††institutetext: Department of Physics, Brown University,
Providence, RI 02912, USA
<EMAIL_ADDRESS>
# Celestial Locality and the Jacobi Identity

Adam Ball

###### Abstract

We show the equivalence of several different tests of the Jacobi identity for
celestial currents at tree level, in particular finding a simple, practical
condition on hard momentum space 4-point amplitudes in any EFT. Along the way
we clarify the role of the order of soft and collinear limits in obstructing
the Jacobi identity for soft insertions and we argue that, despite their
current-algebra-like properties, soft insertions as formulated in this paper
cannot be interpreted as local operators in celestial conformal field theory.

##  1 Introduction

Taking inspiration from the holographic duality of quantum gravity in anti-de
Sitter space, celestial conformal field theory (CCFT) aims to characterize 4D
asymptotically flat quantum gravity by recasting its amplitudes as 2D
correlation functions on the celestial sphere Pasterski:2016qvg . For recent
reviews, see Raclariu:2021zjz ; Pasterski:2021rjz . The program of celestial
holography has seen remarkable progress in recent years, incorporating, among
other things, universal aspects of scattering such as soft and collinear
limits, which are respectively dual to soft currents and the celestial OPE
Donnay:2018neh ; Fan:2019emx ; Pate:2019mfs ; Adamo:2019ipt ; Puhm:2019zbl ;
Pate:2019lpp ; Himwich:2021dau . The all-orders-soft current algebras for pure
Yang-Mills theory and Einstein gravity were first displayed in Guevara:2021abz
, and it was soon realized in Strominger:2021lvk  that the modes of 2D light
transforms of soft gravitons form a $w_{1+\infty}$-wedge current algebra. This
algebra is unchanged by quantum effects in self-dual gravity Ball:2021tmb .
Despite these first few examples forming consistent algebras, it was shown in
Mago:2021wje ; Ren:2022sws  that the ostensible soft currents in many non-
minimally coupled EFTs fail to satisfy the Jacobi identity. Other works
involving the celestial Jacobi identity include Costello:2022wso ;
Costello:2022upu ; Costello:2022jpg ; Bittleston:2022jeq . In this paper we
continue the study of the Jacobi identity of celestial currents at tree level,
establishing the equivalence of several conditions for it and clarifying the
subtleties regarding the order of limits in the soft case. One of our main
takeaways is that soft insertions as formulated below cannot be interpreted as
local operators in CCFT,111There is a loophole in our argument if one allows
only integer conformal dimensions. despite the resemblance to a current
algebra of their action on hard massless insertions. It may be that the
appropriate local objects in CCFT are shadows of soft insertions, as in
Kapec:2016jld ; Kapec:2017gsg ; Kapec:2021eug ; Kapec:2022axw ; Kapec:2022hih
.

The outline of the paper is as follows. In section 2 we establish conventions.
In section 3 we briefly review the recent use of a double residue condition on
hard momentum space amplitudes to test the Jacobi identity of celestial
currents, and then we show using rather elementary tools that the double
residue condition on hard momentum space amplitudes is fully equivalent to a
simple condition on massless 4-point amplitudes. In section 4 we review some
properties of single soft insertions such as the locality of their poles and
the nonlocality of their relationship with collinear limits. In section 5 we
discuss some properties of multiple soft insertions, emphasizing the non-
commutativity (and therefore nonlocality) of soft limits. Finally in section 6
we discuss how to make the Jacobi identity for celestial soft currents well-
defined, and we show that some reasonable definitions are equivalent to the
aforementioned condition on hard amplitudes. We conclude in section 7.

##  2 Conventions

There are several candidates in the celestial holography literature for the
set of allowed values of the conformal dimension $\Delta$ Pasterski:2017kqt ;
Donnay:2020guq ; Atanasov:2021oyu . In this paper we are not primarily
concerned with the question of completeness, so we allow $\Delta$ to be an
arbitrary complex number. Mellin transforms encounter poles at integer values
of $\Delta$, so when we write a generic $\Delta$ we intend it to be non-
integer. There are many interesting proposals, including those in
Pasterski:2017kqt ; Sharma:2021gcz ; Fan:2021pbp ; Fan:2022vbz , for alternate
bases that involve integral transforms on the particles’ angles, but in this
paper we study only plane waves, conformal primary wavefunctions, and their
soft limits. We refer to these objects as insertions in (celestial)
amplitudes. We do not consider form factors nor amplitudes in nontrivial
backgrounds.

We parametrize our massless momenta as

$p^{\mu}=\epsilon\,\omega\,q^{\mu}$ (2.1)

where $\epsilon=\pm 1$ determines whether the momentum is future- or past-
directed and

$q^{\mu}=(1+z{\bar{z}},z+{\bar{z}},-i(z-{\bar{z}}),1-z{\bar{z}}).$ (2.2)

We parametrize a momentum with mass $m$ as

$p^{\mu}=\epsilon\,m\left(\frac{1+y^{2}+z{\bar{z}}}{2y},\frac{z+{\bar{z}}}{2y},\frac{-i(z-{\bar{z}})}{2y},\frac{1-y^{2}-z{\bar{z}}}{2y}\right).$
(2.3)

We use the all-outgoing convention for amplitudes, where crossing symmetry is
used to trade any incoming particles for outgoing particles with past-directed
momenta. Importantly, we treat $z,{\bar{z}}$ as independent variables. This
paper is only concerned with tree-level amplitudes, so we remain agnostic
about whether our momenta are Lorentzian or Kleinian Atanasov:2021oyu . To
restrict to real Kleinian momenta, choose $z,{\bar{z}}\in{\mathbb{R}}$ and
Wick rotate the third component. Our choice of polarization vectors can be
written simply in terms of $q^{\mu}$:

$\displaystyle\varepsilon_{+}^{\mu}$
$\displaystyle\equiv\frac{1}{\sqrt{2}}\partial_{z}q^{\mu}=\frac{1}{\sqrt{2}}({\bar{z}},1,-i,-{\bar{z}}),$
(2.4) $\displaystyle\varepsilon_{-}^{\mu}$
$\displaystyle\equiv\frac{1}{\sqrt{2}}\partial_{\bar{z}}q^{\mu}=\frac{1}{\sqrt{2}}(z,1,i,-z).$

They obey the usual relations

$0=\varepsilon_{\pm}\cdot
q=\varepsilon_{+}\cdot\varepsilon_{+}=\varepsilon_{-}\cdot\varepsilon_{-}$
(2.5)

and

$\varepsilon_{+}\cdot\varepsilon_{-}=1.$ (2.6)

##  3 Jacobi for hard insertions

A striking feature of the holomorphic celestial OPE Fan:2019emx ; Pate:2019lpp
is that it always comes with a factor of $1/z$ Himwich:2021dau , as opposed to
some non-integer power of $z$, no matter the weights of the operators
involved.222A simple example of an OPE lacking this property is that between
two vertex operators in the free boson CFT. In any CFT we are always free to
analytically continue $z$ and ${\bar{z}}$ separately, but in general there is
no guarantee that the resulting $z$ dependence will be single-valued. The
celestial OPE guarantees that in CCFT it is single-valued, at least at leading
order near massless insertions. Consequently the holomorphic OPEs of generic
massless CCFT operators ${\mathcal{O}}_{\Delta_{i}}(z_{i},{\bar{z}}_{i})$
naïvely resemble a holomorphic current algebra, whether or not the
${\mathcal{O}}_{\Delta_{i}}(z_{i},{\bar{z}}_{i})$ are soft. In light of this
fact, one can ask whether these objects satisfy the current algebra version of
the Jacobi identity, i.e. whether

$\displaystyle 0$
$\displaystyle\stackrel{{\scriptstyle?}}{{=}}\oint_{|z_{23}|=\varepsilon}\frac{dz_{2}}{2\pi
i}{\mathcal{O}}_{\Delta_{3}}(z_{3},{\bar{z}}_{3})\oint_{|z_{12}|=\varepsilon}\frac{dz_{1}}{2\pi
i}{\mathcal{O}}_{\Delta_{1}}(z_{1},{\bar{z}}_{1}){\mathcal{O}}_{\Delta_{2}}(z_{2},{\bar{z}}_{2})$
(3.1) $\displaystyle\qquad-\oint_{|z_{13}|=\varepsilon}\frac{dz_{1}}{2\pi
i}{\mathcal{O}}_{\Delta_{1}}(z_{1},{\bar{z}}_{1})\oint_{|z_{23}|=\varepsilon}\frac{dz_{2}}{2\pi
i}{\mathcal{O}}_{\Delta_{2}}(z_{2},{\bar{z}}_{2}){\mathcal{O}}_{\Delta_{3}}(z_{3},{\bar{z}}_{3})$
$\displaystyle\qquad\qquad+\oint_{|z_{23}|=\varepsilon}\frac{dz_{2}}{2\pi
i}{\mathcal{O}}_{\Delta_{2}}(z_{2},{\bar{z}}_{2})\oint_{|z_{13}|=\varepsilon}\frac{dz_{1}}{2\pi
i}{\mathcal{O}}_{\Delta_{3}}(z_{3},{\bar{z}}_{3}){\mathcal{O}}_{\Delta_{1}}(z_{1},{\bar{z}}_{1})$

where $z_{ij}=z_{i}-z_{j}$. This can be rewritten compactly as a “double
residue condition”:

$0\stackrel{{\scriptstyle?}}{{=}}\left(\underset{z_{2}\shortrightarrow
z_{3}}{\rm Res}\,\underset{z_{1}\shortrightarrow z_{2}}{\rm
Res}-\underset{z_{1}\shortrightarrow z_{3}}{\rm
Res}\,\underset{z_{2}\shortrightarrow z_{3}}{\rm
Res}+\underset{z_{2}\shortrightarrow z_{3}}{\rm
Res}\,\underset{z_{1}\shortrightarrow z_{3}}{\rm
Res}\right){\mathcal{O}}_{\Delta_{1}}(z_{1},{\bar{z}}_{1}){\mathcal{O}}_{\Delta_{2}}(z_{2},{\bar{z}}_{2}){\mathcal{O}}_{\Delta_{3}}(z_{3},{\bar{z}}_{3}).$
(3.2)

In both equations it is to be understood that the full celestial amplitude
contains arbitrary other insertions away from $z_{1},z_{2},z_{3}$. Genuine
currents in a local CFT are guaranteed to satisfy this condition by contour
pulling, but the holomorphy of our celestial “hard currents” is only known at
leading order near massless insertions. Consequently the condition must be
checked directly. Converting the collinear limit to the celestial OPE, as in
Pate:2019lpp , relies on the fact that taking a residue on $z_{i}\to z_{j}$
commutes with Mellin transforming on $\omega_{i},\omega_{j}$.333See appendix A
for a discussion of how to take residues of unstripped amplitudes. This allows
us to convert the double residue condition on celestial amplitudes to the same
one on momentum space amplitudes:

$0\stackrel{{\scriptstyle?}}{{=}}\left(\underset{z_{2}\shortrightarrow
z_{3}}{\rm Res}\,\underset{z_{1}\shortrightarrow z_{2}}{\rm
Res}-\underset{z_{1}\shortrightarrow z_{3}}{\rm
Res}\,\underset{z_{2}\shortrightarrow z_{3}}{\rm
Res}+\underset{z_{2}\shortrightarrow z_{3}}{\rm
Res}\,\underset{z_{1}\shortrightarrow z_{3}}{\rm
Res}\right){\mathcal{O}}_{1}(\omega_{1},z_{1},{\bar{z}}_{1}){\mathcal{O}}_{2}(\omega_{2},z_{2},{\bar{z}}_{2}){\mathcal{O}}_{3}(\omega_{3},z_{3},{\bar{z}}_{3}).$
(3.3)

This is the condition studied in Ren:2022sws . They tested it for a large
family of EFTs and found that it generically fails. We find that this failure
can be traced to three-particle factorization channels and their associated
nonlocal poles in $z_{i}$ in momentum space.444It is unclear what happens to a
nonlocal pole in $z_{i}$ in momentum space upon transforming to Mellin space.
Mellin transforming first on $\omega_{i}$ presumably gives some nonlocal
branch point in $z_{i}$, but after all $\omega_{j}$ have been Mellin
transformed it is unclear what kind of singularities in $z_{i}$ to expect. By
“nonlocal” we mean that the poles in $z_{i}$ are not located at $z_{j}$ for
any other insertion ${\mathcal{O}}_{j}$. These stand in contrast with
collinear poles, which are local.

Figure 1:  Feynman diagrams that can contribute to the double residue
condition on particles $1,2,3$. Dotted lines indicate massless bosons, solid
lines indicate any particle in the theory, and the shaded spheres indicate any
tree-level process.

To be concrete, let $\alpha_{1},\alpha_{2},\alpha_{3}$ label massless bosons
of any helicity in any local, unitary, Lorentz-invariant EFT, and consider the
tree-level $n\geq 6$-point momentum space amplitude
${\mathcal{A}}_{\alpha_{1}\alpha_{2}\alpha_{3}\dots}$. View it as a rational
function of $z_{1},z_{2},z_{3}$.555 To make this well-defined choose two
momenta $p_{i},p_{j}$ with $i,j\geq 4$. If $p_{i}$ is massless use momentum
conservation to eliminate $\omega_{i}$ and ${\bar{z}}_{i}$, and if it is
massive eliminate $y_{i}$ and ${\bar{z}}_{i}$. Do the same for $p_{j}$. See
appendix A for a more detailed discussion. Generically the only poles near
$z_{1}=z_{2}=z_{3}$ are the three $z_{12},z_{13},z_{23}$ poles from collinear
factorization, and the pole from three-particle factorization where
$(p_{1}+p_{2}+p_{3})^{2}=0$. Only Feynman diagrams containing two such poles
can contribute to the double residue condition. The candidates are shown in
figure 1. They all factorize on $(p_{1}+p_{2}+p_{3})^{2}=0$, so they all
contain a factor of (a polynomial times) the following propagator,

$\frac{1}{(p_{1}+p_{2}+p_{3})^{2}}=\frac{-1/4}{\epsilon_{1}\epsilon_{2}\omega_{1}\omega_{2}z_{12}{\bar{z}}_{12}+\epsilon_{1}\epsilon_{3}\omega_{1}\omega_{3}z_{13}{\bar{z}}_{13}+\epsilon_{2}\epsilon_{3}\omega_{2}\omega_{3}z_{23}{\bar{z}}_{23}}.$
(3.4)

This factor provides a pole at some $z_{1}=z_{*}$, but it is nonlocal in the
sense that $z_{*}\neq z_{i}$ for any other $z_{i}$. This provides an
obstruction to the contour pulling argument and can cause the double residue
condition to fail. Before moving on let us briefly establish some notation.
Thinking in terms of Feynman diagrams, write our amplitude as

${\mathcal{A}}_{\alpha_{1}\alpha_{2}\alpha_{3}\dots}=\delta^{(4)}\left(\sum_{i=1}^{n}p_{i}\right)\left(\sum_{\hat{\alpha}_{I}}A^{\hat{\alpha}_{I}}_{L}\frac{1}{(p_{1}+p_{2}+p_{3})^{2}}A^{\hat{\alpha}_{I}}_{R}\right)+\dots$
(3.5)

where the sum is over massless particles contributing to the
$p_{1}+p_{2}+p_{3}$ factorization channel, $A^{\hat{\alpha}_{I}}_{L}$ collects
the Feynman diagram factors to the left of the $\hat{\alpha}_{I}$
propagator,666When $\hat{\alpha}_{I}$ has spin then
$A_{L/R}^{\hat{\alpha}_{I}}$ has suppressed indices.
$A_{R}^{\hat{\alpha}_{I}}$ collects the Feynman diagram factors to the right
of the $\hat{\alpha}_{I}$ propagator, and “$\dots$” indicates the contribution
of diagrams not factorizing on $(p_{1}+p_{2}+p_{3})^{2}=0$, which cannot
contribute to the double residue condition. Note also that
$A^{\hat{\alpha}_{I}}_{R}$ generically contains no poles in the neighborhood
of $z_{1}=z_{2}=z_{3}$.

It was pointed out in Guevara:2022qnm  that the double residue condition (3.3)
is equivalent to a second-order residue. There are always many equivalent ways
to write a second-order residue in terms of first-order residues, and here one
way turns out to be particularly illuminating. We find that

$\underset{z_{2}\shortrightarrow z_{3}}{\rm
Res}\underset{z_{1}\shortrightarrow z_{*}}{\rm
Res}{\mathcal{A}}_{\alpha_{1}\alpha_{2}\alpha_{3}\dots}=-\left(\underset{z_{2}\shortrightarrow
z_{3}}{\rm Res}\,\underset{z_{1}\shortrightarrow z_{2}}{\rm
Res}-\underset{z_{1}\shortrightarrow z_{3}}{\rm
Res}\,\underset{z_{2}\shortrightarrow z_{3}}{\rm
Res}+\underset{z_{2}\shortrightarrow z_{3}}{\rm
Res}\,\underset{z_{1}\shortrightarrow z_{3}}{\rm
Res}\right){\mathcal{A}}_{\alpha_{1}\alpha_{2}\alpha_{3}\dots}.$ (3.6)

This can be confirmed explicitly by plugging in a general expression for
$A_{L}^{\hat{\alpha}_{I}}$. Recall that tree-level Feynman diagrams are
rational functions of momentum components, with the denominators coming only
from propagators. Thus for some polynomials $f^{\hat{\alpha}_{I}}_{ij}$ we can
write

$A^{\hat{\alpha}_{I}}_{L}=\frac{f^{\hat{\alpha}_{I}}_{12}}{(p_{1}+p_{2})^{2}}+\frac{f^{\hat{\alpha}_{I}}_{13}}{(p_{1}+p_{3})^{2}}+\frac{f^{\hat{\alpha}_{I}}_{23}}{(p_{2}+p_{3})^{2}}+\dots$
(3.7)

where “$\dots$” indicates massive exchange and contact terms, which have no
poles in the neighborhood of $z_{1}=z_{2}=z_{3}$.

The left hand side of (3.6) has a very clean interpretation. The first residue
gives

$\underset{z_{1}\shortrightarrow z_{*}}{\rm
Res}{\mathcal{A}}_{\alpha_{1}\alpha_{2}\alpha_{3}\dots}=\delta^{(4)}\left(\sum_{i=1}^{n}p_{i}\right)\sum_{\alpha_{I}}A_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{I}}\frac{-1/4}{\epsilon_{1}\omega_{1}(\epsilon_{2}\omega_{2}{\bar{z}}_{12}+\epsilon_{3}\omega_{3}{\bar{z}}_{13})}A_{-\alpha_{I}\dots}$
(3.8)

where now $A^{\hat{\alpha}_{I}}_{L/R}$ have been replaced by bona fide
(stripped) amplitudes and the sum is over helicities as well as particles. It
is still true that $A_{-\alpha_{I}\dots}$ contains no poles in the
neighborhood of $z_{1}=z_{2}=z_{3}$, so we can bring the second residue inside
the sum as

$\underset{z_{2}\shortrightarrow z_{3}}{\rm
Res}\,\underset{z_{1}\shortrightarrow z_{*}}{\rm
Res}{\mathcal{A}}_{\alpha_{1}\alpha_{2}\alpha_{3}\dots}=\delta^{(4)}\left(\sum_{i=1}^{n}p_{i}\right)\sum_{\alpha_{I}}\left(\underset{z_{2}\shortrightarrow
z_{3}}{\rm
Res}A_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{I}}\right)\frac{-1/4}{\epsilon_{1}\omega_{1}(\epsilon_{2}\omega_{2}{\bar{z}}_{12}+\epsilon_{3}\omega_{3}{\bar{z}}_{13})}A_{-\alpha_{I}\dots}.$
(3.9)

Focus now on the residue term in parentheses. After taking the residue
$z_{1}\to z_{*}$ there is no more $z_{1}$ dependence, and we never introduced
$p_{I}$ for $\alpha_{I}$, so the form of the 4-point amplitude
$A_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{I}}$ is as if we had used momentum
conservation to eliminate $z_{1}$ and all three on-shell parameters of
$p_{I}$. Then the only $z_{ij}$ that can show up in
$A_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{I}}$ is $z_{23}$, and the residue on
$z_{2}\to z_{3}$ just grabs the coefficient of $1/z_{23}$ in the Laurent
expansion in $z_{23}$. In spinor-helicity variables we would say that the only
angle bracket allowed is $\langle 23\rangle$, and the residue grabs from the
$1/\langle 23\rangle$ term. Applications of momentum conservation can make an
amplitude unrecognizable, but they can never change the angle bracket weight,
defined here as the number of angle bracket products in the numerator minus
the number in the denominator.777In a case like $\frac{1}{m^{2}-\langle
23\rangle[23]}$ we should expand as
$\frac{1}{m^{2}}\sum_{n=0}^{\infty}\big{(}\langle
23\rangle[23]/m^{2}\big{)}^{n}$ so that each term has definite angle bracket
weight. So no matter what form of
$A_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{I}}$ we are given, we can always
read off the angle bracket weight $-1$ part as the obstruction to the double
residue condition. We note that this is the lowest possible weight for a tree-
level 4-point amplitude.

If the angle bracket weight $-1$ part of
$A_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{I}}$ vanishes for all $\alpha_{I}$,
then (3.9) shows that the double residue condition will be satisfied for
arbitrary “…” in ${\mathcal{A}}_{\alpha_{1}\alpha_{2}\alpha_{3}\dots}$.
Conversely, suppose there exists an $\alpha_{I}$ such that the angle bracket
weight $-1$ part of $A_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{I}}$ is nonzero.
Then there exists an $n\geq 6$-point amplitude failing the double residue
condition, as follows. The idea is just to use CPT conjugates to get a sum
over manifestly non-negative terms. Consider the 6-point amplitude
${\mathcal{A}}_{\alpha_{1}\alpha_{2}\alpha_{3}\bar{\alpha}_{1}\bar{\alpha}_{2}\bar{\alpha}_{3}}$,
where $\bar{\alpha}_{1},\bar{\alpha}_{2},\bar{\alpha}_{3}$ are the conjugate
particles to $\alpha_{1},\alpha_{2},\alpha_{3}$. Use primes to denote the
momenta of the $\bar{\alpha}_{i}$. We have

$\underset{z_{2}\shortrightarrow z_{3}}{\rm
Res}\,\underset{z_{1}\shortrightarrow z_{*}}{\rm
Res}{\mathcal{A}}_{\alpha_{1}\alpha_{2}\alpha_{3}\bar{\alpha}_{1}\bar{\alpha}_{2}\bar{\alpha}_{3}}=\frac{-\frac{1}{4}\delta^{(4)}\left(\sum_{i=1}^{3}p_{i}+p^{\prime}_{i}\right)}{\epsilon_{1}\omega_{1}(\epsilon_{2}\omega_{2}{\bar{z}}_{12}+\epsilon_{3}\omega_{3}{\bar{z}}_{13})}\sum_{\alpha_{I}}\left(\underset{z_{2}\shortrightarrow
z_{3}}{\rm
Res}A_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{I}}\right)A_{\bar{\alpha}_{I}\bar{\alpha}_{1}\bar{\alpha}_{2}\bar{\alpha}_{3}}.$
(3.10)

We are free to take a further residue on $z^{\prime}_{2}\to
z^{\prime}_{3}$888See appendix A for a discussion of how to take residues of
constrained functions. and then set all $p^{\prime}_{i}=-p_{i}$, which gives

$\displaystyle\Big{(}\underset{z^{\prime}_{2}\shortrightarrow
z^{\prime}_{3}}{\rm Res}\underset{z_{2}\shortrightarrow z_{3}}{\rm
Res}\,\underset{z_{1}\shortrightarrow z_{*}}{\rm
Res}{\mathcal{A}}_{\alpha_{1}\alpha_{2}\alpha_{3}\bar{\alpha}_{1}\bar{\alpha}_{2}\bar{\alpha}_{3}}\Big{)}\Big{|}_{p^{\prime}_{i}=-p_{i}}$
(3.11) $\displaystyle\hskip
56.9055pt=\frac{-\frac{1}{4}\delta^{(4)}\left(\sum_{i=1}^{3}p_{i}+p^{\prime}_{i}\right)}{\epsilon_{1}\omega_{1}(\epsilon_{2}\omega_{2}{\bar{z}}_{12}+\epsilon_{3}\omega_{3}{\bar{z}}_{13})}\sum_{\alpha_{I}}\Big{(}\underset{z_{2}\shortrightarrow
z_{3}}{\rm
Res}A_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{I}}\Big{)}\Big{(}\underset{z^{\prime}_{2}\shortrightarrow
z^{\prime}_{3}}{\rm
Res}A_{\bar{\alpha}_{I}\bar{\alpha}_{1}\bar{\alpha}_{2}\bar{\alpha}_{3}}\Big{)}\Big{|}_{p^{\prime}_{i}=-p_{i}}$
$\displaystyle\hskip
56.9055pt=\frac{-\frac{1}{4}\delta^{(4)}(0)}{\epsilon_{1}\omega_{1}(\epsilon_{2}\omega_{2}{\bar{z}}_{12}+\epsilon_{3}\omega_{3}{\bar{z}}_{13})}\sum_{\alpha_{I}}\Big{|}\underset{z_{2}\shortrightarrow
z_{3}}{\rm Res}A_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{I}}\Big{|}^{2}.$

Since we are assuming that at least one $\underset{z_{2}\shortrightarrow
z_{3}}{\rm Res}A_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{I}}$ is nonzero, this
sum cannot vanish. This completes the proof of the following:

Theorem The double residue condition (3.3) on massless bosons
$\alpha_{1},\alpha_{2},\alpha_{3}$ fails if and only if there exists a
massless boson $\alpha_{I}$ such that the angle bracket weight $-1$ part of
the 4-point amplitude $A_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{I}}$ is
nonzero.

This strengthens and generalizes some of the observations in Ren:2022sws . In
particular the angle bracket weight $-1$ part of a 4-point amplitude is
equivalent to its all-line shift constructible part. This condition also
applies directly to the holographic chiral algebras of Monteiro:2022lwm .

Sometimes it is more useful to work with momentum weight rather than angle
bracket weight, where e.g. $p_{i}\cdot p_{j}$ has momentum weight $+2$ and
$\varepsilon_{\pm,i}\cdot p_{j}$ has momentum weight $+1$. Let $W_{a}$ denote
angle bracket weight, $W_{s}$ denote square bracket weight, and $W_{m}$ denote
momentum weight. They are related as $W_{m}=W_{a}+W_{s}$. Furthermore little
group scaling implies $W_{a}-W_{s}=-\sum_{i}s_{i}$, where $s_{i}$ is the
helicity of the $i$th external leg of our amplitude. Combining these gives

$W_{m}=2W_{a}+\sum_{i}s_{i},$ (3.12)

so that angle bracket weight $W_{a}=-1$ corresponds to momentum weight

$W_{m}=-2+\sum_{i}s_{i}.$ (3.13)

##  4 Single soft insertions

In this section we review some of the properties of single soft insertions.
Let ${\mathcal{O}}(\omega,z,{\bar{z}})$ denote a massless insertion in a
momentum space amplitude where all other insertions are hard. It is a well-
defined rational function of $\omega,z,{\bar{z}}$ once we enforce momentum
conservation as discussed in appendix A. In doing so, for convenience choose
not to eliminate any of the $z_{i}$ of the other insertions. We can Laurent
expand near $\omega=0$, giving

${\mathcal{O}}(\omega,z,{\bar{z}})=\sum_{k=-\infty}^{1}\frac{{\mathcal{O}}^{(k)}(z,{\bar{z}})}{\omega^{k}}.$
(4.1)

The ${\mathcal{O}}^{(k)}(z,{\bar{z}})$ are energetically soft insertions. For
later convenience we will also define $S^{(k)}$ to grab the coefficient of
$\omega^{-k}$ in the Laurent series around $\omega=0$ of whatever function it
is acting on. Then by definition

$S^{(k)}{\mathcal{O}}(\omega,z,{\bar{z}})={\mathcal{O}}^{(k)}(z,{\bar{z}}).$
(4.2)

As discussed in Guevara:2019ypd , there is a theorem stating that the Laurent
coefficient ${\mathcal{O}}^{(k)}(z,{\bar{z}})$ is equal to the residue at
$\Delta\to k$ of the Mellin transform of ${\mathcal{O}}(\omega,z,{\bar{z}})$.
That is, the energetically and conformally soft limits give the same objects,
at least in this context of handling one insertion at a time. In particular,
${\mathcal{O}}^{(k)}(z,{\bar{z}})$ is a conformal primary despite being
constructed from momentum space. Note also that its $z$ dependence is
manifestly single-valued around momentum space insertions. Sometimes we will
abuse notation and let $S^{(k)}$ act on a function of $\Delta$ rather than
$\omega$, in which case it should be understood that it corresponds to the
residue at $\Delta\to k$. As a final preliminary comment, note that rational
functions are equal to their Laurent expansions, so ${\mathcal{O}}$ can be
recovered from its soft modes ${\mathcal{O}}^{(k)}$ and in this sense they
form a complete set of insertions.

Since $\omega{\mathcal{O}}(\omega,z,{\bar{z}})$ is analytic in $\omega$ near
$\omega=0$, we can compute the Laurent coefficient
$S^{(k)}{\mathcal{O}}(\omega,z,{\bar{z}})$ by acting on
${\mathcal{O}}(\omega,z,{\bar{z}})$ with the differential operator
$\frac{1}{(1-k)!}\partial_{\omega}^{1-k}\omega$ and then setting $\omega=0$.
We know that $\omega{\mathcal{O}}(\omega,z,{\bar{z}})$ is a rational function
of $\omega,z,{\bar{z}}$, so it can be written as a ratio of polynomials in
$\omega,z,{\bar{z}}$. Differentiating by $\omega$ cannot produce any new
factors in the denominator; it can only increase the powers of factors that
are already there. So the only possible $z$ poles (of any order) in
${\mathcal{O}}^{(k)}(z,{\bar{z}})$ are those corresponding to $z$ poles in
$\omega{\mathcal{O}}(\omega,z,{\bar{z}})|_{\omega=0}$. But $z$ poles in
$\omega{\mathcal{O}}(\omega,z,{\bar{z}})$ only come from two-particle
factorization, higher-particle factorization, and places where the coordinates
degenerate. The latter two types of poles go away when $\omega=0$ since
$p(\omega,z,{\bar{z}})$ simply drops out of the expressions. Therefore the
only poles in ${\mathcal{O}}^{(k)}(z,{\bar{z}})$ are those coming from two-
particle factorization. In the massless case these are just the familiar
collinear poles. In the massless-massive case they come from the massive
propagator

$\frac{1}{m^{2}+\big{(}p(\omega,z,{\bar{z}})+p(y,z^{\prime},{\bar{z}}^{\prime};m)\big{)}^{2}}=\frac{1}{2p(\omega,z,{\bar{z}})\cdot
p(y,z^{\prime},{\bar{z}}^{\prime};m)}=\frac{y}{-2m\epsilon\epsilon^{\prime}\omega(y^{2}+|z-z^{\prime}|^{2})}.$
(4.3)

We see this has a pole at
$z=z^{\prime}-\frac{y^{2}}{{\bar{z}}-{\bar{z}}^{\prime}}$. Technically this is
nonlocal in this paper’s parlance, but its location does not depend on any of
the other massless $z_{i}$, so it is irrelevant to computing the double
residue condition on massless insertions. Now that we have deduced all the $z$
poles of the rational function ${\mathcal{O}}^{(k)}(z,{\bar{z}})$, we can
write it as a sum over poles plus a polynomial in $z$,999This expression is
for a momentum space amplitude, but changing to a conformal primary basis for
the hard insertions would leave much of it qualitatively unchanged. In
particular there would still be a sum over massless poles, a sum over terms
from massless-massive two-particle factorization, and a polynomial term
collecting the rest of the $z$ dependence.

${\mathcal{O}}^{(k)}(z,{\bar{z}})=\sum_{i}\frac{f_{i}^{(k)}({\bar{z}})}{z-z_{i}}+\sum_{j}\frac{g_{j}^{(k)}({\bar{z}})}{z-z_{j}+\frac{y_{j}^{2}}{{\bar{z}}-{\bar{z}}_{j}}}+P^{(k)}(z,{\bar{z}}).$
(4.4)

Here $i$ ranges over the massless hard momenta, and $f_{i}^{(k)}({\bar{z}})$
is determined by soft-collinear limits. Likewise $j$ ranges over the massive
momenta, and $g_{i}^{(k)}({\bar{z}})$ is determined by massless-massive two-
particle factorization channels. Although $P^{(k)}(z,{\bar{z}})$ is polynomial
in $z$, it is not necessarily polynomial in ${\bar{z}}$. In the cases where
universal soft theorems apply we will have $P^{(k)}(z,{\bar{z}})=0$, but this
is not the case in general. The point here is that we have substantial
analytic control over the $z$ dependence of single soft insertions.

The soft and collinear limits $S^{(k)}$ and $\underset{z\shortrightarrow
z_{i}}{\rm Res}$ (where particle $i$ is massless) commute since
$\omega(z-z_{i}){\mathcal{O}}(\omega,z,{\bar{z}})$ is analytic in $\omega,z$
in a neighborhood of $\omega=0,z=z_{i}$, so the differential operator
implementing $S^{(k)}$ commutes with setting $z=z_{i}$. This commutativity is
implicit in Guevara:2021abz ; Himwich:2021dau ; Mago:2021wje ; Ren:2022sws .
Note that the ${\bar{z}}$ modes of ${\mathcal{O}}^{(k)}(z,{\bar{z}})$ will be
holomorphic with simple poles coming only from collinear limits (and from
massless-massive two-particle factorization that we do not care about in this
context). As argued in Guevara:2021abz , these
${\mathcal{O}}^{(k)}(z,{\bar{z}})$ are tantamount to symmetry-generating
currents. However we will argue below that they cannot actually be interpreted
as local operators in CCFT.

As an example, consider two positive-helicity outgoing gluons
${\mathcal{O}}^{+,a}(\omega_{1},z_{1},{\bar{z}}_{1})$ and
${\mathcal{O}}^{+,b}(\omega_{2},z_{2},{\bar{z}}_{2})$ in pure Yang-Mills.
Assume the variables
$\omega_{1},z_{1},{\bar{z}}_{1},\omega_{2},z_{2},{\bar{z}}_{2}$ have not been
eliminated. The collinear limit is

${\mathcal{O}}^{+,a}(\omega_{1},z_{1},{\bar{z}}_{1}){\mathcal{O}}^{+,b}(\omega_{2},z_{2},{\bar{z}}_{2})\sim\frac{-i{f^{ab}}_{c}}{z_{12}}\frac{\omega_{1}+\omega_{2}}{\omega_{1}\omega_{2}}{\mathcal{O}}^{+,c}\big{(}\omega_{1}+\omega_{2},z_{2},\frac{\omega_{1}{\bar{z}}_{1}+\omega_{2}{\bar{z}}_{2}}{\omega_{1}+\omega_{2}}\big{)}.$
(4.5)

To compute the collinear limit with the soft insertion
$[{\mathcal{O}}^{+,a}]^{(0)}$ we expand in $\omega_{1}$ and grab the
$O(\omega_{1}^{0})$ piece, giving

$[{\mathcal{O}}^{+,a}]^{(0)}(z_{1},{\bar{z}}_{1}){\mathcal{O}}^{+,b}(\omega_{2},z_{2},{\bar{z}}_{2})\sim\frac{-i{f^{ab}}_{c}}{z_{12}}\left(\partial_{\omega_{2}}+\frac{1}{\omega_{2}}+\frac{{\bar{z}}_{12}}{\omega_{2}}\partial_{{\bar{z}}_{2}}\right){\mathcal{O}}^{+,c}(\omega_{2},z_{2},{\bar{z}}_{2}).$
(4.6)

The term in parentheses is the subleading soft gluon factor Casali:2014xpa .
The dependence on ${\bar{z}}_{1}$ is linear, and in general the
${\bar{z}}_{1}$ dependence of the collinear limit of
$[{\mathcal{O}}^{+,a}]^{(k)}(z_{1},{\bar{z}}_{1}){\mathcal{O}}^{+,b}(\omega_{2},z_{2},{\bar{z}}_{2})$
will be polynomial of degree $1-k$. Consequently if we expand
$[{\mathcal{O}}^{+,a}]^{(k)}(z_{1},{\bar{z}}_{1})$ in ${\bar{z}}_{1}$ then the
holomorphic coefficients of ${\bar{z}}_{1}^{m}$ with $m<0$ and $m>1-k$ will
not see this collinear pole. We sometimes say that such modes are “outside the
wedge”. Very similar statements hold for soft-collinear limits in general
EFTs. Finally we note that Mellin transforming on $\omega_{2}$ recovers the
soft-hard celestial OPE Guevara:2021abz , which takes the following closed
form for general $k$,

$[{\mathcal{O}}^{+,a}]^{(k)}(z_{1},{\bar{z}}_{1}){\mathcal{O}}^{+,b}_{\Delta_{2}}(z_{2},{\bar{z}}_{2})\sim\frac{-i{f^{ab}}_{c}}{z_{12}}\sum_{m=0}^{1-k}{2-k-\Delta_{2}-m\choose
1-\Delta_{2}}\frac{{\bar{z}}_{12}^{m}}{m!}\partial_{{\bar{z}}_{2}}^{m}{\mathcal{O}}^{+,c}_{\Delta_{2}+k-1}(z_{2},{\bar{z}}_{2}),$
(4.7)

where the binomial coefficient is defined as ${x\choose
y}\equiv\frac{\Gamma(x+1)}{\Gamma(y+1)\Gamma(x-y+1)}$.

Despite the striking resemblance of the $z_{1}$ dependence of
${\mathcal{O}}^{(k)}(z_{1},{\bar{z}}_{1})$ to that of a holomorphic current in
CFT, ${\mathcal{O}}^{(k)}(z_{1},{\bar{z}}_{1})$ cannot be interpreted as a
local operator in CCFT. If it were local then the OPE of two operators far
from it could not be affected by it, meaning that in pure Yang-Mills we would
have

$\displaystyle\underset{z_{1}\shortrightarrow z_{2}}{\rm Res}$
$\displaystyle\left({\mathcal{O}}^{+,a_{1}}_{\Delta_{1}}(z_{1},{\bar{z}}_{1}){\mathcal{O}}^{+,a_{2}}_{\Delta_{2}}(z_{2},{\bar{z}}_{2})[{\mathcal{O}}^{+,a_{3}}]^{(k)}(z_{3},{\bar{z}}_{3})\right)\stackrel{{\scriptstyle?}}{{=}}$
(4.8)
$\displaystyle-i{f^{a_{1}a_{2}}}_{b}\sum_{m=0}^{\infty}B(\Delta_{1}-1+m,\Delta_{2}-1)\frac{{\bar{z}}_{12}^{m}}{m!}\partial_{{\bar{z}}_{2}}^{m}{\mathcal{O}}^{+,b}_{\Delta_{1}+\Delta_{2}-1}(z_{2},{\bar{z}}_{2})[{\mathcal{O}}^{+,a_{3}}]^{(k)}(z_{3},{\bar{z}}_{3}).$

But the left hand side is defined by taking the soft limit before the residue,
while the right hand side is what we would get if we took the residue first.
Equality would imply that the residue and the soft limit commute, which is not
true even in pure Yang-Mills. If we consider a celestial MHV amplitude with
color ordering
$\dots\mathcal{O}^{+}_{\Delta_{1}}(z_{1},{\bar{z}}_{1}){\mathcal{O}}^{+,(k)}(z_{3},{\bar{z}}_{3}){\mathcal{O}}^{+}_{\Delta_{2}}(z_{2},{\bar{z}}_{2})\dots$
then taking ${\rm Res}_{z_{1}\shortrightarrow z_{2}}$ first gives zero whereas
taking the soft limit first gives something nonzero. In general this happens
because when a particle goes soft its three-particle factorization channels
become two-particle factorization channels for the remaining hard particles.
But the physical simplicity does not change the fact that it is inconsistent
with ${\mathcal{O}}^{(k)}$ being a local operator in CCFT.101010If one allows
only integer values of $\Delta$ then the soft limit is part of the definition
of CCFT, as opposed to being a limit of an object within CCFT, and so the
preceding argument may not apply. We will say more about the nonlocality of
soft insertions in the following sections.

##  5 Multiple soft insertions

In this section we discuss properties of amplitudes in which multiple
insertions have been taken soft. Once we choose an order of limits for a
momentum space amplitude, e.g.

$S_{3}^{(k_{3})}S_{2}^{(k_{2})}S_{1}^{(k_{1})}{\mathcal{O}}_{1}(\omega_{1},z_{1},{\bar{z}}_{1}){\mathcal{O}}_{2}(\omega_{2},z_{2},{\bar{z}}_{2}){\mathcal{O}}_{3}(\omega_{3},z_{3},{\bar{z}}_{3}),$
(5.1)

then many of the properties of single soft insertions carry through. In
particular we have rational dependence on $z_{1},z_{2},z_{3}$ and poles only
from two-particle factorization, although now they need not be simple poles.
In general the soft limits do not commute, i.e.

$S_{1}^{(k)}S_{2}^{(\ell)}{\mathcal{O}}_{1}(\omega_{1},z_{1},{\bar{z}}_{1}){\mathcal{O}}_{2}(\omega_{2},z_{2},{\bar{z}}_{2})\neq
S_{2}^{(\ell)}S_{1}^{(k)}{\mathcal{O}}_{1}(\omega_{1},z_{1},{\bar{z}}_{1}){\mathcal{O}}_{2}(\omega_{2},z_{2},{\bar{z}}_{2}).$
(5.2)

Once again this presents an obstacle to interpreting soft insertions as local
operators in CCFT, as there is no sense of the “order of insertion” of local
operators Kapec:2022hih . The commutators of some of the most leading soft
limits for gluons and gravitons have been studied in detail in
Lipstein:2015rxa ; Klose:2015xoa ; Anupam:2018vyu ; Distler:2018rwu ;
Fotopoulos:2019vac ; Fotopoulos:2020bqj ; Campiglia:2021bap . In particular,
for positive-helicity gluons in pure Yang-Mills we have
$[S_{1}^{(1)},S_{2}^{(1)}]=0$ and $[S_{1}^{(1)},S_{2}^{(0)}]=0$, with
$[S_{1}^{(0)},S_{2}^{(0)}]$ and $[S_{1}^{(1)},S_{2}^{(-1)}]$ being nonzero.
All further subleading commutators are nonzero as well. Opposite-helicity
gluons in Yang-Mills fail to commute even at leading order,
$[S_{1}^{(1)},S_{2}^{(1)}]\neq 0$. This latter fact was appreciated in
He:2015zea , where it was recognized as an obstruction to defining the OPE of
two opposite-helicity soft gluon currents. Recently some authors have found it
convenient to use the simultaneous conformally soft limit Mago:2021wje ;
Ren:2022sws

$\lim_{\varepsilon\to
0}\varepsilon^{2}{\mathcal{O}}_{\Delta_{1}=k+\varepsilon}(z_{1},{\bar{z}}_{1}){\mathcal{O}}_{\Delta_{2}=\ell+\varepsilon}(z_{2},{\bar{z}}_{2}),$
(5.3)

which treats the two insertions symmetrically. More generally one could use
arbitrary relative rates of softness,

$\lim_{\varepsilon\to
0}\eta_{1}\eta_{2}\varepsilon^{2}{\mathcal{O}}_{k+\eta_{1}\varepsilon}(z_{1},{\bar{z}}_{1}){\mathcal{O}}_{\ell+\eta_{2}\varepsilon}(z_{2},{\bar{z}}_{2}).$
(5.4)

One can recover either consecutive limit by sending $\eta_{2}/\eta_{1}$ to
zero or infinity, but for generic $\eta_{1},\eta_{2}$ it is not clear how this
limit is related to energetically soft limits. It is instructive to apply this
limit to the celestial OPE in pure Yang-Mills. We will see that the non-
commutativity is rather tame. For two positive-helicity outgoing gluons one
finds

$\displaystyle\lim_{\varepsilon\to
0}\eta_{1}\eta_{2}\varepsilon^{2}{\mathcal{O}}^{+,a}_{k+\eta_{1}\varepsilon}(z_{1},{\bar{z}}_{1}){\mathcal{O}}^{+,b}_{\ell+\eta_{2}\varepsilon}(z_{2},{\bar{z}}_{2})\sim$
(5.5)
$\displaystyle\frac{-i{f^{ab}}_{c}}{z_{12}}\left[\sum_{m=0}^{1-k}{2-k-\ell-m\choose
1-\ell}+\frac{(-)^{1-\ell}}{1+\eta_{2}/\eta_{1}}\sum_{m=3-k-\ell}^{\infty}{k-2+m\choose
1-\ell}\right]\frac{{\bar{z}}_{12}^{m}}{m!}\partial_{{\bar{z}}_{2}}^{m}[{\mathcal{O}}^{+,c}]^{(k+\ell-1)}(z_{2},{\bar{z}}_{2}).$

The $\eta_{2}/\eta_{1}$-dependent part is not necessarily zero, but it cannot
have any poles at all in $z_{2}$ since all collinear poles would have
polynomial ${\bar{z}}_{2}$ dependence of degree $1-(k+\ell-1)=2-k-\ell$, and
this is always killed by $\partial_{{\bar{z}}_{2}}^{m}$ for $m\geq 3-k-\ell$.
For two opposite-helicity outgoing gluons one finds

$\displaystyle\lim_{\varepsilon\to
0}\eta_{1}\eta_{2}\varepsilon^{2}{\mathcal{O}}^{+,a}_{k+\eta_{1}\varepsilon}(z_{1},{\bar{z}}_{1}){\mathcal{O}}^{-,b}_{\ell+\eta_{2}\varepsilon}(z_{2},{\bar{z}}_{2})\sim$
(5.6)
$\displaystyle\frac{-i{f^{ab}}_{c}}{z_{12}}\left[\sum_{m=0}^{1-k}{-k-\ell-m\choose-1-\ell}+\frac{(-)^{-1-\ell}}{1+\eta_{2}/\eta_{1}}\sum_{m=1-k-\ell}^{\infty}{k-2+m\choose-1-\ell}\right]\frac{{\bar{z}}_{12}^{m}}{m!}\partial_{{\bar{z}}_{2}}^{m}[{\mathcal{O}}^{-,c}]^{(k+\ell-1)}(z_{2},{\bar{z}}_{2}).$

Similar comments apply, with the $\eta_{2}/\eta_{1}$-dependent part having no
poles in $z_{2}$.

##  6 Jacobi for soft insertions

The soft double residue condition might naïvely be written as

$\left(\underset{z_{2}\shortrightarrow z_{3}}{\rm
Res}\,\underset{z_{1}\shortrightarrow z_{2}}{\rm
Res}-\underset{z_{1}\shortrightarrow z_{3}}{\rm
Res}\,\underset{z_{2}\shortrightarrow z_{3}}{\rm
Res}+\underset{z_{2}\shortrightarrow z_{3}}{\rm
Res}\,\underset{z_{1}\shortrightarrow z_{3}}{\rm
Res}\right){\mathcal{O}}_{1}^{(k_{1})}(z_{1},{\bar{z}}_{1}){\mathcal{O}}_{2}^{(k_{2})}(z_{2},{\bar{z}}_{2}){\mathcal{O}}_{3}^{(k_{3})}(z_{3},{\bar{z}}_{3}),$
(6.1)

but this expression is ill-defined due to the non-commutativity of soft limits
with each other and more importantly with collinear limits. We discuss three
ways to make sense of this expression. First we address the one that has
already been discussed in the literature, which is to choose some definition
for the soft-soft OPE of two insertions and apply it in succession to define
(6.1). In Mago:2021wje  the soft-soft OPE was defined using the simultaneous
soft limit (5.3), which treats the two insertions symmetrically. But with this
definition, even positive-helicity gluons in pure Yang-Mills fail the double
residue condition.111111The offending terms are only beyond the wedge. In
Guevara:2021abz  the soft-soft OPE of ${\mathcal{O}}_{1}{\mathcal{O}}_{2}$ was
defined by starting from the hard OPE centered on $z_{2}$, taking $\Delta_{1}$
and $\Delta_{2}$ soft, and discarding the beyond-wedge part by hand, which is
equivalent to taking $\Delta_{1}$ soft first. This treats the two insertions
asymmetrically (even after resummation), but it does lead to positive-helicity
gluons in pure Yang-Mills satisfying (6.1).

Another way to define (6.1) is by choosing some order of soft limits and
taking them before the residues, e.g.

$\left(\underset{z_{2}\shortrightarrow z_{3}}{\rm
Res}\,\underset{z_{1}\shortrightarrow z_{2}}{\rm
Res}-\underset{z_{1}\shortrightarrow z_{3}}{\rm
Res}\,\underset{z_{2}\shortrightarrow z_{3}}{\rm
Res}+\underset{z_{2}\shortrightarrow z_{3}}{\rm
Res}\,\underset{z_{1}\shortrightarrow z_{3}}{\rm
Res}\right)S_{3}^{(k_{3})}S_{2}^{(k_{2})}S_{1}^{(k_{1})}{\mathcal{O}}(\omega_{1},z_{1},{\bar{z}}_{1}){\mathcal{O}}(\omega_{2},z_{2},{\bar{z}}_{2}){\mathcal{O}}(\omega_{3},z_{3},{\bar{z}}_{3}).$
(6.2)

As discussed above, the $z_{1},z_{2},z_{3}$ dependence is rational with only
local poles near $z_{1}=z_{2}=z_{3}$. (The massless-massive two-particle
factorization poles are absent in the neighborhoods of generic points on the
$z_{1}=z_{2}=z_{3}$ submanifold of parameter space.) Then by contour pulling,
the double residue condition must be satisfied, even in EFTs with arbitrary
non-minimal couplings. One naturally wonders what the corresponding
modification to the celestial soft current algebra is in such theories, but
this question appears ill-posed. If we tried to read off commutators, then the
commutators of modes of ${\mathcal{O}}_{2}$ and ${\mathcal{O}}_{3}$ would
depend on the properties of ${\mathcal{O}}_{1}$. This is a manifestation of
the nonlocality of soft insertions discussed above.

Finally we can define (6.1) by taking soft limits after the residues. Note
then that if the hard double residue condition is satisfied, we must get zero.
Consider arbitrary relative rates of conformal softness for the three
insertions,

$\lim_{\varepsilon\to
0}\eta_{1}\eta_{2}\eta_{3}\varepsilon^{3}\bigg{(}\underset{z_{2}\shortrightarrow
z_{3}}{\rm Res}\underset{z_{1}\shortrightarrow z_{2}}{\rm
Res}-\underset{z_{1}\shortrightarrow z_{3}}{\rm
Res}\underset{z_{2}\shortrightarrow z_{3}}{\rm
Res}+\underset{z_{2}\shortrightarrow z_{3}}{\rm
Res}\underset{z_{1}\shortrightarrow z_{3}}{\rm
Res}\bigg{)}{\mathcal{O}}_{k_{1}+\eta_{1}\varepsilon}(z_{1},{\bar{z}}_{1}){\mathcal{O}}_{k_{2}+\eta_{2}\varepsilon}(z_{2},{\bar{z}}_{2}){\mathcal{O}}_{k_{3}+\eta_{3}\varepsilon}(z_{3},{\bar{z}}_{3}).$
(6.3)

Each residue splits into “wedge” and “beyond-wedge” parts identical to the
simultaneous soft OPE up to $\eta_{i}$ dependence. We leave this $\eta_{i}$
dependence explicit in the following but otherwise adopt a very compact
notation:

$\displaystyle\lim_{\varepsilon\to
0}\eta_{1}\eta_{2}\eta_{3}\varepsilon^{3}\underset{z_{2}\shortrightarrow
z_{3}}{\rm Res}\underset{z_{1}\shortrightarrow z_{2}}{\rm
Res}{\mathcal{O}}_{1}{\mathcal{O}}_{2}{\mathcal{O}}_{3}$
$\displaystyle=\left(W_{1,2}+\frac{\eta_{1}}{\eta_{1}+\eta_{2}}B_{1,2}\right)\left(W_{12,3}+\frac{\eta_{1}+\eta_{2}}{\eta_{1}+\eta_{2}+\eta_{3}}B_{12,3}\right),$
(6.4) $\displaystyle\lim_{\varepsilon\shortrightarrow
0}\eta_{1}\eta_{2}\eta_{3}\varepsilon^{3}\underset{z_{1}\shortrightarrow
z_{3}}{\rm Res}\underset{z_{2}\shortrightarrow z_{3}}{\rm
Res}{\mathcal{O}}_{1}{\mathcal{O}}_{2}{\mathcal{O}}_{3}$
$\displaystyle=\left(W_{2,3}+\frac{\eta_{2}}{\eta_{2}+\eta_{3}}B_{2,3}\right)\left(W_{1,23}+\frac{\eta_{1}}{\eta_{1}+\eta_{2}+\eta_{3}}B_{1,23}\right),$
$\displaystyle\lim_{\varepsilon\shortrightarrow
0}\eta_{1}\eta_{2}\eta_{3}\varepsilon^{3}\underset{z_{2}\shortrightarrow
z_{3}}{\rm Res}\underset{z_{1}\shortrightarrow z_{3}}{\rm
Res}{\mathcal{O}}_{1}{\mathcal{O}}_{2}{\mathcal{O}}_{3}$
$\displaystyle=\left(W_{1,3}+\frac{\eta_{1}}{\eta_{1}+\eta_{3}}B_{1,3}\right)\left(W_{2,13}+\frac{\eta_{2}}{\eta_{1}+\eta_{2}+\eta_{3}}B_{2,13}\right).$

Note that there is no choice of $\eta_{1},\eta_{2},\eta_{3}$ such that all of
the $\eta_{i}$ ratios are equal to one half, which would correspond to using
the simultaneous soft OPE. Terms in the double residue condition with linearly
independent $\eta_{i}$ dependence must vanish separately. In this way one
finds six independent equations satisfied by the $W$’s and $B$’s. These same
six equations are already implied by the six possible consecutive soft limits
on $\Delta_{1},\Delta_{2},\Delta_{3}$, so it turns out that the more general
limit considered here did not give any extra information. The consecutive soft
limit $S^{(k_{3})}_{3}S_{2}^{(k_{2})}S_{1}^{(k_{1})}$ is particularly
interesting because it involves only the wedge terms,121212There are many ways
to rewrite the double residue condition by swapping
$\underset{z_{i}\shortrightarrow z_{j}}{\rm Res}$ for
$\underset{z_{j}\shortrightarrow z_{i}}{\rm Res}$, and different choices will
lead to different consecutive soft limits being associated with the wedge
terms.

$\displaystyle W_{1,2}W_{12,3}-W_{2,3}W_{1,23}+W_{1,3}W_{2,13}=$ (6.5)
$\displaystyle
S_{3}^{(k_{3})}S_{2}^{(k_{2})}S_{1}^{(k_{1})}\left(\underset{z_{2}\shortrightarrow
z_{3}}{\rm Res}\,\underset{z_{1}\shortrightarrow z_{2}}{\rm
Res}-\underset{z_{1}\shortrightarrow z_{3}}{\rm
Res}\,\underset{z_{2}\shortrightarrow z_{3}}{\rm
Res}+\underset{z_{2}\shortrightarrow z_{3}}{\rm
Res}\,\underset{z_{1}\shortrightarrow z_{3}}{\rm
Res}\right){\mathcal{O}}_{\Delta_{1}}(z_{1},{\bar{z}}_{1}){\mathcal{O}}_{\Delta_{2}}(z_{2},{\bar{z}}_{2}){\mathcal{O}}_{\Delta_{3}}(z_{3},{\bar{z}}_{3}).$

Since the the soft limits (once they are moved inside the Mellin transforms)
are just grabbing Laurent coefficients, the only way for
$S^{(k_{3})}_{3}S_{2}^{(k_{2})}S_{1}^{(k_{1})}$ to vanish for all
$k_{1},k_{2},k_{3}$ is if it acts on zero. Therefore the hard double residue
condition is satisfied if and only if the wedge part of the soft double
residue condition is satisfied. This is consistent with the results of
Mago:2021wje ; Ren:2022sws , which found for a large family of EFTs that the
constraints of the hard double residue condition are the same as those of one
formulation of the soft double residue condition. Comparing with (6.2), which
vanishes, shows that any failure of (LABEL:eq:softwedge) can be traced to non-
commutativity of soft and collinear limits.

It turns out that $B_{1,2}W_{12,3}$ always vanishes whether or not the hard
double residue condition is satisfied, simply because the powers of
$\partial_{{\bar{z}}_{2}}$ in $B_{1,2}$ annihilate the ${\bar{z}}_{2}$
dependence of $W_{12,3}$. This means that no matter what $\eta_{1},\eta_{2}$
are, if we take $\eta_{3}\to\infty$ first (i.e. take ${\mathcal{O}}_{3}$ soft
last) then we will get the left hand side of (LABEL:eq:softwedge). This is
still somewhat unsatisfactory because it treats ${\mathcal{O}}_{3}$
differently from ${\mathcal{O}}_{1}$ and ${\mathcal{O}}_{2}$, despite all
three nominally being the same type of soft object. We advocate for simply
leaving ${\mathcal{O}}_{3}$ hard, in which case the double residue condition
amounts to equivariance of the action of
${\mathcal{O}}_{1}^{(k_{1})},{\mathcal{O}}_{2}^{(k_{2})}$ on
${\mathcal{O}}_{3}$. That is, if $X_{1}$ is a soft mode of ${\mathcal{O}}_{1}$
and $X_{2}$ is a soft mode of ${\mathcal{O}}_{2}$, then the double residue
condition informs us about whether

$X_{1}\cdot(X_{2}\cdot{\mathcal{O}}_{3})-X_{2}\cdot(X_{1}\cdot{\mathcal{O}}_{3})\stackrel{{\scriptstyle?}}{{=}}[X_{1},X_{2}]\cdot{\mathcal{O}}_{3},$
(6.6)

where $X_{i}\cdot{\mathcal{O}}_{3}$ is defined through the soft-hard OPE. The
commutator $[X_{1},X_{2}]$ is computed with the soft-soft OPE, and any
ambiguity from the order of soft limits drops out of
$[X_{1},X_{2}]\cdot{\mathcal{O}}_{3}$. This is similar to the approach taken
in Himwich:2021dau  in the context of $w_{1+\infty}$ generators. The upshot is
that, when the double residue condition is satisfied, the consecutive action
of soft insertions on a hard insertion is consistent with that of a current
algebra, despite the fact that soft insertions as formulated in this paper are
not truly local operators in CCFT.

##  7 Discussion

In this paper we studied the Jacobi identity for holomorphic celestial
currents at tree level in the form of the double residue condition. We showed
that the question of its satisfaction has the same answer for hard insertions
in Mellin space, hard insertions in momentum space, and (suitably defined)
soft insertions. We further established its equivalence with a simple,
practical condition on massless 4-point momentum space amplitudes: the
vanishing of the angle bracket weight $-1$ part. This condition facilitates
the application of known amplitudes results to questions in celestial
holography.

We also highlighted the important role of the order of limits in obstructing
the Jacobi identity for soft insertions. We discussed three different
approaches to defining the soft double residue condition, and advocated for
simply leaving one insertion hard and viewing the condition as a statement
about equivariance of the action of the soft insertions on this hard
insertion. This involves only terms within the wedge. When this double residue
condition is satisfied, the action on the hard insertion is consistent with 2D
locality. But other properties of the soft insertions, related to non-
commutativity of limits, are simply incompatible with 2D locality, even for
the relatively simple case of positive-helicity gluons in pure Yang-
Mills.131313As discussed above, there is a loophole if one only allows integer
values of $\Delta$. This observation is supported by the results of
Kapec:2016jld ; Kapec:2017gsg ; Kapec:2021eug ; Kapec:2022axw ; Kapec:2022hih
, which suggest that it is actually the 2D shadows of soft insertions that
play the role of local currents. It is argued in Kapec:2022hih  that, at least
for the leading soft theorems, soft gluons and gravitons should be thought of
as integrated operators whose insertion deforms the CCFT in conformal
perturbation theory. Then the commutators of (leading) soft limits correspond
to curvature on the conformal manifold of CCFTs. We leave the study of
multiple insertions of soft shadows to the future, but we note that
Fan:2021pbp ; Hu:2022syq ; De:2022gjn ; Chang:2022jut  constitute preliminary
work in this direction.

###### Acknowledgements.

We thank Akshay Yelleshpur Srikant for invaluable discussions during the
development of this work, and Andrew Strominger for comments on the draft. We
also thank Eduardo Casali, Alfredo Guevara, Elizabeth Himwich, Lecheng Ren,
and Anastasia Volovich for useful conversations. We gratefully acknowledge
support from Simons Investigator Award #376208 of A. Volovich.

##  Appendix A Enforcing momentum conservation

We denote an unstripped momentum space amplitude with $m$ massless legs and
$n-m$ massive legs as

${\mathcal{A}}_{n}(\omega_{1},z_{1},{\bar{z}}_{1},\dots,\omega_{m},z_{m},{\bar{z}}_{m},y_{m+1},z_{m+1},{\bar{z}}_{m+1},\dots,y_{n},z_{n},{\bar{z}}_{n}).$
(A.1)

Discrete labels such as particle type, helicity, and in/out are suppressed. It
is related to the stripped amplitude $A_{n}$ as

${\mathcal{A}}_{n}=\delta^{(4)}\Big{(}\sum_{i=1}^{n}p_{i}\Big{)}A_{n}.$ (A.2)

An $n$-point amplitude is naïvely described by $3n$ continuous variables, but
the momentum conserving delta function restricts us to a codimension-four
locus. We can attempt to use $3n-4$ of our variables as coordinates on this
locus, and generically this will be well-behaved. Furthermore most, though not
all, choices of elimination will involve rational expressions of the other
$3n-4$ variables. One choice giving rational substitutions is eliminating
$\omega_{i},{\bar{z}}_{i},\omega_{j},{\bar{z}}_{j}$ for some $i,j$, which is
essentially the same as the standard spinor-helicity approach of eliminating
two square bracket spinors Cachazo:2014fwa . Other rational choices include
eliminating $\omega_{i},{\bar{z}}_{i},y_{j},{\bar{z}}_{j}$, or
$y_{i},{\bar{z}}_{i},y_{j},{\bar{z}}_{j}$, or
$\omega_{i},\omega_{j},\omega_{k},\omega_{\ell}$. For simplicity assume we
make some such rational choice, and denote our $3n-4$ coordinates by
$\xi^{I}$. These coordinates will not describe the locus globally, but they
will be valid almost everywhere, which is sufficient for our purposes. The
amplitude $A_{n}(\xi^{I})$ will be a rational function whose only poles come
from internal propagators going on shell, and places where the coordinate
system breaks down, which we avoid. We now move on to a discussion of
coordinate independence.

Suppose we want to compute the residue ${\rm Res}_{z_{1}\shortrightarrow
z_{*}}\,{\mathcal{A}}_{n}$ of the unstripped amplitude, where $z_{*}$ is any
function of the $\xi^{I}$.141414Assume $z_{*}$ can be approached such that the
coordinates remain valid. There are no double or higher poles, so let us
define

$\underset{z_{1}\shortrightarrow z_{*}}{\rm
Res}{\mathcal{A}}_{n}\equiv\big{[}(z_{1}-z_{*}){\mathcal{A}}_{n}\big{]}\big{|}_{z_{1}=z_{*}}.$
(A.3)

The right hand side (RHS) makes no reference to the choice of coordinates
$\xi^{I}$. Even if $z_{1}$ is eliminated, the RHS is still well-defined.
Furthermore it is easy to see that if $z_{1}$ is one of our coordinates and
$z_{*}$ is independent of $z_{1}$, then the RHS is equivalent to a
conventional residue on the stripped amplitude $A_{n}$,

$\big{[}(z_{1}-z_{*}){\mathcal{A}}_{n}\big{]}\big{|}_{z_{1}=z_{*}}=\delta^{(4)}\Big{(}\sum_{i=1}^{n}p_{i}\Big{)}\underset{z_{1}\shortrightarrow
z_{*}}{\rm Res}A_{n}.$ (A.4)

This is the version used in the body of this paper, but the equivalence to the
RHS of (A.3) guarantees that the result is independent of the choice of
coordinates.

An insertion ${\mathcal{O}}(\omega,z,{\bar{z}})$ is a function of
$\omega,z,{\bar{z}}$, along with the other $3n-7$ implicit coordinates. If we
change our choice of elimination then the new
${\mathcal{O}}(\omega,z,{\bar{z}})$ will be related to the old one simply by
substitution on the newly eliminated coordinates. The same is not true of
${\mathcal{O}}^{(k)}(z,{\bar{z}})$. If we change coordinates and substitute
for the newly eliminated implicit coordinates, they will generically acquire
$\omega$ dependence which must be expanded. In this way
${\mathcal{O}}^{(k)}(z,{\bar{z}})$ is sensitive not only to its own implicit
substitutions, but also those of ${\mathcal{O}}^{(\ell)}(z,{\bar{z}})$ with
$\ell>k$ (recall ${\mathcal{O}}^{(k)}$ comes from the $\omega^{-k}$ term). It
is very satisfying to see how this behavior is consistent with the soft
theorems. In practice we will simply fix a choice of coordinates and not have
to worry about these subtleties.

##  Appendix B Summary of commutativity of limits

In this appendix we summarize some useful relations between Mellin transforms,
soft limits, and residues in $z_{ij}$. As noted in section 4, soft limits are
equivariant with the Mellin transform in the sense that taking an
energetically soft limit on $\omega_{i}$ is equivalent to Mellin transforming
on $\omega_{i}$ and then taking a conformally soft limit on $\Delta_{i}$. This
means that consecutive energetically soft limits are equivalent to consecutive
conformally soft limits. As discussed in section 3, the residue ${\rm
Res}_{z_{i}\shortrightarrow z_{j}}$ commutes with all Mellin transforms and
with soft limits on $\omega_{i}$ and $\omega_{j}$ (but not other
$\omega_{k}$). Soft limits on $\omega_{i},\omega_{j}$ do not commute in
general.

The Mellin transform is simply a change of basis taking us from an unstripped
momentum space amplitude to a celestial amplitude. In fact, there is no need
to use the same basis for each leg. We are free to work with mixed amplitudes
where some legs are momentum eigenstates and some are conformal primaries,
although then the dependence on the momentum space insertions may no longer be
rational.

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