Shadows and Soft Exchange in Celestial CFT

Daniel Kapeca and Prahar Mitraa,b

a School of Natural Sciences, Institute for Advanced Study, Princeton, NJ
08540, USA
b DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK

<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study exponentiated soft exchange in $d+2$ dimensional gauge and
gravitational theories using the celestial CFT formalism. These models exhibit
spontaneously broken asymptotic symmetries generated by gauge transformations
with non-compact support, and the effective dynamics of the associated
Goldstone “edge” mode is expected to be $d$-dimensional. The introduction of
an infrared regulator also explicitly breaks these symmetries so the edge mode
in the regulated theory is really a $d$-dimensional pseudo-Goldstone boson.
Symmetry considerations determine the leading terms in the effective action,
whose coefficients are controlled by the infrared cutoff. Computations in this
model reproduce the abelian infrared divergences in $d=2$, and capture the re-
summed (infrared finite) soft exchange in higher dimensions. The model also
reproduces the leading soft theorems in gauge and gravitational theories in
all dimensions. Interestingly, we find that it is the shadow transform of the
Goldstone mode that has local $d$-dimensional dynamics: the effective action
expressed in terms of the Goldstone mode is non-local for $d>2$. We also
introduce and discuss new magnetic soft theorems. Our analysis demonstrates
that symmetry principles suffice to calculate soft exchange in gauge theory
and gravity.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 $S$-matrix as a Conformal Correlator
    2. 2.2 Soft Theorems and Conserved Currents
    3. 2.3 Infrared Divergences in Abelian Theories
  3. 3 Magnetic Charges
    1. 3.1 Abelian Gauge Theories
    2. 3.2 Non-abelian Gauge Theories and Chromodyons
    3. 3.3 Soft Theorem for Magnetic Branes in $d>2$
  4. 4 A Boundary Model for Bulk Infrared Physics
    1. 4.1 Broken Symmetries in Celestial CFT
    2. 4.2 $U(1)$ Gauge Theory
      1. 4.2.1 Magnetic Charges and Winding Modes in $d=2$
    3. 4.3 Non-abelian Gauge Theory
    4. 4.4 Gravity: Supertranslation Mode
  5. 5 Conclusions and Future Work
  6. A Derivation of Current Ward Identities

##  1 Introduction

Gauge theories and gravitational theories in asymptotically flat spacetimes
exhibit well-known universal infrared behavior. Soft theorems govern the form
of scattering amplitudes at the boundaries of kinematic space, and repeated
soft exchange exponentiates in a way that is independent of the microscopic
details of the scattering process. Universality follows from symmetry, and
both of these phenomena ultimately derive from the asymptotic symmetry
structure of gauge theories in flat spacetime.

In the Coulomb phase, gauge transformations with non-compact support act non-
trivially on physical states. These “large gauge transformations” are broken
by the perturbative vacuum, and correspond to flat directions in field space
protected by symmetry. The unbroken global part of the gauge group gives rise
to global charge conservation, but the local transformations are more akin to
Goldstone bosons for an infinite-dimensional symmetry group. These universal
features find natural expressions in the celestial CFT formalism. The
isomorphism between the Lorentz group in $d+2$ dimensions and the Euclidean
conformal group in $d$ dimensions can be exploited to recast standard
$S$-matrix elements as “celestial correlators” that resemble those of a CFTd.
In this language, the unbroken symmetry group ensures the existence of a
conserved current which is the shadow transform of a soft operator [1] and
which satisfies the local Ward identities required of a CFTd. In this paper,
we demonstrate that the shadow transform of the Goldstone mode (corresponding
to the broken symmetries) also has local $d-$dimensional dynamics, and that
those dynamics fully reproduce the effects of soft exchange. Given that soft
exchange occurs over large scales in the bulk, it is somewhat surprising that
it can be captured by a local action in a lower dimension. We believe that our
result lends support to the holographic interpretation of the celestial CFT
formalism. There is a small body of previous work addressing soft exchange in
celestial CFT, primarily in four dimensions [2, 3, 4, 5, 6, 7, 8]. The methods
and rationale in these treatments differ significantly from ours but the
results for the case $d=2$ are equivalent.

Our analysis relies on the well known (dimension-independent) exponentiation
of soft exchange in abelian gauge theory and gravity. If
$\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\mu}$ denotes the
$S$-matrix element calculated with an infrared (IR) cutoff $\mu$ on loop-
momenta and $\Lambda$ is the scale chosen to separate hard from soft, then

$\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\mu}=e^{-{\Gamma}(\mu,\Lambda)}\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\Lambda}\;.$

The quantity $\Gamma$ can be calculated from one-loop exchange diagrams and
depends on the two scales $\mu$, $\Lambda$ and the quantum numbers of the
external hard particles. The hard amplitude
$\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\Lambda}$ is
model dependent, but the exponential factor $e^{-{\Gamma}}$ is universal and
one would like to compute it using symmetry principles. To do this, we
separate fields into hard and soft pieces, integrate out the hard modes, and
consider the effective action for the slow modes:

$\begin{split}\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\mu}&=\int[d\varphi_{s}]\int[d\varphi_{h}]e^{iS_{\text{bulk}}[\varphi_{s},\varphi_{h}]}{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\\\
&=\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\Lambda}\int[d\varphi_{s}]e^{-S_{\text{soft}}[\varphi_{s}]-S_{\text{int}}[\varphi_{s},j]}\;.\end{split}$

The form of the second equality relies on the identification of the soft modes
$\varphi_{s}$ with the Goldstone edge modes associated to asymptotic
symmetries. These degrees of freedom are effectively $d$-dimensional, and we
expect their “dynamics” to be Euclidean. At this stage of the calculation, the
external particles are accounted for by classical sources $j$ for
$\varphi_{s}$ and the matching condition reads

$\begin{split}\int[d\varphi_{s}]e^{-S_{\text{soft}}[\varphi_{s}]-S_{\text{int}}[\varphi_{s},j]}=e^{-{\Gamma}}\;.\end{split}$

The form of the interaction term $S_{\text{int}}[\varphi_{s},j]$ is completely
fixed by the symmetry transformation properties of the external states under
the broken symmetries. In order to determine $S_{\text{soft}}[\varphi_{s}]$ it
is essential to understand the pattern of symmetry breaking in the IR
regulated theory. The models that we consider exhibit infinite-dimensional
groups of spontaneously broken symmetries $\mathcal{G}$ whose unbroken finite-
dimensional subgroups we denote by $G$. For instance, in the case of $U(1)$
gauge theory on $\mathbb{R}^{1,d+1}$, the group $\mathcal{G}$ consists of maps
$S^{d}\to U(1)$ and $G=U(1)$ corresponds to the constant maps. If the
$\mathcal{G}/G$ edge modes were true Goldstones, then the effective action
$S_{\text{soft}}[\varphi_{s}]$ would be invariant under a non-linear
realization of the full infinite-dimensional group of $\mathcal{G}$
transformations. In the $U(1)$ example, a simple model for such an action is a
$d$-dimensional $U(1)$ gauge theory in which one only integrates over (and
does not quotient by) the trivial flat connections $C_{a}=\partial_{a}C(x)$.
However, the introduction of the infrared regulator breaks these symmetries,
so the effective action can also contain symmetry breaking terms that give an
action to the Goldstone modes and $S_{\text{soft}}[\varphi_{s}]$ is the
leading symmetry breaking term. It takes the form of a (non-local) mass term
for the flat connection $C_{a}$ in the case of abelian gauge theory. In the
models that we consider the resulting path integral is Gaussian and can be
evaluated explicitly. The coefficient of $S_{\text{soft}}[\varphi_{s}]$ is
then determined by matching onto the infrared divergence and it depends
explicitly on the infrared cutoff. The resulting action can be used to compute
the resummed soft exchange in any dimension. Interestingly, it is the shadow
mode that has local dynamics: the action expressed in terms of the Goldstone
variable is non-local when $d>2$. “Integrating in” a second degree of freedom
related to external soft insertions results in a model that also reproduces
the leading soft theorems in gauge and gravitational theories in all
dimensions.

Although we do not treat the non-abelian case completely, our symmetry
considerations do lead to a natural conjecture for the infrared action that
fits in well with known analogies between non-linear sigma models in 2D and
non-abelian gauge theories in 4D. In two dimensions the fluctuations of the
non-abelian Goldstone bosons are strongly coupled and tend to disorder the
system and restore the symmetry. A similar statement applies to the four-
dimensional soft gluon when viewed as the order parameter for a spontaneously
broken non-abelian large gauge symmetry: the soft gluon is strongly coupled in
four and fewer dimensions, and the strong dynamics generically leads to the
confining (unbroken) phase of gauge theory. In fact, a motivating point for
this work is the observation that the infrared phases of Goldstone bosons in
$d$ dimensions match those of gauge theories in $d+2$ dimensions. Both are
infrared free (in the broken phase) when $d>2$ and generically strongly
coupled when $d\leq 2$. We hope to return to this case in future work and to
continue to explore what is two-dimensional about four-dimensional gauge
theory.

The outline of this paper is as follows. In section 2, we review soft theorems
and soft exchange in gauge and gravitational theories in the language of
conformal correlators. In section 3, we extend these results to include
magnetic and dyonic states, and introduce a variety of new magnetic soft
theorems. In section 4 we use symmetry principles to derive the
$d$-dimensional actions that reproduce soft exchange and soft theorems in
$(d+2)$-dimensional models. Section 5 concludes with open questions, and
appendix A contains derivations of important integral formulas.

##  2 Preliminaries

In this section we review known results on the infrared sector of gauge theory
and gravity, beginning with a discussion of the relationship between
$S$-matrix elements and conformal correlation functions. We then review soft
theorems and infrared divergences in the language of these correlators.

###  2.1 $S$-matrix as a Conformal Correlator

The Lorentz group in $d+2$ dimensions is isomorphic to the Euclidean conformal
group in $d$ dimensions. This correspondence can be exploited to recast
scattering amplitudes on $\mathbb{R}^{1,d+1}$ in terms of the correlation
functions of a $d$-dimensional conformal field theory (CFTd).

#### Poincaré and Conformal Algebra

The Poincaré algebra is generated by the operators $P_{\mu}$ and $M_{\mu\nu}$
($\mu,\nu=0,\cdots,d+1$) satisfying the commutation relations

$\begin{split}[P_{\mu},P_{\nu}]&=0\;,\qquad[M_{\mu\nu},P_{\rho}]=2i\eta_{\rho[\mu}P_{\nu]}\;,\qquad[M_{\mu\nu},M_{\rho{\sigma}}]=4i\eta_{[\underline{\rho}[\mu}M_{\nu]\underline{{\sigma}}]}\;,\end{split}$
(2.1)

where $\eta=\textrm{diag}(-1,+1,\cdots,+1)$. The Lorentz generators
$M_{\mu\nu}$ are linear combinations of the conformal generators $J_{ab}$,
$D$, $T_{a}$ and $K_{a}$ ($a,b=1,\cdots,d$)

$\begin{split}J_{ab}=M_{ab}\;,\qquad D=M_{d+1,0}\;,\qquad
T_{a}=M_{0,a}-M_{d+1,a}\;,\qquad K_{a}=M_{0,a}+M_{d+1,a}\;.\end{split}$ (2.2)

The Lorentz algebra implies that these generators satisfy the Euclidean
conformal algebra

$\begin{split}[J_{ab},J_{cd}]&=4i{\delta}_{[\underline{c}[a}J_{b]\underline{d}]}\;,\qquad[J_{ab},T_{c}]=2i{\delta}_{c[a}T_{b]}\;,\qquad[J_{ab},K_{c}]=2i{\delta}_{c[a}K_{b]}\;,\\\
[T_{a},D]&=+iT_{a}\;,\qquad[K_{a},D]=-iK_{a}\;,\qquad[T_{a},K_{b}]=-2i({\delta}_{ab}D+J_{ab})\;.\\\
\end{split}$ (2.3)

The $J_{ab}$ generate $SO(d)$ rotations, $T_{a}$ and $K_{a}$ generate
translations and special conformal transformations, and $D$ is the dilation
operator. The mutually commuting generators $P^{\mu}$ decompose into two
scalars $P^{\pm}=P^{0}\pm P^{d+1}$ and a vector $P^{a}$ which mix under the
action of the conformal algebra:

$\begin{split}&[J_{ab},P^{+}]=0\;,\qquad[P^{+},D]=+iP^{+}\;,\qquad[P^{+},T_{a}]=0\;,\qquad[P^{+},K_{a}]=-2iP_{a}\;,\\\
&[J_{ab},P^{-}]=0\;,\qquad[P^{-},D]=-iP^{-}\;,\qquad[P^{-},T_{a}]=-2iP_{a}\;,\qquad[P^{-},K_{a}]=0\;,\\\
&[J_{ab},P_{c}]=2i{\delta}_{c[a}P_{b]}\;,\qquad[P_{a},D]=0\;,\qquad[P_{a},T_{b}]=-i\delta_{ab}P^{+}\;,\qquad[P_{a},K_{b}]=-i\delta_{ab}P^{-}\;.\end{split}$
(2.4)

#### Momenta and Polarizations

The $S$-matrix computes an overlap between multi-particle _in_ and _out_
states. It is a function of the momenta, spins and other quantum numbers
associated to these states. In order to recast the $S$-matrix as a conformal
correlator, it is useful to parameterize the momenta and polarization vectors
in a way that makes the connection between the Lorentz group and the conformal
group manifest. A convenient parameterization for the momenta is given by

$\begin{split}p^{\mu}({\omega},x)={\omega}{\hat{p}}^{\mu}({\omega},x)\;,\qquad{\hat{p}}^{\mu}({\omega},x)={\hat{q}}^{\mu}(x)+(m^{2}/{\omega}^{2})n^{\mu}\;,\end{split}$
(2.5)

where ${\hat{q}}$ and $n$ are both null vectors given by

$\begin{split}{\hat{q}}^{\mu}(x)=\frac{1}{2}(1+x^{2},2x^{a},1-x^{2})\;,\qquad
n^{\mu}=\frac{1}{2}(1,0^{a},-1)\;.\end{split}$ (2.6)

Momenta appearing in the scattering amplitude will be denoted by
$p^{\mu}_{i}=p^{\mu}({\omega}_{i},x_{i})$. We will adopt the convention in
which outgoing momenta have $\eta_{i}=\text{sign}({\omega}_{i})=+1$ and
incoming momenta have $\eta_{i}=-1$. In these variables the Lorentz invariant
inner product takes the form

$\begin{split}{\hat{p}}_{i}\cdot{\hat{p}}_{j}=-\frac{1}{2}[x_{ij}^{2}+m_{i}^{2}/{\omega}_{i}^{2}+m_{j}^{2}/{\omega}_{j}^{2}]\;,\qquad
x_{ij}^{a}=x_{i}^{a}-x_{j}^{a}\;.\end{split}$ (2.7)

In addition to the momenta, spinning states are also labeled by their
polarizations (little group representation). The $d$ independent polarization
vectors for a massless spin one gauge boson will be taken to be

${\varepsilon}_{a}^{\mu}(x)\equiv{\partial}_{a}\hat{q}^{\mu}(x)=(x_{a},\delta_{a}^{b},-x_{a})\;.$
(2.8)

These satisfy

$\begin{split}n\cdot{\varepsilon}_{a}(x)=0\;,\qquad{\hat{p}}_{i}\cdot{\varepsilon}_{a}(x_{j})=(x_{ij})_{a}\;,\qquad{\varepsilon}_{a}(x_{i})\cdot{\varepsilon}_{b}(x_{j})={\delta}_{ab}\;.\end{split}$
(2.9)

It is also useful to define the polarization sum

$\begin{split}\Pi^{\mu\nu}(x)\equiv{\varepsilon}^{\mu}_{a}(x){\varepsilon}^{a,\nu}(x)=\eta^{\mu\nu}+2n^{\mu}{\hat{q}}^{\nu}(x)+2n^{\nu}{\hat{q}}^{\mu}(x)\;.\end{split}$
(2.10)

The polarization tensor for a massless spin two graviton is given by

$\begin{split}{\varepsilon}^{\mu\nu}_{ab}(x)\equiv\frac{1}{2}[{\varepsilon}_{a}^{\mu}(x){\varepsilon}^{\nu}_{b}(x)+{\varepsilon}_{a}^{\nu}(x){\varepsilon}^{\mu}_{b}(x)]-\frac{1}{d}{\delta}_{ab}\Pi^{\mu\nu}(x)\;,\end{split}$
(2.11)

and the corresponding polarization sum is

$\begin{split}\Pi^{\mu\nu,\rho{\sigma}}(x)&\equiv{\varepsilon}^{\mu\nu}_{ab}(x){\varepsilon}^{ab,\rho{\sigma}}(x)=\frac{1}{2}\left[\Pi^{\mu\rho}(x)\Pi^{\nu{\sigma}}(x)+\Pi^{\mu{\sigma}}(x)\Pi^{\nu\rho}(x)\right]-\frac{1}{d}\Pi^{\mu\nu}(x)\Pi^{\rho{\sigma}}(x)\;.\end{split}$
(2.12)

The Lorentz invariant phase space measure in this parameterization takes the
form

$\begin{split}\int\frac{d^{d+1}p}{p^{0}}=\int
d^{d}x\int_{0}^{\infty}d{\omega}{\omega}^{d-1}\;.\end{split}$ (2.13)

In section 2.3 we will discuss virtual soft exchange and off-shell loop
momenta in loop integrals. A convenient parameterization is
$\ell^{\mu}={\omega}[{\hat{q}}^{\mu}(x)+\kappa n^{\mu}]$. Integrals over off-
shell momenta are given by

$\begin{split}\int d^{d+2}\ell=\frac{1}{2}\int
d^{d}x\int_{-\infty}^{\infty}d{\omega}|{\omega}|^{d+1}\int_{-\infty}^{\infty}d\kappa\;.\end{split}$
(2.14)

#### Scattering States and Conformal Correlators

A scattering amplitude with $m$ outgoing particles and $n-m$ incoming
particles is given by

$\begin{split}{\mathcal{A}}_{n}&=\langle\,p_{1},\cdots,p_{m}\,|\,p_{m+1},\cdots,p_{n}\,\rangle\\\
&=\langle\,0\,|T\\{a_{1}^{\text{out}}(p_{1})\cdots
a_{m}^{\text{out}}(p_{m})a_{m+1}^{\text{in}\dagger}(p_{m+1})\cdots
a_{n}^{\text{in}\dagger}(p_{n})\\}|\,0\,\rangle\;,\end{split}$ (2.15)

where $a_{i}^{\text{in}/\text{out}}(p_{i})$ are the _in_ and _out_
annihilation operators. We will suppress the additional spin, color and/or
flavor indices of these operators when they play no role. We can rewrite this
amplitude in a suggestive way by defining

$\begin{split}{\mathcal{O}}_{i}({\omega}_{i},x_{i})\equiv
a_{i}^{\text{out}}(p({\omega}_{i},x_{i})){\theta}({\omega}_{i})+{\bar{a}}_{i}^{\text{in}\dagger}(-p({\omega}_{i},x_{i})){\theta}(-{\omega}_{i})\;.\end{split}$
(2.16)

According to this definition, when ${\omega}_{i}<0$ the operator inserts the
CPT conjugate incoming particle. The scattering amplitude then takes the form

$\begin{split}{\mathcal{A}}_{n}&=\langle\,{\mathcal{O}}_{1}({\omega}_{1},x_{1})\cdots{\mathcal{O}}_{n}({\omega}_{n},x_{n})\,\rangle\;.\end{split}$
(2.17)

Equation (2.17) is simply a rewriting of the scattering amplitude, and it is
not immediately obvious why this representation is advantageous. Its utility
derives from a reinterpretation of the Lorentz transformation properties of
the operators (2.16). Lorentz invariance fixes the transformation properties
for an annihilation operator

$\begin{split}[a_{i}^{\text{out}}(p),M_{\mu\nu}]={\mathcal{J}}^{(i)}_{\mu\nu}a_{i}^{\text{out}}(p)\;,\qquad{\mathcal{J}}^{(i)}_{\mu\nu}\equiv{\mathcal{L}}_{\mu\nu}+{\mathcal{S}}^{(i)}_{\mu\nu}\;.\end{split}$
(2.18)

${\mathcal{L}}_{\mu\nu}=-2ip_{[\mu}{\partial}_{p^{\nu]}}$ is the orbital
angular momentum and ${\mathcal{S}}^{(i)}_{\mu\nu}$ is the spin angular
momentum operator for the $i$th particle. Combining (2.18) with the
definitions (2.2), (2.16) and the parameterization (2.5), one can determine
the transformation properties of ${\mathcal{O}}_{i}$ under the Euclidean
conformal group. The transformations take a particularly simple form for
massless operators (see [1] for details):

$\begin{split}[{\mathcal{O}}_{i}({\omega},x),T_{a}]&=i{\partial}_{a}{\mathcal{O}}_{i}({\omega},x)\;,\\\
[{\mathcal{O}}_{i}({\omega},x),J_{ab}]&=-i(x_{a}{\partial}_{b}-x_{b}{\partial}_{a}){\mathcal{O}}_{i}({\omega},x)+{\mathcal{S}}^{(i)}_{ab}\,{\mathcal{O}}_{i}({\omega},x)\;,\\\
[{\mathcal{O}}_{i}({\omega},x),D]&=i(x^{a}{\partial}_{a}-{\omega}{\partial}_{\omega}){\mathcal{O}}_{i}({\omega},x)\;,\\\
[{\mathcal{O}}_{i}({\omega},x),K_{a}]&=i[x^{2}{\partial}_{a}-2x_{a}x^{b}{\partial}_{b}+2x_{a}{\omega}{\partial}_{\omega}]{\mathcal{O}}_{i}({\omega},x)+2x^{b}{\mathcal{S}}^{(i)}_{ab}\,{\mathcal{O}}_{i}({\omega},x)\;.\end{split}$
(2.19)

Here, the $SO(d)$ little group generators ${\mathcal{S}}^{(i)}_{ab}$ act in
the representation determined by the spin of ${\mathcal{O}}_{i}$. Equation
(2.19) is precisely the transformation law for a $d$-dimensional conformal
primary operator, with the important caveat that $D$ is not diagonal in this
basis. Instead, the momentum eigenstate operators have a formal scaling
dimension ${\Delta}=-{\omega}{\partial}_{\omega}$. This is simply a reflection
of the fact that momentum eigenstates are not simultaneously boost
eigenstates. By performing a change of basis – namely a Mellin transform with
respect to ${\omega}$ – it is possible to define an equivalent set of
operators which do diagonalize $D$,

$\begin{split}\widehat{{\mathcal{O}}}^{\pm}(\Delta,x)=\int_{0}^{\infty}d{\omega}{\omega}^{{\Delta}-1}{\mathcal{O}}(\pm{\omega},x)\;,\qquad{\Delta}\in\frac{d}{2}+i{\mathbb{R}}\;.\end{split}$
(2.20)

The $\pm$ superscript distinguishes between _in_ and _out_ states. The
restriction of ${\Delta}$ to the principal series is necessary in order for
the operators ${\widehat{\mathcal{O}}}$ to be normalizable. The inverse Mellin
transform is used to map the operators back

$\begin{split}{\mathcal{O}}(\pm|{\omega}|,x)=\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d{\Delta}}{2\pi
i}\,|{\omega}|^{-{\Delta}}\widehat{{\mathcal{O}}}^{\pm}(\Delta,x)\;.\end{split}$
(2.21)

Certain momentum eigenstate operators have universal behavior in the soft
(${\omega}\to 0$) limit. Scattering amplitudes involving these special
operators exhibit poles in this regime, and the residues can be isolated using
a compact contour ${\mathcal{C}}$ surrounding the origin in the complex
$\omega$ plane:

$\widehat{{\mathcal{O}}}(n,x)=\oint_{\mathcal{C}}\frac{d{\omega}}{2\pi
i}\,{\omega}^{n-1}{\mathcal{O}}({\omega},x)\;.$ (2.22)

It is not obvious whether this special class of operators is independent from
those of the form (2.20).

Operators corresponding to massive particles in momentum eigenstates also
transform improperly (not as primaries) under special conformal
transformations. For a massive scalar operator, the commutation relations are

$\begin{split}[{\mathcal{O}}_{i}({\omega},x),T_{a}]&=i{\partial}_{a}{\mathcal{O}}_{i}({\omega},x)\;,\\\
[{\mathcal{O}}_{i}({\omega},x),J_{ab}]&=-i(x_{a}{\partial}_{b}-x_{b}{\partial}_{a}){\mathcal{O}}_{i}({\omega},x)\;,\\\
[{\mathcal{O}}_{i}({\omega},x),D]&=i(x^{a}{\partial}_{a}-{\omega}{\partial}_{\omega}){\mathcal{O}}_{i}({\omega},x)\;,\\\
[{\mathcal{O}}_{i}({\omega},x),K_{a}]&=i[(x^{2}+m_{i}^{2}/{\omega}^{2}){\partial}_{a}-2x_{a}x^{b}{\partial}_{b}+2x_{a}{\omega}{\partial}_{\omega}]{\mathcal{O}}_{i}({\omega},x)\;.\\\
\end{split}$ (2.23)

Despite the more complicated form of this transformation law, there exists a
basis transformation which maps the massive momentum eigenstate
${\mathcal{O}}({\omega},x)$ to a conformal primary operator

$\begin{split}\widehat{{\mathcal{O}}}^{\pm}(\Delta,x)=m^{{\Delta}-d}\int
d^{d}y\int_{0}^{\infty}d{\omega}{\omega}^{d-1}{\mathcal{K}}_{\Delta}(m/{\omega},y;x){\mathcal{O}}(\pm{\omega},y)\;,\qquad{\Delta}\in\frac{d}{2}+i{\mathbb{R}}\;.\end{split}$
(2.24)

The kernel of this integral transform is the bulk-to-boundary propagator for a
field of dimension $\Delta$ in Euclidean hyperbolic space ${\mathbb{H}}_{d+1}$

$\begin{split}{\mathcal{K}}_{\Delta}(z,y;x)=C_{\Delta}\left[\frac{z}{(x-y)^{2}+z^{2}}\right]^{\Delta}\;,\qquad
C_{\Delta}=\frac{{\Gamma}({\Delta})}{\pi^{\frac{d}{2}}{\Gamma}({\Delta}-\frac{d}{2})}\;.\end{split}$
(2.25)

As in the massless case, the scaling dimensions are restricted to the
principle series so that the operators are normalizable. To find the inverse
transform, we use the split representation of the Dirac delta function in
${\mathbb{H}}_{d+1}$

$\begin{split}z^{d+1}{\delta}(z-z^{\prime}){\delta}^{(d)}(x-x^{\prime})=\frac{1}{2}\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d{\Delta}}{2\pi
i}\int
d^{d}y{\mathcal{K}}_{\Delta}(z,x;y){\mathcal{K}}_{d-{\Delta}}(z^{\prime},x^{\prime};y)\;.\end{split}$
(2.26)

Using this identity, the inverse transformation takes the form

$\begin{split}{\mathcal{O}}(\pm|{\omega}|,x)&=\frac{1}{2}\int_{\frac{d}{2}-i\infty}^{\frac{d}{2}+i\infty}\frac{d{\Delta}}{2\pi
i}\int
d^{d}ym^{-{\Delta}}{\mathcal{K}}_{d-{\Delta}}(m/|{\omega}|,x;y)\widehat{{\mathcal{O}}}^{\pm}(\Delta,y)\;.\end{split}$
(2.27)

The operators $\widehat{{\mathcal{O}}}^{\pm}(\Delta,x)$ varied over ${\Delta}$
are not independent of each other. Instead, they satisfy

$\begin{split}\int
d^{d}y\frac{C_{d-{\Delta}}}{[(x-y)^{2}]^{d-{\Delta}}}{\widehat{\mathcal{O}}}^{\pm}(\Delta,y)=\widehat{{\mathcal{O}}}^{\pm}(d-{\Delta},x)\;.\end{split}$
(2.28)

This integral transform defines the shadow transform for scalar operators. The
definition for more general operators is given in (2.37).

The basis transformation for spinning massive states is more complicated since
the massive little group is $SO(d+1)$ and its representations must be
decomposed into representations of $SO(d)$. We will not need the precise form
of this generalized Mellin transform.

#### Summary

The result of this section is that the Mellin-transformed $n$-point scattering
amplitude on $\mathbb{R}^{1,d+1}$

$\begin{split}\widehat{{\mathcal{A}}}_{n}&=\langle\,\widehat{{\mathcal{O}}}^{\eta_{1}}_{1}(\Delta_{1},x_{1})\cdots\widehat{{\mathcal{O}}}^{\eta_{n}}_{n}(\Delta_{n},x_{n})\,\rangle\end{split}$
(2.29)

transforms as a $d$-dimensional conformal correlation function. The operators
appearing on the RHS transform as conformal primaries: massless particles
correspond to local operator insertions given by (2.20) while massive
particles correspond to operator insertions given by (2.24).

The operators ${\mathcal{O}}_{i}$ are generically labelled by additional spin,
flavor, and/or color indices. We only exhibit the spin indices when the
particle is a gluon or graviton since they play a special role in what
follows. The corresponding operators are denoted by ${\mathcal{O}}_{a}$ and
${\mathcal{O}}_{ab}$. These operators create one-particle states with
polarization ${\varepsilon}_{a}$ given by (2.8) or ${\varepsilon}_{ab}$ given
by (2.11), respectively.

###  2.2 Soft Theorems and Conserved Currents

Soft theorems describe the universal behavior of scattering amplitudes
involving one or more energetically soft (long wavelength) external particles.
Long wavelength states are unable to resolve short-distance scattering
processes, so amplitudes with soft insertions typically factorize into lower
point amplitudes acted on by soft factors that are only sensitive to long
distance data (like quantum numbers). This factorization is (semi-)universal
and its form is independent of the microscopic details of the theory.
Universal behavior is usually a consequence of an underlying symmetry, and
work over the past several years has demonstrated that most of these universal
formulas are reflections of the asymptotic symmetry structures in gauge and
gravitational theories in flat space[9, 10, 11, 12, 13, 14]. In this section,
we review the soft photon, soft gluon and the leading soft graviton theorems
in the language of conformal correlators.

#### Leading Soft Photon Theorem

In any dimension, the soft photon theorem implies111This is only formally true
in four dimensions, since both sides of (2.30) generically vanish for
scattering states with a finite number of photons due to the infrared
divergence (see section 2.3). In that case, the information content of the
soft theorem resides in the form of the infrared finite dressed states.

$\begin{split}\langle\,{\mathcal{O}}_{a}({\omega},x){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{C=0}~{}~{}\stackrel{{\scriptstyle{\omega}\to
0}}{{\longrightarrow}}~{}~{}\frac{1}{{\omega}}e\sum_{i}Q_{i}\frac{{\hat{p}}_{i}\cdot{\varepsilon}_{a}(x)}{{\hat{p}}_{i}\cdot{\hat{q}}(x)}\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{C=0}\;,\end{split}$
(2.30)

where ${\mathcal{O}}_{i}\equiv{\mathcal{O}}_{i}({\omega}_{i},x_{i})$ and
$C=A|_{{\partial}{\mathscr{I}}}$ is the asymptotic connection. The subscript
$C=0$ in (2.30) indicates that the connection used to compare the electric
charges at different points on the celestial sphere is taken to be trivial in
standard quantum field theory (the electron has charge $-1$ no matter where it
appears on the celestial sphere) so we will suppress this notation in what
follows.222The role of this asymptotic connection is more fundamental in the
non-abelian case, where it poses a potential obstruction to the definition of
non-abelian dyonic charge.

The soft theorem states that in the limit where the energy of the photon is
small compared to the energies of the hard particles, the amplitude factorizes
into a simple soft factor multiplied by a reduced scattering amplitude
involving the remaining $n$ hard states. The gauge coupling constant is $e$
and $Q_{i}$ is the electric charge of the $i$th particle. In our conventions,
the electric charge is defined by

$\begin{split}Q=\frac{1}{e^{2}}\int_{S^{2}}\star F\;.\end{split}$ (2.31)

We will assume that the gauge group is compact ($U(1)$ as opposed to
${\mathbb{R}}$), so the charges are quantized

$\begin{split}Q_{i}=n_{i}\in{\mathbb{Z}}\;.\end{split}$ (2.32)

We define the leading ($n=1$) soft photon operator according to (2.22)

$S_{a}(x)\equiv\frac{1}{e}\oint_{\mathcal{C}}\frac{d{\omega}}{2\pi
i}{\mathcal{O}}_{a}({\omega},x)\;.$ (2.33)

The contour integral isolates the residue of ${\mathcal{O}}_{a}({\omega},x)$
at ${\omega}=0$, and $S_{a}(x)$ has scaling dimension ${\Delta}=1$. Insertions
of the soft operator are therefore fixed completely by the soft theorem

$\begin{split}\langle\,S_{a}(x){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle={\mathcal{J}}_{a}(x)\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\;,\qquad{\mathcal{J}}_{a}(x)={\partial}_{a}\sum_{i}Q_{i}\log[-{\hat{p}}_{i}\cdot{\hat{q}}(x)]\;.\end{split}$
(2.34)

Multiple insertions of this soft operator simply factorize

$\begin{split}\langle\,S_{a_{1}}(y_{1})\cdots
S_{a_{m}}(y_{m}){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle={\mathcal{J}}_{a_{1}}(y_{1})\cdots{\mathcal{J}}_{a_{m}}(y_{m})\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\;.\end{split}$
(2.35)

This formula is obtained by taking an $n+m$ point amplitude with $m$ external
photons and then taking consecutive soft limits
${\omega}_{1},\cdots,{\omega}_{m}\to 0$. Since the photon does not carry
charge, the order of these limits is irrelevant. Equations (2.34)-(2.35)
together demonstrate that $S_{a}$ behaves as a flat abelian connection. In the
absence of magnetic charges, the gauge field edge mode $C$ is flat as well

$\begin{split}dS=0\;,\qquad dC=0\quad\implies\quad S=d\phi\;,\qquad
C=d{\theta}\;.\end{split}$ (2.36)

It was demonstrated in [1] that the soft photon theorem implies the existence
of a conserved $U(1)$ current in the celestial CFTd. The definition of this
operator involves the shadow transform, which maps a primary operator of
weight ${\Delta}$ in the representation $\mathcal{R}$ of $SO(d)$ to a
conformal primary operator of weight $d-{\Delta}$ and representation
$\mathcal{R}$,

$\begin{split}{\widetilde{\mathcal{O}}}(x)\equiv\int
d^{d}y\frac{1}{[(x-y)^{2}]^{d-{\Delta}}}\mathcal{R}({\mathcal{I}}(x-y))\cdot{\mathcal{O}}(y)\;.\end{split}$
(2.37)

The double shadow transform is proportional to the identity,

$\begin{split}{\widetilde{\widetilde{\mathcal{O}}}}(x)=c_{{\Delta},\mathcal{R}}{\mathcal{O}}(x)\;,\qquad\int
d^{d}z\frac{\mathcal{R}({\mathcal{I}}(x-z){\mathcal{I}}(z-y))}{[(x-z)^{2}]^{\Delta}[(z-y)^{2}]^{d-{\Delta}}}=c_{{\Delta},\mathcal{R}}{\delta}^{(d)}(x-y)\;.\end{split}$
(2.38)

For the spin $s$ representation, this coefficient is given by

$\begin{split}c_{{\Delta},s}=\frac{\pi^{d}({\Delta}-1)(d-{\Delta}-1){\Gamma}(\frac{d}{2}-{\Delta}){\Gamma}({\Delta}-\frac{d}{2})}{({\Delta}-1+s)(d-{\Delta}-1+s){\Gamma}({\Delta}){\Gamma}(d-{\Delta})}\;.\end{split}$
(2.39)

The $U(1)$ conserved current is the shadow transform of the soft operator
(2.33)

$J_{a}(x)=\frac{1}{2c_{1,1}}{\widetilde{S}}_{a}(x)\;.$ (2.40)

In $d>2$, $c_{1,1}$ is given by (2.39) while for $d=2$ direct computation
gives $c_{1,1}=4\pi^{2}$ (instead of $\pi^{2}$).333$c_{1,1}$ vanishes in odd
dimensions, but the shadow integral in (2.40) vanishes as well. A finite
result is obtained by evaluating the RHS of (2.40) for general ${\Delta}$ and
then taking the ${\Delta}\to 1$ limit. Note that $S_{a}(x)$ has dimension 1,
so that its shadow has $\Delta=d-1$ which is appropriate for a conserved
current. Insertions of this current take the form

$\begin{split}\langle\,J_{a}(x){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle=j_{a}(x)\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\;,\qquad
j_{a}(x)=\frac{1}{2c_{1,1}}{\widetilde{\mathcal{J}}}_{a}(x)\;,\end{split}$
(2.41)

and the divergence is given by (see appendix A for details)

$\begin{split}\langle\,{\partial}^{a}J_{a}(x){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle=\sum_{i}Q_{i}{\mathcal{K}}_{d}(m_{i}/{\omega}_{i},x_{i};x)\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\;.\end{split}$
(2.42)

The Ward identity (2.42) associates a non-local charge distribution to the
massive charged states. Massless states have a localized charge distribution
which can be seen using the property

$\begin{split}\lim_{z\to
0}{\mathcal{K}}_{d}(z,x;y)={\delta}^{(d)}(x-y)\;.\end{split}$ (2.43)

#### Leading Soft Gluon Theorem

The generalization of the soft photon theorem (2.30) to non-abelian gauge
theories is straightforward. The soft limit of a single gluon with
polarization $a$ and color $I$ takes the form

$\begin{split}\langle\,{\mathcal{O}}^{I}_{a}({\omega},x){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{C=0}~{}~{}\stackrel{{\scriptstyle{\omega}\to
0}}{{\longrightarrow}}~{}~{}\frac{1}{{\omega}}ig_{\scriptscriptstyle
YM}\sum_{i}\frac{{\hat{p}}_{i}\cdot{\varepsilon}_{a}(x)}{{\hat{p}}_{i}\cdot{\hat{q}}(x)}T^{I}_{i}\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{C=0}\;.\end{split}$
(2.44)

$T^{I}_{i}$ is a generator in the Lie algebra ${\mathfrak{g}}$ (for the gauge
group $G$) acting in the representation $R_{i}$ under which the $i$th particle
transforms, and $g_{{\scriptscriptstyle YM}}$ is the gauge coupling. The
generators are normalized so that

$\begin{split}\big{[}T^{I},T^{J}\big{]}=f^{IJK}T^{K}\;,\qquad
f^{IKL}f^{JKL}={\delta}^{IJ}\;,\qquad(T_{adj}^{I})^{JK}=-f^{IJK}\;.\end{split}$
(2.45)

The notation $C=0$ signifies that the asymptotic connection used to compare
color states at infinity is set to zero in conventional scattering amplitude
calculations. This simplification is benign in the absence of chromo-magnetic
flux. However, as we will discuss in Section 3.2, this background connection
is non-zero in the presence of chromo-dyons and significantly complicates the
definition of the asymptotic symmetry group in non-abelian models. In this
section we will only consider electrically charged states, so we suppress this
notation.

The leading soft gluon operator is defined as

$\begin{split}S^{I}_{a}(x)\equiv\frac{1}{g_{\scriptscriptstyle
YM}}\oint_{\mathcal{C}}\frac{d{\omega}}{2\pi
i}{\mathcal{O}}^{I}_{a}({\omega},x)\;.\end{split}$ (2.46)

Its correlation functions are completely controlled by the soft gluon theorem

$\langle\,S^{I}_{a}(x){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle={\mathcal{J}}^{I}_{a}(x)\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\;,\qquad{\mathcal{J}}^{I}_{a}(x)=i{\partial}_{a}\sum_{i}\log[-{\hat{p}}_{i}\cdot{\hat{q}}(x)]\,T^{I}_{i}\;.$
(2.47)

Note that ${\mathcal{J}}_{a}^{I}(x)$ is a matrix which acts on the color
indices of the amplitude ${\mathcal{A}}_{n}$.

Multiple insertions of the operator $S_{a}^{I}$ are defined by taking an $n+m$
point amplitude with $m$ gluons and then taking the soft limits
${\omega}_{1},\cdots{\omega}_{m}\to 0$. However, since the gluon is itself
charged under the gauge group, the order in which these limits are performed
_does_ matter. We will adopt the prescription that the soft limits are taken
in the “left-to-right” order

$\begin{split}&\langle\,S_{a_{1}}^{I_{1}}(y_{1})\cdots
S_{a_{m}}^{I_{m}}(y_{m}){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\\\
&\qquad\qquad\qquad\equiv\frac{1}{g^{m}_{\scriptscriptstyle
YM}}\oint_{{\mathcal{C}}_{m}}\frac{d{\omega}_{m}}{2\pi
i}\cdots\oint_{{\mathcal{C}}_{1}}\frac{d{\omega}_{1}}{2\pi
i}\langle\,{\mathcal{O}}^{I_{1}}_{a_{1}}({\omega}_{1},y_{1})\cdots{\mathcal{O}}^{I_{m}}_{a_{m}}({\omega}_{m},y_{m}){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\;,\end{split}$
(2.48)

where
${\mathcal{C}}_{1}\subset{\mathcal{C}}_{2}\subset\cdots\subset{\mathcal{C}}_{m}$
are nested contours in the complex $\omega$ plane. The soft operator $S$ and
gauge field edge mode $C$ satisfy a non-abelian flatness condition

$\begin{split}dS+C{\wedge}S+S{\wedge}C=0\;,\quad
dC+C{\wedge}C=0\quad\implies\quad S=d\phi+[C,\phi]\;,\quad
C=UdU^{-1}\;.\end{split}$ (2.49)

We can also construct a non-abelian conserved current

$\begin{split}J_{a}^{I}(x)=\frac{1}{2c_{1,1}}{\widetilde{S}}^{I}_{a}(x)\;,\qquad\langle\,J_{a}^{I}(x){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle=j^{I}_{a}(x)\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\end{split}$
(2.50)

where $j^{I}_{a}(x)=\frac{1}{2c_{1,1}}{\widetilde{\mathcal{J}}}^{I}_{a}(x)$.
Its divergence is given by

$\begin{split}\langle\,{\partial}^{a}J^{I}_{a}(x){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle=i\sum_{i}{\mathcal{K}}_{d}(m_{i}/{\omega}_{i},x_{i};x)T^{I}_{i}\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\;.\end{split}$
(2.51)

#### Leading Soft Graviton Theorem

The leading soft graviton theorem was first derived by Weinberg [15]. The soft
limit of a single graviton with polarization $\varepsilon_{ab}$ takes the form

$\begin{split}\langle\,{\mathcal{O}}_{ab}({\omega},x){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{C=0}~{}~{}\stackrel{{\scriptstyle{\omega}\to
0}}{{\longrightarrow}}~{}~{}\frac{\kappa}{2\omega}\sum_{i}\frac{p_{i\mu}p_{i\nu}{\varepsilon}^{\mu\nu}_{ab}(q)}{p_{i}\cdot\hat{q}}\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{C=0}\;.\end{split}$
(2.52)

Here $\kappa^{2}=32\pi G$ and $C_{ab}=r^{-1}h_{ab}|_{{\partial}{\mathscr{I}}}$
($h_{\mu\nu}=g_{\mu\nu}-\eta_{\mu\nu}$ is the metric fluctuation). The
subscript $C=0$ in (2.52) describes the BMS frame in which the scattering
amplitudes are evaluated and with respect to which the energy-momentum of the
particles are defined. This is the gravitational analogue of the flat
connection $C=0$ used in gauge theory scattering amplitudes. We will suppress
this notation in the rest of this section.

The leading soft graviton operator is defined according to (2.22)

$\begin{split}N_{ab}(x)\equiv\frac{2}{\kappa}\oint_{\mathcal{C}}\frac{d{\omega}}{2\pi
i}{\mathcal{O}}_{ab}({\omega},x)\;\end{split}$ (2.53)

and has scaling dimension ${\Delta}=1$. Insertions of this operator take the
form

$\langle\,N_{ab}(x){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle={\mathcal{J}}_{ab}(x)\,\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\;,$
(2.54)

where

$\begin{split}{\mathcal{J}}_{ab}(x)&\equiv-\left({\partial}_{a}{\partial}_{b}-\frac{1}{d}{\delta}_{ab}{\partial}^{2}\right)\sum_{i}{\omega}_{i}[-{\hat{p}}_{i}\cdot{\hat{q}}(x)]\log[-{\hat{p}}_{i}\cdot{\hat{q}}(x)]\;.\\\
\end{split}$ (2.55)

Multiple insertions of $N_{ab}$ are calculated by taking simultaneous soft
limits of a scattering amplitude involving multiple external gravitons.
Although general relativity is non-linear, its infrared dynamics is
effectively abelian since the charge of the theory is energy-momentum. The
order of soft limits is therefore immaterial and one finds

$\begin{split}&\langle\,N_{a_{1}b_{1}}(y_{1})\cdots
N_{a_{m}b_{m}}(y_{m}){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle={\mathcal{J}}_{a_{1}b_{1}}(y_{1})\cdots{\mathcal{J}}_{a_{m}b_{m}}(y_{m})\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\;.\end{split}$
(2.56)

From this formula, it follows that $N_{ab}$ satisfies the following “flatness
condition”

$\begin{split}{\partial}_{[a}N_{b]c}(x)-\frac{1}{d-1}{\delta}_{c[a}{\partial}^{d}N_{b]d}(x)=0\quad\implies\quad
N_{ab}(x)=2\left({\partial}_{a}{\partial}_{b}-\frac{1}{d}{\delta}_{ab}{\partial}^{2}\right)N(x)\;.\end{split}$
(2.57)

A similar property is also satisfied by the gravitational connection
$C_{ab}(x)$ which then implies

$\begin{split}C_{ab}(x)=2\left({\partial}_{a}{\partial}_{b}-\frac{1}{d}{\delta}_{ab}{\partial}^{2}\right)C(x)\;.\end{split}$
(2.58)

This condition is the higher-dimensional analogue of the four-dimensional
Christodoulou-Klainerman constraint [16].

The leading soft graviton theorem implies the existence of $d+2$ conserved
currents $P^{+}_{a}(x)$, $P^{b}_{a}(x)$ and $P^{-}_{a}(x)$ whose charges are
the translation generators in (LABEL:eq:transAlgebra). However, not all of
these currents are independent. Rather, they sit in a single conformal
multiplet that mixes under Lorentz (conformal) transformations. Up to
improvement terms, (LABEL:eq:transAlgebra) implies

$\begin{split}P^{b}_{a}(x)=\frac{i}{2}[P^{+}_{a}(x),K^{b}]\;,\qquad
P^{-}_{a}(x)=\frac{i}{d}[P^{b}_{a}(x),K_{b}]\;.\end{split}$ (2.59)

The current $P^{+}_{a}(x)$ can be constructed from the shadow transform of the
soft graviton operator as

$\begin{split}P^{+}_{a}(x)=\frac{1}{4c_{1,2}}{\partial}^{b}{\widetilde{N}}_{ab}(x)\;.\end{split}$
(2.60)

In $d>2$, $c_{1,2}$ is given by (2.39) while for $d=2$ direct computation
gives $c_{1,2}=\pi^{2}$ (instead of $\pi^{2}/4$). The soft operator $N_{ab}$
has dimension $1$ so the current has ${\Delta}=d$. The unusual dimension for
this current arises because the conserved charge $P^{+}$ is itself
dimensionful according to (LABEL:eq:transAlgebra). Using this definition, we
find

$\begin{split}\langle\,P^{+}_{a}(x){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle=p_{a}^{+}(x)\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\;,\end{split}$
(2.61)

where

$\begin{split}p^{+}_{a}(x)={\partial}^{b}j_{ab}(x),\qquad
j_{ab}(x)=-\frac{1}{4c_{1,2}}{\widetilde{\mathcal{J}}}_{ab}(x)\;.\end{split}$
(2.62)

The divergence of this current is then given by (see appendix A)

$\begin{split}\langle\,{\partial}^{a}P^{+}_{a}(x){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle=\sum_{i}m_{i}{\mathcal{K}}_{d+1}(m_{i}/{\omega}_{i},x_{i};x)\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\;.\end{split}$
(2.63)

As in the abelian case, the Ward identity (2.63) associates a non-local
energy-momentum distribution to massive charged states. Massless states have a
localized energy-momentum as can be seen from

$\begin{split}\lim_{z\to
0}z\,{\mathcal{K}}_{d+1}(z,x;y)={\delta}^{(d)}(x-y)\;.\end{split}$ (2.64)

###  2.3 Infrared Divergences in Abelian Theories

In any theory with long-range forces, the naive scattering amplitude defined
in (2.15) vanishes in four dimensions due to infrared divergences. Formally,
these infrared divergences arise due to low energy virtual particle exchange
in diagrams like figure 1. Physically, the divergences signify that the
charged asymptotic scattering states are not single particle states but rather
coherent states of charged particles and bremsstrahlung.

$+$$+$$+$$+~{}~{}...$ Figure 1: Some typical diagrams contributing to infrared
divergences

Although each diagram is separately divergent, the infinite sum exponentiates
and the amplitude actually vanishes: ${\mathcal{A}}_{n}\sim e^{-\infty}=0$.
There is simply zero probability to scatter into a state with a finite number
of photons or gravitons in four dimensions: the power radiated in a generic
scattering process does not vanish at zero frequency, and so infinitely many
low energy particles are produced [17, 18]. In higher dimensions this is no
longer the case and exclusive scattering states with finite numbers of
massless quanta can be defined. The soft contributions depicted in figure 1
can be calculated and exponentiated in any number of dimensions, but it is
only in four dimensions that the contribution is divergent.

Standard treatments of this subject restrict attention to inclusive
observables defined by a trace over “unobservable soft quanta” in the initial
and final states. Order by order in perturbation theory, the infinite volume
of phase space available to soft external quanta cancels the vanishing
probability to scatter into an exclusive state with a fixed number of photons.
Although perfectly adequate for practical purposes, this approach to the
problem conflates two separate issues and obscures the underlying physics. It
is certainly the case that any conceivable experimental apparatus has a finite
energy sensitivity. In quantum mechanics, when an observer lacks access to a
particular component of the Hilbert space, he performs a partial trace in
order to obtain a reduced density matrix which can be used to compute
observables. In a realistic scenario, any collider observable in any number of
dimensions should include a trace over the unobservable soft quanta. However,
the “unobservable” scale relevant to the trace is set by the detector
sensitivity and is not related to the infrared divergence. An inadequate
higher-dimensional detector might require a trace over states with relatively
short wavelengths which are certainly produced with nonzero probability. This
inclusive calculation, while relevant to the experiment, is not related to any
infrared divergence since the exclusive observable is well defined in higher
dimensions.

Taking a partial trace generically turns a pure density matrix into a mixed
state. In order to discuss fine-grained questions like unitarity or the
information content of soft quanta, one needs access to the actual quantum
mechanical transition amplitudes in Hilbert space. The difficulty in defining
an $S$-matrix in four-dimensional asymptotically flat space highlights a
deficiency in a particular formalism but does not preclude the calculation of
exact transition probabilities. The key physical observation relevant to the
definition of an infrared finite $S$-matrix was made by Chung, Kibble and
Faddeev-Kulish [19, 20, 21, 22, 23, 24]. The asymptotic Hamiltonian, which
determines the early and late time evolution of scattering states, is not a
free Hamiltonian. In the charged sectors of the Hilbert space, this
Hamiltonian has matrix elements connecting states with different numbers of
photons. A stationary state invariant under early/late time evolution must
therefore contain an infinite number of photons corresponding to
bremsstrahlung clouds. Computations involving these dressed states are
cumbersome and appear to be plagued with ambiguities. One cannot help but feel
that the ultimate treatment has yet to be discovered, but there are
indications that the celestial scattering amplitudes and the conformal basis
contain the appropriate formalism to address the problem[25, 4]. In this paper
we focus on understanding the soft virtual exchange in exclusive amplitudes
and leave the proper determination of the scattering states to future work.

There are several equivalent methods for regulating and cancelling infrared
divergences in scattering amplitudes (see Chapter 13 of Weinberg [26] for a
thorough discussion of the abelian case). Each of these methods breaks or
alters a symmetry of the problem. One common approach introduces a hard IR
cutoff $\mu$ for the photon. Loop integrals over virtual momenta are to be
performed only in the range $|{\omega}|>\mu$. A second scale ${\Lambda}$ is
chosen in order to separate out the contribution of soft particles which are
defined to have $\mu<|{\omega}|<{\Lambda}$. Provided that $\mu$ and
${\Lambda}$ are taken to be much smaller than the typical energy scale of the
scattering amplitude, the contributions of these quanta can be computed
exactly and exponentiated. Hard particles with $|{\omega}|>{\Lambda}$
contribute IR finite (but generally incalculable and non-exponentiating)
corrections. The general structure of a scattering amplitude is then

$\begin{split}\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\mu}=e^{-{\Gamma}}\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\Lambda}\;.\end{split}$
(2.65)

Here $\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\mu}$
denotes the full scattering amplitude with IR cutoff $\mu$ and loop integrals
performed with $|{\omega}|>\mu$, while
$\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\Lambda}$ denotes
the amplitude with loop integrals performed in the range
$|{\omega}|>{\Lambda}$. The soft contribution resides entirely in the
exponential factor $e^{-{\Gamma}}$ which depends on the scales $\mu$,
${\Lambda}$ and on the quantum numbers of the hard states.

#### Abelian Gauge Theories

In an abelian gauge theory, the infrared divergent piece of the full
scattering amplitude is the exponentiation of the single-exchange contribution
shown in the first diagram of figure 1. This dramatic simplification occurs
because soft photons do not emit other soft particles and fails in non-abelian
models (and in models with massless charged matter). The diagram is calculated
using the soft photon theorem. Summing over the propagating internal states,
one finds444The $i{\epsilon}$ prescription for $i=j$ is a bit different and
$-p_{j}\cdot\ell-i{\epsilon}$ is replaced with $-p_{i}\cdot\ell+i{\epsilon}$.

$\begin{split}{\Gamma}_{\text{ph}}=\frac{ie^{2}}{2}\sum_{i,j}Q_{i}Q_{j}\int_{\mu}^{\Lambda}\frac{d^{d+2}\ell}{(2\pi)^{d+2}}\frac{p^{\mu}_{i}p^{\nu}_{j}\Pi_{\mu\nu}(\ell)}{(\ell^{2}-i{\epsilon})(p_{i}\cdot\ell-i{\epsilon})(-p_{j}\cdot\ell-i{\epsilon})}\;.\end{split}$
(2.66)

The integral is taken over the range $\mu<|{\omega}|<{\Lambda}$ and represents
only the contribution of the “soft quanta.” Note that this piece of the
integral can be separated out in any dimension even though it is not infrared
divergent when $d>2$. The polarization sum $\Pi_{\mu\nu}(\ell)$ defined in
(2.10) arises from the numerator of the photon propagator. This is of the form
$\Pi_{\mu\nu}(\ell)=\eta_{\mu\nu}+\text{\emph{gauge-dependent terms}}$ so we
could further simplify the expression by sending
$\Pi_{\mu\nu}(\ell)\to\eta_{\mu\nu}$.

This integral is evaluated explicitly in [26]. For our purposes it is more
useful to retain $\Gamma_{\text{ph}}$ as a $d$-dimensional integral. To do
this, we parameterize all momenta using (2.5) and use (2.14) to find

$\begin{split}{\Gamma}_{\text{ph}}&=-\frac{e^{2}}{4\pi}\sum_{i,j}Q_{i}Q_{j}\int\frac{d^{d}x}{(2\pi)^{d}}\int\limits_{\mu<|{\omega}|<{\Lambda}}d{\omega}|{\omega}|^{d-3}\\\
&\qquad\qquad\qquad\times\int_{-\infty}^{\infty}\frac{d\kappa}{2\pi
i}\frac{2{\hat{p}}^{\mu}_{i}{\hat{p}}^{\nu}_{j}\Pi_{\mu\nu}(x)}{(\kappa+i{\epsilon})(\kappa-2{\hat{p}}_{i}\cdot{\hat{q}}(x)+i\eta\eta_{i}{\epsilon})(\kappa-2{\hat{p}}_{j}\cdot{\hat{q}}(x)-i\eta\eta_{j}{\epsilon})}\;.\end{split}$
(2.67)

We perform the integral over $\kappa$ via contour integration. When
$\eta_{i}=\eta_{j}$ we close the contour in the upper half plane. When
$\eta_{i}=-\eta_{j}=-\eta$, we close the contour in the lower half plane and
we pick up the pole at $\kappa=-i{\epsilon}$. Finally, when
$\eta_{i}=-\eta_{j}=\eta$ we close the contour in the upper half plane. This
contribution vanishes since there are no poles in the upper half plane. We can
then perform the integral over ${\omega}$ and we are left with a
$d$-dimensional integral. We can also simplify further by setting
$\Pi_{\mu\nu}(x)={\varepsilon}^{a}_{\mu}(x){\varepsilon}_{a\nu}(x)$. Plugging
all of this into (2.66) and using the language of conformal correlators, we
find

$\begin{split}{\Gamma}_{\text{ph}}={\alpha}(A_{1}+2\pi
iA_{2})\;,\qquad{\alpha}=\frac{e^{2}}{8\pi}\int_{\mu}^{\Lambda}d{\omega}{\omega}^{d-3}\;,\end{split}$
(2.68)

and

$\begin{split}A_{1}&=\int\frac{d^{d}x}{(2\pi)^{d}}[{\mathcal{J}}_{a}(x)]^{2}\;,\qquad
A_{2}=\int_{-\infty}^{\infty}\frac{d\nu}{2\pi}\int\frac{d^{d}x}{(2\pi)^{d}}[|{\mathcal{J}}_{a}^{+}(\nu,x)|^{2}+|{\mathcal{J}}_{a}^{-}(\nu,x)|^{2}]_{r}\;.\end{split}$
(2.69)

${\mathcal{J}}_{a}(x)$ is defined in (2.34) and

$\begin{split}{\mathcal{J}}_{a}^{\pm}(\nu,x)\equiv{\partial}_{a}\sum_{i\in{\text{out}(+)\atop\text{in}(-)}}Q_{i}\frac{[-{\hat{p}}_{i}\cdot{\hat{q}}(x)]^{i\nu}}{i\nu}\;,\qquad{\mathcal{J}}_{a}(x)=\lim_{\nu\to
0}[{\mathcal{J}}_{a}^{+}(\nu,x)+{\mathcal{J}}_{a}^{-}(\nu,x)]\;.\end{split}$
(2.70)

The $[~{}]_{r}$ symbol removes any $i=j$ terms in the integrand.

In $d>2$, ${\Gamma}_{\text{ph}}\to 0$ polynomially as ${\Lambda},\mu\to 0$
which shows that soft photons do not contribute any divergences to the
scattering amplitude. On the other hand, in $d=2$,
${\Gamma}_{\text{ph}}\to\infty$ logarithmically implying that the scattering
amplitude itself vanishes as a power law as $\mu\to 0$.

#### Gravitational Theories

As in abelian gauge theory, the total graviton contribution to the infrared
divergence is the exponentiation of the single virtual soft graviton
contribution. This simplification holds because, although general relativity
is strongly nonlinear, the charge of the theory is energy-momentum and long
wavelength modes are weakly interacting. The infrared divergent phase in this
case is

$\begin{split}{\Gamma}_{\text{gr}}={\alpha}_{\text{gr}}(A_{1}^{\text{gr}}+2\pi
iA_{2}^{\text{gr}})\;,\qquad{\alpha}_{\text{gr}}=\frac{\kappa^{2}}{32\pi}\int_{\mu}^{\Lambda}d{\omega}{\omega}^{d-3}\;.\end{split}$
(2.71)

The integrals in the exponent are

$\begin{split}A_{1}^{\text{gr}}&=\int\frac{d^{d}x}{(2\pi)^{d}}[{\mathcal{J}}_{ab}(x)]^{2}\;,\qquad
A_{2}^{\text{gr}}=\int_{-\infty}^{\infty}\frac{d\nu}{2\pi}\int\frac{d^{d}x}{(2\pi)^{d}}[|{\mathcal{J}}_{ab}^{+}(\nu,x)|^{2}+|{\mathcal{J}}_{ab}^{-}(\nu,x)|^{2}]_{r}\;,\end{split}$
(2.72)

where ${\mathcal{J}}_{ab}(x)$ is defined in (2.55) and

$\begin{split}{\mathcal{J}}_{ab}^{\pm}(\nu,x)&=\left[{\partial}_{a}{\partial}_{b}-\frac{1}{d}{\delta}_{ab}{\partial}^{2}\right]\sum_{i\in{\text{out}\atop\text{in}}}{\omega}_{i}\frac{[-{\hat{p}}_{i}\cdot{\hat{q}}(x)]^{1+i\nu}}{i\nu(1+i\nu)}\;,\\\
{\mathcal{J}}_{ab}(x)&=\lim_{\nu\to
0}[{\mathcal{J}}^{+}_{ab}(\nu,x)+{\mathcal{J}}^{-}_{ab}(\nu,x)]\;.\end{split}$
(2.73)

##  3 Magnetic Charges

The previous section reviewed soft theorems and infrared divergences for
electrically charged states in gauge theories. In this section, we generalize
those results to include magnetically charged or dyonic states. The first
discussion of an abelian magnetic soft theorem appeared in [27]. The results
on the chromo-magnetic soft theorem and the soft theorem for extended objects
are new.

###  3.1 Abelian Gauge Theories

In four dimensions, zero-dimensional particles can carry both electric and
magnetic charges. The magnetic charge is defined by the surface integral

$\begin{split}P=\frac{1}{2\pi}\int_{S^{2}}F\;\end{split}$ (3.1)

taken at spatial infinity. The generalized soft theorems and infrared
divergences for magnetic states can be obtained using the electromagnetic
duality transformation

$\begin{split}F\to{\widetilde{F}}=-\frac{2\pi}{e^{2}}\star F\;,\qquad
e\to{\widetilde{e}}=\frac{2\pi}{e}\;,\qquad Q\to{\widetilde{Q}}=P\;,\qquad
P\to{\widetilde{P}}=-Q\;.\end{split}$ (3.2)

The allowed set of magnetic charges is constrained by the Dirac quantization
condition $Q_{i}P_{j}\in{\mathbb{Z}}$. This implies that magnetic charge is
quantized in multiples of $m_{0}\in{\mathbb{Z}}$, the smallest magnetic charge
in the spectrum:

$P_{i}=m_{0}w_{i}\;,\qquad w_{i}\in{\mathbb{Z}}\;.$ (3.3)

The integer $m_{0}$ is theory-dependent.

To write down the soft theorem for magnetically charged particles, we Fourier
transform the first equation of (3.2) to determine the dual polarization
vector:

$\begin{split}q_{[\mu}{\widetilde{\varepsilon}}_{\nu]}(q)=-\frac{\pi}{e^{2}}{\epsilon}_{\mu\nu\rho{\sigma}}q^{[\rho}{\varepsilon}^{{\sigma}]}(q)\quad\implies\quad{\widetilde{\varepsilon}}_{\mu}(q)=\frac{2\pi}{e^{2}}{\epsilon}_{\mu\nu\rho{\sigma}}\frac{n^{\nu}q^{\rho}}{n\cdot
q}{\varepsilon}^{\sigma}(q)\;.\end{split}$ (3.4)

In the basis (2.8), this explicitly evaluates to

$\begin{split}{\widetilde{\varepsilon}}_{a}(x)=\frac{2\pi}{e^{2}}{\epsilon}_{ab}{\varepsilon}^{b}(x)\;,\end{split}$
(3.5)

where $\epsilon_{ab}$ is the 2D Levi-Civita tensor normalized as
$\epsilon_{12}=\epsilon^{12}=1$. Magnetically charged particles couple to the
dual polarization of the photon so the soft photon theorem (2.30) becomes [27]

$\begin{split}\langle\,{\mathcal{O}}_{a}({\omega},x){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle~{}~{}\stackrel{{\scriptstyle{\omega}\to
0}}{{\longrightarrow}}~{}&~{}e\sum_{i}{\widetilde{Q}}_{i}\frac{{\hat{p}}_{i}\cdot{\widetilde{\varepsilon}}_{a}(x)}{{\hat{p}}_{i}\cdot{\hat{q}}(x)}\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\\\
&=\frac{2\pi}{e}{\epsilon}_{ab}\sum_{i}P_{i}\frac{{\hat{p}}_{i}\cdot{\varepsilon}^{b}(x)}{{\hat{p}}_{i}\cdot{\hat{q}}(x)}\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\;.\end{split}$
(3.6)

In the language of conformal correlators, this reads

$\begin{split}\langle\,S_{a}(x){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle={\widetilde{\mathcal{J}}}_{a}(x)\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\;,\qquad{\widetilde{\mathcal{J}}}_{a}(x)\equiv\frac{2\pi}{e^{2}}\epsilon_{ab}{\partial}^{b}\sum_{i}P_{i}\log[-{\hat{p}}_{i}\cdot{\hat{q}}(x)]\;.\end{split}$
(3.7)

This analysis extends to dyonic states in a straightforward manner. For any
two pairs of dyons with charges $(Q_{i},P_{i})$ and $(Q_{j},P_{j})$, the
Dirac-Schwinger-Zwanziger quantization condition (derived by quantizing the
total angular momentum of the electromagnetic field of a two-dyon state [28])
implies

$Q_{i}P_{j}-Q_{j}P_{i}\in{\mathbb{Z}}\;.$ (3.8)

It follows that the charge spectrum for dyons takes the form [29]555The
generalized $SL(2,{\mathbb{Z}})$ transformation acts as
$\tau=\tau_{1}+i\tau_{2}\to\frac{a\tau+b}{c\tau+d}$, $n_{i}\to an_{i}-bw_{i}$,
$w_{i}\to-cn_{i}+dw_{i}$ with $a,b,c,d\in{\mathbb{Z}}$ and $ad-bc=1$. On the
field strength this acts as $F+i\star F\to(c\tau+d)(F+i\star F)$.

$\begin{split}Q_{i}=n_{i}+\frac{{\vartheta}}{2\pi}w_{i}\;,\qquad
P_{i}=m_{0}w_{i}\;,\qquad
n_{i},w_{i}\in{\mathbb{Z}}\;,\qquad{\vartheta}\in[0,2\pi)\;.\end{split}$ (3.9)

The soft theorem for these dyons simply combines (2.34) and (3.7) in a duality
covariant way

$\begin{split}\langle\,S_{a}(x){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle&=[{\mathcal{J}}_{a}(x)+{\widetilde{\mathcal{J}}}_{a}(x)]\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\\\
&=\sum_{i}(n_{i}{\delta}_{ab}+w_{i}\tau_{ab}){\partial}^{b}\log[-{\hat{p}}_{i}\cdot{\hat{q}}(x)]\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\;,\end{split}$
(3.10)

where

$\tau_{ab}\equiv\tau_{1}{\delta}_{ab}+\tau_{2}{\epsilon}_{ab}\;,\qquad\tau_{1}\equiv\frac{{\vartheta}}{2\pi}\;,\qquad\tau_{2}\equiv\frac{2\pi
m_{0}}{e^{2}}\;.$ (3.11)

In addition to the electric current defined in (2.40), we can now also define
a magnetic current which couples to magnetic particles,

$\begin{split}K_{a}\equiv-(\star J)_{a}=-{\epsilon}_{ab}J^{b}\;.\end{split}$
(3.12)

This satisfies

$\begin{split}\langle\,{\partial}^{a}K_{a}(x){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle&=\frac{2\pi}{e^{2}}\sum_{i}P_{i}\,{\mathcal{K}}_{2}(m_{i}/{\omega}_{i},x_{i};x)\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\;.\end{split}$
(3.13)

The structure of infrared divergences in the presence of magnetic charges is
similarly obtained by replacing
${\mathcal{J}}_{a}\to{\mathcal{J}}_{a}+{\widetilde{\mathcal{J}}}_{a}$ in the
integral (2.69). The infrared factor is
${\Gamma}_{\text{ph}}={\alpha}(A_{1}+2\pi iA_{2})$ with

$\begin{split}A_{1}&=\int\frac{d^{d}x}{(2\pi)^{d}}\left[{\mathcal{J}}_{a}(x)+{\widetilde{\mathcal{J}}}_{a}(x)\right]^{2}\;,\\\
A_{2}&=\int_{-\infty}^{\infty}\frac{d\nu}{2\pi}\int\frac{d^{d}x}{(2\pi)^{d}}\bigg{(}|{\mathcal{J}}_{a}^{+}(\nu,x)+{\widetilde{\mathcal{J}}}^{+}_{a}(\nu,x)|^{2}+|{\mathcal{J}}_{a}^{-}(\nu,x)+{\widetilde{\mathcal{J}}}^{-}_{a}(\nu,x)|^{2}\bigg{)}_{r}\;,\end{split}$
(3.14)

where

$\begin{split}{\widetilde{\mathcal{J}}}_{a}^{\pm}(\nu,x)\equiv\frac{2\pi}{e^{2}}{\epsilon}_{ab}{\partial}^{b}\sum_{i\in{\text{out}\atop\text{in}}}P_{i}\frac{[-{\hat{p}}_{i}\cdot{\hat{q}}(x)]^{i\nu}}{i\nu}\;.\end{split}$
(3.15)

This formula is straightforward for purely magnetic scattering and follows
from an application of the duality transformation (3.2). The case with both
electric and magnetic charges is slightly more subtle. The quantization
condition (3.8) implies that there is no small expansion parameter when an
electron scatters off of a monopole: the effective coupling is always of order
one. Weinberg also noted [30] that the single photon exchange diagram between
an electron and a monopole appears to violate Lorentz invariance. Both of
these complications are resolved by explicitly exponentiating the diagrams in
figure 1. Although the coupling is of order one, it is possible to treat the
infrared exchange to all orders to obtain a reliable approximation. Similarly,
since all photon exchange diagrams contribute at the same order in the $e^{2}$
expansion, it is only their sum which must be (and is [31], for appropriately
quantized charges) Lorentz and gauge invariant.

###  3.2 Non-abelian Gauge Theories and Chromodyons

The analog of the Dirac quantization condition in non-abelian gauge theory is
the GNO constraint [32]. For a given “electric gauge group” $G$, this
condition restricts the allowed magnetic charges of the theory to be weights
for the dual group $G^{\vee}$ obtained from $G$ by taking the dual root
lattice.

In a deconfined non-abelian gauge theory, there is no difficulty in
identifying the electric gauge group $G$ in the spectrum of electrically
charged states. Similarly, the magnetic gauge group $G^{\vee}$ has an
unambiguous action in the sector of purely magnetically charged states. The
essential complication arises in understanding the action of $G$ and
$G^{\vee}$ in the dyonic sector of the theory. Indeed, semiclassical
quantization of a single isolated non-abelian monopole does not yield states
transforming in representations of $G$ [33, 34] as one would expect if the
asymptotic symmetry group was truly $G\times G^{\vee}$. Chromo-magnetic charge
apparently spoils the action of $G$ in the dyonic sector of the Hilbert space
and the global part of the electric gauge group is ill-defined [35, 36, 37,
38, 39].

There is a simple physical explanation of this mathematical fact. The long-
range chromo-magnetic field of a monopole is controlled by a Gauss law. In the
magnetically charged sector of the theory, the magnetic charge

$Q^{I}=\frac{1}{2\pi}\int_{i^{0}}F^{I}$ (3.16)

is non-vanishing which implies that parallel transport on the sphere at
spatial infinity is path dependent. The “global part of the electric gauge
group” is defined by performing the _same_ large gauge transformation
everywhere on the celestial sphere. To compare the gauge parameter at two
separate points on the sphere, we are required to parallel transport from one
point to the other. In the presence of asymptotic chromo-magnetic flux the
parallel transport is path dependent (for directions in color space that do
not commute with the asymptotic value of $F$) and so the definition of the
global gauge group is ambiguous.666This is essentially the gauge theory
analogue of the “problem of angular momentum” in general relativity and one
might hope that it is resolved analogously. However, if there is no net
magnetic charge then the electric group $G$ can be unambiguously defined.
Indeed, semiclassical quantization of a monopole-antimonopole pair produces
states that do form representations of the electric color group [39].
Therefore, provided that the long-range magnetic field vanishes in the _in_
and _out_ states (i.e. both have net zero magnetic charge), it is possible to
consider a $S$-matrix involving magnetically charged states (the non-abelian
magnetic analog of $e^{+}e^{-}$ scattering). It is in this restricted case
that we expect the following chromo-dyon soft theorem to apply.

We will explore the magnetic soft gluon theorem guided by analogy with the
abelian case777Since the equations of motion in non-abelian gauge theory
explicitly involve the electric gauge potential, the simple duality
transformation (3.2) must be more complicated but might hold asymptotically or
in the soft limit. For this reason we consider the formulas in this subsection
conjectural.  and the electromagnetic duality transformation (3.2). We
conjecture the following form of the magnetic soft gluon theorem

$\langle\,S^{I}_{a}(x){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle=\frac{2\pi
i}{g_{{\scriptscriptstyle
YM}}^{2}}{\epsilon}_{ab}{\partial}^{b}\sum_{i}\log[-{\hat{p}}_{i}\cdot{\hat{q}}(x)]{\hat{T}}_{i}^{I}\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\;.$
(3.17)

Here ${\hat{T}}^{I}$ is a Lie algebra generator of $G^{\vee}$ in
representation $R_{i}$ under which the particle transforms. Generalizing to
chromodyons and including the ${\vartheta}$-term contribution, the generalized
soft gluon theorem can be written as

$\begin{split}\langle\,S^{I}_{a}(x){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle&=i\sum_{i}(T_{i}^{I}{\delta}_{ab}+{\hat{T}}^{I}_{i}\tau_{ab}){\partial}^{b}\log[-{\hat{p}}_{i}\cdot{\hat{q}}(x)]\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\;.\end{split}$
(3.18)

We are unaware of any results regarding the scattering of electric and
magnetic states in non-abelian gauge theory beyond the non-relativistic
approximation of motion on moduli space. It would be interesting to understand
the IR divergence structure of such amplitudes.

###  3.3 Soft Theorem for Magnetic Branes in $d>2$

In four-dimensional gauge theory, both the vector potential $F\equiv dA$ and
the dual vector potential $\star F\equiv d\tilde{A}$ are one-forms and they
couple to zero-dimensional excitations with one-dimensional world-lines. In
higher dimensions the vector potential remains a one-form and couples to
particles, while the dual potential couples to extended objects with
$(d-1)$-dimensional world-volumes which we will refer to as magnetic branes.
In $\mathbb{R}^{1,d+1}$, the dual magnetic potential is a $(d-1)$-form defined
by

$\begin{split}{\tilde{F}}_{(d)}=d{\tilde{A}}_{(d-1)}=\star F_{(2)}=\star
dA_{(1)}\;.\end{split}$ (3.19)

Here, we have introduced a subscript to make the rank of the differential form
explicit. The magnetic brane couples to the gauge field through the Bianchi
equation

$\begin{split}dF_{(2)}=2\pi\star{\tilde{J}}_{(d-1)}\end{split}$ (3.20)

where ${\tilde{J}}_{(d-1)}$ is the conserved $(d-1)$-form current.
Classically, this current is given by888See for example eq. (18.40) in [40].

$\begin{split}{\tilde{J}}_{(d-1)}^{\mu_{1}\dots\mu_{d-1}}(x)=\sum_{i}P_{i}\int_{\Sigma_{i}}dX_{{\Sigma}_{i}}^{\mu_{1}}{\wedge}\cdots{\wedge}dX_{{\Sigma}_{i}}^{\mu_{d-1}}\delta^{(d+2)}(x,X_{{\Sigma}_{i}})\;,\qquad
d\star{\tilde{J}}_{(d-1)}=0\;.\end{split}$ (3.21)

The integral is over the world-volume ${\Sigma}_{i}$ of the branes and
$X_{{\Sigma}_{i}}:{\Sigma}_{i}\hookrightarrow{\mathcal{M}}$ describes the
embedding of the world volume ${\Sigma}_{i}$ in spacetime. $P_{i}$ is the
magnetic charge of the branes.

The soft theorem in the presence of such branes can be derived from asymptotic
symmetries [41]. In any dimension, the usual soft photon theorem (2.30) is
related to the conservation of a $(d+1)$-form

$\begin{split}\star
J_{\varepsilon}\equiv\frac{1}{e^{2}}d({\varepsilon}_{(0)}\star
F_{(2)})\;,\qquad d\star J_{\varepsilon}=0\;.\end{split}$ (3.22)

Using Maxwell’s equation, we can write the current as

$\begin{split}\star
J_{\varepsilon}\equiv\frac{1}{e^{2}}d{\varepsilon}_{(0)}{\wedge}\star
F_{(2)}+{\varepsilon}_{(0)}\star J_{(1)}\;,\end{split}$ (3.23)

where $J_{(1)}$ is the usual electric current of the model. The charge
associated to this current generates electric large gauge transformations
under which $A_{(1)}\to A_{(1)}+d{\varepsilon}_{(0)}$. Evaluating this charge
on the _in_ -state and the _out_ -state, one can derive the leading soft
photon theorem [10, 42]. The first term above inserts a soft photon with
polarization $d{\varepsilon}_{(0)}$ in the $S$-matrix whereas the second term
generates the soft factor in (2.30) via a Fourier transform

$\begin{split}ie{\varepsilon}_{\mu}(q)\int d^{d+2}xe^{-iq\cdot
x}J_{(1)}^{\mu}(x)~{}\stackrel{{\scriptstyle q\to
0}}{{\longrightarrow}}~{}e\sum_{i}Q_{i}\frac{p_{i}\cdot{\varepsilon}(q)}{p_{i}\cdot
q}\;.\end{split}$ (3.24)

To derive the magnetic soft theorem, we define the magnetic conserved current

$\begin{split}\star{\tilde{J}}_{\tilde{\varepsilon}}\equiv\frac{1}{2\pi}d({\tilde{\varepsilon}}_{(d-2)}{\wedge}F_{(2)})\;,\qquad
d\star{\tilde{J}}_{\tilde{\varepsilon}}=0\;.\end{split}$ (3.25)

Using the Bianchi equation (3.20), this can be written as

$\begin{split}\star{\tilde{J}}_{\tilde{\varepsilon}}=\frac{1}{2\pi}d{\tilde{\varepsilon}}_{(d-2)}{\wedge}F_{(2)}+(-1)^{d}{\tilde{\varepsilon}}_{(d-2)}{\wedge}\star{\tilde{J}}_{(d-1)}\;.\end{split}$
(3.26)

The charge associated to this current generates magnetic large gauge
transformations under which
${\tilde{A}}_{(d-1)}\to{\tilde{A}}_{(d-1)}+d{\tilde{\varepsilon}}_{(d-2)}$
[43, 44, 45]. The corresponding Ward identity for this symmetry is obtained by
inserting the associated charge into the $S$-matrix. The first term in (3.26)
inserts a dual soft photon with the $(d-1)$ form polarization
$d{\tilde{\varepsilon}}_{(d-2)}$. This polarization is related to the
polarization vector as

$({\tilde{\varepsilon}}_{a})_{\mu_{1}\dots\mu_{d-1}}(q)=(-1)^{d}\frac{2\pi}{e^{2}}{\epsilon}_{\mu_{1}\dots\mu_{d-1}\nu\rho{\sigma}}\frac{n^{\nu}q^{\rho}}{n\cdot
q}{\varepsilon}_{a}^{\sigma}(q)\;.$ (3.27)

The second term generates the soft factor via a Fourier transform

$\begin{split}&ie({\tilde{\varepsilon}}_{a})_{\mu_{1}\dots\mu_{d-1}}(q)\int
d^{d+2}xe^{-iq\cdot
x}{\tilde{J}}_{(d-1)}^{\mu_{1}\cdots\mu_{d-1}}(x)~{}\stackrel{{\scriptstyle
q\to
0}}{{\longrightarrow}}~{}\frac{2\pi}{e}\sum_{i}P_{i}\,S^{\text{brane}}_{i,a}(q)\;,\end{split}$
(3.28)

where

$\begin{split}S^{\text{brane}}_{i,a}(q)=i(-1)^{d}(d-1)!\frac{{\varepsilon}_{a\mu}(q)n_{\nu}q_{\rho}}{n\cdot
q}\int_{\Sigma_{i}}\star(dX_{{\Sigma}_{i}}^{\mu}{\wedge}dX_{{\Sigma}_{i}}^{\nu}{\wedge}dX_{{\Sigma}_{i}}^{\rho})\,e^{-iq\cdot
X_{{\Sigma}_{i}}}\;.\end{split}$ (3.29)

We evaluate this quantity in a special case. At late times, we can assume that
the magnetic branes are non-interacting and are freely moving. We can describe
a brane in terms of a timelike parameter $\tau$ and $d-2$ spatial parameters
${\sigma}^{\alpha}$, ${\alpha}=1,\cdots,d-2$ as

$\begin{split}X_{{\Sigma}_{i}}^{\mu}=\chi_{i}^{\mu}+\tau
p_{i}^{\mu}+r^{\mu}_{i}({\sigma})\;.\end{split}$ (3.30)

$\chi_{i}$ describes the center of mass of the brane and $p_{i}^{\mu}$
describes the center of mass momentum. The soft factor for such a brane takes
the form

$\begin{split}S^{\text{brane}}_{i,a}(x)=-\frac{1}{{\omega}}\frac{2(d-1)!}{{\hat{p}}_{i}\cdot{\hat{q}}(x)}{\hat{p}}_{i}^{\mu}n^{\nu}{\hat{q}}^{\rho}{\partial}_{a}{\hat{q}}^{\sigma}{\epsilon}_{\mu\nu\rho{\sigma}\mu_{1}\cdots\mu_{d-2}}\int
d^{d-2}{\sigma}e^{-iq\cdot
r_{i}({\sigma})}{\partial}_{{\sigma}^{1}}r_{i}^{\mu_{1}}\cdots{\partial}_{{\sigma}^{d-2}}r_{i}^{\mu_{d-2}}\;.\end{split}$
(3.31)

This suggests that the soft factor has a simple pole at ${\omega}=0$. However,
if the brane has a non-compact spatial volume then the integral can produce
additional factors of ${\omega}^{-1}$. To see this, consider a brane localized
along the $(x^{3},\cdots,x^{d})$ directions. We describe this by the spatial
vector $r^{\mu}({\sigma})=(0,0,0,{\sigma}^{1},\cdots,{\sigma}^{d-2},0)$. In
this case, the soft factor works out to be

$\begin{split}S^{\text{brane}}_{a_{\perp}}(x)=\frac{(d-1)!}{{\omega}^{d-1}}{\epsilon}_{a_{\perp}b_{\perp}}{\partial}^{b_{\perp}}\log[-{\hat{p}}\cdot{\hat{q}}(x_{\perp})]{\delta}^{(d-2)}(x_{\|})\;,\qquad
S^{\text{brane}}_{a_{\|}}(x)=0\;,\end{split}$ (3.32)

where $x^{a}_{\|}=(0,0,x^{3},\cdots,x^{d})$ and
$x^{a}_{\perp}=(x^{1},x^{2},0,\cdots,0)$ describe the longitudinal and
transverse directions to the brane and $a_{\perp}=(1,2)$,
$a_{\|}=(3,\cdots,d)$ are the corresponding indices.

The soft factor therefore diverges as ${\omega}^{-(d-1)}$. This leading
divergence can be extracted by defining a new soft mode

$\begin{split}S^{\text{dual}}_{a}(x)\equiv\frac{1}{e}\oint_{\mathcal{C}}\frac{d{\omega}}{2\pi
i}{\omega}^{d-2}{\mathcal{O}}_{a}({\omega},x)\;.\end{split}$ (3.33)

This is the $n=d-1$ mode in the class of Mellin transformed operators defined
by the compact contour (2.22) and has scaling dimension ${\Delta}=d-1$. The
magnetic soft theorem in $d>2$ reads

$\begin{split}\langle\,S^{\text{dual}}_{a}(x){\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle=\frac{2\pi}{e^{2}}\sum_{i\,\in\,\text{brane}}P_{i}\,S^{(0)\text{brane}}_{i,a}(x)\,\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle\;,\end{split}$
(3.34)

with

$\begin{split}S^{(0)\text{brane}}_{i,a}(x)=\lim_{{\omega}\to
0}{\omega}^{d-1}S^{\text{brane}}_{i,a}(x)\;.\end{split}$ (3.35)

It should be possible to construct a higher-form current (sourced by extended
objects in CFTd) from this soft operator following the construction of (2.41).
We leave this for future work.

##  4 A Boundary Model for Bulk Infrared Physics

In this section, we exploit the asymptotic symmetry structure of gauge and
gravitational theories to construct a boundary ($d$-dimensional) theory whose
correlators reproduce bulk ($(d+2)$-dimensional) infrared physics. Scattering
amplitudes in the bulk theory are calculated by a path integral

$\begin{split}\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\mu}=\int[d\varphi]e^{iS_{\text{bulk}}[\varphi]}{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\;,\end{split}$
(4.1)

where $\varphi$ collectively denotes all the fields in the theory (including
any massless photons, gluons or gravitons). The subscript $\mu$ denotes the IR
cut-off used to regulate and evaluate the path integral on the RHS (all fields
in the path integral are taken to have ${\omega}>\mu$).

In order to isolate the infrared physics in the path integral, we separate all
fields into a soft piece $\varphi_{s}$ (which has $\mu<|{\omega}|<{\Lambda}$)
and a hard piece $\varphi_{h}$ (which has $|{\omega}|>{\Lambda}$). The scales
$\mu$, ${\Lambda}$ are taken to be much smaller than the gap in the massive
sector so the soft fields $\varphi_{s}$ only consist of low energy massless
fields. In models with hard-soft factorization we can integrate out the hard
fields

$\begin{split}\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\mu}&=\int[d\varphi_{s}]\int[d\varphi_{h}]e^{iS_{\text{bulk}}[\varphi_{s},\varphi_{h}]}{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\\\
&=\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\Lambda}\int[d\varphi_{s}]e^{-S_{\text{soft}}[\varphi_{s}]-S_{\text{int}}[\varphi_{s},j]}\;.\end{split}$
(4.2)

The integral over the hard modes produces the hard amplitude
$\langle\,{\mathcal{O}}_{1}\cdots{\mathcal{O}}_{n}\,\rangle_{\Lambda}$ which
does not depend on any of the soft fields or the infrared cutoff $\mu$. The
left-over integral over the soft modes contains an effective action which we
have separated into two parts – $S_{\text{soft}}$ describes the self-
interactions of the soft modes and $S_{\text{int}}$ describes the coupling of
the long-wavelength fluctuations to the hard external particles in the
scattering amplitude. This interaction is encoded through a classical (non-
fluctuating) hard matter current $j$ whose structure is independent of the
microscopic details of the theory. Comparing to the general structure of
infrared divergences (2.65), one finds

$\begin{split}\int[d\varphi_{s}]e^{-S_{\text{soft}}[\varphi_{s}]-S_{\text{int}}[\varphi_{s},j]}=e^{-{\Gamma}}\;.\end{split}$
(4.3)

The above analysis assumes that the operators ${\mathcal{O}}_{i}$ are all
chosen from the hard sector. Scattering amplitudes with external soft
particles require additional soft insertions $S_{i}(x)$ in the path integral.
The multi-soft theorems guarantee that the corresponding amplitude factorizes
so we require

$\begin{split}\int[d\varphi_{s}]e^{-S_{\text{soft}}[\varphi_{s}]-S_{\text{int}}[\varphi_{s},j]}S_{1}\cdots
S_{m}=e^{-{\Gamma}}{\mathcal{J}}_{1}\cdots{\mathcal{J}}_{m}\;,\end{split}$
(4.4)

where $S_{i}$ denotes a generic soft insertion and ${\mathcal{J}}_{i}$ denotes
the corresponding soft factor.

As we will show in the rest of this section, the action for the soft modes and
their coupling to the matter sources is essentially fixed by symmetry
considerations. The coupling constants of the path integral on the LHS of
(4.4) are completely fixed by the soft theorem and the one-loop infrared
divergence. It appears nontrivial that the intrinsically Euclidean path
integral (4.4) is capable of reproducing intrinsically Lorentzian infrared
dynamics.

###  4.1 Broken Symmetries in Celestial CFT

Symmetries in quantum field theory can be approximate or exact. If they are
exact, they can be spontaneously broken or unbroken, depending on the nature
of the vacuum state. In some cases, it is useful to treat an approximate
symmetry as if it were exact. If the model with the exact symmetry lies in the
broken phase and if the source of explicit symmetry breaking is small, then
the pseudo-Goldstone bosons (along with any other massless fields) can be used
to reliably approximate the low-energy dynamics.

There is by now a standard set of claims in the literature (see [41] and
references therein) regarding the symmetry structure of gauge theory and
quantum gravity in asymptotically flat space. Gauge transformations or
diffeomorphisms with non-compact support that respect appropriate boundary
conditions act non-trivially on the Hilbert space in these models. We denote
this infinite-dimensional asymptotic symmetry group of the theory by
${\mathcal{G}}$. Gauge transformations or diffeomorphisms which are
asymptotically “constant” (in an appropriate sense) represent the global part
of ${\mathcal{G}}$ which we denote by $G$. The existence of the
${\mathcal{G}}$ symmetry implies an infinite degeneracy of vacuum states in
the model. These vacuum states are preserved by $G$ but not by
${\mathcal{G}}$. It follows that in these models, the exact symmetry
${\mathcal{G}}$ is spontaneously broken down to $G$. The low energy dynamics
of these models are described by the corresponding Goldstone bosons.

It is important to remember that the group ${\mathcal{G}}$ and its subgroup
$G$ are only defined in spacetimes with non-compact Cauchy slices. There are
no charged states for $G$ or Goldstone bosons for ${\mathcal{G}}/G$ on a
compact manifold, just as there is no spontaneous symmetry breaking of
ordinary symmetries in finite volume. In order to recover ${\mathcal{G}}$ and
$G$ from an infinite volume limit, it is important to take a limit of space-
like slices with boundary. In this limit, the corresponding edge modes become
the Goldstone modes for ${\mathcal{G}}/G$. This point is important precisely
because gauge theories suffer from infrared issues. Perturbative calculations
in gauge theory and gravity require the regulation of long-wavelength
fluctuations. Any method of regulating the infrared divergences (e.g. a small
photon mass, $d>4$ dimensional regularization, finite volume regularization,
etc.) explicitly breaks the ${\mathcal{G}}$ symmetry.999In the case of
dimensional regularization, the group itself changes. The effective action for
these Goldstone modes in the regulated theory will therefore contain symmetry
breaking terms, and we are forced to consider pseudo-Goldstone bosons
associated to a symmetry with both spontaneous and explicit
breaking.101010This is also the case in JT gravity, which describes the
pseudo-Goldstone bosons associated to the IR divergence in AdS2 [46]. If the
symmetry breaking terms vanish as the regulator is removed, then we expect to
land on a model with an exact $G$ symmetry and Goldstone modes for the coset
${\mathcal{G}}/G$. This will be the case when $d>2$ and when the gauge theory
is in the Coulomb phase. However, as we will see, when $d=2$ the pseudo-
Goldstone modes become strongly coupled as the regulator is removed and we
must keep the leading symmetry breaking terms in the effective description.
This strong coupling behavior is a two-dimensional reflection of the infrared
divergence in $D=4$ spacetime dimensions. The four-dimensional Goldstone
bosons are strongly coupled because two-dimensional Goldstone bosons are
strongly interacting.

Since the celestial CFTd formalism is a repackaging of the bulk
$(d+2)$-dimensional dynamics, we expect the asymptotic symmetry structures to
constrain the $d$-dimensional model. When the bulk theories are in the Coulomb
phase, we expect a global $G$ symmetry in the CFTd and a corresponding
conserved current. These are precisely the operators (2.40), (2.50) and (2.60)
constructed from the shadow transforms of the soft operators. Their existence
is required in order to match global symmetries and the universal nature of
soft limits guarantees that they generate the correct symmetry
transformations. We also expect to be able to phrase the dynamics of the
${\mathcal{G}}/G$ Goldstone bosons simply in $d$-dimensional language since it
is also controlled by symmetries.

###  4.2 $U(1)$ Gauge Theory

We will begin by applying the above discussion to the case of $U(1)$ gauge
theories. Goldstone actions for infinite-dimensional symmetry groups like
$\mathcal{G}$ are somewhat exotic, so we begin by reviewing the finite-
dimensional case of global $U(1)$ symmetry breaking and then proceed by
analogy.

The simplest model of a spontaneously broken global $U(1)$ symmetry involves a
scalar field with a Lagrangian invariant under the transformation $\phi(x)\to
e^{iQ{\varepsilon}}\phi(x)$. If the scalar potential has a global minimum with
$|\phi_{\text{min}}|=v\neq 0$, then the boundary value of the field at spatial
infinity cannot vanish and the field acquires a vacuum expectation value set
by the boundary condition at infinity: $\langle\,\phi(x)\,\rangle=v\neq 0$. In
this case $\phi(x)$ is the order parameter, and its boundary condition
spontaneously breaks the $U(1)$ symmetry of the model. The normalizable
fluctuations about this vacuum are described by the field redefinition
$\phi(x)=(v+h(x))e^{iQ{\theta}(x)/v}$. Excitations of $h(x)$ are gapped while
the Goldstone mode ${\theta}(x)$ is massless and protected by the non-linearly
realized $U(1)$ symmetry ${\theta}(x)\to{\theta}(x)+{\varepsilon}$. At low
energies, only the Goldstone mode survives and the dynamics is described by an
action of the form $\int[-\frac{1}{2}({\partial}{\theta})^{2}+\cdots]$ which
is constrained to be invariant under the $U(1)$ symmetry. If the symmetry is
also (weakly) explicitly broken in the ultraviolet, then one must add small
symmetry breaking terms to this action, e.g. a mass term for ${\theta}$.

In the case of a $U(1)$ gauge theory, the relevant symmetry group is the
“asymptotic symmetry group” $\mathcal{G}$, also known as the group of “large”
gauge transformations with non-compact support [10, 47, 48, 49, 50]. Elements
of $\mathcal{G}$ correspond to maps $S^{d}\to U(1)$ and the relevant order
parameter is a Wilson line on the celestial sphere

$\begin{split}W(x)\equiv\exp\left(i\int_{x_{0}}^{x}C\right)\;,\qquad
C=A|_{{\partial}{\mathscr{I}}}\;.\end{split}$ (4.5)

Here $x_{0}$ is an arbitrarily chosen base point. This order parameter is
invariant under constant gauge transformations, but rotates with a phase under
generic elements of $\mathcal{G}$:

$\begin{split}W(x)~{}\stackrel{{\scriptstyle{\mathcal{G}}}}{{\longrightarrow}}~{}e^{i({\varepsilon}(x)-{\varepsilon}(x_{0}))}W(x)\;,\qquad{\varepsilon}(x)\sim{\varepsilon}(x)+2\pi\;.\end{split}$
(4.6)

When the quantum theory is in the Coulomb phase, this order parameter is non-
vanishing, $\langle\,W(x)\,\rangle\neq 0$, which indicates that the
${\mathcal{G}}$ symmetry is spontaneously broken down to $G$. The Goldstone
mode for the broken symmetry is $C$ and it transforms non-linearly as $C\to
C+d{\varepsilon}$.

We now turn to the structure of the low energy action, $S_{\text{soft}}$,
which describes the self-interactions of the soft modes. The first thing to
note is that the soft photon operator $S$ in (2.33) is independent of the
Goldstone mode $C$. In particular, while $C$ transforms under large gauge
transformations, $S$ is related to gauge-invariant matrix elements and
therefore does not transform. Since the operator $S$ creates a soft photon, it
survives in the low energy limit so $S_{\text{soft}}$ depends on both $C$ and
$S$. We will begin by considering the case in which all external states in the
scattering amplitude are hard. In this case, we can integrate out $S$ and
obtain an effective action for the mode $C$. Having determined this effective
action, we will then “integrate in” the mode $S$ by consideration of the soft
photon theorem.

In the absence of magnetic charges the gauge field edge mode $C$ is flat and
realizes the $\mathcal{G}$ symmetry $C\to C+d\varepsilon$ non-linearly. When
the $\mathcal{G}$ symmetry is exact but spontaneously broken, the effective
action for $C$ must be invariant under this transformation. What sort of
action is invariant under the infinite-dimensional symmetry group
$\mathcal{G}$? The simplest example is of course a model with $d$-dimensional
abelian gauge invariance in which we do not quotient by gauge transformations.
Since $C$ is flat, we do not want to integrate over all abelian connections,
only those which are flat. A $d$-dimensional $BF$ theory has the appropriate
domain of integration and the correct symmetries111111This $BF$ theory differs
from the models usually considered in both form and function. It does not
involve the usual quotient by “small gauge transformations,” but is viewed as
part of a higher-dimensional theory.

$\begin{split}S_{BF}[B,C]=i\int B\wedge F\;,\qquad F=dC\;.\end{split}$ (4.7)

Integrating out $B$ sets $F=0$ and the soft dynamics are effectively trivial.
However, as we have noted previously, the infrared regulator explicitly breaks
the ${\mathcal{G}}$ symmetry so the effective action can also contain symmetry
breaking terms. These correspond to non-gauge invariant interactions that
depend explicitly on $C$ rather than $F$.

In standard constructions of effective actions for pseudo-Goldstone bosons,
locality is an organizing principle. In the case at hand, we are attempting to
construct an exotic “interdimensional effective field theory” that calculates
bulk $(d+2)$-dimensional quantities in terms of $d$-dimensional field
variables. While the bulk theory must be local, the boundary theory need not
be. Indeed, given that the boundary theory is required to reproduce long
wavelength effects in the bulk, the theory could indeed be non-local. With
this in mind, the most general leading symmetry breaking term that we can add
to the action takes the form

$\begin{split}S_{\text{soft}}[C]=\int
d^{d}xd^{d}y\,(P^{-1})^{ab}(x-y)C_{a}(x)C_{b}(y)\;.\end{split}$ (4.8)

$P_{ab}(x-y)$ is the propagator for $C$ and will be determined in the next
section. The choice $P_{ab}=\delta_{ab}$ would be a mass term for the
$d$-dimensional gauge field. Integrating out the $B$ field still enforces the
flatness constraint

$\begin{split}C=d{\theta}\;,\qquad{\theta}(x)\sim{\theta}(x)+c\;,\end{split}$
(4.9)

and the mass term for the gauge field becomes a kinetic term for the gauge
parameter $\theta(x)$. The identification is required because the global $G$
symmetry is unbroken on the vacuum: $C$ is a Goldstone with target
$\mathcal{G}/G$ so the zero mode of $\theta(x)$ is a redundant
parameterization of the space of flat connections and should not be integrated
over. Under large gauge transformations,

$\begin{split}{\theta}(x)~{}~{}\stackrel{{\scriptstyle{\mathcal{G}}}}{{\longrightarrow}}~{}~{}{\theta}(x)+{\varepsilon}(x)\;.\end{split}$
(4.10)

The action (4.8) explicitly breaks the $\mathcal{G}$ symmetry but is invariant
under the global $U(1)$ transformations.

Next, we turn to the form of the interaction term $S_{\text{int}}$. The
factorized structure of scattering amplitudes suggests that the scattering
operators ${\mathcal{O}}_{i}$ factorize into a soft and a hard piece:

$\begin{split}{\mathcal{O}}_{i}({\omega}_{i},x_{i})=U_{i}[{\theta}]{\mathcal{O}}_{i}^{H}({\omega}_{i},x_{i})\;.\end{split}$
(4.11)

The hard operators ${\mathcal{O}}^{H}$ depend only on the hard fields
$\varphi_{h}$ and decouple from the soft path integral (4.2), although they
obviously contribute to the hard scattering amplitude. The soft operator
$U_{i}[C]$ can be determined by considering the large gauge transformation of
${\mathcal{O}}_{i}$ which takes the form121212We assume for now that
${\mathcal{O}}_{i}$ is purely electrically charged. The generalization to
incorporate dyons will be considered in section 4.2.1.

$\begin{split}{\mathcal{O}}_{i}({\omega}_{i},x_{i})~{}~{}\stackrel{{\scriptstyle{\mathcal{G}}}}{{\longrightarrow}}~{}~{}\exp\left[iQ_{i}\int
d^{d}x{\varepsilon}(x){\mathcal{K}}_{d}(m_{i}/{\omega}_{i},x_{i};x)\right]{\mathcal{O}}_{i}({\omega}_{i},x_{i})\;.\end{split}$
(4.12)

This transformation law is determined in [10, 47] from a spacetime analysis.
It can also be determined from (2.42). This suggests the following explicit
form of $U_{i}$

$\begin{split}U_{i}[{\theta}]\equiv\exp\left[iQ_{i}\int
d^{d}x{\theta}(x){\mathcal{K}}_{d}(m_{i}/{\omega}_{i},x_{i};x)\right]\;.\end{split}$
(4.13)

Note that this operator is not single-valued under the identification in
(4.9). Instead, we identify

$\begin{split}U_{i}[{\theta}]\sim
U_{i}[{\theta}]e^{iQ_{i}{\theta}_{0}}\end{split}$ (4.14)

which follows from the property

$\begin{split}\int
d^{d}x{\mathcal{K}}_{d}(m_{i}/{\omega}_{i},x_{i};x)=1\;.\end{split}$ (4.15)

Including one such soft operator insertion for each of the operators
${\mathcal{O}}_{i}$ appearing in the scattering amplitude, we find

$\begin{split}S_{\text{int}}[{\theta},j]&=-i\int
d^{d}x{\theta}(x)\sum_{i}Q_{i}{\mathcal{K}}_{d}(m_{i}/{\omega}_{i},x_{i};x)\;.\end{split}$
(4.16)

Note that due to conservation of charge, the interaction action is single-
valued under (4.9), even though the operators $U_{i}$ are not. Using the
definition of $j^{a}(x)$ in (2.41), the action can be written

$\begin{split}S_{\text{int}}[{\theta},j]&=i\int
d^{d}xC_{a}(x)j^{a}(x)\;,\qquad C=d{\theta}\;.\\\ \end{split}$ (4.17)

Having fixed the form of $S_{\text{soft}}$ and $S_{\text{int}}$ we can
evaluate the path integral in (4.3). The propagator $P_{ab}$ will then be
fixed by matching onto the RHS of (4.3). For $U(1)$ gauge theories, the
condition is

$\begin{split}\int[d{\theta}]e^{-S_{\text{soft}}[{\theta}]-S_{\text{int}}[{\theta},j]}=e^{-{\Gamma}_{\text{ph}}}=e^{-{\alpha}(A_{1}+2\pi
iA_{2})}\;,\end{split}$ (4.18)

where

$\begin{split}A_{1}=\int\frac{d^{d}x}{(2\pi)^{d}}{\mathcal{J}}^{a}(x){\mathcal{J}}_{a}(x)\;,\qquad{\alpha}=\frac{e^{2}}{8\pi}\int_{\mu}^{\Lambda}d{\omega}{\omega}^{d-3}\;.\end{split}$
(4.19)

At this stage, we are unable to reproduce the imaginary term $A_{2}$ from this
construction and consequently we postpone a discussion of this to future work.
The path integral on the LHS is Gaussian and can be evaluated explicitly

$\begin{split}e^{-{\Gamma}_{\text{ph}}}&=\int[d{\theta}]\exp\left[-\int
d^{d}xd^{d}y\,(P^{-1})^{ab}(x-y)C_{a}(x)C_{b}(y)-i\int
d^{d}xC_{a}(x)j^{a}(x)\right]\;,\\\ &=\exp\left[-\frac{1}{4}\int
d^{d}xd^{d}yP_{ab}(x-y)j^{a}(x)j^{b}(y)\right]\;.\end{split}$ (4.20)

Comparing this to (4.18) and using (2.41), we find that we can reproduce
$A_{1}$ correctly if

$\begin{split}P_{ab}(x-y)=\frac{16{\alpha}}{(2\pi)^{d}}\int
d^{d}w\frac{{\mathcal{I}}_{ac}(w-x)}{(w-x)^{2}}\frac{{\mathcal{I}}^{c}{}_{b}(w-y)}{(w-y)^{2}}\;.\end{split}$
(4.21)

This follows from the formal properties of the inverse shadow transform
combined with (2.41). To construct the action, we need to invert the
propagator which satisfies

$\begin{split}\int
d^{d}w(P^{-1})^{ac}(w-x)P_{cb}(w-y)={\delta}^{a}_{b}{\delta}^{(d)}(x-y)\;.\end{split}$
(4.22)

It follows that

$\begin{split}(P^{-1})^{ab}(x-y)=\frac{(2\pi)^{d}}{16c_{1,1}^{2}{\alpha}}\int
d^{d}w\frac{{\mathcal{I}}^{ac}(w-x)}{[(w-x)^{2}]^{d-1}}\frac{{\mathcal{I}}_{c}{}^{b}(w-y)}{[(w-y)^{2}]^{d-1}}\;.\end{split}$
(4.23)

The soft action is therefore

$\begin{split}S_{\text{soft}}[{\theta}]&=\frac{(2\pi)^{d}}{16c_{1,1}^{2}{\alpha}}\int
d^{d}w\int
d^{d}xd^{d}y\frac{{\mathcal{I}}^{ac}(w-x){\mathcal{I}}_{c}{}^{b}(w-y)}{[(w-x)^{2}]^{d-1}[(w-y)^{2}]^{d-1}}C_{a}(x)C_{b}(y)\\\
&=\frac{(2\pi)^{d}}{16c_{1,1}^{2}{\alpha}}\int
d^{d}x{\widetilde{C}}^{a}(x){\widetilde{C}}_{a}(x)\;.\end{split}$ (4.24)

Remarkably, although the pseudo-Goldstone action appears non-local when
written in terms of the edge mode $C$, the shadow edge mode ${\widetilde{C}}$
behaves like a local degree of freedom. This may suggest that the
“appropriate” set of local operators in celestial CFTd are the shadow
transforms of bulk operators, and we hope to explore this possibility in
future work. To conclude, we can also express the soft-interaction term using
the shadow mode

$\begin{split}S_{\text{int}}[{\theta},j]=i\int
d^{d}xC_{a}(x)j^{a}(x)=\frac{i}{2c_{1,1}}\int
d^{d}x{\widetilde{C}}_{a}(x){\mathcal{J}}^{a}(x)\;.\end{split}$ (4.25)

This local action for the shadow edge mode reproduces the infrared divergences
in abelian gauge theory scattering amplitudes.

To end this discussion, we now consider scattering amplitudes with external
soft photons. For this purpose, we are required to “integrate in” the soft
photon operator $S=d\phi$. More precisely, we wish to determine an action
$S_{\text{soft}}[{\theta},\phi]$ such that

$\begin{split}\int[d\phi]e^{-S_{\text{soft}}[{\theta},\phi]}=e^{-S_{\text{soft}}[{\theta}]}\;,\qquad\int[d\phi][d{\theta}]e^{-S_{\text{soft}}[{\theta},\phi]-S_{\text{int}}[{\theta},j]}S_{1}\cdots
S_{m}=e^{-{\Gamma}}{\mathcal{J}}_{1}\cdots{\mathcal{J}}_{m}\;.\end{split}$
(4.26)

Note that the zero mode of $\phi$ does not appear in the physical operator $S$
and so is not part of the system described by
$S_{\text{soft}}[{\theta},\phi]$. We therefore gauge the symmetry
$\phi(x)\sim\phi(x)+c$.

The soft action can be determined as follows. The shadow transform of $S$ is
an abelian current (2.40). This operator must generate the global $U(1)$
transformations $\theta\to\theta+\varepsilon$. The usual Noether procedure
then implies an off-diagonal coupling of the form

$\begin{split}S_{\text{soft}}[{\theta},\phi]&=S^{\prime}_{\text{soft}}[\phi]-i\int
d^{d}xC_{a}(x)J^{a}(x)=S^{\prime}_{\text{soft}}[\phi]-\frac{i}{2c_{1,1}}\int
d^{d}x{\widetilde{C}}_{a}(x)S^{a}(x)\;.\end{split}$ (4.27)

$S_{\text{soft}}^{\prime}$ is fixed by imposing (4.26). The required term is
Gaussian:

$\begin{split}S^{\prime}_{\text{soft}}[\phi]=\frac{{\alpha}}{(2\pi)^{d}}\int
d^{d}xS^{a}(x)S_{a}(x)\;,\qquad S=d\phi\;.\end{split}$ (4.28)

The full soft action is therefore

$\begin{split}S_{\text{soft}}[{\theta},\phi]+S_{\text{int}}[{\theta},j]=\frac{{\alpha}}{(2\pi)^{d}}\int
d^{d}xS^{a}(x)S_{a}(x)-\frac{i}{2c_{1,1}}\int
d^{d}x{\widetilde{C}}_{a}(x)[S^{a}(x)-{\mathcal{J}}^{a}(x)]\;,\end{split}$
(4.29)

with

$\begin{split}C=d{\theta}\;,\qquad
S=d\phi\;,\qquad\phi(x)\sim\phi(x)+c\;,\qquad{\theta}(x)\sim{\theta}(x)+c\;.\end{split}$
(4.30)

This path integral correctly reproduces insertions of the soft photon operator

$\begin{split}\langle\,S_{a_{1}}(y_{1})\cdots
S_{a_{m}}(y_{m})\,\rangle=\int[d{\theta}][d\phi]e^{-S_{\text{soft}}[{\theta},\phi]-S_{\text{int}}[{\theta},j]}{\partial}_{a_{1}}\phi(y_{1})\cdots{\partial}_{a_{m}}\phi(y_{m})\;.\end{split}$
(4.31)

The integral over ${\theta}(x)$ gives a Dirac delta function
${\delta}(S-{\mathcal{J}})$ which then localizes the integral over $\phi$. We
find

$\begin{split}\langle\,S_{a_{1}}(y_{1})\cdots
S_{a_{m}}(y_{m})\,\rangle=\exp\left[-\frac{{\alpha}}{(2\pi)^{d}}\int
d^{d}x{\mathcal{J}}^{a}(x){\mathcal{J}}_{a}(x)\right]{\mathcal{J}}_{a_{1}}(y_{1})\cdots{\mathcal{J}}_{a_{m}}(y_{m})\end{split}$
(4.32)

which is (almost131313As mentioned previously, we do not yet understand how
the imaginary part of ${\Gamma}_{\text{ph}}$ is reproduced by the soft action
and we leave this to future work.) the expected result (4.4)!

####  4.2.1 Magnetic Charges and Winding Modes in $d=2$

We have so far demonstrated that the action (4.29) reproduces all infrared
divergences _and_ soft theorems under the assumption that all the particles in
the scattering process are purely electrically charged. In this section, we
generalize the result to include dyonic scattering states in four dimensions.
In order to simplify the discussion, we will assume that all hard particles
are massless.

In $d=2$ the shadow transform of a flat connection localizes, and the
Goldstone mode is proportional to its shadow:

$\begin{split}\widetilde{{\partial}_{a}{\theta}}(x)=\int
d^{2}x\frac{{\mathcal{I}}_{ab}(x-y)}{(x-y)^{2}}{\partial}^{b}{\theta}(y)=2\pi{\partial}_{a}{\theta}(x)\;.\end{split}$
(4.33)

The action (4.29) then reduces to

$\begin{split}S_{\text{soft}}[{\theta},\phi]&=\frac{1}{4\pi^{2}}\int[{\alpha}d\phi{\wedge}\star
d\phi-\pi i\,d{\theta}{\wedge}\star d\phi]\;.\end{split}$ (4.34)

The soft correlators for dyonic charges are given by (3.10) and (3.14) :

$\begin{split}\langle\,S_{a_{1}}(y_{1})\cdots
S_{a_{m}}(y_{m})\,\rangle&=[{\mathcal{J}}_{a_{1}}(y_{1})+{\widetilde{\mathcal{J}}}_{a_{1}}(y_{1})]\cdots[{\mathcal{J}}_{a_{m}}(y_{m})+{\widetilde{\mathcal{J}}}_{a_{m}}(y_{m})]\\\
&\qquad\qquad\qquad\qquad\times\exp\left[-\frac{{\alpha}}{(2\pi)^{d}}\int
d^{d}x\left[{\mathcal{J}}_{a}(x)+{\widetilde{\mathcal{J}}}_{a}(x)\right]^{2}\right]\end{split}$
(4.35)

where

${\mathcal{J}}_{a}(x)={\partial}_{a}\sum_{i}Q_{i}\log[-{\hat{p}}_{i}\cdot{\hat{q}}(x)]\;,\qquad{\widetilde{\mathcal{J}}}_{a}(x)\equiv\frac{2\pi}{e^{2}}{\epsilon}_{ab}{\partial}^{b}\sum
P_{i}\log[-{\hat{p}}_{i}\cdot{\hat{q}}(x)]\;.$ (4.36)

It is easy to see that the soft action (4.29) with an additional source

$\begin{split}S_{\text{soft}}[{\theta},\phi]+S_{\text{int}}[{\theta},j]=\frac{{\alpha}}{(2\pi)^{2}}\int
d^{2}xS^{a}(x)S_{a}(x)-\frac{i}{4\pi}\int
d^{2}xC_{a}(x)[S^{a}(x)-{\mathcal{J}}^{a}(x)-{\widetilde{\mathcal{J}}}_{a}(x)]\;\end{split}$
(4.37)

correctly reproduces these correlation functions. The new interaction term
$\widetilde{{\mathcal{J}}}^{a}C_{a}(x)$ can be interpreted as a background
source for the topological winding current
$\epsilon_{ab}{\partial}^{b}\theta$, just as the original term
${\partial}_{a}\theta{\mathcal{J}}^{a}$ may be interpreted as a background
source for the momentum current ${\partial}_{a}\theta$. In this way, the
electric and magnetic charges of the abelian gauge theory are mapped to the
momentum and winding charges of the vertex operators in the compact boson
theory. The irrational dyonic spectrum in the presence of the $\vartheta$ term
is simply reproduced by a background Kalb-Ramond field. The coupling
$iB_{\theta\phi}d\theta\wedge d\phi$ simply changes the spectrum of allowed
charges in the soft sector, but does not alter the dynamics. The final soft
action takes the form

$\begin{split}S_{\text{soft}}[{\theta},\phi]&=-\frac{1}{4\pi^{2}}\int\left[{\alpha}d\phi{\wedge}\star
d\phi-\pi i\,d{\theta}{\wedge}\star\left(d\phi+\frac{\tau_{1}}{\tau_{2}}\star
d\phi\right)\right]\;.\end{split}$ (4.38)

The structure of this term can actually be determined using the bulk
${\vartheta}$-term

$\begin{split}iS_{\text{bulk}}\ni\frac{i{\vartheta}}{8\pi^{2}m_{0}}\int
d(A\wedge F)\;.\end{split}$ (4.39)

This is a boundary term and its contribution to $S_{\text{soft}}$ can be
determined by integrating over ${\mathscr{I}}$

$\begin{split}S_{\text{soft}}\ni\int_{{\mathscr{I}}}A{\wedge}F\sim\int
d{\theta}{\wedge}d\phi\;.\end{split}$ (4.40)

###  4.3 Non-abelian Gauge Theory

The symmetry-based part of the previous discussion generalizes
straightforwardly to non-abelian gauge theories. The group $\mathcal{G}$ of
maps $S^{d}\to G$ is spontaneously broken on the perturbative vacuum, and the
corresponding order parameter is a non-abelian boundary Wilson line

$\begin{split}W(x)={\mathcal{P}}\left(\exp\int_{x_{0}}^{x}C\right)\;,\qquad
C=A|_{{\partial}{\mathscr{I}}}\;.\end{split}$ (4.41)

Under large gauge transformations

$\begin{split}W(x)\to g(x)W(x)g(x_{0})^{-1}\;,\qquad g(x)\in G\;.\end{split}$
(4.42)

In the quantum theory, the Wilson line has a non-vanishing vacuum expectation
value $\langle\,W(x)\,\rangle\neq 0$ when the gauge theory is in the Coulomb
phase. Global gauge transformations leave the trace of the order parameter
invariant, so the large gauge symmetry is broken down to global $G$ symmetry
and the corresponding Goldstone mode is the flat connection $C=UdU^{-1}$. In
the absence of explicit breaking by the infrared regulator, the Goldstone
action must be invariant under the $d$-dimensional local invariance group
$\mathcal{G}$ and is therefore a gauge theory without a quotient by the orbit
of the gauge group. The flatness condition is imposed by a non-abelian $BF$
term of the form $i\int\text{tr}\left[B{\wedge}F\right]$. The symmetry
breaking term is the non-abelian generalization of the non-local vector mass
term (4.8)

$\begin{split}S_{\text{soft}}[U]=\int
d^{d}xd^{d}y(P^{-1})^{ab}(x-y)\text{tr}\left[C_{a}(x)C_{b}(y)\right]\;,\qquad
C=UdU^{-1}\;.\end{split}$ (4.43)

In $d=2$, there is another possible symmetry breaking term that would give an
action to the pseudo-Goldstones, namely the WZW term, and we expect such a
term to follow from the addition of a non-abelian $\vartheta$-term in the bulk
Lagrangian. Higher-dimensional $\vartheta$-terms similarly correspond to
higher-dimensional WZW terms.

The interaction term $S_{\text{int}}$ is also determined by symmetries. To
describe this, we define the group element $U(x)=e^{-{\theta}(x)}$ where
${\theta}(x)\equiv{\theta}^{I}(x)T^{I}\in{\mathfrak{g}}$. We define the
operator $U_{i}[{\theta}]$ as

$\begin{split}U_{i}[{\theta}]=R_{i}\left(\exp\left[\int
d^{d}x{\theta}(x){\mathcal{K}}_{d}(m_{i}/{\omega}_{i},x_{i};x)\right]\right)\;.\end{split}$
(4.44)

The interaction term $e^{-S_{\text{int}}}$ is obtained as a product of these
operators, one for each of the asymptotic states in the amplitude.

The discussion so far is completely general and used only the symmetries of
the problem. The main complication arises in the “interdimensional effective
field theory” matching condition. In order to determine the propagator
entering in (4.43), we need to match on to the non-abelian infrared
divergence, which is not known exactly and is severely more complicated than
the abelian case. Similarly, the soft gluon theorem receives perturbative
corrections so it is not as simple to “integrate in” the external soft
operator. To make progress we have to revert to perturbation theory, keeping
in mind that the soft dynamics is ultimately non-perturbative. It is well
known that the model defined by (4.43) with a local propagator (the principal
chiral model in two dimensions) is also strongly coupled, and that
perturbation theory about the free point does not accurately capture the
dynamics and symmetry restoration (Coleman-Wagner theorem). Ultimately, one
hopes to apply non-perturbative knowledge of the strong coupling behavior in
the principal chiral model to the much less understood strong coupling problem
in Yang Mills. This is left to future work. Upon linearizing the problem

$U(x)=1-{\theta}(x)+\dots$ (4.45)

the action (4.43) describes a theory of (free) scalars

$\begin{split}S_{\text{soft}}[{\theta}]=\int
d^{d}xd^{d}y(P^{-1})^{ab}(x-y)\text{tr}\left[{\partial}_{a}{\theta}(x){\partial}_{b}{\theta}(y)\right]\;.\end{split}$
(4.46)

This linearized action (with a local propagator) was recently used in [5] to
match onto and reproduce low orders in four-dimensional Yang-Mills
perturbation theory. It seems natural to conjecture that the non-linear
completion of that model

$S_{\text{soft}}[{\theta},\phi]=\int
d^{d}x\,\text{tr}\left[\frac{{\alpha}}{(2\pi)^{d}}S^{a}(x)S_{a}(x)-\frac{i}{2c_{1,1}}S_{a}(x){\widetilde{C}}^{a}(x)\right]$
(4.47)

describes higher orders in perturbation theory. The interaction term would
then be obtained through insertions of the operators (4.44). This path
integral cannot be performed exactly, but could be compared at any order in
perturbation theory. We hope to investigate this possibility in future work.

###  4.4 Gravity: Supertranslation Mode

The asymptotic symmetry group for gravitational theories is the BMS group [51,
52, 9, 53]. The implementation of this symmetry in the celestial CFT is
slightly complicated given that one is trying to realize higher-dimensional
spacetime symmetries as internal symmetries in a lower-dimensional model. The
BMS group consists of $SO(1,d+1)$ Lorentz transformations as well as
supertranslations, which are an infinite-dimensional extension of the $(d+2)$
translation group.141414There is evidence that the Lorentz group should be
viewed as the unbroken subgroup of diffeomorphisms of the celestial sphere,
although a proper asymptotic analysis has not yet been performed and the
question remains open. The BMS symmetries are spontaneously broken down to the
Poincaré group by the perturbative vacuum. The action for the corresponding
Goldstone modes is expected to be BMS invariant in the absence of explicit
breaking.

The standard asymptotic analysis identifies the supertranslation Goldstone
mode as the leading traceless boundary metric fluctuation
$C_{ab}=(h_{ab}-\frac{1}{d}{\delta}_{ab}h)|_{{\partial}{\mathscr{I}}}$ which
satisfies the flatness condition151515As in gauge theory, (4.48) is derived
from the vanishing of the asymptotic field strength (in this case, the Weyl
tensor component $C_{arbc}|_{{\partial}{\mathscr{I}}}$).

$\begin{split}{\omega}_{abc}\equiv{\partial}_{[b}C_{c]a}-\frac{1}{d-1}{\delta}_{a[b}{\partial}^{d}C_{c]d}=0\;.\end{split}$
(4.48)

Supertranslations act non-linearly on the Goldstone mode

$\begin{split}C_{ab}(x)\to
C_{ab}(x)+2\left({\partial}_{a}{\partial}_{b}-\frac{1}{d}{\delta}_{ab}{\partial}^{2}\right)f(x)\;\end{split}$
(4.49)

but leave $\omega_{abc}$ invariant. Here $f(x)$ is any function on $S^{d}$ and
satisfies no periodicity condition. The infinite-dimensional supertranslation
symmetry is spotaneously broken to a finite-dimensional translation symmetry
which is generated by functions of the form

$\begin{split}f_{tr}(x)=\chi^{0}(1+x^{2})-2\chi^{a}x_{a}-\chi^{d+1}(1-x^{2})\end{split}$
(4.50)

which leave $C_{ab}(x)$ invariant.

Following the gauge theory construction, our first step is to find an action
which is invariant under the full supertranslation symmetry. The field
strength ${\omega}_{abc}$ is invariant under supertranslations (4.49) so the
simplest model is a “supertranslation BF theory”

$\begin{split}S=i\int d^{d}xB^{abc}{\omega}_{abc}\;.\end{split}$ (4.51)

Integrating out the $B$ field imposes the flatness condition
${\omega}_{abc}=0$ so we can write

$\begin{split}C_{ab}(x)=2\left({\partial}_{a}{\partial}_{b}-\frac{1}{d}{\delta}_{ab}{\partial}^{2}\right)C(x)\;,\qquad
C(x)\sim C(x)+f_{tr}(x)\;.\end{split}$ (4.52)

The identification is required because the global translation symmetry is
unbroken on the vacuum. Under supertranslations

$\begin{split}C(x)~{}~{}\stackrel{{\scriptstyle{\mathcal{G}}}}{{\longrightarrow}}~{}~{}C(x)+f(x)\;.\end{split}$
(4.53)

The leading symmetry breaking term gives an action to pure supertranslations
and takes the form

$S_{\text{soft}}[C]=\int
d^{d}xd^{d}y(P^{-1})^{ab,cd}(x-y)C_{ab}(x)C_{cd}(y)\;.$ (4.54)

The interaction term follows from the symmetry transformation properties of
the plane wave161616For the considerations of this section, we do not work in
the Mellin-transformed basis.  creation and annihilation operators [54]

$\begin{split}{\mathcal{O}}_{i}({\omega}_{i},x_{i})~{}~{}\stackrel{{\scriptstyle{\mathcal{G}}}}{{\longrightarrow}}~{}~{}\exp\left[\frac{i}{2}m_{i}\int
d^{d}xf(x){\mathcal{K}}_{d+1}(m_{i}/{\omega}_{i},x_{i};x)\right]{\mathcal{O}}_{i}({\omega}_{i},x_{i})\;.\end{split}$
(4.55)

In particular, under the unbroken symmetry transformations (translations)
generated by (4.50), these operators transform as plane waves

$\begin{split}{\mathcal{O}}_{i}({\omega}_{i},x_{i})~{}~{}\stackrel{{\scriptstyle
G}}{{\longrightarrow}}~{}~{}e^{-ip_{i}\cdot\chi}{\mathcal{O}}_{i}({\omega}_{i},x_{i})\;.\end{split}$
(4.56)

It then follows from (4.53) that the operator serving as a classical source
for $C(x)$ is

$\begin{split}U_{i}[C]\equiv\exp\left[\frac{i}{2}m_{i}\int
d^{d}xC(x){\mathcal{K}}_{d+1}(m_{i}/{\omega}_{i},x_{i};x)\right]\;,\qquad
U_{i}[C]\sim U_{i}[C]e^{-ip_{i}\cdot\chi}\;.\end{split}$ (4.57)

The soft-interaction term therefore takes the form

$\begin{split}S_{\text{int}}[C,j]&=-\frac{i}{2}\int
d^{d}xC(x)\sum_{i}m_{i}{\mathcal{K}}_{d+1}(m_{i}/{\omega}_{i},x_{i};x)\;.\end{split}$
(4.58)

Using the definition of $j_{ab}(x)$ in (2.62), we can rewrite this as

$\begin{split}S_{\text{int}}[C,j]&=-\frac{i}{4}\int
d^{d}xC^{ab}(x)j_{ab}(x)\;.\end{split}$ (4.59)

Note that due to momentum conservation, the interaction action is invariant
under (4.52) even though the operators $U_{i}$ are not.

Having fixed both the soft action and the interaction term, we can now
determine the propagator $P_{ab,cd}$ by imposing the constraint (4.3). For
gravitational theories this reads

$\begin{split}\int[dC]e^{-S_{\text{soft}}[C]-S_{\text{int}}[C,j]}=e^{-{\Gamma}_{\text{gr}}}=e^{-{\alpha}_{\text{gr}}(A_{1}^{\text{gr}}+2\pi
iA_{2}^{\text{gr}})}\;,\end{split}$ (4.60)

where

$\begin{split}A_{1}^{\text{gr}}=\int\frac{d^{d}x}{(2\pi)^{d}}{\mathcal{J}}^{ab}(x){\mathcal{J}}_{ab}(x)\;,\qquad{\alpha}_{\text{gr}}=\frac{\kappa^{2}}{32\pi}\int_{\mu}^{\Lambda}d{\omega}{\omega}^{d-3}\;.\end{split}$
(4.61)

As in the abelian case, we leave a discussion of the imaginary term
$A_{2}^{\text{gr}}$ to future work. The soft path integral is Gaussian and we
can evaluate it explicitly. The condition (4.60) then implies

$\begin{split}\frac{1}{64}\int
d^{d}xd^{d}yP_{ab,cd}(x-y)j^{ab}(x)j^{cd}(y)={\alpha}_{\text{gr}}\int\frac{d^{d}x}{(2\pi)^{d}}{\mathcal{J}}^{ab}(x){\mathcal{J}}_{ab}(x)\;.\end{split}$
(4.62)

The propagator $P$ satisfies

$\begin{split}\int
d^{d}wP_{ab,ef}(w-y)(P^{-1})^{ab,cd}(w-x)={\delta}^{\\{c}_{\\{e}{\delta}^{d\\}}_{f\\}}{\delta}^{(d)}(x-y)\;,\end{split}$
(4.63)

where the curly braces denote traceless symmetrization. Using the relationship
between $j_{ab}$ and ${\mathcal{J}}_{ab}$ in (2.62), we determine the
propagator to be

$\begin{split}P_{ab,cd}(x-y)=\frac{1024{\alpha}_{\text{gr}}}{(2\pi)^{d}}\int
d^{d}w\frac{{\mathcal{I}}_{e\\{a}(x-w){\mathcal{I}}_{b\\}f}(x-w)}{(x-w)^{2}}\frac{{\mathcal{I}}^{e}{}_{\\{c}(y-w){\mathcal{I}}^{f}{}_{d\\}}(y-w)}{(y-w)^{2}}\;.\end{split}$
(4.64)

Consequently,

$\begin{split}(P^{-1})^{ab,cd}(x-y)=\frac{(2\pi)^{d}}{1024c_{1,2}^{2}{\alpha}_{\text{gr}}}\int
d^{d}w\frac{{\mathcal{I}}^{e\\{a}(x-w){\mathcal{I}}^{b\\}f}(x-w)}{[(x-w)^{2}]^{d-1}}\frac{{\mathcal{I}}_{e}{}^{\\{c}(y-w){\mathcal{I}}_{f}{}^{d\\}}(y-w)}{[(y-w)^{2}]^{d-1}}\;.\end{split}$
(4.65)

The gravitational soft action is therefore

$\begin{split}S_{\text{soft}}[C]&=\frac{(2\pi)^{d}}{1024c_{1,2}^{2}{\alpha}_{\text{gr}}}\int
d^{d}x{\widetilde{C}}_{ab}(x){\widetilde{C}}^{ab}(x)\;.\end{split}$ (4.66)

As in gauge theory, the shadow gravity mode behaves like a local degree of
freedom. This provides further evidence that the set of local operators in the
celestial CFT are shadow transforms of bulk operators. In terms of the shadow
mode, the soft interaction term takes the form

$\begin{split}S_{\text{int}}[C,j]&=-\frac{i}{16c_{1,2}}\int
d^{d}x{\widetilde{C}}^{ab}(x){\mathcal{J}}_{ab}(x)\;.\end{split}$ (4.67)

To finish this discussion, we ‘integrate in’ the soft graviton mode $N_{ab}$
following the same procedure as in the abelian case. The resultant soft action
is

$\begin{split}S_{\text{soft}}[C,N]=\frac{{\alpha}_{\text{gr}}}{(2\pi)^{d}}\int
d^{d}xN_{ab}(x)N^{ab}(x)+\frac{i}{16c_{1,2}}\int
d^{d}xN_{ab}(x){\widetilde{C}}^{ab}(x)\;,\end{split}$ (4.68)

where
$N_{ab}(x)=2({\partial}_{a}{\partial}_{b}-\frac{1}{d}{\delta}_{ab}{\partial}^{2})N(x)$.
Insertions of the soft graviton operator are then given by

$\begin{split}\langle\,N_{a_{1}b_{1}}(y_{1})\cdots
N_{a_{m}b_{m}}(y_{m})\,\rangle=\int[dN][dC]e^{-S_{\text{soft}}[C,N]-S_{\text{int}}[C,j]}N_{a_{1}b_{1}}(y_{1})\cdots
N_{a_{m}b_{m}}(y_{m})\;.\end{split}$ (4.69)

The integral over $C(x)$ gives a Dirac delta functional
${\delta}(N-{\mathcal{J}})$ which then localizes the integral over $N$. The
result is

$\begin{split}\langle\,N_{a_{1}b_{1}}(y_{1})\cdots
N_{a_{m}b_{m}}(y_{m})\,\rangle={\mathcal{J}}_{a_{1}b_{1}}(y_{1})\cdots{\mathcal{J}}_{a_{m}b_{m}}(y_{m})\exp\left[-\frac{{\alpha}_{\text{gr}}}{(2\pi)^{d}}\int
d^{d}x{\mathcal{J}}_{ab}(x){\mathcal{J}}^{ab}(x)\right]\;\end{split}$ (4.70)

which is expected from (4.4).

##  5 Conclusions and Future Work

We have demonstrated that symmetry principles are sufficient in order to
describe soft dynamics in gauge theory and gravity. In any dimension, the
exponentiated abelian soft exchange arises straightforwardly from a pattern of
spontaneous and explicit symmetry breaking. The effective dynamics of the
long-wavelength Goldstone bosons associated to gauge transformations with non-
compact support are effectively lower-dimensional, and their interactions are
captured by lower-dimensional gauge theories (without a gauge quotient)
deformed by symmetry breaking mass terms. Although the effective action
expressed in terms of the Goldstone edge mode appears non-local, the shadow
Goldstone has local dynamics and reproduces the effects of soft exchange in
the bulk model.

In addition to providing a description of abelian soft exchange in the
celestial CFT formalism, this paper raises two unanswered questions which we
hope to explore in future work. The first question regards the celestial CFT
interpretation of the phase of the exponentiated soft exchange. The imaginary
part of ${\Gamma}$ differs significantly from its real part since it only
receives contributions from pairs of incoming particles and pairs of outgoing
particles. It is not clear to us how to reproduce this qualitative structure
from a soft action.

The second problem that requires further investigation is the non-abelian soft
exchange amplitude. The long-wavelength gluon and the associated pseudo-
Goldstone boson are strongly coupled in $d=2$, and both path integrals can
only be approximated perturbatively. Much more is known and proven in the 2D
non-linear sigma model than in 4D Yang-Mills, and ultimately one would like to
connect the well understood strong coupling behavior of the 2D model to
confinement in the 4D gauge theory. Since it is precisely the long-wavelength
fluctuations of the chromo-electric field that are strongly coupled, this
connection with the lower-dimensional pseudo-Goldstone seems like a promising
way to proceed beyond perturbation theory. We hope to pursue this direction in
future work. We also feel that the soft theorems for non-abelian chromodyons
and magnetic branes conjectured in Section 3 deserve further exploration.

### Acknowledgments

We would like to thank Amr Ahmadain, Elizabeth Himwich, Sruthi Narayan, Monica
Pate, Ana-Maria Raclariu and Andrew Strominger for many useful conversations.
DK gratefully acknowledges support from U.S. Department of Energy grant DE-
SC0009988 and the Adler Family Fund. PM gratefully acknowledges support from
U.S. Department of Energy grant DE-SC0009988, the Infosys Fellowship and STFC
consolidated grants ST/P000681/1, ST/T000694/1.

##  Appendix A Derivation of Current Ward Identities

In this section, we derive the current Ward identities (2.42), (2.51) and
(2.63). We start with the abelian Ward identity. To prove the result, we need
to show that

$\begin{split}{\partial}^{a}j_{a}(x)=\sum_{i}Q_{i}{\mathcal{K}}_{d}(m_{i}/{\omega}_{i},x_{i};x)\end{split}$
(A.1)

where

$\begin{split}j_{a}(x)=\lim_{{\Delta}\to 1}\frac{1}{2c_{{\Delta},1}}\int
d^{d}y\frac{{\mathcal{I}}_{ab}(x-y)}{[(x-y)^{2}]^{d-{\Delta}}}{\mathcal{J}}^{b}(y)\;,\qquad{\mathcal{J}}_{a}(x)={\partial}_{a}\sum_{i}Q_{i}\log[-{\hat{p}}_{i}\cdot{\hat{q}}(x)]\;.\end{split}$
(A.2)

Note that we have regulated the shadow integral so that it is well-defined in
any dimension. Using this definition, the LHS of (A.1) is

$\begin{split}{\partial}^{a}j_{a}(x)=\lim_{{\Delta}\to
1}\frac{1}{2c_{{\Delta},1}}\int
d^{d}y{\partial}^{a}\frac{{\mathcal{I}}_{ab}(x-y)}{[(x-y)^{2}]^{d-{\Delta}}}{\mathcal{J}}^{b}(y)\;.\end{split}$
(A.3)

Using the identity

$\begin{split}{\partial}^{a}\frac{{\mathcal{I}}_{ab}(x)}{(x^{2})^{\lambda}}=\frac{1}{{\lambda}}(d-1-{\lambda}){\partial}_{b}\frac{1}{(x^{2})^{\lambda}}\end{split}$
(A.4)

we find

$\begin{split}{\partial}^{a}j_{a}(x)=\lim_{{\Delta}\to
1}\frac{{\Delta}-1}{2c_{{\Delta},1}(d-{\Delta})}\int
d^{d}y\frac{1}{[(x-y)^{2}]^{d-{\Delta}}}{\partial}^{a}{\mathcal{J}}_{a}(y)\;.\end{split}$
(A.5)

Using the definition (A.2), we find

$\begin{split}{\partial}^{a}{\mathcal{J}}_{a}(x)=\sum_{i}Q_{i}\left(\frac{d-2}{[-{\hat{p}}_{i}\cdot{\hat{q}}(x)]}+\frac{z_{i}^{2}}{[-{\hat{p}}_{i}\cdot{\hat{q}}(x)]^{2}}\right),\qquad
z_{i}=m_{i}/{\omega}_{i}\end{split}$ (A.6)

which implies

$\begin{split}{\partial}^{a}j_{a}(x)=\lim_{{\Delta}\to
1}\frac{{\Delta}-1}{2c_{{\Delta},1}(d-{\Delta})}\sum_{i}Q_{i}[2(d-2)I_{{\Delta},1}(z_{i},x-x_{i})+4z_{i}^{2}I_{{\Delta},2}(z_{i},x-x_{i})]\end{split}$
(A.7)

where

$\begin{split}I_{a,b}(z,x)=\int
d^{d}y\frac{1}{[y^{2}]^{d-a}[(y+x)^{2}+z^{2}]^{b}}\;.\end{split}$ (A.8)

To evaluate this integral we use the Feynman-Schwinger parameterization

$\begin{split}\frac{1}{X_{1}^{a_{1}}\cdots X_{n}^{a_{n}}}&=\frac{{\Gamma}(\sum
a_{i})}{{\Gamma}(a_{1})\cdots{\Gamma}(a_{n})}\int_{0}^{1}\frac{dt_{1}}{t_{1}}\cdots\frac{dt_{n}}{t_{n}}{\delta}\big{(}1-\sum_{i}t_{i}\big{)}\frac{t_{1}^{a_{1}}\cdots
t_{n}^{a_{n}}}{(\sum t_{i}X_{i})^{\sum a_{i}}}\end{split}$ (A.9)

which implies

$\begin{split}I_{a,b}(z,x)&=\frac{{\Gamma}(d-a+b)}{{\Gamma}(d-a){\Gamma}(b)}\int_{0}^{1}dt(1-t)^{d-a-1}t^{b-1}\int
d^{d}y\frac{1}{[(1-t)y^{2}+t(y+x)^{2}+tz^{2}]^{d-a+b}}\\\
&=\frac{{\Gamma}(d-a+b)}{{\Gamma}(d-a){\Gamma}(b)}\int_{0}^{1}dt(1-t)^{d-a-1}t^{b-1}\int
d^{d}y\frac{1}{[(y+tx)^{2}+t(1-t)x^{2}+tz^{2}]^{d-a+b}}\;.\end{split}$ (A.10)

We can now shift $y\to y-tx$ and then move to spherical coordinates to
evaluate the $y$ integral,

$\begin{split}\int
d^{d}y\frac{1}{[(y+tx)^{2}+t(1-t)x^{2}+tz^{2}]^{d-a+b}}&=\frac{2\pi^{d/2}}{{\Gamma}(d/2)}\int_{0}^{\infty}dr\frac{r^{d-1}}{[r^{2}+t(1-t)x^{2}+tz^{2}]^{d-a+b}}\\\
&=\frac{\pi^{d/2}{\Gamma}(\frac{d}{2}-a+b)}{{\Gamma}(d-a+b)}[t(1-t)x^{2}+tz^{2}]^{a-b-\frac{d}{2}}\;.\end{split}$
(A.11)

The integral takes the form

$\begin{split}I_{a,b}(z,x)&=\frac{\pi^{d/2}{\Gamma}(\frac{d}{2}-a+b)}{{\Gamma}(d-a){\Gamma}(b)}\int_{0}^{1}dtt^{a-\frac{d}{2}-1}(1-t)^{d-a-1}[(1-t)x^{2}+z^{2}]^{a-b-\frac{d}{2}}\;.\end{split}$
(A.12)

The integral over $t$ is related to the integral representation of the Gauss
hypergeometric function

$\begin{split}\,_{2}F_{1}(a,b,c;z)=\frac{{\Gamma}(c)}{{\Gamma}(b){\Gamma}(c-b)}\int_{0}^{1}dtt^{b-1}(1-t)^{c-b-1}(1-tz)^{-a}\;.\end{split}$
(A.13)

Using this, we have

$\begin{split}I_{a,b}(z,x)&=\frac{\pi^{d/2}{\Gamma}(\frac{d}{2}-a+b){\Gamma}(a-\frac{d}{2})}{{\Gamma}(b){\Gamma}(d/2)}[x^{2}+z^{2}]^{a-b-\frac{d}{2}}\,_{2}F_{1}\left(\frac{d}{2}-a+b,a-\frac{d}{2};\frac{d}{2};\frac{x^{2}}{x^{2}+z^{2}}\right)\;.\end{split}$
(A.14)

Plugging this into (A.7) and taking ${\Delta}\to 1$, we find

$\begin{split}{\partial}^{a}j_{a}(x)=\sum_{i}Q_{i}{\mathcal{K}}_{d}(m_{i}/{\omega}_{i},x_{i};x)\;.\end{split}$
(A.15)

The derivation of the non-abelian current Ward identity (2.51) immediately
follows from the above calculation as well. We now turn to the gravitational
Ward identity (2.63). To prove this, we need to show that

$\begin{split}{\partial}^{a}{\partial}^{b}j_{ab}(x)=\sum_{i}m_{i}{\mathcal{K}}_{d+1}(m_{i}/{\omega}_{i},x_{i};x)\;,\end{split}$
(A.16)

where

$\begin{split}j_{ab}(x)=-\lim_{{\Delta}\to 1}\frac{1}{4c_{{\Delta},2}}\int
d^{d}y\frac{{\mathcal{I}}_{ac}(x-y){\mathcal{I}}_{bd}(x-y)}{[(x-y)^{2}]^{d-{\Delta}}}{\mathcal{J}}^{cd}(y)\end{split}$
(A.17)

and

$\begin{split}{\mathcal{J}}_{ab}(x)&\equiv-\left({\partial}_{a}{\partial}_{b}-\frac{1}{d}{\delta}_{ab}{\partial}^{2}\right)\sum_{i}{\omega}_{i}[-{\hat{p}}_{i}\cdot{\hat{q}}(x)]\log[-{\hat{p}}_{i}\cdot{\hat{q}}(x)]\;.\end{split}$
(A.18)

As in the abelian case, we have regulated the shadow integral to make it well
defined.

Using the identity

$\begin{split}{\partial}^{a}{\partial}^{b}\frac{{\mathcal{I}}_{a\\{c}(x){\mathcal{I}}_{d\\}b}(x)}{(x^{2})^{\lambda}}=\frac{({\lambda}-d)({\lambda}-d+1)}{{\lambda}({\lambda}+1)}{\partial}_{\\{c}{\partial}_{d\\}}\frac{1}{(x^{2})^{\lambda}}\end{split}$
(A.19)

we find

$\begin{split}{\partial}^{a}{\partial}^{b}j_{ab}(x)&=-\lim_{{\Delta}\to
1}\frac{1}{4c_{{\Delta},2}}\frac{{\Delta}({\Delta}-1)}{(d-{\Delta})(d-{\Delta}+1)}\int
d^{d}y\frac{1}{[(x-y)^{2}]^{d-{\Delta}}}{\partial}_{c}{\partial}_{d}{\mathcal{J}}^{cd}(y)\;.\end{split}$
(A.20)

To simplify this further, we use

$\begin{split}{\partial}^{a}{\partial}^{b}{\mathcal{J}}_{ab}(x)=-\frac{d-1}{d}\sum_{i=1}^{n}{\omega}_{i}\left(\frac{d(d-2)}{[-{\hat{p}}_{i}\cdot{\hat{q}}(x)]}+\frac{2(d-2)z_{i}^{2}}{[-{\hat{p}}_{i}\cdot{\hat{q}}(x)]^{2}}+\frac{2z_{i}^{4}}{[-{\hat{p}}_{i}\cdot{\hat{q}}(x)]^{3}}\right)\;.\end{split}$
(A.21)

It follows that

$\begin{split}{\partial}^{a}{\partial}^{b}j_{ab}(x)&=\lim_{{\Delta}\to
1}\frac{(d-1){\Delta}({\Delta}-1)}{4c_{{\Delta},2}d(d-{\Delta})(d-{\Delta}+1)}\sum_{i}{\omega}_{i}[2d(d-2)I_{{\Delta},1}(z_{i},x-x_{i})\\\
&\qquad\qquad\qquad\qquad\qquad+8(d-2)z_{i}^{2}I_{{\Delta},2}(z_{i},x-x_{i})+16z_{i}^{4}I_{{\Delta},3}(z_{i},x-x_{i})]\;.\end{split}$
(A.22)

Finally, we use (A.14) and take the ${\Delta}\to 1$ limit to find

$\begin{split}{\partial}^{a}{\partial}^{b}j_{ab}(x)&=\sum_{i=1}^{n}m_{i}{\mathcal{K}}_{d+1}(m_{i}/{\omega}_{i},x_{i};x)\;.\end{split}$
(A.23)

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