${\rm w}_{1+\infty}$ in 4D Gravitational Scattering

Elizabeth Himwich∗ and Monica Pate†

∗Princeton Center for Theoretical Science, Princeton University, Princeton, NJ
08544
†The Center for Cosmology and Particle Physics, New York University, New York,
NY 10003

In four-dimensional asymptotically flat spacetimes, an infinite tower of soft
graviton modes is known to generate the symmetry algebra of ${\rm
w}_{1+\infty}$ at tree-level. Here we demonstrate that the symmetry action
follows from soft graviton theorems and acts non-trivially on massive scalar
particles. By generalizing previous analyses that were specifically tailored
to the scattering of massless particles, our results clarify that ${\rm
w}_{1+\infty}$ symmetry is a universal feature of tree-level gravitational
scattering in four-dimensional asymptotically flat spacetimes and originates
from minimally-coupled gravitational interactions. In addition, we show that
the ${\rm w}_{1+\infty}$ symmetry acts non-diagonally on massive states by
mixing an infinite number of conformal families. We also present a concrete
example of non-local behavior on the celestial sphere in the presence of
massive scattering states.

###### Contents

  1. 1 Introduction
  2. 2 Review of celestial amplitudes and ${\rm SL}(2,\mathbb{C})$ transformations
  3. 3 From soft theorems to ${\rm w}_{1+\infty}$
    1. 3.1 Massless particles
    2. 3.2 Massive particles
  4. 4 Momentum space action of ${\rm w}_{1+\infty}$
  5. 5 Generating functions
    1. 5.1 Massless particles
    2. 5.2 Massive particles
  6. 6 Action of ${\rm w}_{1+\infty}$ on massive celestial conformal primaries
  7. A Non-local behavior in conformally soft theorems
    1. A.1 Three-point interaction
    2. A.2 Four-point conformally soft theorem
  8. B ${\rm SL}(2,\mathbb{C})$ action on bulk and boundary points
  9. C Derivation of massive generating function
  10. D Details of massive momentum-space action

##  1 Introduction

Quantum field theory has proven to be a remarkably successful framework for
simplifying and unifying fundamental particle physics. Nevertheless, because
gravitational quantum field theories are non-renormalizable, they ostensibly
do not predict a unique theory of quantum gravity. Holography, a proposed
correspondence between theories of quantum gravity and non-gravitational
quantum field theories in fewer spacetime dimensions, offers an alternative
way to constrain quantum gravity using quantum field theory. The holographic
correspondence thus reframes quantum field theory as the central tool for
describing microscopic physics, including in the context of quantum gravity.

In the decades following the discovery of the AdS/CFT correspondence [1, 2],
holography has been put to the test both within AdS/CFT [3, 4, 5] and by a
number of new holographic proposals [6, 7, 8, 9]. In recent years, celestial
holography has emerged as an important new type of holography. Celestial
holography posits a duality between scattering in asymptotically flat four-
dimensional (4D) spacetime and a two-dimensional conformal field theory (2D
CFT) on the celestial sphere. Reviews and references on the subject can be
found in [10, 11, 12]. The celestial duality is motivated by the isomorphism
between the Lorentz group in four dimensions ${\rm SO}(1,3)$ and the global
conformal group in two dimensions ${\rm SL}(2,\mathbb{C})$. This isomorphism
ensures that scattering amplitudes for particles in highest weight
representations of ${\rm SL}(2,\mathbb{C})$ transform under Lorentz
transformations like correlation functions of primary operators in a 2D CFT
under global conformal transformations [7, 13, 14, 15, 16]. The basis of
states in highest weight representations of ${\rm SL}(2,\mathbb{C})$ is
referred to as the conformal primary basis and scattering amplitudes in this
basis are known as celestial amplitudes.

Holographic proposals stipulate that the defining properties of a quantum
field theory are imprinted in a corresponding dual gravitational theory, and
vice versa. However, identifying these signatures appropriately can be a
difficult task. Locality, in particular, is a key property of quantum field
theory with a nontrivial holographic imprint even in the well-studied example
of AdS/CFT. In celestial holography, the full implications of locality of the
dual 2D CFT are yet to be determined. 2D locality may turn out to require
properties of celestial amplitudes that appear nontrivial or are obscure from
the perspective of scattering in 4D.

A simple notion of 2D locality does readily appear in the 4D scattering of
massless particles. Two essential ingredients underlie this result. First, the
wavefunction of a massless particle in a momentum eigenstate localizes to a
point on a spatial cross section of null infinity that can be identified with
the spatial direction of the momentum four-vector. Second, a state in a
conformal primary basis for massless particles can be constructed from the
subset of momentum eigenstates with momenta pointing in a fixed direction. The
resulting state transforms like a primary operator located at the associated
point on a spatial cross section of null infinity. In this construction, the
location of operators in 2D thus coincides with a projective location of
particles in 4D. Consequently, 2D local behavior, such as singularities when
operators approach one another, has a simply origin and interpretation in 4D
physics.

On the other hand, a notion of 2D locality for massive particles is less
straightforward. In this case, the wavefunctions of momentum eigenstates
localize to points on the resolution of timelike infinity to a spacelike
hyperboloid (Euclidean ${\rm AdS}_{3}$) [17, 18]. States in a conformal
primary basis have been constructed by convolving momentum eigenstates with
${\rm AdS}_{3}/{\rm CFT}_{2}$ bulk-to-boundary propagators and therefore
receive contributions from all momenta, not just a momentum subset [15, 16].
This can be understood geometrically as the statement that the subset of ${\rm
SL}(2,\mathbb{C})$ transformations that preserve a point on the boundary of
${\rm AdS}_{3}$ preserve no proper subset of points in the bulk of ${\rm
AdS}_{3}$. A derivation of this statement is provided in Appendix B. As a
result, for massive particles, there is no direct analog of the massless
correspondence between 2D and 4D locality.111A similar conclusion seems to
apply to the conformal primary of basis of massless particles whose
construction involves an additional shadow transformation. It would very
interesting, but beyond the scope of this paper, to investigate whether
locality is restored when working with a discrete basis for massive particles
analogous to the massless basis in [19, 20].

While such a correspondence is not necessary for 2D locality to exist –
indeed, in AdS/CFT the regimes of bulk and boundary locality do not coincide –
its absence nevertheless calls into question the realization of 2D locality
for massive particles. So far, there are very few examples of explicit
formulas for celestial amplitudes containing massive external
particles.222This is especially true beyond three-point correlators, where the
singularity structure is no longer tightly constrained by 2D global conformal
symmetry. [21, 22, 23] consider amplitudes with a massive source that breaks
translation invariance. [24] considers a $\Delta=0$ massive celestial scalar
coupled to gravitons. Several works have considered partial wave
decompositions and factorization of four-point celestial amplitudes with an
internal massive exchange [25, 26, 27, 28, 29]. The possibility therefore
remains that local behavior emerges only when computing a full celestial
amplitude.

In the hope of inspiring more rigorous studies of 2D locality, we now present
an explicit example to illustrate that a local 2D description of massive
particles must involve modifications to the current prescription. Consider the
scattering of a conformally soft photon $J$, two massless scalars $\phi$ of
charge $Q$, and a massive scalar $\Phi$ of charge $-2Q$. The celestial
amplitude for this process, which to our knowledge has not yet appeared in the
literature and we derive in Appendix A, is given by

$\begin{split}\langle\phi_{\Delta_{1}}&(z_{1},\bar{z}_{1})J(z_{2},\bar{z}_{2})\phi_{\Delta_{3}}(z_{3},\bar{z}_{3})\Phi_{\Delta_{4}}(z_{4},\bar{z}_{4})\rangle\\\
&=Q\left[\frac{1}{z_{21}}+\frac{1}{z_{23}}-\frac{2}{z_{24}}\left(1+\frac{z_{34}}{z_{23}}\frac{z-1+\bar{z}\,{}_{2}F_{1}(1,\frac{\Delta_{1}-\Delta_{3}+\Delta_{4}}{2},\Delta_{4};1-\frac{z}{z-1}\frac{\bar{z}}{\bar{z}-1})}{z-1+\bar{z}}\right)\right]\\\
&\quad\quad\quad\times\langle\phi_{\Delta_{1}}(z_{1},\bar{z}_{1})\phi_{\Delta_{3}}(z_{3},\bar{z}_{3})\Phi_{\Delta_{4}}(z_{4},\bar{z}_{4})\rangle,\end{split}$
(1.1)

where the three-point celestial amplitude is

$\begin{split}\langle\phi_{\Delta_{1}}(z_{1},&\bar{z}_{1})\phi_{\Delta_{3}}(z_{3},\bar{z}_{3})\Phi_{\Delta_{4}}(z_{4},\bar{z}_{4})\rangle\\\
&=\frac{g}{8}\left(\frac{m}{2}\right)^{\Delta_{1}+\Delta_{3}-4}\frac{B\left(\frac{\Delta_{4}+\Delta_{1}-\Delta_{3}}{2},\frac{\Delta_{4}-\Delta_{1}+\Delta_{3}}{2}\right)}{(z_{41}{\bar{z}}_{41})^{\frac{1}{2}(\Delta_{4}+\Delta_{1}-\Delta_{3})}(z_{13}{\bar{z}}_{13})^{\frac{1}{2}(\Delta_{1}+\Delta_{3}-\Delta_{4})}(z_{34}{\bar{z}}_{34})^{\frac{1}{2}(\Delta_{3}+\Delta_{4}-\Delta_{1})}}\end{split}$
(1.2)

and $z$ is the conformal cross ratio

$z=\frac{z_{12}z_{34}}{z_{13}z_{24}}.$ (1.3)

The asymmetry in scalars 1 and 3 results from the assumption that ${\rm
Re}(\Delta_{4})>{\rm Re}(\frac{\Delta_{1}-\Delta_{3}+\Delta_{4}}{2})>0$, which
is needed for the integral expression (A.31) for the hypergeometric function
to converge. Appendix A includes a full derivation of this formula, as well as
more detailed explanations of the statements that follow.

The hypergeometric function in the amplitude (1.1) presents a striking
difference from the corresponding all-massless celestial amplitude.
Momentarily ignoring the additional complexity of this hypergeometric
function, note that the first two terms contain simple poles in $z_{2}$ at
$z_{1}$ and $z_{3}$ with residues $Q_{1}=Q$ and $Q_{3}=Q$, respectively. These
two terms, along with only the first contribution in the round parentheses
(i.e. the term $-\frac{2}{z_{24}}$), would be the complete expression in a
standard local 2D conformal field theory, which would have a third simple pole
at $z_{2}\to z_{4}$ with residue $Q_{4}=-2Q$. By contrast, in (1.1) we find an
additional contribution given by the second term in round parentheses.
Together, the third term (including both contributions in round parentheses)
naively appears to contain simple poles in $z_{2}$ at $z_{3}$, $z_{4}$, and
$z_{\star}$, where $z_{\star}$ is the value of $z_{2}$ for which $z+\bar{z}-1$
vanishes. However, a more careful analysis of residues reveals that the third
term actually contains no simple poles. The celestial amplitude therefore only
contains simple poles at the locations of each of the massless particles (from
the first two terms, at $z_{1}$ and $z_{3}$) but not the massive particle
($z_{4}$). To summarize, the simple poles in this celestial amplitude differ
from those in a traditional conformal field theory with a $U(1)$ current by
the absence of a pole at the location of the charged massive particle. Still,
the residues at $z_{1}$ and $z_{3}$ are each $Q$, as in a standard conformal
field theory.

To reconcile the discrepancy between the singularity structure of the
celestial amplitude and that of standard CFT correlation functions, recall
that the $(1,0)$ conformal weight carried by $J$ guarantees that it must
respect a conservation law of the form

$\begin{split}\oint_{\infty}dz~{}\langle
J(z)\mathcal{O}_{1}(z_{1},\bar{z}_{1})\cdots\mathcal{O}_{n}(z_{n},\bar{z}_{n})\rangle=0.\end{split}$
(1.4)

Since the sum of the residues of the simple poles in the celestial amplitude
(1.1) is $2Q\neq 0$, this implies that there must be some other singularity in
the amplitude that cancels the simple pole contribution. Indeed, the
hypergeometric function ${}_{2}F_{1}(a,b,c;x)$ contains a branch cut in $x$
extending from $x=1$ to $x=\infty$, which we choose to extend along the
positive real axis. In the $z_{2}$ plane, this corresponds to a branch cut
extending between the locations of the massless particle insertions, $z_{1}$
to $z_{3}$. In Appendix A, we show that this branch cut accounts for precisely
the missing contributions that allow (1.4) to hold.

The example (1.1) is especially instructive because it is a particularly clean
and robust result, for the following reasons. First, all of the momentum-
conserving delta functions are absorbed by the integral transformations to the
conformal primary basis. The resulting celestial amplitudes are consequently
smooth in the sense that they do not contain delta-function singularities,
unlike their massless counterparts. Second, (1.1) is essentially fixed by
kinematics and thus unaffected by loop corrections. In particular, neither the
leading soft photon theorem [30] nor the kinematic dependence of three-point
amplitudes receive loop corrections [31]. Furthermore, as detailed in Appendix
A, the contour arguments and other components of the analysis treat $z$ and
$\bar{z}$ as independent complex variables, and so it is not obvious how a
different choice of spacetime signature could resolve this non-locality.

Massive particles are a definitive feature of our universe, and more generally
are often instrumental to the consistency of theories of quantum gravity,
including celestial holography. For example, in the context of celestial
holography, an exchange of massive particle states was found to appear in the
conformal block decomposition of a four-point celestial amplitude with only
massless external particles [32]. This result is likely quite general for the
following reason. Celestial holography is a manifestly conformally covariant
framework that naturally involves physics at all energies and therefore may
require regulators. Because the number of dimensions remains fixed, it would
not be surprising if all manifestly conformally covariant regulators are of
“Pauli-Villars” form, i.e. involve the addition of massive particles. Outside
of celestial holography, massive particles are not only ubiquitous in string
theory, our known example of a consistent theory of quantum gravity, but also
frequently feature in general consistency conditions for quantum gravity [33,
34, 35, 36, 37, 38, 39].

This compelling evidence from both theory and observation suggests that
massive particles should be included in celestial holography, although their
celestial amplitudes appear to admit nonlocal behavior. Therefore – especially
without 2D locality – it is important to establish general properties of
massive states that control their dynamics. Symmetries naturally supply such
dynamical constraints and are the subject of our work.333Indeed, celestial
amplitudes are known to admit other forms of non-local behavior, but some of
these instances are regarded with less concern precisely because they are
understood as result of symmetry. Specifically, four-point celestial
amplitudes for massless particles are known to admit a delta-function
singularity constraining the conformal cross ratio to be real. However, this
constraint can be identified as a remnant of 4D translation symmetry.
Additional forms of non-locality are discussed in [40, 41].  In particular, we
establish that massive particles transform non-trivially under action of the
${\rm w}_{1+\infty}$ symmetry generated by an infinite tower of soft
gravitons.

The existence of a ${\rm w}_{1+\infty}$ symmetry in celestial holography was
first discovered in [42, 43] by analyzing the algebra of an infinite
collection of soft graviton currents that are universal to any celestial
holographic dual. The non-trivial action of ${\rm w}_{1+\infty}$ on generic
massless particles was subsequently determined in the conformal primary basis
in [44]. Both analyses rely heavily on the 2D local behavior of massless
particles described above as well as a conformally-invariant notion of left
and right-moving coordinates that permits an asymmetric treatment of $z$ and
$\bar{z}$.444Recently, this asymmetry has been elevated in celestial
holography to the extent that the left-moving coordinate $z$ is treated as a
point in the 2D spacetime, while the anti-holomorphic coordinate $\bar{z}$ is
treated as an auxiliary parameter, akin to a structure constant [45, 46, 47].
Several of these works were inspired by investigations in twistor space, where
this perspective arises naturally [48, 49]. [50] first established the
connection of twistor space to celestial holography.  By contrast, as
mentioned earlier in the introduction, massive particles do not obviously
enjoy a notion of 2D locality. Likewise, there is no conformally-invariant
notion of left and right-moving coordinates in the bulk of ${\rm AdS}_{3}$ and
consequently the massive analysis demands a uniform treatment of all
coordinates. While these may appear to be mere technical differences between
massive and massless particles, they inhibit a simple generalization of the
massless analysis to the massive case. Indeed, while the number of
investigations regarding ${\rm w}_{1+\infty}$ in celestial holography has
risen rapidly [44, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 45, 46, 47,
63, 64, 65, 66, 67, 68, 69, 70, 71], none yet pertain directly to massive
particles.555There is a construction [72] of a massive self-dual black hole
state with a tower of $\rm{w}_{1+\infty}$ charges, although this state is
built as a coherent sum of massless graviton celestial operators and relies on
the asymmetry in $z$ and $\bar{z}$ in the self-dual context.

In this work, we generalize the analysis of [44] to incorporate massive
particles and thereby establish that the symmetry acts non-trivially on
massive particles. The generalization involves modifying the formulas for the
infinite tower of momentum-space soft theorems found in [73, 74] to facilitate
a simple transformation to the conformal primary basis. In particular, we
demand that the associated soft factors transform like conformal primaries
under ${\rm SL}(2,\mathbb{C})$. We find that these improved soft factors admit
descendants that also transform like primaries (i.e. primary-descendants). Our
analysis thereby affirms that the shortened conformal multiplet structure
found in [42, 75, 76] is unaffected by the specific particle content of the
theory (i.e. massive versus massless) and thus is an intrinsic of property of
the soft gravitons and their ${\rm SL}(2,\mathbb{C})$ representations. We then
find that the symmetry action on momentum-space massive scalar particles takes
a particularly simple form, which we use to establish that it respects the
algebra of ${\rm w}_{1+\infty}$. Finally, we transform our results to the
conformal primary basis and recover the known action of Poincaré generators on
massive scalar particles in [77, 78]. We discover that the additional ${\rm
w}_{1+\infty}$ generators beyond those that generate the Poincaré algebra act
in a highly non-diagonal manner on massive particles in the conformal primary
basis. Specifically, they transform a state of definite conformal weight into
a combination of a infinite number of states with conformal weights differing
by integer values from that of the original state.

The paper is organized as follows. First, we present our conventions and
carefully review the ${\rm SL}(2,\mathbb{C})$ transformation properties of
celestial amplitudes in Section 2. In Section 3, we derive the action of ${\rm
w}_{1+\infty}$ on hard particles directly from the infinite tower of momentum
space soft theorems in [74]. We begin by synthesizing various results from the
existing literature to present a systematic derivation of the symmetry action
on massless hard particles in $(3,1)$ signature. We demonstrate that this
analysis yields the same results as were obtained from $(2,2)$ analysis in
[44]. Then, we generalize the $(3,1)$ analysis to massive hard particles. In
Section 4, we construct charges that generate the symmetries associated to the
infinite tower of soft theorems and show that their action on massive
particles in momentum space respects the algebra of ${\rm w}_{1+\infty}$. We
construct a generating function for the symmetry action at each order in the
soft expansion in Section 5, which elucidates the differences between the
massless and massive case and facilitates the transformation of the symmetry
action to the conformal primary basis. In Section 6, we transform our results
to the conformal primary basis. Appendix A is dedicated to the derivation of
(1.1) and further explanation of the surrounding discussion in the
introduction. Appendix B provides a derivation of the statement in the
introduction that the subset of ${\rm SL}(2,\mathbb{C})$ transformations that
preserve a point on the boundary of ${\rm AdS}_{3}$ preserve no proper subset
of points in the bulk of ${\rm AdS}_{3}$. Further details of the generating
function derivation and proof of the ${\rm w}_{1+\infty}$ commutators are
presented in Appendix C and Appendix D, respectively.

##  2 Review of celestial amplitudes and ${\rm SL}(2,\mathbb{C})$
transformations

In this section, we review the construction of celestial amplitudes from
scattering amplitudes in momentum space. In preparation for subsequent
sections, we also include the ${\rm SL}(2,\mathbb{C})$ transformation
properties of expressions at various intermediate stages of the analysis.

The first step in constructing conformal primary states of massless particles
is to parametrize the scattering data, consisting of momenta $p_{k}$ and
polarizations ${\varepsilon}_{k}$, by an energy scale $\omega_{k}$, a point on
the 2D celestial plane $(z_{k},\bar{z}_{k})$, and a sign $\epsilon_{k}$ to
distinguish outgoing ($\epsilon_{k}=+1$) from incoming ($\epsilon_{k}=-1$)
particles:

$\begin{split}p^{\mu}_{k}=\epsilon_{k}\omega_{k}\hat{q}^{\mu}(z_{k},\bar{z}_{k}),\quad\quad\quad\varepsilon_{k+}^{\mu}=\frac{1}{\sqrt{2}}\frac{1}{z_{0k}}\hat{q}^{\mu}(z_{0},\bar{z}_{k}),\quad\quad\varepsilon_{k-}^{\mu}=\frac{1}{\sqrt{2}}\frac{1}{\bar{z}_{0k}}\hat{q}^{\mu}(z_{k},\bar{z}_{0}),\end{split}$
(2.1)

where

$\begin{split}\hat{q}^{\mu}(z,\bar{z})=\left(1+z\bar{z},z+\bar{z},-i(z-\bar{z}),1-z\bar{z}\right).\end{split}$
(2.2)

Here we have introduced an auxiliary point $(z_{0},\bar{z}_{0})$ so that the
polarizations transform covariantly under ${\rm SL}(2,\mathbb{C})$. This is
equivalent to the introduction of a reference vector in the spinor helicity
formalism. Explicitly, under the transformation

$\begin{split}z\to
z^{\prime}=\frac{az+b}{cz+d},\quad\quad\quad\bar{z}\to\bar{z}^{\prime}=\frac{\bar{a}\bar{z}+\bar{b}}{\bar{c}\bar{z}+\bar{d}},\quad\quad\quad
ad-bc=\bar{a}\bar{d}-\bar{b}\bar{c}=1,\end{split}$ (2.3)

we find

$\begin{split}\hat{q}^{\mu}(z,\bar{z})&\to\hat{q}^{\mu}(z^{\prime},\bar{z}^{\prime})=\frac{1}{(cz+d)(\bar{c}\bar{z}+\bar{d})}\Lambda^{\mu}{}_{\nu}\hat{q}^{\nu}(z,\bar{z}),\\\
\varepsilon_{+}^{\mu}(z,\bar{z};z_{0})&\to\varepsilon_{+}^{\mu}(z^{\prime},\bar{z}^{\prime};z_{0}^{\prime})=\frac{(cz+d)}{(\bar{c}\bar{z}+\bar{d})}\Lambda^{\mu}{}_{\nu}\varepsilon_{+}^{\nu}(z,\bar{z};z_{0}),\\\
\varepsilon_{-}^{\mu}(z,\bar{z};\bar{z}_{0})&\to\varepsilon_{-}^{\mu}(z^{\prime},\bar{z}^{\prime};\bar{z}_{0}^{\prime})=\frac{(\bar{c}\bar{z}+\bar{d})}{(cz+d)}\Lambda^{\mu}{}_{\nu}\varepsilon_{-}^{\nu}(z,\bar{z};\bar{z}_{0}),\\\
p^{\mu}_{k}(\omega_{k},z_{k},\bar{z}_{k})&\to
p^{\mu}_{k}(\omega^{\prime}_{k},z^{\prime}_{k},\bar{z}^{\prime}_{k})=\Lambda^{\mu}{}_{\nu}p^{\nu}_{k}(\omega_{k},z_{k},\bar{z}_{k}),\end{split}$
(2.4)

where $\Lambda$ is the corresponding Lorentz transformation in the vector
representation and in the last line we use

$\omega^{\prime}_{k}=(cz_{k}+d)(\bar{c}\bar{z}_{k}+\bar{d})\omega_{k}.$ (2.5)

Note that in the limit $z_{0},\bar{z}_{0}\to\infty$, we recover the familiar
expressions for polarization vectors:

$\begin{split}\varepsilon_{k+}^{\mu}=\frac{1}{\sqrt{2}}\partial_{z_{k}}\hat{q}^{\mu}(z_{k},\bar{z}_{k}),\quad\quad\quad\quad\varepsilon_{k-}^{\mu}=\frac{1}{\sqrt{2}}\partial_{\bar{z}_{k}}\hat{q}^{\mu}(z_{k},\bar{z}_{k}).\end{split}$
(2.6)

Polarization tensors for gravitons are built out of the polarization vectors
in (2.1) by
$\varepsilon_{k\pm}^{\mu\nu}=\varepsilon_{k\pm}^{\mu}\varepsilon_{k\pm}^{\nu}$.

After parametrizing the scattering data as described above, conformal primary
states of massless particles can be obtained by performing a Mellin
transformation with respect to energy scale $\omega_{k}$:

$\int_{0}^{\infty}\frac{d\omega_{k}}{\omega_{k}}~{}\omega_{k}^{\Delta_{k}}.$
(2.7)

Massive conformal primary states are likewise constructed using a
parametrization of massive momenta $p_{k}$ by a mass $m_{k}$, a point
$\hat{p}_{k}(y_{k},w_{k},\bar{w}_{k})$ on the unit hyperboloid
$\hat{p}_{k}^{2}=-1$, and a sign $\epsilon_{k}$:

$\begin{split}p_{k}^{\mu}=\epsilon_{k}m_{k}\hat{p}_{k}^{\mu},\quad\quad\quad\hat{p}_{k}^{\mu}=\frac{1}{2y_{k}}\left(y_{k}^{2}n^{\mu}+\hat{q}^{\mu}(w_{k},\bar{w}_{k})\right),\end{split}$
(2.8)

where

$\begin{split}n^{\mu}\equiv\partial_{z}\partial_{\bar{z}}\hat{q}^{\mu}(z,\bar{z}).\end{split}$
(2.9)

${\rm SL}(2,\mathbb{C})$ acts on these parameters like the isometries of ${\rm
AdS}_{3}$ on coordinate charts,

$\begin{split}w_{k}\to
w^{\prime}_{k}&=\frac{(aw_{k}+b)(\bar{c}\bar{w}_{k}+\bar{d})+a\bar{c}y_{k}^{2}}{(cw_{k}+d)(\bar{c}\bar{w}_{k}+\bar{d})+c\bar{c}y_{k}^{2}},\\\
\bar{w}_{k}\to\bar{w}^{\prime}_{k}&=\frac{(\bar{a}\bar{w}_{k}+\bar{b})(cw_{k}+d)+\bar{a}cy_{k}^{2}}{(cw_{k}+d)(\bar{c}\bar{w}_{k}+\bar{d})+c\bar{c}y_{k}^{2}},\\\
y_{k}\to
y^{\prime}_{k}&=\frac{y_{k}}{(cw_{k}+d)(\bar{c}\bar{w}_{k}+\bar{d})+c\bar{c}y_{k}^{2}},\end{split}$
(2.10)

and generates Lorentz transformations on massive momenta:

$\begin{split}p^{\mu}_{k}(y_{k},w_{k},\bar{w}_{k})\to
p^{\mu}_{k}(y^{\prime}_{k},w^{\prime}_{k},\bar{w}^{\prime}_{k})=\Lambda^{\mu}{}_{\nu}p^{\nu}_{k}(y_{k},w_{k},\bar{w}_{k}).\end{split}$
(2.11)

Conformal primary states of massive scalars are then obtained by integrating
the result against an ${\rm AdS}_{3}$ bulk-to-boundary propagator

$\begin{split}\mathcal{G}_{\Delta}(\hat{p}_{k};\hat{q}_{k})\equiv\frac{1}{(-\hat{p}_{k}\cdot\hat{q}(z_{k},\bar{z}_{k}))^{\Delta}}=\left(\frac{y_{k}}{y_{k}^{2}+(w_{k}-z_{k})(\bar{w}_{k}-\bar{z}_{k})}\right)^{\Delta}\end{split}$
(2.12)

with the ${\rm SL}(2,\mathbb{C})$-invariant measure on the unit three-
dimensional hyperboloid

$\begin{split}\int[d\hat{p}_{k}]=\int_{0}^{\infty}\frac{dy_{k}}{y_{k}^{3}}\int
d^{2}w_{k}.\end{split}$ (2.13)

This construction readily generalizes to massive spinning particles by simply
replacing the scalar bulk-to-boundary propagator with its appropriate spinning
counterpart [79, 80, 81].

Putting everything together, we obtain the following relation between
celestial amplitudes $\langle\mathcal{O}\cdots\rangle$ and momentum-space
amplitudes $\mathcal{A}$:

$\begin{split}\langle\mathcal{O}_{\Delta_{1}}(z_{1},\bar{z}_{1})\cdots\mathcal{O}_{\Delta_{n}}(z_{n},\bar{z}_{n})\rangle=\prod_{\begin{subarray}{c}j~{}\rm
massless\\\ k~{}\rm
massive\end{subarray}}\int_{0}^{\infty}\frac{d\omega_{j}}{\omega_{j}}\omega_{j}^{\Delta_{j}}\int[d\hat{p}_{k}]~{}\mathcal{G}_{\Delta_{k}}(\hat{p}_{k};\hat{q}_{k})~{}\mathcal{A}(p_{1},\cdots,p_{n}),\end{split}$
(2.14)

where we have suppressed all spin labels on both celestial operators
$\mathcal{O}_{\Delta}$ and bulk-to-boundary propagators
$\mathcal{G}_{\Delta}$. Under the action of ${\rm SL}(2,\mathbb{C})$ (2.3),
celestial amplitudes transform according to

$\begin{split}\langle\mathcal{O}_{\Delta_{1}}(z^{\prime}_{1},\bar{z}^{\prime}_{1})\cdots\mathcal{O}_{\Delta_{n}}(z^{\prime}_{n},\bar{z}^{\prime}_{n})\rangle=\left[\prod_{k=1}^{n}(cz_{k}+d)^{2h_{k}}(\bar{c}\bar{z}_{k}+\bar{d})^{2\bar{h}_{k}}\right]\langle\mathcal{O}_{\Delta_{1}}(z_{1},\bar{z}_{1})\cdots\mathcal{O}_{\Delta_{n}}(z_{n},\bar{z}_{n})\rangle,\end{split}$
(2.15)

where $(h_{k},\bar{h}_{k})$ are the left and right conformal weights, related
to $\Delta_{k}$ and $s_{k}$ by

$\begin{split}(h_{k},\bar{h}_{k})=\left(\frac{\Delta_{k}+s_{k}}{2},\frac{\Delta_{k}-s_{k}}{2}\right).\end{split}$
(2.16)

The analysis in the following sections will focus on amplitudes for scalar
particles and positive helicity (conformally) soft gravitons. We introduce
$G_{\Delta}$ to denote a positive helicity graviton of conformal weight
$\Delta$ and $\Phi_{\Delta}$ ($\phi_{\Delta}$) to denote massive (massless)
conformal primary scalars of weight $\Delta$. In expressions where mass is
irrelevant, $\Phi_{\Delta}$ and $\phi_{\Delta}$ are collectively represented
by $\mathcal{O}_{\Delta}$. We expect that all non-trivial aspects of the
analysis appear in the coupling of (conformally) soft gravitons to scalar
particles and thus the generalization of the following analysis to spinning
particles will be straightforward.

##  3 From soft theorems to ${\rm w}_{1+\infty}$

In [74], the authors use gauge invariance to show that an amplitude for
graviton emission admits universal behavior to all orders in the low-energy
expansion

$\begin{split}\mathcal{A}(\omega\hat{q};p_{1},\cdots,p_{n})=\frac{\kappa}{2}\sum_{\ell=-1}^{\infty}\omega^{\ell}\mathcal{A}^{(\ell)}(\hat{q};p_{1},\cdots,p_{n}),\end{split}$
(3.1)

where $\kappa^{2}=32\pi G$. The leading and subleading terms in the expansion
take the previously-discovered [30, 82] form

$\begin{split}\mathcal{A}^{(-1)}(\hat{q};p_{1},\cdots,p_{n})=\sum_{k=1}^{n}S^{\prime\prime(-1)}_{k}\mathcal{A}(p_{1},\cdots,p_{n}),\quad\quad\quad
S^{\prime\prime(-1)}_{k}=\frac{{\varepsilon}_{\mu\nu}p_{k}^{\mu}p_{k}^{\nu}}{\hat{q}\cdot
p_{k}},\end{split}$ (3.2)

and

$\begin{split}\mathcal{A}^{(0)}(\hat{q};p_{1},\cdots,p_{n})=\sum_{k=1}^{n}S^{\prime\prime(0)}_{k}\mathcal{A}(p_{1},\cdots,p_{n}),\quad\quad\quad
S^{\prime\prime(0)}_{k}=\frac{{\varepsilon}_{\mu\nu}p_{k}^{\mu}(i\hat{q}_{\sigma}\mathcal{J}^{\nu\sigma}_{k})}{\hat{q}\cdot
p_{k}},\end{split}$ (3.3)

respectively. Here the soft graviton has momentum $\omega\hat{q}$ and
polarization $\varepsilon_{\mu\nu}$ and $p_{k}$, $\mathcal{J}_{k}$ are the
momentum and total angular momentum of the $k$th particle. Both orbital
$\mathcal{L}_{k}$ and intrinsic $\mathcal{S}_{k}$ angular momentum contribute
to $\mathcal{J}_{k}$:

$\begin{split}\mathcal{J}_{k}=\mathcal{L}_{k}+\mathcal{S}_{k},\quad\quad\quad\mathcal{L}_{k}{}_{\mu\nu}=-i\left(p_{k\mu}\frac{\partial}{\partial
p_{k}^{\nu}}-p_{k\nu}\frac{\partial}{\partial p_{k}^{\mu}}\right).\end{split}$
(3.4)

Beyond subleading order, the soft theorems were found to take the form [73,
74]

$\begin{split}\mathcal{A}^{(\ell>0)}(\hat{q};p_{1},\cdots,p_{n})&=\sum_{k=1}^{n}S^{\prime\prime(\ell)}_{k}\mathcal{A}(p_{1},\cdots,p_{n})+{\varepsilon}_{\mu\nu}\mathcal{B}^{\mu\nu}_{\ell}(\hat{q};p_{1},\cdots,p_{n}),\end{split}$
(3.5)

where

$\begin{split}S^{\prime\prime(\ell)}_{k}=\frac{1}{(\ell+1)!}\frac{{\varepsilon}_{\mu\nu}(i\hat{q}_{\rho}\mathcal{J}^{\mu\rho}_{k})(i\hat{q}_{\sigma}\mathcal{J}^{\nu\sigma}_{k})}{\hat{q}\cdot
p_{k}}\left(\hat{q}\cdot\frac{\partial}{\partial
p_{k}}\right)^{\ell-1},\quad\quad\quad\ell\in\mathbb{Z}_{\geq 1}.\end{split}$
(3.6)

Here $\mathcal{B}^{\mu\nu}_{\ell}$ is a non-universal contribution, which is
generally non-vanishing. The non-universal corrections at subsubleading order
($\ell=1$) are controlled at tree level by a finite number of curvature
couplings (including $RF^{2}$ and $R^{2}\Phi$) [83]. Analogous structure
beyond subsubleading order is not known nor necessarily expected.

Equivalent statements for celestial amplitudes are obtained by performing the
appropriate change of basis, reviewed in the previous section. Using the
mathematical identity666(3.7) converges provided that the amplitude falls off
at high energies as $\mathcal{A}(\omega)\sim\omega^{-m}$ where
$m+\ell>\epsilon$.

$\begin{split}\lim_{\omega\to
0}\partial_{\omega}^{\ell+1}\left(\omega\mathcal{A}(\omega)\right)=(\ell+1)!\lim_{\epsilon\to
0}\epsilon\int_{0}^{\infty}\frac{d\omega}{\omega}~{}\omega^{-\ell+\epsilon}\mathcal{A}(\omega),\end{split}$
(3.7)

we recall that the soft limit takes the following form in the conformal
primary basis [14, 84, 85, 27, 86, 87, 88]

$\begin{split}\langle
H_{-\ell}(z,\bar{z})\mathcal{O}_{\Delta_{1}}(z_{1},\bar{z}_{1})\cdots&\mathcal{O}_{\Delta_{n}}(z_{n},\bar{z}_{n})\rangle\\\
&=\prod_{\begin{subarray}{c}j~{}\rm massless\\\ k~{}\rm
massive\end{subarray}}\int_{0}^{\infty}\frac{d\omega_{j}}{\omega_{j}}\omega_{j}^{\Delta_{j}}\int[d\hat{p}_{k}]~{}\mathcal{G}_{\Delta_{k}}(\hat{p}_{k};\hat{q}_{k})\mathcal{A}^{(\ell)}(\hat{q};p_{1},\cdots,p_{n}),\end{split}$
(3.8)

where

$\begin{split}H_{-\ell}(z,\bar{z})\equiv\lim_{\epsilon\to 0}\epsilon
G_{-\ell+\epsilon}(z,\bar{z}),\end{split}$ (3.9)

is a conformally soft graviton that transforms under ${\rm SL}(2,\mathbb{C})$
with conformal weight

$\begin{split}(h,\bar{h})=\left(\frac{-\ell+2}{2},\frac{-\ell-2}{2}\right).\end{split}$
(3.10)

Notice that the soft graviton in (3.8) is already in a state with definite
conformal weight. This equivalence between momentum-space soft limits and
emissions of particles of definite conformal weight has several non-trivial
consequences.

First, it clarifies the conditions under which non-universal corrections
$\mathcal{B}^{\mu\nu}_{\ell}$ can appear. To see this, treat $z$ and $\bar{z}$
as independent complex variables and consider the following celestial
amplitude involving only massless celestial operators $\phi_{\Delta}$:

$\begin{split}\langle
H_{-\ell}&(z,\bar{z})\phi_{\Delta_{1}}(z_{1},\bar{z}_{1})\cdots\rangle\\\
&=\oint_{z}\frac{dw}{2\pi i}\frac{1}{w-z}\langle
H_{-\ell}(w,\bar{z})\phi_{\Delta_{1}}(z_{1},\bar{z}_{1})\cdots\rangle\\\
&=-\sum_{j=1}^{m}\oint_{z_{j}}\frac{dw}{2\pi i}\frac{1}{w-z}\langle
H_{-\ell}(w,\bar{z})\phi_{\Delta_{1}}(z_{1},\bar{z}_{1})\cdots\rangle+\oint_{\infty}\frac{dw}{2\pi
i}\frac{1}{w-z}\langle
H_{-\ell}(w,\bar{z})\phi_{\Delta_{1}}(z_{1},\bar{z}_{1})\cdots\rangle.\end{split}$
(3.11)

To reach the last line, we have exploited the fact that in the soft limit, the
delta-function for momentum conservation no longer constrains the position $w$
of the soft graviton and the amplitude is thus meromorphic in $w$. This allows
us to deform the contour to infinity at the price of including contributions
from singularities at other points, denoted collectively by $z_{j}$. As
mentioned in the introduction and detailed in Appendix A, the assumption of
meromorphicity is only strictly justified for massless scattering.

Next, we exploit ${\rm SL}(2,\mathbb{C})$ symmetry to study the contribution
at infinity. Using (2.15), we find that under a general conformal
transformation (2.3), the celestial amplitude transforms as

$\begin{split}\langle
H_{-\ell}(z^{\prime},\bar{z}^{\prime})\phi_{\Delta_{1}}(z_{1}^{\prime},\bar{z}_{1}^{\prime})\cdots\phi_{\Delta_{n}}(z_{n}^{\prime},\bar{z}_{n}^{\prime})\rangle&=(cz+d)^{-\ell+2}(\bar{c}\bar{z}+\bar{d})^{-\ell-2}\prod_{k=1}^{n}[(cz_{k}+d)(\bar{c}\bar{z}_{k}+\bar{d})]^{\Delta_{k}}\\\
&\quad\quad\quad\quad\quad\times\langle
H_{-\ell}(z,\bar{z})\phi_{\Delta_{1}}(z_{1},\bar{z}_{1})\cdots\phi_{\Delta_{n}}(z_{n},\bar{z}_{n})\rangle.\end{split}$
(3.12)

Notice that the limit $z\to\infty$ sends $z^{\prime}\to a/c$. Since $a$ and
$c$ are generic, we expect the right-hand side of (3.12) to be finite when
evaluated in this limit at $z^{\prime}=a/c$. Hence, we deduce that

$\begin{split}\lim_{z\to\infty}\langle
H_{-\ell}(z,\bar{z})\phi_{\Delta_{1}}(z_{1},\bar{z}_{1})\cdots\phi_{\Delta_{n}}(z_{n},\bar{z}_{n})\rangle\sim\frac{1}{z^{-\ell+2}}.\end{split}$
(3.13)

The leading, subleading, and subsubleading soft limits have $\ell=-1,0,1$,
respectively. In these cases the contribution from the contour around infinity
in (3.11) thus vanishes and the conformally soft limit is entirely determined
by the singularities in $z$. In celestial amplitudes for massless scattering,
the singularities in $z$ are entirely dictated by the collinear singularities
in momentum space amplitudes or equivalently the singular terms in the
celestial operator product expansion (OPE). Indeed, one can verify that these
precisely reproduce the soft factors in (3.2) and (3.3), the leading and
subleading case respectively. At subsubleading order, the celestial OPE can
receive corrections from a finite number of curvature couplings (including
$RF^{2}$ and $R^{2}\Phi$), but still precisely reproduces the corrections to
the subsubleading soft graviton theorem that arise from these couplings. Thus,
in purely massless scattering, the form of the soft theorems up to
subsubleading order is tightly controlled by ${\rm SL}(2,\mathbb{C})$
symmetry. In particular, ${\rm SL}(2,\mathbb{C})$ symmetry implies
factorization, namely that the amplitude for soft emission can be written as a
soft factor (or operator) acting on the amplitude for a hard scattering
process. As we saw in the introduction, the singularity structure is more
subtle when massive particles are present. Nevertheless, conformal symmetry
ensures a sufficiently fast fall-off in the limit $z\to\infty$ so that the
contribution from $z=\infty$ to a contour integral of $H_{-\ell}$ vanishes for
$\ell=-1,0,1$. Therefore, up to and including subsubleading order, the
conformally soft limit is still fully determined by the singularity structure
in $z$.777Taking a contour integral in $z$ of (3.11) that surrounds all of the
hard insertions, a similar argument implies that the leading and subleading
soft graviton insertions with $\ell=-1,0$ correspond to globally conserved
charges (the familiar global translations and Lorentz transformations). The
analog of this statement for global $U(1)$ charge conservation is presented in
Appendix A in the derivation of (A.23). However, at subsubleading order and
beyond ($\ell\geq 1$) the contour integrals include terms from infinity and
therefore the soft insertions at these orders do not correspond to globally
conserved charges.

On the other hand, when $\ell\geq 2$ (equivalently beyond subsubleading
order), the contribution from the contour around infinity in (3.11) need not
vanish. This means that the conformally soft limit is not entirely determined
by the singularities in $z$, and thus for massless particles not entirely
determined by the collinear singularities of momentum-space amplitudes, nor
equivalently by the celestial OPE. This result is fully consistent with the
non-universal corrections in (3.5) that were found in [74] to appear beyond
subsubleading order and need not admit the factorizing form described above.

This result further demonstrates that statements about universal soft behavior
beyond subsubleading order require more care. In particular, the presence of a
non-universal contribution introduces an ambiguity in defining the universal
contribution because terms can in principle be freely moved between the two.
The authors in [74] address this ambiguity by defining the universal
contribution as the image of a particular projection operator.

In this work, we demonstrate that the ambiguity can be resolved such that the
infinite tower of soft graviton theorems organizes according to a ${\rm
w}_{1+\infty}$ symmetry. The manifest ${\rm SL}(2,\mathbb{C})$-covariance of
scattering amplitudes in a conformal primary basis was instrumental in the
discovery of the original formulation of ${\rm w}_{1+\infty}$ in celestial
holography [42, 44, 43]. We therefore seek expressions for the soft factor
that render the partition into universal and non-universal pieces invariant
under the action of ${\rm SL}(2,\mathbb{C})$. Since the conformally soft
graviton transforms like a primary of weight
$(h,\bar{h})=\left(\frac{-\ell+2}{2},\frac{-\ell-2}{2}\right)$, a conformally-
invariant partition thus requires the associated soft factor also to transform
as a primary with the same conformal weight. Upon identifying expressions that
transform like ${\rm SL}(2,\mathbb{C})$ primaries, we further demonstrate that
these generate a ${\rm w}_{1+\infty}$ symmetry action on hard massless and
massive particles. Our derivation of a ${\rm w}_{1+\infty}$ symmetry action on
massless hard particles from momentum-space soft theorems merely synthesizes a
variety of existing results from the literature into a systematic procedure.
On the other hand, the generalization to massive particles reveals new
expressions for universal soft behavior that to our knowledge have not yet
appeared in the literature.

###  3.1 Massless particles

The soft factors appearing in (3.6) do not transform like primaries.
Specifically, the factors of $\hat{q}\cdot\partial_{p_{k}}$ are problematic.
However, a natural proposal is to complete these factors into angular momentum
generators:

$\begin{split}\hat{q}^{\mu}\frac{\partial}{\partial
p_{k}^{\mu}}\to\frac{1}{\varepsilon_{+}\cdot p_{k}}\left((\varepsilon_{+}\cdot
p_{k})~{}\hat{q}^{\mu}\frac{\partial}{\partial p_{k}^{\mu}}-(\hat{q}\cdot
p_{k})~{}\varepsilon_{+}^{\mu}\frac{\partial}{\partial
p_{k}^{\mu}}\right)=\frac{1}{\varepsilon_{+}\cdot
p_{k}}~{}F_{+}^{\mu\nu}\mathcal{L}_{k}{}_{\mu\nu},\end{split}$ (3.14)

where $F_{\pm}$ is the field strength

$\begin{split}F_{\pm}^{\mu\nu}=i\varepsilon_{\pm}^{[\mu}\hat{q}^{\nu]}.\end{split}$
(3.15)

Notice that division by $\varepsilon_{+}\cdot p_{k}$ renders the expression
gauge-dependent (i.e. non-invariant under
$\varepsilon_{+}\to\varepsilon_{+}+\hat{q}$). We will show that this gauge
dependence ultimately drops out from the symmetry action.

Under ${\rm SL}(2,\mathbb{C})$ transformations (2.3) in which the reference
point $z_{0}$ is also transformed, the modified soft factors

$\begin{split}S^{\prime(\ell)}_{k}(z,\bar{z})=\frac{\varepsilon_{+}{}_{\mu\nu}p^{\mu}_{k}p^{\nu}_{k}}{\hat{q}\cdot
p_{k}}\frac{1}{(\ell+1)!}\left(\frac{F_{+}\cdot\mathcal{J}_{k}}{\varepsilon_{+}\cdot
p_{k}}\right)^{\ell+1}\end{split}$ (3.16)

transform like primaries, which we demonstrate shortly. In (3.16),

$\hat{q}^{\mu}=\hat{q}^{\mu}(z,\bar{z}),\quad\quad\quad\varepsilon^{\mu}_{+}=\varepsilon^{\mu}_{+}(z,\bar{z};z_{0}),\quad\quad\quad
F_{+}\cdot\mathcal{J}_{k}=F_{+}^{\mu\nu}\mathcal{J}_{k}{}_{\mu\nu},$ (3.17)

and we have further replaced the orbital angular momentum $\mathcal{L}_{k}$
with the total angular momentum $\mathcal{J}_{k}$. These are equivalent for
scalars, which are the focus of the present analysis, but we expect that
(3.16) is also the correct generalization for spinning particles. In fact,
(3.16) has appeared before in the literature [89, 53]. The authors of [89]
observe that this expression implies that the full soft factor expansion
resums into an exponential:

$\sum_{\ell=-1}^{\infty}\omega^{\ell}S^{\prime(\ell)}_{k}(z,\bar{z})=\frac{\varepsilon_{+}{}_{\mu\nu}p^{\mu}_{k}p^{\nu}_{k}}{\omega\hat{q}\cdot
p_{k}}\exp\left(\omega\frac{F_{+}\cdot\mathcal{J}_{k}}{\varepsilon_{+}\cdot
p_{k}}\right).$ (3.18)

An additional advantage of (3.16) is that it subsumes the leading and
subleading cases and applies to every order in the soft expansion
($\ell\geq-1$). Therefore, we can verify the transformation of the tower of
soft factors under ${\rm SL}(2,\mathbb{C})$ all in one go. In particular,
using (2.4) and (2.11), one finds

$\begin{split}\frac{\varepsilon_{+}{}_{\mu\nu}p^{\mu}_{k}p^{\nu}_{k}}{\hat{q}\cdot
p_{k}}&\to\frac{(cz+d)^{3}}{(\bar{c}\bar{z}+\bar{d})}~{}\frac{\varepsilon_{+}{}_{\mu\nu}p^{\mu}_{k}p^{\nu}_{k}}{\hat{q}\cdot
p_{k}},\\\ \varepsilon_{+}\cdot
p_{k}&\to\frac{(cz+d)}{(\bar{c}\bar{z}+\bar{d})}~{}\varepsilon_{+}\cdot
p_{k},\\\
F_{+}\cdot\mathcal{L}_{k}&\to\frac{1}{(\bar{c}\bar{z}+\bar{d})^{2}}~{}F_{+}\cdot\mathcal{L}_{k}.\end{split}$
(3.19)

Putting these together, the total transformation of the proposed soft factor

$\begin{split}S^{\prime(\ell)}_{k}(z,\bar{z})\to(cz+d)^{-\ell+2}(\bar{c}\bar{z}+\bar{d})^{-\ell-2}S^{\prime(\ell)}_{k}(z,\bar{z}),\end{split}$
(3.20)

precisely matches the transformation of a weight
$(\frac{-\ell+2}{2},\frac{-\ell-2}{2})$ primary.

Notice that (3.20) follows from transforming the reference point $z_{0}$ in
addition to the points that specify the momenta of the scattering particles.
Since the reference point is arbitrary and the full amplitude is independent
of its choice, the “non-universal” piece beyond subsubleading order must also
depend on the reference point. At leading and subleading order, momentum and
angular momentum conservation can be used to show that the choice of reference
point drops out in the sum over $k$ in (3.2) and (3.3), respectively, while at
subsubleading order $S^{\prime(\ell=1)}_{k}$ is simply independent of the
reference point.

Strictly speaking, the non-trivial dependence on the reference vector for
$\ell>1$ indicates that the partition is not conformally invariant.
Explicitly, when the reference vector is not transformed, it is clear that the
partition is not preserved. This is equivalent to the previous observation
that the factors of $\varepsilon_{+}\cdot p_{k}$ appearing in the denominator
of $S^{\prime(\ell)}_{k}(z,\bar{z})$ when $\ell>1$ render the soft factor non-
invariant under gauge transformations
$\varepsilon_{+}\to\varepsilon_{+}+\hat{q}$. Nevertheless, we now show that
our analysis of massless hard particles tolerates this subtle non-invariance
because it can be removed in a conformally covariant manner.

To do so, we exploit the property of conformally soft positive-helicity
gravitons that they admit right-moving descendants that are also primaries
(i.e. primary-descendants) [90, 91, 42, 75, 76]. By virtue of respecting ${\rm
SL}(2,\mathbb{C})$ transformations identical to those of conformally soft
gravitons, $S^{\prime(\ell)}_{k}(z,\bar{z})$ is likewise guaranteed to admit a
primary descendant. That is, under ${\rm SL}(2,\mathbb{C})$, the soft factor
descendant $\partial^{\ell+3}_{\bar{z}}S^{\prime(\ell)}_{k}(z,\bar{z})$
transforms like a primary of weight $(\frac{-\ell+2}{2},\frac{\ell+4}{2})$:

$\begin{split}\partial^{\ell+3}_{\bar{z}}S^{\prime(\ell)}_{k}(z,\bar{z})\to(cz+d)^{-\ell+2}(\bar{c}\bar{z}+\bar{d})^{\ell+4}\partial^{\ell+3}_{\bar{z}}S^{\prime(\ell)}_{k}(z,\bar{z}),\end{split}$
(3.21)

which follows from the following identity for any function $f(\bar{z})$:

$\begin{split}\left[(\bar{c}\bar{z}+\bar{d})^{2}\partial_{\bar{z}}\right]^{\ell+3}\left[\frac{f({\bar{z}})}{(\bar{c}\bar{z}+\bar{d})^{\ell+2}}\right]=(\bar{c}\bar{z}+\bar{d})^{\ell+4}\partial_{\bar{z}}^{\ell+3}f(\bar{z}).\end{split}$
(3.22)

Rewriting (3.16) using (3.17) and the parametrization for massless particles
(2.1), we obtain the following explicit expression:

$\begin{split}S^{\prime(\ell)}_{k}(z,\bar{z})&=-\frac{\epsilon_{k}\omega_{k}}{(\ell+1)!}\frac{\bar{z}-\bar{z}_{k}}{z-z_{k}}\left(\frac{z-z_{0}}{z_{k}-z_{0}}\right)^{\ell-1}\left(\frac{1}{\epsilon_{k}\omega_{k}}\left[\omega_{k}\partial_{\omega_{k}}-(\bar{z}_{k}-\bar{z})\partial_{\bar{z}_{k}}\right]\right)^{\ell+1}\\\
&=\frac{(-1)^{\ell}}{(\ell+1)!}\frac{\bar{z}-\bar{z}_{k}}{z-z_{k}}\left(\frac{z-z_{0}}{z_{k}-z_{0}}\right)^{\ell-1}\left(\sum_{m=0}^{\ell+1}{\ell+1\choose
m}\frac{\Gamma(-\omega_{k}\partial_{\omega_{k}}+1)}{\Gamma(-\omega_{k}\partial_{\omega_{k}}-\ell+m)}(\bar{z}_{k}-\bar{z})^{m}\partial_{\bar{z}_{k}}^{m}\right)\left(\epsilon_{k}\omega_{k}\right)^{-\ell}.\end{split}$
(3.23)

To arrive at this expression, it is helpful to notice that

$\begin{split}F_{+}\cdot\mathcal{L}_{k}&=-\sqrt{2}\left[(\bar{z}_{k}-\bar{z})^{2}\partial_{\bar{z}_{k}}-(\bar{z}_{k}-\bar{z})\omega_{k}\partial_{\omega_{k}}\right],\\\
F_{-}\cdot\mathcal{L}_{k}&=-\sqrt{2}\left[(z_{k}-z)^{2}\partial_{z_{k}}-(z_{k}-z)\omega_{k}\partial_{\omega_{k}}\right].\end{split}$
(3.24)

Then, expression for the associated primary descendant takes the explicit form

$\begin{split}\partial_{\bar{z}}^{\ell+3}S^{\prime(\ell)}_{k}(z,\bar{z})=\sum_{m=0}^{\ell+1}\frac{(-1)^{\ell+m}(m+1)}{(\ell+1-m)!}\frac{\Gamma(-\omega_{k}\partial_{\omega_{k}}+1)}{\Gamma(-\omega_{k}\partial_{\omega_{k}}-\ell+m)}2\pi\partial_{\bar{z}}^{\ell+1-m}\delta^{(2)}(z-z_{k})~{}\partial_{\bar{z}_{k}}^{m}\left(\epsilon_{k}\omega_{k}\right)^{-\ell}.\end{split}$
(3.25)

Importantly, this descendant no longer depends on the reference point $z_{0}$.
This implies that – so long as the modes of the primary descendant can be
shown to generate the action ${\rm w}_{1+\infty}$ – we need not identify a
conformally invariant partition at the level of the soft factor. In other
words, we have obtained a conformally invariant partition at the level of the
primary descendant of the soft factor. Of course, it may be possible to modify
the massless soft factor from [74] to be strictly conformally covariant or
equivalently independent of choice of reference point $z_{0}$. However, as we
will now demonstrate, it is unnecessary to find such an expression for the
sole purpose of establishing a ${\rm w}_{1+\infty}$ symmetry action.

A central lesson from the equivalence between soft theorems and Ward
identities for infinite-dimensional symmetries is the interpretation of the
universal soft factor as the generator of an infinitesimal symmetry
transformation on hard, single-particle states:

$\begin{split}S^{(\ell)}_{k}(z,\bar{z})|p_{k}\rangle\sim\delta^{(\ell)}_{(z,\bar{z})}|p_{k}\rangle.\end{split}$
(3.26)

The underlying symmetry algebra can be extracted by decomposing these local
symmetry transformations in a basis of modes and calculating the commutation
relations between the modes. A proper labelling of these modes was essential
to the previous discovery [43] of an underlying ${\rm w}_{1+\infty}$ symmetry.

To the same end, we now show that modes of the primary descendant correctly
reproduce the action of ${\rm w}_{1+\infty}$ on massless particles found in
[44]. Explicitly, consider

$\begin{split}\delta^{p}_{m}|p_{k}\rangle\equiv-\frac{1}{2}\int\frac{d^{2}z}{2\pi}~{}\bar{z}^{p+m-1}\partial_{\bar{z}}^{2p-1}S^{\prime(2p-4)}_{k}(z,\bar{z})|p_{k}\rangle,\end{split}$
(3.27)

where

$\begin{split}p=1,\frac{3}{2},2,\frac{5}{2},\cdots,\quad\quad\quad 1-p\leq
m\leq p-1.\end{split}$ (3.28)

Here, in anticipation of the results in [43] and [44], we have labelled the
transformations by the right-moving conformal weight
$p=\bar{h}=\frac{\ell+4}{2}$ and right-moving ${\rm SL}(2,\mathbb{C})$ mode
$m$ of the primary descendant.

To compare directly with the results in [44], we need to work in a conformal
primary basis. Using (3.25), we immediately find the action on outgoing
particles ($\epsilon_{k}=1$) to be

$\begin{split}\delta^{q}_{n}\phi_{\Delta_{k}}(z_{k},\bar{z}_{k})&=\int_{0}^{\infty}\frac{d\omega_{k}}{\omega_{k}}~{}\omega_{k}^{\Delta_{k}}~{}\delta^{q}_{n}|p_{k}\rangle\\\
&=\frac{1}{2}\sum_{m=0}^{2q-3}{q+n-1\choose
m}\frac{(2q-2-m)\Gamma(\Delta_{k}+1)}{\Gamma(\Delta_{k}+1-m)}\bar{z}_{k}^{q+n-1-m}\partial_{\bar{z}_{k}}^{2q-3-m}\phi_{\Delta_{k}-2q+4}(z_{k},\bar{z}_{k}),\end{split}$
(3.29)

which precisely matches the action of ${\rm w}_{1+\infty}$ found in [44].
Then, by directly importing the results in [44], we reach the conclusion that
the symmetry transformations defined in (3.27) respect the algebra

$\left[\delta^{p}_{m},\delta^{q}_{n}\right]=\left[m(q-1)-n(p-1)\right]\delta^{p+q-2}_{m+n}.$
(3.30)

###  3.2 Massive particles

The massless analysis in the previous subsection can only be directly extended
to massive particles provided that analogous simplifications arise. In
particular, the observation that the primary descendant was independent of the
reference point $z_{0}$, and thereby constituted a conformally covariant
expression for universal soft behavior, was a critical component of the
argument.

However, when $p_{k}$ is timelike, the primary descendant of (3.16) is
independent of $z_{0}$ only in the leading few cases. Specifically, when
$\ell=-1,0,1$, we find

$\begin{split}\partial_{\bar{z}}^{\ell+3}S^{\prime(\ell)}_{k}(z,\bar{z})&=\mathcal{N}_{\ell}\frac{p_{k}^{4}}{(\hat{q}\cdot
p_{k})^{\ell+4}}\left(F_{-}\cdot\mathcal{J}_{k}\right)^{\ell+1},\end{split}$
(3.31)

where $\mathcal{N}_{\ell}$ is just a normalization factor

$\mathcal{N}_{\ell}=(-\sqrt{2})^{\ell-1}(\ell+3)(\ell+2).$ (3.32)

To verify $z_{0}$-independence of (3.31), note that with the parametrization
of massive momenta (2.8), the angular momentum generators are
$z_{0}$-independent and take the form

$\begin{split}F_{+}\cdot\mathcal{L}_{k}&=-\sqrt{2}\left[(\bar{w}_{k}-\bar{z})^{2}\partial_{\bar{w}_{k}}+(\bar{w}_{k}-\bar{z})y_{k}\partial_{y_{k}}-y_{k}^{2}\partial_{w_{k}}\right],\\\
F_{-}\cdot\mathcal{L}_{k}&=-\sqrt{2}\left[(w_{k}-z)^{2}\partial_{w_{k}}+(w_{k}-z)y_{k}\partial_{y_{k}}-y_{k}^{2}\partial_{\bar{w}_{k}}\right].\end{split}$
(3.33)

It then directly follows that (3.31) is independent of $z_{0}$.

On the other hand, when $\ell>1$, the primary descendant of (3.16) depends on
$z_{0}$ and thus is not conformally covariant. To see this explicitly,
consider $\ell=2$. In this case, using

$1=\frac{p_{k}^{2}}{p_{k}^{2}}=\frac{(\partial_{z}\hat{q}\cdot
p_{k})(\partial_{\bar{z}}\hat{q}\cdot p_{k})-(\hat{q}\cdot p_{k})(n\cdot
p_{k})}{p_{k}^{2}}$ (3.34)

we can split $S^{\prime(2)}_{k}(z,\bar{z})$ into a sum of individually
conformally covariant terms, one of which depends on the reference point and
one which does not:

$\begin{split}S^{\prime(2)}_{k}(z,\bar{z})&=\frac{\mathcal{N}_{2}}{6!}\left[\frac{1}{p_{k}^{2}}(\hat{q}\cdot
p_{k})^{6}\partial_{\bar{z}}\left(\frac{(F_{+}\cdot\mathcal{L}_{k})^{3}}{(\hat{q}\cdot
p_{k})^{6}}\right)-\frac{1}{p_{k}^{2}}(\hat{q}_{0}\cdot
p_{k})^{6}\partial_{\bar{z}}\left(\frac{(F_{+}\cdot\mathcal{L}_{k})^{3}}{(\hat{q}_{0}\cdot
p_{k})^{6}}\right)\right],\end{split}$ (3.35)

where

$\begin{split}\hat{q}_{0}\equiv\hat{q}(z_{0},\bar{z}).\end{split}$ (3.36)

The conformal covariance of each individual term can be verified using the
following ${\rm SL}(2,\mathbb{C})$ transformations of the constituent pieces:

$\begin{split}\hat{q}\cdot p_{k}\to\frac{\hat{q}\cdot
p_{k}}{(cz+d)(\bar{c}\bar{z}+\bar{d})},\quad\quad\quad
F_{+}\cdot\mathcal{J}_{k}\to\frac{F_{+}\cdot\mathcal{J}_{k}}{(\bar{c}\bar{z}+\bar{d})^{2}}.\end{split}$
(3.37)

Since each term individually transforms like a primary, each term individually
admits a primary descendant. Explicitly,

$\begin{split}\partial^{5}_{\bar{z}}\left[\frac{\mathcal{N}_{2}}{6!}\frac{1}{p_{k}^{2}}(\hat{q}\cdot
p_{k})^{6}\partial_{\bar{z}}\left(\frac{(F_{+}\cdot\mathcal{L}_{k})^{3}}{(\hat{q}\cdot
p_{k})^{6}}\right)\right]=\mathcal{N}_{2}\frac{p_{k}^{4}}{(\hat{q}\cdot
p_{k})^{6}}\left(F_{-}\cdot\mathcal{L}_{k}\right)^{3},\end{split}$ (3.38)

which is the natural generalization of (3.31) to $\ell=2$. Similarly, the
descendant relation for the second term in (3.35) takes the same form only
with $\hat{q}$ replaced by $\hat{q}_{0}$.

More generally, at any order $(\ell\geq 1)$ the soft expression from the
previous subsection can be written as a sum over individually conformally
covariant terms, only one of which is independent of the reference point:

$\begin{split}S^{\prime(\ell)}_{k}(z,\bar{z})&=\frac{\mathcal{N}_{\ell}}{(2\ell+2)!}f^{2\ell+2}\frac{1}{p_{k}^{2\ell-2}}\partial_{\bar{z}}^{\ell-1}\left[\frac{(F_{+}\cdot\mathcal{L}_{k})^{\ell+1}}{f^{\ell+4}}\right]\\\
&\quad-\frac{\mathcal{N}_{\ell}}{(\ell+3)!(\ell-2)!}\frac{1}{p_{k}^{2\ell-2}}\sum_{j=0}^{\ell-2}{\ell-2\choose
j}\frac{(-1)^{j}}{j+\ell+4}f^{\ell-2-j}f_{0}^{\ell+4+j}\partial_{{\bar{z}}}^{\ell-1}\left[\frac{f^{j}(F_{+}\cdot\mathcal{L}_{k})^{\ell+1}}{f_{0}^{\ell+4+j}}\right].\end{split}$
(3.39)

Here we have introduced the shorthand

$\begin{split}f\equiv\hat{q}\cdot p_{k},\quad\quad\quad
f_{0}\equiv\hat{q}_{0}\cdot p_{k}.\end{split}$ (3.40)

As in (3.35), the explicit division by $p_{k}^{2}$ indicates that the massless
limit requires some care. Specifically, it requires that we assemble various
contractions of $p_{k}$ as in the numerator in (3.34) and evaluate
$p_{k}^{2}/p_{k}^{2}=1$ before taking $p_{k}^{2}\to 0$. Also as in (3.35), the
conformal covariance of each term in (3.39) can be verified using (3.37) and
(3.22).

Since the first term in (3.39) is conformally covariant by itself and
independent of the reference point, it is the natural proposal for the
universal soft behavior at subℓ+1leading order for massive particles:

$\begin{split}S^{(\ell)}_{k}(z,\bar{z})&=\frac{\mathcal{N}_{\ell}}{(2\ell+2)!}\frac{(\hat{q}\cdot
p_{k})^{2\ell+2}}{p_{k}^{2\ell-2}}\partial_{\bar{z}}^{\ell-1}\left[\frac{(F_{+}\cdot\mathcal{L}_{k})^{\ell+1}}{(\hat{q}\cdot
p_{k})^{\ell+4}}\right].\end{split}$ (3.41)

With a strictly conformally covariant soft factor at hand, we can now proceed
exactly as in the massless case. In particular, the primary descendant is now
guaranteed to be independent of the reference point and generalizes the
simplification found in (3.31) to $\ell>1$:

$\begin{split}\partial^{\ell+3}_{\bar{z}}S^{(\ell)}_{k}(z,\bar{z})=\mathcal{N}_{\ell}\frac{p_{k}^{4}}{(\hat{q}\cdot
p_{k})^{\ell+4}}\left(F_{-}\cdot\mathcal{L}_{k}\right)^{\ell+1}.\end{split}$
(3.42)

In deriving (3.42), it is helpful first to exchange powers of
$F_{+}\cdot\mathcal{L}_{k}$ for powers of $F_{-}\cdot\mathcal{L}_{k}$ by using
the identity

$\displaystyle(F_{+}\cdot\mathcal{L}_{k})^{n}=\frac{(\hat{q}\cdot
p_{k})^{2n+1}}{(p_{k}^{2})^{n}(2n)!}\partial_{z}^{2n}\left(\frac{\left(F_{-}\cdot\mathcal{L}_{k}\right)^{n}}{\hat{q}\cdot
p_{k}}\right),$ (3.43)

which can be proved by induction. Combining this result with the result (3.31)
for $\ell=-1,0,1$, we arrive at the following set of transformations of
massive particles, each now associated with a fixed order of the soft
expansion:

$\begin{split}\delta^{p}_{m}|p_{k}\rangle&\equiv-\frac{1}{2}\int\frac{d^{2}z}{2\pi}\bar{z}^{p+m-1}\partial_{\bar{z}}^{2p-1}\left\\{\begin{array}[]{cl}S^{\prime(2p-4)}_{k}(z,\bar{z})|p_{k}\rangle,&\quad
p=\frac{3}{2},2,\frac{5}{2}\vspace{1ex}\\\
S^{(2p-4)}_{k}(z,\bar{z})|p_{k}\rangle,&\quad
p>\frac{5}{2}\end{array}\right.\\\
&=-\frac{\mathcal{N}_{2p-4}}{2}\int\frac{d^{2}z}{2\pi}\bar{z}^{p+m-1}\frac{p_{k}^{4}}{(\hat{q}\cdot
p_{k})^{2p}}\left(F_{-}\cdot\mathcal{L}_{k}\right)^{2p-3}|p_{k}\rangle.\end{split}$
(3.44)

Note the last line applies for $p\geq\frac{3}{2}$, but omits the case when
$p=1$. As observed in [43], $p=1$ corresponds to a central term in the wedge
algebra. No soft theorem analysis has yet provided a non-vanishing value for
this central extension so we leave the question of an appropriate expression
for massive scalars to a future investigation. To complete the same analysis
as for massless particles in the previous subsection, it remains to be shown
that the transformations (3.44) generate a ${\rm w}_{1+\infty}$ symmetry
action on massive particles. This is the subject of the next section.

##  4 Momentum space action of ${\rm w}_{1+\infty}$

In this section, we establish that the transformation

$\begin{split}\delta^{p}_{m}|p_{k}\rangle&=-\frac{\mathcal{N}_{2p-4}}{2}\int\frac{d^{2}z}{2\pi}\bar{z}^{p+m-1}\frac{p_{k}^{4}}{(\hat{q}\cdot
p_{k})^{2p}}\left(F_{-}\cdot\mathcal{L}_{k}\right)^{2p-3}|p_{k}\rangle,\end{split}$
(4.1)

respects the algebra

$\left[\delta^{p}_{m},\delta^{q}_{n}\right]=\left[m(q-1)-n(p-1)\right]\delta^{p+q-2}_{m+n}.$
(4.2)

This result follows from an induction proof that mirrors the one presented in
Appendix C of [44]. Here we will sketch the logic and point the reader to [44]
for explicit details omitted here.

In [44], an abstract symmetry action $\delta^{p}_{m}$ with
$p=1,\frac{3}{2},2,\frac{5}{2},\cdots$ and $1-p\leq m\leq p-1$ was proven
generally to respect the algebra (4.2), provided that it respects base cases
of (4.2) with $(p,m)=(1,0)$, $(\frac{3}{2},-\frac{1}{2})$, $(2,\pm 1)$,
$(2,0)$ and $(\frac{5}{2},-\frac{3}{2})$ and $(q,n)$ compatible with (3.28)
but otherwise arbitrary. The proof in [44] of (4.2) for generic $p,m,q,n$
given these base cases depended solely on the abstract algebra (4.2) and not
on the explicit representation of $\delta^{p}_{m}$. Hence we can directly
import that result, thus reducing the general proof to establishing these base
cases for the explicit transformation in (4.1). Also note that we will not
treat the case $(p,m)=(1,0)$ since our expression does not extend to this
case, as discussed in the previous section. Importantly, this base case was
not used in proving the induction step in [44] and thus its omission will not
affect the above argument.

First consider $(p,m)=(\frac{3}{2},-\frac{1}{2})$. Evaluating (4.1), we find

$\begin{split}\delta^{\frac{3}{2}}_{-\frac{1}{2}}|p_{k}\rangle&=\frac{\epsilon_{k}m_{k}}{4y_{k}}|p_{k}\rangle=-\frac{1}{4}(n\cdot
p_{k})|p_{k}\rangle.\end{split}$ (4.3)

To evaluate the commutator, we will need the identity

$\begin{split}\left[p_{k}^{\mu},(F_{-}\cdot\mathcal{L}_{k})^{\ell}\right]=-2i\ell
F_{-}^{\mu\nu}p_{k}{}_{\nu}(F_{-}\cdot\mathcal{L}_{k})^{\ell-1}.\end{split}$
(4.4)

Then, we find888Note to compute the commutators, we employ the standard
interpretation of the ordering so that for example ${\rm SL}(2,\mathbb{C})$
transformations of the form
$\delta^{2}_{m}\mathcal{O}=\bar{z}^{m}\left[(m+1)\bar{h}+\bar{z}\partial_{\bar{z}}\right]\mathcal{O}$
obey the familiar commutation relation
$\left[\delta^{2}_{m},\delta^{2}_{n}\right]=(m-n)\delta^{2}_{m+n}$. This
requires that
$\delta^{2}_{m}\delta^{2}_{n}\mathcal{O}=\bar{z}^{n}\left((n+1)\bar{h}+\bar{z}\partial_{\bar{z}}\right)\delta^{2}_{m}\mathcal{O}=\bar{z}^{n}\left((n+1)\bar{h}+\bar{z}\partial_{\bar{z}}\right)\bar{z}^{m}\left((m+1)\bar{h}+\bar{z}\partial_{\bar{z}}\right)\mathcal{O}.$

$\begin{split}\left[\delta^{\frac{3}{2}}_{-\frac{1}{2}},\delta^{q}_{n}\right]|p_{k}\rangle&=-\frac{1}{4}\frac{\mathcal{N}_{2q-4}}{2}\int\frac{d^{2}z}{2\pi}\bar{z}^{q+n-1}\frac{p_{k}^{4}}{(\hat{q}\cdot
p_{k})^{2q}}\left[n\cdot
p_{k},\left(F_{-}\cdot\mathcal{L}_{k}\right)^{2q-3}\right]|p_{k}\rangle\\\
&=-\frac{1}{2}\frac{\mathcal{N}_{2q-4}}{2}\int\frac{d^{2}z}{2\pi}\bar{z}^{q+n-1}p_{k}^{4}~{}\frac{(-i)(2q-3)n_{\mu}p_{k\nu}F_{-}^{\mu\nu}}{(\hat{q}\cdot
p_{k})^{2q}}\left(F_{-}\cdot\mathcal{L}_{k}\right)^{2q-4}|p_{k}\rangle.\end{split}$
(4.5)

Next, we note that

$\begin{split}-i\mathcal{N}_{2q-4}(2q-3)\frac{n_{\mu}p_{k\nu}F_{-}^{\mu\nu}}{(\hat{q}\cdot
p_{k})^{2q}}=\mathcal{N}_{2q-5}\partial_{\bar{z}}\left(\frac{1}{(\hat{q}\cdot
p_{k})^{2q-1}}\right).\end{split}$ (4.6)

Integrating by parts with respect to $\bar{z}$ and using the fact that
$\partial_{\bar{z}}\left(F_{-}\cdot\mathcal{L}_{k}\right)=0$, we find

$\begin{split}\left[\delta^{\frac{3}{2}}_{-\frac{1}{2}},\delta^{q}_{n}\right]|p_{k}\rangle&=\frac{1}{2}\left(q+n-1\right)\frac{\mathcal{N}_{2q-5}}{2}\int\frac{d^{2}z}{2\pi}\bar{z}^{q+n-2}\frac{p_{k}^{4}}{(\hat{q}\cdot
p_{k})^{2q-1}}\left(F_{-}\cdot\mathcal{L}_{k}\right)^{2q-4}|p_{k}\rangle\\\
&=-\frac{1}{2}\left(q+n-1\right)\delta^{q-\frac{1}{2}}_{n-\frac{1}{2}}|p_{k}\rangle.\end{split}$
(4.7)

To establish (4.2) when $p=2$ and $m=-1,0,1$, we again explicitly evaluate
(4.1) with these values:

$\begin{split}\delta^{2}_{m}|p_{k}\rangle&=\underbrace{\frac{1}{2}\bar{w}_{k}^{m-1}\left(2\bar{w}_{k}^{2}\partial_{\bar{w}_{k}}+(m+1)\bar{w}_{k}y_{k}\partial_{y_{k}}-m(m+1)y_{k}^{2}\partial_{w_{k}}\right)}_{\equiv\bar{L}_{m}}|p_{k}\rangle.\end{split}$
(4.8)

Notice these are just three of the Killing vectors on the unit hyperboloid
parametrized by $(y_{k},w_{k},\bar{w}_{k})$. Hence, their commutators are just
familiar ${\rm SL}(2,\mathbb{C})$ expressions. We will need the following

$\begin{split}\left[\bar{L}_{m},\frac{1}{(\hat{q}\cdot
p_{k})^{2q}}\right]&=-{\bar{z}}^{m}\left((m+1)q+{\bar{z}}\partial_{\bar{z}}\right)\frac{1}{(\hat{q}\cdot
p_{k})^{2q}},\\\
\left[\bar{L}_{m},F_{-}\cdot\mathcal{L}_{k}\right]&=0.\end{split}$ (4.9)

Using these, we readily find

$\begin{split}\left[\delta^{2}_{m},\delta^{q}_{n}\right]|p_{k}\rangle&=\frac{\mathcal{N}_{2q-4}}{2}\int\frac{d^{2}z}{2\pi}\bar{z}^{q+n-1}p_{k}^{4}\left[\bar{L}_{m},\frac{(F_{-}\cdot\mathcal{L}_{k})^{2q-3}}{(\hat{q}\cdot
p_{k})^{2q}}\right]|p_{k}\rangle\\\
&=-\frac{\mathcal{N}_{2q-4}}{2}\int\frac{d^{2}z}{2\pi}\bar{z}^{q+n+m-1}\left((m+1)q+{\bar{z}}\partial_{\bar{z}}\right)p_{k}^{4}\frac{(F_{-}\cdot\mathcal{L}_{k})^{2q-3}}{(\hat{q}\cdot
p_{k})^{2q}}|p_{k}\rangle.\end{split}$ (4.10)

Once again integrating by parts in $\bar{z}$, this becomes

$\begin{split}\left[\delta^{2}_{m},\delta^{q}_{n}\right]|p_{k}\rangle&=-\left(m(q-1)-n\right)\frac{\mathcal{N}_{2q-4}}{2}\int\frac{d^{2}z}{2\pi}\bar{z}^{q+n+m-1}(p_{k})^{4}\frac{(F_{-}\cdot\mathcal{L}_{k})^{2q-3}}{(\hat{q}\cdot
p_{k})^{2q}}|p_{k}\rangle\\\
&=\left(m(q-1)-n\right)\delta^{q}_{m+n}|p_{k}\rangle.\end{split}$ (4.11)

Finally, we treat the case when $p=\frac{5}{2}$ and $m=-\frac{3}{2}$.
Proceeding as before, we evaluate (4.1) for these values and find

$\begin{split}\delta^{\frac{5}{2}}_{-\frac{3}{2}}|p_{k}\rangle&=\frac{3y_{k}}{\epsilon_{k}m_{k}}\partial_{\bar{w}_{k}}^{2}|p_{k}\rangle.\end{split}$
(4.12)

Using this expression, one can verify by brute force that this generator
likewise respects

$\begin{split}\left[\delta^{\frac{5}{2}}_{-\frac{3}{2}},\delta^{q}_{n}\right]|p_{k}\rangle&=-\frac{3}{2}\left(q+n-1\right)\delta^{q+\frac{1}{2}}_{n-\frac{3}{2}}|p_{k}\rangle.\end{split}$
(4.13)

The details of this calculation are outlined in Appendix D. Having thus
established the base cases (4.7), (4.11), and (4.13), we can directly apply
the induction proof in Appendix C of [44] to establish that the symmetry
transformation respects the algebra (4.2) when acting on massive particles.

##  5 Generating functions

Before turning to the conformal primary basis, it is helpful to compare the
results of our momentum-space analysis in the massless and massive cases. The
${\rm w}_{1+\infty}$ symmetry transformations of massless particles in the
conformal primary basis can be easily deduced from momentum space expressions.
In particular, the coordinates $z_{k},\bar{z}_{k}$ are preserved and the
energy scale $\omega_{k}$ raises the conformal weight $\Delta_{k}$ by one. By
contrast, for massive particles, momentum becomes a non-trivial differential
operator in the conformal primary basis. Therefore, simple expressions for the
charges like (4.1) involving inverse powers of momentum must be treated with
some care.

To derive expressions in the conformal primary basis, we proceed in two steps.
First, we re-express the integral expression in sum form in this section.
Second, in the next section, we transform to the conformal primary basis.
Rather than dealing with each mode $n$ of a transformation $\delta^{q}_{n}$
individually, we find it convenient to work with a generating function that
neatly packages all the ${\rm SL}(2,\mathbb{C})$-related symmetry
transformations associated to a particular order of the soft expansion. We
begin with a discussion of the generating function for massless particles.
Previous analyses in $(2,2)$ signature [92, 93, 94, 95, 96, 42] essentially
treat the soft factor itself as a generating function. In order to generalize
to the massive case, we explain how the same massless generating function can
be equivalently derived directly from the charges in $(1,3)$, as opposed to
the soft factor in $(2,2)$. Finally, we present the massive generalization and
use the generating function to re-express the integral in (4.1) in sum form,
which will facilitate the transformation to the conformal primary basis in the
next section.

###  5.1 Massless particles

We begin with the definition of the generators (3.27). We can find a
generating function by considering

$\displaystyle\mathbf{G}_{p}(\bar{a})|p_{k}\rangle$
$\displaystyle\equiv\sum_{m=1-p}^{p-1}\frac{\Gamma(2p-1)}{\Gamma(p+m)\Gamma(p-m)}(-\bar{a})^{p-m-1}\delta^{p}_{m}|p_{k}\rangle$
(5.1)
$\displaystyle=-\frac{1}{2}\int\frac{d^{2}z}{2\pi}\left(\bar{z}-\bar{a}\right)^{2p-2}\partial_{\bar{z}}^{2p-1}S^{\prime(2p-4)}_{k}(z,\bar{z})|p_{k}\rangle.$

Now, integrating by parts, using the explicit form of the massless soft factor
primary descendant (3.25), the form of $F_{+}\cdot\mathcal{L}_{k}$ (3.24) and
$\partial_{z}\hat{q}\cdot p_{k}$ for massless particles, and relabelling $a\to
z$, we find

$\displaystyle\mathbf{G}_{p}(\bar{z})|p_{k}\rangle$
$\displaystyle=\frac{1}{2}(2p-2)(-1)^{2p-3}\epsilon_{k}\omega_{k}(\bar{z}_{k}-\bar{z})\left(\frac{1}{\epsilon_{k}\omega_{k}}\left[\omega_{k}\partial_{\omega_{k}}-(\bar{z}_{k}-\bar{z})\partial_{\bar{z}_{k}}\right]\right)^{2p-3}|p_{k}\rangle$
(5.2)
$\displaystyle=\frac{\mathcal{N}_{2p-4}}{2(2p-1)}\partial_{z}\hat{q}\cdot
p_{k}\left(\frac{F_{+}\cdot\mathcal{L}_{k}}{\partial_{z}\hat{q}\cdot
p_{k}}\right)^{2p-3}|p_{k}\rangle.$

Note that this expression is a polynomial in positive powers of
$(\bar{z}-\bar{z}_{k})$. The powers of $F_{+}\cdot\mathcal{L}_{k}$ naturally
produce the expected action in terms of $\bar{z}$ descendants. Although the
same powers of $F_{+}\cdot\mathcal{L}_{k}$ appear in the soft factor (3.16)
and the right-hand side of (5.2), the two expressions fundamentally differ
because the right-hand side of (5.2) is independent of the reference point
$z_{0}$ while the soft factor (for $p>\frac{5}{2}$) is not, as emphasized in
Section 3.1.

This same generating function equivalently can be found by taking a contour
integral of the massless soft factor, which is a similar prescription to that
in a standard 2D CFT and to that in [44] for determining the ${\rm
w}_{1+\infty}$ symmetry action on massless particles. In particular, when $z$
and $\bar{z}$ are regarded as independent complex variables, the contour
integral of the soft factor (3.16) reproduces (5.2) up to normalization:

$\oint_{z_{k}}\frac{dz}{2\pi
i}S^{\prime(2p-4)}_{k}(z,\bar{z})|p_{k}\rangle=\frac{(-1)^{2p}\mathcal{N}_{2p-4}}{(2p-1)!}\partial_{z}\hat{q}\cdot
p_{k}\left(\frac{F_{+}\cdot\mathcal{L}_{k}}{\partial_{z}\hat{q}\cdot
p_{k}}\right)^{2p-3}|p_{k}\rangle.$ (5.3)

Note that the reference point $z_{0}$ is also projected out by the contour
integral. Recognizing that (5.3) is related to the sum form of
$\mathbf{G}_{p}(\bar{z})$ by (5.2), we recover precisely the form of the
discrete light transform that appears in [44, 43]:

$\displaystyle\oint_{z_{k}}\frac{dz}{2\pi
i}S^{\prime(2p-4)}_{k}(z,\bar{z})|p_{k}\rangle=\sum_{m=1-p}^{p-1}\frac{(-1)^{2p}}{\frac{1}{2}\Gamma(p+m)\Gamma(p-m)}(-\bar{z})^{p-m-1}\delta^{p}_{m}|p_{k}\rangle.$
(5.4)

The coefficients in the sum are precisely the ones that were previously
introduced by hand in [44, 43] to make the ${\rm w}_{1+\infty}$ symmetry
manifest. Here we see that they appear naturally in the expansion of the
contour integral of the soft factor in powers of $\bar{z}$.

###  5.2 Massive particles

In this subsection, we present a generating function for the momentum-space
action of ${\rm w}_{1+\infty}$ on massive particles, which can be directly
compared with the generating function for the massless case in the previous
subsection.

In the massless case, the generators involve descendants of the soft factor
(3.25) that can be naturally rewritten in terms of
$F_{+}\cdot\mathcal{L}_{k}$, as we saw in the previous subsection. In the
massive case, however, we begin from an integral expression involving angular
momentum generators of the opposite helicity. To write a generating function
in a form analogous with that of the massless case, we proceed in two steps:
first, find a sum (rather than integral) expression for the generating
function for the generators (4.1), and second, rewrite that expression in
terms of the right-moving angular momentum generators.

First, in analogy with the massless case, we consider the integral

$\mathbf{G}_{p}(\bar{a})|p_{k}\rangle\equiv-\frac{\mathcal{N}_{2p-4}}{2}\int\frac{d^{2}z}{2\pi}(\bar{z}-\bar{a})^{2p-2}\frac{p_{k}^{4}}{(\hat{q}\cdot
p_{k})^{2p}}\left(F_{-}\cdot\mathcal{L}_{k}\right)^{2p-3}|p_{k}\rangle.$ (5.5)

To turn this integral into a sum, we recognize that it can be recast in terms
of a shadow transform [97] (recall that $\hat{q}\cdot p_{k}$ has weight
$(h,\bar{h})=\left(-\frac{1}{2},-\frac{1}{2}\right)$):

$\int\frac{d^{2}z}{2\pi}\frac{\left[(z-a)(\bar{z}-\bar{a})\right]^{2p-2}}{\left(\hat{q}\cdot
p_{k}\right)^{2p}}=\mathcal{C}_{p}\frac{\left(\hat{q}(a)\cdot
p_{k}\right)^{2p-2}}{m_{k}^{4p-2}},$ (5.6)

where $\mathcal{C}_{p}$ is a normalization given by

$\mathcal{C}_{p}=\frac{\Gamma(2p-1)}{\Gamma(2p)}=\frac{1}{(2p-1)}.$ (5.7)

Expanding $F_{-}\cdot\mathcal{L}_{k}$ in powers of $(z-a)$, we have

$\begin{split}\mathbf{G}_{p}(\bar{a})|p_{k}\rangle=-\frac{\mathcal{N}_{2p-4}}{2}\sum_{n=0}^{4p-6}\frac{1}{n!}\left(\int\frac{d^{2}z}{2\pi}(\bar{z}-\bar{a})^{2p-2}(z-a)^{n}\frac{p_{k}^{4}}{(\hat{q}\cdot
p_{k})^{2p}}\right)\partial_{a}^{n}\left(F_{-}(a)\cdot\mathcal{L}_{k}\right)^{2p-3}|p_{k}\rangle.\end{split}$
(5.8)

Then using

$\begin{split}(-\partial_{a})^{B}(z-a)^{A}=\frac{A!}{(A-B)!}(z-a)^{A-B},\end{split}$
(5.9)

which remains formally true for inverse derivatives interpreted as definite
integrals from $z$ to $a$, we find

$\begin{split}\mathbf{G}_{p}(\bar{a})|p_{k}\rangle=-\frac{\mathcal{N}_{2p-4}}{2(2p-2)!}\sum_{n=0}^{4p-6}\Big{(}(-\partial_{a})^{2p-2-n}\int\frac{d^{2}z}{2\pi}(\bar{z}-\bar{a})^{2p-2}&(z-a)^{2p-2}\frac{p_{k}^{4}}{(\hat{q}\cdot
p_{k})^{2p}}\Big{)}\\\
&\times\partial_{a}^{n}\left(F_{-}(a)\cdot\mathcal{L}_{k}\right)^{2p-3}|p_{k}\rangle.\end{split}$
(5.10)

Finally, using the shadow transform (5.6) and relabelling $a\to z$, we have

$\displaystyle\mathbf{G}_{p}(\bar{z})|p_{k}\rangle=-\frac{\mathcal{N}_{2p-4}\mathcal{C}_{p}}{2(2p-2)!m_{k}^{4p-6}}\sum_{n=0}^{4p-6}(-\partial_{z})^{2p-2-n}(\hat{q}\cdot
p_{k})^{2p-2}~{}\partial_{z}^{n}\left(F_{-}(z)\cdot\mathcal{L}_{k}\right)^{2p-3}|p_{k}\rangle.$
(5.11)

To compare this expression to the massless case, and also to rewrite the
generating function in an explicit form that does not involve formal inverse
powers of derivatives, we use the identity

$\displaystyle(F_{-}\cdot\mathcal{L}_{k})^{n}=\frac{(-1)^{n}}{(2n)!m_{k}^{2n}}\sum_{m=0}^{2n}\left[(-\partial_{\bar{z}})^{2n-m}(p_{k}\cdot\hat{q})^{2n}\right]~{}\partial_{\bar{z}}^{m}\left(F_{+}\cdot\mathcal{L}_{k}\right)^{n},$
(5.12)

which follows from the identity (3.43). Now, using the completeness relation
(3.34) for massive particles to expand the factors of $\hat{q}\cdot p_{k}$ and
evaluating the derivatives, we ultimately find

$\displaystyle\mathbf{G}_{p}(\bar{z})|p_{k}\rangle=\frac{\mathcal{N}_{2p-4}\mathcal{C}_{p}}{2\left(-n\cdot
p_{k}\right)^{2p-4}}\sum_{\ell=0}^{4p-6}\frac{1}{\ell!}\left(\sum_{j=2p-4}^{4p-6}(-1)^{j}{\ell\choose
j}\right)\left(-\frac{\partial_{z}\hat{q}\cdot p_{k}}{n\cdot
p_{k}}\right)^{\ell-2p+4}\partial_{\bar{z}}^{\ell}\left(F_{+}\cdot\mathcal{L}_{k}\right)^{2p-3}|p_{k}\rangle.$
(5.13)

More details of the derivation of this result are presented in Appendix (C).
Note that this expression is a polynomial in positive powers of
$(\bar{z}-\bar{w}_{k})$. For particular values of $p$, the generating
functions become

$\displaystyle\mathbf{G}_{\frac{3}{2}}(\bar{z})|p_{k}\rangle$
$\displaystyle=\frac{1}{4}\partial_{z}\hat{q}\cdot p_{k}|p_{k}\rangle,$ (5.14)
$\displaystyle\mathbf{G}_{2}(\bar{z})|p_{k}\rangle$
$\displaystyle=-\frac{\sqrt{2}}{2}\left(F_{+}\cdot\mathcal{L}_{k}\right)|p_{k}\rangle,$
$\displaystyle\mathbf{G}_{p>2}(\bar{z})|p_{k}\rangle$
$\displaystyle=\frac{\mathcal{N}_{2p-4}\mathcal{C}_{p}}{2\left(-n\cdot
p_{k}\right)^{2p-4}}\frac{(-1)^{2p}}{(2p{-}5)!}\sum_{\ell=2p-4}^{4p-6}\frac{1}{\ell(\ell{-}2p{+}4)!}\left(-\frac{\partial_{z}\hat{q}\cdot
p_{k}}{n\cdot
p_{k}}\right)^{\ell{-}2p{+}4}\partial_{\bar{z}}^{\ell}\left(F_{+}\cdot\mathcal{L}_{k}\right)^{2p-3}|p_{k}\rangle.$

For $p=\frac{3}{2}$ and $p=2$ these exactly match the expressions found in the
previous subsection. For $p>2$, the massless limit of these expressions
reduces to the form found in the previous subsection. For massive momenta when
$p>2$, (5.14) is not equivalent to the massless expression (5.2) because the
massive soft factor was modified as described in Section 3.2. This
modification was ultimately required to obtain conformally covariant
expressions.

##  6 Action of ${\rm w}_{1+\infty}$ on massive celestial conformal primaries

In this section, we formulate the action of ${\rm w}_{1+\infty}$ on massive
scalar celestial primaries by transforming the results of the previous section
to the conformal primary basis.

Although we now have expressions for the symmetry action in terms of simple
powers of $\hat{q}\cdot p_{k}$, $n\cdot p_{k}$, and
$F_{+}\cdot\mathcal{L}_{k}$, one subtlety still remains in determining the
associated action on massive particles in the conformal primary basis: the
momentum-space expressions for the charges involve inverse powers of the four-
momentum $p_{k}$. In the conformal primary basis, the four-momentum $P$
becomes a non-trivial differential operator [78, 79] that acts on massive
scalars as

$\begin{split}P_{k}^{\mu}\Phi_{\Delta_{k}}&(z_{k},\bar{z}_{k})\\\
&=\frac{\epsilon_{k}m_{k}}{2}\Bigg{[}\left(\partial_{z_{k}}\partial_{\bar{z}_{k}}\hat{q}^{\mu}_{k}+\frac{\left(\partial_{\bar{z}_{k}}\hat{q}^{\mu}_{k}\right)\partial_{z_{k}}+\left(\partial_{z_{k}}\hat{q}^{\mu}_{k}\right)\partial_{\bar{z}_{k}}}{\Delta_{k}-1}+\frac{\hat{q}_{k}^{\mu}\partial_{z_{k}}\partial_{\bar{z}_{k}}}{(\Delta_{k}-1)^{2}}\right)e^{-\partial_{\Delta_{k}}}\\\
&\hskip
245.4159pt+\frac{\Delta_{k}\hat{q}^{\mu}_{k}}{\Delta_{k}-1}e^{\partial_{\Delta_{k}}}\Bigg{]}\Phi_{\Delta_{k}}(z_{k},\bar{z}_{k}),\end{split}$
(6.1)

so in the case of massive external particles, inverse powers of momentum must
be treated with some care. This is in contrast to the massless conformal
primary basis, in which the action of four-momentum is simply weight-raising
and so inverse powers of momentum can be easily interpreted as weight-
lowering.

One benefit of the generating function form (5.13) as a sum, rather than an
integral, is that it makes manifest the fact that inverse powers of the
differential operator $P_{k}$, as opposed to just the scalar multiplier
$m_{k}$, are necessarily involved in the expressions for the massive
generators for $p>2$. To work with these expressions in the conformal primary
basis, it is therefore helpful to provide a prescription for performing
calculations.

A natural (albeit formal) interpretation of the inverse powers is to replace
them with a Schwinger parametrization: specifically, recalling the action
(4.3) of $n\cdot p_{k}$, consider making the replacement

$\frac{4^{m}}{(-n\cdot
p_{k})^{m}}|p_{k}\rangle\mapsto\left(\delta^{\frac{3}{2}}_{-\frac{1}{2}}\right)^{-m}|p_{k}\rangle\equiv\frac{1}{(m-1)!}\int_{0}^{\infty}dx\
x^{m-1}\sum_{j=0}^{\infty}\frac{1}{j!}(-x\
\delta^{\frac{3}{2}}_{-\frac{1}{2}})^{j}|p_{k}\rangle.$ (6.2)

The Schwinger parametrization is a popular trick for working with formal
expressions involving inverted differential operators and has even appeared
before in studies of the matrix elements of momentum generators in a Lorentz
basis [98]. To prove that the parametrization of the symmetry action with
(6.2) will give the same algebra, we must simply check that this replacement
gives the same commutators with the $\delta^{\frac{3}{2}}_{n}$ and
$\delta^{2}_{n}$ generators, because $\hat{q}\cdot p_{k}$ and
$F_{+}\cdot\mathcal{L}_{k}$ are constructed from these modes and all of the
generators in (5.13) are constructed from powers of these operators. Since the
momenta commute, the only nontrivial cases to check are $\delta^{2}_{0}$ and
$\delta^{2}_{1}$. Before making the replacement (6.2), we compute the
commutators explicitly from their derivative form (4.8):

$\displaystyle\left[\frac{1}{2}y_{k}\partial_{y_{k}}+\bar{w}_{k}\partial_{\bar{w}_{k}},\left(\frac{y_{k}}{\epsilon_{k}m_{k}}\right)^{m}\right]|p_{k}\rangle$
$\displaystyle=\frac{m}{2}\left(\frac{y_{k}}{\epsilon_{k}m_{k}}\right)^{m}|p_{k}\rangle,$
(6.3)
$\displaystyle\left[\bar{w}_{k}^{2}\partial_{\bar{w}_{k}}+\bar{w}_{k}y_{k}\partial_{y_{k}}-y_{k}^{2}\partial_{w_{k}},\left(\frac{y_{k}}{\epsilon_{k}m_{k}}\right)^{m}\right]|p_{k}\rangle$
$\displaystyle=m\bar{w}_{k}\left(\frac{y_{k}}{\epsilon_{k}m_{k}}\right)^{m}=m\left(\frac{\bar{w}_{k}\epsilon_{k}m_{k}}{y_{k}}\right)\left(\frac{y_{k}}{\epsilon_{k}m_{k}}\right)^{m+1}|p_{k}\rangle.$

After making the replacement, it is straightforward to check using the
commutators $\left[\delta^{2}_{n},\delta^{\frac{3}{2}}_{-\frac{1}{2}}\right]$
and the expression in (6.2) that

$\displaystyle\left[\delta^{2}_{0},\left(\delta^{\frac{3}{2}}_{-\frac{1}{2}}\right)^{-m}\right]|p_{k}\rangle$
$\displaystyle=-\frac{m}{2}\left(\delta^{\frac{3}{2}}_{-\frac{1}{2}}\right)^{-m}|p_{k}\rangle,$
(6.4)
$\displaystyle\left[\delta^{2}_{1},\left(\delta^{\frac{3}{2}}_{-\frac{1}{2}}\right)^{-m}\right]|p_{k}\rangle$
$\displaystyle=-m\delta^{\frac{3}{2}}_{\frac{1}{2}}\left(\delta^{\frac{3}{2}}_{-\frac{1}{2}}\right)^{-m-1}|p_{k}\rangle.$

Recalling the sign difference in derivative ordering (see footnote 8), the
commutators before and after the replacement precisely match and confirm that
$(\delta^{\frac{3}{2}}_{-\frac{1}{2}})^{-m}$ has right-weight $\frac{m}{2}$
and that $\delta^{2}_{1}$ has the proper raising action. Thus, to transform to
the conformal primary basis, we simply use the replacement

$\bar{w}_{k}=-\frac{\partial_{z}\hat{q}\cdot p_{k}|_{\bar{z}=0}}{n\cdot
p_{k}}\mapsto\delta^{\frac{3}{2}}_{\frac{1}{2}}\left(\delta^{\frac{3}{2}}_{-\frac{1}{2}}\right)^{-1},\
\ \ \frac{1}{-n\cdot
p_{k}}\mapsto\frac{1}{4}\left(\delta^{\frac{3}{2}}_{-\frac{1}{2}}\right)^{-1}.$
(6.5)

With this new interpretation of the generators in (5.13) involving positive
powers of momentum, the action in the conformal primary basis now readily
follows from the action in momentum space:

$\begin{split}\delta^{q}_{n}\Phi_{\Delta_{k}}(z_{k},\bar{z}_{k})\equiv\int[d\hat{p}_{k}]~{}\mathcal{G}_{\Delta_{k}}(\hat{p}_{k};\hat{q}_{k})\delta^{q}_{n}|p_{k}\rangle.\end{split}$
(6.6)

To evaluate this action, first note that the bulk-to-boundary propagator
satisfies

$\begin{split}p_{k}^{\mu}\mathcal{G}_{\Delta_{k}}&(\hat{p}_{k};\hat{q}_{k})\\\
&=\frac{\epsilon_{k}m_{k}}{2}\Bigg{[}\left(\partial_{z_{k}}\partial_{\bar{z}_{k}}\hat{q}^{\mu}_{k}+\frac{\left(\partial_{\bar{z}_{k}}\hat{q}^{\mu}_{k}\right)\partial_{z_{k}}+\left(\partial_{z_{k}}\hat{q}^{\mu}_{k}\right)\partial_{\bar{z}_{k}}}{\Delta_{k}-1}+\frac{\hat{q}_{k}^{\mu}\partial_{z_{k}}\partial_{\bar{z}_{k}}}{(\Delta_{k}-1)^{2}}\right)e^{-\partial_{\Delta_{k}}}\\\
&\hskip
245.4159pt+\frac{\Delta_{k}\hat{q}^{\mu}_{k}}{\Delta_{k}-1}e^{\partial_{\Delta_{k}}}\Bigg{]}\mathcal{G}_{\Delta_{k}}(\hat{p}_{k};\hat{q}_{k}),\end{split}$
(6.7)

(which is how one verifies (6.1)). Collecting the transformations (6.6) in the
generating function from the previous section, we find

$\displaystyle\mathbf{G}_{p}(\bar{z})\Phi_{\Delta_{k}}(z_{k},\bar{z}_{k})=\frac{\mathcal{N}_{2p-4}\mathcal{C}_{p}}{2\left(-n\cdot
P_{k}\right)^{2p-4}}\sum_{\ell=0}^{4p-6}$
$\displaystyle\frac{1}{\ell!}\left(\sum_{j=2p-4}^{4p-6}(-1)^{j}{\ell\choose
j}\right)\left(-\frac{\partial_{z}\hat{q}\cdot P_{k}}{n\cdot
P_{k}}\right)^{\ell-2p+4}$ (6.8)
$\displaystyle\times\partial_{\bar{z}}^{\ell}\int[d\hat{p}_{k}]~{}\mathcal{G}_{\Delta_{k}}(\hat{p}_{k};\hat{q}_{k})\left(F_{+}\cdot\mathcal{L}_{k}\right)^{2p-3}|p_{k}\rangle,$

where in the above expression capital $P_{k}$ is used to indicate that
momentum is now the operator appearing in (6.1) and inverse powers are
evaluated using the Schwinger parametrization as in (6.2).

Next, note that the bulk-to-boundary propagator is an intertwiner and thus
obeys

$\begin{split}\left(F_{+}\cdot\mathcal{L}_{k}\right)\mathcal{G}_{\Delta_{k}}(\hat{p}_{k};\hat{q}_{k})=\sqrt{2}\left[\Delta_{k}(\bar{z}_{k}-\bar{z})+(\bar{z}_{k}-\bar{z})^{2}\partial_{\bar{z}_{k}}\right]\mathcal{G}_{\Delta_{k}}(\hat{p}_{k};\hat{q}_{k}).\end{split}$
(6.9)

As a result, we can integrate by parts and then further pull this action
outside the momentum integral to find

$\begin{split}\mathbf{G}_{p}(\bar{z})\Phi_{\Delta_{k}}(z_{k},\bar{z}_{k})=\frac{\mathcal{N}_{2p-4}\mathcal{C}_{p}}{2\left(-n\cdot
P_{k}\right)^{2p-4}}\sum_{\ell=0}^{4p-6}&\frac{1}{\ell!}\left(\sum_{j=2p-4}^{4p-6}(-1)^{j}{\ell\choose
j}\right)\left(-\frac{\partial_{z}\hat{q}\cdot P_{k}}{n\cdot
P_{k}}\right)^{\ell-2p+4}\\\
&\times\partial_{\bar{z}}^{\ell}\left(-\sqrt{2}\left[\Delta_{k}(\bar{z}_{k}-\bar{z})+(\bar{z}_{k}-\bar{z})^{2}\partial_{\bar{z}_{k}}\right]\right)^{2p-3}\Phi_{\Delta_{k}}(z_{k},\bar{z}_{k}),\end{split}$
(6.10)

where we have identified

$\begin{split}\int[d\hat{p}_{k}]~{}\mathcal{G}_{\Delta_{k}}(\hat{p}_{k};\hat{q}_{k})~{}|p_{k}\rangle=\Phi_{\Delta_{k}}(z_{k},\bar{z}_{k}).\end{split}$
(6.11)

As in the massless case, we identify the leading $q=\frac{3}{2}$ generators as
simply reproducing the action of translations (6.1) in the conformal primary
basis. Likewise, the $q=2$ charges generate the action of (half of) ${\rm
SL}(2,\mathbb{C})$. Explicitly,

$\begin{split}\delta^{2}_{m}\Phi_{\Delta_{k}}(z_{k},\bar{z}_{k})&=\bar{z}_{k}^{m}\left((m+1)\frac{\Delta_{k}}{2}+\bar{z}_{k}\partial_{\bar{z}_{k}}\right)\Phi_{\Delta_{k}}(z_{k},\bar{z}_{k})\\\
&=\bar{L}_{m}\Phi_{\Delta_{k}}(z_{k},\bar{z}_{k}).\end{split}$ (6.12)

When $p>2$, the inverse powers of $n\cdot p_{k}$ in the momentum-space
generating function become an infinite number of weight-shifting operators in
the conformal primary basis, as is made explicit by the Schwinger parameter
representation. As such, the action of ${\rm w}_{1+\infty}$ on massive
celestial primaries is notably more complicated than on massless celestial
primaries, where the action maps a single conformal family to a single other
conformal family. By contrast, in the massive case, a single conformal family
is mapped by generators with $p>2$ to an infinite number of other conformal
families. It would interesting to investigate whether some form of
simplification or organizing structure materializes when working with a
discrete conformal primary basis involving only integer-valued conformal
dimensions.

## Acknowledgements

We are grateful to Nick Agia, Alfredo Guevara, Hofie Hannesdottir, Aidan
Herderschee, Y.T. Albert Law, Donal O’Connell, and Ana Raclariu for insightful
conversations. This work was supported by NSF grant 2310633.

##  Appendix A Non-local behavior in conformally soft theorems

Consider a $U(1)$ gauge theory with three species of charged scalar particles,
$\phi_{1}$, $\phi_{3}$ and $\Phi_{4}$, which couple via the three-point
interaction

$\begin{split}\mathcal{L}_{\rm int}\sim g\phi_{1}\phi_{3}\Phi_{4}.\end{split}$
(A.1)

Let $\phi_{1}$ and $\phi_{3}$ be massless and $\Phi_{4}$ be massive with mass
$m$. Their respective charges, $Q_{1}$, $Q_{3}$, and $Q_{4}$, are required by
the $U(1)$ gauge symmetry to obey charge conservation:

$\begin{split}Q_{1}+Q_{3}+Q_{4}=0.\end{split}$ (A.2)

In this appendix, we will see that a contour integral of a 2D $U(1)$ current
encircling the operator insertions will vanish, as we expect in a standard
conformal field theory. However, unlike in a standard conformal field theory,
there are non-local contributions to the contour integral that are due to the
presence of a massive particle.

###  A.1 Three-point interaction

Here we review the celestial three-point amplitude coupling one massive and
two massless charged particles. Note that this amplitude has previously
appeared in [26]. The three-point celestial amplitude takes the general form

$\begin{split}\langle\phi_{\Delta_{1}}(z_{1},\bar{z}_{1})&\phi_{\Delta_{3}}(z_{3},\bar{z}_{3})\Phi_{\Delta_{4}}(z_{4},\bar{z}_{4})\rangle\\\
&=\int_{0}^{\infty}\frac{d\omega_{1}}{\omega_{1}}~{}\omega_{1}^{\Delta_{1}}\int_{0}^{\infty}\frac{d\omega_{3}}{\omega_{3}}~{}\omega_{3}^{\Delta_{3}}\int[d\hat{p}_{4}]~{}\mathcal{G}_{\Delta_{4}}(\hat{p}_{4};\hat{q}_{4})~{}\mathcal{A}_{3}(\epsilon_{1}\omega_{1}\hat{q}_{1},\epsilon_{3}\omega_{3}\hat{q}_{3},\epsilon_{4}m\hat{p}_{4}),\end{split}$
(A.3)

where $\mathcal{A}_{3}$ is the momentum-space three-point amplitude

$\mathcal{A}_{3}(p_{1},p_{2},p_{3})=g\delta^{(4)}(p_{1}+p_{2}+p_{3})$ (A.4)

and

$\hat{q}_{k}^{\mu}\equiv\hat{q}^{\mu}(z_{k},\bar{z}_{k}).$ (A.5)

Using ${\rm SL}(2,\mathbb{C})$ covariance to send $z_{1}\to 0$, $z_{3}\to 1$
and $z_{4}\to\infty$, (A.3) becomes

$\begin{split}\langle\phi_{\Delta_{1}}(z_{1},\bar{z}_{1})\phi_{\Delta_{3}}(z_{3},\bar{z}_{3})\Phi_{\Delta_{4}}(z_{4},\bar{z}_{4})\rangle=\frac{\lim_{\tau,\bar{\tau}\to\infty}\tau^{\Delta_{4}}\bar{\tau}^{\Delta_{4}}\langle\phi_{\Delta_{1}}(0,0)\phi_{\Delta_{3}}(1,1)\Phi_{\Delta_{4}}(\tau,\bar{\tau})\rangle}{(z_{41}\bar{z}_{41})^{\frac{1}{2}(\Delta_{1}+\Delta_{4}-\Delta_{3})}(z_{13}\bar{z}_{13})^{\frac{1}{2}(\Delta_{1}+\Delta_{3}-\Delta_{4})}(z_{34}\bar{z}_{34})^{\frac{1}{2}(\Delta_{3}+\Delta_{4}-\Delta_{1})}},\end{split}$
(A.6)

where

$\begin{split}\lim_{\tau,\bar{\tau}\to\infty}\tau^{\Delta_{4}}\bar{\tau}^{\Delta_{4}}&\langle\phi_{\Delta_{1}}(0,0)\phi_{\Delta_{3}}(1,1)\Phi_{\Delta_{4}}(\tau,\bar{\tau})\rangle\\\
&=\int_{0}^{\infty}\frac{d\omega_{1}}{\omega_{1}}~{}\omega_{1}^{\Delta_{1}}\int_{0}^{\infty}\frac{d\omega_{3}}{\omega_{3}}~{}\omega_{3}^{\Delta_{3}}\int_{0}^{\infty}\frac{dy_{4}}{y_{4}^{3}}\int
d^{2}w_{4}~{}y_{4}^{\Delta_{4}}\\\ &\quad\quad\times
g\delta^{(4)}\left(\epsilon_{1}\omega_{1}\hat{q}(0,0)+\epsilon_{3}\omega_{3}\hat{q}(1,1)+\epsilon_{4}\frac{m}{2y_{4}}(y_{4}^{2}n+\hat{q}(w_{4},\bar{w}_{4}))\right).\end{split}$
(A.7)

The solution to the delta function constraint is

$w_{4}=\bar{w}_{4}=\frac{m^{2}}{m^{2}+4\omega_{1}^{2}},\quad\quad\quad
y_{4}=-\epsilon_{1}\epsilon_{4}\frac{2m\omega_{1}}{m^{2}+4\omega_{1}^{2}},\quad\quad\quad\omega_{3}=\epsilon_{1}\epsilon_{3}\frac{m^{2}}{4\omega_{1}}.$
(A.8)

Since both $y_{4}$ and $\omega_{3}$ are positive, notice that scattering is
only nontrivial when $\epsilon_{1}=\epsilon_{3}=-\epsilon_{4}$. Assuming this,
the delta function can be written as

$\begin{split}\delta^{(4)}&\left(\epsilon_{1}\omega_{1}\hat{q}(0,0)+\epsilon_{3}\omega_{3}\hat{q}(1,1)+\epsilon_{4}\frac{m}{2y_{4}}(y_{4}^{2}n+\hat{q}(w_{4},\bar{w}_{4}))\right)\\\
&\quad\quad\quad\quad\quad=\frac{8m\omega_{1}^{2}}{(m^{2}+4\omega_{1}^{2})^{3}}\delta^{(2)}\left(w_{4}-\frac{m^{2}}{m^{2}+4\omega_{1}^{2}}\right)\delta\left(y_{4}-\frac{2m\omega_{1}}{m^{2}+4\omega_{1}^{2}}\right)\delta\left(\omega_{3}-\frac{m^{2}}{4\omega_{1}}\right).\end{split}$
(A.9)

Using this result to perform the integrals, we find

$\begin{split}\lim_{\tau,\bar{\tau}\to\infty}\tau^{\Delta_{4}}\bar{\tau}^{\Delta_{4}}&\langle\phi_{\Delta_{1}}(0,0)\phi_{\Delta_{3}}(1,1)\Phi_{\Delta_{4}}(\tau,\bar{\tau})\rangle=\frac{g}{8}\left(\frac{m}{2}\right)^{\Delta_{1}+\Delta_{3}-4}B\left(\frac{\Delta_{4}+\Delta_{1}-\Delta_{3}}{2},\frac{\Delta_{4}-\Delta_{1}+\Delta_{3}}{2}\right).\end{split}$
(A.10)

Substituting this into (A.6), we obtain the following closed form expression
for the three-point celestial amplitude:

$\begin{split}\langle\phi_{\Delta_{1}}(z_{1},\bar{z}_{1})\phi_{\Delta_{3}}(z_{3},\bar{z}_{3})\Phi_{\Delta_{4}}(z_{4},\bar{z}_{4})\rangle=\frac{\frac{g}{8}\left(\frac{m}{2}\right)^{\Delta_{1}+\Delta_{3}-4}B\left(\frac{\Delta_{4}+\Delta_{1}-\Delta_{3}}{2},\frac{\Delta_{4}-\Delta_{1}+\Delta_{3}}{2}\right)}{(z_{41}\bar{z}_{41})^{\frac{1}{2}(\Delta_{1}+\Delta_{4}-\Delta_{3})}(z_{13}\bar{z}_{13})^{\frac{1}{2}(\Delta_{1}+\Delta_{3}-\Delta_{4})}(z_{34}\bar{z}_{34})^{\frac{1}{2}(\Delta_{3}+\Delta_{4}-\Delta_{1})}}.\end{split}$
(A.11)

###  A.2 Four-point conformally soft theorem

Now consider the celestial amplitude for the emission of a conformally soft
photon $J$. Once again, we use conformal invariance to write this as

$\begin{split}\langle\phi_{\Delta_{1}}(z_{1},\bar{z}_{1})&J(z_{2},\bar{z}_{2})\phi_{\Delta_{3}}(z_{3},\bar{z}_{3})\Phi_{\Delta_{4}}(z_{4},\bar{z}_{4})\rangle\\\
&=\frac{z_{41}z_{34}}{z_{13}z_{24}^{2}}\frac{\lim_{\tau,\bar{\tau}\to\infty}\tau^{\Delta_{4}}\bar{\tau}^{\Delta_{4}}\langle\phi_{\Delta_{1}}(0,0)J(z,\bar{z})\phi_{\Delta_{3}}(1,1)\Phi_{\Delta_{4}}(\tau,\bar{\tau})\rangle}{(z_{41}\bar{z}_{41})^{\frac{1}{2}(\Delta_{1}+\Delta_{4}-\Delta_{3})}(z_{13}\bar{z}_{13})^{\frac{1}{2}(\Delta_{1}+\Delta_{3}-\Delta_{4})}(z_{34}\bar{z}_{34})^{\frac{1}{2}(\Delta_{3}+\Delta_{4}-\Delta_{1})}}.\end{split}$
(A.12)

Here

$z=\frac{z_{12}z_{34}}{z_{13}z_{24}}$ (A.13)

is the conformal cross ratio and

$\begin{split}&\lim_{\tau,\bar{\tau}\to\infty}\tau^{\Delta_{4}}\bar{\tau}^{\Delta_{4}}\langle\phi_{\Delta_{1}}(0,0)J(z,\bar{z})\phi_{\Delta_{3}}(1,1)\Phi_{\Delta_{4}}(\tau,\bar{\tau})\rangle\\\
&\quad=\sqrt{2}\int_{0}^{\infty}\frac{d\omega_{1}}{\omega_{1}}~{}\omega_{1}^{\Delta_{1}}\int_{0}^{\infty}\frac{d\omega_{3}}{\omega_{3}}~{}\omega_{3}^{\Delta_{3}}\int_{0}^{\infty}\frac{dy_{4}}{y_{4}^{3}}\int
d^{2}w_{4}~{}y_{4}^{\Delta_{4}}\mathcal{A}_{4}(\epsilon_{1}\omega_{1}\hat{q}(0,0),\hat{q}(z,\bar{z}),\epsilon_{3}\omega_{3}\hat{q}(1,1),\epsilon_{4}m\hat{p}_{4}).\end{split}$
(A.14)

The four-point amplitude $\mathcal{A}_{4}$ is determined in momentum space by
the soft theorem:

$\begin{split}\mathcal{A}_{4}(p_{1},\hat{q},p_{3},p_{4})&=\sum_{k=1,3,4}Q_{k}\frac{\varepsilon_{+}\cdot
p_{k}}{\hat{q}\cdot
p_{k}}~{}\mathcal{A}_{3}(p_{1},p_{3},p_{4})=\sum_{k=1,3,4}Q_{k}\frac{\varepsilon_{+}\cdot
p_{k}}{\hat{q}\cdot p_{k}}~{}g\delta^{(4)}(p_{1}+p_{3}+p_{4}).\end{split}$
(A.15)

Parametrizing the momenta as in (A.14), we find

$\begin{split}\mathcal{A}_{4}&(\epsilon_{1}\omega_{1}\hat{q}(0,0),\hat{q}(z,\bar{z}),\epsilon_{3}\omega_{3}\hat{q}(1,1),\epsilon_{4}m\hat{p}_{4})\\\
&=\frac{1}{\sqrt{2}}\left[\frac{Q_{1}}{z}+\frac{Q_{3}}{z-1}+\frac{Q_{4}(\bar{z}-\bar{w}_{4})}{y_{4}^{2}+|z-w_{4}|^{2}}\right]g\delta^{(4)}\left(\epsilon_{1}\omega_{1}\hat{q}(0,0)+\epsilon_{3}\omega_{3}\hat{q}(1,1)+\frac{\epsilon_{4}m}{2y_{4}}(y_{4}^{2}n+\hat{q}(w_{4},\bar{w}_{4}))\right).\end{split}$
(A.16)

Notice, the same delta function appears here as in the expression for the
three-point function (A.7) and thus is also solved by (A.8). Using (A.9) to
evaluate the integrals, (A.14) becomes

$\begin{split}&\lim_{\tau,\bar{\tau}\to\infty}\tau^{\Delta_{4}}\bar{\tau}^{\Delta_{4}}\langle\phi_{\Delta_{1}}(0,0)J(z,\bar{z})\phi_{\Delta_{3}}(1,1)\Phi_{\Delta_{4}}(\tau,\bar{\tau})\rangle\\\
&\quad=\frac{g}{4}\left(\frac{m}{2}\right)^{\Delta_{1}+\Delta_{3}-4}\int_{0}^{\infty}\frac{d\omega_{1}}{\omega_{1}}~{}\omega_{1}^{\Delta_{1}-\Delta_{3}+\Delta_{4}}\left(1+\omega_{1}^{2}\right)^{-\Delta_{4}}\left[\frac{Q_{1}}{z}+\frac{Q_{3}}{z-1}+\frac{Q_{4}(\bar{z}(1+\omega_{1}^{2})-1)}{(z-1)(\bar{z}-1)+\omega_{1}^{2}z\bar{z}}\right].\end{split}$
(A.17)

Including the prefactors in (A.12), making the additional change of variables

$\begin{split}t=\frac{\omega_{1}^{2}}{1+\omega_{1}^{2}},\end{split}$ (A.18)

and recalling the expression for the three-point amplitude (A.11), we arrive
at the following expression for the conformally soft theorem:

$\begin{split}\langle\phi_{\Delta_{1}}&(z_{1},\bar{z}_{1})J(z_{2},\bar{z}_{2})\phi_{\Delta_{3}}(z_{3},\bar{z}_{3})\Phi_{\Delta_{4}}(z_{4},\bar{z}_{4})\rangle\\\
&=\frac{z_{41}z_{34}}{z_{13}z_{24}^{2}}\frac{\langle\phi_{\Delta_{1}}(z_{1},\bar{z}_{1})\phi_{\Delta_{3}}(z_{3},\bar{z}_{3})\Phi_{\Delta_{4}}(z_{4},\bar{z}_{4})\rangle}{B\left(\frac{\Delta_{4}+\Delta_{1}-\Delta_{3}}{2},\frac{\Delta_{4}-\Delta_{1}+\Delta_{3}}{2}\right)}\int_{0}^{1}\frac{dt}{t(1-t)}~{}t^{\frac{1}{2}(\Delta_{4}+\Delta_{1}-\Delta_{3})}(1-t)^{\frac{1}{2}(\Delta_{4}-\Delta_{1}+\Delta_{3})}\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\times\left[\frac{Q_{1}}{z}+\frac{Q_{3}}{z-1}+\frac{Q_{4}}{(z-1)(\bar{z}-1)}\frac{\bar{z}-1+t}{1-(1-\frac{z\bar{z}}{(z-1)(\bar{z}-1)})t}\right].\end{split}$
(A.19)

Before performing the remaining integral, let us pause for a moment and
examine the singularity structure in $z_{2}$. First, note that the explicit
dependence on $\bar{z}$ reveals that in the presence of massive particles,
correlators of $J$ are no longer meromorphic functions, despite the fact that
$J$ is a weight $(1,0)$ current. As a result, in $(3,1)$ signature where
$z_{k}$ and $\bar{z}_{k}$ are related by complex conjugation, one can no
longer freely deform contours of integration with respect to $z_{2}$. Thus, if
we construct charges from line integrals of $J$, then, strictly speaking,
standard arguments about contour deformations do not apply in this context.

Nevertheless, if we relax the signature and allow $z_{k}$ and $\bar{z}_{k}$ to
be independent complex variables, then the correlator is meromorphic in
$z_{2}$ and ${\rm SL}\times{\rm SL}$ symmetry guarantees that the physics of
charge conservation is still encoded in the singularity structure in $z_{2}$.
To see this, consider the following contour integral:

$\begin{split}\oint_{\mathcal{C}_{\infty}}\frac{dz}{2\pi i}\langle
J(z,\bar{z})\mathcal{O}_{1}(z_{1},\bar{z}_{1})\cdots\mathcal{O}_{n}(z_{n},\bar{z}_{n})\rangle,\end{split}$
(A.20)

where $\mathcal{C}_{\infty}$ is a closed contour encircling the point at
infinity. We can evaluate this integral in two ways. First, we can deform the
integral to encircle the singularities in the correlator, denoted collectively
by $z_{j}$:

$\begin{split}\oint_{\mathcal{C}_{\infty}}\frac{dz}{2\pi i}\langle
J(z,\bar{z})\mathcal{O}_{1}(z_{1},\bar{z}_{1})\cdots\mathcal{O}_{n}(z_{n},\bar{z}_{n})\rangle&=\sum_{j}\oint_{\mathcal{C}_{z_{j}}}\frac{dz}{2\pi
i}\langle
J(z,\bar{z})\mathcal{O}_{1}(z_{1},\bar{z}_{1})\cdots\mathcal{O}_{n}(z_{n},\bar{z}_{n})\rangle.\end{split}$
(A.21)

Note that generically there may be both pointlike and extended singularities,
so it is a slight abuse of notation to label them by individual points $z_{j}$
on the complex plane. Next, we can use the requirement from conformal symmetry
that any correlator involving $J$ must fall off as999A derivation of (A.22) is
presented in the main text. See the discussion leading up to (3.13).

$\lim_{z\to\infty}\langle
J(z,\bar{z})\mathcal{O}_{1}(z_{1},\bar{z}_{1})\cdots\mathcal{O}_{n}(z_{n},\bar{z}_{n})\rangle\sim\frac{1}{z^{2}}$
(A.22)

to argue that this integral must vanish:

$\begin{split}\oint_{\mathcal{C}_{\infty}}\frac{dz}{2\pi i}\langle
J(z,\bar{z})\mathcal{O}_{1}(z_{1},\bar{z}_{1})\cdots\mathcal{O}_{n}(z_{n},\bar{z}_{n})\rangle=0.\end{split}$
(A.23)

Thus, (A.21) and (A.23) together imply a constraint on the singularity
structure in $z$.

It is instructive to verify this behavior explicitly in our example. Let us
focus on the following contribution to the integrand in (A.19):

$\begin{split}&\frac{z_{41}z_{34}}{z_{13}z_{24}^{2}}\left[\frac{Q_{1}}{z}+\frac{Q_{3}}{z-1}+\frac{Q_{4}}{(z-1)(\bar{z}-1)}\frac{\bar{z}-1+t}{1-(1-\frac{z\bar{z}}{(z-1)(\bar{z}-1)})t}\right],\end{split}$
(A.24)

which contains all the non-trivial $z_{2}$ dependence. We focus on the
singularity structure in $z_{2}$, holding $\bar{z}_{2}$ fixed. Note that
studying the singularity structure before taking the integral with respect to
$t$ effectively amounts to working in momentum space. Specifically, a similar
argument directly applies to $\mathcal{A}_{4}$ in (A.15).

The integrand (A.24) contains simple poles in $z_{2}$ at $z_{1}$ and $z_{3}$.
Near these points, the conformal cross ratio $z$ approaches $0$ and $1$,
respectively, and we find

$\begin{split}\lim_{z_{2}\to
z_{1}}\frac{z_{41}z_{34}}{z_{13}z_{24}^{2}}\left[\frac{Q_{1}}{z}+\frac{Q_{3}}{z-1}+\frac{Q_{4}}{(z-1)(\bar{z}-1)}\frac{\bar{z}-1+t}{1-(1-\frac{z\bar{z}}{(z-1)(\bar{z}-1)})t}\right]&\to\frac{Q_{1}}{z_{21}},\\\
\lim_{z_{2}\to
z_{3}}\frac{z_{41}z_{34}}{z_{13}z_{24}^{2}}\left[\frac{Q_{1}}{z}+\frac{Q_{3}}{z-1}+\frac{Q_{4}}{(z-1)(\bar{z}-1)}\frac{\bar{z}-1+t}{1-(1-\frac{z\bar{z}}{(z-1)(\bar{z}-1)})t}\right]&\to\frac{Q_{3}}{z_{23}}.\end{split}$
(A.25)

Here, it is also helpful to notice that

$\frac{z_{41}z_{34}}{z_{13}z_{24}^{2}}\left[\frac{Q_{1}}{z}+\frac{Q_{3}}{z-1}\right]=\frac{Q_{1}}{z_{21}}+\frac{Q_{3}}{z_{23}}-\frac{Q_{1}+Q_{3}}{z_{24}}.$
(A.26)

In a standard local 2D conformal field theory with charge conservation such
that $Q_{4}=-(Q_{1}+Q_{3})$, these terms would constitute the complete
singularity structure. In particular, as $z_{2}$ approaches $z_{4}$, there
would be a simple pole with residue $Q_{4}$. However, in (A.24) there is an
additional term involving $Q_{4}$, which also contributes a simple pole in the
limit as $z_{2}$ approaches $z_{4}$:

$\begin{split}\lim_{z_{2}\to
z_{4}}\frac{z_{41}z_{34}}{z_{13}z_{24}^{2}}\left[\frac{Q_{1}}{z}+\frac{Q_{3}}{z-1}+\frac{Q_{4}}{(z-1)(\bar{z}-1)}\frac{\bar{z}-1+t}{1-(1-\frac{z\bar{z}}{(z-1)(\bar{z}-1)})t}\right]\to-\frac{Q_{1}+Q_{3}+Q_{4}}{z_{24}}=0.\end{split}$
(A.27)

Thus, unlike in a standard 2D conformal field theory, the full expression is
regular at $z_{4}$. Notice that (A.23) implies that (A.24) must contain
another singularity in $z_{2}$, since the residues considered so far do not
add to zero. Indeed, the final singularity is the solution to

$1-\left[1-\frac{z\bar{z}}{(z-1)(\bar{z}-1)}\right]t=0.$ (A.28)

Denoting the solution by $z_{*}$, where

$z_{*}=\frac{t\bar{z}z_{1}z_{34}-(1-t)(\bar{z}-1)z_{3}z_{41}}{t\bar{z}z_{34}-(1-t)(\bar{z}-1)z_{41}},$
(A.29)

we find

$\lim_{z_{2}\to
z_{*}}\frac{z_{41}z_{34}}{z_{13}z_{24}^{2}}\left[\frac{Q_{1}}{z}+\frac{Q_{3}}{z-1}+\frac{Q_{4}}{(z-1)(\bar{z}-1)}\frac{\bar{z}-1+t}{1-(1-\frac{z\bar{z}}{(z-1)(\bar{z}-1)})t}\right]\to\frac{Q_{4}}{z_{2*}}.$
(A.30)

The explicit dependence of $z_{*}$ on $t$ in (A.29) reflects the fact that in
complexified momentum space, a null momentum $\hat{q}$ (satisfying
$\hat{q}^{2}=0$) can have vanishing inner product with a timelike momentum
$p_{k}$ (satisfying $p_{k}^{2}<0$). That is, the simple pole at $z_{*}$ is the
pole that arises when the inner product between $\hat{q}(z_{2},\bar{z}_{2})$
and $p_{4}=m\hat{p}_{4}$ vanishes.

Consistency requires that the singularity structure of the celestial amplitude
in $z_{2}$ also conspires to respect (A.23) after performing the final
integral in $t$. To investigate this behavior, we use the integral definition
of the hypergeometric function

$\begin{split}B(b,c-b){}_{2}F_{1}(a,b,c;x)=\int_{0}^{1}dt~{}t^{b-1}(1-t)^{c-b-1}(1-xt)^{-a},\end{split}$
(A.31)

and the identity

${}_{2}F_{1}(a,b,c;x)-{}_{2}F_{1}(a-1,b,c;x)-\frac{b}{c}x~{}{}_{2}F_{1}(a,b+1,c+1;x)=0$
(A.32)

to obtain the following closed form expression for the conformally soft
theorem:

$\begin{split}\langle\phi_{\Delta_{1}}&(z_{1},\bar{z}_{1})J(z_{2},\bar{z}_{2})\phi_{\Delta_{3}}(z_{3},\bar{z}_{3})\Phi_{\Delta_{4}}(z_{4},\bar{z}_{4})\rangle\\\
&=\Bigg{[}\frac{Q_{1}}{z_{21}}+\frac{Q_{3}}{z_{23}}+\frac{Q_{4}}{z_{24}}\Bigg{(}1+\frac{z_{34}}{z_{23}}\frac{1}{z{-}1{+}\bar{z}}\left[z{-}1{+}\bar{z}\,{}_{2}F_{1}\Big{(}1,\frac{\Delta_{1}{-}\Delta_{3}{+}\Delta_{4}}{2},\Delta_{4};1{-}\frac{z}{z{-}1}\frac{\bar{z}}{\bar{z}{-}1}\Big{)}\right]\Bigg{)}\Bigg{]}\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\times\langle\phi_{\Delta_{1}}(z_{1},\bar{z}_{1})\phi_{\Delta_{3}}(z_{3},\bar{z}_{3})\Phi_{\Delta_{4}}(z_{4},\bar{z}_{4})\rangle.\end{split}$
(A.33)

The hypergeometric integral (A.31) converges provided ${\rm Re}(c)>{\rm
Re}(b)>0$ so here we assume that the conformal weights respect

${\rm Re}(\Delta_{4})>{\rm
Re}\left(\frac{\Delta_{1}-\Delta_{3}+\Delta_{4}}{2}\right)>0.$ (A.34)

Notice that the spectrum of conformal weights presented in [16] satisfies this
condition. In the conformal primary basis, (A.33) retains the simple poles in
$z_{2}$ at $z_{1}$ and $z_{3}$ with residues proportional to $Q_{1}$ and
$Q_{3}$, respectively:

$\begin{split}\lim_{z_{2}\to
z_{1}}z_{21}\frac{\langle\phi_{\Delta_{1}}(z_{1},\bar{z}_{1})J(z_{2},\bar{z}_{2})\phi_{\Delta_{3}}(z_{3},\bar{z}_{3})\Phi_{\Delta_{4}}(z_{4},\bar{z}_{4})\rangle}{\langle\phi_{\Delta_{1}}(z_{1},\bar{z}_{1})\phi_{\Delta_{3}}(z_{3},\bar{z}_{3})\Phi_{\Delta_{4}}(z_{4},\bar{z}_{4})\rangle}=Q_{1},\\\
\lim_{z_{2}\to
z_{3}}z_{23}\frac{\langle\phi_{\Delta_{1}}(z_{1},\bar{z}_{1})J(z_{2},\bar{z}_{2})\phi_{\Delta_{3}}(z_{3},\bar{z}_{3})\Phi_{\Delta_{4}}(z_{4},\bar{z}_{4})\rangle}{\langle\phi_{\Delta_{1}}(z_{1},\bar{z}_{1})\phi_{\Delta_{3}}(z_{3},\bar{z}_{3})\Phi_{\Delta_{4}}(z_{4},\bar{z}_{4})\rangle}=Q_{3}.\end{split}$
(A.35)

For the second limit, we use $z_{23}\to 0$, which corresponds to $z\to 1$ and
sends the argument of the hypergeometric function to infinity. Note that the
hypergeometric function has branch points at the arguments $x=1$ and
$x=\infty$, and a branch cut extends between them. Generic paths $z_{23}\to 0$
avoid this branch cut and in this limit

$\begin{split}\lim_{|x|\to\infty}&B\left(\frac{\Delta_{1}-\Delta_{3}+\Delta_{4}}{2},\frac{\Delta_{3}-\Delta_{1}+\Delta_{4}}{2}\right){}_{2}F_{1}\left(1,\frac{\Delta_{1}-\Delta_{3}+\Delta_{4}}{2},\Delta_{4};x\right)\\\
&\quad\quad\quad=\lim_{|x|\to\infty}\int_{0}^{1}\frac{dt}{t(1-t)}~{}t^{\frac{1}{2}(\Delta_{4}+\Delta_{1}-\Delta_{3})}(1-t)^{\frac{1}{2}(\Delta_{4}-\Delta_{1}+\Delta_{3})}\frac{1}{1-xt}=0.\end{split}$
(A.36)

Similarly, for generic paths $z_{21}\to 0$ that avoid the branch cut, the
hypergeometric function is regular. Also, as before, there is no simple pole
in (A.33) at $z_{4}$. To see this, recall that $z_{2}\to z_{4}$ implies
$z\to\infty$, so we find

$\begin{split}\lim_{z_{2}\to
z_{4}}&\frac{Q_{4}}{z_{24}}\Bigg{(}1+\frac{z_{34}}{z_{23}}\frac{1}{z{-}1{+}\bar{z}}\left[z{-}1{+}\bar{z}\,{}_{2}F_{1}\Big{(}1,\frac{\Delta_{1}{-}\Delta_{3}{+}\Delta_{4}}{2},\Delta_{4};1{-}\frac{z}{z{-}1}\frac{\bar{z}}{\bar{z}{-}1}\Big{)}\right]\Bigg{)}\\\
&=\lim_{z_{2}\to
z_{4}}\frac{Q_{4}}{z_{24}}\left(1+\frac{z_{34}}{z_{23}}\right)=\frac{Q_{4}}{z_{43}},\end{split}$
(A.37)

which is manifestly regular. Finally, as in momentum space, there naively
appears to be another pole in $z_{2}$, this time where

$z-1+\bar{z}\to 0,$ (A.38)

or equivalently,

$z_{2}\to\frac{z_{1}z_{34}+z_{4}z_{13}(1-\bar{z})}{z_{34}+z_{13}(1-\bar{z})}.$
(A.39)

However, in this limit, the hypergeometric function approaches unity

$\lim_{z\to
1-\bar{z}}{}_{2}F_{1}\Big{(}1,\frac{\Delta_{1}-\Delta_{3}+\Delta_{4}}{2},\Delta_{4};1-\frac{z}{z-1}\frac{\bar{z}}{\bar{z}-1}\Big{)}={}_{2}F_{1}\Big{(}1,\frac{\Delta_{1}-\Delta_{3}+\Delta_{4}}{2},\Delta_{4};0\Big{)}=1,$
(A.40)

so there is no simple pole in $z_{2}$ at this point.

Specifying (A.23) to the four-point amplitude (A.33) and deforming the
contour, we again reach the conclusion that there must be some other
singularity in $z_{2}$. We will now demonstrate that the contribution from the
third simple pole in momentum space (A.30) is replaced by a contribution from
the discontinuity along the branch cut of the hypergeometric function, so that

$\begin{split}0&=\oint_{\mathcal{C}_{\infty}}\frac{dz_{2}}{2\pi
i}\langle\phi_{\Delta_{1}}(z_{1})J(z_{2})\phi_{\Delta_{3}}(z_{3})\Phi_{\Delta_{4}}(z_{4})\rangle\\\
&={\rm
Res}_{z_{1}}\left[\langle\phi_{\Delta_{1}}(z_{1})J(z_{2})\phi_{\Delta_{3}}(z_{3})\Phi_{\Delta_{4}}(z_{4})\rangle\right]+{\rm
Res}_{z_{3}}\left[\langle\phi_{\Delta_{1}}(z_{1})J(z_{2})\phi_{\Delta_{3}}(z_{3})\Phi_{\Delta_{4}}(z_{4})\rangle\right]\\\
&\quad\quad+\int_{\mathcal{C}}\frac{dz_{2}}{2\pi i}~{}{\rm
disc}\left[\langle\phi_{\Delta_{1}}(z_{1})J(z_{2})\phi_{\Delta_{3}}(z_{3})\Phi_{\Delta_{4}}(z_{4})\rangle\right]\\\
&=\left(Q_{1}+Q_{3}\right)\langle\phi_{\Delta_{1}}(z_{1})\phi_{\Delta_{3}}(z_{3})\Phi_{\Delta_{4}}(z_{4})\rangle+\int_{\mathcal{C}}\frac{dz_{2}}{2\pi
i}~{}{\rm
disc}\left[\langle\phi_{\Delta_{1}}(z_{1})J(z_{2})\phi_{\Delta_{3}}(z_{3})\Phi_{\Delta_{4}}(z_{4})\rangle\right],\end{split}$
(A.41)

where $\mathcal{C}$ is a contour along the branch cut in $z_{2}$ between
$z_{1}$ and $z_{3}$. Choosing the contour around infinity to have clockwise
orientation as usual, it can be deformed to contours around each simple pole
and the branch cut with counterclockwise orientation.

To determine the contribution from the cut, first note that we can take the
branch cut in the hypergeometric function to extend along the positive real
$x$-axis from $x=1$ to $x=\infty$ and the discontinuity to be defined as the
negative imaginary contour subtracted from the positive imaginary contour
along this axis. The contribution to contour integral is101010The orientation
of the $x$ contour can be deduced from the counterclockwise contour
orientation in $z_{2}$ by mapping $z_{1}\to 0,z_{3}\to 1,z_{4}\to\infty$.

$\begin{split}\int_{\mathcal{C}}\frac{dz_{2}}{2\pi i}~{}{\rm
disc}&\left[\frac{z_{34}}{z_{23}z_{24}}\frac{Q_{4}\bar{z}}{z-1+\bar{z}}{}_{2}F_{1}\Big{(}1,\frac{\Delta_{1}-\Delta_{3}+\Delta_{4}}{2},\Delta_{4};1-\frac{z}{z-1}\frac{\bar{z}}{\bar{z}-1}\Big{)}\right]\\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad=\frac{Q_{4}}{2\pi
i}\int_{1}^{\infty}\frac{dx}{x}~{}{\rm
disc}\left[{}_{2}F_{1}\Big{(}1,\frac{\Delta_{1}-\Delta_{3}+\Delta_{4}}{2},\Delta_{4};x\Big{)}\right],\end{split}$
(A.42)

where we simplified using the change of variables

$x=1-\frac{z}{z-1}\frac{\bar{z}}{\bar{z}-1},\quad\quad
dx=\frac{z_{13}z_{34}}{z_{41}z_{23}^{2}}\frac{\bar{z}}{\bar{z}-1}dz_{2}.$
(A.43)

The value of the discontinuity can be determined from the integral form
(A.31):

$\begin{split}{\rm
disc}\left[{}_{2}F_{1}(1,b,c;x)\right]&=\frac{1}{B(b,c-b)}\int_{0}^{1}\frac{du}{u(1-u)}~{}u^{b}(1-u)^{c-b}\lim_{\epsilon\to
0}\left[\frac{1}{1-ux-i\epsilon}-\frac{1}{1-ux+i\epsilon}\right]\\\
&=\frac{1}{B(b,c-b)}\int_{0}^{1}\frac{du}{u(1-u)}~{}u^{b}(1-u)^{c-b}2\pi
i\delta(1-ux)\\\ &=\frac{2\pi
i}{B(b,c-b)}\Theta(x-1)~{}x^{-c+1}(x-1)^{c-b-1}.\end{split}$ (A.44)

Substituting back into (A.42), we find

$\begin{split}\int_{\mathcal{C}}\frac{dz_{2}}{2\pi i}~{}{\rm
disc}&\left[\frac{z_{34}}{z_{23}z_{24}}\frac{Q_{4}\bar{z}}{z-1+\bar{z}}{}_{2}F_{1}\Big{(}1,\frac{\Delta_{1}-\Delta_{3}+\Delta_{4}}{2},\Delta_{4};1-\frac{z}{z-1}\frac{\bar{z}}{\bar{z}-1}\Big{)}\right]=Q_{4},\end{split}$
(A.45)

which is precisely the contribution needed for the consistency of (A.41).

The main lesson to take away from this appendix is that celestial operators
representing massive particles can admit non-local behavior. In particular, in
theories with charged massive particles, it appears that charge conservation
in celestial holography is still consistent with contour integrals of soft
photon currents, but massive particles may produce branch cut singularities
instead of the simple poles that are familiar from the context of a local
conformal field theory.

##  Appendix B ${\rm SL}(2,\mathbb{C})$ action on bulk and boundary points

In this appendix, we justify the statement in the introduction that the set of
${\rm SL}(2,\mathbb{C})$ transformations that preserve a point on the
celestial sphere preserve no proper subset of points in the bulk of ${\rm
AdS}_{3}$. For simplicity and without loss of generality, consider the
conformal transformations that preserve the origin on the celestial plane
$(z,\bar{z})=(0,0)$. These take the form of (2.3) with

$\begin{split}\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)=\left(\begin{matrix}1/d&0\\\
c&d\end{matrix}\right).\end{split}$ (B.1)

Selecting an arbitrary bulk point $(y_{k},w_{k},\bar{w}_{k})$, notice that the
transformation with

$\begin{split}\left(c,d\right)=\sqrt{\frac{y_{k}}{y_{k}^{2}+w_{k}\bar{w}_{k}}}\left(\bar{w}_{k}/y_{k},1\right),\end{split}$
(B.2)

maps the bulk point $(1,0,0)$ to this point $(y_{k},w_{k},\bar{w}_{k})$.
Likewise, the inverse transformation maps the arbitrarily point
$(y_{k},w_{k},\bar{w}_{k})$ to $(1,0,0)$. Therefore by suitably combining such
transformations, we can find an ${\rm SL}(2,\mathbb{C})$ transformation that
preserves the boundary point $(z,\bar{z})=(0,0)$, but maps between any pair of
bulk points.

##  Appendix C Derivation of massive generating function

In this appendix, we include more details about the derivation of (5.13).
Starting from (5.11) and using (5.12) we find:

$\displaystyle\mathbf{G}_{p}(\bar{z})$
$\displaystyle=-\frac{\mathcal{N}_{2p-4}\mathcal{C}_{p}}{2(2p-2)!m_{k}^{4p-6}}\frac{(-1)^{2p-3}}{(4p-6)!m_{k}^{4p-6}}$
(C.1)
$\displaystyle\qquad\times\sum_{n=0}^{4p-6}\left(-\partial_{z}\right)^{2p-2-n}(\hat{q}\cdot
p_{k})^{2p-2}\partial_{z}^{n}\left[\sum_{\ell=0}^{4p-6}\left[(-\partial_{\bar{z}})^{4p-6-\ell}(p_{k}\cdot\hat{q})^{4p-6}\right]~{}\partial_{\bar{z}}^{\ell}\left(F_{+}\cdot\mathcal{L}_{k}\right)^{2p-3}\right].$

Now consider the sum

$S=\sum_{n=0}^{4p-6}\left(-\partial_{z}\right)^{2p-2-n}\left(\hat{q}\cdot
p_{k}\right)^{2p-2}\partial_{z}^{n}\left[\sum_{\ell=0}^{4p-6}\left(-\partial_{\bar{z}}\right)^{4p-6-\ell}\left(\hat{q}\cdot
p_{k}\right)^{4p-6}\partial_{\bar{z}}^{\ell}\left(F_{+}\cdot\mathcal{L}_{k}\right)^{2p-3}\right].$
(C.2)

Using completeness (3.34) and the fact that

$\partial_{z}^{n}\left(\partial_{\bar{z}}\hat{q}\cdot
p_{k}\right)^{m}=\frac{m!}{(m-n)!}\left(\partial_{\bar{z}}\hat{q}\cdot
p_{k}\right)^{m-n}\left(n\cdot p_{k}\right)^{n},$ (C.3)

which is formally true for both positive and negative $n$, we can rewrite this
as

$\displaystyle S$
$\displaystyle=\sum_{\ell=0}^{4p-6}(m_{k}^{2})^{4p-6}(-1)^{\ell}\frac{\left(\partial_{z}\hat{q}\cdot
p_{k}\right)^{\ell-2p+4}}{(n\cdot
p_{k})^{\ell}}\partial_{\bar{z}}^{\ell}\left(F_{+}\cdot\mathcal{L}_{k}\right)^{2p-3}$
(C.4)
$\displaystyle\qquad\times\left[\sum_{n=0}^{4p-6}\sum_{m=0}^{2p-2}\sum_{j=0}^{4p-6}{2p-2\choose
m}{4p-6\choose
j}\frac{(-1)^{6p-8-n}m!j!j!\left(\frac{\partial_{\bar{z}}\hat{q}\cdot
p_{k}\partial_{z}\hat{q}\cdot
p_{k}}{m_{k}^{2}}\right)^{j+m-2p+2}}{(m-(2p-2-n))!(j-(4p-6-\ell))!(j-n)!}\right].$

To simplify the sum in brackets, consider the sum

$S^{\prime}=\sum_{n=0}^{4p-6}\sum_{m=0}^{2p-2}\sum_{j=0}^{4p-6}(-1)^{6p-8-n}{2p-2\choose
m}{4p-6\choose
j}\frac{m!j!j!A^{j+m-2p+2}}{(m-(2p-2-n))!(j-(4p-6-\ell))!(j-n)!},$ (C.5)

where $A$ does not depend on the indices in the sum. We re-index with $k=m+j$
and evaluate the sum in $n$ first:

$\displaystyle S^{\prime}$
$\displaystyle=\sum_{j=0}^{4p-6}\sum_{k=j}^{2p-2+j}\frac{(-1)^{2p}}{(k-2p+2)}\frac{(2p-2)!(4p-6)!A^{k-2p+2}}{(k-j-2p+1)!(2p-2-k+j)!(4p-6-j)!(j-4p+6+\ell)!}.$
(C.6)

Notice that if $k-j<2p+1$, the first factorial in the denominator diverges,
while if $k-j>2p-2$, the second factorial in the denominator diverges. This
will only be cancelled by the divergence in $\frac{1}{(k-2p+2)}$ if $k=2p-2$,
which also enforces $j\leq 2p-2$ since $k=m+j$. Therefore, the sum collapses
and we find, including normalization, that

$\displaystyle\mathbf{G}_{p}(\bar{z})=\frac{\mathcal{N}_{2p-4}\mathcal{C}_{p}}{2}\frac{1}{\left(-n\cdot
p_{k}\right)^{2p-4}}\sum_{\ell=0}^{4p-6}\frac{1}{\ell!}\left(\sum_{j=2p-4}^{4p-6}(-1)^{j}{\ell\choose
j}\right)\left(-\frac{\partial_{z}\hat{q}\cdot p_{k}}{n\cdot
p_{k}}\right)^{\ell-2p+4}\partial_{\bar{z}}^{\ell}\left(F_{+}\cdot\mathcal{L}_{k}\right)^{2p-3},$
(C.7)

which is manifestly zero when $\ell<2p-4$. Note that for $p>2$ and $\ell\leq
4p-6$ we have the simplification

$\sum_{j=2p-4}^{4p-6}(-1)^{j}{\ell\choose
j}=(-1)^{2p}\frac{(2p-4)}{\ell}{\ell\choose 2p-4},\ \ \ p>2\ {\rm and}\
\ell\leq 4p-6,$ (C.8)

which is used to obtain the simplified form (5.14).

##  Appendix D Details of massive momentum-space action

In this appendix, we provide details of the proof of the base case (4.13).
First, note that if we demonstrate that

$\begin{split}\left[\delta^{\frac{5}{2}}_{m},\delta^{q}_{n}\right]|p_{k}\rangle&=\left(m(q-1)-\frac{3}{2}n\right)\delta^{q+\frac{1}{2}}_{m+n}|p_{k}\rangle\end{split}$
(D.1)

holds for all modes $-\frac{3}{2}\leq m\leq\frac{3}{2}$ and a single mode
$n\in\left[1-q,q-1\right]$, then using the action of $\delta^{2}_{k}$ (4.11),
it can be shown that it also holds for all $n\in\left[1-q,q-1\right]$. This
statement is a consequence of $\rm{SL}(2,\mathbb{C})$ symmetry and the proof
is exactly as in Appendix C.2 of [44]. Then $\eqref{eq:w52proof}$ will follow
as a special case of (D.1) with $m=-\frac{3}{2}$. We will therefore be done if
we show that

$\begin{split}\left[\delta^{\frac{5}{2}}_{m},\delta^{q}_{1-q}\right]|p_{k}\rangle&=(q-1)\left(m+\frac{3}{2}\right)\delta^{q+\frac{1}{2}}_{1+m-q}|p_{k}\rangle\end{split}$
(D.2)

for $-\frac{3}{2}\leq m\leq\frac{3}{2}$. To do so, we use the explicit form of
$\delta^{q+\frac{1}{2}}_{1-q+m}$ for $-\frac{3}{2}\leq m\leq\frac{3}{2}$,
which can be derived from the integral form (4.1) or equivalently from the
generating function (5.13). The modes relevant to the commutator (D.2) have
action

$\displaystyle\delta^{q}_{1-q}|p_{k}\rangle$
$\displaystyle=\frac{1}{2}\left(\frac{2y_{k}}{\epsilon_{k}m_{k}}\right)^{2q-4}(2q-2)\partial_{\bar{w}_{k}}^{2q-3}|p_{k}\rangle,$
(D.3) $\displaystyle\delta^{q}_{2-q}|p_{k}\rangle$
$\displaystyle=\frac{1}{2}\left[(2q-2)\bar{w}_{k}\partial_{\bar{w}_{k}}+(2q-3)y_{k}\partial_{y_{k}}\right]\left(\frac{2y_{k}}{\epsilon_{k}m_{k}}\right)^{2q-4}\partial_{\bar{w}_{k}}^{2q-4}|p_{k}\rangle,$
$\displaystyle\delta^{q}_{3-q}|p_{k}\rangle$
$\displaystyle=\frac{1}{2}\left[(2q-2)\bar{w}_{k}^{2}\partial_{\bar{w}_{k}}^{2}+2(2q-3)y_{k}\partial_{y_{k}}\bar{w}_{k}\partial_{\bar{w}_{k}}+(2q-4)y_{k}\partial_{y_{k}}(y_{k}\partial_{y_{k}}-1)-2y_{k}^{2}\partial_{w_{k}}\partial_{\bar{w}_{k}}\right]$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\left(\frac{2y_{k}}{\epsilon_{k}m_{k}}\right)^{2q-4}\partial_{\bar{w}_{k}}^{2q-5}|p_{k}\rangle,$
$\displaystyle\delta^{q}_{4-q}|p_{k}\rangle$
$\displaystyle=\frac{1}{2}\Big{[}(2q-2)\bar{w}_{k}^{3}\partial_{\bar{w}_{k}}^{3}+3(2q-3)y_{k}\partial_{y_{k}}\bar{w}_{k}^{2}\partial_{\bar{w}_{k}}^{2}+3(2q-4)y_{k}\partial_{y_{k}}(y_{k}\partial_{y_{k}}-1)\bar{w}_{k}\partial_{\bar{w}_{k}}$
$\displaystyle\qquad+(2q-5)y_{k}\partial_{y_{k}}(y_{k}\partial_{y_{k}}-1)(y_{k}\partial_{y_{k}}-2)-6y_{k}^{2}\bar{w_{k}}\partial_{\bar{w}_{k}}^{2}\partial_{w_{k}}-6y_{k}^{3}\partial_{y_{k}}\partial_{w_{k}}\partial_{\bar{w}_{k}}\Big{]}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\left(\frac{2y_{k}}{\epsilon_{k}m_{k}}\right)^{2q-4}\partial_{\bar{w}_{k}}^{2q-6}|p_{k}\rangle.$

Using this form, one can verify (D.2) for $-\frac{3}{2}\leq m\leq\frac{3}{2}$
by brute force, completing the proof.

## References

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