11institutetext: Institute of Physics, Sachivalaya Marg, Bhubaneshwar,
India-751005
and Homi Bhabha National Institute, Anushakti Nagar, Mumbai,
India-40008522institutetext: Okinawa Institute of Science and
Technology,1919-1 Tancha, Onna-son, Okinawa 904-0495,Japan

# MHV Gluon Scattering Amplitudes from Celestial Current Algebras

Shamik Banerjee 2 Sudip Ghosh<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We show that the Mellin transform of an $n$-point tree level MHV gluon
scattering amplitude, also known as the celestial amplitude in pure Yang-Mills
theory, satisfies a system of $(n-2)$ linear first order partial differential
equations corresponding to $(n-2)$ positive helicity gluons. Although these
equations closely resemble Kniznik-Zamolodchikov equations for $SU(N)$ current
algebra there is also an additional “correction” term coming from the
subleading soft gluon current algebra. These equations can be used to compute
the leading term in the gluon-gluon OPE on the celestial sphere. Similar
equations can also be written down for the momentum space tree level MHV
scattering amplitudes. We also propose a way to deal with the non closure of
subleading current algebra generators under commutation. This is then used to
compute some subleading terms in the mixed helicity gluon OPE.

##  1 Introduction

It is generally believed that any consistent theory of quantum gravity on a
space-time with asymptotic boundary should have a holographic dual description
in terms of a theory living on the boundary at infinity. The dual theory can
compute all the observables which make sense in the bulk theory of quantum
gravity. In the case of asymptotically flat space-time the observables are the
$S$-matrix elements and it has been proposed that the dual theory is a
conformal field theory Strominger:2017zoo ; Kapec:2014opa ; Kapec:2016jld ;
He:2017fsb ; Ball:2019atb ; Kapec:2017gsg ; Banerjee:2019aoy ;
Banerjee:2019tam ; Campiglia:2014yka ; Donnay:2020guq ; Barnich:2009se ;
Barnich:2011ct ; Pasterski:2016qvg ; Pasterski:2017kqt ; Cheung:2016iub ;
deBoer:2003vf ; Banerjee:2018gce ; Banerjee:2018fgd ; Banerjee:2019prz ;
Pasterski:2017ylz ; Schreiber:2017jsr ; Cardona:2017keg ; Lam:2017ofc ;
Banerjee:2017jeg ; Stieberger:2018edy ; Stieberger:2018onx ; Bagchi:2016bcd ;
Fotopoulos:2019tpe ; Albayrak:2020saa ; Casali:2020vuy ; Law:2020xcf ;
Gonzalez:2020tpi ; Muck:2020wtx ; Narayanan:2020amh , dubbed “Celestial
conformal field theory (CCFT)”, which lives on the celestial sphere. The
Lorentz group acts on the celestial sphere as the group of global conformal
transformations and when the bulk space-time is four dimensional, one can
show, starting from the subleading soft graviton theorem Cachazo:2014fwa  that
CCFT should have a stress tensor Kapec:2014opa ; Kapec:2016jld ; He:2017fsb
which generates local conformal transformations on the two dimensional
celestial sphere. So CCFT should have the full Virasoro symmetry just like a
more conventional two dimensional CFT. But this is not the end of the story.
Since the asymptotic symmetry group in four dimensions contains
supertranslations Sachs:1962zza ; Bondi:1962px , CCFT should also have
supertranslation symmetry Strominger:2013jfa ; He ; Strominger:2014pwa ;
Fotopoulos:2019vac ; Banerjee:2020kaa ; Banerjee:2020zlg . On top of that,
there are various other infinite dimensional current algebra symmetries,
coming from soft factorisation theorems, under which CCFT should be invariant
Strominger:2013lka ; Pate:2019lpp ; Banerjee:2020zlg ; Campiglia:2014yka ;
Campiglia:2020qvc ; Compere:2018ylh ; Donnay:2020guq ; He:2014cra ;
Kapec:2014zla ; Kapec:2015ena ; Campiglia:2015qka ; Lysov:2014csa ;
Campiglia:2016hvg ; Conde:2016csj ; Himwich:2019dug ; He:2020ifr ;
Nande:2017dba . Now, the CCFT is supposed to compute bulk scattering
amplitudes. So a natural question arises as to how the bulk scattering
amplitudes in four dimensions are constrained by the infinite dimensional
symmetries of the dual CCFT. For recent developments in this direction please
see Pate:2019lpp ; Law:2019glh ; Banerjee:2020zlg ; Himwich:2020rro ;
Fotopoulos:2019vac ; Banerjee:2020kaa ; Ebert:2020nqf .

The study of CCFT is facilitated by going to the Mellin space
Pasterski:2016qvg ; Pasterski:2017kqt ; Cheung:2016iub ; deBoer:2003vf ;
Banerjee:2018gce ; Banerjee:2018fgd ; Banerjee:2019prz ; Pasterski:2017ylz ;
Schreiber:2017jsr ; Cardona:2017keg ; Lam:2017ofc ; Banerjee:2017jeg ;
Stieberger:2018edy ; Stieberger:2018onx ; Fotopoulos:2019tpe ;
Albayrak:2020saa ; Casali:2020vuy ; Law:2020xcf ; Gonzalez:2020tpi ;
Muck:2020wtx ; Narayanan:2020amh . In Mellin space scattering amplitudes can
be written as the correlation functions of conformal primaries 111For a brief
review of Mellin amplitudes for gluons please see the Appendix A. In this
appendix we also make some comments on the notation used in the paper..
Conformal primaries are Mellin transform of Fock space creation (annihilation)
operators which create (annihilate) asymptotic free particle states in a
scattering event. They are called conformal primaries because under Lorentz
transformations, which act on the celestial sphere as global conformal
transformations, they transform like primary operators in a conformal field
theory. Let us now briefly describe the main results of this paper.

In this paper we focus on the Maximal Helicity Violating (MHV) gluon
scattering amplitudes in pure Yang-Mills (YM) theory with gauge group
$G=SU(N)$. These are amplitudes of the form $\langle{--+++\cdots}\rangle$,
where two gluons have negative helicity and the rest have positive helicity.
Their explicit expressions are given by the famous Parke-Taylor formula
Parke:1986gb . Now this is the first non-trivial helicity amplitude because at
tree level in pure YM theory, amplitudes of the form
$\langle{-+++\cdots}\rangle$ with only one negative helicity gluon and
$\langle{+++\cdots}\rangle$ with all positive helicity gluons vanish. As a
result of this at tree-level there is no negative helicity soft gluon in the
MHV-sector and also, the set of MHV amplitudes is closed under taking
collinear limits. This allows us to define, just like in the case of gravity
Banerjee:2020zlg , an autonomous MHV-sector of the CCFT which computes the MHV
gluon scattering amplitudes. The gluon MHV-sector is characterised by the fact
that it is governed by the $G$ current algebra Strominger:2013lka ; He:2014cra
; Kapec:2014zla ; Kapec:2015ena ; Campiglia:2015qka ; He:2020ifr ;
Nande:2017dba  at level zero which arises from the leading positive helicity
soft gluon theorem. There is also the subleading positive helicity soft gluon
theorem Pate:2019lpp ; Lysov:2014csa ; Campiglia:2016hvg ; Conde:2016csj ;
Himwich:2019dug  which gives rise to another infinite set of currents which
play an equally important role in the theory. The significance of the
autonomous MHV sectors for gluons and gravitons Banerjee:2020zlg  is that they
behave somewhat like Minimal models of two dimensional conformal field
theories and hence are exactly solvable.

Let us now consider an $n$-point MHV amplitude with two negative helicity
gluons and $(n-2)$ positive helicity gluons. The main result of this paper is
a system of $(n-2)$ coupled first order linear partial differential equations
satisfied by the tree-level MHV amplitudes. In Mellin space they are given by,

$\begin{split}&\bigg{[}\frac{C_{A}}{2}\hskip 1.42271pt\frac{\partial}{\partial
z_{i}}-h_{i}\hskip 1.42271pt\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{n}\frac{T^{a}_{i}\hskip 1.42271ptT^{a}_{j}}{z_{i}-z_{j}}\\\
&+\frac{1}{2}\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{n}\frac{\epsilon_{j}\left(2\bar{h}_{j}-1-(\bar{z}_{i}-\bar{z}_{j})\frac{\partial}{\partial\bar{z}_{j}}\right)}{z_{i}-z_{j}}\hskip
1.42271ptT^{a}_{j}P^{-1}_{j}T^{a}_{i}P_{-1,-1}(i)\bigg{]}\bigg{\langle}\prod_{k=1}^{n}\mathcal{O}^{a_{k}}_{h_{k},\bar{h}_{k}}(z_{k},\bar{z}_{k})\bigg{\rangle}_{\text{MHV}}=0\end{split}$
(1)

where $i\in\left(1,2,\cdots,n-2\right)$ runs over the $(n-2)$ positive
helicity gluons in the MHV amplitude. In the above equation,
$\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z})$ is a gluon conformal primary with
scaling dimension $(h,\bar{h})$ and $a$ is a Lie algebra index. $T^{a}$ is the
Lie algebra generator in the adjoint representation and $C_{A}$ is the
quadratic Casimir. We have also introduced the symbol $\epsilon_{j}$ which is
$\pm 1$ depending on whether
$\mathcal{O}^{a_{j}}_{h_{j},\bar{h}_{j}}(z_{j},\bar{z}_{j})$ is outgoing or
incoming. The operator $P_{j}^{-1}$ acts on a gluon conformal primary
$\mathcal{O}^{a_{k}}_{h_{k},\bar{h}_{k}}(z_{k},\bar{z}_{k})$ as

$P_{j}^{-1}\mathcal{O}^{a_{k}}_{h_{k},\bar{h}_{k}}(z_{k},\bar{z}_{k})=\mathcal{O}^{a_{k}}_{h_{k}-\frac{1}{2},\bar{h}_{k}-\frac{1}{2}}(z_{k},\bar{z}_{k})\delta_{jk}$
(2)

Similarly the global time translation generator $P_{-1,-1}(i)$ acts on the
$i$-th positive helicity gluon conformal primary according to

$P_{-1,-1}(i)\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})=\epsilon_{i}\mathcal{O}^{a_{i}}_{h_{i}+\frac{1}{2},\bar{h}_{i}+\frac{1}{2}}(z_{i},\bar{z}_{i})$
(3)

Equation (1) can be easily transformed to momentum space and for the momentum
space MHV scattering amplitude it can be written as,

$\begin{split}&\bigg{[}C_{A}\hskip 1.42271pt\frac{\partial}{\partial
z_{i}}+\left(\omega_{i}\frac{\partial}{\partial\omega_{i}}-1\right)\sum_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{n}\frac{T^{a}_{i}\hskip
1.42271ptT^{a}_{j}}{z_{i}-z_{j}}\\\ &+\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{n}\frac{\epsilon_{i}\omega_{i}}{\epsilon_{j}\omega_{j}}\frac{\left(\sigma_{j}+\omega_{j}\frac{\partial}{\partial\omega_{j}}+(\bar{z}_{i}-\bar{z}_{j})\frac{\partial}{\partial\bar{z}_{j}}\right)}{z_{i}-z_{j}}T^{a}_{i}T^{a}_{j}\bigg{]}\bigg{\langle}\prod_{k=1}^{n}A^{a_{k}}(\epsilon_{k}\omega_{k},z_{k},\bar{z}_{k},\sigma_{k})\bigg{\rangle}_{\text{MHV}}=0\end{split}$
(4)

Here the null momentum of an on-shell gluon has been parametrised as

$p=\epsilon\hskip
1.13791pt\omega(1+z\bar{z},z+\bar{z},-i(z-\bar{z}),1-z\bar{z})$

and $\epsilon=\pm 1$ for an outgoing (incoming) gluon. We have also denoted
the momentum space creation (annihilation) operators by
$A^{a}(\epsilon\omega,z,\bar{z},\sigma)$ where $\sigma$ is the helicity of the
gluon. Note that the MHV amplitude in (4) includes the overall momentum
conserving delta function.

Now (1) or (4) are examples of (holographic) constraints on (hard) scattering
amplitudes coming from the infinite dimensional symmetries of CCFT. They are
obtained in the same way as differential equations for MHV graviton scattering
amplitudes were obtained in Banerjee:2020zlg . Presumably (1) or (4), together
with the Ward identities coming from Poincare invariance, can be solved to
obtain the MHV gluon scattering amplitudes. Along this line we make a
preliminary check in which we determine the leading gluon-gluon OPE
coefficients from the differential equations and our results match with those
of Pate:2019lpp ; Fan:2019emx .

The origin of equations (1) and (4) will be discussed in great detail in
section (6) but let us mention a few things before we close this section.

  1. 1.

The first two terms of (1) closely resemble the Knizhnik-Zamolodchikov (KZ)
equation Knizhnik:1984nr  satisfied by the correlation functions of current
algebra primaries in WZW model. The only difference is the prefactor $h_{i}$
in the second term in (1) and we also have to set the level $k$ of the current
algebra to $0$.

  2. 2.

In WZW model the scaling dimensions of the primaries are determined in terms
of the level of the current algebra and the representation of the zero mode
algebra under which the primary transforms. But, this is not the case here.
(1) holds for any value of the scaling dimension $(h,\bar{h})$ of the gluon
primary. This is consistent with the fact that in CCFT the scaling dimension
$\Delta=h+\bar{h}$ of a (hard) primary is a continuously varying (complex)
number and therefore should not be constrained in any way.

  3. 3.

The third term in (1) is an additional contribution coming from the (local)
subleading soft gluon symmetry. This has no analog in the usual KZ equation
and is most likely related to the fact that there is no Sugawara stress tensor
in CCFT. This is also very different from pure gravity where the corresponding
differential equations for MHV amplitudes do not have any contribution from
the subsubleading soft graviton theorem.

It will be very interesting to understand the origin of the differential
equations (1) or (4) from the point of view of Twistor string theory
Witten:2003nn ; Nair:1988bq ; Cachazo:2004kj ; Cachazo:2004zb ; Cachazo:2004by
; Roiban:2004yf ; Cachazo:2013hca ; Cachazo:2013iea ; Adamo:2014yya ;
Geyer:2014lca ; Adamo:2015fwa . This will also shed more light on the true
nature of the CCFT. We leave these questions for future research.

An outline of this paper is as follows. We begin in Section (2) by discussing
the action of Poincare generators on gluon primary operators on the celestial
sphere. The definition of a conformal primary operator is given explicitly in
Section (2.1). In Section (3) we consider the leading conformal soft gluon
theorem which is equivalent to the Ward identity for a level zero Kac-Moody
algebra on the celestial sphere. We specify here the commutators involving
modes of the Kac-Moody current and the Poincare generators. Using the current
algebra Ward identity we also relate here the celestial correlators involving
Kac-Moody descendants to correlation functions of gluon primary operators.
Section (3.1) contains the definition of a primary operator under the current
algebra. In Section (4) we consider a set of currents $(J^{a},K^{a})$ on the
celestial sphere which arise from the subleading conformal soft gluon theorem.
In Section (4.1) we derive the OPE between a subleading soft gluon and a hard
gluon primary which yields an important constraint on the OPE of hard gluons
in the subleading conformal soft limit. From this OPE we also extract the
definition of descendants created by modes of $(J^{a},K^{a})$ and use the Ward
identity, corresponding to the subleading soft gluon theorem, to obtain
correlation functions with insertions of these descendants. In Section
(LABEL:subleadprim) the definition of a primary under the subleading soft
symmetry algebra is provided. Section (4.2) lists various useful commutation
relations. In Section (5), we discuss the interpretation of the commutation
relations between modes of the subleading soft symmetry generators in the
light of the fact that these generators do not close to form a Lie algebra in
the conventional sense. In Sections (6) and (7) we derive the differential
equations (1) and (4) for tree-level MHV gluon amplitudes in Mellin space and
Fock space respectively. In Section (8) we use the differential equation (1)
to determine the structure of the leading OPE for gluon primaries in Yang-
Mills theory. In particular, Section (8.1) deals with the case where both
gluons in the OPE are either incoming or outgoing and in Section (8.2) we
consider the case where one of the gluons in the OPE is outgoing and the other
is incoming. In Section (9), we illustrate how some descendant OPE
coefficients in the OPE between gluons of opposite helicities can be
determined using the underlying infinite dimensional symmetry algebras. We end
the paper with a set of Appendices. Appendix A contains a brief review of
celestial amplitudes and comments on some of the notation used in this paper.
In Appendix B we present a detailed calculation of the first subleading
correction to the leading celestial OPE of positive helicity gluons using the
Mellin transform of the $5$-point tree level MHV gluon amplitude in Yang-Mills
theory. In Appendix C we use the $4$-point MHV Mellin amplitude to extract the
first set of subleading terms in the OPE between opposite helicity gluon
primaries.

##  2 Poincare invariance

Since the scattering amplitudes are Poincare invariant, generators of the
Poincare group act on the conformal primaries which live on the celestial
sphere222For a brief review of conformal primaries and Mellin amplitudes for
gluons and some comments on notations used in this paper, please see the
Appendix A.. The Lorentz group $SL(2,\mathbb{C})$ acts on the celestial sphere
as the global conformal group and we denote its generators by
$\left(L_{0},L_{\pm 1},\bar{L}_{0},\bar{L}_{\pm 1}\right)$. Their commutation
relations are given by,

$\left[L_{m},L_{n}\right]=(m-n)L_{m+n},\quad\left[\bar{L}_{m},\bar{L}_{n}\right]=(m-n)\bar{L}_{m+n},\quad\left[L_{m},\bar{L}_{n}\right]=0,\quad
m,n=0,\pm 1$ (5)

They act on a gluon conformal primary $\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z})$
as,

$\left[L_{1},\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z})\right]=\left(z^{2}\partial+2hz\right)\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z}),\hskip
5.69046pt\left[L_{0},\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z})\right]=h\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z}),\hskip
5.69046pt\left[L_{-1},\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z})\right]=\partial\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z})$
(6)

$\left[\bar{L}_{1},\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z})\right]=\left(\bar{z}^{2}\bar{\partial}+2\bar{h}\bar{z}\right)\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z}),\hskip
5.69046pt\left[\bar{L}_{0},\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z})\right]=\bar{h}\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z}),\hskip
5.69046pt\left[\bar{L}_{-1},\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z})\right]=\bar{\partial}\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z})$
(7)

The four global space-time translation generators will be denoted by $P_{m,n}$
where $m,n=0,\pm 1$ and they are mutually commuting

$\left[P_{m,n},P_{m^{\prime},n^{\prime}}\right]=0$ (8)

The commutation relations between Lorentz and global space-time translation
generators are given by,

$[L_{n},P_{m^{\prime},n^{\prime}}]=\bigg{(}\frac{n-1}{2}-m^{\prime}\bigg{)}P_{m^{\prime}+n,n^{\prime}}\qquad[\bar{L}_{n},P_{m^{\prime},n^{\prime}}]=\bigg{(}\frac{n-1}{2}-n^{\prime}\bigg{)}P_{m^{\prime},n^{\prime}+n}$
(9)

The translation generators $P_{m,n}$ act on conformal primaries according to

$\left[P_{m,n},\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z})\right]=\epsilon
z^{m+1}\bar{z}^{n+1}\mathcal{O}^{a}_{h+\frac{1}{2},\bar{h}+\frac{1}{2}}(z,\bar{z})$
(10)

where $\epsilon=\pm 1$ for an outgoing (incoming) gluon.

###  2.1 Poincare primary

It follows from the definition of a conformal primary operator that the
following standard relations hold

$L_{1}\mathcal{O}^{a}_{h,\bar{h}}(0)=\bar{L}_{1}\mathcal{O}^{a}_{h,\bar{h}}(0)=0,\quad
L_{0}\mathcal{O}^{a}_{h,\bar{h}}(0)=h\mathcal{O}^{a}_{h,\bar{h}}(0),\quad\bar{L}_{0}\mathcal{O}^{a}_{h,\bar{h}}(0)=\bar{h}\mathcal{O}^{a}_{h,\bar{h}}(0)$
(11)

but a Poincare primary Banerjee:2020kaa  must also satisfy the additional
conditions

$P_{0,-1}\mathcal{O}^{a}_{h,\bar{h}}(0)=P_{-1,0}\mathcal{O}^{a}_{h,\bar{h}}(0)=P_{0,0}\mathcal{O}^{a}_{h,\bar{h}}(0)=0$
(12)

which follow from (10).

##  3 Leading soft gluon

The leading conformally soft Donnay:2018neh ; Pate:2019mfs ; Fan:2019emx ;
Nandan:2019jas ; Adamo:2019ipt  333Conformally soft graviton theorems have
been studied in Donnay:2018neh ; Puhm:2019zbl ; Guevara:2019ypd . gluon
operator $j^{a}(z)$ with positive helicity is defined as,

$j^{a}(z)=\lim_{\Delta\rightarrow
1}(\Delta-1)\mathcal{O}^{a}_{\Delta,+}(z,\bar{z})$ (13)

where $\mathcal{O}^{a}_{\Delta,+}(z,\bar{z})$ is a positive helicity gluon
primary with scaling dimension $\Delta$. The soft operator $j^{a}(z)$ is a
Kac-Moody current Strominger:2013lka ; He:2014cra ; Kapec:2014zla ;
Kapec:2015ena ; Campiglia:2015qka ; He:2020ifr ; Nande:2017dba  whose
correlation function with a collection of gluon primaries is given by the
leading soft gluon theorem and has the standard form in Mellin space,

$\bigg{\langle}{j^{a}(z)\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\bigg{\rangle}=-\sum_{k=1}^{n}\frac{T^{a}_{k}}{z-z_{k}}\bigg{\langle}{\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\bigg{\rangle}$
(14)

where

$T^{a}_{k}O^{a_{i}}_{h_{i},\bar{h}_{i}}(z,\bar{z})=if^{aa_{i}b}\mathcal{O}^{b}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})\delta_{ik}$
(15)

as gluons transform in the adjoint representation. Now let us consider the
modes $j^{a}_{n}$ of the above current. They satisfy the algebra

$\left[j^{a}_{m},j^{b}_{n}\right]=-if^{abc}j^{c}_{m+n}$ (16)

and act on a gluon primary as

$\left[j^{a}_{m},\mathcal{O}^{b}_{h,\bar{h}}(z,\bar{z})\right]=-if^{abc}z^{n}\mathcal{O}^{c}_{h,\bar{h}}(z,\bar{z})$
(17)

We note that the level of the current algebra here is zero which will be
further justified by the form of the gluon-gluon OPE Ebert:2020nqf  in the MHV
sector.

The commutation relations with the (Lorentz) global conformal generators are
given by,

$\left[L_{m},j^{a}_{n}\right]=-nj^{a}_{m+n},\quad\left[\bar{L}_{m},j^{a}_{n}\right]=0,\qquad
m=0,\pm 1$ (18)

Similarly, the commutators with the generators $\\{P_{m,n},m,n=0,\pm 1\\}$ of
global space-time translations are given by,

$\left[P_{m,n},j^{a}_{p}\right]=0$ (19)

For our purposes an important role is played by the correlation functions of
the descendants $j^{a}_{-p}\mathcal{O}^{b}_{h,\bar{h}}(z,\bar{z}),p\geq 1$
with a collection of gluon primaries. These are given by

$\bigg{\langle}{j^{a}_{-p}\mathcal{O}^{b}_{h,\bar{h}}(z,\bar{z})\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\bigg{\rangle}=\mathscr{J}^{a}_{-p}(z)\bigg{\langle}\mathcal{O}^{b}_{h,\bar{h}}(z,\bar{z})\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})\bigg{\rangle}$
(20)

where the operator $\mathscr{J}^{a}_{-p}(z)$ is defined as,

$\begin{gathered}\mathscr{J}^{a}_{-p}(z)\bigg{\langle}\mathcal{O}^{b}_{h,\bar{h}}(z,\bar{z})\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})\bigg{\rangle}=\sum_{k=1}^{n}\frac{T^{a}_{k}}{(z_{k}-z)^{p}}\bigg{\langle}\mathcal{O}^{b}_{h,\bar{h}}(z,\bar{z})\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})\bigg{\rangle},\quad
p\geq 1\end{gathered}$ (21)

###  3.1 Leading current algebra primary

A current algebra primary $\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z})$ is defined
by the standard conditions

$j^{a}_{n}\mathcal{O}^{b}_{h,\bar{h}}(0)=0,\quad\forall n\geq 1$ (22)

and

$j^{a}_{0}\mathcal{O}^{b}_{h,\bar{h}}(0)=-T^{a}\mathcal{O}^{b}_{h,\bar{h}}(0)=-if^{abc}\mathcal{O}^{c}_{h,\bar{h}}(0)$
(23)

##  4 Subleading soft gluon

The subleading conformally soft Donnay:2018neh ; Pate:2019mfs ; Fan:2019emx ;
Nandan:2019jas ; Adamo:2019ipt  gluon operator $S^{+a}_{1}(z,\bar{z})$ with
positive helicity is defined as,

$S^{+a}_{1}(z,\bar{z})=\lim_{\Delta\rightarrow
0}\Delta\mathcal{O}^{a}_{\Delta,+}(z,\bar{z})$ (24)

where $\mathcal{O}^{a}_{\Delta,+}(z,\bar{z})$ is a positive helicity gluon
primary with scaling dimension $\Delta$. The correlation function of
$S^{+a}_{1}(z,\bar{z})$ with a collection of primary gluon operators is given
by the subleading soft gluon theorem in Mellin space Pate:2019lpp ,

$\begin{gathered}\bigg{\langle}{S^{+a}_{1}(z,\bar{z})\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\bigg{\rangle}\\\
=-\sum_{k=1}^{n}\frac{\epsilon_{k}}{z-z_{k}}\left(-2\bar{h}_{k}+1+(\bar{z}-\bar{z}_{k})\bar{\partial}_{k}\right)T^{a}_{k}P_{k}^{-1}\bigg{\langle}\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})\bigg{\rangle}\end{gathered}$
(25)

where

$T^{a}_{k}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})=if^{aa_{i}b}\
\mathcal{O}^{b}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})\ \delta_{ki}$ (26)

and

$P_{k}^{-1}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})=\mathcal{O}^{a_{i}}_{h_{i}-\frac{1}{2},\bar{h}_{i}-\frac{1}{2}}(z_{i},\bar{z}_{i})\
\delta_{ki}$ (27)

Here $\epsilon_{k}=\pm 1$ depending on whether the gluon primary
$\mathcal{O}^{a_{k}}_{h_{k},\bar{h}_{k}}(z_{k},\bar{z}_{k})$ is outgoing or
incoming. For simplicity of notation we keep the additional label $\epsilon$
implicit when we write the correlation functions of the gluons.

Now following Pate:2019lpp ; Banerjee:2020zlg  we expand the R.H.S of (25) in
powers of the coordinate $\bar{z}$ of the subleading conformally soft gluon
operator $S^{+a}_{1}(z,\bar{z})$ and define two currents $J^{a}(z)$ and
$K^{a}(z)$ whose Ward identities are given by,

$\begin{gathered}\bigg{\langle}{J^{a}(z)\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\bigg{\rangle}=-\sum_{k=1}^{n}\frac{\epsilon_{k}}{z-z_{k}}\left(-2\bar{h}_{k}+1-\bar{z}_{k}\bar{\partial}_{k}\right)T^{a}_{k}P_{k}^{-1}\bigg{\langle}\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})\bigg{\rangle}\end{gathered}$
(28)

and

$\begin{gathered}\bigg{\langle}K^{a}(z)\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})\bigg{\rangle}=-\sum_{k=1}^{n}\frac{\epsilon_{k}}{z-z_{k}}\bar{\partial}_{k}\
T^{a}_{k}P_{k}^{-1}\bigg{\langle}\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})\bigg{\rangle}\end{gathered}$
(29)

This is equivalent to expanding the soft operator $S^{+a}_{1}(z,\bar{z})$ as,

$S^{+a}_{1}(z,\bar{z})=J^{a}(z)+\bar{z}\hskip 1.42271ptK^{a}(z)$ (30)

We can now define the modes of the currents, $J^{a}_{n}(z)$ and
$K^{a}_{n}(z)$, in the standard way. Their actions on a gluon primary are
given by the following commutation relations,

$\begin{split}\left[J^{a}_{n},\mathcal{O}^{b}_{h,\bar{h}}(z,\bar{z})\right]&=-\epsilon
z^{n}\left(-2\bar{h}+1-\bar{z}\bar{\partial}\right)T^{a}P^{-1}\mathcal{O}^{b}_{h,\bar{h}}(z,\bar{z})\\\
&=-i\epsilon
f^{abc}z^{n}\left(-2\bar{h}+1-\bar{z}\bar{\partial}\right)\mathcal{O}^{c}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(z,\bar{z})\end{split}$
(31)

and

$\begin{gathered}\left[K^{a}_{n},\mathcal{O}^{b}_{h,\bar{h}}(z,\bar{z})\right]=-\epsilon
z^{n}\bar{\partial}\
T^{a}P^{-1}\mathcal{O}^{b}_{h,\bar{h}}(z,\bar{z})=-i\epsilon
f^{abc}z^{n}\bar{\partial}\
\mathcal{O}^{c}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(z,\bar{z})\end{gathered}$
(32)

Although the generators $\left(J^{a}_{n},K^{a}_{n}\right)$ do not form a Lie
algebra under commutation, we will see in section (5) that when the
commutators $\left[J^{a}_{m},J^{b}_{n}\right]$,
$\left[J^{a}_{m},K^{b}_{n}\right]$ and $\left[K^{a}_{m},K^{b}_{n}\right]$ act
on a gluon conformal primary or its descendants, the results are given by
simple expressions which look almost like closure. This is crucial for our
purpose in this paper.

###  4.1 OPE between $S^{+a}_{1}(z,\bar{z})$ and a hard gluon conformal
primary

Suppose we want to compute the OPE between $S^{+a}_{1}(z,\bar{z})$ and the
gluon primary $\mathcal{O}^{a_{1}}_{h_{1},\bar{h}_{1}}(z_{1},\bar{z}_{1})$.
For this we have to expand the R.H.S of (25) in powers of $(z-z_{1})$ and
$(\bar{z}-\bar{z}_{1})$. So as $(z,\bar{z})\rightarrow(z_{1},\bar{z}_{1})$ we
can write,

$\begin{gathered}\bigg{\langle}S^{+a}_{1}(z,\bar{z})\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})\bigg{\rangle}\\\
=\left(-\frac{\epsilon_{1}}{z-z_{1}}\left(-2\bar{h}_{1}+1\right)T^{a}_{1}P^{-1}_{1}+\sum_{p=1}^{\infty}(z-z_{1})^{p-1}\mathcal{J}^{a}_{-p}(1)\right)\bigg{\langle}{\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\bigg{\rangle}\\\
+(\bar{z}-\bar{z}_{1})\left(-\frac{\epsilon_{1}}{z-z_{1}}\bar{\partial}_{1}T^{a}_{1}P_{1}^{-1}+\sum_{p=1}^{\infty}(z-z_{1})^{p-1}\mathcal{K}^{a}_{-p}(1)\right)\bigg{\langle}{\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\bigg{\rangle}\end{gathered}$
(33)

where the differential operators $\mathcal{J}^{a}_{-p}(1)$ and
$\mathcal{K}^{a}_{-p}(1)$ are defined as,

$\begin{gathered}\mathcal{J}^{a}_{-p}(1)\bigg{\langle}{\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\bigg{\rangle}=-\left(\sum_{j=2}^{n}\epsilon_{j}\frac{2\bar{h}_{j}-1+(\bar{z}_{j}-\bar{z}_{1})\bar{\partial}_{j}}{(z_{j}-z_{1})^{p}}T^{a}_{j}P^{-1}_{j}\right)\bigg{\langle}{\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\bigg{\rangle},\quad
p\geq 1\end{gathered}$ (34)

and

$\begin{gathered}\mathcal{K}^{a}_{-p}(1)\bigg{\langle}{\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\bigg{\rangle}=\left(\sum_{j=2}^{n}\epsilon_{j}\frac{\bar{\partial}_{j}}{(z_{j}-z_{1})^{p}}T^{a}_{j}P^{-1}_{j}\right)\bigg{\langle}{\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\bigg{\rangle},\quad
p\geq 1\end{gathered}$ (35)

From (33) the OPE between $S^{a}_{1}(z,\bar{z})$ and
$\mathcal{O}^{a_{1}}_{h_{1},\bar{h}_{1}}(z_{1},\bar{z}_{1})$ can be extracted
to be,

$\begin{gathered}S^{+a}_{1}(z,\bar{z})\mathcal{O}^{a_{1}}_{h_{1},\bar{h}_{1}}(z_{1},\bar{z}_{1})\\\
=-\frac{\epsilon_{1}}{z-z_{1}}\left(-2\bar{h}_{1}+1\right)T^{a}_{1}P^{-1}_{1}\mathcal{O}^{a_{1}}_{h_{1},\bar{h}_{1}}(z_{1},\bar{z}_{1})+\sum_{p=1}^{\infty}(z-z_{1})^{p-1}\left(J^{a}_{-p}\mathcal{O}^{a_{1}}_{h_{1},\bar{h}_{1}}\right)(z_{1},\bar{z}_{1})\\\
-(\bar{z}-\bar{z}_{1})\left(\epsilon_{1}\frac{\bar{\partial}_{1}}{z-z_{1}}T^{a}_{1}P_{1}^{-1}\mathcal{O}^{a_{1}}_{h_{1},\bar{h}_{1}}(z_{1},\bar{z}_{1})+\sum_{p=1}^{\infty}(z-z_{1})^{p-1}\left(K^{a}_{-p}\mathcal{O}^{a_{1}}_{h_{1},\bar{h}_{1}}\right)(z_{1},\bar{z}_{1})\right)\end{gathered}$
(36)

where the correlation functions with the insertion of the descendants
$J^{a}_{-p}\mathcal{O}^{a_{1}}_{h_{1},\bar{h}_{1}}(z_{1},\bar{z}_{1})$ and
$K^{a}_{-p}\mathcal{O}^{a_{1}}_{h_{1},\bar{h}_{1}}(z_{1},\bar{z}_{1})$ are
given by (34) and (35), respectively.

Now (36) acts as a boundary condition on the OPE of two hard gluon primaries
one which is positive helicity. Using the definition (24) of the subleading
conformally soft gluon $S^{+a}_{1}(z,\bar{z})$ we can write,

$\boxed{\lim_{\Delta\rightarrow
0}\Delta\mathcal{O}^{a}_{\Delta,+}(z,\bar{z})\mathcal{O}^{a_{1}}_{h_{1},\bar{h}_{1}}(z_{1},\bar{z}_{1})=\eqref{ope1}}$
(37)

As we will see (37) is a nontrivial constraint on the OPE of two hard gluons.
Inside correlation functions (37) means that we first take the OPE limit
$(z,\bar{z})\rightarrow(z_{1},\bar{z}_{1})$ and then take the subleading
conformal soft limit. In that limit we should always get back (33).

We also note that the operator product expansion (36) leads to the following
conditions which are satisfied by any gluon primary
$\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z})$,

$J^{a}_{n}\mathcal{O}^{a}_{h,\bar{h}}(0)=0,\quad
J^{a}_{0}\mathcal{O}^{b}_{h,\bar{h}}(0)=-i\epsilon
f^{abc}(-2\bar{h}+1)\mathcal{O}^{c}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(0),\quad\forall
n>0$ (38)

and

$K^{a}_{n}\mathcal{O}^{a}_{h,\bar{h}}(0)=0,\quad
K^{a}_{0}\mathcal{O}^{b}_{h,\bar{h}}(0)=-i\epsilon
f^{abc}\bar{\partial}\mathcal{O}^{c}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(0),\quad\forall
n>0$ (39)

###  4.2 Commutation relations involving subleading generators

In this section we collect some useful commutation relations. These are the
”classical” commutators which can be easily obtained from the action of the
generators on a primary operator. For the convenience of the reader we gather
here the actions of the Poincare and current algebra generators on a gluon
primary,

$\left[L_{1},\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z})\right]=\left(z^{2}\partial+2hz\right)\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z}),\quad\left[L_{-1},\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z})\right]=\partial\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z})$
(40)
$\left[\bar{L}_{1},\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z})\right]=\left(\bar{z}^{2}\bar{\partial}+2\bar{h}\bar{z}\right)\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z}),\quad\left[\bar{L}_{-1},\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z})\right]=\bar{\partial}\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z})$
(41) $\left[P_{m,n},\mathcal{O}^{a}_{h,\bar{h}}(z,\bar{z})\right]=\epsilon
z^{m+1}\bar{z}^{n+1}\mathcal{O}^{a}_{h+\frac{1}{2},\bar{h}+\frac{1}{2}}(z,\bar{z})$
(42)
$\left[j^{a}_{n},\mathcal{O}^{b}_{h,\bar{h}}(z,\bar{z})\right]=-iz^{n}f^{abc}\mathcal{O}^{c}_{h,\bar{h}}(z,\bar{z})$
(43) $\left[J^{a}_{n},\mathcal{O}^{b}_{h,\bar{h}}(z,\bar{z})\right]=-i\epsilon
f^{abc}z^{n}\left(-2\bar{h}+1-\bar{z}\bar{\partial}\right)\mathcal{O}^{c}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(z,\bar{z})$
(44) $\left[K^{a}_{n},\mathcal{O}^{b}_{h,\bar{h}}(z,\bar{z})\right]=-i\epsilon
f^{abc}z^{n}\bar{\partial}\mathcal{O}^{c}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(z,\bar{z})$
(45)

where $\epsilon=\pm 1$ depending on whether the gluon is outgoing or incoming.
Using these we arrive at the following commutation relations between
generators,

$\left[L_{1},J^{a}_{n}\right]=-(n+1)J^{a}_{n+1},\quad\left[L_{0},J^{a}_{n}\right]=\left(n-\frac{1}{2}\right)J^{a}_{n},\quad\left[L_{-1},J^{a}_{n}\right]=-nJ^{a}_{n-1}$
(46)
$\left[\bar{L}_{1},J^{a}_{n}\right]=0,\quad\left[\bar{L}_{0},J^{a}_{n}\right]=-\frac{1}{2}J^{a}_{n},\quad\left[\bar{L}_{-1},J^{a}_{n}\right]=-K^{a}_{n}$
(47)
$\left[L_{1},K^{a}_{n}\right]=-(n+1)K^{a}_{n+1},\quad\left[L_{0},K^{a}_{n}\right]=\left(n-\frac{1}{2}\right)K^{a}_{n},\quad\left[L_{-1},K^{a}_{n}\right]=-nK^{a}_{n-1}$
(48)
$\left[\bar{L}_{1},K^{a}_{n}\right]=J^{a}_{n},\quad\left[\bar{L}_{0},K^{a}_{n}\right]=\frac{1}{2}K^{a}_{n},\quad\left[\bar{L}_{-1},K^{a}_{n}\right]=0$
(49)
$\left[P_{m,-1},J^{a}_{n}\right]=j^{a}_{m+n+1},\quad\left[P_{m,0},J^{a}_{n}\right]=0,\quad
m=0,-1$ (50)
$\left[P_{m,-1},K^{a}_{n}\right]=0,\quad\left[P_{m,0},K^{a}_{n}\right]=j^{a}_{m+n+1},\quad
m=0,-1$ (51)
$\left[j^{a}_{m},J^{b}_{n}\right]=-if^{abc}J^{c}_{m+n},\quad\left[j^{a}_{m},K^{b}_{n}\right]=-if^{abc}K^{c}_{m+n}$
(52)

##  5 How to interpret the commutators of subleading symmetry generators

It is well known Lysov:2014csa  that the subleading symmetry generators
$J^{a}_{n}$ and $K^{a}_{n}$ do not close under commutation. So they are not
the generators of a Lie algebra symmetry in the ordinary sense. To see this we
note that the scaling dimensions of $J^{a}_{n}$ and $K^{a}_{n}$ are given by
$(-n-1/2,-1/2)$ and $(-n-1/2,1/2)$, respectively. Therefore the
antiholomorphic scaling dimension of the commutator amongst these generators
must be an integer. But there is no generator with integer antiholomorphic
scaling dimension that can appear here and so the generators
$(J^{a}_{m},K^{a}_{m})$ cannot form a Lie algebra.

Now, at least for the purposes of this paper, what we really need to know is
how the commutator of two subleading generators acts on a gluon primary or its
descendants. For example, in the OPE of two outgoing gluon primaries of
opposite helicities given by, $\mathcal{O}^{a}_{\Delta_{1},+}$ and
$\mathcal{O}^{b}_{\Delta_{1},-}$, one gets a subleading term of the form
(235), $J^{a}_{-1}\mathcal{O}^{b}_{\Delta_{1}+\Delta_{2},-}$. In order to
calculate the OPE coefficient multiplying this operator, one has to know the
structure of the term

$J^{c}_{1}J^{a}_{-1}\mathcal{O}^{b}_{\Delta_{1}+\Delta_{2},-}=\left[J^{c}_{1},J^{a}_{-1}\right]\mathcal{O}^{b}_{\Delta_{1}+\Delta_{2},-}$
(53)

In (53) we have used (38) which gives
$J^{c}_{1}O^{b}_{\Delta_{1}+\Delta_{2},-}=0$. More generally, we will get
terms like

$\left(\left[J^{a}_{m},J^{b}_{n}\right],\left[J^{a}_{m},K^{b}_{n}\right],\left[K^{a}_{m},K^{b}_{n}\right]\right)\prod_{i}J^{c_{i}}_{p_{i}}\prod_{j}K^{d_{j}}_{q_{j}}\prod_{k}j^{e_{k}}_{r_{k}}\mathcal{O}^{f}_{h,\bar{h}}(z,\bar{z})$
(54)

from the OPE and we need to get simplified expressions for them. In order to
do this we start by computing the following commutators

$\left[\left[J^{a}_{m},J^{b}_{n}\right],\mathcal{O}^{c}_{h,\bar{h}}(z,\bar{z})\right],\quad\left[\left[J^{a}_{m},K^{b}_{n}\right],\mathcal{O}^{c}_{h,\bar{h}}(z,\bar{z})\right],\quad\left[\left[K^{a}_{m},K^{b}_{n}\right],\mathcal{O}^{c}_{h,\bar{h}}(z,\bar{z})\right]$
(55)

Let us focus on the first commutator. Using (44) and the Jacobi identity we
get,

$\begin{gathered}\left[\left[J^{a}_{m},J^{b}_{n}\right],\mathcal{O}^{c}_{h,\bar{h}}(z,\bar{z})\right]\\\
=-f^{abd}f^{dce}z^{m+n}\left(-2\bar{h}+1-\bar{z}\bar{\partial}\right)\left(-2\bar{h}+2-\bar{z}\bar{\partial}\right)\mathcal{O}^{e}_{h-1,\bar{h}-1}(z,\bar{z})\end{gathered}$
(56)

Again using (44) we can write this as

$\left[\left[J^{a}_{m},J^{b}_{n}\right],\mathcal{O}^{c}_{h,\bar{h}}(z,\bar{z})\right]=-i\epsilon
f^{abd}\left(-2\bar{h}+1-\bar{z}\bar{\partial}\right)\left[J^{d}_{m+n},\mathcal{O}^{c}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(z,\bar{z})\right]$
(57)

where we have used $\epsilon^{2}=1$. Now we take the limit
$(z,\bar{z})\rightarrow(0,0)$ and from (57) we get

$\left[\left[J^{a}_{m},J^{b}_{n}\right],\mathcal{O}^{c}_{h,\bar{h}}(0)\right]=-i\epsilon
f^{abd}\left(-2\bar{h}+1\right)\left[J^{d}_{m+n},\mathcal{O}^{c}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(0)\right]$
(58)

Since the mode $J^{a}_{n}$ of the current $J^{a}$ is defined with respect to
the point $(0,0)$, (58) can be interpreted as the relation between the
descendants

$\boxed{\left[J^{a}_{m},J^{b}_{n}\right]\mathcal{O}^{c}_{h,\bar{h}}(0)=-i\epsilon
f^{abd}\left(-2\bar{h}+1\right)J^{d}_{m+n}\mathcal{O}^{c}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(0)}$
(59)

Now we can apply the same argument to get the other two relations

$\boxed{\left[J^{a}_{m},K^{b}_{n}\right]\mathcal{O}^{c}_{h,\bar{h}}(0)=-i\epsilon
f^{abd}\left(-2\bar{h}+1\right)K^{d}_{m+n}\mathcal{O}^{c}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(0)}$
(60)

and

$\boxed{\left[K^{a}_{m},K^{b}_{n}\right]\mathcal{O}^{c}_{h,\bar{h}}(0)=-i\epsilon
f^{abd}K^{d}_{m+n}\bar{\partial}\mathcal{O}^{c}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(0)}$
(61)

These are the relations that will be used in the following sections to obtain
recursion relations for subleading OPE coefficients and to define null states
or primary descendants.

We can see that the L.H.S of (59), (60) and (61) are linear in the subleading
symmetry generators. But, they also depend on the scaling dimension of the
gluon primary on which the commutators are acting and also the gluon primary
appearing on the R.H.S has shifted dimension compared to the one appearing on
the L.H.S. This is a signature of the fact that the generators do not form a
Lie algebra. This is perhaps the closest one can come towards forming a closed
algebra out of $J^{a}_{n}$ and $K^{a}_{n}$ when they act on states in a
Hilbert space and, as we will see, this is sufficient for finding (subleading)
OPE coefficients as we will see in section (9).

Before we end this section we would like to mention the result when the
commutator acts on a level-$1$ descendant. They are given by,

$\left[J^{a}_{m},J^{b}_{n}\right]J^{c}_{p}\mathcal{O}^{d}_{h,\bar{h}}(0)=-i\epsilon
f^{abe}(-2\bar{h}+1)J^{e}_{m+n}J^{c}_{p}\mathcal{O}^{d}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(0)$
(62)

$\left[J^{a}_{m},K^{b}_{n}\right]J^{c}_{p}\mathcal{O}^{d}_{h,\bar{h}}(0)=-i\epsilon
f^{abe}(-2\bar{h}+1)K^{e}_{m+n}J^{c}_{p}\mathcal{O}^{d}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(0)$
(63)

$\left[K^{a}_{m},K^{b}_{n}\right]J^{c}_{p}\mathcal{O}^{d}_{h,\bar{h}}(0)=-i\epsilon
f^{abe}(-2\bar{h}+1)K^{e}_{m+n}J^{c}_{p}\bar{\partial}\mathcal{O}^{d}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(0)$
(64)

$\left[J^{a}_{m},J^{b}_{n}\right]K^{c}_{p}\mathcal{O}^{d}_{h,\bar{h}}(0)=-i\epsilon
f^{abe}(-2\bar{h}+1)J^{e}_{m+n}K^{c}_{p}\mathcal{O}^{d}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(0)$
(65)

$\left[J^{a}_{m},K^{b}_{n}\right]K^{c}_{p}\mathcal{O}^{d}_{h,\bar{h}}(0)=-i\epsilon
f^{abe}(-2\bar{h}+1)K^{e}_{m+n}K^{c}_{p}\mathcal{O}^{d}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(0)$
(66)

$\left[K^{a}_{m},K^{b}_{n}\right]K^{c}_{p}\mathcal{O}^{d}_{h,\bar{h}}(0)=-i\epsilon
f^{abe}(-2\bar{h}+1)K^{e}_{m+n}K^{c}_{p}\bar{\partial}\mathcal{O}^{d}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(0)$
(67)

$\left[J^{a}_{m},J^{b}_{n}\right]j^{c}_{p}\mathcal{O}^{d}_{h,\bar{h}}(0)=-i\epsilon
f^{abe}(-2\bar{h}+1)J^{e}_{m+n}j^{c}_{p}\mathcal{O}^{d}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(0)$
(68)

$\left[J^{a}_{m},K^{b}_{n}\right]j^{c}_{p}\mathcal{O}^{d}_{h,\bar{h}}(0)=-i\epsilon
f^{abe}(-2\bar{h}+1)K^{e}_{m+n}j^{c}_{p}\mathcal{O}^{d}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(0)$
(69)

$\left[K^{a}_{m},K^{b}_{n}\right]j^{c}_{p}\mathcal{O}^{d}_{h,\bar{h}}(0)=-i\epsilon
f^{abe}(-2\bar{h}+1)K^{e}_{m+n}j^{c}_{p}\bar{\partial}\mathcal{O}^{d}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(0)$
(70)

The above relations, say for example (62), can be obtained by starting from
the commutator

$\left[\left[J^{a}_{m},J^{b}_{n}\right],\left[J^{c}_{p},\mathcal{O}^{d}_{h,\bar{h}}(z,\bar{z})\right]\right]$
(71)

The above relations have obvious generalisations to a general descendant and
are given by,

$\left[J^{a}_{m},J^{b}_{n}\right]\prod_{i}J^{c_{i}}_{p_{i}}\prod_{j}K^{d_{j}}_{q_{j}}\prod_{k}j^{e_{k}}_{r_{k}}\mathcal{O}^{f}_{h,\bar{h}}(0)=-i\epsilon
f^{abx}(-2\bar{h}+1)J^{x}_{m+n}\prod_{i}J^{c_{i}}_{p_{i}}\prod_{j}K^{d_{j}}_{q_{j}}\prod_{k}j^{e_{k}}_{r_{k}}\mathcal{O}^{f}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(0)$
(72)

$\left[J^{a}_{m},K^{b}_{n}\right]\prod_{i}J^{c_{i}}_{p_{i}}\prod_{j}K^{d_{j}}_{q_{j}}\prod_{k}j^{e_{k}}_{r_{k}}\mathcal{O}^{f}_{h,\bar{h}}(0)=-i\epsilon
f^{abx}(-2\bar{h}+1)K^{x}_{m+n}\prod_{i}J^{c_{i}}_{p_{i}}\prod_{j}K^{d_{j}}_{q_{j}}\prod_{k}j^{e_{k}}_{r_{k}}\mathcal{O}^{f}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(0)$
(73)

$\left[K^{a}_{m},K^{b}_{n}\right]\prod_{i}J^{c_{i}}_{p_{i}}\prod_{j}K^{d_{j}}_{q_{j}}\prod_{k}j^{e_{k}}_{r_{k}}\mathcal{O}^{f}_{h,\bar{h}}(0)=-i\epsilon
f^{abx}(-2\bar{h}+1)K^{x}_{m+n}\prod_{i}J^{c_{i}}_{p_{i}}\prod_{j}K^{d_{j}}_{q_{j}}\prod_{k}j^{e_{k}}_{r_{k}}\bar{\partial}\mathcal{O}^{f}_{h-\frac{1}{2},\bar{h}-\frac{1}{2}}(0)$
(74)

We do not need these more general relations in this paper but, it will be
interesting to check their consistency with explicit calculations performed
using scattering amplitudes. For example, a good check of this will be to
compute subleading OPE coefficients in the gluon-gluon OPE directly from the
(MHV) scattering amplitude and compare them with the results from the
recursion relations obtained using (72), (73) and (74). Our derivation of
these relations has been somewhat hand waving. We leave a more rigorous
derivation to future work.

##  6 Differential equation for MHV gluon amplitudes in Mellin space

In this section we will derive a differential equation for Mellin transformed
tree level $n$-point MHV gluon amplitudes in Yang-Mills theory.

Consider the celestial OPE between two positive helicity outgoing gluon
primaries in Yang-Mills theory. We will denote these operators below as
$\mathcal{O}^{a}_{\Delta,+}(z,\bar{z})$ and
$\mathcal{O}^{a_{1}}_{\Delta_{1},+}(z_{1},\bar{z}_{1})$ where the subscript
$(+)$ denotes that both operators have positive helicity. In order to arrive
at the proposed differential equation, we will be interested in the
contribution from descendants which constitute the first subleading correction
to the leading singular term in this OPE. This was recently obtained in
Ebert:2020nqf  and is given by

$\begin{split}\mathcal{O}^{a}_{\Delta,+}(z,\bar{z})\mathcal{O}^{a_{1}}_{\Delta_{1},+}(z_{1},\bar{z}_{1})&=-i\hskip
1.42271ptB(\Delta-1,\Delta_{1}-1)\bigg{[}\frac{f^{aa_{1}x}}{z-z_{1}}+\frac{\Delta-1}{\Delta+\Delta_{1}-2}\hskip
1.42271ptf^{aa_{1}x}L_{-1}\\\
&+i\left(\frac{\Delta-1}{\Delta+\Delta_{1}-2}\hskip
1.42271pt\delta^{ax}\delta^{a_{1}y}+\frac{\Delta_{1}-1}{\Delta+\Delta_{1}-2}\hskip
1.42271pt\delta^{ay}\delta^{a_{1}x}\right)j^{y}_{-1}\bigg{]}\mathcal{O}^{x}_{\Delta+\Delta_{1}-1,+}(z_{1},\bar{z}_{1})+\cdots\end{split}$
(75)

where $\mathcal{O}^{x}_{\Delta+\Delta_{1}-1,+}(z_{1},\bar{z}_{1})$ is a
positive helicity (outgoing) gluon primary. The leading primary OPE
coefficient is given by the Euler beta function, $B(\Delta-1,\Delta_{1}-1)$.
The dots above denote contributions from descendants at further subleading
orders in $(z-z_{1}),(\bar{z}-\bar{z}_{1})$. In Ebert:2020nqf  the above OPE
was extracted from the Mellin transform of the tree-level $4$-point MHV gluon
amplitude. We refer the reader to Section B of the Appendix in this paper for
a derivation of (75) from the Mellin transform of the $5$-point MHV gluon
amplitude.

Now let us take the subleading conformal soft limit $\Delta\rightarrow 0$ in
the above OPE. We then obtain

$\begin{gathered}\lim_{\Delta\to 0}\Delta\hskip
1.42271pt\mathcal{O}^{a}_{\Delta,+}(z,\bar{z})\mathcal{O}^{a_{1}}_{\Delta_{1},+}(z_{1},\bar{z}_{1})\\\
=\bigg{[}\frac{(\Delta_{1}-2)}{z-z_{1}}\hskip
1.42271ptif^{aa_{1}x}-if^{aa_{1}x}L_{-1}+\left(\delta^{ax}\delta^{a_{1}y}-(\Delta_{1}-1)\delta^{ay}\delta^{a_{1}x}\right)j^{y}_{-1}\bigg{]}O^{x}_{\Delta_{1}-1,+}(z_{1},\bar{z}_{1})+\cdots\end{gathered}$
(76)

According to our discussion in Section (4.1), in the subleading conformal soft
limit the OPE should obey the general constraint given by equation (37).
Therefore as $\Delta\rightarrow 0$ in (75) we should get

$\begin{split}&\lim_{\Delta\to 0}\Delta\hskip
1.42271pt\mathcal{O}^{a}_{\Delta,+}(z,\bar{z})\mathcal{O}^{a_{1}}_{\Delta_{1},+}(z_{1},\bar{z}_{1})=\bigg{[}\frac{(\Delta_{1}-2)}{z-z_{1}}\hskip
1.42271ptT^{a}_{1}P_{1}^{-1}+J^{a}_{-1}\bigg{]}\mathcal{O}^{a_{1}}_{\Delta_{1},+}(z_{1},\bar{z}_{1})+\cdots\end{split}$
(77)

where

$\begin{split}T^{a}P_{1}^{-1}\mathcal{O}^{a_{1}}_{\Delta_{1},+}(z_{1},\bar{z}_{1})=if^{aa_{1}x}\mathcal{O}^{x}_{\Delta_{1}-1,+}(z_{1},\bar{z}_{1})\end{split}$
(78)

Then comparing (76) and (77) we see that the leading singular terms in the OPE
match. But the subleading terms appear to be different. Therefore in order for
the OPE in (75) to be consistent with the subleading soft gluon theorem we
must have the following relation

$\begin{split}&\left[\delta^{a_{1}x}J^{a}_{-1}+if^{aa_{1}x}L_{-1}P_{1}^{-1}-\left(\delta^{ax}\delta^{a_{1}y}-(\Delta_{1}-1)\delta^{ay}\delta^{a_{1}x}\right)j^{y}_{-1}P_{1}^{-1}\right]\mathcal{O}^{x}_{\Delta_{1},+}(z_{1},\bar{z}_{1})=0\end{split}$
(79)

Multiplying the above by $if^{aa_{1}b}$ and using the relation

$\begin{split}&f^{aa_{1}b}f^{aa_{1}c}=C_{A}\hskip
1.42271pt\delta^{bc}\end{split}$ (80)

where $C_{A}$ is the quadratic Casimir of the adjoint representation, we can
express (79) as

$\begin{split}&\bigg{[}C_{A}L_{-1}+\Delta_{1}\hskip
1.42271ptj^{a}_{-1}T^{a}_{1}+J^{a}_{-1}T^{a}_{1}P_{1}\bigg{]}\mathcal{O}^{a_{1}}_{\Delta_{1}-1,+}(z_{1},\bar{z}_{1})=0\end{split}$
(81)

Further shifting $\Delta_{1}\rightarrow\Delta_{1}+1$ in (81) we get

$\begin{split}&\bigg{[}C_{A}L_{-1}+(\Delta_{1}+1)\hskip
1.42271ptj^{a}_{-1}T^{a}_{1}+J^{a}_{-1}T^{a}_{1}P_{1}\bigg{]}\mathcal{O}^{a_{1}}_{\Delta_{1},+}(z_{1},\bar{z}_{1})=0\end{split}$
(82)

Up to this point we have been considering the gluon primary in (81) to be
outgoing. But this can be easily generalised to the case of an incoming
positive helicity gluon. In that case we simply get an additional minus sign
before the third term in (82). Thus for an incoming positive helicity gluon we
have

$\begin{split}&\bigg{[}C_{A}L_{-1}+(\Delta_{1}+1)\hskip
1.42271ptj^{a}_{-1}T^{a}_{1}-J^{a}_{-1}T^{a}_{1}P_{1}\bigg{]}\mathcal{O}^{a_{1}}_{\Delta_{1},+}(z_{1},\bar{z}_{1})=0\end{split}$
(83)

Therefore in general we have the condition

$\begin{split}&\boxed{\Psi^{a}(z,\bar{z})=\bigg{[}C_{A}L_{-1}-(\Delta+1)\hskip
1.42271ptj^{b}_{-1}j^{b}_{0}-J^{b}_{-1}j^{b}_{0}P_{-1,-1}\bigg{]}\mathcal{O}^{a}_{\Delta,+}(z,\bar{z})=0}\end{split}$
(84)

where $P_{-1,-1}$ is the global time translation generator which acts on a
gluon primary as $P_{-1,-1}\mathcal{O}^{b}_{\Delta,+}=\epsilon\hskip
1.13791pt\mathcal{O}^{b}_{\Delta+1,+}$. Here $\epsilon=\pm 1$ for an outgoing
(incoming) gluon. Note that in obtaining (84) we have used the definition (23)
of a current algebra primary according to which

$\begin{split}j^{a}_{0}\mathcal{O}^{b}_{\Delta,+}(z,\bar{z})=-if^{abc}\mathcal{O}^{c}_{\Delta,+}(z,\bar{z})=-T^{a}\mathcal{O}^{b}_{\Delta,+}(z,\bar{z})\end{split}$
(85)

Now consider the linear combination of descendants denoted as
$\Psi^{a}(z,\bar{z})$ in (84). Applying the definition of a Poincare and
current algebra primary from Sections (2.1) and (3.1) and using the
commutation relations given in Section (4.2), it can be easily checked that
444In (86) and (87), the modes of the symmetry generators have been defined
with respect to the point $(z,\bar{z})$.

$\begin{split}L_{1}\Psi^{a}(z,\bar{z})=\bar{L}_{1}\Psi^{a}(z,\bar{z})=P_{0,-1}\Psi^{a}(z,\bar{z})=P_{-1,0}\Psi^{a}(z,\bar{z})=0\end{split}$
(86)
$\begin{split}j^{a}_{0}\Psi^{b}(z,\bar{z})=-if^{abc}\Psi^{c}(z,\bar{z}),\quad
j^{a}_{m}\Psi^{b}(z,\bar{z})=0,\quad\forall m\geq 1\end{split}$ (87)

Equations (86) and (87) together imply that $\Psi^{a}(z,\bar{z})$ is in fact a
primary operator with respect to the Poincare group and the current algebra
associated to the leading soft gluon theorem. 555Since the subleading symmetry
generators do not form a closed (Lie) algebra we do not impose “primary-state”
condition under subleading soft symmetry. This will require further study and
we hope to come back to this in future. In fact, one can easily check that the
null-state $\Psi^{a}$ is uniquely determined by the primary-state conditions
under the Poincare group and leading soft gluon $SU(N)$ current algebra. Thus
$\Psi^{a}(z,\bar{z})$ is a null field and we can consistently set
$\Psi^{a}(z,\bar{z})$ to zero within celestial MHV gluon amplitudes.

Now let us insert (84) within Mellin transformed tree-level MHV gluon
amplitudes. Below we will denote the gluon primaries as
$\mathcal{O}^{a_{k}}_{h_{k},\bar{h}_{k}}(z_{k},\bar{z}_{k})$. The negative
helicity gluons in the MHV amplitude will be labelled by $(n-1)$ and $n$. Then
for every positive helicity gluon $i\in(1,2,\cdots n-2)$ we get a decoupling
relation

$\begin{split}&\bigg{\langle}\left[C_{A}L_{-1}(i)-2\hskip 1.42271pth_{i}\hskip
1.42271ptj^{a}_{-1}(i)j^{a}_{0}(i)-J^{a}_{-1}(i)j^{a}_{0}(i)P_{-1,-1}(i)\right]\prod_{k=1}^{n}\mathcal{O}^{a_{k}}_{h_{k},\bar{h}_{k}}(z_{k},\bar{z}_{k})\bigg{\rangle}_{\text{MHV}}=0\end{split}$
(88)

where we have used $2\hskip 1.42271pth_{i}=\Delta_{i}+1$ for a positive
helicity gluon primary. The index $(i)$ accompanying
$L_{-1},j^{a}_{0},j^{a}_{-1},J^{a}_{-1}$ and $P_{-1,-1}$ above denotes that
these modes act on the $i$-th positive helicity gluon. Then using the
representation of $L_{-1},j^{a}_{-1},J^{a}_{-1}$ in terms of differential
operators, we obtain from (88) with $i\in(1,2,\cdots,n-2)$

$\begin{split}&\bigg{[}\frac{C_{A}}{2}\hskip 1.42271pt\frac{\partial}{\partial
z_{i}}-h_{i}\hskip 1.42271pt\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{n}\frac{T^{a}_{i}\hskip 1.42271ptT^{a}_{j}}{z_{i}-z_{j}}\\\
&+\frac{1}{2}\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{n}\frac{\epsilon_{j}\left(2\bar{h}_{j}-1-(\bar{z}_{i}-\bar{z}_{j})\frac{\partial}{\partial\bar{z}_{j}}\right)}{z_{i}-z_{j}}\hskip
1.42271ptT^{a}_{j}P^{-1}_{j}T^{a}_{i}P_{-1,-1}(i)\bigg{]}\bigg{\langle}\prod_{k=1}^{n}\mathcal{O}^{a_{k}}_{h_{k},\bar{h}_{k}}(z_{k},\bar{z}_{k})\bigg{\rangle}_{\text{MHV}}=0\end{split}$
(89)

We have thus obtained $(n-2)$ linear first order partial differential
equations (PDEs) for tree-level $n$-point gluon MHV amplitudes in Mellin
space. The $(n-2)$ PDEs correspond to the $(n-2)$ positive helicity gluons in
the $n$-point MHV amplitude.

Now let us note that the structure of (89) is similar to the Knizhnik-
Zamolodchikov (KZ) equation Knizhnik:1984nr  obeyed by correlation functions
of primary operators in WZW theory. The KZ equation is given by

$\begin{split}&\bigg{[}\left(k+\frac{C_{A}}{2}\right)\frac{\partial}{\partial
z_{i}}-\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{n}\frac{T^{a}_{i}\hskip
1.42271ptT^{a}_{j}}{z_{i}-z_{j}}\bigg{]}\left\langle\phi^{a_{1}}_{h_{1},\bar{h}_{1}}(z_{1},\bar{z}_{1})\cdots\phi^{a_{n}}_{h_{n},\bar{h}_{n}}(z_{n},\bar{z}_{n})\right\rangle=0\end{split}$
(90)

where $\phi^{a_{i}}_{h_{i},\bar{h}_{i}}$ are the primary operators and $k$ is
the level of the current algebra. In the context of our paper, we will take
these operators to transform in the adjoint representation of the zero mode
algebra and so the superscript $a_{i}$ is a Lie algebra index. Let us now
compare the differential equation (89) and the KZ equation (90).

First of all, for an $n$-point correlation function of primaries in WZW theory
there are $n$ differential equations because every primary in WZW theory is
degenerate. This should be contrasted with the case of MHV amplitudes where an
$n$-point MHV amplitude satisfies $(n-2)$ differential equations corresponding
to $(n-2)$ positive helicity gluons. This is related to the fact that within
the MHV sector, governed by leading and subleading current algebras coming
from positive helicity soft gluon, negative helicity gluons have no null
states. This is a major difference.

Secondly, note that the coefficient of the $\partial_{z_{i}}$ term in (89) is
$C_{A}/2$, where $C_{A}$ is the quadratic Casimir of the adjoint
representation. In the KZ equation (90), this coefficient is given by
$(k+C_{A}/2)$. At a superficial level this is consistent with the fact that in
our case the $SU(N)$ current algebra has level $k=0$.

Now let us consider the second term within the square brackets in (89). This
arises from the $j^{a}_{-1}(i)j^{a}_{0}(i)$ piece in (88). The analogous term
is also present in the KZ equation (90) but, the coefficient of this term in
our case depends on the holomorphic conformal weight $h_{i}$ of the primary
operator $\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})$ whose
null state gives rises to (89). This is an important difference which plays a
crucial role due to the following reason.

In WZW theory, the KZ equation follows from the existence of a Sugawara stress
tensor. From the expression of the Sugawara stress tensor it also follows that
the holomorphic conformal weight of a current algebra primary is given by
Knizhnik:1984nr ; Ginsparg:1988ui

$\begin{split}h_{r}=\frac{C_{r}}{2\hskip 1.13791ptk+C_{A}}\end{split}$ (91)

where $C_{r}$ is the quadratic Casimir of the representation $r$, under which
the primary operator transforms. Here we are considering $r$ to be the adjoint
representation and so $C_{r}=C_{A}$.

The null state relation which gives rise to the usual KZ equation holds only
when the primary operator, with respect to which the null state is defined,
has (holomorphic) weight given by (91). But (88) and consequently (89) hold
for arbitrary values of the scaling dimensions for the positive helicity gluon
primaries in the MHV amplitude666Here we are assuming that the dimensions
$\Delta_{i}$ of primary operators can be analytically continued off the
principal series where $\Delta_{i}=1+i\lambda_{i}$ with
$\lambda_{i}\in\mathbb{R}$. For tree-level amplitudes, the fact that such an
analytic continuation is possible is evident from the explicit expressions of
the corresponding Mellin amplitudes. See Donnay:2020guq  for a discussion on
how conformal primaries with general dimensions can be expressed in terms of
contour integrals over the principal series.. The coefficient $h_{i}$ in front
of the second term in (88), (89) plays a crucial role in ensuring this.

Finally let us discuss the third term within the square brackets in (89). This
arises due to the subleading soft gluon symmetry. There is no counterpart of
this term in the KZ equation (90). Thus compared to the usual KZ equation,
this can be regarded as a correction term. Another consequence of this term is
that unlike the KZ equation, the differential operators acting on the
celestial MHV amplitude are not purely holomorphic.

Recently in Fan:2020xjj  a Sugawara construction of the stress tensor was
performed for celestial CFTs by studying Mellin transformed gluon amplitudes
in Yang-Mills theory in the limit where a pair of gluons become conformally
soft. However it was observed that within correlation functions, this stress
tensor generates the correct conformal transformations only for the (leading)
conformally soft gluons but fails to do so for the hard gluon primaries. This
indicates that the Sugawara construction does not yield the full stress tensor
in the celestial CFT putatively dual to Yang-Mills theory. The possibility
that the full stress tensor may include contributions in addition to the
Sugawara stress tensor was also pointed out in McLoughlin:2016uwa ;
Cheung:2016iub . The additional term coming from the subleading soft gluon
symmetry in the differential equation (89) that we have obtained, further
suggests that the standard form of the Sugawara construction involving only
the leading current $j^{a}(z)$, may not apply to celestial CFTs and most
likely the subleading currents $(J^{a}(z),K^{a}(z))$ play an important role in
any such construction.

In the following sections we will study the implications of this differential
equation for the celestial OPE of gluon primary operators in Yang-Mills
theory. In particular we will show that the leading celestial OPE of gluons
can be determined using this equation.

##  7 Differential equation for MHV gluon amplitudes in momentum space

The differential equation (89) was derived for the Mellin transformed gluon
amplitude. We can also write down an equivalent form of this equation for the
amplitude in Fock space. Let us denote the tree-level Fock space MHV amplitude
as

$\begin{split}\bigg{\langle}\prod_{k=1}^{n}A^{a_{k}}(\epsilon_{k}\omega_{k},z_{k},\bar{z}_{k},\sigma_{k})\bigg{\rangle}_{\text{MHV}}\end{split}$
(92)

where $\epsilon=\pm 1$. $A^{a_{k}}(\omega_{k},z_{k},\bar{z}_{k},\sigma_{k})$
and $A^{a_{k}}(-\omega_{k},z_{k},\bar{z}_{k},\sigma_{k})$ are annihilation and
creation operators respectively for the external gluons with helicity
$\sigma_{k}$ in the S-matrix. In (92) we will take gluons $(1,2,\cdots,n-2)$
to have positive helicity. Then gluons $(n-1)$ and $n$ have negative helicity.

Now in order to recast (89) to momentum space, we make the following
substitutions

$\begin{split}&\Delta_{i}\rightarrow-\omega_{i}\partial_{\omega_{i}},\quad
P_{-1,-1}(i)\rightarrow\epsilon_{i}\omega_{i},\quad
P^{-1}_{i}\rightarrow\omega_{i}^{-1}\end{split}$ (93)

Applying an inverse Mellin transform we can replace the correlation function
in (89) with the Fock space amplitude (92). Thus we obtain the differential
equation

$\begin{split}&\bigg{[}C_{A}\hskip 1.42271pt\frac{\partial}{\partial
z_{i}}+\left(\omega_{i}\frac{\partial}{\partial\omega_{i}}-1\right)\sum_{\begin{subarray}{c}j=1\\\
j\neq i\end{subarray}}^{n}\frac{T^{a}_{i}\hskip
1.42271ptT^{a}_{j}}{z_{i}-z_{j}}\\\ &+\sum_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{n}\frac{\epsilon_{i}\omega_{i}}{\epsilon_{j}\omega_{j}}\frac{\left(\sigma_{j}+\omega_{j}\frac{\partial}{\partial\omega_{j}}+(\bar{z}_{i}-\bar{z}_{j})\frac{\partial}{\partial\bar{z}_{j}}\right)}{z_{i}-z_{j}}T^{a}_{i}T^{a}_{j}\bigg{]}\bigg{\langle}\prod_{k=1}^{n}A^{a_{k}}(\epsilon_{k}\omega_{k},z_{k},\bar{z}_{k},\sigma_{k})\bigg{\rangle}_{\text{MHV}}=0\end{split}$
(94)

As in (89), here we have $(n-2)$ partial differential equations labelled by
the index $i$ in (94) which runs over the $(n-2)$ positive helicity gluons in
the MHV amplitude. Also note that the amplitude appearing above is the full
tree level scattering amplitude and so explicitly includes the delta function
which imposes overall energy-momentum conservation. Therefore the partial
derivative with respect to $\bar{z}_{j}$ in (94) has a nontrivial action on
the amplitude (92).

##  8 Leading OPE from differential equation

In this section we will show that the leading structure of the celestial OPE
of gluons can be determined using the differential equation (89).

###  8.1 OPE for outgoing (incoming) gluons

Let us first consider the celestial OPE between a positive helicity gluon
primary $\mathcal{O}^{a_{1}}_{\Delta_{1},+}(z_{1},\bar{z}_{1})$ and another
gluon primary denoted by
$\mathcal{O}^{a_{2}}_{\Delta_{2},\sigma_{2}}(z_{2},\bar{z}_{2})$. The spin
$\sigma_{2}$ of the second gluon primary will be left unspecified for now. We
will also take both these gluons to be outgoing. Then let us assume that the
leading OPE in this case takes the following general form

$\begin{split}&\mathcal{O}^{a_{1}}_{\Delta_{1},+}(z_{1},\bar{z}_{1})\mathcal{O}^{a_{2}}_{\Delta_{2},\sigma_{2}}(z_{2},\bar{z}_{2})=-i\hskip
1.42271ptz_{12}^{p}\bar{z}_{12}^{q}\hskip 1.42271ptf^{a_{1}a_{2}x}\hskip
1.42271ptC_{p,q}\left(\Delta_{1},\Delta_{2},\sigma_{2}\right)\mathcal{O}^{x}_{\Delta,\sigma}(z_{2},\bar{z}_{2})+\cdots\end{split}$
(95)

where $z_{12}=(z_{1}-z_{2}),\bar{z}_{12}=(\bar{z}_{1}-\bar{z}_{2})$.
$\mathcal{O}^{x}_{\Delta,\sigma}(z_{2},\bar{z}_{2})$ is the leading primary
operator that can appear in the OPE and
$C_{p,q}\left(\Delta_{1},\Delta_{2},\sigma_{2}\right)$ is the associated OPE
coefficient. The dots denote possible contributions from descendants. The
conformal dimension and spin of
$\mathcal{O}^{x}_{\Delta,\sigma}(z_{2},\bar{z}_{2})$ are given by

$\begin{split}\Delta=\Delta_{1}+\Delta_{2}+p+q,\quad\sigma=p-q+\sigma_{2}+1\end{split}$
(96)

Our objective now is to determine the values of $p,q$ and the OPE coefficient
$C_{p,q}\left(\Delta_{1},\Delta_{2},\sigma_{2}\right)$. In carrying out this
analysis, we will assume that the structure of the OPE in (95) holds for
arbitrary values of the dimensions $\Delta_{1}$ and $\Delta_{2}$ with $p,q$
and $\sigma_{2}$ fixed. This was also done in Banerjee:2020zlg  in the context
of the celestial OPE of gravitons in Einstein gravity. As in Banerjee:2020zlg
, our results below will further justify this assumption. We will see that the
values of $p,q$ and $C_{p,q}\left(\Delta_{1},\Delta_{2},\sigma_{2}\right)$
obtained using the differential equation (89) precisely match with the
corresponding results of Fan:2019emx , where the leading celestial OPE was
derived from the Mellin transform of the splitting function which appears in
the leading collinear limit in gluon scattering amplitudes in Yang-Mills
theory Bern:1998sv .

Let us now write the differential equation (89) as

$\begin{split}&\bigg{[}\frac{C_{A}}{2}\frac{\partial}{\partial
z_{1}}-h_{1}\sum_{j=2}^{n}\frac{T^{a}_{j}\hskip
1.42271ptT^{a}_{1}}{z_{1}-z_{j}}+\frac{1}{2}\sum_{j=2}^{n}\frac{\epsilon_{j}(2\bar{h}_{j}-1-(\bar{z}_{1}-\bar{z}_{j})\frac{\partial}{\partial\bar{z}_{j}})}{(z_{1}-z_{j})}\hskip
1.42271ptT^{a}_{j}P^{-1}_{j}T^{a}_{1}P_{-1,-1}(1)\bigg{]}\\\
&\times\bigg{\langle}\mathcal{O}^{a_{1}}_{\Delta_{1},+}(z_{1},\bar{z}_{1})\mathcal{O}^{a_{2}}_{\Delta_{2},\sigma_{2}}(z_{2},\bar{z}_{2})\prod_{k=3}^{n}\mathcal{O}^{a_{k}}_{h_{k},\bar{h}_{k}}(z_{k},\bar{z}_{k})\bigg{\rangle}_{\text{MHV}}=0\end{split}$
(97)

where $\epsilon_{1}=\epsilon_{2}=1$. Using (97) we can derive a recursion
relation for the leading OPE coefficient as follows. Let us substitute the
leading OPE (95) inside the correlator in (97). Then it is evident that at
leading order in the OPE limit, the $z_{12},\bar{z}_{12}$ dependence on the
L.H.S. of (97) will be of the form $z_{12}^{p-1}\bar{z}_{12}^{q}$. Assuming
that $\eqref{ymde1}$ is satisfied order by order in the OPE regime
$(z_{12}\rightarrow 0,\bar{z}_{12}\rightarrow 0)$, we can then set the
coefficient of the $z_{12}^{p-1}\bar{z}_{12}^{q}$ term to zero. Consequently
we obtain the following recursion relation

$\begin{split}&\left(p\hskip 1.42271ptC_{A}\hskip
1.42271ptf^{a_{1}a_{2}b}+2\hskip
1.42271pth_{1}f^{aa_{1}x}f^{xby}f^{ya_{2}a}\right)C_{p,q}(\Delta_{1},\Delta_{2},\sigma_{2})\\\
&=(2\bar{h}_{2}-1+q)f^{aa_{1}x}f^{xby}f^{ya_{2}a}C_{p,q}(\Delta_{1}+1,\Delta_{2}-1,\sigma_{2})\end{split}$
(98)

where $2h_{1}=\Delta_{1}-1$ and $2\bar{h}_{2}=\Delta_{2}-\sigma_{2}$. Then
applying the identity

$\begin{split}f^{aa_{1}x}f^{xby}f^{ya_{2}a}=\frac{1}{2}\hskip
1.42271ptC_{A}\hskip 1.42271ptf^{a_{1}a_{2}b}\end{split}$ (99)

where $C_{A}$ is the quadratic Casimir of the adjoint representation, we get
from (98)

$\begin{split}&\left(\Delta_{1}+2p+1\right)C_{p,q}(\Delta_{1},\Delta_{2},\sigma_{2})=(\Delta_{2}-\sigma_{2}+q-1)C_{p,q}(\Delta_{1}+1,\Delta_{2}-1,\sigma_{2})\end{split}$
(100)

In the ensuing discussion, it will be convenient for us to express (100) in
another form. For this purpose, let us first note the following relation due
to the invariance of the OPE under global time translations Pate:2019lpp

$\begin{split}&C_{p,q}(\Delta_{1},\Delta_{2},\sigma_{2})=C_{p,q}(\Delta_{1}+1,\Delta_{2},\sigma_{2})+C_{p,q}(\Delta_{1},\Delta_{2}+1,\sigma_{2})\end{split}$
(101)

Then shifting $\Delta_{2}\rightarrow\Delta_{2}+1$ in (100) and using (101) we
get

$\begin{split}&(\Delta_{1}+2p+1)C_{p,q}(\Delta_{1},\Delta_{2},\sigma_{2})=\left(\Delta_{1}+\Delta_{2}+2p+q-\sigma_{2}+1\right)C_{p,q}(\Delta_{1}+1,\Delta_{2},\sigma_{2})\end{split}$
(102)

We shall now derive another recursion relation for the leading OPE coefficient
by appealing to the subleading soft gluon theorem in a similar fashion as in
Pate:2019lpp . Consider the action of the subleading soft symmetry generator
$J^{a}_{0}$ on a gluon primary. This is given by (31)

$\begin{split}&J^{a}_{0}\mathcal{O}^{b,\sigma}_{\Delta}(z,\bar{z})=-i\epsilon
f^{abc}(-2\bar{h}+1-\bar{z}\partial_{\bar{z}})\mathcal{O}^{c,\sigma}_{\Delta-1}(z,\bar{z})\end{split}$
(103)

where $\epsilon=\pm 1$ for an outgoing (incoming) gluon. Then requiring both
sides of the OPE (95) to transform in the same way under the action (103) we
get777Note that in order to arrive at (104) we have set $z_{2}=\bar{z}_{2}=0$
in the OPE (95).

$\begin{split}&f^{aa_{1}x}f^{xa_{2}y}(\Delta_{1}+q-2)C_{p,q}(\Delta_{1}-1,\Delta_{2},\sigma_{2})+f^{aa_{2}x}f^{a_{1}xy}(\Delta_{2}-\sigma_{2}-1)C_{p,q}(\Delta_{1},\Delta_{2}-1,\sigma_{2})\\\
&=f^{axy}f^{a_{1}a_{2}x}(\Delta_{1}+\Delta_{2}+2q-\sigma_{2}-2)C_{p,q}(\Delta_{1},\Delta_{2},\sigma_{2})\end{split}$
(104)

Multiplying both sides of (104) by $f^{yba}$ and using the identity (99) we
then obtain

$\begin{split}&(\Delta_{1}+q-2)C_{p,q}(\Delta_{1}-1,\Delta_{2},\sigma_{2})+(\Delta_{2}-\sigma_{2}-1)C_{p,q}(\Delta_{1},\Delta_{2}-1,\sigma_{2})\\\
&=2(\Delta_{1}+\Delta_{2}+2q-\sigma_{2}-2)C_{p,q}(\Delta_{1},\Delta_{2},\sigma_{2})\end{split}$
(105)

In order to easily determine the values of $p$ and $q$ let us bring (105) into
the same form as (102). To achieve this, we shift
$\Delta_{1}\rightarrow\Delta_{1}+1$ in (105). This gives,

$\begin{split}&(\Delta_{1}+q-1)C_{p,q}(\Delta_{1},\Delta_{2},\sigma_{2})+(\Delta_{2}-\sigma_{2}-1)C_{p,q}(\Delta_{1}+1,\Delta_{2}-1,\sigma_{2})\\\
&=2(\Delta_{1}+\Delta_{2}+2q-\sigma_{2}-1)C_{p,q}(\Delta_{1}+1,\Delta_{2},\sigma_{2})\end{split}$
(106)

Then using (100) we can eliminate
$C_{p,q}(\Delta_{1}+1,\Delta_{2}-1,\sigma_{2})$ from (106) and obtain

$\begin{split}&\bigg{[}\Delta_{1}+q-1+\frac{(\Delta_{1}+2p+1)(\Delta_{2}-\sigma_{2}-1)}{\Delta_{2}-\sigma_{2}+q-1}\bigg{]}C_{p,q}(\Delta_{1},\Delta_{2},\sigma_{2})\\\
&=2(\Delta_{1}+\Delta_{2}+2q-\sigma_{2}-1)C_{p,q}(\Delta_{1}+1,\Delta_{2},\sigma_{2})\end{split}$
(107)

Now we can solve for $p,q$ using equations (102) and (107). These equations
admit nontrivial solutions provided we have

$\begin{split}&\frac{\Delta_{1}+q-1}{2p+\Delta_{1}+1}+\frac{\Delta_{2}-\sigma_{2}-1}{\Delta_{2}-\sigma_{2}+q-1}=\frac{2(\Delta_{1}+\Delta_{2}+2q-\sigma_{2}-1)}{\Delta_{1}+\Delta_{2}+2p+q-\sigma_{2}+1}\end{split}$
(108)

Note that the differential equation (97) holds for any value of $\Delta_{1}$.
Further as mentioned before, the values of $p,q$ in the celestial OPE (95) do
not depend on $\Delta_{1},\Delta_{2}$. Consequently we can vary $\Delta_{1}$
and $\Delta_{2}$ independently in (108). Thereby the only non-trivial solution
of the above equation is888A simple way to see this is by taking $\Delta_{1}$
or $\Delta_{2}$ to be large in (108). For example, keeping $\Delta_{2}$ fixed
and taking $\Delta_{1}$ large in (108) we immediately get $q$=0. Similarly in
the limit where $\Delta_{2}$ becomes large we easily find that $p=-1$. The
assumption that $\Delta_{1},\Delta_{2}$ can be analytically continued off the
principal series is again implicit here.

$\begin{split}&p=-1,\quad q=0\end{split}$ (109)

This is precisely what we expect in pure Yang-Mills theory. Then substituting
(109) in (96) we immediately get

$\begin{split}\Delta=\Delta_{1}+\Delta_{2}-1,\quad\sigma=\sigma_{2}\end{split}$
(110)

The leading OPE (95) for outgoing gluon primaries then takes the form

$\begin{split}&\mathcal{O}^{a_{1}}_{\Delta_{1},+}(z_{1},\bar{z}_{1})\mathcal{O}^{a_{2}}_{\Delta_{2},\sigma_{2}}(z_{2},\bar{z}_{2})\sim-\frac{if^{a_{1}a_{2}x}}{z_{12}}\hskip
1.42271ptC_{-1,0}\left(\Delta_{1},\Delta_{2},\sigma_{2}\right)\mathcal{O}^{x}_{\Delta_{1}+\Delta_{2}-1,\sigma_{2}}(z_{2},\bar{z}_{2})\end{split}$
(111)

where $\sigma_{2}=\pm 1$. Now we can determine the OPE coefficient as follows.
After substituting $p=-1,q=0$, equation (102) as well as (107) reduces to

$\begin{split}&C_{-1,0}(\Delta_{1}+1,\Delta_{2},\sigma_{2})=\frac{\Delta_{1}-1}{\Delta_{1}+\Delta_{2}-\sigma_{2}-1}\
C_{-1,0}(\Delta_{1},\Delta_{2},\sigma_{2})\end{split}$ (112)

Then using the above in the recursion relation (101) we obtain

$\begin{split}&C_{-1,0}(\Delta_{1},\Delta_{2}+1,\sigma_{2})=\frac{\Delta_{2}-\sigma_{2}}{\Delta_{1}+\Delta_{2}-\sigma_{2}-1}\
C_{-1,0}(\Delta_{1},\Delta_{2},\sigma_{2})\end{split}$ (113)

Recursion relations of the form in (112) and (113) were also obtained using
time translation invariance and the global subleading soft gluon symmetry in
Pate:2019lpp . The solution of these equations is given by

$\begin{split}&C_{-1,0}(\Delta_{1},\Delta_{2},\sigma_{2})=\alpha\hskip
1.42271ptB(\Delta_{1}-1,\Delta_{2}-\sigma_{2})\end{split}$ (114)

where $B(x,y)$ is the Euler-Beta function. The constant $\alpha$ is as of yet
undetermined. We can fix it by using the leading conformal soft limit.
Consider $\Delta_{1}\rightarrow 1$ in (111). Then matching with the Ward
identity (14) gives $\alpha=1$. Thus for outgoing gluons, we get

$\begin{split}&C_{-1,0}(\Delta_{1},\Delta_{2},\sigma_{2})=B(\Delta_{1}-1,\Delta_{2}-\sigma_{2})\end{split}$
(115)

In the case where both gluon primaries are incoming, an identical analysis
again gives $p=-1,q=0$. The OPE coefficient also takes the same form as in
(114). However in order to determine the overall constant we should note that
the Kac-Moody current for an incoming gluon is given by

$\begin{split}&j^{a,-}(z,\bar{z})=-\lim_{\Delta\to
1}(\Delta-1)\mathcal{O}^{a,-}_{\Delta,\sigma}(z,\bar{z})\end{split}$ (116)

where the superscript $(-)$ above denotes an incoming gluon. Due to this minus
sign in comparison to (13) for an outgoing gluon, the leading OPE coefficient
for incoming gluons is given by

$\begin{split}&C_{-1,0}(\Delta_{1},\Delta_{2},\sigma_{2})=-B(\Delta_{1}-1,\Delta_{2}-\sigma_{2})\end{split}$
(117)

The OPE coefficients in (115) and (117) precisely match with the corresponding
results derived in Pate:2019lpp ; Fan:2019emx .

###  8.2 Outgoing-incoming OPE

We will now deal with the case where one of the gluon primaries in the
celestial OPE is outgoing and the other is incoming. Here we will take the
outgoing gluon primary to have positive helicity and denote it as
$\mathcal{O}^{a_{1}}_{\Delta_{1},+}(z_{1},\bar{z}_{1})$. The incoming gluon
primary will be denoted by
$\mathcal{O}^{a_{2},-}_{\Delta_{2},\sigma_{2}}(z_{2},\bar{z}_{2})$ where the
superscript $(-)$ denotes that it is incoming. We will not fix the spin of the
incoming gluon and so $\sigma_{2}=\pm 1$.

In this case, both an outgoing and an incoming gluon primary can contribute to
the OPE at leading order. Then as in the previous subsection we begin by
assuming the following general form of the leading OPE

$\begin{split}&\mathcal{O}^{a_{1}}_{\Delta_{1},+}(z_{1},\bar{z}_{1})\mathcal{O}^{a_{2},-}_{\Delta_{2},\sigma_{2}}(z_{2},\bar{z}_{2})\\\
&=-if^{a_{1}a_{2}x}z_{12}^{p}\bar{z}_{12}^{q}\bigg{[}C^{+}_{p,q}\left(\Delta_{1},\Delta_{2},\sigma_{2}\right)\mathcal{O}^{x,+}_{\Delta,\sigma}(z_{2},\bar{z}_{2})+C^{-}_{p,q}\left(\Delta_{1},\Delta_{2},\sigma_{2}\right)\mathcal{O}^{x,-}_{\Delta,\sigma}(z_{2},\bar{z}_{2})\bigg{]}+\cdots\end{split}$
(118)

where $\mathcal{O}^{x,+}_{\Delta,\sigma}$ and
$\mathcal{O}^{x,-}_{\Delta,\sigma}$ in the R.H.S. of (118) respectively denote
the outgoing and incoming primaries which contribute to the leading OPE. Their
conformal dimension and spin are

$\begin{split}\Delta=\Delta_{1}+\Delta_{2}+p+q,\quad\sigma=p-q+\sigma_{2}+1\end{split}$
(119)

In (118) $C^{\pm}_{p,q}\left(\Delta_{1},\Delta_{2},\sigma_{2}\right)$ is the
OPE coefficient corresponding to the outgoing (incoming) primary that appears
in the OPE. The dots denote possible contributions from descendants.

Now following exactly the same steps as in the previous subsection (8.1) we
can obtain a recursion relation for the leading OPE coefficients using the
differential equation (97). Here we get

$\begin{split}(2p+\Delta_{1}+1)C^{\pm}_{p,q}(\Delta_{1},\Delta_{2},\sigma_{2})+(\Delta_{2}-\sigma_{2}-1+q)C^{\pm}_{p,q}(\Delta_{1}+1,\Delta_{2}-1,\sigma_{2})=0\end{split}$
(120)

Then applying the global subleading soft symmetry generator $J^{a}_{0}$ to the
OPE (118) we get another recursion relation analogous to (105)

$\begin{split}&(\Delta_{1}+q-2)C^{\pm}_{p,q}(\Delta_{1}-1,\Delta_{2},\sigma_{2})-(\Delta_{2}-\sigma_{2}-1)C^{\pm}_{p,q}(\Delta_{1},\Delta_{2}-1,\sigma_{2})\\\
&=\pm
2(\Delta_{1}+\Delta_{2}+2q-\sigma_{2}-2)C^{\pm}_{p,q}(\Delta_{1},\Delta_{2},\sigma_{2})\end{split}$
(121)

Let us also note that invariance of the OPE (118) under global time
translations yields

$\begin{split}C^{\pm}_{p,q}(\Delta_{1},\Delta_{2},\sigma_{2})=\pm\bigg{[}C^{\pm}_{p,q}(\Delta_{1}+1,\Delta_{2},\sigma_{2})-C^{\pm}_{p,q}(\Delta_{1},\Delta_{2}+1,\sigma_{2})\bigg{]}\end{split}$
(122)

Then using (122) and performing similar manipulations as before we can rewrite
(120) and (121) as follows

$\begin{split}(2p+\Delta_{1}+1)C^{\pm}_{p,q}(\Delta_{1},\Delta_{2},\sigma_{2})=\pm(\Delta_{1}+\Delta_{2}+2p+q-\sigma_{2}+1)C^{\pm}_{p,q}(\Delta_{1}+1,\Delta_{2},\sigma_{2})\end{split}$
(123)

and

$\begin{split}&\bigg{[}\Delta_{1}+q-1+\frac{(\Delta_{1}+2p+1)(\Delta_{2}-\sigma_{2}-1)}{\Delta_{2}-\sigma_{2}+q-1}\bigg{]}C^{\pm}_{p,q}(\Delta_{1},\Delta_{2},\sigma_{2})\\\
&=\pm
2(\Delta_{1}+\Delta_{2}+2q-\sigma_{2}-1)C^{\pm}_{p,q}(\Delta_{1}+1,\Delta_{2},\sigma_{2})\end{split}$
(124)

The system of equations (123) and (124) have the same form as the analogous
equations (102) and (107) obtained in the case of the OPE between two outgoing
(incoming) gluons. It then follows by similar arguments that they admit non-
trivial solutions iff

$\begin{split}&p=-1,\quad q=0\end{split}$ (125)

This is again the expected result in Yang-Mills theory. The OPE coefficients
can now be obtained by solving (122) and (123) with $p=-1,q=0$ in the same way
as shown in the previous subsection (8.1). We then get

$\begin{split}&C^{+}_{-1,0}(\Delta_{1},\Delta_{2},\sigma_{2})=\alpha\hskip
1.42271ptB(\Delta_{2}-\sigma_{2},2-\Delta_{1}-\Delta_{2}+\sigma_{2})\\\
&C^{-}_{-1,0}(\Delta_{1},\Delta_{2},\sigma_{2})=\beta\hskip
1.42271ptB(\Delta_{1}-1,2-\Delta_{1}-\Delta_{2}+\sigma_{2})\end{split}$ (126)

where $\alpha,\beta$ are constants. These can be fixed by using the leading
conformal soft limit. In order to determine $\beta$ we can put $p=-1,q=0$ in
the OPE (118) and then take $\Delta_{1}\rightarrow 1$. The leading conformal
soft theorem (14) then implies $\beta=1$. Similarly considering the limit
$\Delta_{2}\rightarrow 1$ in the OPE (118) and comparing with the Ward
identity (14) gives $\alpha=-1$. Thus finally the leading OPE coefficients in
the case of outgoing-incoming gluon OPE are given by

$\begin{split}&C^{+}_{-1,0}(\Delta_{1},\Delta_{2},\sigma_{2})=-B(\Delta_{2}-\sigma_{2},2-\Delta_{1}-\Delta_{2}+\sigma_{2})\\\
&C^{-}_{-1,0}(\Delta_{1},\Delta_{2},\sigma_{2})=B(\Delta_{1}-1,2-\Delta_{1}-\Delta_{2}+\sigma_{2})\end{split}$
(127)

Again the above results for the OPE coefficients are in perfect agreement with
Pate:2019lpp .

##  9 Subleading OPE coefficients from symmetry

In this section we will illustrate how the OPE coefficients of descendants in
the celestial OPE between gluon primaries can be determined using the
underlying symmetries. For the OPE between positive helicity gluons, some of
the descendant OPE coefficients were obtained in Ebert:2020nqf  using
translation, global conformal and leading soft gluon current algebra
symmetries. Here we will consider the mixed helicity case, i.e., the OPE
between a positive helicity and a negative helicity gluon primary. We will see
that the subleading soft gluon symmetry plays a crucial role here.

Let us denote the gluon primaries whose OPE we want to consider as
$\mathcal{O}^{a}_{\Delta_{1},+}(z_{1},\bar{z}_{1})$ and
$\mathcal{O}^{b}_{\Delta_{2},-}(z_{2},\bar{z}_{2})$. We will also consider
both of these to be outgoing. Then as shown in Section C of the Appendix in
this paper, upto the first subleading order this OPE is given by

$\begin{split}&\mathcal{O}^{a}_{\Delta_{1},+}(z,\bar{z})\mathcal{O}^{b}_{\Delta_{2},-}(z_{2},\bar{z}_{2})\\\
&\sim
B(\Delta_{1}-1,\Delta_{2}+1)\bigg{[}-\frac{if^{abx}}{z_{12}}+\Delta_{1}\hskip
1.42271pt\delta^{bx}\hskip
1.42271ptj^{a}_{-1}+\frac{(\Delta_{1}-1)}{(\Delta_{1}+\Delta_{2})}\hskip
1.13791pt\delta^{bx}J^{a}_{-1}P_{-1,-1}\bigg{]}\mathcal{O}^{x}_{\Delta_{1}+\Delta_{2}-1,-}(z_{2},\bar{z}_{2})\end{split}$
(128)

where
$P_{-1,-1}\mathcal{O}^{x}_{\Delta_{1}+\Delta_{2}-1,-}=\mathcal{O}^{x}_{\Delta_{1}+\Delta_{2},-}$.
In the Appendix, Section C we have derived this result from the Mellin
transform of the $4$-point MHV gluon amplitude in Yang-Mills theory. Although
we have obtained this from the $4$-point Mellin amplitude, the above form of
the mixed helicity OPE is expected to hold within any $n$-point tree level MHV
gluon amplitude in Yang-Mills.

Now it is important to note that in the OPE (128), at order
$\mathcal{O}(z_{12}^{0}\bar{z}_{12}^{0})$, we encounter descendants associated
to both the leading soft gluon current algebra as well the subleading soft
gluon symmetry algebra. These are given by the operators
$j^{a}_{-1}\mathcal{O}^{x}_{\Delta_{1}+\Delta_{2}-1,-}$ and
$J^{a}_{-1}\mathcal{O}^{x}_{\Delta_{1}+\Delta_{2},-}$ respectively in (128).
We will now show that the OPE coefficients for these descendants can also
determined using symmetries as follows.

In general, we can have the following descendants appearing at
$\mathcal{O}(1)$ in the mixed-helicity OPE

$\begin{split}&L_{-1}\mathcal{O}^{a}_{\Delta,-},\quad
j^{a}_{-1}\mathcal{O}^{b}_{\Delta,-},\quad
J^{a}_{-1}P_{-1,-1}\mathcal{O}^{b}_{\Delta,-}\end{split}$ (129)

The above operators are linearly independent. This is because the vanishing
condition (84) holds only for a positive helicity gluon primary. This will be
further justified by our analysis below. Then the general form of the
$\mathcal{O}(1)$ term in the mixed helicity OPE can be written as

$\begin{split}&\mathcal{O}^{a}_{\Delta_{1},+}(z,\bar{z})\mathcal{O}^{b}_{\Delta_{2},-}(0)\supset
B(\Delta_{1}-1,\Delta_{2}+1)\bigg{[}\alpha_{1}f^{abx}\hskip
1.42271ptL_{-1}+\alpha_{2}\hskip
1.42271pt\delta^{bx}j^{a}_{-1}+\alpha_{3}\hskip
1.42271pt\delta^{bx}J^{a}_{-1}P_{-1,-1}\bigg{]}\mathcal{O}^{x}_{\Delta_{1}+\Delta_{2}-1,-}(0)\end{split}$
(130)

where $\alpha_{1},\alpha_{2},\alpha_{3}$ are constants which we want to
determine. The leading OPE coefficient $B(\Delta_{1}-1,\Delta_{2}+1)$ can be
obtained using the differential equation (89) as shown in Section 8. Also note
that we have placed the operator $\mathcal{O}^{b}_{\Delta_{2},-}$ at the
origin, without loss of generality.

Now let us apply the subleading soft symmetry mode $J^{c}_{1}$ to the OPE in
(130). Then using (38) and applying the commutation relations listed in
Sections (4.2) and (5) we get the recursion relation

$\begin{split}&i\hskip
1.13791pt\alpha_{1}f^{cxd}f^{abx}+\left(\alpha_{2}-(\Delta_{1}+\Delta_{2})\alpha_{3}-1\right)f^{cax}f^{xbd}=0\end{split}$
(131)

The coefficients $\alpha_{1},\alpha_{2},\alpha_{3}$ in the above equation do
not carry any Lie algebra indices. This equation should then hold for any
allowed values of the free indices $(a,b,c,d)$. Thus we can set for example
$a=c$ in (131). The structure constants being antisymmetric then immediately
gives us

$\begin{split}&\alpha_{1}=0\end{split}$ (132)

Similarly setting $a=b$ in (131) we get

$\begin{split}&\alpha_{2}-(\Delta_{1}+\Delta_{2})\alpha_{3}-1=0\end{split}$
(133)

Now it can be easily checked that applying the current algebra mode
$j^{c}_{1}$ yields exactly the same recursion relation as in (131). Then in
order to fix $\alpha_{2}$ we can apply $L_{1}$ to both sides of (130). Again
using the relevant commutation relations from Section 4.2 we obtain

$\begin{split}&i\hskip
1.13791pt\alpha_{1}(\Delta_{1}+\Delta_{2})f^{aby}+(\alpha_{2}-\Delta_{1})f^{aby}=0\end{split}$
(134)

Substituting $\alpha_{1}=0$ from (132) into the above equation we get

$\begin{split}&\alpha_{2}=\Delta_{1}\end{split}$ (135)

Finally we can solve for $\alpha_{3}$ from (131) by putting in the value of
$\alpha_{2}$ obtained above. This yields

$\begin{split}&\alpha_{3}=\frac{\alpha_{2}-1}{\Delta_{1}+\Delta_{2}}=\frac{\Delta_{1}-1}{\Delta_{1}+\Delta_{2}}\end{split}$
(136)

We thus find that the values of $\alpha_{1},\alpha_{2},\alpha_{3}$ obtained in
(132), (135) and (136) precisely agree with those extracted directly from the
Mellin amplitude. Now it is easy to see that (130) is already invariant under
the action of the translation generator $P_{-1,0}$. It is also straightforward
to check that these values of the descendant OPE coefficients satisfy the
recursion relations that follow from applying the translation generator
$P_{0,-1}$ to the OPE (130). The fact that all these recursion relations are
mutually consistent and admit a unique solution further justifies the absence
of the null state relation (84) for a negative helicity gluon primary.

## Acknowledgements

The work of SB is partially supported by the Science and Engineering Research
Board (SERB) grant MTR/2019/000937 (Soft-Theorems, S-matrix and Flat-Space
Holography). S.G. would like to thank Yasha Neiman for useful discussions. The
work of S.G. is supported by the Quantum Gravity Unit of the Okinawa Institute
of Science and Technology Graduate University (OIST), Japan.

##  10 Note Added

In a previous version of the paper we demanded that the conditions (38) and
(39), satisfied by gluon primaries, should also hold for any primary
descendant or null-state of a gluon primary. In other words, we demanded that
any primary descendant should also be primary under the subleading soft gluon
symmetry. We feel that a naive imposition of this condition is somewhat
restrictive and in this version we have removed this. A more precise statement
will appear elsewhere.

The main results of the paper remain unchanged. For example, the null-state
(84) is uniquely determined by the primary-state conditions with respect to
the Poinacre group and the leading soft gluon $SU(N)$ current algebra.

##  Appendix A Brief review of Celestial or Mellin amplitudes for massless
particles

The Celestial or Mellin amplitude for $n$ gluons in four dimensions is defined
as the Mellin transformation of the $n$-particle $S$-matrix element, given by
Pasterski:2016qvg ; Pasterski:2017kqt

$\mathcal{M}_{n}\big{(}\\{z_{i},\bar{z}_{i},h_{i},\bar{h}_{i},a_{i}\\}\big{)}=\prod_{i=1}^{n}\int_{0}^{\infty}d\omega_{i}\
\omega_{i}^{\Delta_{i}-1}S_{n}\big{(}\\{\omega_{i},z_{i},\bar{z}_{i},\sigma_{i},a_{i}\\}\big{)}$
(137)

where $\sigma_{i}=\pm 1$ denotes the helicity of the $i$-th gluon and the on-
shell momenta are parametrized as,

$p_{i}=\omega_{i}(1+z_{i}\bar{z}_{i},z_{i}+\bar{z}_{i},-i(z_{i}-\bar{z}_{i}),1-z_{i}\bar{z}_{i}),\quad
p_{i}^{2}=0$ (138)

$a_{i}$ denotes the Lie algebra index carried by the $i$-th gluon. The scaling
dimensions $(h_{i},\bar{h}_{i})$ are defined as,

$h_{i}=\frac{\Delta_{i}+\sigma_{i}}{2},\quad\bar{h}_{i}=\frac{\Delta_{i}-\sigma_{i}}{2}$
(139)

The Lorentz group $SL(2,\mathbb{C})$ acts on the celestial sphere as the group
of global conformal transformations and the Mellin amplitude $\mathcal{M}_{n}$
transforms as,

$\mathcal{M}_{n}\big{(}\\{z_{i},\bar{z}_{i},h_{i},\bar{h}_{i},a_{i}\\}\big{)}=\prod_{i=1}^{n}\frac{1}{(cz_{i}+d)^{2h_{i}}}\frac{1}{(\bar{c}\bar{z}_{i}+\bar{d})^{2\bar{h}_{i}}}\mathcal{M}_{n}\bigg{(}\frac{az_{i}+b}{cz_{i}+d}\
,\frac{\bar{a}\bar{z}_{i}+\bar{b}}{\bar{c}\bar{z}_{i}+\bar{d}}\
,h_{i},\bar{h}_{i},a_{i}\bigg{)}$ (140)

This is the familiar transformation law for the correlation function of
primary operators of weight $(h_{i},\bar{h}_{i})$ in a $2$-D CFT under the
global conformal group $SL(2,\mathbb{C})$.

We can also define a modified Mellin amplitude 999The exponentials in (141)
can also be thought of as convergence factors. It is good to have them
because, as discussed in Pate:2019mfs , the Mellin amplitude for gluons, as
defined in (137), is only marginally convergent. as in Banerjee:2018gce ;
Banerjee:2019prz ,

$\mathcal{M}_{n}\big{(}\\{u_{i},z_{i},\bar{z}_{i},h_{i},\bar{h}_{i},a_{i}\\}\big{)}=\prod_{i=1}^{n}\int_{0}^{\infty}d\omega_{i}\
\omega_{i}^{\Delta_{i}-1}e^{-i\sum_{i=1}^{n}\epsilon_{i}\omega_{i}u_{i}}S_{n}\big{(}\\{\omega_{i},z_{i},\bar{z}_{i},\sigma_{i},a_{i}\\}\big{)}$
(141)

where $u$ can be thought of as a time coordinate and $\epsilon_{i}=\pm 1$ for
an outgoing (incoming) particle. Under (Lorentz) conformal tranansformation
the modified Mellin amplitude $\mathcal{M}_{n}$ transforms as,

$\mathcal{M}_{n}\big{(}\\{u_{i},z_{i},\bar{z}_{i},h_{i},\bar{h}_{i},a_{i}\\}\big{)}=\prod_{i=1}^{n}\frac{1}{(cz_{i}+d)^{2h_{i}}}\frac{1}{(\bar{c}\bar{z}_{i}+\bar{d})^{2\bar{h}_{i}}}\mathcal{M}_{n}\bigg{(}\frac{u_{i}}{|cz_{i}+d|^{2}}\
,\frac{az_{i}+b}{cz_{i}+d}\
,\frac{\bar{a}\bar{z}_{i}+\bar{b}}{\bar{c}\bar{z}_{i}+\bar{d}}\
,h_{i},\bar{h}_{i},a_{i}\bigg{)}$ (142)

Under global space-time translation, $u\rightarrow
u+A+Bz+\bar{B}\bar{z}+Cz\bar{z}$, the modified amplitude is invariant, i.e,

$\mathcal{M}_{n}\big{(}\\{u_{i}+A+Bz_{i}+\bar{B}\bar{z}_{i}+Cz_{i}\bar{z}_{i},z_{i},\bar{z}_{i},h_{i},\bar{h}_{i},a_{i}\\}\big{)}=\mathcal{M}_{n}\big{(}\\{u_{i},z_{i},\bar{z}_{i},h_{i},\bar{h}_{i},a_{i}\\}\big{)}$
(143)

Now in order to make manifest the conformal nature of the dual theory living
on the celestial sphere it is useful to write the (modified) Mellin amplitude
as a correlation function of conformal primary operators. So let us define a
generic conformal primary operator as,

$\mathcal{O}^{a,\epsilon}_{h,\bar{h}}(z,\bar{z})=\int_{0}^{\infty}d\omega\
\omega^{\Delta-1}A^{a}(\epsilon\omega,z,\bar{z},\sigma)$ (144)

where $\epsilon=\pm 1$ for an annihilation (creation) operator of a massless
gluon of helicity $\sigma$ and Lie algebra index $a$. Under (Lorentz)
conformal transformation the conformal primary transforms like a primary
operator of scaling dimension $(h,\bar{h})$

$\mathcal{O}^{\prime
a,\epsilon}_{h,\bar{h}}(z,\bar{z})=\frac{1}{(cz+d)^{2h}}\frac{1}{(\bar{c}\bar{z}+\bar{d})^{2\bar{h}}}\mathcal{O}^{a,\epsilon}_{h,\bar{h}}\bigg{(}\frac{az+b}{cz+d}\
,\frac{\bar{a}\bar{z}+\bar{b}}{\bar{c}\bar{z}+\bar{d}}\bigg{)}$ (145)

Similarly in the presence of the time coordinate $u$ we have,

$\mathcal{O}^{\prime
a,\epsilon}_{h,\bar{h}}(u,z,\bar{z})=\int_{0}^{\infty}d\omega\
\omega^{\Delta-1}e^{-i\epsilon\omega u}A^{a}(\epsilon\omega,z,\bar{z},\sigma)$
(146)

Under (Lorentz) conformal transformations

$\mathcal{O}^{\prime
a,\epsilon}_{h,\bar{h}}(u,z,\bar{z})=\frac{1}{(cz+d)^{2h}}\frac{1}{(\bar{c}\bar{z}+\bar{d})^{2\bar{h}}}\mathcal{O}^{a,\epsilon}_{h,\bar{h}}\bigg{(}\frac{u}{|cz+d|^{2}},\frac{az+b}{cz+d}\
,\frac{\bar{a}\bar{z}+\bar{b}}{\bar{c}\bar{z}+\bar{d}}\bigg{)}$ (147)

In terms of (144), the Mellin amplitude can be written as the correlation
function of conformal primary operators

$\mathcal{M}_{n}=\bigg{\langle}{\prod_{i=1}^{n}\mathcal{O}^{a_{i},\epsilon_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\bigg{\rangle}$
(148)

Similarly using (146), the modified Mellin amplitude can be written as,

$\mathcal{M}_{n}=\bigg{\langle}{\prod_{i=1}^{n}\mathcal{O}^{a_{i},\epsilon_{i}}_{h_{i},\bar{h}_{i}}(u_{i},z_{i},\bar{z}_{i})}\bigg{\rangle}$
(149)

###  A.1 Comments on notation in the paper

Note that conformal primaries carry an additional index $\epsilon$ which
distinguishes between an incoming and an outgoing particle. In the paper, for
notational simplicity, we omit this additional index unless this plays an
important role. So in most places we simply write the (modified) Mellin
amplitude as,

$\mathcal{M}_{n}=\bigg{\langle}{\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(z_{i},\bar{z}_{i})}\bigg{\rangle}$
(150)

or

$\mathcal{M}_{n}=\bigg{\langle}{\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{h_{i},\bar{h}_{i}}(u_{i},z_{i},\bar{z}_{i})}\bigg{\rangle}$
(151)

Similarly in many places in the paper we denote a gluon conformal primary of
weight $\Delta=h+\bar{h}$ by $\mathcal{O}^{a}_{\Delta,\sigma}$ where
$\sigma=\pm 1$ is the helicity (= $h-\bar{h}$). Since we are considering pure
Yang-Mills, we can further simplify the notation to
$\mathcal{O}^{a}_{\Delta,\pm}$ by omitting the $\sigma=\pm 1$.

##  Appendix B OPE from $5$-point MHV gluon amplitude

In this section of the Appendix we will consider the Mellin transform of the
$5$-point tree-level MHV gluon amplitude in Yang-Mills theory. Our main
objective here is to show that the OPE in (75) which was obtained in
Ebert:2020nqf  from the $4$-point MHV amplitude also holds within the
$5$-point Mellin amplitude.

###  B.1 $5$-point MHV gluon amplitude

The tree-level $5$-point gluon amplitude in Yang-Mills theory can be expressed
as DelDuca:1999rs

$\begin{split}\mathcal{A}(1^{a_{1}},2^{a_{2}},3^{a_{3}},4^{a_{4}},5^{a_{5}})=(ig)^{3}\sum_{\rho\in
S_{3}}f^{a_{1}a_{\sigma_{2}}x_{1}}f^{x_{1}a_{\sigma_{3}}x_{2}}f^{x_{2}a_{\sigma_{4}}a_{5}}A(1,\rho(2),\rho(3),\rho(4),5)\end{split}$
(152)

where $g$ is the Yang-Mills coupling. $f^{abc}$’s are Lie algebra structure
constants . The sum above runs over a basis of $3!$ color-stripped partial
amplitudes denoted by $A(1,\rho(2),\rho(3),\rho(4),5)$. This is an over-
complete basis. Owing to BCJ relations Bern:2008qj , any $4$ of the sub-
amplitudes in (152) can be written in terms of $2$ linearly independent sub-
amplitudes. Let us choose this BCJ basis to be given by the sub-amplitudes

$\begin{split}A(1,2,3,4,5),\quad A(1,4,3,2,5)\end{split}$ (153)

Then the remaining sub-amplitudes are given by Bern:2008qj

$\begin{split}&A(1,2,4,3,5)=\alpha_{1}A(1,2,3,4,5)+\alpha_{2}A(1,4,3,2,5)\\\
&A(1,3,2,4,5)=\alpha_{3}A(1,2,3,4,5)+\alpha_{4}A(1,4,3,2,5)\\\
&A(1,3,4,2,5)=\alpha_{5}A(1,2,3,4,5)+\alpha_{6}A(1,4,3,2,5)\\\
&A(1,4,2,3,5)=\alpha_{7}A(1,2,3,4,5)+\alpha_{8}A(1,4,3,2,5)\end{split}$ (154)

where the $\alpha_{i}$’s are given by the following kinematic factors

$\begin{split}&\alpha_{1}=\frac{s_{45}(s_{12}+s_{24})}{s_{24}s_{35}},\quad\alpha_{2}=-\frac{s_{14}s_{25}}{s_{24}s_{35}},\quad\alpha_{3}=\frac{s_{12}(s_{24}+s_{45})}{s_{13}s_{24}},\quad\alpha_{4}=-\frac{s_{14}s_{25}}{s_{13}s_{24}},\\\
&\alpha_{5}=-\frac{s_{12}s_{45}}{s_{13}s_{24}},\quad\alpha_{6}=\frac{s_{14}(s_{24}+s_{25})}{s_{13}s_{24}},\quad\alpha_{7}=-\frac{s_{12}s_{45}}{s_{35}s_{24}},\quad\alpha_{8}=\frac{s_{25}(s_{14}+s_{24})}{s_{35}s_{24}}\end{split}$
(155)

Here $s_{ij}=(p_{i}+p_{j})^{2}$, with $p_{i}$ being the momenta of the
external particles. Now our interest here will be in the MHV configuration.
For this we will take the helicities of the gluons in (152) to be
$\sigma_{1}=\sigma_{2}=-1,\sigma_{3}=\sigma_{4}=\sigma_{5}=1$. Then using
(154) we obtain

$\begin{split}&\mathcal{A}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},4^{+a_{4}},5^{+a_{5}})\\\
&=(ig)^{3}\bigg{[}\left(c_{1}+c_{2}\alpha_{1}+c_{3}\alpha_{3}+c_{4}\alpha_{5}+c_{5}\alpha_{7}\right)A(1^{-},2^{-},3^{+},4^{+},5^{+})\\\
&\hskip
34.14322pt+\left(c_{6}+c_{2}\alpha_{2}+c_{3}\alpha_{4}+c_{4}\alpha_{6}+c_{5}\alpha_{8}\right)A(1^{-},4^{+},3^{+},2^{-},5^{+})\bigg{]}\end{split}$
(156)

where the $c_{i}$’s denote the following color structures

$\begin{split}&c_{1}=f^{a_{1}a_{2}x_{1}}f^{x_{1}a_{3}x_{2}}f^{x_{2}a_{4}a_{5}},\quad
c_{2}=f^{a_{1}a_{2}x_{1}}f^{x_{1}a_{4}x_{2}}f^{x_{2}a_{3}a_{5}},\quad
c_{3}=f^{a_{1}a_{3}x_{1}}f^{x_{1}a_{2}x_{2}}f^{x_{2}a_{4}a_{5}}\\\
&c_{4}=f^{a_{1}a_{3}x_{1}}f^{x_{1}a_{4}x_{2}}f^{x_{2}a_{2}a_{5}},\quad
c_{5}=f^{a_{1}a_{4}x_{1}}f^{x_{1}a_{2}x_{2}}f^{x_{2}a_{3}a_{5}},\quad
c_{6}=f^{a_{1}a_{4}x_{1}}f^{x_{1}a_{3}x_{2}}f^{x_{2}a_{2}a_{5}}\end{split}$
(157)

Now parametrising the null momenta of the external gluons in the amplitude as

$\begin{split}&p^{\mu}_{i}=\epsilon_{i}\hskip
1.13791pt\omega_{i}q^{\mu}_{i},\quad
q^{\mu}_{i}=\left(1+z_{i}\bar{z}_{i},z_{i}+\bar{z}_{i},-i(z_{i}-\bar{z}_{i}),1-z_{i}\bar{z}_{i}\right),\quad
i=1,2,3,4,5\end{split}$ (158)

where $\epsilon_{i}=\pm 1$ and using the Parke-Taylor formula for MHV
amplitudes Parke:1986gb , we can write (156) as

$\begin{split}&\mathcal{A}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},4^{+a_{4}},5^{+a_{5}})\\\
&=-\frac{(ig)^{3}}{2}\hskip
1.42271pt\frac{\omega_{1}\omega_{2}}{\omega_{3}\omega_{4}\omega_{5}}\hskip
1.42271pt\frac{z_{12}^{3}}{z_{15}z_{23}z_{35}}\bigg{[}\frac{\mathcal{X}_{1}}{z_{45}}+\frac{z_{12}}{z_{15}z_{25}}\left(1-\frac{z_{45}}{z_{15}}\right)^{-1}\hskip
1.42271pt\mathcal{X}_{2}\bigg{]}\left(1-\frac{z_{45}}{z_{35}}\right)^{-1}\end{split}$
(159)

where $\mathcal{X}_{1},\mathcal{X}_{2}$ are given by

$\begin{split}\mathcal{X}_{1}&=c_{1}+c_{2}\
\frac{z_{45}\bar{z}_{45}}{z_{35}\bar{z}_{35}}\
\frac{\epsilon_{4}\omega_{4}}{\epsilon_{3}\omega_{3}}+c_{3}\
\frac{z_{12}\bar{z}_{12}}{z_{13}\bar{z}_{13}}\
\frac{\epsilon_{2}\omega_{2}}{\epsilon_{3}\omega_{3}}\\\
&+\frac{z_{12}\bar{z}_{12}z_{45}\bar{z}_{45}}{z_{25}\bar{z}_{25}}\left(1-\frac{z_{45}}{z_{25}}\right)^{-1}\left(1-\frac{\bar{z}_{45}}{\bar{z}_{25}}\right)^{-1}\bigg{[}(c_{3}-c_{4})\
\frac{1}{z_{13}\bar{z}_{13}}\
\frac{\epsilon_{5}\omega_{5}}{\epsilon_{3}\omega_{3}}+(c_{2}-c_{5})\
\frac{1}{z_{35}\bar{z}_{35}}\
\frac{\epsilon_{1}\omega_{1}}{\epsilon_{3}\omega_{3}}\bigg{]}\end{split}$
(160)

and

$\begin{split}\mathcal{X}_{2}&=c_{6}+c_{4}\
\frac{\epsilon_{4}\omega_{4}}{\epsilon_{3}\omega_{3}}\
\frac{z_{15}\bar{z}_{15}}{z_{13}\bar{z}_{13}}\left(1-\frac{z_{45}}{z_{15}}\right)\left(1-\frac{\bar{z}_{45}}{\bar{z}_{15}}\right)+c_{5}\
\frac{\epsilon_{2}\omega_{2}}{\epsilon_{3}\omega_{3}}\
\frac{z_{25}\bar{z}_{25}}{z_{35}\bar{z}_{35}}\\\ &+\bigg{[}(c_{5}-c_{2})\
\frac{\epsilon_{1}\omega_{1}}{\epsilon_{3}\omega_{3}}\
\frac{z_{15}\bar{z}_{15}}{z_{35}\bar{z}_{35}}+(c_{4}-c_{3})\frac{\epsilon_{5}\omega_{5}}{\epsilon_{3}\omega_{3}}\
\frac{z_{15}\bar{z}_{15}}{z_{13}\bar{z}_{13}}\bigg{]}\left(1-\frac{z_{45}}{z_{15}}\right)\left(1-\frac{\bar{z}_{45}}{\bar{z}_{15}}\right)\left(1-\frac{z_{45}}{z_{25}}\right)^{-1}\left(1-\frac{\bar{z}_{45}}{\bar{z}_{25}}\right)^{-1}\end{split}$
(161)

Note that we have written the amplitude in this particular form (159) to
facilitate the extraction of the celestial OPE between gluons $4^{+a_{4}}$ and
$5^{+a_{5}}$ in the next subsection.

###  B.2 Mellin transform of $5$-point gluon MHV amplitude

Let us now consider the Mellin transform of the $5$-point gluon MHV amplitude
in (156). Here we will use the modified Mellin transform prescription which is
reviewed in Section A101010 The Mellin transform prescription of
Pasterski:2017kqt  can also be used here since it is convergent for tree-level
gluon amplitudes. The celestial OPE finally will be the same irrespective of
wether it is obtained from the usual Mellin amplitude or its modified version
involving the time coordinates, $u_{i}$.. For convenience in what follows we
will set the Yang-Mills coupling constant $g=2$. The modified Mellin amplitude
is given by

$\begin{split}&\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},4^{+a_{4}},5^{+a_{5}})\\\
&=\left\langle\mathcal{O}^{a_{1}}_{\Delta_{1},-}(1)\mathcal{O}^{a_{2}}_{\Delta_{2},-}(2)\mathcal{O}^{a_{3}}_{\Delta_{3},+}(3)\mathcal{O}^{a_{4}}_{\Delta_{4},+}(4)\mathcal{O}^{a_{5}}_{\Delta_{5},+}(5)\right\rangle\\\
&=\int_{0}^{\infty}\prod_{i=1}^{5}d\omega_{i}\hskip
1.42271pt\omega_{i}^{\Delta_{i}-1}\hskip
1.42271pte^{-i\sum\limits_{i=1}^{5}\epsilon_{i}\omega_{i}u_{i}}\hskip
1.42271pt\mathcal{A}1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},4^{+a_{4}},5^{+a_{5}})\hskip
1.42271pt\delta^{(4)}\left(\sum_{i=1}^{5}\epsilon_{i}\omega_{i}q_{i}^{\mu}\right)\end{split}$
(162)

where $\mathcal{O}^{a_{i}}_{\Delta_{i},\pm}(i)$ denotes a gluon primary
operator with dimension $\Delta_{i}=1+i\lambda_{i}$. The subscript $(\pm)$
here denotes the spin of the operator. The label $(i)$ collectively denotes
the coordinates $(z_{i},\bar{z}_{i},u_{i})$ at null infinity where the $i$-th
gluon primary is inserted.

Now we are interested in extracting the celestial OPE between the gluons
$(4^{+a_{4}},5^{+a_{5}})$. We will further take them to outgoing and so
$\epsilon_{4}=\epsilon_{5}=1$. Then let us define

$\begin{split}\omega_{4}=\omega_{p}t,\quad\omega_{5}=(1-t)\omega_{p}\end{split}$
(163)

where $t\in[0,1]$. The delta function imposing overall energy-momentum
conservation in (162) can be expressed in the following form

$\begin{split}&\delta^{(4)}\left(\sum_{i=1}^{5}\varepsilon_{i}\omega_{i}q_{i}^{\mu}\right)\\\
&=\frac{1}{4\ \omega_{P}}\delta\left(x-\bar{x}-t\hskip
1.42271ptz_{45}\left(\frac{x}{z_{35}}-\frac{\bar{x}}{z_{25}}\right)-t\hskip
1.42271pt\bar{z}_{45}\left(\frac{x}{\bar{z}_{25}}-\frac{\bar{x}}{\bar{z}_{35}}\right)+t\hskip
1.42271ptz_{45}\bar{z}_{45}\left(\frac{x}{z_{35}\bar{z}_{25}}-\frac{\bar{x}}{z_{25}\bar{z}_{35}}\right)\right)\prod_{i=1}^{3}\delta(\omega_{i}-\omega_{i}^{*})\end{split}$
(164)

where we have defined

$\begin{split}\omega_{i}^{*}=\varepsilon_{i}\hskip
1.13791pt\omega_{P}(\sigma_{i,1}+t\ z_{45}\ \sigma_{i,2}+t\ \bar{z}_{45}\
\sigma_{i,3}+t\ z_{45}\bar{z}_{45}\ \sigma_{i,4}),\quad
i\in(1,2,3)\end{split}$ (165)

and

$\begin{split}x=z_{12}z_{35}\bar{z}_{13}\bar{z}_{25},\quad\bar{x}=z_{13}z_{25}\bar{z}_{12}\bar{z}_{35}\end{split}$
(166)

The $\sigma_{i,j}$’s in (165) are given by

$\begin{split}&\sigma_{1,1}=-\frac{z_{25}\bar{z}_{35}}{z_{12}\bar{z}_{13}},\quad\sigma_{1,2}=\frac{\bar{z}_{35}}{z_{12}\bar{z}_{13}},\quad\sigma_{1,3}=\frac{z_{25}}{z_{12}\bar{z}_{13}},\quad\sigma_{1,4}=-\frac{1}{z_{12}\bar{z}_{13}}\\\
&\sigma_{2,1}=\frac{z_{15}\bar{z}_{35}}{z_{12}\bar{z}_{23}},\quad\sigma_{2,2}=-\frac{\bar{z}_{35}}{z_{12}\bar{z}_{23}},\quad\sigma_{2,3}=-\frac{z_{15}}{z_{12}\bar{z}_{23}},\quad\sigma_{2,4}=\frac{1}{z_{12}\bar{z}_{23}}\\\
&\sigma_{3,1}=-\frac{z_{25}\bar{z}_{15}}{z_{23}\bar{z}_{13}},\quad\sigma_{3,2}=\frac{\bar{z}_{15}}{z_{23}\bar{z}_{13}},\quad\sigma_{3,3}=\frac{z_{25}}{z_{23}\bar{z}_{13}},\quad\sigma_{3,4}=-\frac{1}{z_{23}\bar{z}_{13}}\end{split}$
(167)

The representation of the delta function in (164) enables us to localise the
integrals with respect to $\omega_{1},\omega_{2},\omega_{3}$ in (162). This
leaves us with integrals over $\omega_{P}$ and $t$. The $\omega_{P}$-integral
can be easily done and we get

$\begin{split}&\int_{0}^{\infty}d\omega_{P}\hskip
2.84544pt\omega_{P}^{i\Lambda-1}\exp\left[-i\hskip
0.85355pt\omega_{P}\left(\mathcal{U}_{1}+z_{45}\hskip 0.85355ptt\hskip
1.13791pt\mathcal{U}_{2}+\bar{z}_{45}\hskip 0.85355ptt\hskip
1.13791pt\mathcal{U}_{3}+z_{45}\bar{z}_{45}\hskip 0.85355ptt\hskip
1.13791pt\mathcal{U}_{4}+t\hskip 1.13791ptu_{45}\right)\right]\\\
&=\frac{\Gamma(i\Lambda)}{(i\hskip
1.42271pt\mathcal{U}_{1})^{i\Lambda}}\left[1+\frac{t}{\mathcal{U}_{1}}\left(z_{45}\hskip
1.13791pt\mathcal{U}_{2}+\bar{z}_{45}\hskip
1.13791pt\mathcal{U}_{3}+z_{45}\bar{z}_{45}\hskip
1.13791pt\mathcal{U}_{4}+u_{45}\right)\right]^{-i\Lambda}\end{split}$ (168)

where

$\begin{split}&\Lambda=\sum_{i=1}^{5}\lambda_{i},\quad\mathcal{U}_{1}=\sum_{i=1}^{3}\sigma_{i,1}u_{i5},\quad\mathcal{U}_{2}=\sum_{i=1}^{3}\sigma_{i,2}u_{i5},\quad\mathcal{U}_{3}=\sum_{i=1}^{3}\sigma_{i,3}u_{i5},\quad\mathcal{U}_{4}=\sum_{i=1}^{3}\sigma_{i,4}u_{i5}\end{split}$
(169)

Then using (168) and the expression of the $5$-point MHV amplitude obtained in
(159) we finally get

$\begin{split}&\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},4^{+a_{4}},5^{+a_{5}})\\\
&=-i\hskip 1.42271pt\mathcal{N}\int_{0}^{1}dt\hskip
2.84544ptt^{i\lambda_{4}-1}(1-t)^{i\lambda_{5}-1}\
\prod_{i=1}^{3}\Theta(\epsilon_{i}(\sigma_{i,1}+z_{45}\hskip 0.85355ptt\hskip
0.85355pt\sigma_{i,2}+\bar{z}_{45}\hskip 0.85355ptt\hskip
0.85355pt\sigma_{i,3}+z_{45}\bar{z}_{45}\hskip 0.85355ptt\hskip
0.85355pt\sigma_{i,4}))\ \mathcal{I}_{1}(t)\hskip
1.70709pt\mathcal{I}_{2}(t)\end{split}$ (170)

In the above integral the theta functions

$\begin{split}\Theta(\epsilon_{i}(\sigma_{i,1}+z_{45}\hskip 0.85355ptt\hskip
0.85355pt\sigma_{i,2}+\bar{z}_{45}\hskip 0.85355ptt\hskip
0.85355pt\sigma_{i,3}+z_{45}\bar{z}_{45}\hskip 0.85355ptt\hskip
0.85355pt\sigma_{i,4})),\quad i=1,2,3\end{split}$ (171)

simply impose the condition that $(\frac{\omega_{i}^{*}}{\omega_{P}})\geq 0$.
This is required because in the original integral (162), the energy variables
satisfy $\omega_{i}\geq 0$. The prefactor $\mathcal{N}$ in (170) is given by

$\begin{split}\mathcal{N}&=\frac{z_{12}^{3}}{z_{15}z_{23}z_{35}}\
(\epsilon_{1}\sigma_{1,1})^{1+i\lambda_{1}}(\epsilon_{2}\sigma_{2,1})^{1+i\lambda_{2}}(\epsilon_{3}\sigma_{3,1})^{i\lambda_{3}-1}\
\frac{\Gamma(i\Lambda)}{(i\hskip
1.13791pt\mathcal{U}_{1})^{i\Lambda}}\end{split}$ (172)

and $\mathcal{I}_{1}(t),\mathcal{I}_{2}(t)$ in the integrand in (170) are

$\begin{split}\mathcal{I}_{1}(t)=&\left(1+z_{45}\hskip 0.85355ptt\hskip
0.85355pt\frac{\sigma_{1,2}}{\sigma_{1,1}}+\bar{z}_{45}\hskip 0.85355ptt\hskip
0.85355pt\frac{\sigma_{1,3}}{\sigma_{1,1}}+z_{45}\bar{z}_{45}\hskip
0.85355ptt\hskip
0.85355pt\frac{\sigma_{1,4}}{\sigma_{1,1}}\right)^{1+i\lambda_{1}}\left(1+z_{45}\hskip
0.85355ptt\hskip 0.85355pt\frac{\sigma_{2,2}}{\sigma_{2,1}}+\bar{z}_{45}\hskip
0.85355ptt\hskip
0.85355pt\frac{\sigma_{2,3}}{\sigma_{2,1}}+z_{45}\bar{z}_{45}\hskip
0.85355ptt\hskip
0.85355pt\frac{\sigma_{2,4}}{\sigma_{2,1}}\right)^{1+i\lambda_{2}}\times\\\
&\left(1+z_{45}\hskip 0.85355ptt\hskip
0.85355pt\frac{\sigma_{3,2}}{\sigma_{3,1}}+\bar{z}_{45}\hskip 0.85355ptt\hskip
0.85355pt\frac{\sigma_{3,3}}{\sigma_{3,1}}+z_{45}\bar{z}_{45}\hskip
0.85355ptt\hskip
0.85355pt\frac{\sigma_{3,4}}{\sigma_{3,1}}\right)^{i\lambda_{3}-1}\left[1+\frac{t}{\mathcal{U}_{1}}\left(z_{45}\
\mathcal{U}_{2}+\bar{z}_{45}\ \mathcal{U}_{3}+z_{45}\bar{z}_{45}\
\mathcal{U}_{4}+u_{45}\right)\right]^{-i\Lambda}\times\\\
&\delta\left(x-\bar{x}-t\
z_{45}\left(\frac{x}{z_{35}}-\frac{\bar{x}}{z_{25}}\right)-t\
\bar{z}_{45}\left(\frac{x}{\bar{z}_{25}}-\frac{\bar{x}}{\bar{z}_{35}}\right)+t\
z_{45}\bar{z}_{45}\left(\frac{x}{z_{35}\bar{z}_{25}}-\frac{\bar{x}}{z_{25}\bar{z}_{35}}\right)\right)\end{split}$
(173)

and

$\begin{split}\mathcal{I}_{2}(t)=\bigg{[}\frac{\mathcal{X}_{1}(t)}{z_{45}}+\frac{z_{12}}{z_{15}z_{25}}\left(1-\frac{z_{45}}{z_{15}}\right)^{-1}\mathcal{X}_{2}(t)\bigg{]}\left(1-\frac{z_{45}}{z_{35}}\right)^{-1}\end{split}$
(174)

In (174), $\mathcal{X}_{1}(t),\mathcal{X}_{2}(t)$ are given by (160) and (161)
respectively after substituting the parametrisation of $\omega_{4},\omega_{5}$
in (163) and setting $\omega_{i}=\omega_{i}^{*},i=1,2,3$.

###  B.3 Mellin transform of $4$-point gluon MHV amplitude

Here we will obtain the modified Mellin transform of the tree-level $4$-point
MHV gluon amplitude which is required in order to extract the OPE from the
$5$-point Mellin amplitude (170). Since we shall be interested in taking the
OPE between gluons $(4^{+a_{4}},5^{+a_{5}})$ in (170), it will be convenient
to denote the momentum space gluon amplitude here as
$\mathcal{A}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},5^{+x})$. We will also take the
gluon labelled by $5^{+x}$ to be outgoing.

The color-dressed $4$-point MHV amplitude is given by

$\begin{split}&\mathcal{A}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},5^{+x})\\\
&=(ig)^{2}\left(f^{a_{1}a_{2}y}f^{ya_{3}x}A(1^{-},2^{-},3^{+},5^{+})+f^{a_{1}a_{3}y}f^{ya_{2}x}A(1^{-},3^{+},2^{-},5^{+})\right)\end{split}$
(175)

where $A(1^{-},2^{-},3^{+},5^{+})$ and $A(1^{-},3^{+},2^{-},5^{+})$ are color-
stripped partial amplitudes. Now applying the BCJ relation Bern:2008qj

$\begin{split}&A(1^{-},3^{+},2^{-},5^{+})=\frac{s_{12}}{s_{13}}A(1^{-},2^{-},3^{+},5^{+})\end{split}$
(176)

we can express (175) as

$\begin{split}&\mathcal{A}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},5^{+x})=(ig)^{2}\left(f^{a_{1}a_{2}y}f^{ya_{3}x}+f^{a_{1}a_{3}y}f^{ya_{2}x}\
\frac{s_{12}}{s_{13}}\right)A(1^{-},2^{-},3^{+},5^{+})\end{split}$ (177)

Then parametrising the null momenta as

$\begin{split}&p^{\mu}_{i}=\epsilon_{i}\hskip
1.13791pt\omega_{i}q^{\mu}_{i},\quad
q^{\mu}_{i}=\left(1+z_{i}\bar{z}_{i},z_{i}+\bar{z}_{i},-i(z_{i}-\bar{z}_{i}),1-z_{i}\bar{z}_{i}\right),\quad
i=1,2,3,5\end{split}$ (178)

we get from (177)

$\begin{split}&\mathcal{A}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},5^{+x})\\\
&=-(ig)^{2}\left(f^{a_{1}a_{2}y}f^{ya_{3}x}+f^{a_{1}a_{3}y}f^{ya_{2}x}\
\frac{\epsilon_{2}\omega_{2}}{\epsilon_{3}\omega_{3}}\
\frac{z_{12}\bar{z}_{12}}{z_{13}\bar{z}_{13}}\right)\frac{\omega_{1}\omega_{2}}{\omega_{3}\omega_{5}}\
\frac{z_{12}^{3}}{z_{15}z_{23}z_{35}}\end{split}$ (179)

Now let us compute the modified Mellin transform of (179). In order to relate
this to the factorisation of the $5$-point Mellin amplitude in the
$(4^{+},5^{+})$ OPE channel we will take the dimension of the primary
corresponding to the gluon $5^{+x}$ in (179) to be $\Delta_{4}+\Delta_{5}-1$.
Then the Mellin amplitude is

$\begin{split}&\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},5^{+x})\\\
&=\left\langle\mathcal{O}^{a_{1}}_{\Delta_{1},-}(1)\mathcal{O}^{a_{2}}_{\Delta_{2},-}(2)\mathcal{O}^{a_{3}}_{\Delta_{3},+}(3)\mathcal{O}^{x}_{\Delta_{4}+\Delta_{5}-1,+}(5)\right\rangle\\\
&=\int_{0}^{\infty}d\omega_{5}\hskip
1.42271pt\omega_{5}^{\Delta_{5}-1}e^{-i\omega_{5}u_{5}}\prod_{i=1}^{3}d\omega_{i}\hskip
1.42271pt\omega_{i}^{\Delta_{i}-1}\hskip
1.42271pte^{-i\sum\limits_{i=1}^{3}\epsilon_{i}\omega_{i}u_{i}}\hskip
1.42271pt\mathcal{A}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},5^{+x})\hskip
1.42271pt\delta^{(4)}\left(\sum_{i=1}^{3}\epsilon_{i}\omega_{i}q_{i}^{\mu}+\omega_{5}q^{\mu}_{5}\right)\end{split}$
(180)

The momentum conservation imposing delta function in the above integral can be
represented as

$\begin{split}&\delta^{(4)}\left(\sum_{i=1}^{3}\epsilon_{i}\omega_{i}q_{i}^{\mu}+\omega_{5}q^{\mu}_{5}\right)=\frac{1}{4\
\omega_{5}}\prod_{i=1}^{3}\delta(\omega_{i}-\omega_{i}^{*})\hskip
1.13791pt\delta\left(x-\bar{x}\right)\end{split}$ (181)

where

$\begin{split}&\omega_{i}^{*}=\epsilon_{i}\hskip 1.42271pt\omega_{5}\hskip
1.42271pt\sigma_{i,1},\quad i=1,2,3.\end{split}$ (182)

The $\sigma_{i,1}$’s in (182) were specified in (167). Note that using the
definition of $x,\bar{x}$ given in (166) it follows that the delta function
$\delta(x-\bar{x})$ above imposes the constraint

$\begin{split}&r=\bar{r}\end{split}$ (183)

where $r$ and $\bar{r}$ are the cross-ratios

$\begin{split}r=\frac{z_{12}z_{35}}{z_{13}z_{25}},\quad\bar{r}=\frac{\bar{z}_{12}\bar{z}_{35}}{\bar{z}_{13}\bar{z}_{25}}\end{split}$
(184)

Now using (181) in (180) localises the integrals with respect to
$\omega_{i},i=1,2,3$ . Then doing the remaining integral over $\omega_{5}$ we
get

$\begin{split}&\int_{0}^{\infty}d\omega_{5}\hskip
2.84544pt\omega_{5}^{i\Lambda-1}\exp\left(-i\hskip 0.85355pt\omega_{5}\hskip
0.85355pt\mathcal{U}_{1}\right)=\frac{\Gamma(i\Lambda)}{(i\hskip
1.42271pt\mathcal{U}_{1})^{i\Lambda}}\end{split}$ (185)

where

$\begin{split}&\Lambda=\sum_{i=1}^{5}\lambda_{i},\quad\mathcal{U}_{1}=\sum_{i=1}^{3}\sigma_{i,1}u_{i5}\end{split}$
(186)

Therefore the $4$-pt Mellin amplitude takes the form (with $g=2$)

$\begin{split}\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},5^{+x})&=\frac{z_{12}^{3}}{z_{14}z_{23}z_{34}}\
(\epsilon_{1}\sigma_{1,1})^{1+i\lambda_{1}}(\epsilon_{2}\sigma_{2,1})^{1+i\lambda_{2}}(\epsilon_{3}\sigma_{3,1})^{i\lambda_{3}-1}\frac{\Gamma(i\Lambda)}{(i\hskip
1.13791pt\mathcal{U}_{1})^{i\Lambda}}\times\\\ &\hskip
28.45274pt\left(f^{a_{1}a_{2}x}f^{xa_{3}a_{4}}-r\hskip
1.70709ptf^{a_{1}a_{3}x}f^{xa_{2}a_{4}}\right)\delta(x-\bar{x})\
\prod_{i=1}^{3}\Theta(\epsilon_{i}\sigma_{i,1})\end{split}$ (187)

Let us also note that $\mathcal{N}$ defined in (172) which appears as a
prefactor in the $5$-point Mellin amplitude in (170) is simply related to the
$4$-point Mellin amplitude as

$\begin{split}\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},5^{+x})&=\left(f^{a_{1}a_{2}x}f^{xa_{3}a_{4}}-r\hskip
1.70709ptf^{a_{1}a_{3}x}f^{xa_{2}a_{4}}\right)\mathcal{N}\ \delta(x-\bar{x})\
\prod_{i=1}^{3}\Theta(\epsilon_{i}\sigma_{i,1})\end{split}$ (188)

###  B.4 OPE decomposition of $5$-point Mellin amplitude

We shall now extract the celestial OPE between gluon primaries
$\mathcal{O}^{a_{4}}_{\Delta_{4},+}$ and $\mathcal{O}^{a_{5}}_{\Delta_{5},+}$
from the $5$-point Mellin amplitude. For this purpose we will set $u_{45}=0$
and expand (170) around $z_{45}=0,\bar{z}_{45}=0$. Now note that expanding the
theta functions in (170) generates delta function contributions whose
arguments are functions of $z_{ij},\bar{z}_{ij}$ with $(i,j)\in(1,2,3,5)$. We
will assume that the operators which do not participate in the OPE are all
inserted at separated points at null infnity and thereby such contact terms
can be ignored. The contributions of interest here will then only come from
expanding the integrands $\mathcal{I}_{1}(t),\mathcal{I}_{2}(t)$ in (170).

Here we will restrict attention to the leading and the first subleading terms
in the OPE regime. The leading term corresponds to the Mellin transform of the
collinear splitting function Bern:1998sv . The subleading terms in the Mellin
amplitude can likely be related to the subleading corrections to the leading
collinear limit in the momentum space amplitude which has been studied in
Nandan:2016ohb .

####  B.4.1 Leading term

Using (173) and (174), it is easy to see that the terms from
$\mathcal{I}_{1}(t)$ and $\mathcal{I}_{2}(t)$ which contribute at leading
order in the OPE limit $(z_{45}\rightarrow 0,\bar{z}_{45}\rightarrow 0)$ are
given by

$\begin{split}\mathcal{I}_{1}(t)\Big{|}_{\mathcal{O}(1)}=\delta(x-\bar{x})\end{split}$
(189)

and

$\begin{split}\mathcal{I}_{2}(t)\Big{|}_{\mathcal{O}(z_{45}^{-1})}&=\frac{1}{z_{45}}\
\mathcal{X}_{1}(t)\Big{|}_{\mathcal{O}(1)}=\frac{1}{z_{45}}\left(c_{1}-\frac{\bar{r}(1-r)}{(1-\bar{r})}\hskip
1.42271ptc_{3}\right)=(c_{1}-rc_{3})\end{split}$ (190)

where in obtaining the last equality in (190) we have used the fact on the
support of $\delta(x-\bar{x})$ we get $r=\bar{r}$. Then doing the $t$-integral
and using (188), we find that the $5$-point Mellin amplitude factorises as

$\begin{split}&\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},4^{+a_{4}},5^{+a_{5}})\\\
&=-\frac{if^{a_{4}a_{5}x}}{z_{45}}\hskip
1.42271ptB(\Delta_{4}-1,\Delta_{5}-1)\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},5^{+x})+\cdots\end{split}$
(191)

where $B(x,y)$ is the Euler Beta function and the dots denote subleading
terms. Then (191) implies that the leading celestial OPE is

$\begin{split}&\mathcal{O}^{a_{4}}_{\Delta_{4},+}(z_{4},\bar{z}_{4})\mathcal{O}^{a_{5}}_{\Delta_{5},+}(z_{5},\bar{z}_{5})\sim-\frac{if^{a_{4}a_{5}x}}{z_{45}}\hskip
1.42271ptB(\Delta_{4}-1,\Delta_{5}-1)\hskip
1.42271pt\mathcal{O}^{x}_{\Delta_{4}+\Delta_{5}-1,+}(z_{5},\bar{z}_{5})\end{split}$
(192)

This of course agrees with the expected result Pate:2019lpp ; Fan:2019emx .

####  B.4.2 Subleading terms: $\mathcal{O}(z_{45}^{0}\bar{z}_{45}^{0})$

We now want to consider the subleading term of order
$\mathcal{O}(z_{45}^{0}\bar{z}_{45}^{0})$ in the OPE decomposition. This will
correspond to the contribution of descendants in the celestial OPE. Let us
first gather the relevant terms from $\mathcal{I}_{1}(t)$ and
$\mathcal{I}_{2}(t)$ which can contribute at this order. From (173) we get

$\begin{split}\mathcal{I}_{1}(t)\Big{|}_{\mathcal{O}(z_{45})}=z_{45}\
t\bigg{[}&\mathcal{Z}\hskip
1.42271pt\delta(x-\bar{x})+\partial_{z_{5}}\delta(x-\bar{x})\bigg{]}\end{split}$
(193)

where $\mathcal{Z}$ is given by

$\begin{split}\mathcal{Z}=\bigg{(}(1+i\lambda_{1})\frac{\sigma_{1,2}}{\sigma_{1,1}}+(1+i\lambda_{2})\frac{\sigma_{2,2}}{\sigma_{2,1}}+(i\lambda_{3}-1)\frac{\sigma_{3,2}}{\sigma_{3,1}}-\frac{i\Lambda}{\mathcal{U}_{1}}\hskip
1.42271pt\mathcal{U}_{2}\bigg{)}\end{split}$ (194)

Then from (174) we have

$\begin{split}\mathcal{I}_{2}(t)\Big{|}_{\mathcal{O}(z_{45}^{-1})}&=\frac{1}{z_{45}}\
\mathcal{X}_{1}(t)\Big{|}_{\mathcal{O}(1)}=\frac{1}{z_{45}}\left(c_{1}-\frac{\bar{r}(1-r)}{(1-\bar{r})}\hskip
1.42271ptc_{3}\right)\end{split}$ (195)

and

$\begin{split}\mathcal{I}_{2}(t)\Big{|}_{\mathcal{O}(1)}&=\frac{1}{z_{45}}\
\mathcal{X}_{1}(t)\Big{|}_{\mathcal{O}(z_{45})}+\frac{1}{z_{35}}\
\mathcal{X}_{1}(t)\Big{|}_{\mathcal{O}(1)}+\frac{z_{12}}{z_{15}z_{25}}\
\mathcal{X}_{2}(t)\Big{|}_{\mathcal{O}(1)}\\\ &=\frac{(c_{1}-r\
c_{3})}{z_{35}}+\left(-\frac{1}{z_{15}}+\frac{1}{z_{25}}\right)(c_{6}-c_{5}+(1-r)(c_{3}-c_{4}))+c_{2}\left(\frac{1}{z_{25}}-\frac{1}{z_{35}}\right)\\\
&+t\left(\frac{1}{z_{15}}-\frac{1}{z_{25}}\right)c_{3}\end{split}$ (196)

Using (193), (195), (196) and performing the $t$-integral in (170), we find
the $\mathcal{O}(z_{45}^{0}\bar{z}_{45})$ contribution to the $5$-point Mellin
amplitude in the OPE regime to be given by

$\begin{split}&\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},4^{+a_{4}},5^{+a_{5}})\Big{|}_{\mathcal{O}(z_{45}^{0}\bar{z}_{45}^{0})}\\\
&=-i\hskip 1.42271ptB(\Delta_{4}-1,\Delta_{5}-1)\hskip
1.42271pt\mathcal{N}\bigg{[}\frac{\Delta_{4}-1}{(\Delta_{4}+\Delta_{5}-2)}\left(\mathcal{Z}\hskip
1.42271pt\delta(x-\bar{x})+\partial_{z_{5}}\delta(x-\bar{x})\right)\left(c_{1}-\frac{\bar{r}(1-r)}{(1-\bar{r})}\hskip
1.42271ptc_{3}\right)+\\\ &\left(\frac{(c_{1}-r\
c_{3})}{z_{35}}+\left(-\frac{1}{z_{15}}+\frac{1}{z_{25}}\right)(c_{6}-c_{5}+(1-r)(c_{3}-c_{4}))+c_{2}\left(\frac{1}{z_{25}}-\frac{1}{z_{35}}\right)\right)\delta(x-\bar{x})\\\
&+\frac{\Delta_{4}-1}{(\Delta_{4}+\Delta_{5}-2)}\left(\frac{1}{z_{15}}-\frac{1}{z_{25}}\right)c_{3}\hskip
1.42271pt\delta(x-\bar{x})\bigg{]}\prod_{i=1}^{3}\Theta(\epsilon_{i}\sigma_{i,1})\end{split}$
(197)

We can simplify the above further by noting the following identity

$\begin{split}&\frac{\bar{r}(1-r)}{(1-\bar{r})}\
\partial_{z_{5}}\delta(x-\bar{x})=r\hskip
1.42271pt\partial_{z_{5}}\delta(x-\bar{x})+\left(\frac{1}{z_{15}}-\frac{1}{z_{25}}\right)\delta(x-\bar{x})\end{split}$
(198)

Applying (198) in (197) then gives

$\begin{split}&\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},4^{+a_{4}},5^{+a_{5}})\Big{|}_{\mathcal{O}(z_{45}^{0}\bar{z}_{45}^{0})}\\\
&=-i\hskip 1.42271ptB(\Delta_{4}-1,\Delta_{5}-1)\hskip
1.42271pt\mathcal{N}\bigg{[}\frac{\Delta_{4}-1}{(\Delta_{4}+\Delta_{5}-2)}\left(\mathcal{Z}\hskip
1.42271pt\delta(x-\bar{x})+\partial_{z_{5}}\delta(x-\bar{x})\right)\left(c_{1}-r\hskip
1.42271ptc_{3}\right)\\\ &+\left(\frac{(c_{1}-r\
c_{3})}{z_{35}}+\left(-\frac{1}{z_{15}}+\frac{1}{z_{25}}\right)(c_{6}-c_{5}+(1-r)(c_{3}-c_{4}))+c_{2}\left(\frac{1}{z_{25}}-\frac{1}{z_{35}}\right)\right)\delta(x-\bar{x})\bigg{]}\prod_{i=1}^{3}\Theta(\epsilon_{i}\sigma_{i,1})\end{split}$
(199)

Now at this order in the OPE we expect to find descendants created by the
action of $L_{-1}$ and the current algebra mode $j^{a}_{-1}$. In order to
demonstrate this we need to know the action of these modes on the $4$-point
Mellin amplitude. We will now evaluate each of these in turn. First let us
consider the correlator

$\begin{split}&\left\langle\mathcal{O}^{a_{1}}_{\Delta_{1},-}(1)\mathcal{O}^{a_{2}}_{\Delta_{2},-}(2)\mathcal{O}^{a_{3}}_{\Delta_{3},+}(3)\left(L_{-1}\mathcal{O}^{x}_{\Delta_{4}+\Delta_{5}-1,+}(5)\right)\right\rangle\\\
&=\mathcal{L}_{-1}(5)\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},5^{+x})\\\
&=\partial_{z_{5}}\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},5^{+x})\end{split}$
(200)

Using the expression of the $4$-point Mellin amplitude obtained in (188) the
above can be straightforwardly evaluated and we obtain111111Since we are
interested in the $4$-point correlator at separated points on the celestial
sphere, we do not differentiate the theta functions in (188) with respect to
$z_{5}$ while evaluating (200).

$\begin{split}&f^{a_{4}a_{5}x}\mathcal{L}_{-1}\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},5^{+x})\\\
&=\bigg{[}(c_{1}-r\hskip 1.13791ptc_{3})\mathcal{N}\left[\mathcal{Z}\hskip
1.42271pt\delta(x-\bar{x})+\partial_{z_{5}}\delta(x-\bar{x})\right]+(c_{1}-r\hskip
1.13791ptc_{3})\left(\frac{1}{z_{15}}+\frac{1}{z_{35}}\right)\mathcal{N}\
\delta(x-\bar{x})\\\
&-r\left(\frac{1}{z_{25}}-\frac{1}{z_{35}}\right)c_{3}\hskip
1.42271pt\mathcal{N}\
\delta(x-\bar{x})\bigg{]}\prod_{i=1}^{3}\Theta(\epsilon_{i}\sigma_{i,1})\end{split}$
(201)

where $\mathcal{Z}$ is given by (194). The other correlation function of
interest here is

$\begin{split}&\left\langle\mathcal{O}^{a_{1}}_{\Delta_{1},-}(1)\mathcal{O}^{a_{2}}_{\Delta_{2},-}(2)\mathcal{O}^{a_{3}}_{\Delta_{3},+}(3)\left(j^{a_{4}}_{-1}\mathcal{O}^{a_{5}}_{\Delta_{4}+\Delta_{5}-1,+}(5)\right)\right\rangle\\\
&=\mathscr{J}^{a_{4}}_{-1}(5)\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},5^{+a_{5}})\end{split}$
(202)

This can be easily calculated using (21). We then get

$\begin{split}&\mathscr{J}^{a_{4}}_{-1}(5)\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},5^{+a_{5}})\\\
&=-i\bigg{[}\frac{1}{z_{15}}\
f^{a_{4}a_{1}x}(f^{xa_{2}y}f^{ya_{3}a_{5}}-r\hskip
1.42271ptf^{xa_{3}y}f^{ya_{2}a_{5}})+\frac{1}{z_{25}}\
f^{a_{4}a_{2}x}(f^{a_{1}xy}f^{ya_{3}a_{5}}-r\hskip
1.42271ptf^{a_{1}a_{3}y}f^{yxa_{5}})\\\ &\hskip
34.14322pt+\frac{1}{z_{35}}f^{a_{4}a_{3}x}(f^{a_{1}a_{2}y}f^{yxa_{5}}-r\hskip
1.42271ptf^{a_{1}xy}f^{ya_{2}a_{5}})\bigg{]}\mathcal{N}\ \delta(x-\bar{x})\
\prod_{i=1}^{3}\Theta(\epsilon_{i}\sigma_{i,1})\end{split}$ (203)

For our purposes it is more convinient to write the above in terms of the
independent color structures $c_{i}$ given in (157). We then arrive at

$\begin{split}&\mathscr{J}^{a_{4}}_{-1}(5)\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},5^{+a_{5}})\\\
&=i\bigg{[}\frac{1}{z_{15}}(-c_{5}+r\hskip
1.42271ptc_{6})+\frac{1}{z_{25}}(c_{5}-c_{2}-r\hskip
1.42271pt(c_{4}-c_{3}))+\frac{1}{z_{35}}(c_{2}-c_{1}-r\hskip
1.42271pt(c_{6}-c_{4}))\bigg{]}\mathcal{N}\ \delta(x-\bar{x})\
\prod_{i=1}^{3}\Theta(\epsilon_{i}\sigma_{i,1})\end{split}$ (204)

Now it can be easily shown that the term appearing in the second line of (199)
is simply

$\begin{split}&\bigg{[}\frac{(c_{1}-r\
c_{3})}{z_{35}}+\left(-\frac{1}{z_{15}}+\frac{1}{z_{25}}\right)(c_{6}-c_{5}+(1-r)(c_{3}-c_{4}))+c_{2}\left(\frac{1}{z_{25}}-\frac{1}{z_{35}}\right)\bigg{]}\mathcal{N}\
\delta(x-\bar{x})\ \prod_{i=1}^{3}\Theta(\epsilon_{i}\sigma_{i,1})\\\
&=-\bigg{[}\frac{1}{z_{15}}(-c_{5}+r\hskip
1.42271ptc_{6})+\frac{1}{z_{25}}(c_{5}-c_{2}-r\hskip
1.42271pt(c_{4}-c_{3}))+\frac{1}{z_{35}}(c_{2}-c_{1}-r\hskip
1.42271pt(c_{6}-c_{4}))\bigg{]}\mathcal{N}\ \delta(x-\bar{x})\
\prod_{i=1}^{3}\Theta(\epsilon_{i}\sigma_{i,1})\\\
&=i\mathscr{J}^{a_{4}}_{-1}(5)\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},5^{+a_{5}})\end{split}$
(205)

In order to arrive at our desired result let us also note the following useful
relation

$\begin{split}&i\left(\mathscr{J}^{a_{5}}_{-1}(5)\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},5^{+a_{4}})-\mathscr{J}^{a_{4}}_{-1}(5)\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},5^{+a_{5}})\right)\\\
&=\left[-(c_{1}-r\hskip
1.13791ptc_{3})\left(\frac{1}{z_{15}}+\frac{1}{z_{35}}\right)+r\
c_{3}\left(\frac{1}{z_{25}}-\frac{1}{z_{35}}\right)\right]\mathcal{N}\
\delta(x-\bar{x})\ \prod_{i=1}^{3}\Theta(\epsilon_{i}\sigma_{i,1})\end{split}$
(206)

Then applying (201), (204) and (206) in equation (199) we finally get

$\begin{split}&\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},4^{+a_{4}},5^{+a_{5}})\Big{|}_{\mathcal{O}(z_{45}^{0}\bar{z}_{45}^{0})}\\\
&=-i\hskip
1.42271ptB(\Delta_{4}-1,\Delta_{5}-1)\bigg{[}\frac{\Delta_{4}-1}{(\Delta_{4}+\Delta_{5}-2)}\
f^{a_{4}a_{5}x}\mathcal{L}_{-1}(5)\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},5^{+x})\\\
&+\frac{\Delta_{4}-1}{(\Delta_{4}+\Delta_{5}-2)}\
i\mathscr{J}^{a_{5}}_{-1}(5)\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},5^{+a_{5}})+\frac{\Delta_{5}-1}{(\Delta_{4}+\Delta_{5}-2)}\
i\mathscr{J}^{a_{4}}_{-1}(5)\mathcal{M}(1^{-a_{1}},2^{-a_{2}},3^{+a_{3}},5^{+a_{5}})\bigg{]}\end{split}$
(207)

The above result implies that upto the order under consideration here, the
celestial OPE for positive helicity outgoing gluon primaries is given by

$\begin{split}\mathcal{O}^{a_{4}}_{\Delta_{4},+}(z_{4},\bar{z}_{4})\mathcal{O}^{a_{5}}_{\Delta_{5},+}(z_{5},\bar{z}_{5})&\sim-i\hskip
1.42271ptB(\Delta_{4}-1,\Delta_{5}-1)\bigg{[}\frac{f^{a_{4}a_{5}x}}{z_{45}}+\frac{(\Delta_{4}-1)}{(\Delta_{4}+\Delta_{5}-2)}f^{a_{4}a_{5}x}L_{-1}\\\
&+i\left(\frac{(\Delta_{4}-1)}{(\Delta_{4}+\Delta_{5}-2)}\delta^{a_{4}x}\delta^{a_{5}y}+\frac{(\Delta_{4}-1)}{(\Delta_{4}+\Delta_{5}-2)}\delta^{a_{4}y}\delta^{a_{5}x}\right)j^{y}_{-1}\bigg{]}\mathcal{O}^{x}_{\Delta_{4}+\Delta_{5}-1,+}(z_{5},\bar{z}_{5})\end{split}$
(208)

This matches with the corresponding result obtained in Ebert:2020nqf .

##  Appendix C Mixed Helicity OPE from $4$-pt MHV gluon amplitude

In this section, using the $4$-point Mellin amplitude, we will obtain the
first subleading correction to the leading OPE between a positive helicity and
a negative helicity gluon primary.

###  C.1 $4$-point Mellin amplitude

Consider the color-dressed $4$-point MHV gluon amplitude
$\mathcal{A}(1^{-a_{1}},2^{+a_{2}},3^{+a_{3}},4^{-a_{4}})$. This is given by

$\begin{split}&\mathcal{A}(1^{-a_{1}},2^{+a_{2}},3^{+a_{3}},4^{-a_{4}})=(ig)^{2}\left(f^{a_{1}a_{2}x}f^{xa_{3}a_{4}}+f^{a_{1}a_{3}x}f^{xa_{2}a_{4}}\
\frac{s_{34}}{s_{24}}\right)A(1^{-},2^{+},3^{+},4^{-})\end{split}$ (209)

where

$\begin{split}&A(1^{-},2^{+},3^{+},4^{-})=-\frac{\langle
14\rangle^{3}}{\langle 12\rangle\langle 23\rangle\langle
34\rangle}\end{split}$ (210)

Using the $(\omega_{i},z_{i},\bar{z}_{i})$ parametrisation of null momenta, we
get from (209)

$\begin{split}&\mathcal{A}(1^{-a_{1}},2^{+a_{2}},3^{+a_{3}},4^{-a_{4}})\\\
&=(ig)^{2}\left(f^{a_{1}a_{2}x}f^{xa_{3}a_{4}}+f^{a_{1}a_{3}x}f^{xa_{2}a_{4}}\
\frac{\epsilon_{3}\omega_{3}}{\epsilon_{2}\omega_{2}}\
\frac{z_{34}\bar{z}_{34}}{z_{24}\bar{z}_{24}}\right)\frac{\omega_{1}\omega_{4}}{\omega_{2}\omega_{3}}\
\frac{z_{14}^{3}}{z_{12}z_{23}z_{34}}\end{split}$ (211)

The modified Mellin transform of (211) is then

$\begin{split}&\mathcal{M}(1^{-a_{1}},2^{+a_{2}},3^{+a_{3}},4^{-a_{4}})\\\
&=\left\langle\mathcal{O}^{a_{1}}_{\Delta_{1},-}(1)\mathcal{O}^{a_{2}}_{\Delta_{2},+}(2)\mathcal{O}^{a_{3}}_{\Delta_{3},+}(3)\mathcal{O}^{a_{4}}_{\Delta_{4},-}(4)\right\rangle\\\
&=\int_{0}^{\infty}\prod_{i=1}^{4}d\omega_{i}\hskip
1.42271pt\omega_{i}^{\Delta_{i}-1}\hskip
1.42271pte^{-i\sum\limits_{i=1}^{4}\epsilon_{i}\omega_{i}u_{i}}\hskip
1.42271pt\mathcal{A}(1^{-a_{1}},2^{+a_{2}},3^{+a_{3}},4^{-a_{4}})\hskip
1.42271pt\delta^{(4)}\left(\sum_{i=1}^{4}\epsilon_{i}\omega_{i}q_{i}^{\mu}\right)\end{split}$
(212)

where $\Delta_{i}=1+i\lambda_{i}$. Now we shall be interested in extracting
the celestial OPE between $\mathcal{O}^{a_{3}}_{\Delta_{3},+}(3)$ and
$\mathcal{O}^{a_{4}}_{\Delta_{4},-}(4)$ from the Mellin amplitude (212). We
will also take both of them to be outgoing, i.e.,
$\epsilon_{3}=\epsilon_{4}=1$. Then let us make the following change of
variables in (212).

$\begin{split}\omega_{3}=\omega_{P}\hskip
1.13791ptt,\quad\omega_{4}=\omega_{P}(1-t)\end{split}$ (213)

The momentum conservation imposing delta function can be written as

$\begin{split}\delta^{(4)}\left(\sum_{i=1}^{4}\varepsilon_{i}\omega_{i}q_{i}^{\mu}\right)=&\frac{1}{4\epsilon_{1}\omega_{1}\epsilon_{2}\omega_{2}z_{12}^{2}}\prod_{i=1}^{2}\delta\left(\omega_{i}-\omega_{i}^{*}\right)\delta\left(\bar{z}_{14}-\frac{\omega_{P}t}{\epsilon_{1}\omega_{1}}\frac{z_{23}}{z_{12}}\bar{z}_{34}\right)\delta\left(\bar{z}_{24}+\frac{\omega_{P}t}{\epsilon_{2}\omega_{2}}\frac{z_{13}}{z_{12}}\bar{z}_{34}\right)\end{split}$
(214)

where121212In this section for simplicity of notation, we will often use the
same symbols to denote similar quantities which also occur in Section B. In
all such cases, to avoid confusion, we urge the reader to note the relevant
definitions which have been explicitly provided in the respective sections.

$\begin{split}&\omega^{*}_{i}=\epsilon_{i}\omega_{P}\left(\sigma_{i,1}+z_{34}\hskip
1.13791ptt\hskip 1.13791pt\sigma_{i,2}\right),\quad i=1,2\end{split}$ (215)

and

$\begin{split}\sigma_{1,1}=\frac{z_{24}}{z_{12}},\quad\sigma_{2,1}=-\frac{z_{14}}{z_{12}},\quad\sigma_{1,2}=-\sigma_{2,2}=-\frac{1}{z_{12}}\end{split}$
(216)

Applying the delta function (214), the $\omega_{1},\omega_{2}$ integrals can
be trivially done. Then using (211) and performing the integral over
$\omega_{P}$, the Mellin amplitude can be expressed as

$\begin{split}&\mathcal{M}(1^{-a_{1}},2^{+a_{2}},3^{+a_{3}},4^{-a_{4}})=-i\hskip
1.42271pt\mathcal{N}\int_{0}^{1}dt\hskip
2.84544ptt^{i\lambda_{3}-1}(1-t)^{i\lambda_{4}+1}\
\prod_{i=1}^{3}\Theta(\epsilon_{i}(\sigma_{i,1}+z_{34}\hskip 0.85355ptt\hskip
0.85355pt\sigma_{i,2}))\ \mathcal{I}(t)\end{split}$ (217)

where the theta functions above impose
$(\frac{\omega_{i}^{*}}{\omega_{P}})\geq 0$. The prefactor $\mathcal{N}$ and
the integrand $\mathcal{I}(t)$ are given by

$\begin{split}\mathcal{N}&=i\hskip 1.13791pt\epsilon_{1}\epsilon_{2}\
\frac{z_{14}^{3}}{z_{12}^{3}z_{24}}\
(\epsilon_{1}\sigma_{1,1})^{i\lambda_{1}}(\epsilon_{2}\sigma_{2,1})^{i\lambda_{2}-2}\
\frac{\Gamma(i\Lambda)}{(i\hskip
1.13791pt\mathcal{U}_{1})^{i\Lambda}}\end{split}$ (218)

with $\Lambda=\sum\limits_{i=1}^{4}\lambda_{i},\hskip
2.84544pt\mathcal{U}_{1}=\sum\limits_{i=1}^{2}\sigma_{i,1}u_{i4}$ and

$\begin{split}\mathcal{I}(t)=&\frac{1}{z_{34}}\left(1-\frac{z_{34}}{z_{24}}\right)^{-1}\bigg{[}f^{a_{1}a_{2}x}f^{xa_{3}a_{4}}-\frac{z_{12}z_{34}}{z_{14}z_{24}}\left(1-\frac{z_{34}}{z_{14}}\right)^{-1}f^{a_{1}a_{3}x}f^{xa_{2}a_{4}}\bigg{]}\times\\\
&\left(1+z_{34}\hskip 0.85355ptt\hskip
0.85355pt\frac{\sigma_{1,2}}{\sigma_{1,1}}\right)^{i\lambda_{1}}\times\left(1+z_{34}\hskip
0.85355ptt\hskip
0.85355pt\frac{\sigma_{2,2}}{\sigma_{2,1}}\right)^{i\lambda_{2}-2}\left[1+\frac{t}{\mathcal{U}_{1}}\left(z_{34}\
\mathcal{U}_{2}+u_{34}\right)\right]^{-i\Lambda}\times\\\
&\delta\left(\bar{z}_{14}-\frac{t}{\sigma_{1,1}}\left(1+z_{34}\ t\
\frac{\sigma_{1,2}}{\sigma_{1,1}}\right)^{-1}\frac{z_{23}}{z_{12}}\bar{z}_{34}\right)\delta\left(\bar{z}_{24}+\frac{t}{\sigma_{2,1}}\left(1+z_{34}\
t\
\frac{\sigma_{2,2}}{\sigma_{2,1}}\right)^{-1}\frac{z_{13}}{z_{12}}\bar{z}_{34}\right)\end{split}$
(219)

In (217) we have again set the Yang-Mills coupling $g=2$. Now let us also note
the expression of the $3$-point Mellin amplitude in terms of which the
$4$-point Mellin amplitude factorises in the $(3^{+},4^{-})$ OPE channel. This
can be easily derived and is given by

$\begin{split}\mathcal{M}(1^{-a_{1}},2^{+a_{2}},4^{-x})&=\left\langle\mathcal{O}^{a_{1}}_{\Delta_{1},-}(1)\mathcal{O}^{a_{2}}_{\Delta_{2},+}(2)\mathcal{O}^{x}_{\Delta_{3}+\Delta_{4}-1,-}(4)\right\rangle\\\
&=f^{a_{1}a_{2}x}\hskip 1.13791pt\mathcal{N}\hskip
1.13791pt\delta(\bar{z}_{14})\delta(\bar{z}_{24})\
\prod_{i=1}^{3}\Theta(\epsilon_{i}(\sigma_{i,1}))\end{split}$ (220)

with $\mathcal{N}$ defined in (218).

###  C.2 OPE decomposition of $4$-point Mellin amplitude

We can now extract the mixed helicity celestial OPE in the $(3^{+},4^{-})$
channel by setting $u_{34}=0$ in (217) and expanding the integrand
$\mathcal{I}(t)$ around $z_{34}=0,\bar{z}_{34}=0$. As before, in order to
avoid irrelevant contact terms we will simply set $z_{34}=0$ inside the
argument of the theta functions in the Mellin integral in (217).

####  C.2.1 Leading term

The leading term in (217) as $z_{34}\rightarrow 0$ is

$\begin{split}\mathcal{M}(1^{-a_{1}},2^{+a_{2}},3^{+a_{3}},4^{-a_{4}})&\approx-i\hskip
1.42271pt\mathcal{N}\int_{0}^{1}dt\hskip
2.84544ptt^{i\lambda_{3}-1}(1-t)^{i\lambda_{4}+1}\hskip
1.42271ptf^{a_{1}a_{2}x}f^{xa_{3}a_{4}}\hskip
1.13791pt\frac{\delta(\bar{z}_{14})\delta(\bar{z}_{24})}{z_{34}}\
\prod_{i=1}^{3}\Theta(\epsilon_{i}(\sigma_{i,1}))\\\
&=-\frac{if^{a_{3}a_{4}x}}{z_{34}}\hskip
1.42271ptB(\Delta_{3}-1,\Delta_{4}+1)\hskip
1.42271pt\mathcal{M}(1^{-a_{1}},2^{+a_{2}},4^{-x})\end{split}$ (221)

From the above result we get the leading celestial OPE to be

$\begin{split}\mathcal{O}^{a_{3}}_{\Delta_{3},+}(z_{3},\bar{z}_{3})\mathcal{O}^{a_{4}}_{\Delta_{4},-}(z_{4},\bar{z}_{4})&\sim-\frac{if^{a_{3}a_{4}x}}{z_{34}}\hskip
1.42271ptB(\Delta_{3}-1,\Delta_{4}+1)\hskip