Multi-Soft gluon limits and extended current algebras at null-infinity

In this note we consider aspects of the current algebra interpretation of multi-soft limits of tree-level gluon scattering amplitudes in four dimensions. Building on the relation between a positive helicity gluon soft-limit and the Ward identity for a level-zero Kac-Moody current, we use the double-soft limit to define the Sugawara energy-momentum tensor and, by using the triple- and quadruple-soft limits, show that it satisfies the correct OPEs for a CFT. We study the resulting Knizhnik-Zamolodchikov equations and show that they hold for positive helicity gluons in MHV amplitudes. Turning to the sub-leading soft-terms we define a one-parameter family of currents whose Ward identities correspond to the universal tree-level sub-leading soft-behaviour. We compute the algebra of these currents formed with the leading currents and amongst themselves. Finally, by parameterising the ambiguity in the double-soft limit for mixed helicities, we introduce a non-trivial OPE between the holomorphic and anti-holomorphic currents and study some of its implications.


Introduction
The simple geometric fact that a null vector in Minkowski space defines a point on the sphere at null infinity naturally suggests interpreting scattering amplitudes for massless particles as twodimensional correlation functions. An early and prominent example of such an interpretation was given by Nair in [1] where certain N = 4 super-Yang-Mills (SYM) amplitudes were constructed using the operator product expansion (OPE) of the Kac-Moody currents which define a two-dimensional Wess-Zumino-Witten model. This construction in part lead to the development of the twistor string [2] which gives a complete description of tree-level N = 4 SYM in terms of a two-dimensional world-sheet theory.
Kac-Moody structures have also recently appeared in the study of the soft-limits of gluon amplitudes [3,4]. Scattering amplitudes of massless particles such as gluons can be formally expanded in powers of a small scaling parameter δ multiplying the momentum of a particle which is thus taken to be soft, p µ = δ q µ , and which schematically gives where A n depends only on the remaining hard momenta. That scattering amplitudes of photons and gravitons have a universal, leading divergent behaviour as the external leg becomes soft has been long understood [5][6][7]. It is also known that at tree-level the behaviour at the sub-leading order is universal, which is sometimes referred to as the Low theorem [5,6,8,9], while for tree-level graviton amplitudes there is further universal behaviour at sub-sub-leading order [10]. Analogous sub-leading behaviour for gluon amplitudes has been recently studied [11][12][13][14]. In [4] He et al showed that one could identify the tree-level leading soft theorem for positive helicity gluons with the Ward identity for holomorphic two-dimensional Kac-Moody currents. These currents were shown to be related to the asymptotic symmetries of Yang-Mills theory given by large CPT-invariant gauge transformations [3] 1 . The OPE of the currents was extracted from the limit of two soft positive helicity gluons and it was shown that the level is zero in this identification, that is the algebra is a standard current algebra (1. 2) It was further noted that the limit where one positive and one negative helicity gluon are both taken soft is ambiguous with the result depending on the order in which the limit is taken. In [4] the authors gave the prescription of first taking all positive helicity gluons soft which resulted in one copy of the holomorphic Kac-Moody algebra but without a second anti-holomorphic copy arising from the negative helicity gluons.
Multi-soft limits of gluon amplitudes have been studied more generally in a number of recent works and universal behaviour in the limit where all gluons are taken soft simultaneous was found for tree-level amplitudes using the BCFW [18] and CHY [19] formalisms in [20,21]. These limits have also been studied using the CSW [22] formalism by [23] and by considering the field theory limit of string theory [24]. At leading order in the soft parameter this simultaneous double-and multi-soft limit differs from the consecutive case when there are particles of both positive and negative helicity. There is also universal behaviour in the sub-leading order of the multi-soft limit however the order in which the limit is taken is important even when all gluons have the same helicity.
We use these multi-soft limits to study further aspects of the current algebra interpretation in four dimensions. Starting with the leading soft-limit for positive helicity gluons we show that, by analogy with the Sugawara construction [25], one can use the double soft-limit to define a holomorphic energy-momentum tensor. Moreover we show, by analyzing the triple-and quadruplesoft limits of positive helicity gluons that this energy-momentum tensor has the standard OPE with both the holomorphic current and with itself. In a conformal field theory with a Kac-Moody symmetry the correlation functions of primary fields should satisfy certain differential equations, the Knizhnik-Zamolodchikov (KZ) equations [26]. We consider the KZ equation for the amplitudes and show that while they hold for positive helicity gluons in MHV amplitudes this is not the case for the negative helicity gluons or for non-MHV amplitudes. While this suggests that the holomorphic currents do not provide a complete description of the putative CFT it motivates further analysing the sub-leading and negative helicity soft-limit. We thus study the soft limit at sub-leading order and give these sub-leading soft-theorems an interpretation as Ward identities for two-dimensional currents which depend on a continuous parameter corresponding to the soft gluon energy, J a sub (z; ω z ). The OPE of these currents with themselves and with the leading order currents is then extracted from the sub-leading double-soft limits. As the algebra depends on exactly how the double soft limit is taken we introduce parameters to characterise this ambiguity. In the language of the two-dimensional field theory the leading terms of the OPEs are given by As in [4] the anti-holomorphic currents are defined using the soft-limits of negative helicity gluons and to study the algebra with the holomorphic currents we consider double-soft limits involving both a positive and negative helicity gluon. However rather than picking a specific ordering of soft-limits we consider a linear combination and introduce variables with which we parameterise the ambiguity. The resulting algebra of currents is found to be Using (1.5) one can compute the OPE of the anti-holomorphic currents with the holomorphic stressenergy tensor. We do this in Sec. 5.2 and show that it is consistent with the triple-soft limit involving two positive helicity gluons and one negative. Finally, using the leading order holomorphic and antiholomorphic currents, we define a conjugation operator C(z) ∝ : J aJ a : (z) and compute its OPE with the currents. While we don't attempt to give a more fundamental interpretation of these two-dimensional currents the renewed focus on understanding the asymptotic symmetries has lead to a number of interesting proposals for a holographic description of flat space scattering amplitudes. In [29], building on [30], a world-sheet theory with the null infinity boundary of flat space as its target space was proposed. An alternative approach based on the ambi-twistor string, [31], was studied in [32]. Recently, Cheung et al [33] have proposed a holographic description which following, [34], makes use of a slicing of flat space into a family of AdS 3 sub-spaces to interpret four-dimensional amplitudes in terms of a two-dimensional CFT. They identify the conserved currents and energy-momentum tensor with the asymptotic symmetries of gluon and graviton amplitudes. While there are a number of differences with our current work, for example the central charge of the current algebra, it would be very interesting to understand better if there is a holographic interpretation of the sub-leading and anti-holomorphic currents.

Soft limits for gluon amplitudes
The central objects of our study are the soft limits of gluon scattering amplitudes. While we will be interested in the full colour dressed amplitudes it is convenient to start with colour stripped amplitudes. Consider an n+1-leg amplitude where we take the first leg to be a soft particle of helicity h q with momentum δq, or in terms of spinor variables, { √ δλ q , √ δλ q } 2 . The soft limit corresponds to expanding the amplitude in powers of δ 1 and we keep only those terms in the expansion which are universal. As it will make subsequent formulae simpler we will often simply denote the non-soft legs with momenta p i , i = 1, . . . , n by 1, 2, . . . , n.

Single-soft limits
The single-soft limits for gluon amplitudes were found, up to the first sub-leading term, in [6,8,11] A n+1 (δ 1 q hq , 1, . . . , n) → 1 where, for a positive helicity gluon, h q = +, between neighboring particles n and 1, the universal soft-factors can be written using the spinor-helicity formalism as n,1 (q + ) = 1 q1 Here we use the conventions and similarly for the dotted indices, [ab] = αβλα aλβ b . Correspondingly we will write the contraction of a spinor with a derivative asλα For a negative helicity gluon the soft factors are given by conjugation of the spinor variables, λ i ↔λ i .

Multi-soft limits
One can similarly consider the limit of gluon amplitudes with multiple soft gluons. However as mentioned there is in general an ambiguity in the result which depends on the order in which the gluons are taken to be soft. Starting with the case of two soft gluons we consider the n + 2-leg amplitude where we will take the first and second legs to be soft particles with helicities h q 1 and h q 2 and momenta { √ In particular, if one takes the gluons soft sequentially we call this a consecutive soft limit in contradistinction to the simultaneous or double soft limit. These consecutive soft limits can be calculated straightforwardly from repeated action of the above single soft factors.
We define the consecutive soft factor, CSL n,1 (q where we take the particle q 1 soft first and then q 2 . As it will be of interest later, let us give the explicit expression: If we take the reverse consecutive limit, i.e. take leg q 1 soft and then leg q 2 , the leading term in CSL(q + 1 , q + 2 ) is unchanged. However when the particles have different helicity, or when we extract sub-leading multi-soft terms, the order of limits will be important and so we consider linear combinations by defining more general multi-soft limits with the ellipses denoting all further permutations. Thus we can define a multi-parameter family of consecutive soft limits The effect of this general limit can be seen in the sub-leading term in the expansion of two soft, positive helicity gluons which a short calculation shows is given by It is perhaps worthwhile to note that this expression is only valid for generic external momenta as we have neglected holomorphic anomaly terms that can arise when external legs are collinear with soft legs, however it will turn out that they are not necessary for our considerations.
For the case of mixed helicity the two orderings of limits already differ at leading order in the soft expansion and so we can define a two parameter double-soft limit where αCSL (0) has the overall (δ 1 δ 2 ) −1 dependence stripped off. As described previously, we can of course also consider the simultaneous double-soft limit where both particles are taken soft together. The case of two gluons of the same helicity is identical to the consecutive limit but the case of one negative helicity and one positive is different. In four space-time dimensions the leading order double-soft mixed helicity factor is given in the spinor helicity formalism by This expression differs from the consecutive limit due to the sum of soft momenta in the denominator but it is closest to the symmetric α 12 = α 21 = 1 2 case. A similar ambiguity appears at sub-leading order when we consider the double-soft limit in the mixed helicity case. Explicit formulae can be found in [20] but in this case we will focus on the consecutive limits which can be calculated by repeated use of the single-soft limits (2.2).

Current algebra interpretation of Yang-Mills soft limits
Let us consider the single and double soft limits of Yang-Mills amplitudes but now instead of just focussing on colour-ordered partial amplitudes we examine the full amplitude where the sum runs over all permutations, S n , modulo those which are cyclic, Z n , and T a are the generators of the colour algebra which we will take to be su(N ). We will drop the factors of g YM but as we are only concerned with tree-level amplitudes they can be trivially restored. If we take the soft-limit of the n + 1-particle amplitude we have at leading order in the soft-expansion We can rewrite this soft limit in terms of two-dimensional position space variables by using the parameterisation for the soft-gluon momentum and for the hard momenta the parameterisation Essentially the complex z-variable describes the position of the intersection of the soft gluon's trajectory with the sphere at asymptotic infinity and the u i 's are the corresponding intersection points for the hard gluons, see Fig. 1. We take all gluons to be outgoing and so this is in fact the anti-sky mapping. In these variables we have for the leading single soft factor (2.2) , where we see that it has simple poles as the soft momenta approach the hard momentum in position space i.e. z → u i for each i. Now we define, and so where in the last line T a is understood to act (in the adjoint representation) only on the -th particle colour factor in A n . This is the result of He et al [4] who compared it to the OPE for a Kac-Moody current J a and a primary field in a representation R and the consequent Ward Identity Note that we use the notation J a in (3.5) to denote the expression derived from a four-dimensional scattering amplitude perspective and the J a in (3.7) to be the corresponding quantity from the two dimensional CFT perspective. It is the formal similarity of these two expressions that suggests the interpretation of the soft-gluon limits as corresponding to the insertion of the current operator and it is interesting to ask to what extent this analogy can be extended.

An su(N ) current algebra
As a first step, it is natural to ask whether one can reproduce the current-current OPE At the level of correlation functions we would expect to find, That the corresponding result can be found by analyzing the double-soft limits of amplitudes was shown in [4] and as it is useful for our later results we rederive this fact in our notations. To this end we consider the double soft limit which we can write as in a slightly condensed notation with Tr(q 1 q 2 σ 1 . . . σ n ) = Tr(T aq 1 T aq 2 T aσ 1 . . . T aσ n ) etc. As we first want to find the OPE of two holomorphic currents we consider the leading order double soft limit of two positive helicity gluons. As the gluons have the same helicity the order of limits is not relevant and so we simply take the simultaneous limit with a single soft parameter δ 1 = δ 2 = δ. We use the identity for consecutive soft-limits here with both helicities positive h q 1 = h q 2 = +, and so we can combine the first two terms to find Re-writing the soft-factors in terms of polarisation vectors one recovers the result quoted in [4]. The singularities of these double soft factors in the collinear limit, as the soft momenta approach each other, can be made clear by rewriting them in position space and examining the z 1 → z 2 limit. We can see that there will be no singularities from the terms involving the products of single soft factors; these terms instead give poles when the soft momenta approach the hard momenta. However the term involving the consecutive double soft limit does have such a singularity and gives rise to the non-trivial algebra between the holomorphic currents. More explicitly, defining (3.14) Hence comparing with (3.9) we see that the double soft-limit reproduces the structure of level-zero Kac-Moody or simply a current algebra as described in [4].

Sugawara construction for amplitudes
If this is a correct interpretation, i.e. there is a level-zero Kac-Moody symmetry acting on amplitudes, we may try to construct a holomorphic energy-momentum tensor via the Sugawara construction. Namely, we define where γ is a constant related to the dual Coxeter number, h ∨ , by γ = 1 2h ∨ , and the normal ordering is missing the usual singular term that occurs in the Kac-Moody case as the current algebra has no central extension. At the level of correlation functions this leads us to again consider double insertions of holomorphic currents and so, as the analogue for amplitudes is the double soft limit, we again consider (3.13) but now with contracted adjoint indices. That is we define We could in principle work with a general Lie group but for simplicity we focus on the case of su(N ) colour where we can contract the indices a and b using the identity 17) and in this case h ∨ = N . To calculate the scattering amplitude analogue of the insertion of of an energy-momentum tensor into a correlation function we consider the singularities as either of the soft momenta approach one of the hard momenta, say u m , and then expand the z 2 dependence near z 1 . Using the notation z 1m = z 1 − u m for m = 1, . . . , n we write the result Tr(mσ 1 . . . ) A n (mσ 1 . . . ) . (3.18) While this definition seems sensible we should ask whether it satisfies the basic properties of a energy-momentum tensor for a conformal field theory.

Energy-momentum tensor/current OPE
In two-dimensional field theory a defining characteristic of T S (z 1 ) is its OPE with the holomorphic current operator The analogous object for the amplitude is a specific case of the triple soft-limit We take all gluons to have positive helicity, such that the triple and double soft limits are the same regardless of whether we take the consecutive or simultaneous soft limit and so, as in the double-soft case, we set all soft parameters equal. The product of three holomorphic currents is then defined to be In the standard two-dimensional field theory calculation, to compute the OPE of the energymomentum tensor with the current operator we collect the terms which are singular as z 3 approaches either z 1 or z 2 . In the soft-limit such terms correspond to the soft momentum q 3 becoming collinear with either q 1 or q 2 . In particular the contributions from the triple soft terms split into two groups, those where the soft particles q 1 and q 2 are adjacent which come with a colour factor (N − 1 N )Tr(cσ 1 . . . σ n ) while those where q 1 and q 2 are split by particle q 3 have (− 1 N )Tr(cσ 1 . . . σ n ). Combining these terms we find that the (− 1 N ) factors cancel. There are further contributions from terms involving double soft limits times single soft factors which have colour structures f ca d Tr(a . . . σ r d . . . ) however these terms cancel between themselves. This makes use of the identity (3.11) as well as the fact that Following the conformal field theory calculation, we then expand this in powers of z 21 = z 2 − z 1 and keep the leading term (in principle there could be a more singular term however this term is absent which corresponds to the level of the current algebra being zero ). We finally have which after expanding the z 1 dependence on the right hand side is the required result.

Energy-momentum OPE
Using the OPE of holomorphic currents it can be shown that the Sugawara energy-momentum tensor for a current algebra (k = 0) satisfies the OPE and so defines a conformal field theory with vanishing central charge. Correspondingly, inserted into a correlator of primary fields Φ(u i ), we have that In terms of the soft-limits of amplitudes this Ward identity should be reproduced by the collinear part of the quadruple soft limit for positive helicity gluons. Once again as we are only considering positive helicity gluons there is no ambiguity in the order of limits. In the two-dimensional field theory construction one starts with pairs of currents, J a 1 (z 1 )J a 2 (z 2 ) and J a 3 (z 3 )J a 4 (z 4 ), and to calculate the OPE one first extracts the terms which become singular as either z 1 or z 2 approaches either z 3 or z 4 and then takes the limits z 2 → z 1 and z 4 → z 3 . For the amplitude one similarly considers those terms which are singular as either q 1 or q 2 become collinear with q 3 or q 4 . The singular terms get contributions from quadruple soft terms, where all soft particles are adjacent, from triple-soft terms and from the "double" double-soft terms. Carefully combining all these terms we find, after taking the z 2 → z 1 limit, that the singular terms in the quadruple soft limit are This can be compared with the definition of a single insertion of the energy-momentum tensor in (3.18) and, after further expanding the z 1 dependence near z 3 , we see that we indeed reproduce the OPE in (3.24) as expected. While we restrict our considerations to gluon amplitudes it is interesting to note that the subleading soft theorem for a graviton has been identified with the Ward identity for a two-dimensional energy-momentum tensor [36] which suggests interpreting the collinear limit of double-soft gluons as a graviton. This is reminiscent of recent work [37] showing that amplitudes describing the interactions of gravitons with gluons can be written as linear combinations of amplitudes in which the graviton is replaced by a pair of collinear gluons.

Comments on a Knizhnik-Zamolodchikov equation
Given the above construction for the energy-momentum tensor it is interesting to ask if we can derive conformal Ward identities by considering correlation functions with insertions of the energymomentum tensor. However, a priori, even if there is a sensible conformal field theory interpretation of the asymptotic states the current algebra may only form part of it, which is to say the full energymomentum tensor T (z) would be given by (3.26) If we consider the individual terms in the mode expansion In conformal field theory we can insert this inside a correlation functions of primary fields and in particular if we consider the case where T (z) is absent we can derive the Knizhnik-Zamolodchikov equation.
Turning now to the scattering amplitudes, we consider the individual terms in the expression for an insertion of the energy-momentum tensor following from the double-soft limit, (3.18). We have at leading order where C um is a contour surrounding the point u m . This has the expected form for an insertion of the energy-momentum tensor into a correlation function of primary fields with weight one. More non-trivially the sub-leading term is given by The usual conformal Ward identity would then imply where L −1 is the residue of T (z 1 ) at z 1 = u m . It is tempting to ask what would happen if the L −1 term would be absent, that is, if the current algebra described the complete CFT. As an example one can consider the MHV amplitudes. It is useful to define a rescaled quantity G n as follows In the simplest three-particle case and it is immediately obvious that (3.31) with L −1 absent is satisfied for u 3 but not u 1 or u 2 . In fact it is easy to check explicitly that the KZ-equation with L −1 absent continues to hold for the positive helicity gluons in higher point MHV amplitudes (currently done with Mathematica to seven-points). It is slightly non-trivial to check that the double trace terms, which arise when m and in (3.31) are not adjacent in a specific colour-ordered term, vanish. This requires that the colour ordered amplitudes satisfy a number of identities. For example, for the m = 3 KZ equation one such relation at four points is the vanishing of the Tr(T a 1 T a 4 )Tr(T a 2 T a 3 ) double-trace terms, which requires that where cyc{σ , . . . , σ n−1 } denotes the sum over all cyclic permutations. We checked all such identities hold up to seven points for MHV amplitudes. However, starting with the six-point NMHV amplitude it no longer appears that the KZ equations holds even for the positive helicity gluons. This suggests, unsurprisingly, that if there is a CFT interpretation there is additional structure and as there is further universal behaviour in the soft-limits we attempt to also interpret these in the context of a two-dimensional description. Let us round off this section with an interesting connection of the above amplitude relations with the so called Bern-Carrasco-Johansson (BCJ) [38] relations resulting from the color-kinematic duality of gauge theory amplitudes. We can re-write the previous example of the four point amplitude relation (3.34) in a different way using Kleiss-Kuiff and the photon decoupling relations, which gives A(1, 2, 3, 4) = A(1, 4, 3, 2) and A(1, 3, 2, 4) = A (1,4,2,3). Using these relations we can rewrite the left-hand side of (3.34) as, which is remarkably the 4−point BCJ relation. Thus the MHV amplitude relation derived from demanding the vanishing of the double-trace terms in our KZ equations leads to the BCJ relation at least for 4 points. This connection is harder to see for higher number of points and we will report on this in a future publication.

Sub-leading currents
So far we have only been considering the leading soft terms, however it is known that the sub-leading soft terms for Yang-Mills amplitudes are also universal. This behaviour, the gluon version of the Low theorem [5,6,8,9] described in Sec. 2.1, is understood to be valid only at tree-level and for generic configurations of the remaining external hard legs [13]. We will only consider such configurations and so neglect boundary terms in the kinematic space where additional particles become collinear.
At loop level such corrections could no longer be avoided and so it would be interesting to carefully understand these terms, and more generally to make contact with the SCET [39] description used in [13]. We thus repeat the above analysis at the sub-leading order by defining the sub-leading currents, J a sub , as where the (1 + δ∂ δ ) factor picks out the sub-leading soft term. We have that where∂ = ∂ ∂z and ∂ ω = ∂ ∂ω . If we again attempt to interpret this in terms of the OPE of a current with a primary field we must consider the primary fields as depending on an auxiliary variable ν and we have The appearance of an auxilliary parameter and derivatives in the representation is (very vaguely) reminiscent of the Cartan element of the non-compact affine sl 2 current algebra in the principle continuous series representations, Φ j, (w), for which we introduce the complex parameter x, and the OPE is given by where t 3 = x ∂ ∂x . In the case at hand we see that, in addition to having a one-parameter family of such currents, there is no extension as k = 0. Adamo and Casali [29] have, in their world-sheet theory with null infinity as its target space, given the definition of a charge with reproduces the sub-leading soft factor. It would be interesting to understand if there is a relation between the currents introduced here and their charges which have the interpretation as rotating the space of null generators. It is also known that these sub-leading soft terms in four-dimensions are intimately connected to the conformal invariance of the theory [13].

Sub-leading algebra
To continue the two-dimensional interpretation of the collinear soft limits to sub-leading order and given these new currents we analyse the algebra that they form with the leading currents and amongst themselves.
J sub J : To find the OPE of the Kac-Moody current J a (z) with J a sub (z) we must consider part of the double soft limit at sub-leading order. The order of limits in this case is relevant and a specific prescription must be given with the results dependent on the prescription. We start with the consecutive limit and take leg q 2 soft before leg q 1 : An important feature of this formula is that, as previously, the singularities as q 1 and q 2 become collinear only occur in the third set of terms on the right-hand side as those that appear in the first set of terms cancel amongst themselves Using the coordinate representation (3.2) we can find the most singular part as If we had instead taken δ 2 to zero after δ 1 we would find σn,q 2 (q 2 ) Tr(q 1 q 2 σ 1 . . . σ n ) + · · · + iS (1) q 2 ,σ 1 (q hq 1 1 )S (0) σn,σ 1 (q hq 2 2 )f aq 2 aq 1 c Tr(cσ 1 . . . σ n ) A n (σ 1 , . . . , σ n ) . (4.8) Again one can show that in the collinear limit the potentially singular terms only come from the last line above, however in this case there are in fact no non-vanishing singular terms in the collinear limit. Such terms potentially could have arisen from the holomorphic anomaly ∂ ∂λȧ 1 λµ = π ȧḃμḃ δ (2) ( λµ ) (1 + δ 1 ∂ δ 1 )δ 2 A n+2 = − σ∈Sn · · · + iπf aq 2 aq 1 c σ n σ 1 q 2 q 1 − [q 1 σ n ] q 2 σ 1 δ (2) ( σ n q 2 ) + [q 1 σ 1 ] q 2 σ n δ (2) ( q 2 σ 1 ) Tr(cσ 1 . . . σ n ) A n (σ 1 , . . . , σ n ) . However these terms are also of order one when q 1 becomes collinear with q 2 . Thus taking a combination with parameters α 12 and α 21 only the α 21 parameter contributes and from the definition J a sub (z 1 )J b (z 2 ) · A n ({u i ,ū i }) ≡ (4.11) lim α (1 + δ 1 ∂ δ 1 )δ 2 ω z 1 ω z 2 A n+2 (δ 1 q 1 , +, a; δ 2 q 2 , +, b; {p i , h i , a i }) we find

Anti-holomorphic currents
We can of course repeat the previous calculations for the negative helicity gluon and find the antiholomorphic currents. We definē J a (z) · A n ({u i ,ū i , ν i }) ≡ lim δ→0 (δω)A n+1 (δp, −, a; {p i , h i , a i }), (5.1) so that the current still acts with the adjoint action but because of the missing minus sign it acts from the right rather than the left or, alternatively, with the complex conjugate generator

OPE for anti-holomorphic currents
Slightly more non-trivially we can consider the mixed helicity double soft limit and attempt to reproduce the action of the holomorphic currents on the anti-holomorphic discussed in [4]. We can again start from (3.10) however now h q 1 = + while h q 2 = −. In this case there is an ambiguity in the double soft limit which we parameterise in case of consecutive limits by using a specific case of the general multi-parameter limit lim α ≡ (α 12 lim ) . (5. 3) The corresponding two-parameter family of consecutive double-soft factors, αCSL σn,σ 1 , still satisfies the identity analogous to (3.11) αCSL (0) σn,σ 1 (1 + , 2 − ) + αCSL (0) σn,σ 1 (2 − , 1 + ) = S (0) σn,σ 3 (1 + )S (0) σn,σ 3 (2 − ) . where the constants d 1 are those appearing in the JJ-OPE (1.5) and we see that C essentially acts as a charge conjugation operator. Just as for the energy-momentum tensor we can analyze the same OPE by examining the triple soft-limit of two positive helicity gluons and one negative but now h q 2 = − rather than h q 3 . The calculation is identical to that above and the final result can be written as where we again see that with the appropriate choice of parameter in the multi-soft limit that we find the expression calculated directly from the JJ OPE.