††institutetext: 1Center for Quantum Mathematics and Physics (QMAP),
University of California, Davis, CA 95616, USA††institutetext: 2School of
Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA

# Asymptotic Symmetries and Weinberg’s Soft Photon Theorem in Minkd+2

Temple He1 and Prahar Mitra2<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We show that Weinberg’s leading soft photon theorem in massless abelian gauge
theories implies the existence of an infinite-dimensional large gauge symmetry
which acts non-trivially on the null boundaries ${\mathscr{I}}^{\pm}$ of
$(d+2)$-dimensional Minkowski spacetime. These symmetries are parameterized by
an arbitrary function ${\varepsilon}(x)$ of the $d$-dimensional celestial
sphere living at ${\mathscr{I}}^{\pm}$. This extends the previously
established equivalence between Weinberg’s leading soft theorem and asymptotic
symmetries from four and higher even dimensions to _all_ higher dimensions.

##  1 Introduction

Soft theorems are equations that describe how scattering amplitudes in
Minkowski spacetime involving one or many low energy, or soft, massless
particles factorizes into a term involving the soft gauge particles (known as
the soft factor) and one involving the amplitude without the soft particles,
i.e.

$\begin{split}{\mathcal{A}}_{n}~{}~{}\xrightarrow{m~{}\text{soft
particles}}~{}~{}S_{m}{\mathcal{A}}_{n-m},\end{split}$ (1.1)

where $S_{m}$ is the soft factor involving $m$ soft particles. Such theorems
were established over half a century ago in the works of Bloch, Nordseick,
Low, Yennie, Frautschi, Suura Bloch:1937pw ; Nordsieck:1937zz ; Low:1954kd ;
GellMann:1954kc ; Low:1958sn ; Yennie:1961ad  and in a more modern language by
Weinberg Weinberg:1965nx ; Weinberg:1995mt . A particularly interesting
feature of all the soft theorems is that the soft factor is _universal_ , i.e.
it does not depend on the details of the theory under consideration. Rather,
it depends only on the quantum numbers (linear momenta, angular momenta and
charges) of the external particles involved in the scattering amplitude. Such
universality naturally suggests that perhaps the soft theorems arise from some
underlying symmetry, and indeed, it was relatively recently discovered that
these theorems are precisely the Ward identities associated to the asymptotic
symmetries of scattering amplitudes (see Strominger:2017zoo  for a review and
a comprehensive list of references).

Largely in part due to this discovery, in the last several years, there has
been a resurgence of activity in the study of soft theorems, and many
interesting directions have been explored. The equivalence was originally
conceived for four-dimensional theories where it has been established to hold
for gauge theories He:2014cra ; Mohd:2014oja ; He:2015zea ; Campiglia:2015qka
; Kapec:2015ena ; Strominger:2015bla ; Gabai:2016kuf , gravity He:2014laa ;
Campiglia:2015yka ; Campiglia:2015kxa , as well as their supersymmetric
counterparts Dumitrescu:2015fej . The equivalence has proved useful in the
exploration of not just the original soft theorems, but also new subleading
and sub-subleading soft theorems Cachazo:2014fwa ; Schwab:2014fia ;
Kalousios:2014uva ; Luo:2014wea ; DiVecchia:2016amo ; Sen:2017nim ;
Laddha:2017vfh ; Hirai:2018ijc . Some of these were discovered by studying
known asymptotic symmetries Barnich:2011ct ; Kapec:2014opa ; Campiglia:2014yka
while others led to the discovery of new asymptotic symmetries Lysov:2014csa ;
Campiglia:2016jdj ; Campiglia:2016hvg ; Conde:2016csj ; Campiglia:2016efb ;
Conde:2016rom ; Laddha:2017ygw . Finally, this equivalence has also been
extended to theories living in higher even dimensions Schwab:2014xua ;
Kapec:2014zla ; Kapec:2015vwa , which is certainly relevant in the context of
string theory.111More generally, soft theorems and asymptotic symmetries are
also equivalent to memory effects which have been explored in Thorne:1992sdb ;
Strominger:2014pwa ; Pasterski:2015zua ; Bieri:2015yia ; Susskind:2015hpa ;
Hotta:2016qtv ; Hollands:2016oma ; Pate:2017vwa ; Pate:2017fgt ; Ball:2018prg
, but this will be beyond the scope of this paper (see Strominger:2017zoo  for
a review).

The plethora of successes in these endeavors has given confidence that perhaps
every soft theorem in Minkowski spacetime can be interpreted as the Ward
identity for some asymptotic symmetry. However, there is a noticeable class of
theories in which the bridge between soft theorems and asymptotic symmetries
is missing – these are theories in odd spacetime dimensions. Soft theorems are
true and have the same universal form in both even and odd dimensions.222The
standard derivation of these theorems in perturbation theory relies largely on
the fact that the free field propagator has the form $p^{-2}$, which is true
in any dimension. Yet, to our knowledge, there has been no successful attempt
in showing the equivalence of soft theorems and asymptotic symmetries in odd
dimensional theories. The fundamental issue lies in the qualitatively
different properties of massless waves in even and odd dimensions. For
instance, the massless Green’s function is supported entirely on the light
cone in even dimensions, whereas in odd dimensions it is supported everywhere
inside the lightcone Balazs1955 ; Soodak1993 . This implies that for a
complete understanding of waves in odd dimensions, one needs to study timelike
infinity $i^{\pm}$ as well as null infinity ${\mathscr{I}}^{\pm}$. Another
symptom of the same feature is that the near ${\mathscr{I}}^{\pm}$ expansions
of massless fields is analytic in even dimensions (i.e. admits a Taylor series
expansion in $r^{-1}$ where $r$ is the radius of the transverse sphere) and
non-analytic in odd dimensions. These issues make the discussion of asymptotic
symmetries particularly difficult in odd dimensions. Despite this, it would be
very strange if no such relationship exists, implying that soft theorems in
odd dimensions have a wholly different origin than those in even dimensions.

In this paper, we resolve the issues discussed in the previous paragraph and
show that, at least in the simple example of a $U(1)$ gauge theory with
massless charged matter, the leading soft photon theorem is the Ward identity
associated to large gauge symmetry in _all_ dimensions, thereby unifying and
generalizing the previous discussions from four and higher even dimensions to
all dimensions. We organize the paper so that in §2, we study $U(1)$ gauge
theories in $d+2$ dimensions and discuss asymptotic fall-offs of both free and
interacting gauge fields. In §3, we impose a matching condition across spatial
infinity $i^{0}$ and demonstrate that the resulting Ward identity implies the
leading soft photon theorem. Finally, in §4, we show that the matching
condition implies conservation of a charge which generates large gauge
transformations on the fields at ${\mathscr{I}}^{\pm}$.

##  2 $U(1)$ Gauge Theories in $d+2$ Dimensions

In this section, we introduce the conventions used in this paper and discuss
the necessary background information regarding the equations of motion and the
asymptotics of the $U(1)$ gauge field.

###  2.1 Classical Equations

We work in flat null coordinates $x^{\mu}=(u,r,x^{a})$, $a=1,\ldots,d$ which
are related to Cartesian coordinates $X^{A}$ by333These coordinates were
utilized in the case of four dimensions in Dumitrescu:2015fej  and in
Kapec:2017gsg  in higher dimensions.

$\displaystyle
X^{A}=\frac{r}{2}\left(1+x^{2}+\frac{u}{r},2x^{a},1-x^{2}-\frac{u}{r}\right),\qquad
x^{2}=\delta_{ab}x^{a}x^{b}.$ (2.1)

Lowercase Latin indices are raised and lowered by the Kronecker delta
$\delta^{ab}$ and $\delta_{ab}$, respectively. Using these coordinates, the
metric of Minkowski spacetime becomes

$\begin{split}ds^{2}&=\eta_{AB}dX^{A}dX^{B}=-dudr+r^{2}\delta_{ab}dx^{a}dx^{b},\end{split}$
(2.2)

where $\eta_{AB}=(-1,1,\ldots,1)$ is the standard Minkowski metric in
Cartesian coordinates. The null boundaries ${\mathscr{I}}^{\pm}$ are located
at $r\to\pm\infty$ while keeping $(u,x^{a})$ fixed. On this hypersurface,
$x^{a}$ is the stereographic coordinate on the celestial sphere. The point
labeled by $x^{a}$ on ${\mathscr{I}}^{+}$ is antipodal to the point with the
same label on ${\mathscr{I}}^{-}$. The boundaries of ${\mathscr{I}}^{+}$
(${\mathscr{I}}^{-}$) are at $u\to\pm\infty$ and are denoted by
${\mathscr{I}}^{+}_{\pm}$ (${\mathscr{I}}^{-}_{\pm}$). In Appendix A.1, we
provide more details about this coordinate system.

A $U(1)$ gauge theory is described by a field strength $F_{\mu\nu}$ and matter
fields $\Psi_{i}$. The matter fields couple to the field strength via a
conserved $U(1)$ current $J_{\mu}$. Dynamics of the field strength is
described by Maxwell’s equations and the Bianchi identity, given respectively
by

$\displaystyle e^{2}J_{\nu}$ $\displaystyle=\nabla^{\mu}F_{\mu\nu}$ (2.3)
$\displaystyle 0$
$\displaystyle=\partial_{{\alpha}}F_{\mu\nu}+\partial_{\nu}F_{{\alpha}\mu}+\partial_{\mu}F_{\nu{\alpha}}.$
(2.4)

In flat null coordinates, (2.3) takes the form

$\displaystyle\begin{split}e^{2}J_{u}&=2\partial_{u}F_{ur}-\frac{1}{r^{2}}\partial^{a}F_{ua}\\\
e^{2}J_{r}&=-2\partial_{r}F_{ur}-\frac{1}{r^{2}}\partial^{a}F_{ra}-\frac{2d}{r}F_{ur}\\\
e^{2}J_{a}&=-2\partial_{r}F_{ua}-2\partial_{u}F_{ra}-\frac{2(d-2)}{r}F_{ua}-\frac{1}{r^{2}}\partial^{b}F_{ab}.\end{split}$
(2.5)

Furthermore, the Bianchi identity (2.4) is solved by

$\begin{split}F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu},\qquad
A_{\mu}\sim
A_{\mu}+\partial_{\mu}{\varepsilon},\qquad{\varepsilon}\sim{\varepsilon}+2\pi.\end{split}$
(2.6)

Note that the theory is invariant under gauge transformations

$\begin{split}A_{\mu}~{}\to~{}A_{\mu}+\partial_{\mu}{\varepsilon},\qquad\Psi_{i}~{}\to~{}e^{iQ_{i}{\varepsilon}}\Psi_{i},\end{split}$
(2.7)

where $Q_{i}\in{\mathbb{Z}}$ is the $U(1)$ charge of the field $\Psi_{i}$.
Gauge transformations that vanish at infinity correspond to redundant
descriptions of the same physical state and can be eliminated by a choice of
gauge.

###  2.2 Asymptotics

In this section, we study the structure of the field strength near
${\mathscr{I}}^{\pm}$. The general solution to (2.3) and (2.4) has the form

$\begin{split}F_{\mu\nu}=F_{\mu\nu}^{(R)}+F_{\mu\nu}^{(C)}.\end{split}$ (2.8)

The radiative field $F_{\mu\nu}^{(R)}$ satisfies the homogeneous Maxwell’s
equations ($J_{\nu}=0$) and $F_{\mu\nu}^{(C)}$ satisfies (2.3). The separation
of the field strength into a radiative part and a Coulombic part is fixed by
either choosing a Green’s function or by imposing boundary conditions on the
fields. We will be interested in incoming ${(-)}$ and outgoing ${(+)}$
solutions to Maxwell’s equations, whose radiative and Coulombic parts are
denoted respectively by $F_{\mu\nu}^{(R\pm)}$ and $F_{\mu\nu}^{(C\pm)}$. These
fields have the following large $r$ fall-offs near ${\mathscr{I}}^{\pm}$:444As
we shall see in (2.23), in odd dimensions, the radiative field has an
expansion in both integer and half-integer powers of $|r|$, as indicated in
the first column of (2.9). In even dimensions, we obtain later in
(LABEL:Furlargereven1) that the radiative field has an expansion in integer
powers of $|r|$, though non-analytic terms of the form $\log r$ are also
present (not shown in (2.9)).

$\begin{split}F_{ur}^{(R\pm)}&=O\left(|r|^{-\frac{d}{2}-1}\right)+O\left(|r|^{-d}\right),\qquad\quad
F_{ur}^{(C\pm)}=O\left(|r|^{-d}\right)\\\
F_{ra}^{(R\pm)}&=O\left(|r|^{-\frac{d}{2}}\right)+O\left(|r|^{-d}\right),\qquad\qquad
F_{ra}^{(C\pm)}=O\left(|r|^{-d}\right)\\\
F_{ua}^{(R\pm)}&=O\left(|r|^{-\frac{d}{2}+1}\right)+O\left(|r|^{-d+1}\right),\qquad
F_{ua}^{(C\pm)}=O\left(|r|^{-d+1}\right)\\\
F_{ab}^{(R\pm)}&=O\left(|r|^{-\frac{d}{2}+1}\right)+O\left(|r|^{-d+1}\right),\qquad
F_{ab}^{(C\pm)}=O\left(|r|^{-d+1}\right).\end{split}$ (2.9)

The leading fall-offs for the radiative part are determined by noting that the
electromagnetic stress tensor components $T_{uu}$ and $T_{ua}$ must fall off
as $|r|^{-d}$ in order to have finite energy-momentum and angular-momentum
flux through ${\mathscr{I}}^{\pm}$. Then, using the fact that $T_{uu}\sim
r^{-2}F_{ua}F_{u}{}^{a}$ and $T_{ua}\sim r^{-2}F_{ab}F_{u}{}^{b}$, we can
determine the fall-offs for $F_{ua}$ and $F_{ab}$. The remaining ones are then
determined by Lorentz invariance. Similarly, the fall-offs for the Coulombic
part are determined by Gauss’s law, which also determines the fall-offs for
the matter current

$\begin{split}J_{u}=O\big{(}|r|^{-d}\big{)},\qquad
J_{a}=O\big{(}|r|^{-d}\big{)},\qquad
J_{r}=O\big{(}|r|^{-d-2}\big{)}.\end{split}$ (2.10)

We would now like to determine the precise large $r$ expansion for both the
radiative and the Coulombic fields.

####  2.2.1 Radiative Field

The radiative gauge field can be decomposed into creation and annihilation
operators $\mathcal{O}^{(\pm){\dagger}}_{a}$ and $\mathcal{O}^{(\pm)}_{a}$ as
follows:

$\begin{split}F^{(R\pm)}_{AB}(X)=e\int\frac{d^{d+1}q}{(2\pi)^{d+1}}\frac{1}{2q^{0}}\left[{\varepsilon}^{a}_{AB}({\vec{q}}\,){\mathcal{O}}^{(\pm)}_{a}(\vec{q}\,)e^{iq\cdot
X}+{\varepsilon}^{a}_{AB}({\vec{q}}\,)^{*}{\mathcal{O}}^{{(\pm)}\dagger}_{a}(\vec{q}\,)e^{-iq\cdot
X}\right],\end{split}$ (2.11)

where $q^{0}=|\vec{q}\,|$ and ${\varepsilon}_{AB}^{a}(\vec{q}\,)$ are the $d$
polarization tensors labeled by $a$ and defined via

$\begin{split}{\varepsilon}^{a}_{AB}(\vec{q}\,)=i\left[q_{A}{\varepsilon}^{a}_{B}({\vec{q}}\,)-q_{B}{\varepsilon}^{a}_{A}({\vec{q}}\,)\right],\qquad
q^{A}{\varepsilon}_{A}^{a}({\vec{q}}\,)=0,\qquad\eta^{AB}{\varepsilon}^{a}_{A}({\vec{q}}\,){\varepsilon}_{B}^{b}({\vec{q}}\,)^{*}=\delta^{ab}.\end{split}$
(2.12)

In a quantum theory, the creation and annihilation modes satisfy the canonical
commutation relation

$\begin{split}\left[{\mathcal{O}}^{(\pm)}_{a}(\vec{q}\,)\,,\,{\mathcal{O}}^{{(\pm)}\dagger}_{b}(\vec{q}\,^{\prime})\right]=\big{(}2q^{0}\big{)}\delta_{ab}(2\pi)^{d+1}\delta^{(d+1)}(\vec{q}-\vec{q}\,^{\prime}).\end{split}$
(2.13)

The equations in (2.12) remain invariant under
${\varepsilon}^{a}_{A}(\vec{q}\,)\to{\varepsilon}^{a}_{A}(\vec{q}\,)+f^{a}(\vec{q}\,)q_{A}$
for any arbitrary functions $f^{a}$. We use this to parameterize the on-shell
photon momentum and polarization as555We discuss a more general
parameterization of off-shell momenta in Appendix A.2.

$\begin{split}q^{A}={\omega}{\hat{q}}^{A},\qquad{\hat{q}}^{A}=\left(\frac{1+y^{2}}{2}\,,\,y^{a}\,,\,\frac{1-y^{2}}{2}\right),\qquad{\varepsilon}^{a}_{A}({\vec{q}}\,)=\partial^{a}{\hat{q}}_{A}=\left(-y^{a},\delta^{a}_{b},-y^{a}\right).\end{split}$
(2.14)

Substituting (2.14) into (2.11) and writing the integral over flat null
coordinates, we obtain666Similar mode expansions exist for the other
components of the field strength, but we won’t need them here.

$\begin{split}F^{(R\pm)}_{ur}(u,r,x)=\frac{e}{4(2\pi)^{d+1}r}\int_{0}^{\infty}d{\omega}\int
d^{d}y\,{\omega}^{d-1}\left[\partial^{a}{\mathcal{O}}^{(\pm)}_{a}({\omega},x+y)e^{-\frac{i}{2}{\omega}u-\frac{i}{2}{\omega}ry^{2}}+{\text{c.c.}}\right].\end{split}$
(2.15)

To determine the asymptotic expansion of (2.15) at large $|u|$ and $|r|$, we
assume that the creation and annihilation modes admit a soft expansion of the
form

$\begin{split}\partial^{a}{\mathcal{O}}^{(\pm)}_{a}({\omega},x+y)=\sum_{n=0}^{\infty}{\omega}^{n-1}\partial^{a}{\mathcal{O}}^{(\pm,n)}_{a}(x+y)\qquad\text{as}\qquad{\omega}\to
0^{+}.\end{split}$ (2.16)

This soft expansion allows for a simple pole in ${\omega}$ which is consistent
with Weinberg’s soft photon theorem. The corresponding “soft photon operator”
is ${\mathcal{O}}_{a}^{(\pm,0)}(x)$.777This operator was denoted by $S_{a}(x)$
in Kapec:2017gsg . It will play a central role in the following section.

We further assume that the coefficients admit a Fourier transform

$\begin{split}\partial^{a}{\mathcal{O}}^{(\pm,n)}_{a}(x+y)&=\int\frac{d^{d}k}{(2\pi)^{d}}e^{ik\cdot(x+y)}{\mathscr{O}}^{(\pm)}_{n}(k)\\\
{\mathscr{O}}^{(\pm)}_{n}(k)&=\int d^{d}y\,e^{-ik\cdot
y}\partial^{a}{\mathcal{O}}^{(\pm,n)}_{a}(y).\end{split}$ (2.17)

Substituting (2.16) and (2.17) into (2.15), we find

$\begin{split}F^{(R\pm)}_{ur}(u,r,x)=\frac{e}{2(2\pi)^{\frac{d}{2}+1}}\sum_{n=0}^{\infty}\int\frac{d^{d}k}{(2\pi)^{d}}k^{2\nu_{n}}\left[\frac{ie^{ik\cdot
x}}{(ir)^{d+n}}{\mathscr{O}}^{(\pm)}_{n}(k)(kz)^{-\nu_{n}}K_{\nu_{n}}\left(kz\right)+{\text{c.c.}}\right].\end{split}$
(2.18)

where

$\begin{split}z=\frac{\sqrt{iu}}{\sqrt{ir}},\qquad\nu_{n}=\frac{d}{2}-1+n,\end{split}$
(2.19)

and $K_{\nu}$ is the modified Bessel function of the second kind. We now
consider the large $|r|$ expansion of the field strength, or equivalently the
small $z$ limit. This takes a qualitatively different form depending on
whether $d$ is odd or even, so we consider the two cases separately.

#### Odd Dimensions

When $d\in 2{\mathbb{Z}}+1$, the order of the Bessel function is
$\nu_{n}=\frac{d}{2}-1+n\not\in{\mathbb{Z}}$. For non-integer order, the
asymptotic expansion near $z=0$ of the Bessel function is

$\begin{split}K_{\nu}(z)=\frac{\pi}{2}\csc(\pi\nu)\sum_{s=0}^{\infty}\frac{1}{{\Gamma}(s+1)}\left[\frac{\left(z/2\right)^{2s-\nu}}{{\Gamma}(1+s-\nu)}-\frac{\left(z/2\right)^{2s+\nu}}{{\Gamma}(1+s+\nu)}\right],\quad\text{as}\quad
z\to 0.\end{split}$ (2.20)

Substituting (2.20) into (2.18), we find

$\begin{split}F^{(R\pm)}_{ur}(u,r,x)&=\frac{e}{8(2\pi)^{\frac{d}{2}}}\sum_{n=0}^{\infty}\sum_{s=0}^{\infty}\frac{\csc(\pi\nu_{n})}{2^{2s-\nu_{n}}{\Gamma}(s+1){\Gamma}(1+s-\nu_{n})}\\\
&\qquad\qquad\qquad\qquad\times\left[\frac{i(iu)^{s-\nu_{n}}}{(ir)^{\frac{d}{2}+1+s}}\int\frac{d^{d}k}{(2\pi)^{d}}k^{2s}{\mathscr{O}}^{(\pm)}_{n}(k)e^{ik\cdot
x}+{\text{c.c.}}\right]\\\
&\qquad\qquad-\frac{e}{8(2\pi)^{\frac{d}{2}}}\sum_{n=0}^{\infty}\sum_{s=0}^{\infty}\frac{\csc(\pi\nu_{n})}{2^{2s+\nu_{n}}{\Gamma}(s+1){\Gamma}(1+s+\nu_{n})}\\\
&\qquad\qquad\qquad\qquad\times\left[\frac{i(iu)^{s}}{(ir)^{d+n+s}}\int\frac{d^{d}k}{(2\pi)^{d}}k^{2(s+\nu_{n})}{\mathscr{O}}^{(\pm)}_{n}(k)e^{ik\cdot
x}+{\text{c.c.}}\right].\end{split}$ (2.21)

We can now use the inverse Fourier transform (2.17) to go back to the momentum
space variables. Evaluating

$\begin{split}\int\frac{d^{d}k}{(2\pi)^{d}}k^{2{\kappa}}{\mathscr{O}}^{(\pm)}_{n}(k)e^{ik\cdot
x}=\begin{cases}(-\partial^{2})^{\kappa}\partial^{a}{\mathcal{O}}^{(\pm,n)}_{a}(x)&{\kappa}\in{\mathbb{Z}}\\\
\frac{4^{\kappa}{\Gamma}\left(\frac{d}{2}+{\kappa}\right)}{\pi^{\frac{d}{2}}{\Gamma}(-{\kappa})}\displaystyle\int
d^{d}y\frac{\partial^{a}{\mathcal{O}}^{(\pm,n)}_{a}(y)}{\left[(x-y)^{2}\right]^{\frac{d}{2}+{\kappa}}}&{\kappa}\not\in{\mathbb{Z}}\end{cases}\end{split}$
(2.22)

where $\partial^{2}=\partial^{a}\partial_{a}$, and substituting into (2.21),
we obtain

$\begin{split}F^{(R\pm)}_{ur}(u,r,x)&=\frac{e}{8(2\pi)^{\frac{d}{2}}}\sum_{n=0}^{\infty}\sum_{s=0}^{\infty}\frac{2^{\nu_{n}-2s}\csc(\pi\nu_{n})}{{\Gamma}(s+1){\Gamma}(1+s-\nu_{n})}\\\
&\qquad\qquad\qquad\qquad\qquad\times\left[\frac{i(iu)^{s-\nu_{n}}}{(ir)^{\frac{d}{2}+1+s}}(-\partial^{2})^{s}\partial^{a}{\mathcal{O}}^{(\pm,n)}_{a}(x)+{\text{c.c.}}\right]\\\
&\qquad\qquad\qquad+\frac{e}{2^{\frac{d}{2}+3}\pi^{d+1}}\sum_{n=0}^{\infty}\sum_{s=0}^{\infty}(-1)^{s}\frac{{\Gamma}\left(\frac{d}{2}+\nu_{n}+s\right)}{2^{-\nu_{n}}{\Gamma}(s+1)}\\\
&\qquad\qquad\qquad\qquad\qquad\times\left[\frac{i(iu)^{s}}{(ir)^{d+n+s}}\int
d^{d}y\frac{\partial^{a}{\mathcal{O}}^{(\pm,n)}_{a}(y)}{\left[(x-y)^{2}\right]^{\frac{d}{2}+\nu_{n}+s}}+{\text{c.c.}}\right].\end{split}$
(2.23)

#### Even Dimensions

When $d\in 2{\mathbb{Z}}$, the order of the Bessel function is
$\nu_{n}=\frac{d}{2}-1+n\in{\mathbb{Z}}$. For integer order, the asymptotic
expansion near $z=0$ of the Bessel function is given by

$\begin{split}K_{\nu}(z)&=\sum_{s=0}^{\nu-1}\frac{(-1)^{s}{\Gamma}(\nu-s)\left(\frac{z}{2}\right)^{2s-\nu}}{2{\Gamma}(s+1)}\\\
&\qquad\qquad\qquad\qquad\qquad+\sum_{s=0}^{\infty}\frac{(-1)^{\nu-1}\left[\log\left(\frac{z}{2}e^{{\gamma}_{E}}\right)-\frac{1}{2}\left(H_{s}+H_{s+\nu}\right)\right]\left(\frac{z}{2}\right)^{2s+\nu}}{{\Gamma}(s+1){\Gamma}(s+\nu+1)},\end{split}$
(2.24)

where $\gamma_{E}$ is the Euler-Mascheroni constant and $H_{n}$ is the $n$-th
harmonic number (we define $H_{0}=0$). Substituting (2.24) into (2.18), we
find

$\begin{split}&F^{(R\pm)}_{ur}(u,r,x)\\\
&\quad=\frac{e}{2(2\pi)^{\frac{d}{2}+1}}\sum_{n=0}^{\infty}\sum_{s=0}^{\nu_{n}-1}\frac{(-1)^{s}{\Gamma}(\nu_{n}-s)}{2^{2s-\nu_{n}+1}{\Gamma}(s+1)}\left[\frac{i(iu)^{s-\nu_{n}}}{(ir)^{\frac{d}{2}+1+s}}\int\frac{d^{d}k}{(2\pi)^{d}}k^{2s}{\mathscr{O}}^{(\pm)}_{n}(k)e^{ik\cdot
x}+{\text{c.c.}}\right]\\\
&\qquad\qquad\qquad+\frac{e}{2(2\pi)^{\frac{d}{2}+1}}\sum_{n=0}^{\infty}\sum_{s=0}^{\infty}\frac{2^{-2s-\nu_{n}}(-1)^{\nu_{n}-1}}{{\Gamma}(s+1){\Gamma}(s+\nu_{n}+1)}\int\frac{d^{d}k}{(2\pi)^{d}}k^{2\left(s+\nu_{n}\right)}\\\
&\qquad\qquad\qquad\quad\times\left[\frac{i(iu)^{s}\left[\log\left(\frac{k}{2}\frac{\sqrt{iu}}{\sqrt{ir}}e^{{\gamma}_{E}}\right)-\frac{1}{2}\left(H_{s}+H_{s+\nu_{n}}\right)\right]}{(ir)^{d+n+s}}{\mathscr{O}}^{(\pm)}_{n}(k)e^{ik\cdot
x}+{\text{c.c.}}\right].\end{split}$ (2.25)

Again, we want to use the inverse Fourier transform (2.17) to go back to the
momentum space variables. Evaluating for $\kappa\in{\mathbb{Z}}$

$\begin{split}\int\frac{d^{d}k}{(2\pi)^{d}}k^{2{\kappa}}\log
k\,{\mathscr{O}}^{(\pm)}_{n}(k)e^{ik\cdot
x}&=\frac{(4\pi)^{\frac{d}{2}}{\Gamma}({\kappa}+1){\Gamma}\left(\frac{d}{2}+{\kappa}\right)}{(-1)^{{\kappa}+1}2^{1-2{\kappa}}}\int
d^{d}y\frac{\partial^{a}{\mathcal{O}}^{(\pm,n)}_{a}(y)}{\left[(x-y)^{2}\right]^{\frac{d}{2}+{\kappa}}},\end{split}$
(2.26)

we get upon substituting (2.22) and (2.26) into (LABEL:Furlargereven)

$\begin{split}&F^{(R\pm)}_{ur}(u,r,x)\\\
&=\frac{e}{2(2\pi)^{\frac{d}{2}+1}}\sum_{n=0}^{\infty}\sum_{s=0}^{\nu_{n}-1}\frac{(-1)^{s}{\Gamma}(\nu_{n}-s)}{2^{2s-\nu_{n}+1}{\Gamma}(s+1)}\left[\frac{i(iu)^{s-\nu_{n}}}{(ir)^{\frac{d}{2}+1+s}}(-\partial^{2})^{s}\partial^{a}{\mathcal{O}}^{(\pm,n)}_{a}(x)+{\text{c.c.}}\right]\\\
&~{}+\frac{e}{8\pi}\sum_{n=0}^{\infty}\sum_{s=0}^{\infty}\frac{(-1)^{s}2^{\frac{d}{2}+\nu_{n}}{\Gamma}\left(\frac{d}{2}+s+\nu_{n}\right)}{{\Gamma}(s+1)}\left[\frac{i(iu)^{s}}{(ir)^{d+n+s}}\int
d^{d}y\frac{\partial^{a}{\mathcal{O}}^{(\pm,n)}_{a}(y)}{\left[(x-y)^{2}\right]^{\frac{d}{2}+\nu_{n}+s}}+{\text{c.c.}}\right]\\\
&~{}+\frac{e}{2(2\pi)^{\frac{d}{2}+1}}\sum_{n=0}^{\infty}\sum_{s=0}^{\infty}\frac{2^{-2s-\nu_{n}}(-1)^{\nu_{n}-1}}{{\Gamma}(s+1){\Gamma}(s+\nu_{n}+1)}\\\
&\qquad\times\left[\frac{i(iu)^{s}\left[\log\left(\frac{1}{2}\frac{\sqrt{iu}}{\sqrt{ir}}e^{{\gamma}_{E}}\right)-\frac{1}{2}\left(H_{s}+H_{s+\nu_{n}}\right)\right]}{(ir)^{d+n+s}}(-\partial^{2})^{s+\nu_{n}}\partial^{a}{\mathcal{O}}^{(\pm,n)}_{a}(x)+{\text{c.c.}}\right].\end{split}$
(2.27)

####  2.2.2 Coulombic Field

Having performed the asymptotic expansion for the radiative field, we now turn
to the Coulombic field. The large $r$ expansion for this can be performed in
two ways. First, we could follow strategy used in the previous section and
start by writing the explicit solution for the Coulombic field using the
massless scalar Green’s function as

$\begin{split}F_{AB}^{(C\pm)}(X)=e^{2}\int
d^{d+2}Y{\mathfrak{G}}^{(\pm)}(X-Y)\left[\partial_{A}J_{B}(Y)-\partial_{B}J_{A}(Y)\right],\end{split}$
(2.28)

where ${\mathfrak{G}}^{(\pm)}(X)$ is the retarded $(-)$ or advanced $(+)$
massless scalar Green’s function satisfying
$\partial^{2}{\mathfrak{G}}^{(\pm)}(X)=\delta^{(d+2)}(X)$. Explicitly,

$\begin{split}{\mathfrak{G}}^{(\pm)}(X)=2\theta(\mp
X^{0})\text{Re}\,[f_{d}(X)],\qquad
f_{d}(X)=\frac{(-i)^{d+1}\Gamma\left(\frac{d}{2}\right)}{4\pi^{\frac{d+2}{2}}\left[(X^{0})^{2}-|\vec{X}\,|^{2}-i{\epsilon}\right]^{\frac{d}{2}}}.\end{split}$
(2.29)

Then, assuming an appropriate asymptotic expansion (near
${\mathscr{I}}^{\pm}$) for the conserved current (along with some other
assumptions regarding the behaviour of the current near $i^{\pm}$), we can use
(2.28) and (2.29) to determine the large $r$ expansion of the Coulombic field.

While possible, this is a tedious exercise and it is more convenient to
determine the large $r$ expansion of the Coulombic field as follows. We start
by assuming that the Coulombic field admits a Taylor expansion of the
following form near ${\mathscr{I}}^{\pm}$ (consistent with (2.9)):

$\begin{split}F_{ur}^{(C\pm)}(u,r,x)&=\sum_{n=0}^{\infty}\frac{F_{ur}^{(C\pm,d+n)}(u,x)}{|r|^{d+n}},\qquad
F_{ua}^{(C\pm)}(u,r,x)=\sum_{n=0}^{\infty}\frac{F_{ua}^{(C\pm,d-1+n)}(u,x)}{|r|^{d-1+n}}\\\
F_{ra}^{(C\pm)}(u,r,x)&=\sum_{n=0}^{\infty}\frac{F_{ra}^{(C\pm,d+n)}(u,x)}{|r|^{d+n}},\qquad
F_{ab}^{(C\pm)}(u,r,x)=\sum_{n=0}^{\infty}\frac{F_{ab}^{(C\pm,d-1+n)}(u,x)}{|r|^{d-1+n}}.\end{split}$
(2.30)

Similarly, consistent with (2.10), we also assume the expansion

$\begin{split}J_{u}(u,r,x)&=\sum_{n=0}^{\infty}\frac{J_{u}^{(\pm,d+n)}(u,x)}{|r|^{d+n}}\\\
J_{a}(u,r,x)&=\sum_{n=0}^{\infty}\frac{J_{a}^{(\pm,d+n)}(u,x)}{|r|^{d+n}}\\\
J_{r}(u,r,x)&=\sum_{n=0}^{\infty}\frac{J_{r}^{(\pm,d+2+n)}(u,x)}{|r|^{d+2+n}}.\end{split}$
(2.31)

Substituting (2.30) and (2.31) into (2.5), we can obtain equations order by
order in the large $r$ expansion. Solving these yields the full asymptotic
expansion for the Coulombic field strength, though for our purposes, it
suffices to obtain just the following leading order constraint equation
(obtained from the $u$ component of Maxwell’s equations):

$\begin{split}2\partial_{u}F^{(C\pm,d)}_{ur}&=e^{2}J^{(\pm,d)}_{u}.\end{split}$
(2.32)

##  3 Weinberg’s Soft Photon Theorem

###  3.1 Matching Condition

As in four dimensions, we impose a matching condition on the radial electric
field888The unusual sign is due to the coordinate choice. To see that this is
correct, one can verify it for the Liénard-Wiechert potential for a static
charge, given by $A_{0}(\vec{X})=c|\vec{X}|^{1-d}$ and $\vec{A}(\vec{X})=0$.
Moving to flat null coordinates, we find
$F_{ur}=\frac{c}{2}\left[\big{(}1+x^{2}\big{)}\partial_{u}-\partial_{r}\right]\big{|}\vec{X}\big{|}^{1-d}=\frac{2^{d-1}c(d-1)(r+rx^{2}-u)}{\left[u^{2}-2ur(1-x^{2})+r^{2}(1+x^{2})^{2}\right]^{\frac{d+1}{2}}}.$
This implies $\lim\limits_{r\to\pm\infty}\big{(}|r|^{d}F_{ur}\big{)}=\pm
2^{d-1}c(d-1)(1+x^{2})^{-d}$. For more extensive discussions on the antipodal
matching condition, we refer the reader to Campiglia:2017mua ; Prabhu:2018gzs
.

$\begin{split}\left.F_{ur}^{(+,d)}\right|_{{\mathscr{I}}^{+}_{-}}&=-\left.F_{ur}^{(-,d)}\right|_{{\mathscr{I}}^{-}_{+}}.\end{split}$
(3.1)

Recall that coordinate $x$ labels antipodal points on the celestial spheres on
${\mathscr{I}}^{+}$ and ${\mathscr{I}}^{-}$, so the above condition is an
_antipodal_ matching condition. To massage this matching condition into a more
useful form, we define the “charge”

$\begin{split}Q^{\pm}_{\varepsilon}=\pm\frac{2}{e^{2}}\int_{{\mathscr{I}}^{\pm}_{\mp}}d^{d}x\,{\varepsilon}F_{ur}^{(\pm,d)},\end{split}$
(3.2)

where ${\varepsilon}\equiv{\varepsilon}(x)$ is a function defined on the
$d$-dimensional celestial sphere at ${\mathscr{I}}^{\pm}$. As we will show in
§4 using the covariant phase space formalism, $Q^{\pm}_{\varepsilon}$ is the
charge that generates large gauge transformations on ${\mathscr{I}}^{\pm}$. We
can therefore think of $Q^{\pm}_{\varepsilon}$ as measuring the local $U(1)$
charge of the $in$ and $out$ state respectively. For ${\varepsilon}=1$, this
measures the total global $U(1)$ charge. These charges can be determined in
experiment by measuring the electromagnetic memory Yoshida:2017fao ;
Pasterski:2015zua ; Susskind:2015hpa .

The antipodal matching condition (3.1) immediately implies

$\displaystyle Q^{+}_{\varepsilon}=Q^{-}_{\varepsilon}.$ (3.3)

Breaking up the field strength into radiative and Coulombic pieces, we can
write the charges as

$\begin{split}Q^{\pm}_{\varepsilon}=Q^{\pm S}_{\varepsilon}+Q^{\pm
H}_{\varepsilon},\end{split}$ (3.4)

where

$\begin{split}Q^{\pm
S}_{\varepsilon}=\pm\frac{2}{e^{2}}\int_{{\mathscr{I}}^{\pm}_{\mp}}d^{d}x\,{\varepsilon}F_{ur}^{(R\pm,d)},\qquad
Q^{\pm
H}_{\varepsilon}=\pm\frac{2}{e^{2}}\int_{{\mathscr{I}}^{\pm}_{\mp}}d^{d}x\,{\varepsilon}F_{ur}^{(C\pm,d)}.\end{split}$
(3.5)

$Q^{\pm S}_{\varepsilon}$ are the incoming and outgoing soft charges and
$Q^{\pm H}_{\varepsilon}$ are the incoming and outgoing hard charges. Using
(2.32), the hard charges can be written as

$\begin{split}Q^{\pm
H}_{\varepsilon}&=-\int_{{\mathscr{I}}^{\pm}}dud^{d}x\,{\varepsilon}J^{(\pm,d)}_{u}\pm\frac{2}{e^{2}}\int_{{\mathscr{I}}^{\pm}_{\pm}}d^{d}x\,{\varepsilon}F_{ur}^{(C\pm,d)}.\end{split}$
(3.6)

The second term above is the Coulombic field in the far past or far future and
only receives contributions from stable massive particles. In this paper, we
assume that our theory contains only stable massless states so this
contribution vanishes. Thus, the hard charges become

$\begin{split}Q^{\pm
H}_{\varepsilon}&=-\int_{{\mathscr{I}}^{\pm}}dud^{d}x\,{\varepsilon}J^{(\pm,d)}_{u}.\end{split}$
(3.7)

To determine the soft charge in odd and even dimensions, we extract the
coefficient of $|r|^{-d}$ in (2.23) and (LABEL:Furlargereven1), respectively.
In odd dimensions, we get

$\begin{split}F^{(R\pm,d)}_{ur}&=\pm\frac{e(-1)^{\frac{d-1}{2}}{\Gamma}\left(d-1\right)}{16\pi^{d+1}}\int
d^{d}y\frac{\partial^{a}{\mathcal{O}}^{(\pm,0)}_{a}(y)+\partial^{a}{\mathcal{O}}^{(\pm,0)\dagger}_{a}(y)}{\left[(x-y)^{2}\right]^{d-1}},\end{split}$
(3.8)

and for even dimensions, we get

$\begin{split}F^{(R\pm,d)}_{ur}&=-\frac{e}{(4\pi)^{\frac{d}{2}+1}{\Gamma}\left(\frac{d}{2}\right)}\sum_{n=1}^{\infty}2^{n}{\Gamma}(n)\left[i(iu)^{-n}(-\partial^{2})^{\frac{d}{2}-1}\partial^{a}{\mathcal{O}}^{(\pm,n)}_{a}(x)+{\text{c.c.}}\right]\\\
&\qquad\qquad+\frac{ie(-1)^{\frac{d}{2}}{\Gamma}\left(d-1\right)}{2^{4-d}\pi}\int
d^{d}y\frac{\partial^{a}{\mathcal{O}}^{(\pm,0)}_{a}(y)-\partial^{a}{\mathcal{O}}^{(\pm,0)\dagger}_{a}(y)}{\left[(x-y)^{2}\right]^{d-1}}\\\
&\qquad\qquad+\frac{ie\left[\log\left(\frac{|u|e^{2{\gamma}_{E}}}{4|r|}\right)-H_{\frac{d}{2}-1}\right]}{(4\pi)^{\frac{d}{2}+1}{\Gamma}\left(\frac{d}{2}\right)}(-\partial^{2})^{\frac{d}{2}-1}\left[\partial^{a}{\mathcal{O}}^{(\pm,0)}_{a}(x)-\partial^{a}{\mathcal{O}}^{(\pm,0)\dagger}_{a}(x)\right]\\\
&\qquad\qquad-\frac{e\left[{\Theta}(u)-{\Theta}(r)\right]}{8(4\pi)^{\frac{d}{2}}{\Gamma}\left(\frac{d}{2}\right)}(-\partial^{2})^{\frac{d}{2}-1}\left[\partial^{a}{\mathcal{O}}^{(\pm,0)}_{a}(x)+\partial^{a}{\mathcal{O}}^{(\pm,0)\dagger}_{a}(x)\right].\end{split}$
(3.9)

Notice that in even dimensions this coefficient has a logarithmic divergence,
which means the soft charge is potentially ill-defined in even dimensions. To
cancel this divergence, we impose999Note that (3.10) also implies that we can
no longer think of ${\mathcal{O}}^{(\pm,0)\dagger}_{a}$ as a creation
operator. Rather, it is an operator that shifts the vacuum. From the
perspective of the path integral, this condition is equivalent to continuity
of the $S$-matrix at ${\omega}=0$, i.e. the zero energy limit of the operators
does not depend on whether we approach zero from the positive or negative real
axis (see He:2015zea ).

$\begin{split}\partial^{a}{\mathcal{O}}^{(\pm,0)}_{a}(x)=\partial^{a}{\mathcal{O}}^{(\pm,0)\dagger}_{a}(x).\end{split}$
(3.10)

We remark that this constraint implies that in even dimensions
$\left.F_{ur}^{(R\pm,d)}\right|_{{\mathscr{I}}^{\pm}_{\pm}}=0$, an assumption
that was made previously in He:2014cra . The soft charge now takes the form

$\begin{split}Q^{\pm
S}_{\varepsilon}=\begin{cases}\frac{1}{(4\pi)^{\frac{d}{2}}{\Gamma}\left(\frac{d}{2}\right)e}\displaystyle\int
d^{d}x\,{\varepsilon}(x)(-\partial^{2})^{\frac{d}{2}-1}\partial^{a}{\mathcal{O}}^{(\pm,0)}_{a}(x)&d\in
2{\mathbb{Z}}\\\
\frac{(-1)^{\frac{d-1}{2}}{\Gamma}\left(d-1\right)}{4\pi^{d+1}e}\displaystyle\int
d^{d}x\,{\varepsilon}(x)\displaystyle\int
d^{d}y\frac{\partial^{a}{\mathcal{O}}^{(\pm,0)}_{a}(y)}{\left[(x-y)^{2}\right]^{d-1}}&d\in
2{\mathbb{Z}}+1.\end{cases}\end{split}$ (3.11)

We conclude this section by bringing the soft charge into the same form for
both odd and even dimensions by judiciously choosing ${\varepsilon}$ for the
remainder of the paper to be

$\begin{split}{\varepsilon}(z)=f_{x}(z)=(-\partial^{2})\log\left[(x-z)^{2}\right].\end{split}$
(3.12)

Then, using the fact that in even dimensions

$\begin{split}(-\partial^{2})^{\frac{d}{2}-1}f_{x}(z)&=(-\partial^{2})^{\frac{d}{2}}\log\left[(x-z)^{2}\right]=-(4\pi)^{\frac{d}{2}}\Gamma\left(\frac{d}{2}\right)\delta^{(d)}(x-z),\end{split}$
(3.13)

and in odd dimensions

$\begin{split}\int
d^{d}z\frac{f_{x}(z)}{\left[(z-y)^{2}\right]^{d-1}}&=-2(d-2)\int
d^{d}z\frac{1}{\left[(z-y)^{2}\right]^{d-1}(x-z)^{2}}\\\
&=-\frac{4(-1)^{\frac{d-1}{2}}\pi^{d+1}}{\Gamma(d-1)}\delta^{(d)}(x-y),\end{split}$
(3.14)

the soft charge becomes for all spacetime dimensions101010Although we picked
${\varepsilon}=f_{x}$ in (3.12), we can recover the soft charge
$Q_{\varepsilon}^{\pm S}$ for general ${\varepsilon}$ from (3.15) by using
(3.11) to write $Q_{\varepsilon}^{\pm S}$ as an appropriate linear combination
of $Q^{\pm S}_{f_{x}}\sim\partial^{a}{\mathcal{O}}_{a}^{(\pm,0)}$. This means
we didn’t lose any generality by choosing ${\varepsilon}=f_{x}$, and the Ward
identity we derive below is also completely general.

$\displaystyle Q_{f_{x}}^{\pm
S}=-\frac{1}{2e}\left(\partial^{a}{\mathcal{O}}_{a}^{(\pm,0)}(x)+\partial^{a}{\mathcal{O}}_{a}^{(\pm,0){\dagger}}(x)\right)=-\frac{1}{e}\partial^{a}{\mathcal{O}}_{a}^{(\pm,0)}(x).$
(3.15)

###  3.2 Ward Identity

Thus far, our discussion of fields, matching conditions, and charges has been
entirely classical. We now extend the discussion to a semi-classical theory
and derive a Ward identity. In such a theory, the quantity of interest is the
scattering amplitude, which is the overlap between the $in$ state (living on
$i^{-}\cup{\mathscr{I}}^{-}$) and the $out$ state (living on
$i^{+}\cup{\mathscr{I}}^{+}$), i.e.

$\begin{split}{\mathcal{A}}_{n}=\langle\,\text{out}\,|\,\text{in}\,\rangle,\end{split}$
(3.16)

where $n$ is the total number of particles in the $in$ and $out$ state. The
classical matching condition (3.3) then translates to

$\begin{split}\langle\,\text{out}\,|\left(Q^{+}_{\varepsilon}-Q^{-}_{\varepsilon}\right)|\,\text{in}\,\rangle=0.\end{split}$
(3.17)

Using (3.4), we can rewrite this as

$\begin{split}\langle\,\text{out}\,|\left(Q^{+S}_{\varepsilon}-Q^{-S}_{\varepsilon}\right)|\,\text{in}\,\rangle=-\langle\,\text{out}\,|\left(Q^{+H}_{\varepsilon}-Q^{-H}_{\varepsilon}\right)|\,\text{in}\,\rangle.\end{split}$
(3.18)

To simplify this, we evaluate the action of the hard charge on one-particle
states, i.e. the right-hand-side of (3.18). Let
$|\,\Psi_{i},\vec{p}_{i},s_{i}\,\rangle$ be a massless one-particle state with
charge $Q_{i}$, momentum $\vec{p}_{i}$, and spin $s_{i}$. We may parameterize
the momentum as

$\begin{split}p_{i}^{A}={\omega}_{i}\left(\frac{1+x_{i}^{2}}{2},x_{i}^{a},\frac{1-x_{i}^{2}}{2}\right).\end{split}$
(3.19)

The action of the conserved current on the bra and ket states is (see Appendix
B for an explicit calculation for scalar fields)

$\begin{split}\int_{{\mathscr{I}}^{-}}duJ_{u}^{(-,d)}(u,x)|\,\Psi_{i},\vec{p}_{i},s_{i}\,\rangle&=|\,\Psi_{i},\vec{p}_{i},s_{i}\,\rangle
Q_{i}\delta^{(d)}\left(x-x_{i}\right)\\\
\langle\,\Psi_{i},\vec{p}_{i},s_{i}\,|\int_{{\mathscr{I}}^{+}}duJ_{u}^{(+,d)}(u,x)&=Q_{i}\delta^{(d)}\left(x-x_{i}\right)|\,\Psi_{i},\vec{p}_{i},s_{i}\,\rangle.\\\
\end{split}$ (3.20)

It follows using (3.7) that

$\begin{split}Q^{-H}_{\varepsilon}|\,\Psi_{i},\vec{p}_{i},s_{i}\,\rangle=-Q_{i}{\varepsilon}(x_{i})|\,\Psi_{i},\vec{p}_{i},s_{i}\,\rangle,\qquad\langle\,\Psi_{i},\vec{p}_{i}\,|Q^{+H}_{\varepsilon}=-Q_{i}{\varepsilon}(x_{i})\langle\,\Psi_{i},\vec{p}_{i},s_{i}\,|.\\\
\end{split}$ (3.21)

Note that this is simply an infinitesimal large gauge transformation of the
state. Setting ${\varepsilon}=f_{x}$ in (3.18) and using (3.15) and (3.21), we
derive the Ward identity

$\begin{split}&\langle\,\text{out}\,|\left[\partial^{a}{\mathcal{O}}^{(+,0)}_{a}(x)-\partial^{a}{\mathcal{O}}^{(-,0)}_{a}(x)\right]|\,\text{in}\,\rangle=e\sum_{i=1}^{n}\eta_{i}Q_{i}\partial^{2}\log\left[(x-x_{i})^{2}\right]\langle\,\text{out}\,|\,\text{in}\,\rangle.\end{split}$
(3.22)

Here, $\eta_{i}=+1$ for outgoing particles and $\eta_{i}=-1$ for incoming
particles.

###  3.3 Soft Theorem

Finally, in this section we will show that the Ward identity
(LABEL:mainwardid) is implied by Weinberg’s leading soft photon theorem, which
in standard momentum space variables reads

$\begin{split}{\mathcal{A}}^{\text{out}}_{n+1}(\vec{p}_{\gamma},{\varepsilon}_{a};p_{1},\cdots,p_{n})=e\sum_{i=1}^{n}\eta_{i}Q_{i}\frac{p_{i}\cdot{\varepsilon}_{a}(p_{\gamma})}{p_{i}\cdot
p_{\gamma}}{\mathcal{A}}_{n}(p_{1},\cdots,p_{n})+O\big{(}(p_{\gamma})^{0}\big{)},\end{split}$
(3.23)

where ${\varepsilon}_{a}$ is the polarization vector (not to be confused with
the function ${\varepsilon}(z)$ in (3.12)). Note that the leading order term
in the soft expansion is a simple pole in photon momentum $p_{\gamma}$, and
this is known as the Weinberg pole.111111The first subleading term in the soft
expansion is also universal and is related to an asymptotic symmetry, but this
is beyond the scope of this paper. We want to parametrize (3.23) using the
momentum space parameterization employed in the previous sections (see §A.2
for details), i.e.

$\begin{split}p_{\gamma}^{A}&={\omega}\left(\frac{1+x^{2}}{2},x^{a},\frac{1-x^{2}}{2}\right)\\\
p_{i}^{A}&={\omega}_{i}\left(\frac{1+x_{i}^{2}}{2},x_{i}^{a},\frac{1-x_{i}^{2}}{2}\right)\\\
{\varepsilon}^{A}_{a}(p_{\gamma})&=\left(x_{a},\delta^{b}_{a},-x_{a}\right).\end{split}$
(3.24)

In this parametrization, the soft factor corresponding to polarization $a$ is

$\begin{split}e\sum_{i=1}^{n}\eta_{i}Q_{i}\frac{p_{i}\cdot{\varepsilon}_{a}(p_{\gamma})}{p_{i}\cdot
p_{\gamma}}=\frac{e}{\omega}\sum_{i=1}^{n}\eta_{i}Q_{i}\partial_{a}\log\left[(x-x_{i})^{2}\right].\end{split}$
(3.25)

The last step is to recast the scattering amplitudes in (3.23) in terms of the
$in$ and $out$ states. The $n$-point amplitude on the right-hand-side is
simply $\langle\,\text{out}\,|\,\text{in}\,\rangle$. On the other hand, an
outgoing photon corresponds to the insertion of the operator
${\mathcal{O}}^{(+)}_{a}({\omega},x)-{\mathcal{O}}^{{(-)}}_{a}({\omega},x)$
(see Appendix C for details). The coefficient of ${\omega}^{-1}$ in (3.25) is
extracted by first multiplying by ${\omega}$ and then taking a ${\omega}\to 0$
limit. Using (2.16), we find that this operator is

$\displaystyle\lim_{{\omega}\to
0}\left[{\omega}{\mathcal{O}}^{(+)}_{a}({\omega},x)-{\omega}{\mathcal{O}}^{{(-)}}_{a}({\omega},x)\right]={\mathcal{O}}_{a}^{(+,0)}(x)-{\mathcal{O}}_{a}^{(-,0)}(x).$
(3.26)

Substituting this into an $S$-matrix element gives precisely the left-hand-
side of (3.23), and using (3.25), we see that the soft theorem can be written
as

$\begin{split}\langle\,\text{out}\,|\left({\mathcal{O}}_{a}^{(+,0)}(x)-{\mathcal{O}}_{a}^{(-,0)}(x)\right)|\,\text{in}\,\rangle=e\sum_{i=1}^{n}\eta_{i}Q_{i}\partial_{a}\log\left[(x-x_{i})^{2}\right]\langle\,\text{out}\,|\,\text{in}\,\rangle.\end{split}$
(3.27)

Acting on both sides with $\partial^{a}$ and using (3.10), we immediately
reproduce the Ward identity (LABEL:mainwardid), completing our demonstration
that Weinberg’s leading soft photon theorem implies the Ward identity.

We conclude this section with a few comments.

  * •

We have shown above that Weinberg’s leading soft photon theorem for an
outgoing soft photon implies the Ward identity. However, we could have just as
well chosen to work with an incoming soft photon, in which case the soft
theorem reads

$\begin{split}{\mathcal{A}}^{\text{in}}_{n+1}(\vec{p}_{\gamma},{\varepsilon}_{a};p_{1},\cdots,p_{n})=-e\sum_{i=1}^{n}\eta_{i}Q_{i}\frac{p_{i}\cdot{\varepsilon}_{a}(p_{\gamma})}{p_{i}\cdot
p_{\gamma}}{\mathcal{A}}_{n}(p_{1},\cdots,p_{n})+O\big{(}(p_{\gamma})^{0}\big{)},\end{split}$
(3.28)

and differs from (3.23) by only a relative sign. The left-hand-side of the
above equation corresponds to the insertion of
${\mathcal{O}}_{a}^{(-)\dagger}({\omega},x)-{\mathcal{O}}_{a}^{(+)\dagger}({\omega},x)$.
Multiplying by ${\omega}$, taking the ${\omega}\to 0$ limit, and using (3.10),
we immediately find that the outgoing soft photon theorem (3.23) _implies_ the
incoming soft photon theorem, and vice versa.

  * •

We have not yet demonstrated the equivalence between the Ward identity and
Weinberg’s leading soft photon theorem. Instead, we have only established the
fact that the soft theorem implies the Ward identity. To prove the converse,
we need to use the additional constraint that

$\begin{split}\partial_{a}{\mathcal{O}}_{b}^{(\pm,0)}(x)-\partial_{b}{\mathcal{O}}_{a}^{(\pm,0)}(x)=0.\end{split}$
(3.29)

This constraint is obviously true from the standpoint of the soft theorem
(3.27),121212This can be derived from a careful study of the symplectic form,
which will be done in detail in forthcoming work. This constraint was also
discussed in Strominger:2017zoo . and is solved by

$\begin{split}{\mathcal{O}}_{a}^{(\pm,0)}(x)=\partial_{a}{\mathcal{S}}^{(\pm)}(x).\end{split}$
(3.30)

We can then derive the Ward identity for the insertion of
${\mathcal{S}}^{(\pm)}(x)$, which is given by131313To do this, we choose
${\varepsilon}(z)=\log\left[(x-z)^{2}\right]$ in the Ward identity (3.18).

$\begin{split}&\langle\,\text{out}\,|\left[{\mathcal{S}}^{(+)}(x)-{\mathcal{S}}^{(-)}(x)\right]|\,\text{in}\,\rangle=e\sum_{i=1}^{n}\eta_{i}Q_{i}\log\left[(x-x_{i})^{2}\right]\langle\,\text{out}\,|\,\text{in}\,\rangle.\end{split}$
(3.31)

Acting on both sides with $\partial_{a}$ and using (3.30), we immediately
reproduce the soft theorem (3.27).

##  4 Large Gauge Symmetry

In the previous section, we have shown that the Ward identity corresponding to
the charge $Q^{\pm}_{\varepsilon}$ is equivalent Weinberg’s soft photon
theorem. We now show that this charge generates _large gauge transformations_
on ${\mathscr{I}}^{\pm}$ using the covariant phase space formalism Wald:1999wa
.141414A recent analysis of the charge generating large gauge transformations
at _spatial infinity_ in all dimensions $d\geq 4$ is given in Esmaeili:2019hom
. We have already done this for the hard charge, so it remains to show that
the soft charge (i.e. the radiative field) acts on the gauge field to generate
large gauge transformations. For this purpose, we can focus simply on pure
$U(1)$ gauge theory. This is described by the action

$\begin{split}S[A]=-\frac{1}{2e^{2}}\int_{M}F\wedge\ast F,\qquad F=dA,\qquad
A\sim A+d{\varepsilon}.\end{split}$ (4.1)

Varying this, we find

$\begin{split}\delta S=-\frac{1}{e^{2}}\int_{M}\delta A\wedge(d\ast
F)-\frac{1}{e^{2}}\int_{\partial M}\delta A\wedge\ast F.\end{split}$ (4.2)

The bulk term gives us the equations of motion $d\ast F=0$. From the boundary
term we can read off the canonical one-form on a Cauchy surface $\Sigma$ as

$\begin{split}\Theta_{\Sigma}(\delta)&=\frac{1}{e^{2}}\int_{\Sigma}\delta
A\wedge\ast F.\end{split}$ (4.3)

If we restrict $A$ to satisfy the equations of motion and $\delta A$ to
satisfy the linearized equations of motion, then $\Theta_{\Sigma}$ can be
understood as a one-form at the point $A$ in the phase space $\Gamma$. The
presymplectic form on $\Sigma$ is

$\begin{split}\Omega_{\Sigma}(\delta,\delta^{\prime})&=\delta\Theta_{\Sigma}(\delta^{\prime})-\delta^{\prime}\Theta_{\Sigma}(\delta)=\frac{1}{e^{2}}\int_{\Sigma}\left[\delta^{\prime}A\wedge\ast\delta
F-\delta A\wedge\ast\delta^{\prime}F\right].\end{split}$ (4.4)

At this point, we recall that the gauge field $A$ is determined only up to a
gauge choice (4.1). To keep track of this, we separate the gauge field into
two pieces

$\begin{split}A={\tilde{A}}+d\psi,\end{split}$ (4.5)

where ${\tilde{A}}$ satisfies a chosen gauge-fixing condition,151515In our
work, it is often convenient to choose $f[{\tilde{A}}]={\tilde{A}}_{u}$.

$\begin{split}f[{\tilde{A}}]=0,\end{split}$ (4.6)

and gauge transformations simply involve

$\begin{split}\psi\to\psi+{\varepsilon}.\end{split}$ (4.7)

It follows that the symplectic form is

$\begin{split}\Omega_{\Sigma}(\delta,\delta^{\prime})&=\frac{1}{e^{2}}\int_{\Sigma}\left[\delta^{\prime}{\tilde{A}}\wedge\ast\delta{\tilde{F}}-\delta{\tilde{A}}\wedge\ast\delta^{\prime}{\tilde{F}}\right]+\frac{1}{e^{2}}\int_{\partial\Sigma}\left[\delta^{\prime}\psi\ast\delta
F-\delta\psi\ast\delta^{\prime}F\right].\end{split}$ (4.8)

We note that modes for which $\psi|_{\partial\Sigma}=0$ cannot appear in the
symplectic form and need to be fixed in order to have a non-degenerate
symplectic form. We assume that this has been done, and so the remaining
boundary value of $\psi$ symplectically couples to the boundary field
strength.

The charge that generates a particular transformation $\Delta$ on the phase
space satisfies

$\begin{split}\delta
Q^{\Sigma}_{\Delta}=\Omega_{\Sigma}(\delta,\Delta).\end{split}$ (4.9)

In particular, for gauge transformations
$\delta_{\varepsilon}{\tilde{A}}=\delta_{\varepsilon}F=0$ and
$\delta_{\varepsilon}\psi={\varepsilon}$, we get

$\begin{split}Q^{\Sigma}_{\varepsilon}=\frac{1}{e^{2}}\int_{\partial\Sigma}{\varepsilon}\ast
F.\end{split}$ (4.10)

Note that the charge is non-zero only if ${\varepsilon}|_{\partial\Sigma}\neq
0$, so this means $Q^{\Sigma}_{\varepsilon}$ is the generator of _large gauge
transformations_. To determine its form on ${\mathscr{I}}^{\pm}$, we note that
the directed area element on ${\mathscr{I}}_{\mp}^{\pm}$ is

$\begin{split}dS^{\pm}_{\mu\nu}=\pm
d^{d}x|r|^{d}\delta^{r}_{[\mu}\delta^{u}_{\nu]}.\end{split}$ (4.11)

The $\pm$ sign arises from the outward-pointing null vector, which on
${\mathscr{I}}^{\pm}$ is $\mp\partial_{\mu}r$. Thus, we have

$\begin{split}Q^{\pm}_{\varepsilon}=\pm\frac{2}{e^{2}}\int_{{\mathscr{I}}^{\pm}_{\mp}}d^{d}x\,{\varepsilon}F^{(\pm,d)}_{ur},\end{split}$
(4.12)

which is precisely the charge (3.2).

## Acknowledgements

We would like to thank Eduardo Casali, Indranil Halder, Daniel Kapec, Alok
Laddha, Monica Pate, and Kartik Prabhu for useful conversations, and
especially Andrew Strominger for reading a preprint and offering helpful
comments. TH is grateful to be supported by U.S. Department of Energy grant
DE-SC0009999 and by funds from the University of California. PM gratefully
acknowledges support from U.S. Department of Energy grant DE-SC0009988.

##  Appendix A Coordinate Conventions

###  A.1 Position Space

In this subsection, we will gain some intuition regarding the flat null
coordinates utilized throughout this paper by collecting miscellaneous facts
about them. Flat null coordinates $x^{\mu}=(u,x^{a},r)$ are related to
Cartesian coordinates $X^{A}$ via

$\begin{split}X^{0}=\frac{r}{2}\left(1+x^{2}\right)+\frac{u}{2},\qquad
X^{a}=rx^{a},\qquad
X^{d+1}=\frac{r}{2}\left(1-x^{2}\right)-\frac{u}{2}.\end{split}$ (A.1)

The inverse relation is

$\begin{split}u=\frac{\big{(}X^{0}\big{)}^{2}-X^{a}X_{a}-\big{(}X^{d+1}\big{)}^{2}}{X^{0}+X^{d+1}},\qquad
x^{a}=\frac{X^{a}}{X^{0}+X^{d+1}},\qquad r=X^{0}+X^{d+1}.\end{split}$ (A.2)

Though the coordinates are ill-defined on the hypersurface $X^{0}+X^{d+1}=0$
where $r=0$, away from this point, the metric is well defined and takes the
form

$\begin{split}ds^{2}=\eta_{AB}dX^{A}dX^{b}&=-\big{(}dX^{0}\big{)}^{2}+dX^{a}dX_{a}+\big{(}dX^{d+1}\big{)}^{2}\\\
&=-dudr+r^{2}dx^{a}dx_{a},\end{split}$ (A.3)

where $\eta_{AB}=(-1,1,\ldots,1)$ is the usual Minkowski metric in Cartesian
coordinates. The range of the coordinates is

$\begin{split}u\in{\mathbb{R}},\qquad x^{a}\in{\mathbb{R}}^{d},\qquad
r\in{\mathbb{R}}\setminus\\{0\\}.\end{split}$ (A.4)

We note

$\begin{split}\partial_{u}&=\frac{1}{2}\left[\partial_{X^{0}}-\partial_{X^{d+1}}\right]\\\
\partial_{a}&=rx_{a}\left[\partial_{X^{0}}-\partial_{X^{d+1}}\right]+r\partial_{X^{a}}\\\
\partial_{r}&=\frac{1}{2}\left(1+x^{2}\right)\partial_{X^{0}}+x^{a}\partial_{X^{a}}+\frac{1}{2}\left(1-x^{2}\right)\partial_{X^{d+1}},\\\
\end{split}$ (A.5)

and

$\begin{split}dX^{0}&=\frac{1}{2}\left(1+x^{2}\right)dr+\frac{1}{2}du+rx_{a}dx^{a}\\\
dX^{a}&=x^{a}dr+rdx^{a}\\\
dX^{d+1}&=\frac{1}{2}\left(1-x^{2}\right)dr-\frac{1}{2}du-rx_{a}dx^{a}.\\\
\end{split}$ (A.6)

The volume form is

$\begin{split}{\varepsilon}&=dX^{0}\wedge dX^{1}\wedge\cdots\wedge
dX^{d}\wedge dX^{d+1}=\frac{1}{2}r^{d}du\wedge dx^{1}\wedge\cdots\wedge
dx^{d}\wedge dr.\end{split}$ (A.7)

The nonvanishing metric components and its inverse are

$\begin{split}g_{ur}=-\frac{1}{2},\qquad g_{ab}=r^{2}\delta_{ab},\qquad
g^{ur}=-2,\qquad g^{ab}=\frac{1}{r^{2}}\delta^{ab}.\end{split}$ (A.8)

The nonvanishing Christoffel symbols are

$\begin{split}{\Gamma}^{u}_{ab}[g]=2r\delta_{ab},\qquad{\Gamma}^{a}_{rb}[g]=\frac{1}{r}\delta^{a}_{b}.\end{split}$
(A.9)

####  A.1.1 Boundaries of Minkowski Spacetime

To determine the location of the boundaries of Minkowski spacetime in flat
null coordinates, we relate them to Eddington-Finkelstein coordinates
$(U,V,{\lambda},{\Theta})$, which are

$\begin{split}X^{0}=\frac{V+U}{2},\qquad
X^{a}=\frac{V-U}{2}\sqrt{1-{\lambda}^{2}}{\hat{n}}^{a}({\Theta}),\qquad
X^{d+1}=\frac{V-U}{2}{\lambda}.\end{split}$ (A.10)

Note that the radial distance is simply expressed as
$R\equiv\sqrt{X^{a}X_{a}+\big{(}X^{d+1}\big{)}^{2}}=\frac{V-U}{2}$. The
coordinate ranges are

$\begin{split}V\geq U,\qquad{\lambda}\in(-1,1),\qquad{\Theta}\in
S^{d-1}.\end{split}$ (A.11)

Together, the coordinates $({\lambda},{\Theta})$ parametrize $S^{d}$.
${\lambda}\in(0,1)$ is the northern hemisphere and ${\lambda}\in(-1,0)$ is the
southern hemisphere. ${\lambda}=0$ is the equator, ${\lambda}=+1$ is the north
pole and ${\lambda}=-1$ is the south pole. Let
$\big{(}{\lambda}_{\text{ap}},{\Theta}_{\text{ap}}\big{)}$ be the antipodal
point to $({\lambda},{\Theta})$. Then

$\begin{split}{\hat{n}}\big{(}{\Theta}_{\text{ap}}\big{)}=-{\hat{n}}({\Theta}),\qquad{\lambda}_{\text{ap}}=-{\lambda}.\end{split}$
(A.12)

Furthermore, in terms of the flat null coordinates, we have

$\begin{split}u&=\frac{2UV}{V(1+{\lambda})+U(1-{\lambda})}=\frac{\big{(}X^{0}\big{)}^{2}-R^{2}}{X^{0}+{\lambda}R}\\\
x^{a}&=\frac{(V-U)\sqrt{1-{\lambda}^{2}}{\hat{n}}^{a}({\Theta})}{V(1+{\lambda})+U(1-{\lambda})}=\frac{R\sqrt{1-{\lambda}^{2}}{\hat{n}}^{a}({\Theta})}{X^{0}+{\lambda}R}\\\
r&=\frac{V}{2}(1+{\lambda})+\frac{U}{2}(1-{\lambda})=X^{0}+{\lambda}R.\end{split}$
(A.13)

We can now determine the boundaries of Minkowski spacetime in flat null
coordinates.

  1. 1.

Future Null Infinity ${\mathscr{I}}^{+}$: This is located at $V\to+\infty$
while keeping $(U,{\lambda},{\Theta})$ fixed. In this limit,

$\begin{split}u&\to\frac{2U}{1+{\lambda}},\qquad
x^{a}\to\sqrt{\frac{1-{\lambda}}{1+{\lambda}}}{\hat{n}}^{a}({\Theta}),\qquad
r\to\frac{V}{2}(1+{\lambda}).\end{split}$ (A.14)

Thus, we find that ${\mathscr{I}}^{+}$ is located at $r\to\infty$. At this
point, $x^{a}$ are the stereographic coordinates on $S^{d}$. The north pole
($\lambda=1$) is at $|x^{a}|=0$, the south pole ($\lambda=-1$) is at
$|x^{a}|=\infty$, and the equator ($\lambda=0$) is on the unit circle
$x^{a}x_{a}=1$. $u$ is the null generator of this surface.

  2. 2.

Past Null Infinity ${\mathscr{I}}^{-}$: This is located at $U\to-\infty$ while
keeping $(V,{\lambda},{\Theta})$ fixed. In this limit,

$\begin{split}u&\to\frac{2V}{1+{\lambda}_{\text{ap}}},\qquad
x^{a}\to\sqrt{\frac{1-{\lambda}_{\text{ap}}}{1+{\lambda}_{\text{ap}}}}{\hat{n}}^{a}\big{(}{\Theta}_{\text{ap}}\big{)},\qquad
r\to-\frac{|U|}{2}\left(1+{\lambda}_{\text{ap}}\right).\end{split}$ (A.15)

Thus, we find that ${\mathscr{I}}^{-}$ is located at $r\to-\infty$. Again,
$x^{a}$ are the stereographic coordinates on $S^{d}$, but now with an
antipodal identification. The north pole ($\lambda=1$) is at $|x^{a}|=\infty$,
the south pole ($\lambda=-1$) is at $|x^{a}|=0$, and the equator ($\lambda=0$)
is on the unit circle $x^{a}x_{a}=1$. $u$ is the null generator of this
surface.

  3. 3.

Future Timelike Infinity $i^{+}$: This is located at $X^{0}\to+\infty$ while
keeping $(R,{\lambda},{\Theta})$ fixed. In this limit,

$\begin{split}u&\to X^{0}-{\lambda}R,\qquad
x^{a}\to\frac{R\sqrt{1-{\lambda}^{2}}}{X^{0}}{\hat{n}}^{a}({\Theta}),\qquad
r\to X^{0}+{\lambda}R.\end{split}$ (A.16)

Thus, we find that $i^{+}$ located at $u,r\to\infty$ and $x^{a}\to 0$ while
keeping $u-r$ and $rx^{a}$ fixed.

  4. 4.

Past Timelike Infinity $i^{-}$: This is located at $X^{0}\to-\infty$ while
keeping $(R,{\lambda},{\Theta})$ fixed. In this limit,

$\begin{split}u&\to-\left(\big{|}X^{0}\big{|}-{\lambda}_{\text{ap}}R\right),\quad
x^{a}\to\frac{R\sqrt{1-{\lambda}_{\text{ap}}^{2}}}{\big{|}X^{0}\big{|}}{\hat{n}}^{a}\left({\Theta}_{\text{ap}}\right),\quad
r\to-\big{|}X^{0}\big{|}-{\lambda}_{\text{ap}}R.\end{split}$ (A.17)

Thus, we find that $i^{-}$ located at $u,r\to-\infty$ and $x^{a}\to 0$ while
keeping $u-r$ and $rx^{a}$ fixed.

  5. 5.

Spatial Infinity $i^{0}$: This is located at $R\to+\infty$ while keeping
$(X^{0},{\lambda},{\Theta})$ fixed. In this limit,

$\begin{split}u&\to-\frac{R}{{\lambda}}+\frac{X^{0}}{{\lambda}^{2}},\qquad
x^{a}\to\frac{1}{{\lambda}}\sqrt{1-{\lambda}^{2}}{\hat{n}}^{a}({\Theta}),\qquad
r={\lambda}R+X^{0}.\end{split}$ (A.18)

This limit corresponds to $ur\to-\infty$ while keeping $\frac{u}{r}$ and
$x^{a}$ fixed. In particular, on the northern hemisphere, we take
$u\to-\infty$ and $r\to+\infty$, whereas on the southern hemisphere, we take
$u\to+\infty$ and $r\to-\infty$.

####  A.1.2 Isometries

The isometries of ${\mathbb{R}}^{1,d+1}$ are the $d+2$ translations and the
$O(1,d+1)$ rotations. The corresponding Killing vectors in flat null
coordinates are

$\begin{split}\xi_{f}&=f(x)\partial_{u}-\frac{1}{2r}\partial^{a}f(x)\partial_{a}+\frac{1}{2d}\partial^{2}f(x)\partial_{r}\\\
\zeta_{Y}&=\psi(x)\left(u\partial_{u}-r\partial_{r}\right)+\left[Y^{a}(x)-\frac{u}{2r}\partial^{a}\psi(x)\right]\partial_{a},\end{split}$
(A.19)

where $\psi(x)=\frac{1}{d}\partial_{a}Y^{a}(x)$, and $f(x)$ and $Y^{a}(x)$
satisfy the equations

$\begin{split}\partial_{a}\partial_{b}f(x)=\frac{1}{d}\delta_{ab}\partial^{2}f(x),\qquad\partial_{a}Y_{b}(x)+\partial_{b}Y_{a}(x)=\frac{2}{d}\delta_{ab}\partial^{c}Y_{c}(x).\\\
\end{split}$ (A.20)

This implies that $Y^{a}(x)$ is the conformal Killing vector of
${\mathbb{R}}^{d}$ (which manifests the isomorphism
$SO(1,d+1)\cong\mathfrak{conf}({\mathbb{R}}^{d})$). The explicit solutions to
this are

$\begin{split}f(x)=f_{0}+f_{a}x^{a}+f_{d+1}x^{2},\qquad
Y^{a}(x)=\chi^{a}+{\omega}^{a}{}_{b}x^{b}+\lambda
x^{a}+\zeta^{a}x^{2}-2x^{a}\zeta\cdot x.\end{split}$ (A.21)

Thus, $\psi(x)=\lambda-2\zeta\cdot x$. The $SO(d)$ rotations have $\psi(x)=0$
(generated by $\chi^{a}$ and ${\omega}^{a}{}_{b}$), whereas the boosts in
$SO(1,d+1)$ have $\psi(x)\neq 0$ (generated by $\lambda$ and $\zeta^{a}$).
Finally, we remark that the Lie algebra of the Killing vectors (the Poincaré
algebra) takes the form

$\begin{split}\left[\xi_{f},\xi_{f^{\prime}}\right]&=0,\qquad\left[\zeta_{Y},\xi_{f}\right]=\xi_{Y^{a}\partial_{a}f-f\psi},\qquad\left[\zeta_{Y},\zeta_{Y^{\prime}}\right]=\zeta_{[Y,Y^{\prime}]}.\end{split}$
(A.22)

###  A.2 Momentum Space

In this subsection, we observe a general parameterization of off-shell momenta
and list some useful formulae.

####  A.2.1 Off-Shell Momentum

Following closely the relationship between flat null coordinates and Cartesian
coordinates, we parameterize an _off-shell_ momentum as

$\begin{split}q^{0}=\frac{{\omega}}{2}\left(1+y^{2}\right)+\frac{\mu}{2{\omega}},\qquad
q^{a}={\omega}y^{a},\qquad
q^{d+1}=\frac{{\omega}}{2}\left(1-y^{2}\right)-\frac{\mu}{2{\omega}}.\end{split}$
(A.23)

Inversely, $\mu=-q^{A}q_{A}$, $y^{a}=q^{a}(q^{0}+q^{d+1})^{-1}$, and
${\omega}=q^{0}+q^{d+1}$. This parameterization is valid everywhere except
when ${\omega}=q^{0}+q^{d+1}=0$. The range of these coordinates is
$\mu\in{\mathbb{R}}$, $y^{a}\in{\mathbb{R}}^{d}$ and
${\omega}\in{\mathbb{R}}\setminus\\{0\\}$. The momentum space volume form is

$\begin{split}{\varepsilon}&=dq^{0}\wedge dq^{1}\wedge\cdots\wedge
dq^{d}\wedge dq^{d+1}=\frac{1}{2}{\omega}^{d-1}d\mu\wedge
dy^{1}\wedge\cdots\wedge dy^{d}\wedge d{\omega}.\end{split}$ (A.24)

This implies that the off-shell integration measure and Dirac delta function
are

$\begin{split}\int\frac{d^{d+2}q}{(2\pi)^{d+2}}&=\frac{1}{2(2\pi)^{d+2}}\int_{-\infty}^{\infty}d\mu\int_{-\infty}^{\infty}d{\omega}\int_{{\mathbb{R}}^{d}}d^{d}y\,|{\omega}|^{d}\\\
\delta^{(d+2)}\left(q-q^{\prime}\right)&=2|{\omega}|^{1-d}\delta\left(\mu-\mu^{\prime}\right)\delta\left({\omega}-{\omega}^{\prime}\right)\delta^{(d)}\left(y-y^{\prime}\right).\end{split}$
(A.25)

####  A.2.2 On-Shell Momentum and Polarization Vectors

The on-shell condition on a momentum implies

$\begin{split}q^{A}q_{A}=-m^{2},\qquad q^{0}>0\quad\implies\quad\mu=m^{2}\geq
0,\qquad{\omega}>0.\end{split}$ (A.26)

The on-shell integration measure and Dirac delta function are

$\begin{split}\int\frac{d^{d+2}q}{(2\pi)^{d+2}}(2\pi)\delta\big{(}q^{2}+m^{2}\big{)}{\theta}(q^{0})&=\int\frac{d^{d+1}q}{(2\pi)^{d+1}}\frac{1}{2q^{0}}=\frac{1}{2(2\pi)^{d+1}}\int_{0}^{\infty}d{\omega}\int_{{\mathbb{R}}^{d}}d^{d}y\,{\omega}^{d-1}\\\
(2q^{0})\delta^{(d+1)}\left(\vec{q}-\vec{q}\,^{\prime}\right)&=2{\omega}^{1-d}\delta\left({\omega}-{\omega}^{\prime}\right)\delta^{(d)}\left(y-y^{\prime}\right).\end{split}$
(A.27)

The $d+1$ polarizations of a massive particle are taken to be

$\begin{split}{\varepsilon}_{a}^{A}(y)&=\left(y_{a},\delta_{a}^{b},-y_{a}\right),\qquad{\varepsilon}^{A}_{\omega}({\omega},y)=\frac{1}{2}\left(1+y^{2}-\frac{m^{2}}{{\omega}^{2}},2y^{a},1-y^{2}+\frac{m^{2}}{{\omega}^{2}}\right).\end{split}$
(A.28)

In the massless limit ${\varepsilon}^{A}_{\omega}\to{\omega}^{-1}q^{A}$. Thus,
in this limit, this is the pure gauge mode and does not contribute to physical
amplitudes. For future reference, we collect the formulae

$\begin{split}q({\omega},y)\cdot
q({\omega}^{\prime},y^{\prime})&=-\frac{1}{2}{\omega}{\omega}^{\prime}\left[(y-y^{\prime})^{2}+\frac{m^{2}}{{\omega}^{2}}+\frac{m^{\prime
2}}{{\omega}^{\prime 2}}\right]\\\
q({\omega},y)\cdot{\varepsilon}_{a}(y^{\prime})&={\omega}(y-y^{\prime})_{a}\\\
q({\omega},y)\cdot{\varepsilon}_{{\omega}^{\prime}}({\omega}^{\prime},y^{\prime})&=-\frac{{\omega}}{2}\left[(y-y^{\prime})^{2}+\frac{m^{2}}{{\omega}^{2}}-\frac{m^{\prime
2}}{{\omega}^{\prime 2}}\right]\\\
{\varepsilon}_{a}(y)\cdot{\varepsilon}_{b}(y^{\prime})&=\delta_{ab}\\\
{\varepsilon}_{a}(y)\cdot{\varepsilon}_{{\omega}^{\prime}}({\omega}^{\prime},y^{\prime})&=-(y-y^{\prime})_{a}\\\
{\varepsilon}_{{\omega}}({\omega},y)\cdot{\varepsilon}_{{\omega}^{\prime}}({\omega}^{\prime},y^{\prime})&=-\frac{1}{2}\left[(y-y^{\prime})^{2}-\frac{m^{2}}{{\omega}^{2}}-\frac{m^{\prime
2}}{{\omega}^{\prime 2}}\right]\\\ X(u,r,x)\cdot
q({\omega},y)&=-\frac{{\omega}u}{2}-\frac{{\omega}r}{2}\left[(x-y)^{2}+\frac{m^{2}}{{\omega}^{2}}\right]\\\
X(u,r,x)\cdot{\varepsilon}_{a}({\omega},y)&=r(x-y)_{a}\\\
X(u,r,x)\cdot{\varepsilon}_{{\omega}}({\omega},y)&=-\frac{u}{2}-\frac{r}{2}\left[(x-y)^{2}-\frac{m^{2}}{{\omega}^{2}}\right].\\\
\end{split}$ (A.29)

We also require the components of the momentum and polarization vectors in
flat null coordinates

$\begin{split}q_{u}({\omega},y)&=-\frac{{\omega}}{2}\\\
q_{r}({\omega},y)&=-\frac{{\omega}}{2}\left[(x-y)^{2}+\frac{m^{2}}{{\omega}^{2}}\right]\\\
q_{a}({\omega},y)&=-r\omega(x-y)_{a}\\\
({\varepsilon}_{a})_{u}({\omega},y)&=0\\\
({\varepsilon}_{a})_{r}({\omega},y)&=(x-y)_{a}\\\
({\varepsilon}_{a})_{b}({\omega},y)&=r\delta_{ab}\\\
({\varepsilon}_{\omega})_{u}({\omega},y)&=-\frac{1}{2}\\\
({\varepsilon}_{\omega})_{r}({\omega},y)&=-\frac{1}{2}\left[(x-y)^{2}-\frac{m^{2}}{{\omega}^{2}}\right]\\\
({\varepsilon}_{\omega})_{a}({\omega},y)&=-r(x-y)_{a}.\end{split}$ (A.30)

For massless vectors, we note from the formulae above that our polarization
choice is equivalent to the gauge choice $A_{u}=0$. The polarization tensors
for the massless field strength then has the following components in flat null
coordinates

$\begin{split}({\varepsilon}_{a})_{ur}({\omega},y)&=-\frac{i}{2}{\omega}(x-y)_{a}\\\
({\varepsilon}_{a})_{rb}({\omega},y)&=-\frac{i}{2}{\omega}r(x-y)^{2}{\mathcal{I}}_{ab}(x-y)\\\
({\varepsilon}_{a})_{ub}({\omega},y)&=-\frac{i}{2}{\omega}r\delta_{ab}\\\
({\varepsilon}_{c})_{ab}({\omega},y)&=-i{\omega}r^{2}\left[(x-y)_{a}\delta_{bc}-(x-y)_{b}\delta_{ac}\right],\end{split}$
(A.31)

where ${\mathcal{I}}_{ab}(x-y)$ is the conformally invariant tensor

$\begin{split}{\mathcal{I}}_{ab}(x-y)=\delta_{ab}-\frac{2(x-y)_{a}(x-y)_{b}}{(x-y)^{2}}.\end{split}$
(A.32)

##  Appendix B Action of the Hard Charge

In this section, we show that the hard charge generates large gauge
transformations on the matter states by an explicit computation for a
minimally coupled scalar field $\Phi$ with $U(1)$ charge $Q\in{\mathbb{Z}}$.
The corresponding conserved current is

$\begin{split}J_{\mu}=iQ\left(\Phi^{*}D_{\mu}\Phi-(D_{\mu}\Phi)^{*}\Phi\right),\qquad
D_{\mu}\Phi=\partial_{\mu}\Phi-iQA_{\mu}\Phi.\end{split}$ (B.1)

The product of fields is defined via normal ordering.

The mode expansion for the outgoing/incoming scalar field is

$\begin{split}\Phi^{(\pm)}(X)=\int\frac{d^{d+1}q}{(2\pi)^{d+1}}\frac{1}{2q^{0}}\left[{\mathcal{O}}_{\Phi}^{(\pm)}(\vec{q}\,)e^{iq\cdot
X}+{\mathcal{O}}_{{\overline{\Phi}}}^{{(\pm)}\dagger}(\vec{q}\,)e^{-iq\cdot
X}\right],\end{split}$ (B.2)

where

$\begin{split}\left[{\mathcal{O}}_{\Phi}^{(\pm)}(\vec{q}\,),{\mathcal{O}}_{\Phi}^{{(\pm)}\dagger}(\vec{q}\,^{\prime})\right]=\left[{\mathcal{O}}_{{\overline{\Phi}}}^{(\pm)}(\vec{q}\,),{\mathcal{O}}_{{\overline{\Phi}}}^{{(\pm)}\dagger}(\vec{q}\,^{\prime})\right]=(2q^{0})(2\pi)^{d+1}\delta^{(d+1)}\left(\vec{q}-\vec{q}\,^{\prime}\right).\end{split}$
(B.3)

Moving to flat null coordinates and using (2.14), we find

$\begin{split}\Phi^{(\pm)}(u,r,x)&=\frac{1}{2(2\pi)^{d+1}}\int_{{\mathbb{R}}^{d}}d^{d}y\int_{0}^{\infty}d{\omega}\,{\omega}^{d-1}\left[{\mathcal{O}}_{\Phi}^{(\pm)}({\omega},x+y)e^{-\frac{i}{2}{\omega}u-\frac{i}{2}{\omega}ry^{2}}\right.\\\
&\left.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+{\mathcal{O}}_{{\overline{\Phi}}}^{{(\pm)}\dagger}({\omega},x+y)e^{\frac{i}{2}{\omega}u+\frac{i}{2}{\omega}ry^{2}}\right],\end{split}$
(B.4)

where

$\begin{split}\left[{\mathcal{O}}_{\Phi}^{(\pm)}({\omega},x),{\mathcal{O}}_{\Phi}^{{(\pm)}\dagger}({\omega}^{\prime},x^{\prime})\right]&=\left[{\mathcal{O}}_{{\overline{\Phi}}}^{(\pm)}({\omega},x),{\mathcal{O}}_{{\overline{\Phi}}}^{{(\pm)}\dagger}({\omega}^{\prime},x^{\prime})\right]\\\
&=2{\omega}^{1-d}(2\pi)^{d+1}\delta({\omega}-{\omega}^{\prime})\delta^{(d)}\left(x-x^{\prime}\right).\end{split}$
(B.5)

The leading order term in the large $r$ expansion of the scalar field is (this
is determined following precisely the method of §2.2.1)

$\begin{split}\Phi^{(\pm)}~{}~{}\to~{}~{}\frac{1}{2(2\pi)^{\frac{d}{2}+1}}\int_{0}^{\infty}d{\omega}\,{\omega}^{\frac{d}{2}-1}\left[\frac{e^{-\frac{i}{2}{\omega}u}}{(ir)^{\frac{d}{2}}}{\mathcal{O}}_{\Phi}^{(\pm)}({\omega},x)+\frac{e^{\frac{i}{2}{\omega}u}}{(-ir)^{\frac{d}{2}}}{\mathcal{O}}_{{\overline{\Phi}}}^{{(\pm)}\dagger}({\omega},x)\right]+\cdots.\end{split}$
(B.6)

We then determine that

$\begin{split}\int du\,J_{u}^{(\pm,d)}&=\frac{\pi
Q}{(2\pi)^{d+2}}\int_{0}^{\infty}d{\omega}\,{\omega}^{d-1}\left[{\mathcal{O}}_{\Phi}^{{(\pm)}\dagger}({\omega},x){\mathcal{O}}_{\Phi}^{(\pm)}({\omega},x)-{\mathcal{O}}_{{\overline{\Phi}}}^{{(\pm)}\dagger}({\omega},x){\mathcal{O}}_{{\overline{\Phi}}}^{{(\pm)}}({\omega},x)\right].\end{split}$
(B.7)

An outgoing or incoming scalar state of charge $Q$ is

$\begin{split}\langle\,\Phi,{\omega},x\,|=\langle\,0\,|{\mathcal{O}}_{\Phi}^{{(+)}}({\omega},x),\qquad|\,\Phi,{\omega},x\,\rangle={\mathcal{O}}_{\Phi}^{{(-)}\dagger}({\omega},x)|\,0\,\rangle.\end{split}$
(B.8)

Using these definitions, we find

$\begin{split}\langle\,\Phi,{\omega},x^{\prime}\,|\int
du\,J_{u}^{(+,d)}(u,x)&=Q\delta^{(d)}(x-x^{\prime})\langle\,\Phi,{\omega},x^{\prime}\,|\\\
\int
du\,J_{u}^{(-,d)}(u,x)|\,\Phi,{\omega},x^{\prime}\,\rangle&=Q\delta^{(d)}(x-x^{\prime})|\,\Phi,{\omega},x^{\prime}\,\rangle.\end{split}$
(B.9)

The action of the hard charge (3.7) is then

$\begin{split}\langle\,\Phi,{\omega},x\,|Q^{+H}_{\varepsilon}=-Q{\varepsilon}(x)\langle\,\Phi,{\omega},x\,|,\qquad
Q^{-H}_{\varepsilon}|\,\Phi,{\omega},x\,\rangle=-Q{\varepsilon}(x)|\,\Phi,{\omega},x\,\rangle.\end{split}$
(B.10)

##  Appendix C Soft Photon Operator

We describe how to obtain the soft photon insertion operator in an $S$-matrix
using the path integral approach. First, the $n$-point amplitude is

$\begin{split}{\mathcal{A}}_{n}(p_{1},\cdots,p_{n})=\langle\,\text{out}\,|\,\text{in}\,\rangle.\end{split}$
(C.1)

This is computed in the path integral formalism by computing off-shell
momentum space correlators and taking an appropriate on-shell limit. For
instance, for an outgoing scalar particle with momentum $\vec{p}$, we insert

$\begin{split}iZ_{\Phi}\lim_{p^{0}\to\sqrt{|\vec{p}\,|+m^{2}}-i{\epsilon}}\left(p^{2}-i{\epsilon}\right)\int
d^{d+2}X\,e^{-ip\cdot X}\Phi(X).\end{split}$ (C.2)

Here, $\Phi(X)$ is an operator that creates (among other things) a scalar one-
particle state, and $Z_{\Phi}$ is the corresponding wave-function
renormalization factor.

The amplitude involving an extra outgoing photon, i.e.
${\mathcal{A}}_{n+1}^{\text{out}}$ from (3.23), similarly corresponds to the
insertion of the operator

$\begin{split}\frac{iZ}{e}\lim_{p_{\gamma}^{0}\to|\vec{p}_{\gamma}|-i{\epsilon}}\left(p_{\gamma}^{2}-i{\epsilon}\right){\varepsilon}_{a}^{A}(\vec{p}_{\gamma})^{*}\int
d^{d+2}X\,e^{-ip_{\gamma}\cdot X}A_{A}(X).\end{split}$ (C.3)

To understand what this operator is in terms of the $in$ and $out$
creation/annihilation operators, we expand the gauge field $A_{A}(x)$ in terms
of time-dependent mode coefficients:

$\begin{split}A_{A}(X)=e\int\frac{d^{d+1}q}{(2\pi)^{d+1}}\frac{1}{2q^{0}}\left[{\varepsilon}^{a}_{A}({\vec{q}}\,){\mathcal{O}}_{a}(X^{0},\vec{q}\,)e^{iq\cdot
X}+{\varepsilon}^{a}_{A}({\vec{q}}\,)^{*}{\mathcal{O}}^{\dagger}_{a}(X^{0},\vec{q}\,)e^{-iq\cdot
X}\right].\end{split}$ (C.4)

Using this, we find

$\begin{split}\frac{iZ}{e}\lim_{p_{\gamma}^{0}\to|\vec{p}_{\gamma}|-i{\epsilon}}\left(p_{\gamma}^{2}-i{\epsilon}\right){\varepsilon}_{a}^{A}(\vec{p}_{\gamma})^{*}\int
d^{d+2}X\,e^{-ip_{\gamma}\cdot
X}A_{A}(X)={\mathcal{O}}^{(+)}_{a}(\vec{p}_{\gamma})-{\mathcal{O}}^{{(-)}}_{a}(\vec{p}_{\gamma}).\end{split}$
(C.5)

where
$\lim_{X^{0}\to\pm\infty}{\mathcal{O}}_{a}(X^{0},\vec{p}_{\gamma})=Z^{-1}{\mathcal{O}}_{a}^{(\pm)}(\vec{p}_{\gamma})$.161616To
show this, we additionally assume that
$\lim\limits_{X^{0}\to\pm\infty}\partial_{0}{\mathcal{O}}_{a}(X^{0},\vec{p}_{\gamma})=0$
(see Schwartz Schwartz:2013pla  for a discussion on this). For photons with
non-zero energy, the operator ${\mathcal{O}}^{{(-)}}_{a}(\vec{p}_{\gamma})$
annihilates the $in$ vacuum (assuming there are no other incoming photons with
the same momentum, i.e. assuming no forward scattering) so that $S$-matrices
with energetic outgoing photons correspond to insertions of
${\mathcal{O}}^{(+)}_{a}(\vec{p}_{\gamma})$. On the other hand, owing to
(3.10), ${\mathcal{O}}_{a}^{(-,0)}(x)$ no longer annihilates the vacuum;
rather, it creates an incoming soft photon (to be more precise, it produces a
new vacuum state). Therefore, it follows that the operator generating the soft
limit must include both terms from (C.5) and is given by

$\begin{split}\lim_{{\omega}\to
0}\left[{\omega}{\mathcal{O}}^{(+)}_{a}({\omega},x)-{\omega}{\mathcal{O}}^{{(-)}}_{a}({\omega},x)\right]={\mathcal{O}}_{a}^{(+,0)}(x)-{\mathcal{O}}_{a}^{(-,0)}(x).\end{split}$
(C.6)

## References

  * (1) F. Bloch and A. Nordsieck, “Note on the Radiation Field of the electron,” Phys. Rev. 52 (1937) 54–59.
  * (2) A. Nordsieck, “The Low Frequency Radiation of a Scattered Electron,” Phys. Rev. 52 (1937) 59–62.
  * (3) F. E. Low, “Scattering of light of very low frequency by systems of spin 1/2,” Phys. Rev. 96 (1954) 1428–1432.
  * (4) M. Gell-Mann and M. L. Goldberger, “Scattering of low-energy photons by particles of spin 1/2,” Phys. Rev. 96 (1954) 1433–1438.
  * (5) F. E. Low, “Bremsstrahlung of very low-energy quanta in elementary particle collisions,” Phys. Rev. 110 (1958) 974–977.
  * (6) D. R. Yennie, S. C. Frautschi, and H. Suura, “The infrared divergence phenomena and high-energy processes,” Annals Phys. 13 (1961) 379–452.
  * (7) S. Weinberg, “Infrared photons and gravitons,” Phys.Rev. 140 (1965) B516–B524.
  * (8) S. Weinberg, The Quantum theory of fields. Vol. 1: Foundations.  Cambridge University Press, 2005\.
  * (9) A. Strominger, “Lectures on the Infrared Structure of Gravity and Gauge Theory,” arXiv:1703.05448 [hep-th].
  * (10) T. He, P. Mitra, A. P. Porfyriadis, and A. Strominger, “New Symmetries of Massless QED,” JHEP 10 (2014) 112, arXiv:1407.3789 [hep-th].
  * (11) A. Mohd, “A note on asymptotic symmetries and soft-photon theorem,” JHEP 02 (2015) 060, arXiv:1412.5365 [hep-th].
  * (12) T. He, P. Mitra, and A. Strominger, “2D Kac-Moody Symmetry of 4D Yang-Mills Theory,” JHEP 10 (2016) 137, arXiv:1503.02663 [hep-th].
  * (13) M. Campiglia and A. Laddha, “Asymptotic symmetries of QED and Weinberg’s soft photon theorem,” JHEP 07 (2015) 115, arXiv:1505.05346 [hep-th].
  * (14) D. Kapec, M. Pate, and A. Strominger, “New Symmetries of QED,” Adv. Theor. Math. Phys. 21 (2017) 1769–1785, arXiv:1506.02906 [hep-th].
  * (15) A. Strominger, “Magnetic Corrections to the Soft Photon Theorem,” Phys. Rev. Lett. 116 no. 3, (2016) 031602, arXiv:1509.00543 [hep-th].
  * (16) B. Gabai and A. Sever, “Large gauge symmetries and asymptotic states in QED,” JHEP 12 (2016) 095, arXiv:1607.08599 [hep-th].
  * (17) T. He, V. Lysov, P. Mitra, and A. Strominger, “BMS supertranslations and Weinberg’s soft graviton theorem,” JHEP 05 (2015) 151, arXiv:1401.7026 [hep-th].
  * (18) M. Campiglia and A. Laddha, “New symmetries for the Gravitational S-matrix,” JHEP 04 (2015) 076, arXiv:1502.02318 [hep-th].
  * (19) M. Campiglia and A. Laddha, “Asymptotic symmetries of gravity and soft theorems for massive particles,” JHEP 12 (2015) 094, arXiv:1509.01406 [hep-th].
  * (20) T. T. Dumitrescu, T. He, P. Mitra, and A. Strominger, “Infinite-Dimensional Fermionic Symmetry in Supersymmetric Gauge Theories,” arXiv:1511.07429 [hep-th].
  * (21) F. Cachazo and A. Strominger, “Evidence for a New Soft Graviton Theorem,” arXiv:1404.4091 [hep-th].
  * (22) B. U. W. Schwab, “Subleading Soft Factor for String Disk Amplitudes,” arXiv:1406.4172 [hep-th].
  * (23) C. Kalousios and F. Rojas, “Next to subleading soft-graviton theorem in arbitrary dimensions,” JHEP 01 (2015) 107, arXiv:1407.5982 [hep-th].
  * (24) H. Luo, P. Mastrolia, and W. J. Torres Bobadilla, “Subleading soft behavior of QCD amplitudes,” Phys. Rev. D91 no. 6, (2015) 065018, arXiv:1411.1669 [hep-th].
  * (25) P. Di Vecchia, R. Marotta, and M. Mojaza, “Subsubleading soft theorems of gravitons and dilatons in the bosonic string,” JHEP 06 (2016) 054, arXiv:1604.03355 [hep-th].
  * (26) A. Sen, “Subleading Soft Graviton Theorem for Loop Amplitudes,” arXiv:1703.00024 [hep-th].
  * (27) A. Laddha and P. Mitra, “Asymptotic Symmetries and Subleading Soft Photon Theorem in Effective Field Theories,” JHEP 05 (2018) 132, arXiv:1709.03850 [hep-th].
  * (28) H. Hirai and S. Sugishita, “Conservation Laws from Asymptotic Symmetry and Subleading Charges in QED,” JHEP 07 (2018) 122, arXiv:1805.05651 [hep-th].
  * (29) G. Barnich and C. Troessaert, “Supertranslations call for superrotations,” PoS (2010) 010, arXiv:1102.4632 [gr-qc]. [Ann. U. Craiova Phys.21,S11(2011)].
  * (30) D. Kapec, V. Lysov, S. Pasterski, and A. Strominger, “Semiclassical Virasoro symmetry of the quantum gravity $\mathcal{S}$-matrix,” JHEP 08 (2014) 058, arXiv:1406.3312 [hep-th].
  * (31) M. Campiglia and A. Laddha, “Asymptotic symmetries and subleading soft graviton theorem,” Phys. Rev. D90 no. 12, (2014) 124028, arXiv:1408.2228 [hep-th].
  * (32) V. Lysov, S. Pasterski, and A. Strominger, “Low’s Subleading Soft Theorem as a Symmetry of QED,” Phys. Rev. Lett. 113 no. 11, (2014) 111601, arXiv:1407.3814 [hep-th].
  * (33) M. Campiglia and A. Laddha, “Sub-subleading soft gravitons: New symmetries of quantum gravity?,” Phys. Lett. B764 (2017) 218–221, arXiv:1605.09094 [gr-qc].
  * (34) M. Campiglia and A. Laddha, “Subleading soft photons and large gauge transformations,” JHEP 11 (2016) 012, arXiv:1605.09677 [hep-th].
  * (35) E. Conde and P. Mao, “Remarks on asymptotic symmetries and the subleading soft photon theorem,” Phys. Rev. D95 no. 2, (2017) 021701, arXiv:1605.09731 [hep-th].
  * (36) M. Campiglia and A. Laddha, “Sub-subleading soft gravitons and large diffeomorphisms,” JHEP 01 (2017) 036, arXiv:1608.00685 [gr-qc].
  * (37) E. Conde and P. Mao, “BMS Supertranslations and Not So Soft Gravitons,” arXiv:1612.08294 [hep-th].
  * (38) A. Laddha and A. Sen, “Sub-subleading Soft Graviton Theorem in Generic Theories of Quantum Gravity,” arXiv:1706.00759 [hep-th].
  * (39) B. U. W. Schwab and A. Volovich, “Subleading Soft Theorem in Arbitrary Dimensions from Scattering Equations,” Phys. Rev. Lett. 113 no. 10, (2014) 101601, arXiv:1404.7749 [hep-th].
  * (40) D. Kapec, V. Lysov, and A. Strominger, “Asymptotic Symmetries of Massless QED in Even Dimensions,” arXiv:1412.2763 [hep-th].
  * (41) D. Kapec, V. Lysov, S. Pasterski, and A. Strominger, “Higher-Dimensional Supertranslations and Weinberg’s Soft Graviton Theorem,” arXiv:1502.07644 [gr-qc].
  * (42) K. S. Thorne, “Gravitational-wave bursts with memory: The Christodoulou effect,” Phys. Rev. D45 no. 2, (1992) 520–524.
  * (43) A. Strominger and A. Zhiboedov, “Gravitational Memory, BMS Supertranslations and Soft Theorems,” JHEP 01 (2016) 086, arXiv:1411.5745 [hep-th].
  * (44) S. Pasterski, “Asymptotic Symmetries and Electromagnetic Memory,” arXiv:1505.00716 [hep-th].
  * (45) L. Bieri, D. Garfinkle, and S.-T. Yau, “Gravitational Waves and Their Memory in General Relativity,” arXiv:1505.05213 [gr-qc].
  * (46) L. Susskind, “Electromagnetic Memory,” arXiv:1507.02584 [hep-th].
  * (47) M. Hotta, J. Trevison, and K. Yamaguchi, “Gravitational Memory Charges of Supertranslation and Superrotation on Rindler Horizons,” Phys. Rev. D94 no. 8, (2016) 083001, arXiv:1606.02443 [gr-qc].
  * (48) S. Hollands, A. Ishibashi, and R. M. Wald, “BMS Supertranslations and Memory in Four and Higher Dimensions,” arXiv:1612.03290 [gr-qc].
  * (49) M. Pate, A.-M. Raclariu, and A. Strominger, “Color Memory: A Yang-Mills Analog of Gravitational Wave Memory,” Phys. Rev. Lett. 119 no. 26, (2017) 261602, arXiv:1707.08016 [hep-th].
  * (50) M. Pate, A.-M. Raclariu, and A. Strominger, “Gravitational Memory in Higher Dimensions,” JHEP 06 (2018) 138, arXiv:1712.01204 [hep-th].
  * (51) A. Ball, M. Pate, A.-M. Raclariu, A. Strominger, and R. Venugopalan, “Measuring Color Memory in a Color Glass Condensate at Electron-Ion Colliders,” arXiv:1805.12224 [hep-ph].
  * (52) N. L Balazs, “Wave propagation in even and odd dimensional spaces,” Proceedings of the Physical Society. Section A 68 (1955) 521.
  * (53) H. Soodak and M. S. Tiersten, “Wakes and waves in n dimensions,” American Journal of Physics 61 no. 5, (1993) 395–401, https://doi.org/10.1119/1.17230. https://doi.org/10.1119/1.17230.
  * (54) D. Kapec and P. Mitra, “A $d$-Dimensional Stress Tensor for Minkd+2 Gravity,” JHEP 05 (2018) 186, arXiv:1711.04371 [hep-th].
  * (55) M. Campiglia and R. Eyheralde, “Asymptotic $U(1)$ charges at spatial infinity,” arXiv:1703.07884 [hep-th].
  * (56) K. Prabhu, “Conservation of asymptotic charges from past to future null infinity: Maxwell fields,” JHEP 10 (2018) 113, arXiv:1808.07863 [gr-qc].
  * (57) D. Yoshida and J. Soda, “Electromagnetic Memory Effect Induced by Axion Dark Matter,” arXiv:1704.04169 [gr-qc].
  * (58) R. M. Wald and A. Zoupas, “A General definition of ‘conserved quantities’ in general relativity and other theories of gravity,” Phys. Rev. D61 (2000) 084027, arXiv:gr-qc/9911095 [gr-qc].
  * (59) E. Esmaeili, “Asymptotic Symmetries of Maxwell Theory in Arbitrary Dimensions at Spatial Infinity,” arXiv:1902.02769 [hep-th].
  * (60) M. D. Schwartz, Quantum Field Theory and the Standard Model.  Cambridge University Press, 2014\.