativity to the Maxwell action,
so that

$\begin{split}S[A,\Phi]=S_{{\mathcal{M}}}[A,\Phi]+\frac{1}{e^{2}}\int_{{\Sigma}^{+}}A\wedge\star
F-\frac{1}{e^{2}}\int_{{\Sigma}^{-}}A\wedge\star F,\end{split}$ (1.4)

where $S_{\mathcal{M}}$ is the bulk action (including matter fields $\Phi$),
and $\Sigma^{\pm}\equiv{\mathcal{I}}^{\pm}\cup i^{\pm}$ are the non-spacelike
boundaries. Secondly, the edge mode contribution to the entanglement entropy
studied in [33, 34] is an ultraviolet (UV) effect, which arises from degrees
of freedom living close to the entangling surface and is dealt with in the
usual way through renormalization. However, in our analysis, since the surface
of interest lives on the _asymptotic_ boundary of spacetime, we have to deal
with additional IR divergences (at least in four dimensions). Therefore, we
must be more careful about how to evaluate the bulk part of the action (1.4)
on-shell, and appropriately determine the $i{\epsilon}$ prescription in the
Lorentzian path integral.333 This was not an issue for Donnelly and Wall in
[33, 34], as their entanglement entropy calculation was done in Euclidean
signature. Our goal is to show that once these subtleties are dealt with, the
relation between the soft contribution to the on-shell action and the soft
effective action is given by

$\displaystyle\begin{split}S[A,\Phi]\big{|}_{\text{soft}\text{+on-
shell}}=iS_{\text{eff}}[\phi,\theta],\end{split}$ (1.5)

where the extra factor of $i$ is present due to the fact that $S[A,\Phi]$ is a
Lorentzian action whereas $S_{\text{eff}}$ is Euclidean (1.2). This is the
main result of our work.

Because the entanglement edge modes that are studied by Donnelly and Wall [33,
34], albeit using different boundary conditions, are precisely the on-shell
modes living on a codimension-two boundary, our result (1.5) solidifies the
connection between the entanglement edge modes and the soft modes obtained
from a symplectic analysis [18, 19]. This is perhaps not entirely surprising,
as it is natural in many regards to identify the soft modes with entanglement
edge modes, both of which live on codimension-two surfaces. Nevertheless, we
view the novel feature in our analysis to be the determination of precisely
which boundary conditions allow us to establish an equivalence between the two
types of boundary modes.

Relating soft and entanglement edge modes in gauge theory lays the foundation
for doing the same in gravity. Gravitational edge modes enter into the study
of subregions in gravity, where they help answer the question: What are the
degrees of freedom associated with a subregion in gravity? We therefore
anticipate that applying our approach to gravity may connect soft modes in
gravity to entanglement edge modes and, in turn, to objects utilized to
diagnose entanglement, such as the modular Hamiltonian proposed in [36]. For
instance, by determining the appropriate GHY boundary terms needed such that
the soft limit of the on-shell action reproduces the soft effective action in
gravity, we may conclude that the corresponding boundary conditions for the
gravitational edge modes are a “natural” choice. We leave such directions for
future work.

This paper is organized as follows. We will introduce the preliminaries
involving soft theorems and soft factorization of amplitudes in Section 2. In
Section 3, we will perform the computation that establishes the equivalence
between the on-shell action capturing the edge mode degrees of freedom and the
soft effective action. We summarize our results in Section 4. In Appendix A,
we prove a technical identity that relates the matter current to its shadow
transform, which is instrumental in showing the equivalence between the two
actions. In Appendix B, we also take into account massive matter particles
that may be present in the theory.

##  2 Preliminaries

We begin by establishing the necessary prerequisites. In Section 2.1, we
introduce the notation and conventions used throughout this paper. We will be
following those given in Appendix A of both [17] and [19], where more details
can be found. In Section 2.2, we present a brief review of soft factorization
in scattering amplitudes and introduce the soft effective action derived in
[24].

###  2.1 Notations and Conventions

##### Position Space Coordinates:

Our theory lives in $(d+2)$-dimensional Minkowski spacetime,
${\mathcal{M}}={\mathbb{R}}^{1,d+1}$, and for computational simplicity we will
work in flat null coordinates $x^{\mu}=(u,x^{a},r)$, where
$u,r\in{\mathbb{R}}$ and $x^{a}\in{\mathbb{R}}^{d}$. These are related to
Cartesian coordinates $X^{A}$ by

$\begin{split}X^{A}=r{\hat{q}}^{A}(x)+un^{A},\qquad{\hat{q}}^{A}(x)=\left(\frac{1+x^{2}}{2},x^{a},\frac{1-x^{2}}{2}\right),\qquad
n^{A}=\left(\frac{1}{2},0^{a},-\frac{1}{2}\right).\end{split}$ (2.1)

Note that ${\hat{q}}^{A}(x)$ and $n^{A}$ are null and
$n\cdot{\hat{q}}(x)=-\frac{1}{2}$. It follows the Minkowski line element in
flat null coordinates is given by

$\begin{split}{{\mathsf{d}}}s^{2}=\eta_{AB}\,{{\mathsf{d}}}X^{A}\,{{\mathsf{d}}}X^{B}=-{{\mathsf{d}}}u\,{{\mathsf{d}}}r+r^{2}{\delta}_{ab}\,{{\mathsf{d}}}x^{a}\,{{\mathsf{d}}}x^{b}.\end{split}$
(2.2)

We will throughout this paper use lowercase Greek letters $\mu,\nu,\ldots$ to
denote flat null coordinates and capital Latin letters $A,B,\ldots$ to denote
Cartesian coordinates. Lowercase Latin indices denote the transverse
directions along the celestial sphere ${\mathbb{S}}^{d}$ and are raised and
lowered by the Cartesian metric $\delta_{ab}$.

The null boundaries ${\mathcal{I}}^{\pm}$ are located at $r\to\pm\infty$ while
keeping $(u,x)$ fixed, and their topology is given by
${\mathbb{R}}\times{\mathbb{S}}^{d}$. The past (future) boundary of
${\mathcal{I}}^{+}$ (${\mathcal{I}}^{-}$) is located at $u=-\infty$
($u=+\infty$) and is denoted by ${\mathcal{I}}^{+}_{-}$
(${\mathcal{I}}^{-}_{+}$). The point labeled by coordinate $x^{a}$ on
${\mathcal{I}}^{+}$ is antipodal to the point with the same coordinate value
on ${\mathcal{I}}^{-}$.444The antipodal point on ${\mathbb{S}}^{d}$ can be
defined by embedding ${\mathbb{S}}^{d}\hookrightarrow{\mathbb{R}}^{d+1}$,
which maps $x^{a}\mapsto\vec{X}(x)$, where $\vec{X}\cdot\vec{X}=1$. On
${\mathbb{R}}^{d+1}$, the antipodal map is given by $\vec{X}\mapsto-\vec{X}$.
In these coordinates, the integration of forms on ${\mathcal{M}}$,
${\mathcal{I}}^{\pm}$ and ${\mathcal{I}}^{\pm}_{\mp}$ are given by (we follow
the conventions outlined in Appendix A of [19])

$\begin{split}\int_{\mathcal{M}}C_{d+2}&=-\frac{1}{2}\int_{{\mathbb{R}}}{{\mathsf{d}}}u\int_{\mathbb{R}}{{\mathsf{d}}}r\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,|r|^{d}(\star
C_{d+2}),\\\
\int_{{\mathcal{I}}^{\pm}}C_{d+1}&=-\frac{1}{2}\int_{\mathbb{R}}{{\mathsf{d}}}u\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\left(\lim_{r\to\pm\infty}|r|^{d}(\star
C_{d+1})^{r}\right),\\\
\int_{{\mathcal{I}}^{\pm}_{\mp}}C_{d}&=\frac{1}{2}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\left(\lim_{u\to\mp\infty}\lim_{r\to\pm\infty}|r|^{d}(\star
C_{d})^{ur}\right),\end{split}$ (2.3)

where $C_{p}$ denotes a $p$-form.

##### Momentum Space Coordinates:

An off-shell momentum is parametrized by

$\begin{split}\ell^{A}={\omega}\big{(}{\hat{q}}^{A}(x)+{\kappa}n^{A}\big{)},\end{split}$
(2.4)

where $\ell^{2}=-\kappa\omega^{2}$. The off-shell integration measure is

$\begin{split}\int_{\mathcal{M}}\frac{{{\mathsf{d}}}^{d+2}\ell}{(2\pi)^{d+2}}=\frac{1}{4\pi}\int_{|{\omega}|>\mu}{{\mathsf{d}}}{\omega}\,|{\omega}|^{d+1}\int_{{\mathbb{S}}^{d}}\frac{{{\mathsf{d}}}^{d}x}{(2\pi)^{d}}\int_{\mathbb{R}}\frac{{{\mathsf{d}}}{\kappa}}{2\pi},\end{split}$
(2.5)

where to deal with IR divergences, all momentum space integrals are performed
with a cut-off $\mu$, which is taken to be much smaller than all other scales
in the problem. Similarly, we use the following parametrization for on-shell
momenta:

$\begin{split}p^{A}={\omega}\left[{\hat{q}}^{A}(x)+\left(\frac{m^{2}}{{\omega}^{2}}\right)n^{A}\right],\qquad
p^{2}=-m^{2}.\end{split}$ (2.6)

The properties and advantages of using the flat null coordinates for position
and momenta were further expounded in Appendix A of [17].

##### Scattering Amplitudes:

Given an IR cutoff $\mu$, an $n$-point scattering amplitude can be written as
a time-ordered vacuum correlation function, such that

$\begin{split}A_{n}=\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\mu},\end{split}$
(2.7)

where we denoted555This definition for ${\mathcal{O}}_{k}$ is the one
implemented by the LSZ reduction formula (see Section 4.3 of [18]).

$\begin{split}{\mathcal{O}}_{k}\equiv{\theta}({\omega}_{k})\Big{[}{\mathcal{O}}^{+}_{k}({\omega}_{k}{\hat{q}}(x_{k}))-{\mathcal{O}}^{-}_{k}({\omega}_{k}{\hat{q}}(x_{k}))\Big{]}+{\theta}(-{\omega}_{k})\Big{[}{\mathcal{O}}^{-}_{k}(-{\omega}_{k}{\hat{q}}(x_{k}))^{\dagger}-{\mathcal{O}}^{+}_{k}(-{\omega}_{k}{\hat{q}}(x_{k}))^{\dagger}\Big{]}.\end{split}$
(2.8)

Here, ${\theta}({\omega})$ is the Heaviside step function, the $\pm$
superscript corresponds to either the outgoing ($+$) or incoming ($-$) mode,
and ${\mathcal{O}}_{k}^{\pm}$ (${\mathcal{O}}_{k}^{\pm\dagger}$) is the
annihilation (creation) operator for the $k$th particle. Furthermore, we
denote the operator that inserts a photon with momentum
$q^{A}={\omega}{\hat{q}}^{A}(x)$ and polarization $a$ by
${\mathcal{O}}_{a}({\omega},x)$. The corresponding polarization vector is
given by

$\begin{split}{\varepsilon}^{A}_{a}(x)=\partial_{a}{\hat{q}}^{A}(x)=\big{(}x_{a},{\delta}_{a}^{b},-x_{a}\big{)}.\end{split}$
(2.9)

###  2.2 Soft Factorization

####  2.2.1 Real Soft Photons

Weinberg’s leading soft photon theorem [3] states that a scattering amplitude
with $m$ photons, each with momentum $q_{i}$ and polarization $a_{i}$, and $n$
hard particles, each with momentum $p_{k}$ and $U(1)$ charge
$Q_{k}\in{\mathbb{Z}}$, factorizes in the leading soft limit ($q_{i}^{0}\ll
p_{k}^{0}$ for all $i$ and $k$) as666In abelian gauge theories, the soft limit
can be taken either consecutively or simultaneously without any ambiguity.
This is no longer the case for nonabelian gauge theories.

$\begin{split}A_{m+n}\quad&\xrightarrow{q_{i}\to
0}\quad\SS_{m}^{(0)}A_{n},\qquad\SS_{m}^{(0)}\equiv\prod_{i=1}^{m}\left(e\sum_{k=1}^{n}Q_{k}\frac{p_{k}\cdot{\varepsilon}_{a_{i}}(q_{i})}{p_{k}\cdot
q_{i}-i{\epsilon}}\right),\end{split}$ (2.10)

where the superscript on $\SS_{m}^{(0)}$ signifies this is the _leading_ soft
factor. To recast this into a cleaner form, we will utilize the notation
introduced in the previous subsection. First, we define the soft photon
operator

$\begin{split}N_{a}(x)\equiv\frac{1}{2e}\left(\lim_{{\omega}\to
0^{+}}+\lim_{{\omega}\to
0^{-}}\right)\big{(}{\omega}{\mathcal{O}}_{a}({\omega},x)\big{)}=N_{a}^{+}(x)-N_{a}^{-}(x),\end{split}$
(2.11)

where $N_{a}^{\pm}(x)$ are the Hermitian $out$ and $in$ soft photon operators
[18],777In (2.12), we have used ${\theta}(0)=\frac{1}{2}$. This choice is not
arbitrary and is rather precisely what we get if we use the LSZ representation
of the operators defined in (2.8) and then take the limit in (2.11) (see
Section of 4.3 of [18]).

$\begin{split}N_{a}^{\pm}(x)\equiv\lim_{{\omega}\to
0^{+}}\frac{1}{e}{\omega}{\mathcal{O}}^{\pm}_{a}({\omega}{\hat{q}}(x))=N_{a}^{\pm}(x)^{\dagger}.\end{split}$
(2.12)

Note that the factor of ${\omega}$ is needed to cancel the simple pole in the
soft factor at $q_{i}=0$ so that the soft limit is well-defined. Using this,
(2.10) can be written as [24]

$\begin{split}\langle\,N_{a_{1}}(x_{1})\cdots
N_{a_{m}}(x_{m}){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\mu}={\mathcal{J}}_{a_{1}}(x_{1})\cdots{\mathcal{J}}_{a_{m}}(x_{m})\,\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\mu},\end{split}$
(2.13)

where

$\begin{split}{\mathcal{J}}_{a}(x)\equiv\partial_{a}\sum_{k=1}^{n}Q_{k}\ln|p_{k}\cdot{\hat{q}}(x)|=\partial_{a}{\mathcal{J}}(x).\end{split}$
(2.14)

From (2.13) and (2.14), it is clear that when inserted into an $S$-matrix
element, $N_{a}(x)$ satisfies the constraint

$\begin{split}\partial_{[a}N_{b]}(x)=0\quad\implies\quad
N_{a}(x)=\partial_{a}\phi(x).\end{split}$ (2.15)

A more careful derivation of this constraint by demanding the invertibility of
the symplectic form was given in [18].

####  2.2.2 Virtual Soft Photons

Scattering amplitudes in four-dimensional gauge theories formally vanish due
to IR divergences. In the perturbative expansion, these arise from diagrams
involving exchanges of virtual photons. Each diagram is separately divergent,
but the infinite sum exponentiates, and the full amplitude vanishes.
Introducing an IR cutoff $\mu$ to regulate the divergences, one finds that an
$n$-point amplitude has the form (see Chapter 13 of [37] for details)

$\begin{split}A_{n}=e^{-{\Gamma}(\mu,{\Lambda})}{\tilde{A}}_{n},\end{split}$
(2.16)

where ${\Gamma}(\mu,{\Lambda})$ captures the IR divergences, ${\Lambda}$ is
the energy scale demarcating the soft from the hard modes, i.e.,
$\mu\ll{\Lambda}\ll|p_{k}^{0}|$ for all $k$, and ${\tilde{A}}_{n}$ is an IR
finite amplitude. In abelian gauge theories, the explicit form of ${\Gamma}$
can be easily worked out to be888The $i{\epsilon}$ prescription is a bit
different when $k=k^{\prime}$, in which case
$p_{k^{\prime}}\cdot\ell+i{\epsilon}$ is replaced with
$p_{k}\cdot\ell-i{\epsilon}$.

$\begin{split}{\Gamma}(\mu,{\Lambda})=-\frac{ie^{2}}{2}\sum_{k,k^{\prime}=1}^{n}Q_{k}Q_{k^{\prime}}\int_{\mu}^{\Lambda}\frac{{{\mathsf{d}}}^{d+2}\ell}{(2\pi)^{d+2}}\frac{p_{k}\cdot
p_{k^{\prime}}}{(\ell^{2}-i{\epsilon})(p_{k}\cdot\ell-i{\epsilon})(p_{k^{\prime}}\cdot\ell+i{\epsilon})},\end{split}$
(2.17)

where the integration limits above denote integration over the regime
$\mu<|{\omega}|<{\Lambda}$. The explicit form of ${\Gamma}$ was determined in
[24] to be999Actually, ${\Gamma}$ also has an imaginary part that was
calculated in [24], but the soft effective action constructed there (and the
one we discuss here) does not address this piece. Hence, we will also only
focus on the real part here, and leave the imaginary part of ${\Gamma}$ to be
discussed in future work.

$\begin{split}{\Gamma}={\alpha}\int_{{\mathbb{S}}^{d}}\frac{{{\mathsf{d}}}^{d}x}{(2\pi)^{d}}\big{(}{\mathcal{J}}_{a}(x)\big{)}^{2},\qquad{\alpha}=\frac{e^{2}}{8\pi}\int_{\mu}^{\Lambda}{{\mathsf{d}}}{\omega}\,{\omega}^{d-3}.\end{split}$
(2.18)

We can then write (2.16) as

$\begin{split}\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\mu}=\exp\left[-{\alpha}\int_{{\mathbb{S}}^{d}}\frac{{{\mathsf{d}}}^{d}x}{(2\pi)^{d}}\big{(}{\mathcal{J}}_{a}(x)\big{)}^{2}\right]\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\Lambda}.\end{split}$
(2.19)

Note from (2.18) that ${\alpha}\to\infty$ as we remove the IR cutoff by taking
$\mu\to 0$ in four dimensions ($d=2$), from which we find that $A_{n}\to 0$.
On the other hand, there are no IR divergences in dimensions greater than four
($d>2$) since ${\alpha}$ remains finite as $\mu\to 0$, and so amplitudes
$A_{n}$ are not automatically vanishing as $\mu\to 0$.

###  2.3 Soft Effective Action

Celestial holography postulates that scattering amplitudes in $d+2$ dimensions
are correlation functions in a putative holographic conformal field theory in
$d$ dimensions. While there have been a few attempts at constructing explicit
examples of flat holography [38, 39, 40], these are only applicable to a very
special class of four-dimensional theories. A small step towards a formulation
of flat holography in general dimensions was taken in [24], which used
effective field theory techniques to construct a $d$-dimensional action that
partially reproduces the soft factorization described in the previous section
(see also the related works [16, 41, 25, 26, 27, 28, 42, 29]). Essentially,
the analysis in [24] began with the path integral definition of a generic
scattering amplitude with $m$ soft particles and $n$ hard particles, given by

$\begin{split}&\langle\,N_{1}\cdots
N_{m}{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\mu}=\int_{\mu}[{{\mathsf{d}}}\varphi]\,e^{iS[\varphi]}N_{1}(\varphi)\cdots
N_{m}(\varphi){\mathcal{O}}_{1}(\varphi)\cdots{\mathcal{O}}_{n}(\varphi),\end{split}$
(2.20)

where the subscript $\mu$ indicates the path integral is over all fields
$\varphi$ with $|{\omega}|>\mu$. We can now separate $\varphi$ into a hard
piece $\varphi_{h}$ and a soft piece $\varphi_{s}$, which respectively have
support in the momentum range $|{\omega}|>{\Lambda}$ and
$\mu<|{\omega}|<{\Lambda}$. By definition, the soft operators depend only on
the soft fields, so that $N_{i}\equiv N_{i}(\varphi_{s})$, whereas the hard
operators factorize as
${\mathcal{O}}_{k}(\varphi)=U_{k}(\varphi_{s}){\mathcal{O}}_{k}(\varphi_{h})$
[18]. Substituting these results into the soft theorem (2.10) and recalling
(2.16), we recover the soft factorization

$\displaystyle A_{m+n}\xrightarrow{q_{i}\to
0}e^{-{\Gamma}(\mu,{\Lambda})}\SS_{m}^{(0)}\tilde{A}_{n},$ (2.21)

with

$\begin{split}{\tilde{A}}_{n}&=\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\Lambda}=\int_{{\Lambda}}[{{\mathsf{d}}}\varphi]\,e^{iS[\varphi_{h}]}{\mathcal{O}}_{1}(\varphi_{h})\cdots{\mathcal{O}}_{n}(\varphi_{h})\\\
\end{split}$ (2.22)

and

$\begin{split}e^{-{\Gamma}(\mu,{\Lambda})}\SS^{(0)}_{m}&=\langle\,N_{1}\cdots
N_{m}U_{1}\cdots U_{n}\,\rangle_{\mu}\\\
&=\int_{\mu}[{{\mathsf{d}}}\varphi_{s}]\,e^{-S_{\text{soft}}[\varphi_{s}]}N_{1}(\varphi_{s})\cdots
N_{m}(\varphi_{s})U_{1}(\varphi_{s})\cdots U_{n}(\varphi_{s}),\end{split}$
(2.23)

where $S_{\text{soft}}[\varphi_{s}]$ is the effective action for the soft
modes. This can be constructed by integrating out the hard modes explicitly.
However, given the universal IR features that this action is supposed to
reproduce, we expect $S_{\text{soft}}[\varphi_{s}]$ to be universal in any
abelian gauge theory. Motivated by this, the authors of [24] used general
effective field theory ideas to construct the action.

The relevant soft fields in gauge theories are the soft photon operators
$N^{\pm}_{a}(x)$, defined in (2.12), and the Goldstone mode for large gauge
transformations101010Note that the gauge field satisfies an antipodal matching
condition $A_{a}|_{{\mathcal{I}}^{+}_{-}}=A_{a}|_{{\mathcal{I}}^{-}_{+}}$, so
(2.24) could also have been defined on ${\mathcal{I}}^{-}_{+}$.

$\begin{split}C_{a}(x)\equiv
A_{a}|_{{\mathcal{I}}^{+}_{-}}(x)=\partial_{a}{\theta}(x),\qquad{\theta}(x)\sim{\theta}(x)+2\pi.\end{split}$
(2.24)

Substituting (2.17) into (2.22), it was shown in [24] that the effective
action for the soft modes is given by

$\begin{split}S_{\text{soft}}[\phi,{\theta}]={\alpha}\int_{{\mathbb{S}}^{d}}\frac{{{\mathsf{d}}}^{d}x}{(2\pi)^{d}}\big{(}N_{a}(x)\big{)}^{2}-\frac{i}{2c_{1,1}}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,{\widetilde{C}}^{a}(x)N_{a}(x),\end{split}$
(2.25)

where ${\widetilde{C}}_{a}(x)$ is the shadow transform of $C_{a}(x)$. For a
vector field of scaling dimension ${\Delta}$, this is defined by

$\begin{split}{\widetilde{C}}_{a}(x)\equiv\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}y\frac{{\mathcal{I}}_{ab}(x-y)}{[(x-y)^{2}]^{d-{\Delta}}}C^{b}(y),\qquad{\mathcal{I}}_{ab}(x)\equiv{\delta}_{ab}-2\frac{x_{a}x_{b}}{x^{2}}.\end{split}$
(2.26)

Notice that up to a normalization constant, the shadow transform is its
inverse:

$\begin{split}\widetilde{\widetilde{V}}_{a}(x)=c_{{\Delta},1}V_{a}(x),\qquad
c_{{\Delta},1}=\frac{\pi^{d}({\Delta}-1)(d-{\Delta}-1){\Gamma}(\frac{d}{2}-{\Delta}){\Gamma}({\Delta}-\frac{d}{2})}{{\Gamma}({\Delta}+1){\Gamma}(d-{\Delta}+1)}.\end{split}$
(2.27)

In our case of interest, $C_{a}(x)$ has scaling dimension ${\Delta}=1$, and
its shadow transform is evaluated first using (2.26) for generic ${\Delta}$
and then taking the limit ${\Delta}\to 1$.

Lastly, the operators $U_{k}$ are given by

$\begin{split}U_{k}({\theta})=\exp\left[iQ_{k}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,{\theta}(x){\mathcal{K}}_{d}(z_{k},x_{k};x)\right],\qquad
z_{k}\equiv\frac{m_{k}}{|{\omega}_{k}|},\end{split}$ (2.28)

where we have used the momentum parametrization (2.6), and
${\mathcal{K}}_{\Delta}$ is the bulk-to-boundary propagator in Euclidean
$\text{AdS}_{d+1}$, given by

$\begin{split}{\mathcal{K}}_{\Delta}(z,x;y)=\frac{{\Gamma}({\Delta})}{\pi^{\frac{d}{2}}{\Gamma}\big{(}{\Delta}-\frac{d}{2}\big{)}}\left[\frac{z}{(x-y)^{2}+z^{2}}\right]^{\Delta}.\end{split}$
(2.29)

The total product of the operators $U_{k}$ can be written in a nicer form as

$\begin{split}e^{-S_{\text{int}}[\theta]}&\equiv U_{1}({\theta})\cdots
U_{n}({\theta})\\\
&=\exp\left[i\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,{\theta}(x)\sum_{k=1}^{n}Q_{k}{\mathcal{K}}_{d}(z_{k},x_{k};x)\right]\\\
&=\exp\left[-\frac{i}{2c_{1,1}}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,{\widetilde{C}}^{a}(x){\mathcal{J}}_{a}(x)\right],\end{split}$
(2.30)

where we recall ${\mathcal{J}}_{a}(x)$ is the soft factor defined in (2.14),
and in the last equality we used the property

$\begin{split}\sum_{k=1}^{n}Q_{k}{\mathcal{K}}_{d}(z_{k},x_{k};x)=\frac{1}{2c_{1,1}}\partial^{a}\widetilde{{\mathcal{J}}}_{a}(x),\end{split}$
(2.31)

which was derived in [24], as well as the shadow identity

$\begin{split}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,C^{a}(x){\widetilde{\mathcal{J}}}_{a}(x)=\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,{\widetilde{C}}^{a}(x){\mathcal{J}}_{a}(x).\end{split}$
(2.32)

The full soft effective action is then

$\displaystyle\begin{split}S_{\text{eff}}[\phi,\theta]&=S_{\text{soft}}[\phi,\theta]+S_{\text{int}}[\theta]\\\
&={\alpha}\int_{{\mathbb{S}}^{d}}\frac{{{\mathsf{d}}}^{d}x}{(2\pi)^{d}}\big{(}N_{a}(x)\big{)}^{2}-\frac{i}{2c_{1,1}}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,{\widetilde{C}}^{a}(x)\big{(}N_{a}(x)-{\mathcal{J}}_{a}(x)\big{)}.\end{split}$
(2.33)

##  3 Soft On-Shell Action $\to$ Soft Effective Action

In this section, we show that the soft effective action (2.33) can be obtained
by evaluating the bulk gauge theory action on-shell given a specific choice of
boundary conditions, and then extracting the contribution from the soft modes.
Consider a generic model describing an abelian gauge field coupled to charged
matter, which is described by a Lagrangian of the form

$\begin{split}S=\int_{\mathcal{M}}\left(-\frac{1}{2e^{2}}F\wedge\star
F+L_{M}\right)+S_{\text{bdy}},\end{split}$ (3.1)

where $L_{M}$ is the rest of the Lagrangian and includes all the matter field
contributions and any potential higher derivative terms in the Lagrangian.
Generically, it is a polynomial function of the arguments

$\begin{split}L_{M}\equiv
L_{M}\bigg{(}\partial_{A_{1}}\cdots\partial_{A_{n}}F_{AB},D_{(A_{1}}\cdots
D_{A_{n})}\Phi^{i}\bigg{)},\qquad
D_{A}\Phi^{i}\equiv\partial_{A}-iQ_{i}A_{A}\Phi^{i},\end{split}$ (3.2)

where $D_{(A_{1}}\cdots D_{A_{n})}$ denotes $n$ symmetrized covariant
derivatives.111111The commutator of covariant derivatives simplifies to the
field strength, in that $[D_{A},D_{B}]\Phi^{i}=-iQ_{i}F_{AB}\Phi^{i}$. Thus,
without loss of generality, it suffices to consider symmetrized derivatives.
In particular, we are interested in the leading order contribution of the soft
gauge field modes to the on-shell action, which would arise from the lowest
derivative terms in the action. Now, after integrating out the matter fields,
what remains at the lowest derivative order is a term of the form
$A_{A}J^{A}$, where $J_{A}$ is a background conserved current that is
determined from the boundary conditions used for the charged matter fields. To
summarize, as far as the contribution of the soft modes is concerned, we can
restrict ourselves to a simple model described by the action

$\begin{split}S[A]=\int_{\mathcal{M}}\left(-\frac{1}{2e^{2}}F\wedge\star
F+(-1)^{d}A\wedge\star J\right)+\frac{1}{e^{2}}\int_{{\Sigma}^{+}}A\wedge\star
F-\frac{1}{e^{2}}\int_{{\Sigma}^{-}}A\wedge\star F,\end{split}$ (3.3)

where ${\Sigma}^{\pm}={\mathcal{I}}^{\pm}\cup i^{\pm}$. A model of this type
was considered in [43, 44, 45], where it was indeed shown to reproduce all the
IR effects described earlier in Section 2.2.

Let us begin by focusing on the boundary terms in (3.3), which are required so
that the variational principle imposes the relevant boundary conditions for
our model. To see this, note that the variation of the action has the form

$\begin{split}{\delta}S[A]&=-\frac{1}{e^{2}}\int_{\mathcal{M}}{\delta}A\wedge\left({{\mathsf{d}}}\star
F-(-1)^{d}e^{2}\star J\right)\\\
&\qquad+\frac{1}{e^{2}}\int_{{\Sigma}^{+}}A\wedge\star{\delta}F-\frac{1}{e^{2}}\int_{{\Sigma}^{-}}A\wedge\star{\delta}F-\frac{1}{e^{2}}\int_{i^{0}}{\delta}A\wedge\star
F.\end{split}$ (3.4)

The first term in (3.4) gives us Maxwell’s equations121212Notice that (3.5)
also implies current conservation, since acting on both sides by
${{\mathsf{d}}}$ yields ${{\mathsf{d}}}\star J=0$, or $\partial^{A}J_{A}=0$.

$\begin{split}{{\mathsf{d}}}\star F=(-1)^{d}e^{2}\star
J\quad\implies\quad\partial^{A}F_{AB}(X)=e^{2}J_{B}(X).\end{split}$ (3.5)

Furthermore, the variational principle holds only if the terms in the second
line of (3.4) vanishes, which requires us to impose Neumann boundary
conditions on ${\Sigma}^{\pm}$ and Dirichlet boundary conditions on $i^{0}$,
so that

$\begin{split}{\delta}A|_{i^{0}}=0,\qquad\iota_{n}{\delta}F|_{{\Sigma}^{\pm}}=0,\end{split}$
(3.6)

where $\iota_{n}$ is the interior product with respect to the normal vector
$n^{A}$, i.e., $(\iota_{n}F)_{A}=n^{B}F_{BA}$.

Before continuing, we note that for the calculations presented in this
section, we did not need to know the precise form of the background current
$J_{A}$ (aside from the fact that it is conserved). However, to match the
results here to those of Sections 2.2 and 2.3, we will need the current to be
the one corresponding to $n$ charged point-particles (this is the relevant
choice for the scattering problem), so that

$\begin{split}J^{A}(X)&=-\sum_{k=1}^{n}{\theta}(\eta_{k}X^{0})\eta_{k}Q_{k}\frac{p_{k}^{A}}{p^{0}_{k}}{\delta}^{(d+1)}\left(\vec{X}-\frac{\vec{p}_{k}}{p^{0}_{k}}X^{0}\right),\end{split}$
(3.7)

where $\eta_{k}=\pm 1$ distinguishes outgoing ($+$) particles from incoming
($-$) ones.131313Notice that (3.7) assumes that the scattering takes place at
a single point $X^{A}=0$. This is of course not true for a generic scattering
process, but since we are only interested in the leading soft (IR) behavior of
the current, the actual details of the scattering process are not relevant,
and (3.7) is a reasonable approximation.

###  3.1 Solutions to Maxwell’s Equations

Since we aim to evaluate the action on-shell, we start by discussing solutions
to (3.5). We work in axial null gauge, given by141414Notice that
$A_{u}=n^{A}A_{A}$, so the axial null gauge (3.8) is the same as imposing
$A_{u}=0$, which was previously called temporal gauge in [18, 19].

$\begin{split}n^{A}A_{A}(X)=0.\end{split}$ (3.8)

To solve (3.5), we decompose the gauge field into the pieces

$\begin{split}A_{A}(X)={\hat{A}}_{A}(X)+\partial_{A}{\theta}(X),\qquad{\theta}(X)\sim{\theta}(X)+2\pi,\end{split}$
(3.9)

where ${\theta}(X)$ captures the Goldstone mode for large gauge
transformations, and ${\hat{A}}_{A}(X)$ is the part of the gauge field that
admits a Fourier transform, namely

$\begin{split}{\hat{A}}_{A}(X)=e\int_{\mathcal{M}}\frac{{{\mathsf{d}}}^{d+2}\ell}{(2\pi)^{d+2}}e^{i\ell\cdot
X}{\hat{A}}_{A}(\ell).\end{split}$ (3.10)

Similarly, we consider the Fourier transform the current, given by

$\begin{split}J_{A}(X)=\int_{\mathcal{M}}\frac{{{\mathsf{d}}}^{d+2}\ell}{(2\pi)^{d+2}}e^{i\ell\cdot
X}J_{A}(\ell),\qquad\ell^{A}J_{A}(\ell)=0,\end{split}$ (3.11)

where the second equality is due to current conservation
$\partial^{A}J_{A}(X)=0$. Substituting the Fourier modes into (3.5), we obtain

$\begin{split}\ell^{2}{\hat{A}}_{A}(\ell)-\ell_{A}\ell^{B}{\hat{A}}_{B}(\ell)=-eJ_{A}(\ell).\end{split}$
(3.12)

To solve this, we will find it convenient to expand the gauge field and
current using the basis of vectors
$\\{n^{A},\ell^{A},{\varepsilon}^{A}_{a}(\ell)\\}$ on ${\mathbb{R}}^{1,d+1}$,
where the polarization vectors ${\varepsilon}^{A}_{a}(\ell)$ are defined in
(2.9) (with $x^{a}$ related to $\ell^{A}$ via (2.4)) and satisfy the
properties

$\begin{split}n_{A}{\varepsilon}^{A}_{a}(\ell)=\ell_{A}{\varepsilon}^{A}_{a}(\ell)=0,\qquad\eta_{AB}{\varepsilon}^{A}_{a}(\ell){\varepsilon}^{B}_{b}(\ell)={\delta}_{ab},\qquad{\varepsilon}^{A}_{a}(\ell)={\varepsilon}^{A}_{a}(-\ell)={\varepsilon}^{A}_{a}(\ell)^{*}.\end{split}$
(3.13)

Expanding the gauge field and current in this basis, we find

$\begin{split}{\hat{A}}_{A}(\ell)=n_{A}L(\ell)+{\varepsilon}_{A}^{a}(\ell){\hat{A}}_{a}(\ell),\qquad
J_{A}(\ell)=\left(\frac{\ell_{A}}{n\cdot\ell}-\frac{\ell^{2}n_{A}}{(n\cdot\ell)^{2}}\right)J_{n}(\ell)+{\varepsilon}_{A}^{a}(\ell)J_{a}(\ell),\end{split}$
(3.14)

where the coefficients are fixed by imposing the gauge condition (3.8) and
current conservation $\ell^{A}J_{A}(\ell)=0$. Substituting this result into
(3.12), we obtain

$\begin{split}L(\ell)=e\frac{J_{n}(\ell)}{(n\cdot\ell)^{2}},\qquad\ell^{2}{\hat{A}}_{a}(\ell)=-eJ_{a}(\ell).\end{split}$
(3.15)

The second equation above solves to

$\begin{split}{\hat{A}}_{a}(\ell)=2\pi{\mathcal{O}}^{\text{rad}}_{a}(\ell){\delta}(\ell^{2})-\frac{eJ_{a}(\ell)}{\ell^{2}},\end{split}$
(3.16)

where the first term ${\mathcal{O}}^{\text{rad}}_{a}(\ell)$ is the homogeneous
(radiative) solution, and the second term is the Coulombic solution. We would
now like to substitute this result into (3.10) to determine the gauge field in
position space. However, to evaluate this Fourier integral, the pole at
$\ell^{2}=0$ in the second term of (3.16) has to be regulated by an
$i{\epsilon}$ prescription. Depending on how this is done, the corresponding
radiative solution is incoming or outgoing. More precisely, we have

$\begin{split}{\hat{A}}_{a}(\ell)=2\pi{\mathcal{O}}^{\pm}_{a}(\ell){\delta}(\ell^{2})-\frac{eJ_{a}(\ell)}{-(\ell^{0}\mp
i{\epsilon})^{2}+|\vec{\ell}\,|^{2}},\end{split}$ (3.17)

where, as before, the $\pm$ superscript corresponds to the outgoing ($+$) and
incoming ($-$) radiative modes, respectively. Furthermore, depending on the
sign of $\ell^{0}=\pm|\vec{\ell}\,|$, the operator
${\mathcal{O}}_{a}^{\pm}(\ell)$ reduces to a creation or an annihilation
operator in the quantum theory, and we have the identification

$\begin{split}{\mathcal{O}}^{\pm}_{a}(|\vec{\ell}\,|,\vec{\ell}\,)={\mathcal{O}}^{\pm}_{a}(\vec{\ell}\,),\qquad{\mathcal{O}}^{\pm}_{a}(-|\vec{\ell}\,|,-\vec{\ell}\,)={\mathcal{O}}^{\pm}_{a}(\vec{\ell}\,)^{\dagger}.\end{split}$
(3.18)

Finally, we remark that using the identity

$\begin{split}\frac{1}{x\pm
i{\epsilon}}={\mathcal{P}}\left(\frac{1}{x}\right)\mp
i\pi{\delta}(x),\end{split}$ (3.19)

where ${\mathcal{P}}$ is the Cauchy principal value, a useful consequence of
(3.17) is

$\begin{split}\left[J_{a}(\ell)-\frac{i}{e}{\Theta}(\ell^{0})\big{(}{\mathcal{O}}^{+}_{a}(\ell)-{\mathcal{O}}^{-}_{a}(\ell)\big{)}\right]{\delta}(\ell^{2})=0,\end{split}$
(3.20)

where ${\Theta}$ is the sign function.

###  3.2 On-Shell Action

Having constructed the solutions, we now turn to the on-shell action. First,
using the decomposition (3.9), the action (3.3) can be recast into the form

$\begin{split}S[A]&=\int_{\mathcal{M}}\left(-\frac{1}{2e^{2}}{\hat{F}}\wedge\star{\hat{F}}+(-1)^{d}{\hat{A}}\wedge\star
J\right)-\frac{1}{e^{2}}\int_{{\mathcal{I}}_{-}^{+}}{\theta}\star{\hat{F}}-\frac{1}{e^{2}}\int_{{\mathcal{I}}_{+}^{-}}{\theta}\star{\hat{F}}\\\
&\qquad+\frac{1}{e^{2}}\int_{{\Sigma}^{+}}{\hat{A}}\wedge\star\hat{F}-\frac{1}{e^{2}}\int_{{\Sigma}^{-}}{\hat{A}}\wedge\star\hat{F}+(-1)^{d}\int_{i^{0}}{\theta}\star
J\\\
&\qquad-\frac{1}{e^{2}}\int_{{\Sigma}^{+}}{\theta}\left({{\mathsf{d}}}\star\hat{F}-(-1)^{d}e^{2}\star
J\right)+\frac{1}{e^{2}}\int_{{\Sigma}^{-}}{\theta}\left({{\mathsf{d}}}\star\hat{F}-(-1)^{d}e^{2}\star
J\right).\end{split}$ (3.21)

The terms in the last line are proportional to the equations of motion (3.5)
and therefore vanish on-shell. Furthermore, all the terms in the second line
vanish on-shell as well. To see why, first note that there is no charge flux
through $i^{0}$, so the last term in the second line vanishes. Secondly, using
(2.3), the first two terms in the second line can be written as151515We remark
that although (3.22) only includes the contribution from
${\mathcal{I}}^{\pm}$, we are allowed to drop the contribution from $i^{\pm}$.
This is because in the absence of massive particles, the gauge field vanishes
on $i^{\pm}$, while in the presence of massive particles, the gauge field only
receives a Coulombic contribution on $i^{\pm}$ (see Appendix B), which falls
off too quickly to contribute to (3.22).

$\displaystyle\begin{split}\frac{1}{e^{2}}\int_{{\Sigma}^{\pm}}\hat{A}\wedge\star\hat{F}&=-\frac{1}{e^{2}}\int_{\mathbb{R}}{{\mathsf{d}}}u\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\lim_{r\to\pm\infty}\,|r|^{d}\hat{A}_{a}\partial_{u}\hat{A}^{a}.\end{split}$
(3.22)

Following [19], we now decompose the gauge field into radiative and Colulombic
modes, so that

$\displaystyle\hat{A}_{\mu}=\hat{A}_{\mu}^{R\pm}+\hat{A}_{\mu}^{C\pm},$ (3.23)

where the radiative piece ($R$) is the homogeneous solution to Maxwell’s
equations, and the Coluombic piece ($C$) is the inhomogenous solution. The
$\pm$ superscript indicates whether we are taking an advanced Green’s function
($+$) or retarded Green’s function ($-$). The fall-off conditions for these
pieces obey [19]

$\displaystyle\begin{split}\hat{A}_{r}^{R\pm}&={\mathcal{O}}(|r|^{-\frac{d}{2}-1})+{\mathcal{O}}(|r|^{-d}),\qquad\hat{A}_{r}^{C\pm}={\mathcal{O}}(|r|^{-d}),\\\
\hat{A}_{a}^{R\pm}&={\mathcal{O}}(|r|^{-\frac{d}{2}+1})+{\mathcal{O}}(|r|^{-d+1}),\qquad\hat{A}_{r}^{C\pm}={\mathcal{O}}(|r|^{-d+1}).\end{split}$
(3.24)

Given these fall-off conditions, it is clear from the integrand in (3.22) that
only the radiative part of the gauge field contributes to the integral, as the
Coulombic modes fall off too quickly as $|r|\to\infty$. It follows we have

$\displaystyle\begin{split}\frac{1}{e^{2}}\int_{{\Sigma}^{\pm}}\hat{A}\wedge\star\hat{F}=-\frac{1}{e^{2}}\int_{\mathbb{R}}{{\mathsf{d}}}u\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,\hat{A}_{a}^{\pm}\partial_{u}\hat{A}^{\pm
a},\end{split}$ (3.25)

where

$\displaystyle\
\hat{A}_{a}^{\pm}(u,x)=\lim_{r\to\pm\infty}|r|^{\frac{d}{2}-1}\hat{A}_{a}^{\pm}(u,r,x).$
(3.26)

As $\hat{A}_{a}^{\pm}$ only involves the radiative modes, it admits a mode
expansion, which is given on-shell in [19] to be

$\displaystyle\begin{split}\hat{A}_{a}^{\pm}(u,x)&=\pm\frac{e}{2(2\pi)^{\frac{d}{2}+1}}\int_{0}^{\infty}{{\mathsf{d}}}{\omega}\,{\omega}^{\frac{d}{2}-1}\Big{[}{\mathcal{O}}^{\pm}_{a}({\omega}\hat{q}(x))e^{-\frac{i{\omega}u}{2}\mp\frac{i\pi
d}{4}}+{\text{c.c.}}\Big{]}.\end{split}$ (3.27)

Substituting this into (3.25), we obtain

$\displaystyle\begin{split}\frac{1}{e^{2}}\int_{{\Sigma}^{\pm}}\hat{A}\wedge\star\hat{F}&=-\frac{i}{8(2\pi)^{d+2}}\int_{\mathbb{R}}{{\mathsf{d}}}u\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,\int_{0}^{\infty}{{\mathsf{d}}}{\omega}\,{{\mathsf{d}}}{\omega}^{\prime}\,{\omega}^{\frac{d}{2}-1}{\omega}^{\prime\frac{d}{2}}\\\
&\qquad\qquad\Big{[}{\mathcal{O}}^{\pm}_{a}({\omega}\hat{q}(x))e^{-\frac{i{\omega}u}{2}\mp\frac{i\pi
d}{4}}+{\text{c.c.}}\Big{]}\Big{[}{\mathcal{O}}^{\pm
a}({\omega}^{\prime}\hat{q}(x))e^{-\frac{i{\omega}^{\prime}u}{2}\mp\frac{i\pi
d}{4}}-{\text{c.c.}}\Big{]}\\\
&=\frac{i}{8(2\pi)^{d+2}}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,\int_{0}^{\infty}{{\mathsf{d}}}{\omega}\,{{\mathsf{d}}}{\omega}^{\prime}\,{\omega}^{\frac{d}{2}-1}{\omega}^{\prime\frac{d}{2}}\\\
&\qquad\qquad\times\int_{\mathbb{R}}{{\mathsf{d}}}u\,\Big{[}{\mathcal{O}}_{a}^{\pm}({\omega}\hat{q}(x)){\mathcal{O}}^{\pm
a}({\omega}^{\prime}\hat{q}(x))^{\dagger}e^{-\frac{i({\omega}-{\omega}^{\prime})u}{2}}-{\text{c.c.}}\Big{]}\\\
&=0.\end{split}$ (3.28)

This proves the claim that all the terms in the second line of (3.21) vanish.

Thus, we see that on-shell, the only terms that survive in (3.21) are those in
the first line, i.e.,

$\displaystyle S[A]\big{|}_{\text{on-shell}}$
$\displaystyle=\int_{\mathcal{M}}\left(-\frac{1}{2e^{2}}{\hat{F}}\wedge\star{\hat{F}}+(-1)^{d}{\hat{A}}\wedge\star
J\right)-\frac{1}{e^{2}}\int_{{\mathcal{I}}_{-}^{+}}{\theta}\star{\hat{F}}-\frac{1}{e^{2}}\int_{{\mathcal{I}}_{+}^{-}}{\theta}\star{\hat{F}}.$
(3.29)

In the rest of this subsection, we will demonstrate that with a suitable
contour deformation, the bulk integral in (3.29) in the soft and on-shell
limit becomes (see (3.40))

$\displaystyle\begin{split}\int_{\mathcal{M}}\left(-\frac{1}{2e^{2}}{\hat{F}}\wedge\star{\hat{F}}+(-1)^{d}{\hat{A}}\wedge\star
J\right)&\quad\xrightarrow{\text{soft}+\text{on-shell}}\quad
i{\alpha}\int_{{\mathbb{S}}^{d}}\frac{{{\mathsf{d}}}^{d}x}{(2\pi)^{d}}\big{(}N_{a}(x)\big{)}^{2},\end{split}$
(3.30)

and the boundary integrals in (3.29) in the on-shell limit becomes (see
(3.51))161616Only the soft modes are non-vanishing on
${\mathcal{I}}^{\pm}_{\mp}$, so we do not need to take a soft limit.

$\displaystyle-\frac{1}{e^{2}}\int_{{\mathcal{I}}_{-}^{+}}{\theta}\star{\hat{F}}-\frac{1}{e^{2}}\int_{{\mathcal{I}}_{+}^{-}}{\theta}\star{\hat{F}}$
$\displaystyle\quad\xrightarrow{\text{on-
shell}}\quad\frac{1}{2c_{1,1}}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,\widetilde{C}^{a}(x)\big{(}N_{a}(x)-{\mathcal{J}}_{a}(x)\big{)}.$
(3.31)

where $C_{a}(x)$ was defined in (2.24). Substituting (3.30) and (3.31) into
(3.29), we see that the soft limit of the on-shell action (with a suitable
contour deformation) is given by

$\displaystyle S[A]\big{|}_{\text{soft}\text{+on-
shell}}=i{\alpha}\int_{{\mathbb{S}}^{d}}\frac{{{\mathsf{d}}}^{d}x}{(2\pi)^{d}}\big{(}N_{a}(x)\big{)}^{2}+\frac{1}{2c_{1,1}}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,\widetilde{C}^{a}(x)\big{(}N_{a}(x)-{\mathcal{J}}_{a}(x)\big{)}.$
(3.32)

Comparing with the soft effective action (2.33), we see that

$\displaystyle\begin{split}S[A]\big{|}_{\text{soft}\text{+on-
shell}}=iS_{\text{eff}}[\phi,\theta].\end{split}$ (3.33)

This is the main result of our paper, and it proves our claim that the soft
limit of on-shell degrees of freedom localized on the celestial sphere, i.e.,
the soft limit of edge modes, are precisely the soft and Goldstone modes
parametrizing the low-energy Hilbert space of the gauge theory. To understand
the factor of $i$, note that from (2.23), the path integral involves
$e^{-S_{\text{eff}}[\phi,\theta]}$. On the other hand, if we had chosen
instead to insert the on-shell action into the path integral, it would involve
$e^{iS[A]|_{\text{on-shell}}}$, implying (3.33) is indeed correct. In the next
two subsubsections, we will derive both (3.30) and (3.31), which were
necessary to prove the main result (3.33).

####  3.2.1 Bulk Term

We start with the first term in (3.29), which in momentum space can be written
as

$\begin{split}S_{{\text{bulk}}}[{\hat{A}}]=\int_{\mathcal{M}}\frac{{{\mathsf{d}}}^{d+2}\ell}{(2\pi)^{d+2}}\left[-\frac{1}{2}\left(\ell^{2}{\hat{A}}(\ell)\cdot{\hat{A}}(-\ell)-|\ell\cdot{\hat{A}}(\ell)|^{2}\right)-e{\hat{A}}(\ell)\cdot
J(-\ell)\right].\end{split}$ (3.34)

To render the Lorentzian path integral finite, we need to deform the contour
of integration over $\ell$. A simple way to do this is to replace
$\ell^{2}\to\ell^{2}-i{\epsilon}$ above. Applying this deformation and then
evaluating the action on-shell by utilizing the solutions constructed in
Section 3.1, we find

$\begin{split}S_{{\text{bulk}}}[{\hat{A}}]\big{|}_{\text{on-
shell}}=\frac{e^{2}}{2}\int_{\mathcal{M}}\frac{{{\mathsf{d}}}^{d+2}\ell}{(2\pi)^{d+2}}\frac{1}{\ell^{2}-i{\epsilon}}\left[|J_{a}(\ell)|^{2}-\frac{\ell^{2}}{(n\cdot\ell)^{2}}|J_{n}(\ell)|^{2}\right].\end{split}$
(3.35)

Next, using the identity (3.19) and (3.20), we can rewrite the action as

$\begin{split}S_{{\text{bulk}}}[{\hat{A}}]\big{|}_{\text{on-
shell}}&=\int_{\mathcal{M}}\frac{{{\mathsf{d}}}^{d+2}\ell}{(2\pi)^{d+2}}\left[\frac{i\pi}{2}{\delta}(\ell^{2})\big{|}{\mathcal{O}}^{+}_{a}(\ell)-{\mathcal{O}}^{-}_{a}(\ell)\big{|}^{2}+\frac{e^{2}}{2}\left(\frac{{\mathcal{P}}(|J_{a}(\ell)|^{2})}{\ell^{2}}-\frac{|J_{n}(\ell)|^{2}}{(n\cdot\ell)^{2}}\right)\right].\end{split}$
(3.36)

For the soft effective action, we only keep the first term above, as this is
the term that is responsible for the real part of the IR divergence
${\Gamma}$.171717The imaginary part of ${\Gamma}$ is related to the second
term of (3.36), and as mentioned previously was ignored in [24]. Since our
goal in this paper is to reproduce the results of [24], we will ignore the
second term in (3.36) for now, and hope to return to it in future work.
Extracting the soft contribution here, we find

$\begin{split}S_{{\text{bulk}}}[{\hat{A}}]\big{|}_{\text{on-
shell+soft}}&=\frac{i\pi}{2}\int_{\mu}^{\Lambda}\frac{{{\mathsf{d}}}^{d+2}\ell}{(2\pi)^{d+2}}{\delta}(\ell^{2})\big{|}{\mathcal{O}}^{+}_{a}(\ell)-{\mathcal{O}}^{-}_{a}(\ell)\big{|}^{2}.\end{split}$
(3.37)

Using (2.5), this can be rewritten as

$\begin{split}S_{{\text{bulk}}}[{\hat{A}}]\big{|}_{\text{on-
shell+soft}}&=\frac{i}{8\pi}\int_{\mu}^{\Lambda}{{\mathsf{d}}}{\omega}\,{\omega}^{d-1}\int_{{\mathbb{S}}^{d}}\frac{{{\mathsf{d}}}^{d}x}{(2\pi)^{d}}\big{|}{\mathcal{O}}^{+}_{a}({\omega}{\hat{q}}(x))-{\mathcal{O}}^{-}_{a}({\omega}{\hat{q}}(x))\big{|}^{2}.\end{split}$
(3.38)

We now recall that $\mu$ and ${\Lambda}$ are much smaller than any other scale
in the problem, so using (2.12), we can write

$\begin{split}{\mathcal{O}}^{\pm}_{a}({\omega}{\hat{q}}(x))\to\frac{e}{{\omega}}N_{a}^{\pm}(x),\end{split}$
(3.39)

which implies

$\begin{split}S_{{\text{bulk}}}[{\hat{A}}]\big{|}_{\text{on-
shell+soft}}&=i{\alpha}\int_{{\mathbb{S}}^{d}}\frac{{{\mathsf{d}}}^{d}x}{(2\pi)^{d}}\big{(}N_{a}(x)\big{)}^{2},\end{split}$
(3.40)

where $N_{a}$ is defined in (2.11) and ${\alpha}$ in (2.18). This proves our
claim (3.30).

####  3.2.2 Boundary Terms

We now turn to the boundary terms in (3.29), which are

$\displaystyle
S_{{\text{bdy}}}[\hat{A},\theta]=-\frac{1}{e^{2}}\int_{{\mathcal{I}}^{+}_{-}}\theta\star\hat{F}-\frac{1}{e^{2}}\int_{{\mathcal{I}}^{-}_{+}}\theta\star\hat{F}.$
(3.41)

Using (2.3), we can write the terms in coordinate notation as

$\displaystyle\begin{split}\frac{1}{e^{2}}\int_{{\mathcal{I}}^{\pm}_{\mp}}\theta\star\hat{F}&=\frac{2}{e^{2}}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,{\theta}(x)\left(\lim_{u\to\mp\infty}\lim_{r\to\pm\infty}|r|^{d}\hat{F}_{ur}(u,r,x)\right),\end{split}$
(3.42)

where we have used the matching condition on the gauge field
${\theta}|_{{\mathcal{I}}^{+}_{-}}={\theta}|_{{\mathcal{I}}^{-}_{+}}$ (see
Footnote 10). To evaluate (3.42) explicitly, we decompose the field strength
into radiative and Coulombic parts, so that

$\displaystyle\begin{split}{\hat{F}}_{ur}(u,r,x)={\hat{F}}_{ur}^{R\pm}(u,r,x)+{\hat{F}}_{ur}^{C\pm}(u,r,x).\end{split}$
(3.43)

From [19], we have for abelian gauge theories the _on-shell_ identity

$\displaystyle\begin{split}\left(\lim_{u\to\mp\infty}\lim_{r\to\pm\infty}|r|^{d}F_{ur}^{R\pm}(u,r,x)\right)=\pm\frac{e^{2}}{4c_{1,1}}\partial^{a}\widetilde{N}^{\pm}_{a}(x).\end{split}$
(3.44)

Furthermore, Maxwell’s equations imply [46]

$\displaystyle\begin{split}2\partial_{u}\left(\lim_{r\to\pm\infty}|r|^{d}{\hat{F}}_{ur}^{C\pm}(u,r,x)\right)&=e^{2}J_{u}^{\pm}(u,x),\qquad
J_{u}^{\pm}(u,x)\equiv\lim_{r\to\pm\infty}|r|^{d}J_{u}(u,r,x).\end{split}$
(3.45)

Integrating this differential equation in $u$, we obtain

$\begin{split}\left(\lim_{u\to\mp\infty}\lim_{r\to\pm\infty}|r|^{d}{\hat{F}}_{ur}^{C\pm}(u,r,x)\right)&=\mp\frac{e^{2}}{2}\int_{\mathbb{R}}{{\mathsf{d}}}u\,J_{u}^{\pm}(u,x)+\left(\lim_{u\to\pm\infty}\lim_{r\to\pm\infty}|r|^{d}{\hat{F}}_{ur}^{C\pm}(u,r,x)\right).\end{split}$
(3.46)

In the absence of massive particles (the contribution from massive particles
is discussed in Appendix B), the second term above vanishes from the fall-off
condition (3.24). Assuming this and substituting (3.44) and (3.46) into
(3.42), we get

$\displaystyle\begin{split}\frac{1}{e^{2}}\int_{{\mathcal{I}}^{\pm}_{\mp}}\theta\star\hat{F}&=\pm\frac{1}{2c_{1,1}}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,{\theta}(x)\partial^{a}\widetilde{N}^{\pm}_{a}(x)\mp\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,{\theta}(x)\int_{\mathbb{R}}{{\mathsf{d}}}u\,J_{u}^{\pm}(u,x)\\\
&=\mp\frac{1}{2c_{1,1}}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,{\widetilde{C}}^{a}(x)N^{\pm}_{a}(x)\mp\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,{\theta}(x)\int_{\mathbb{R}}{{\mathsf{d}}}u\,J_{u}^{\pm}(u,x),\end{split}$
(3.47)

where in the second equality we integrated by parts in the first term and then
used the shadow identity (2.32). Substituting this into (3.41), it follows
that

$\displaystyle\begin{split}S_{{\text{bdy}}}[\hat{A},\theta]&=\frac{1}{2c_{1,1}}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,\widetilde{C}^{a}(x)N_{a}(x)+\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,\theta(x)\int_{\mathbb{R}}{{\mathsf{d}}}u\,\big{(}J_{u}^{+}(u,x)-J_{u}^{-}(u,x)\big{)},\end{split}$
(3.48)

where we used the definition (2.11). To further simplify this expression, we
use the following identity, which we prove in Appendix A,

$\displaystyle\begin{split}\int_{\mathbb{R}}{{\mathsf{d}}}u\,\big{(}J_{u}^{+}(u,x)-J_{u}^{-}(u,x)\big{)}=\frac{1}{2c_{1,1}}\partial^{a}\widetilde{\mathcal{J}}_{a}(x),\end{split}$
(3.49)

where

$\displaystyle\begin{split}{\mathcal{J}}_{a}(x)\equiv\frac{i}{2}\left(\lim_{{\omega}\to
0^{+}}+\lim_{{\omega}\to
0^{-}}\right){\omega}{\varepsilon}^{A}_{a}(x)\int_{\mathcal{M}}{{\mathsf{d}}}^{d+2}X\,e^{i{\omega}\hat{q}(x)\cdot
X}J_{A}(X),\end{split}$ (3.50)

and importantly reduces precisely to our earlier definition of
${\mathcal{J}}_{a}$ in (2.14) when the spacetime current $J_{A}(X)$ is the
point-particle current (3.7). Using (3.49), we can rewrite (3.48) as

$\displaystyle\begin{split}S_{{\text{bdy}}}[\hat{A},\theta]&=\frac{1}{2c_{1,1}}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,\widetilde{C}^{a}(x)\big{(}N_{a}(x)-{\mathcal{J}}_{a}(x)\big{)},\end{split}$
(3.51)

which proves (3.31).

##  4 Summary

We have in this paper shown that the soft effective action (1.3) can be
derived from a general action for an abelian gauge theory (1.4), taken on-
shell in the soft limit, and this result is summarized in (1.5). Importantly,
our analysis fixes the _type_ of boundary conditions necessary to derive the
soft effective action. In particular, the soft modes are _not_ the
entanglement edge modes studied by Donnelly and Wall in [33, 34], which
analyzed edge modes of Maxwell theory with magnetic conductor boundary
conditions imposed. Rather, they are the edge modes for gauge theory with
Neumann boundary conditions at timelike and null infinity and Dirichlet
boundary conditions at spatial infinity. It would be very interesting to
explore the entanglement properties of soft modes by viewing them as
entanglement edge modes and following, in spirit, the analysis of Donnelly and
Wall. We will leave such explorations for future work.

Furthermore, now that the connection between soft modes and edge modes has
been established in abelian gauge theories, there are natural extensions of
our analysis to both nonabelian gauge theories and gravity. By beginning with
the action in nonabelian gauge theory or gravity with suitable boundary terms
added to impose Neumann boundary condition on $\Sigma^{\pm}$, we can derive
the on-shell action. By then taking the soft limit, it would be interesting to
confirm that we get precisely the soft effective action for nonabelian gauge
theory and gravity given in [27, 24].

Indeed, it would be most interesting to study the IR sector of gravity, which
is expected to have similar behavior as the IR sector of abelian gauge
theories at leading order in small energies. In particular, we would like to
determine what are the appropriate GHY boundary terms to add to the Einstein-
Hilbert action such that in the soft and on-shell limit it becomes the
gravitational soft effective action. This should allow us to appropriately
identify soft modes with entanglement edge modes in gravity and gain insight
into the modular Hamiltonian. The modular Hamiltonian has been an object of
study in connection to quantum fluctuations in spacetime subregions [47, 48,
36], and we expect this will open the possibility of utilizing soft or edge
modes to study subregion spacetime fluctuations. These, and many other
exciting connections between soft modes and entanglement, are current avenues
under exploration.

## Acknowledgements

We would like to thank Sangmin Choi, Laurent Freidel, Per Kraus, Andrea Puhm,
Ana-Maria Raclariu, Antony Speranza, and Andrew Strominger for useful
conversations. T.H., A.S., and K.Z. are supported by the Heising-Simons
Foundation “Observational Signatures of Quantum Gravity” collaboration grant
2021-2817, the U.S. Department of Energy, Office of Science, Office of High
Energy Physics, under Award No. DE-SC0011632, and the Walter Burke Institute
for Theoretical Physics. P.M. is supported by the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 852386). The work of K.Z. is also supported by a
Simons Investigator award.

##  Appendix A Relating Current to its Shadow

In this appendix, we will prove the identity (3.49). Given a conserved current
$J_{A}(X)$, we want to compute the soft limit of its Fourier transform, which
we defined symmetrically to be (see (3.50))

$\displaystyle\begin{split}{\mathcal{J}}_{a}(x)&\equiv\frac{i}{2}\left(\lim_{{\omega}\to
0^{+}}+\lim_{{\omega}\to
0^{-}}\right){\omega}{\varepsilon}^{A}_{a}(x)\int_{\mathcal{M}}{{\mathsf{d}}}^{d+2}X\,e^{i{\omega}\hat{q}(x)\cdot
X}J_{A}(X).\end{split}$ (A.1)

We now compute using flat null coordinates

$\displaystyle\begin{split}\int_{\mathcal{M}}{{\mathsf{d}}}^{d+2}X\,e^{i{\omega}\hat{q}(x)\cdot
X}J_{A}(X)&=\frac{1}{|{\omega}|^{d+1}}\int_{\mathbb{R}}{{\mathsf{d}}}u\int_{{\mathbb{R}}}{{\mathsf{d}}}r\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}y\,\frac{|r|^{d}}{2}e^{-\frac{i{\omega}}{2}u-\frac{i}{2}r{\Theta}({\omega})(x-y)^{2}}J_{A}\left(u,\frac{r}{|{\omega}|},y\right),\end{split}$
(A.2)

where we recall ${\Theta}$ is the sign function. Denoting

$\displaystyle J_{A}^{\pm}(u,y)=\lim_{r\to\pm\infty}|r|^{d}J_{A}(u,r,y),$
(A.3)

we have

$\displaystyle\begin{split}{\mathcal{J}}_{a}(x)&=\frac{i}{2}\bigg{(}\lim_{{\omega}\to
0^{+}}+\lim_{{\omega}\to
0^{-}}\bigg{)}\partial_{a}\hat{q}^{A}(x){\Theta}({\omega})\int_{-\infty}^{0}{{\mathsf{d}}}r\int_{\mathbb{R}}{{\mathsf{d}}}u\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}y\,\frac{1}{2}e^{-\frac{i{\omega}}{2}u-\frac{i}{2}r{\Theta}({\omega})(x-y)^{2}}J_{A}^{-}(u,y)\\\
&\qquad+\frac{i}{2}\bigg{(}\lim_{{\omega}\to 0^{+}}+\lim_{{\omega}\to
0^{-}}\bigg{)}\partial_{a}\hat{q}^{A}(x){\Theta}({\omega})\int_{0}^{\infty}{{\mathsf{d}}}r\int_{\mathbb{R}}{{\mathsf{d}}}u\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}y\,\frac{1}{2}e^{-\frac{i{\omega}}{2}u-\frac{i}{2}r{\Theta}({\omega})(x-y)^{2}}J_{A}^{+}(u,y),\end{split}$
(A.4)

where we used (A.3) and the fact the ${\omega}\to 0^{\pm}$ limit for $J_{A}$
corresponds to the large $r$ limit. We can now perform the $r$ integral
directly, where we have to regulate using the $i{\epsilon}$ prescription:

$\displaystyle\begin{split}\int_{-\infty}^{0}{{\mathsf{d}}}r\,e^{-\frac{i}{2}r{\Theta}({\omega})\left[(x-y)^{2}+i{\Theta}({\omega}){\epsilon}\right]}&=\frac{e^{-\frac{i}{2}r{\Theta}({\omega})\left[(x-y)^{2}+i{\Theta}({\omega}){\epsilon}\right]}}{-\frac{i}{2}{\Theta}({\omega})\left[(x-y)^{2}+i{\Theta}({\omega}){\epsilon}\right]}\bigg{|}_{r=-\infty}^{r=0}=\frac{2i{\Theta}({\omega})}{(x-y)^{2}+i{\Theta}({\omega}){\epsilon}}\\\
\int_{0}^{\infty}{{\mathsf{d}}}r\,e^{-\frac{i}{2}r{\Theta}({\omega})\left[(x-y)^{2}-i{\Theta}({\omega}){\epsilon}\right]}&=\frac{e^{-\frac{i}{2}r{\Theta}({\omega})\left[(x-y)^{2}-i{\Theta}({\omega}){\epsilon}\right]}}{-\frac{i}{2}{\Theta}({\omega})\left[(x-y)^{2}-i{\Theta}({\omega}){\epsilon}\right]}\bigg{|}_{r=0}^{r=\infty}=-\frac{2i{\Theta}({\omega})}{(x-y)^{2}-i{\Theta}({\omega}){\epsilon}}.\end{split}$
(A.5)

It follows

$\displaystyle\begin{split}{\mathcal{J}}_{a}(x)&=\frac{1}{2}\bigg{(}\lim_{{\omega}\to
0^{+}}+\lim_{{\omega}\to
0^{-}}\bigg{)}\partial_{a}\hat{q}^{A}(x)\int_{\mathbb{R}}{{\mathsf{d}}}u\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}y\left[\frac{J_{A}^{+}(u,y)}{(x-y)^{2}-i{\Theta}({\omega}){\epsilon}}-\frac{J_{A}^{-}(u,y)}{(x-y)^{2}+i{\Theta}({\omega}){\epsilon}}\right]\\\
&=\partial_{a}\hat{q}^{A}(x)\int_{\mathbb{R}}{{\mathsf{d}}}u\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}y\,{\mathcal{P}}\left(\frac{1}{(x-y)^{2}}\right)\big{(}J_{A}^{+}(u,y)-J_{A}^{-}(u,y)\big{)},\end{split}$
(A.6)

where in the last equality, we used the definition of the principal value

$\displaystyle{\mathcal{P}}\left(\frac{1}{(x-y)^{2}}\right)\equiv\frac{1}{2}\bigg{[}\frac{1}{(x-y)^{2}+i{\epsilon}}+\frac{1}{(x-y)^{2}-i{\epsilon}}\bigg{]}.$
(A.7)

Now, recalling the on-shell momentum parametrization via flat null coordinates
given in (2.6) with $m=0$, we evaluate

$\displaystyle\begin{split}\partial_{a}\hat{q}^{A}(x)J_{A}^{\pm}(u,y)&=x_{a}\big{(}J_{0}^{\pm}(u,y)-J_{d+1}^{\pm}(u,y)\big{)}+J_{X^{a}}^{\pm}(u,y),\end{split}$
(A.8)

where $J^{\pm}_{X^{a}}(u,y)$ labels the $a$-component of $J^{\pm}_{A}$ in
Cartesian coordinates; we use this notation to distinguish it from the
$a$-component of $J^{\pm}_{A}$ in flat null coordinates, which we denote as
usual by $J^{\pm}_{a}$. To rewrite the right-hand-side of (A.8) in terms of
flat null components $J_{\mu}(u,y)$, we perform the coordinate change

$\displaystyle\begin{split}J_{u}(u,r,y)&=\frac{1}{2}\big{(}J_{0}(u,r,y)-J_{d+1}(u,r,y)\big{)}\\\
J_{a}(u,r,y)&=ry_{a}\big{(}J_{0}(u,r,y)-J_{d+1}(u,r,y)\big{)}+rJ_{X^{a}}(u,r,y).\end{split}$
(A.9)

It immediately follows that

$\displaystyle\begin{split}J_{X^{a}}(u,r,y)&=\frac{1}{r}J_{a}(u,r,y)-2y_{a}J_{u}(u,r,y).\end{split}$
(A.10)

Multiplying both sides of the above equations by $|r|^{d}$ and taking
$r\to\pm\infty$, we have

$\displaystyle\begin{split}J_{u}^{\pm}(u,y)&=\frac{1}{2}\big{(}J^{\pm}_{0}(u,y)-J_{d+1}^{\pm}(u,y)\big{)},\qquad
J_{X^{a}}^{\pm}(u,y)=\pm J_{a}^{\pm}(u,y)-2y_{a}J_{u}^{\pm}(u,y),\end{split}$
(A.11)

where we defined

$\displaystyle
J_{a}^{\pm}(u,y)=\lim_{r\to\pm\infty}|r|^{d-1}J_{a}^{\pm}(u,r,y).$ (A.12)

Substituting (A.11) into (A.8), we obtain

$\displaystyle\partial_{a}\hat{q}^{A}(x)J_{A}^{\pm}(u,y)$
$\displaystyle=2(x_{a}-y_{a})J_{u}^{\pm}(u,y)\pm J_{a}^{\pm}(u,y).$ (A.13)

Substituting this into (A.6), we obtain

$\displaystyle\begin{split}{\mathcal{J}}_{a}(x)&=2\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}y\,\frac{x_{a}-y_{a}}{(x-y)^{2}}\int_{\mathbb{R}}{{\mathsf{d}}}u\,\big{(}J_{u}^{+}(u,y)-J_{u}^{-}(u,y)\big{)}\\\
&\qquad+\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}y\,\frac{1}{(x-y)^{2}}\int_{\mathbb{R}}{{\mathsf{d}}}u\,\big{(}J_{a}^{+}(u,y)+J_{a}^{-}(u,y)\big{)},\end{split}$
(A.14)

where we have implicitly dropped the principal value notation ${\mathcal{P}}$
for simplicity.

We now want to take the shadow transform of (A.14) and then take the
divergence. Observing the fact ${\mathcal{J}}_{a}(x)$ has scaling dimension 1,
we compute

$\displaystyle\begin{split}\partial^{a}\widetilde{{\mathcal{J}}_{a}}(x)&=2\partial^{a}_{x}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}z\,\frac{{\mathcal{I}}_{ab}(x-z)}{[(x-z)^{2}]^{d-1}}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}y\,\frac{x_{a}-y_{a}}{(z-y)^{2}}\int_{\mathbb{R}}{{\mathsf{d}}}u\,\big{(}J_{u}^{+}(u,y)-J_{u}^{-}(u,y)\big{)}\\\
&\qquad+\partial^{a}_{x}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}z\,\frac{{\mathcal{I}}_{ab}(x-z)}{[(x-z)^{2}]^{d-1}}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}y\,\frac{1}{(z-y)^{2}}\int_{\mathbb{R}}{{\mathsf{d}}}u\,\big{(}J_{a}^{+}(u,y)+J_{a}^{-}(u,y)\big{)}.\end{split}$
(A.15)

Let us first focus on the second line. We compute

$\displaystyle\begin{split}&\partial^{a}_{x}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}z\,\frac{{\mathcal{I}}_{ab}(x-z)}{[(x-z)^{2}]^{d-1}}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}y\,\frac{1}{(z-y)^{2}}\int_{\mathbb{R}}{{\mathsf{d}}}u\,\big{(}J_{a}^{+}(u,y)+J_{a}^{-}(u,y)\big{)}\\\
&\qquad=\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}z\,\frac{{\mathcal{I}}_{ab}(z)}{(z^{2})^{d-1}}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}y\,\frac{1}{(z-y)^{2}}\int_{\mathbb{R}}{{\mathsf{d}}}u\,\partial^{a}\big{(}J_{a}^{+}(u,y+x)+J_{a}^{-}(u,y+x)\big{)},\end{split}$
(A.16)

where we set $z\to z+x$ and $y\to y+x$ in the integral and then pulled in the
derivative $\partial^{a}_{x}$. By current conservation, we have
$\partial^{a}J_{a}^{\pm}(u,x)=0$.181818This can be proved by taking the large
$r$ limit of $\partial^{A}J_{A}(X)=0$ and using the fact that
$J_{r}(u,x)={\mathcal{O}}(|r|^{-d-2})$ [17]. Therefore, only the term
involving $J_{u}^{\pm}$ on the right-hand-side of (A.15) survives, and we have

$\displaystyle\begin{split}\partial^{a}\widetilde{{\mathcal{J}}_{a}}(x)&=2\partial^{a}_{x}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}z\,\frac{{\mathcal{I}}_{ab}(x-z)}{[(x-z)^{2}]^{d-1}}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}y\,\frac{x_{a}-y_{a}}{(z-y)^{2}}\int_{\mathbb{R}}{{\mathsf{d}}}u\,\big{(}J_{u}^{+}(u,y)-J_{u}^{-}(u,y)\big{)}\\\
&=\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}z\,\frac{{\mathcal{I}}_{ab}(z)}{(z^{2})^{d-1}}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}y\int_{\mathbb{R}}{{\mathsf{d}}}u\,\partial^{a}_{x}\partial_{z}^{b}\ln\big{[}(z+x-y)^{2}\big{]}\big{(}J_{u}^{+}(u,y)-J_{u}^{-}(u,y)\big{)}.\end{split}$
(A.17)

Using the fact

$\displaystyle\partial^{a}\partial^{b}\ln(x^{2})=\frac{2\,{\mathcal{I}}^{ab}(x)}{x^{2}},$
(A.18)

we can further simplify (A.17) to obtain

$\displaystyle\begin{split}\int_{\mathbb{R}}{{\mathsf{d}}}u\,\big{(}J_{u}^{+}(u,x)-J_{u}^{-}(u,x)\big{)}=\frac{1}{2c_{1,1}}\partial^{a}\widetilde{{\mathcal{J}}_{a}}(x),\end{split}$
(A.19)

which is precisely (3.49).

##  Appendix B Massive Particles

In Section 3.2.2, we proved that the identity (3.51) holds in the absence of
massive particles. To be precise, we ignored the second term in (3.46). In
this appendix, we show that including that term allows us to account for
massive particles in the soft effective action. First, we use the fact that in
the far future, the only source for the gauge field is the Liénard-Wiechert
field strength generated by the massive particles (generalized to arbitrary
dimensions), so that given a set of massive particles with momenta $p_{k}^{A}$
and $U(1)$ charge $Q_{k}$, the field strength is191919We determined this by
covariantizing the four-dimensional Liénard-Wiechert field strength (for
instance, see Chapter 14 of [49]) and then generalizing to all dimensions. The
prefactor is fixed by requiring that Maxwell’s equations are satisfied for the
point-particle current (3.7) (away from $X^{0}=0$).

$\begin{split}{\hat{F}}^{C}_{AB}(X)&=\sum_{k\,\text{massive}}\frac{e^{2}m_{k}^{d}Q_{k}}{{\Omega}_{d}}{\theta}(\eta_{k}X^{0})\frac{p_{kA}X_{B}-p_{kB}X_{A}}{\left[(p_{k}\cdot
X)^{2}+m_{k}^{2}X^{2}\right]^{\frac{1}{2}(d+1)}},\end{split}$ (B.1)

where ${\Omega}_{d}=\frac{2\pi^{(d+1)/2}}{{\Gamma}((d+1)/2)}$ is the volume of
the unit $S^{d}$, ${\theta}$ the Heaviside function, $\eta_{k}$ is positive
(negative) for outgoing (incoming) particles, and the sum is only over massive
particles. The superscript $C$ indicates that this is the Coulombic solution
(recall that there is no radiation for the gauge field through $i^{\pm}$).
Using (2.6) and moving to flat null coordinates, we find

$\begin{split}{\hat{F}}^{C}_{ur}(u,r,x)&=\frac{e^{2}}{2}\sum_{k\,\text{massive}}\frac{2^{d}m_{k}^{d}Q_{k}}{{\Omega}_{d}}\frac{{\theta}\big{[}\eta_{k}\big{(}r(1+x^{2})+u\big{)}\big{]}\left[u{\omega}_{k}-r{\omega}_{k}\left((x-x_{k})^{2}+\frac{m_{k}^{2}}{{\omega}_{k}^{2}}\right)\right]}{\left\\{{\omega}_{k}^{2}\left[u+r\left((x-x_{k})^{2}+\frac{m_{k}^{2}}{{\omega}_{k}^{2}}\right)\right]^{2}-4urm_{k}^{2}\right\\}^{\frac{1}{2}(d+1)}}.\end{split}$
(B.2)

Using this, as well as the Legendre duplication formula

$\displaystyle{\Gamma}(z){\Gamma}\bigg{(}z+\frac{1}{2}\bigg{)}=2^{1-2z}\sqrt{\pi}{\Gamma}(2z),$
(B.3)

the second term in (3.46) evaluates to

$\begin{split}\lim_{u\to\pm\infty}\lim_{r\to\pm\infty}|r|^{d}{\hat{F}}^{C}_{ur}(u,r,x)&=-\frac{e^{2}}{2}\sum_{k\,\text{massive}\atop\eta_{k}=\pm}Q_{k}{\mathcal{K}}_{d}(z_{k},x_{k};x),\qquad
z_{k}\equiv\frac{m_{k}}{|{\omega}_{k}|},\end{split}$ (B.4)

where ${\mathcal{K}}_{\Delta}$ is the bulk-to-boundary propagator (2.29).
Substituting this result into (3.42), we find that the contribution of massive
particles to the boundary action (3.41) is

$\displaystyle S_{{\text{bdy}}}[\hat{A},\theta]\big{|}_{\text{massive}}$
$\displaystyle=\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,{\theta}(x)\sum_{k\,\text{massive}}Q_{k}{\mathcal{K}}_{d}(z_{k},x_{k};x)=-\frac{1}{2c_{1,1}}\int_{{\mathbb{S}}^{d}}{{\mathsf{d}}}^{d}x\,\widetilde{C}^{a}(x){\mathcal{J}}_{a}^{\text{massive}}(x),$
(B.5)

where ${\mathcal{J}}_{a}^{\text{massive}}$ is the soft factor involving only
massive particles, and in the last equality we have used the properties (2.31)
and (2.32). Adding this to (3.51), we reproduce exactly the massive
contribution to (3.31).

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