MHV Gluon Scattering Amplitudes from Celestial Current Algebras

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 Knizhnik-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 and our results match with those obtained from an explicit calculation using the Mellin MHV amplitude.

A Brief review of Celestial or Mellin amplitudes for massless particles 27 A.1 Comments on notation in the paper 28

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 [11-20, 23-25, 30-50], 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 [10] that CCFT should have a stress tensor [12][13][14] 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 [26,27], CCFT should also have supertranslation symmetry [2,5,7,8,28,29]. On top of that, there are various other infinite dimensional current algebra symmetries, coming from soft factorisation theorems, under which CCFT should be invariant [1,3,5,[19][20][21][22][60][61][62][63][64][65][66][67][68][69]. 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 [3][4][5][6][7][8][9]. The study of CCFT is facilitated by going to the Mellin space . In Mellin space scattering amplitudes can be written as the correlation functions of conformal primaries 1 . 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 − − + + + · · · , where two gluons have negative helicity and the rest have positive helicity. Their explicit expressions are given by the famous Parke-Taylor formula [77]. Now this is the first non-trivial helicity amplitude because at tree level in pure YM theory, amplitudes of the form − + + + · · · with only one negative helicity gluon and + + + · · · 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 [5], 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 [1,[60][61][62][63][64][65] at level zero which arises from the leading positive helicity soft gluon theorem. There is also the subleading positive helicity soft gluon theorem [3,[66][67][68][69] 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 [5] 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, where i ∈ (1, 2, · · · , n − 2) runs over the (n − 2) positive helicity gluons in the MHV amplitude. In the above equation, O a h,h (z,z) is a gluon conformal primary with scaling dimension (h,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 ǫ j which is ±1 depending on whether O a j h j ,h j (z j ,z j ) is outgoing or incoming. The operator P −1 j acts on a gluon conformal primary O a k h k ,h k (z k ,z k ) as Similarly the global time translation generator P −1,−1 (i) acts on the i-th positive helicity gluon conformal primary according to Equation (1.1) can be easily transformed to momentum space and for the momentum space MHV scattering amplitude it can be written as, A a k (ǫ k ω k , z k ,z k , σ k ) Here the null momentum of an on-shell gluon has been parametrised as and ǫ = ±1 for an outgoing (incoming) gluon. We have also denoted the momentum space creation (annihilation) operators by A a (ǫω, z,z, σ) where σ is the helicity of the gluon. Note that the MHV amplitude in (1.4) includes the overall momentum conserving delta function. Now (1.1) or (1.4) 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 [5]. Presumably (1.1) or (1.4), 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 [3,53].
The origin of equations (1.1) and (1.4) will be discussed in great detail in section (6) but let us mention a few things before we close this section.
1. The first two terms of (1.1) closely resemble the Knizhnik-Zamolodchikov (KZ) equation [72] 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 (1.1) and we also have to set the level k of the current algebra to 0.
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. (1.1) holds for any value of the scaling dimension (h,h) of the gluon primary. This is consistent with the fact that in CCFT the scaling dimension ∆ = h +h of a (hard) primary is a continuously varying (complex) number and therefore should not be constrained in any way.
3. The third term in (1.1) 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.
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 (??) 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 (1.1) and (1.4) for tree-level MHV gluon amplitudes in Mellin space and Fock space respectively. In Section (8) we use the differential equation (1.1) 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.

Poincare invariance
Since the scattering amplitudes are Poincare invariant, generators of the Poincare group act on the conformal primaries which live on the celestial sphere 2 . The Lorentz group SL(2, C) acts on the celestial sphere as the global conformal group and we denote its generators by L 0 , L ±1 ,L 0 ,L ±1 . Their commutation relations are given by, They act on a gluon conformal primary O a h,h (z,z) as, The four global space-time translation generators will be denoted by P m,n where m, n = 0, ±1 and they are mutually commuting The commutation relations between Lorentz and global space-time translation generators are given by, The translation generators P m,n act on conformal primaries according to where ǫ = ±1 for an outgoing (incoming) gluon.

Poincare primary
It follows from the definition of a conformal primary operator that the following standard relations hold but a Poincare primary [8] must also satisfy the additional conditions which follow from (2.6).

Leading soft gluon
The leading conformally soft [51][52][53][54][55] 3 gluon operator j a (z) with positive helicity is defined as, j a (z) = lim where O a ∆,+ (z,z) is a positive helicity gluon primary with scaling dimension ∆. The soft operator j a (z) is a Kac-Moody current [1,[60][61][62][63][64][65] 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, as gluons transform in the adjoint representation. Now let us consider the modes j a n of the above current. They satisfy the algebra and act on a gluon primary as 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 [9] in the MHV sector.
The commutation relations with the (Lorentz) global conformal generators are given by, [L m , j a n ] = −nj a m+n , L m , j a n = 0, Similarly, the commutators with the generators {P m,n , m, n = 0, ±1} of global space-time translations are given by, P m,n , j a p = 0 (3.7) For our purposes an important role is played by the correlation functions of the descendants j a −p O b h,h (z,z), p ≥ 1 with a collection of gluon primaries. These are given by where the operator J a −p (z) is defined as, 3 Conformally soft graviton theorems have been studied in [51,56,57].

Leading current algebra primary
A current algebra primary O a h,h (z,z) is defined by the standard conditions j a n O b h,h (0) = 0, ∀n ≥ 1 (3.10) and

Subleading soft gluon
The subleading conformally soft [51][52][53][54][55] gluon operator S +a 1 (z,z) with positive helicity is defined as, where O a ∆,+ (z,z) is a positive helicity gluon primary with scaling dimension ∆. The correlation function of S +a 1 (z,z) with a collection of primary gluon operators is given by the subleading soft gluon theorem in Mellin space [3], and Here ǫ k = ±1 depending on whether the gluon primary O a k h k ,h k (z k ,z k ) is outgoing or incoming. For simplicity of notation we keep the additional label ǫ implicit when we write the correlation functions of the gluons. Now following [3,5] we expand the R.H.S of (4.2) in powers of the coordinatez of the subleading conformally soft gluon operator S +a 1 (z,z) and define two currents J a (z) and K a (z) whose Ward identities are given by, This is equivalent to expanding the soft operator S +a 1 (z,z) as, 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, and Although the generators (J a n , K a n ) do not form a Lie algebra under commutation, we will see in section (5) that when the commutators J a m , J b n , J a m , K b n and K a m , K b n 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,z) and a hard gluon conformal primary Suppose we want to compute the OPE between S +a 1 (z,z) and the gluon primary O a 1 h 1 ,h 1 (z 1 ,z 1 ). For this we have to expand the R.H.S of (4.2) in powers of (z − z 1 ) and (z −z 1 ). So as (z,z) → (z 1 ,z 1 ) we can write, where the differential operators J a −p (1) and K a −p (1) are defined as, From (4.10) the OPE between S a 1 (z,z) and O a 1 can be extracted to be, where the correlation functions with the insertion of the descendants (z 1 ,z 1 ) are given by (4.11) and (4.12), respectively. Now (4.13) acts as a boundary condition on the OPE of two hard gluon primaries one which is positive helicity. Using the definition (4.1) of the subleading conformally soft gluon As we will see (4.14) is a nontrivial constraint on the OPE of two hard gluons. Inside correlation functions (4.14) means that we first take the OPE limit (z,z) → (z 1 ,z 1 ) and then take the subleading conformal soft limit. In that limit we should always get back (4.10).
We also note that the operator product expansion (4.13) leads to the following conditions which are satisfied by any gluon primary O a h,h (z,z),

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, where ǫ = ±1 depending on whether the gluon is outgoing or incoming. Using these we arrive at the following commutation relations between generators, L 1 , J a n = 0, L 0 , J a n = − 1 2 J a n , L −1 , J a n = −K a n (4.24) L 1 , K a n = J a n , L 0 , K a n = 1 2 K a n , L −1 , K a n = 0 (4.26)

How to interpret the commutators of subleading symmetry generators
It is well known [66] 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, O a In order to calculate the OPE coefficient multiplying this operator, one has to know the structure of the term from the OPE and we need to get simplified expressions for them. In order to do this we start by computing the following commutators Let us focus on the first commutator. Using (4.21) and the Jacobi identity we get, Again using (4.21) we can write this as where we have used ǫ 2 = 1. Now we take the limit (z,z) → (0, 0) and from (5.5) we get Since the mode J a n of the current J a is defined with respect to the point (0, 0), (5.6) can be interpreted as the relation between the descendants Now we can apply the same argument to get the other two relations 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 (5.7), (5.8) and (5.9) 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, The above relations, say for example (5.10), can be obtained by starting from the commutator The above relations have obvious generalisations to a general descendant and are given by, 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 gluongluon OPE directly from the (MHV) scattering amplitude and compare them with the results from the recursion relations obtained using (5.20), (5.21) and (5.22). Our derivation of these relations has been somewhat hand waving. We leave a more rigorous derivation to future work.

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 O a ∆,+ (z,z) and O a 1 ∆ 1 ,+ (z 1 ,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 [9] and is given by is a positive helicity (outgoing) gluon primary. The leading primary OPE coefficient is given by the Euler beta function, B(∆ − 1, ∆ 1 − 1). The dots above denote contributions from descendants at further subleading orders in (z − z 1 ), (z −z 1 ). In [9] 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 (6.1) from the Mellin transform of the 5-point MHV gluon amplitude. Now let us take the subleading conformal soft limit ∆ → 0 in the above OPE. We then obtain 2) According to our discussion in Section (4.1), in the subleading conformal soft limit the OPE should obey the general constraint given by equation (4.14). Therefore as ∆ → 0 in (6.1) we should get where Then comparing (6.2) and (6.3) 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 (6.1) to be consistent with the subleading soft gluon theorem we must have the following relation Multiplying the above by if aa 1 b and using the relation where C A is the quadratic Casimir of the adjoint representation, we can express (6.5) as Further shifting ∆ 1 → ∆ 1 + 1 in (6.7) we get Up to this point we have been considering the gluon primary in (6.7) 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 (6.8). Thus for an incoming positive helicity gluon we have Therefore in general we have the condition where P −1,−1 is the global time translation generator which acts on a gluon primary as Here ǫ = ±1 for an outgoing (incoming) gluon. Note that in obtaining (6.10) we have used the definition (3.11) of a current algebra primary according to which Now consider the linear combination of descendants denoted as Ψ a (z,z) in (6.10). 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 4 L 1 Ψ a (z,z) =L 1 Ψ a (z,z) = P 0,−1 Ψ a (z,z) = P −1,0 Ψ a (z,z) = 0 (6.12) Equations (6.12) and (6.13) together imply that Ψ a (z,z) is in fact a primary operator with respect to the Poincare group and the current algebra associated to the leading soft gluon theorem. 5 In fact, one can easily check that the null-state Ψ a is uniquely determined by the primary-state conditions under the Poincare group and leading soft gluon SU (N ) current algebra. Thus Ψ a (z,z) is a null field and we can consistently set Ψ a (z,z) to zero within celestial MHV gluon amplitudes. Now let us insert (6.10) within Mellin transformed tree-level MHV gluon amplitudes. Below we will denote the gluon primaries as O a k h k ,h k (z k ,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 ∈ (1, 2, · · · n − 2) we get a decoupling relation where we have used 2 h i = ∆ 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 (6.14) with i ∈ (1, 2, · · · , n − 2) 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 (6.15) is similar to the Knizhnik-Zamolodchikov (KZ) equation [72] obeyed by correlation functions of primary operators in WZW theory. The KZ equation is given by 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 (6.15) and the KZ equation (6.16).
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 ∂ z i term in (6.15) is C A /2, where C A is the quadratic Casimir of the adjoint representation. In the KZ equation (6.16), 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 (6.15). This arises from the j a −1 (i)j a 0 (i) piece in (6.14). The analogous term is also present in the KZ equation (6.16) but, the coefficient of this term in our case depends on the holomorphic conformal weight h i of the primary operator O a i h i ,h i (z i ,z i ) whose null state gives rises to (6.15). 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 [72,73] h r = C r 2k + C A (6.17) 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 (6.17). But (6.14) and consequently (6.15) hold for arbitrary values of the scaling dimensions for the positive helicity gluon primaries in the MHV amplitude 6 . The coefficient h i in front of the second term in (6.14), (6.15) plays a crucial role in ensuring this.
Finally let us discuss the third term within the square brackets in (6.15). This arises due to the subleading soft gluon symmetry. There is no counterpart of this term in the KZ equation (6.16). 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 [70] 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 [32,71]. The additional term coming from the subleading soft gluon symmetry in the differential equation (6.15) 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.

Differential equation for MHV gluon amplitudes in momentum space
The differential equation (6.15) 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 n k=1 A a k (ǫ k ω k , z k ,z k , σ k ) MHV (7.1) where ǫ = ±1. A a k (ω k , z k ,z k , σ k ) and A a k (−ω k , z k ,z k , σ k ) are annihilation and creation operators respectively for the external gluons with helicity σ k in the S-matrix. In (7.1) we will take gluons (1, 2, · · · , n − 2) to have positive helicity. Then gluons (n − 1) and n have negative helicity. Now in order to recast (6.15) to momentum space, we make the following substitutions Applying an inverse Mellin transform we can replace the correlation function in (6.15) with the Fock space amplitude (7.1). Thus we obtain the differential equation As in (6.15), here we have (n − 2) partial differential equations labelled by the index i in (7.3) 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 toz j in (7.3) has a nontrivial action on the amplitude (7.1).

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 (6.15).

OPE for outgoing (incoming) gluons
Let us first consider the celestial OPE between a positive helicity gluon primary O a 1 ∆ 1 ,+ (z 1 ,z 1 ) and another gluon primary denoted by O a 2 ∆ 2 ,σ 2 (z 2 ,z 2 ). The spin σ 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 where z 12 = (z 1 − z 2 ),z 12 = (z 1 −z 2 ). O x ∆,σ (z 2 ,z 2 ) is the leading primary operator that can appear in the OPE and C p,q (∆ 1 , ∆ 2 , σ 2 ) is the associated OPE coefficient. The dots denote possible contributions from descendants. The conformal dimension and spin of O x ∆,σ (z 2 ,z 2 ) are given by Our objective now is to determine the values of p, q and the OPE coefficient C p,q (∆ 1 , ∆ 2 , σ 2 ). In carrying out this analysis, we will assume that the structure of the OPE in (8.1) holds for arbitrary values of the dimensions ∆ 1 and ∆ 2 with p, q and σ 2 fixed. This was also done in [5] in the context of the celestial OPE of gravitons in Einstein gravity. As in [5], our results below will further justify this assumption. We will see that the values of p, q and C p,q (∆ 1 , ∆ 2 , σ 2 ) obtained using the differential equation (6.15) precisely match with the corresponding results of [53], 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 [74].
Let us now write the differential equation (6.15) as where ǫ 1 = ǫ 2 = 1. Using (8.3) we can derive a recursion relation for the leading OPE coefficient as follows. Let us substitute the leading OPE (8.1) inside the correlator in (8.3). Then it is evident that at leading order in the OPE limit, the z 12 ,z 12 dependence on the L.H.S. of (8.3) will be of the form z p−1 12z q 12 . Assuming that (8.3) is satisfied order by order in the OPE regime (z 12 → 0,z 12 → 0), we can then set the coefficient of the z p−1 12z q 12 term to zero. Consequently we obtain the following recursion relation where 2h 1 = ∆ 1 − 1 and 2h 2 = ∆ 2 − σ 2 . Then applying the identity where C A is the quadratic Casimir of the adjoint representation, we get from (8.4) In the ensuing discussion, it will be convenient for us to express (8.6) in another form. For this purpose, let us first note the following relation due to the invariance of the OPE under global time translations [3] Then shifting ∆ 2 → ∆ 2 + 1 in (8.6) and using (8.7) we get 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 [3]. Consider the action of the subleading soft symmetry generator J a 0 on a gluon primary. This is given by (4.8) where ǫ = ±1 for an outgoing (incoming) gluon. Then requiring both sides of the OPE (8.1) to transform in the same way under the action (8.9) we get 7 Multiplying both sides of (8.10) by f yba and using the identity (8.5) we then obtain In order to easily determine the values of p and q let us bring (8.11) into the same form as (8.8). To achieve this, we shift ∆ 1 → ∆ 1 + 1 in (8.11). This gives, Then using (8.6) we can eliminate C p,q (∆ 1 + 1, ∆ 2 − 1, σ 2 ) from (8.12) and obtain Now we can solve for p, q using equations (8.8) and (8.13). These equations admit nontrivial solutions provided we have Note that the differential equation (8.3) holds for any value of ∆ 1 . Further as mentioned before, the values of p, q in the celestial OPE (8.1) do not depend on ∆ 1 , ∆ 2 . Consequently we can vary ∆ 1 and ∆ 2 independently in (8.14). Thereby the only non-trivial solution of the above equation is 8 This is precisely what we expect in pure Yang-Mills theory. Then substituting (8.15) in (8.2) we immediately get The leading OPE (8.1) for outgoing gluon primaries then takes the form where σ 2 = ±1. Now we can determine the OPE coefficient as follows. After substituting p = −1, q = 0, equation (8.8) as well as (8.13) reduces to Then using the above in the recursion relation (8.7) we obtain Recursion relations of the form in (8.18) and (8.19) were also obtained using time translation invariance and the global subleading soft gluon symmetry in [3]. The solution of these equations is given by where B(x, y) is the Euler-Beta function. The constant α is as of yet undetermined. We can fix it by using the leading conformal soft limit. Consider ∆ 1 → 1 in (8.17). Then matching with the Ward identity (3.2) gives α = 1. Thus for outgoing gluons, we get 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 (8.20). However in order to determine the overall constant we should note that the Kac-Moody current for an incoming gluon is given by where the superscript (−) above denotes an incoming gluon. Due to this minus sign in comparison to (3.1) for an outgoing gluon, the leading OPE coefficient for incoming gluons is given by The OPE coefficients in (8.21) and (8.23) precisely match with the corresponding results derived in [3,53].

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 O a 1 ∆ 1 ,+ (z 1 ,z 1 ). The incoming gluon primary will be denoted by O a 2 ,− ∆ 2 ,σ 2 (z 2 ,z 2 ) where the superscript (−) denotes that it is incoming. We will not fix the spin of the incoming gluon and so σ 2 = ±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 where O x,+ ∆,σ and O x,− ∆,σ in the R.H.S. of (8.24) respectively denote the outgoing and incoming primaries which contribute to the leading OPE. Their conformal dimension and spin are In (8.24) C ± p,q (∆ 1 , ∆ 2 , σ 2 ) 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 (8.3). Here we get Then applying the global subleading soft symmetry generator J a 0 to the OPE (8.24) we get another recursion relation analogous to (8.11) Let us also note that invariance of the OPE (8.24) under global time translations yields Then using (8.28) and performing similar manipulations as before we can rewrite (8.26) and (8.27) as follows (2p + ∆ 1 + 1)C ± p,q (∆ 1 , ∆ 2 , σ 2 ) = ±(∆ 1 + ∆ 2 + 2p + q − σ 2 + 1)C ± p,q (∆ 1 + 1, ∆ 2 , σ 2 ) (8.29) and The system of equations (8.29) and (8.30) have the same form as the analogous equations (8.8) and (8.13) 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 This is again the expected result in Yang-Mills theory. The OPE coefficients can now be obtained by solving (8.28) and (8.29) with p = −1, q = 0 in the same way as shown in the previous subsection (8.1). We then get where α, β are constants. These can be fixed by using the leading conformal soft limit. In order to determine β we can put p = −1, q = 0 in the OPE (8.24) and then take ∆ 1 → 1. The leading conformal soft theorem (3.2) then implies β = 1. Similarly considering the limit ∆ 2 → 1 in the OPE (8.24) and comparing with the Ward identity (3.2) gives α = −1. Thus finally the leading OPE coefficients in the case of outgoing-incoming gluon OPE are given by Again the above results for the OPE coefficients are in perfect agreement with [3].

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 [9] 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 O a ∆ 1 ,+ (z 1 ,z 1 ) and O b ∆ 2 ,− (z 2 ,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 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 (9.1), at order O(z 0 12z 0 12 ), 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 respectively in (9.1). 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 O(1) in the mixed-helicity OPE The above operators are linearly independent. This is because the vanishing condition (6.10) holds only for a positive helicity gluon primary. This will be further justified by our analysis below. Then the general form of the O(1) term in the mixed helicity OPE can be written as where α 1 , α 2 , α 3 are constants which we want to determine. The leading OPE coefficient B(∆ 1 − 1, ∆ 2 + 1) can be obtained using the differential equation (6.15) as shown in Section 8. Also note that we have placed the operator O b ∆ 2 ,− at the origin, without loss of generality. Now let us apply the subleading soft symmetry mode J c 1 to the OPE in (9.3). Then using (4.15) and applying the commutation relations listed in Sections (4.2) and (5) we get the recursion relation The coefficients α 1 , α 2 , α 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 (9.4). The structure constants being antisymmetric then immediately gives us Similarly setting a = b in (9.4) we get Now it can be easily checked that applying the current algebra mode j c 1 yields exactly the same recursion relation as in (9.4). Then in order to fix α 2 we can apply L 1 to both sides of (9.3). Again using the relevant commutation relations from Section 4.2 we obtain Substituting α 1 = 0 from (9.5) into the above equation we get Finally we can solve for α 3 from (9.4) by putting in the value of α 2 obtained above. This yields We thus find that the values of α 1 , α 2 , α 3 obtained in (9.5), (9.8) and (9.9) precisely agree with those extracted directly from the Mellin amplitude. Now it is easy to see that (9.3) 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 (9.3). The fact that all these recursion relations are mutually consistent and admit a unique solution further justifies the absence of the null state relation (6.10) for a negative helicity gluon primary.

Note Added
In a previous version of the paper we demanded that the conditions (4.15) and (4.16), 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 (6.10) is uniquely determined by the primary-state conditions with respect to the Poinacre group and the leading soft gluon SU (N ) current algebra.

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 [30,31] where σ i = ±1 denotes the helicity of the i-th gluon and the on-shell momenta are parametrized as, a i denotes the Lie algebra index carried by the i-th gluon. The scaling dimensions (h i ,h i ) are defined as, The Lorentz group SL(2, C) acts on the celestial sphere as the group of global conformal transformations and the Mellin amplitude M n transforms as, This is the familiar transformation law for the correlation function of primary operators of weight (h i ,h i ) in a 2-D CFT under the global conformal group SL(2, C).
We can also define a modified Mellin amplitude 9 as in [34,35], where u can be thought of as a time coordinate and ǫ i = ±1 for an outgoing (incoming) particle. Under (Lorentz) conformal tranansformation the modified Mellin amplitude M n transforms as, 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, where ǫ = ±1 for an annihilation (creation) operator of a massless gluon of helicity σ and Lie algebra index a. Under (Lorentz) conformal transformation the conformal primary transforms like a primary operator of scaling dimension (h,h) Similarly in the presence of the time coordinate u we have, In terms of (A.8), the Mellin amplitude can be written as the correlation function of conformal primary operators Similarly using (A.10), the modified Mellin amplitude can be written as, A.1 Comments on notation in the paper Note that conformal primaries carry an additional index ǫ 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, Similarly in many places in the paper we denote a gluon conformal primary of weight ∆ = h +h by O a ∆,σ where σ = ±1 is the helicity (= h −h). Since we are considering pure Yang-Mills, we can further simplify the notation to O a ∆,± by omitting the σ = ±1.

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 (6.1) which was obtained in [9] 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 [75] A(1 a 1 , 2 a 2 , 3 a 3 , 4 a 4 , 5 a 5 ) = (ig) 3 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, ρ(2), ρ(3), ρ(4), 5). This is an over-complete basis. Owing to BCJ relations [76], any 4 of the sub-amplitudes in (B.1) can be written in terms of 2 linearly independent sub-amplitudes. Let us choose this BCJ basis to be given by the sub-amplitudes 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 (B.1) to be σ 1 = σ 2 = −1, σ 3 = σ 4 = σ 5 = 1. Then using (B.3) we obtain where the c i 's denote the following color structures Now parametrising the null momenta of the external gluons in the amplitude as where ǫ i = ±1 and using the Parke-Taylor formula for MHV amplitudes [77], we can write (B.5) as Mellin amplitude is given by where O a i ∆ i ,± (i) denotes a gluon primary operator with dimension ∆ i = 1 + iλ i . The subscript (±) here denotes the spin of the operator. The label (i) collectively denotes the coordinates (z i ,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 ǫ 4 = ǫ 5 = 1. Then let us define where t ∈ [0, 1]. The delta function imposing overall energy-momentum conservation in (B.11) can be expressed in the following form δ (4) can be easily done and we get Then using (B.17) and the expression of the 5-point MHV amplitude obtained in (B.8) we finally get In the above integral the theta functions simply impose the condition that ( ω * i ω P ) ≥ 0. This is required because in the original integral (B.11), the energy variables satisfy ω i ≥ 0. The prefactor N in (B.19) is given by and I 1 (t), I 2 (t) in the integrand in (B.19) are I 1 (t) = 1 + z 45 t σ 1,2 σ 1,1 +z 45 t σ 1,3 σ 1,1 + z 45z45 t σ 1,4 σ 1,1

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 (B.19).

B.4 OPE decomposition of 5-point Mellin amplitude
We shall now extract the celestial OPE between gluon primaries O a 4 ∆ 4 ,+ and O a 5 ∆ 5 ,+ from the 5-point Mellin amplitude. For this purpose we will set u 45 = 0 and expand (B.19) around z 45 = 0,z 45 = 0. Now note that expanding the theta functions in (B.19) generates delta function contributions whose arguments are functions of z ij ,z ij with (i, j) ∈ (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 I 1 (t), I 2 (t) in (B.19).
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 [74]. 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 [59].

B.4.1 Leading term
Using (B.22) and (B.23), it is easy to see that the terms from I 1 (t) and I 2 (t) which contribute at leading order in the OPE limit (z 45 → 0,z 45 → 0) are given by ) in the OPE decomposition. This will correspond to the contribution of descendants in the celestial OPE. Let us first gather the relevant terms from I 1 (t) and I 2 (t) which can contribute at this order. From (B.22) we get where Z is given by Then from (B.23) we have and and and Thus upto the first the subleading order, the celestial OPE between outgoing gluon primaries of opposite helicities is given by The above OPE manifestly satisfies both the leading as well as the subleading conformal soft gluon theorem.