Algorithmic construction of SYM multiparticle superfields in the BCJ gauge

We write down closed formulas for all necessary steps to obtain multiparticle super Yang--Mills superfields in the so-called BCJ gauge. The superfields in this gauge have obvious applications in the quest for finding BCJ-satisfying representations of amplitudes. As a benefit of having these closed formulas, we identify the explicit finite gauge transformation responsible for attaining the BCJ gauge. To do this, several combinatorial maps on words are introduced and associated identities rigorously proven.


Introduction
The definition and usage of multiparticle superfields [1,2] of supersymmetric Yang-Mills (SYM) theory [3] has proved to be an essential feature in obtaining compact expressions for high-multiplicity amplitudes in superstring [4] and field theories [5] using the pure spinor formalism [6].
In the simplest formulation of multiparticle superfields in the Lorenz gauge, their definition is given by a straightforward recursion over the particle labels [2]. While this recursive definition has its own merits and is certainly useful in relating the new expressions for tree-level amplitudes [7] to the standard Berends-Giele recursions [8], there is an alternative formulation related by a non-linear gauge transformation whose properties have more appeal, the BCJ-gauge representation [1]. As will be reviewed in section 2.3, the superfields in this gauge satisfy generalized Jacobi identities [9] in their particle labels, for example A m 12 = −A m 21 , A m 123 + A m 231 + A m 312 = 0, and so forth. In this gauge, they constitute the natural building blocks used in the expressions of local SYM numerators satisfying the Bern-Carrasco-Johansson numerator identities [10] at tree- [11] and loop-level [12,13].
As explained in [2], the gauge transformations required to go to the BCJ gauge are encoded in so-called redefining superfields H [P,Q] to be reviewed below. Until now, the explicit expressions of these superfields were known only up to multiplicity five [2]. In A m P A n Q F mn R + 1 2 (W P γ m W Q )A m R + cyclic(P, Q, R) .
As a consequence of the quadratic corrections H 2 in these formulas, we will show in section 5.3 that the superfields satisfying the generalized Jacobi identities follow from a standard gauge transformation of SYM theory in its finite form, We note that in [2] only the first three terms of (1.4) were identified.
While in pursuit of finding these formulas we also filled some gaps of the previous discussions. These mostly concern writing down closed formulas for expressing contact terms (in a multitude of different situations) where the multiparticle labels are given in terms of an arbitrary configuration of nested Lie brackets. As will be explained in section 3, we found a novel recursive description of such terms which is universal and whose backbone is given by the solution to a purely combinatorial problem. Several equations relevant to the framework of multiparticle superfields can be written down using this newly found recursion and we prove several associated results.
Finally, in the appendices we write down some longer examples of applications of several recursive maps from the main text, among other things.

Review
In this section we review some aspects of the construction of 10d supersymmetric Yang-Mills superfields following the recent discussions of [2,1] using the framework of perturbiners [14]. For the original references on the covariant description of super Yang-Mills in ten dimensions, see [15,16]  The spinor representation is based on the 16 × 16 Pauli matrices γ m αβ = γ m βα satisfying the Clifford algebra γ (m αβ γ n)βγ = 2η mn δ γ α . In this paper the (anti)symmetrization of n indices does not include a factor of 1 n! .

Multiparticle index notation
In the following discussions we will use a notation based on "words" composed of "letters" from the alphabet of natural numbers. Capital letters from the Latin alphabet are used to represent words (e.g. P = 1423) while their composing letters are represented by lower case letters (e.g. i = 3). The length of a word P is denoted |P | and it is given by the number of its letters. The reversal of a word P = p 1 p 2 . . . p |P| isP = p |P| . . . p 2 p 1 . The word notation is also used in place of arbitrary commutators, such as P = [1, 2] ≡ 12 − 21; the context will disambiguate whether a word denotes a sequence of letters or a bracketing structure. In addition, when the bracketing structure is nested from left to right such as 1,2], 3], 4], 5] we will often write it as P = 12345. Such structures may be referred to as (left-to-right) "Dynkin brackets" The multiparticle momentum for a word with letters (labels) from massless particles (k i · k i ) = 0 and its associated Mandelstam invariant are given by For example k m 123 ≡ k m 1 + k m 2 + k m 3 and s 123 = s 12 + s 13 + s 23 .

Non-linear supersymmetric Yang-Mills
To describe ten-dimensional SYM one introduces Lie algebra-valued superfield connections A α = A α (x, θ) and A m = A m (x, θ) and the supercovariant derivatives [16,15], αβ ∇ m and the associated Bianchi identities imply the following non-linear equations of motion [15], These equations are invariant under the gauge transformations of the superpotentials which in turn induce the gauge transformations of their field-strengths δ Ω W α = Ω, W α , δ Ω F mn = Ω, F mn , and δ Ω W α m = Ω, W α m where Ω ≡ Ω(x, θ) is a Lie algebra-valued gauge parameter superfield. The equations of motion (2.3) can also be rewritten as To solve the wave equations (2.7) we use the perturbiner method of Selivanov [14]. In this approach, one expands the superfields K ∈ {A α , A m , W α , F mn } as a series with respect to the generators t i j of a Lie algebra summed over all possible non-empty words P as After plugging these series in (2.7) one learns that the expansion coefficients K P ∈ {A P α , A m P , W α P , F mn P } turn out to be the Berends-Giele currents, where s P = 1 2 k 2 P arises from the operator acting on plane waves of momentum k m P and Notice that the above Berends-Giele currents are non-local superfields as they contain inverse factors of Mandelstams variables.

Linearized description of 10d SYM
The linearized description of ten-dimensional super-Yang-Mills is obtained by discarding the quadratic terms from the equations of motion (2.6) and yields (2.11) In the context of scattering amplitudes, the superfields are labelled with a distinct natural number i to associate them with the i-th particle taking part in the scattering process.
This association will be generalized below.

Generalized Jacobi identities
As we will discuss below in the context of multiparticle superfields, there is the notion of a superfield satisfying certain symmetries dubbed BCJ symmetries in [2]. These symmetries can be given a precise mathematical characterization in terms of what is called generalized Jacobi identities in the mathematics literature [9,17].
Let A be a word and ℓ(A) its left-to-right bracketing defined in (A.1). The generalized Jacobi identities correspond to the elements in the kernel of ℓ. For example ℓ(12 + 21) = 0 , ℓ(123 + 231 + 312) = 0 , (2.12) which correspond with the antisymmetry and Jacobi identity of the Lie bracket.
Using the identity ℓ(P ℓ(Q)) = [ℓ(P ), ℓ(Q)] it is easy to see that ℓ(Aℓ(B) + Bℓ(A)) = 0 for any words A and B. In addition, due to the recursive definition of ℓ if ℓ(P ) = 0 it also follows that ℓ(P Q) = 0 for any word Q. Therefore, for objects labelled by words, the generalized Jacobi identities can be characterized by an abstract operator £ k We emphasize the arbitrary partition of non-empty words A and B in the above definition (while C can be empty), leading to a non-unique operator £. For instance £ 3 • K 123 = K 123 − K 132 + K 231 , for A = 1, B = 23 and C = ∅ (2.14) Note that if £ 2 • K 123 = 0 then the right-hand side of the expressions in (2.14) agree and can be written as the cyclic sum K 123 + K 231 + K 312 . Definition 1. The objects K P are said to satisfy generalized Jacobi identities iff The generalized Jacobi identities are also called BCJ symmetries.
The defining identities for objects K P of increasing multiplicities can be written as where we have already used the fact that K P satisfies the BCJ symmetries £ k • K P = 0 for all k ≤ |P | to simplify the appearance of the above. This fact in general can be used to show the equivalence of the BCJ symmetries for the various partitions of P = ABC as mentioned after the example (2.14).
It is not hard to be convinced that the BCJ symmetries are equivalent to the symmetries of a concatenated string of structure constants, K 12...p ↔ f 12a 2 f a 2 3a 3 f a 3 4a 4 ...f a p−1 pa p .
If K P satisfies BCJ symmetries then it is convenient to use the notation K ℓ(P ) ≡ K P .
In particular, this implies that for superfields in the BCJ gauge we have [18], For example, K [12,34] = K 1234 − K 1243 . In addition, it follows from the definitions (2.13) and (2.15) that if K P with |P | = n satisfies generalized Jacobi identities then 18) which implies that there is an (n − 1)! basis of K P .

Contact terms for general Lie polynomials
For the purpose of this paper, P is a Lie polynomial if it is a linear combination of words written in terms of (nested) Lie brackets [x, y] ≡ xy − yx. For example P = [ [1,2], 3] = 123 − 213 − 312 + 321 is a Lie polynomial while Q = 123 is not 1 .
In this section we will introduce mathematical maps acting on words and Lie polynomials that will play a central role in later discussions about several aspects of local and non-local multiparticle superfields.

Planar binary tree map on words
A nested Lie bracket can be interpreted as a planar binary tree and vice versa [20]. In the context of tree-level scattering amplitudes one can map each planar binary tree to a product of inverse Mandelstam invariants. For example the two binary trees with three leaves are mapped to Mapping the sum over all binary trees with a given number of leaves will be related to Berends-Giele currents later on, and the explicit expansions can be generated from the following recursion.
Definition 2 (Binary tree map). A word P of length |P | is recursively mapped to a Lie polynomial built from a sum over all planar binary trees with |P | leaves as A theorem by Dynkin-Specht-Wever states that if ℓ(P ) = |P |P then P is a Lie polynomial [17], and this fact can be used to find the expression written in terms of nested Lie brackets [19].  for the diagrammatic representation of b(1234).

Contact terms associated to Lie polynomials
Given the Lie polynomial [1,2] we can associate to it the following contact terms propor- . It is easy to see that this definition leads to a deconcatenation of b(12), We would like to extend this action to an arbitrary Lie polynomial C • [P, Q] such that The following definition does the job, as will be proven below.
Definition 3 (Contact term map). Let C be the coproduct C : Lie → Lie ⊗ Lie that maps a Lie polynomial into the tensor product of two Lie polynomials recursively by and k m P ≡ k m p 1 p 2 ...p |P| , where p i for i = 1 to i = |P | are the letters of P .
As an immediate consistency check, we note that the definitions given in (3.6) imply that Note that when the contact term map is used to generate combinations of superfields, the notation described in (C.5) and (5.1) may be used. For example applications of the C map, see the appendix C.
Proof. The proof is inductive in nature. When the word P has length two the statement has been verified explicitly in (3.3). We now assume that the relation (3.7) is satisfied for any word P of length less than n, and let Q be a word of length n. Then we get where we have used the definition of the contact term algorithm (3.5). Now we separate the above into the three possible cases; both of |X| and |Y | being greater than 1, |X| = 1, and |Y | = 1. We then use that C • b(i) = 0 for i a letter, and that the induction hypothesis (3.7) holds for all C • b(P ) such that |P | < |Q|, so that every application of the map C can be removed from this equation. This leaves us with Absorbing the |X| = 1 and |Y | = 1 summations into the |X| > 1, |Y | > 1 cases we get Now we shall consider the two double sums. First of all we merge them using that, for example, XY =Q,|X|>1 AB=X is the same as ABY =Q . Then we remove the ∧ using the definition (3.6) to get We can now group the terms into two sets of four in a convenient way which we will now look at separately. With the first set of terms, it is clear from relabeling the second sum that it is just which is identically zero. The second set of terms in (3.11) can be simplified using the definition of the b map (3.1) leading to Then, since B and Y are adjacent everywhere they appear in the first sum, we can condense them into a single word, and likewise for X and C in the second sum. This leaves us with 3 We now return to (3.8) and, using that the double sum terms are given by (3.13), we finally Hence the result is proved.
. We then suppose that the relation (3.15) is satisfied for the bracket P , and consider C where q is a single letter.
where δ is the deshuffle map (A.2). Hence if (3.15) is true for the Dynkin bracket P , it is true for the Dynkin bracket [P, q], and so by induction the result is proved.
This result is important, as it shows that the general redefinition formulae of this paper reduce to those previously found in [2] when the multiplicity is less than six.

Contact term-like algorithms for simplifying redefinition terms
In this subsection a further pair of algorithms based around that of contact terms (3.5) will be defined, which will be useful when simplifying the redefinition terms (4.25) in the next section. The first of these will be denotedC, and is defined bỹ (note the C map (3.5) on the right-hand side) where∧ is defined by In addition we define a related algorithmC ′ in terms ofC, The following notation, similar to that of (C.5), will be used with these maps where the double bracket [[·, ·]] is defined in (5.1).

Lemma 2. The mapC satisfies
for any Dynkin brackets P and Q.
Proof. To see this we use the identity (3.15) as follows,

Redefinitions of local multiparticle superfields
In this section we write down the redefinition algorithms to obtain multiparticle superfields in the so-called BCJ gauge starting from both the Lorenz and hybrid gauges with the most general bracketing configurations. The characterization of these redefinitions as a gauge transformation was identified in [2] and it will be reviewed and expanded in the next section.

Multiparticle superfields
It was shown in [1,2] that the single-particle description admits a generalization in terms We will review two different ways to construct them below. At the same time we will seamlessly fill some gaps in the discussions of [1,2] by utilizing the framework developed in the previous section.

Multiparticle superfield in the Lorenz gauge
The generalization of the single-particle linearized superfields of (2.11) to an arbitrary number of labels follows from the local version of the recursive solution to the non-linear wave equations (2.7) and can be summarized by the following definition 4 : Definition 6 (Lorenz gauge). Multiparticle super-Yang-Mills superfields in the Lorenz gauge are defined starting with the multiplicity-one superfieldsÂ i α ,Â i m ,Ŵ α i andF mn i and recursively for arbitrary nested bracketings viâ 4 The Lorenz gauge discussion in [2] is missing the definition of the general field-strengthF mn while the definition ofŴ α [P,Q] is misleading as £ 3 •Ŵ α [12,3] = 0 if one does not use momentum conservation.
and the map C• is defined in (3.5). Alternatively, the field-strength can be written aŝ Identifying the pair of words P and Q for the superfields on the right-hand side of (4.5) leads to further applications of the recursions in (4.2) until eventually all superfields are of single-particle nature.

Multiparticle superfields in the hybrid gauge
Let us assume that all superfields of multiplicities P and Q in K P and K Q have been redefined to satisfy all the BCJ symmetries (2.15) (we will explain how to do this below).
Since multiparticle superfields K P in the BCJ gauge satisfy the same symmetries as the ..], p |P| ] their multiparticle labels will be written as plain words P = p 1 p 2 . . . p |P| . One then defines higher-multiplicity superfields inǨ [P,Q] as follows: Definition 7 (Hybrid gauge). Multiparticle super-Yang-Mills superfields in the hybrid gauge are distinguished by a check accentǨ [P,Q] and are defined by where the superfields in K P and K Q on the right-hand side satisfy the generalized Jacobi identities (2.15) and are the local form of the superfields of higher-mass dimension defined in [2] with the map Note an important difference with respect to the definitions of superfieldsK [P,Q] in the Lorenz gauge (4.2). The definitions in the Lorenz gauge are recursive while in the hybrid gauge they are not -the superfieldsǨ [P,Q] on the left-hand side of (4.7) have to be redefined before they can be used as the input on the right-hand side at the next step. However, from a purely practical perspective, to obtain the explicit expressions of the superfields in the BCJ gauge it is more convenient to use the hybrid gauge.

From hybrid gauge to BCJ gauge
The general formula to redefine the superfieldsǨ [P,Q] ∈ {Ǎ α ,Ǎ m ,W α } from the hybrid gauge (4.7) to superfields K [P,Q] ∈ {A α , A m , W α } in the BCJ gauge is given by Alternatively, the identity (3.21) can be used to rewrite (4.9) more succinctly as 5 (4.10) These redefinitions introduce new superfields H [P,Q] whose purpose is to make the resulting linear combinations satisfy the BCJ symmetries. For example, the first instances of the redefinition (4.9) for A m [P,Q] up to multiplicity |P | + |Q| = 5 are given by (recall thať . In general, the effect of the above redefinitions is such that K [P,Q] = K P ℓ(Q) , as shown in (2.17).
We have not yet discussed how the field strength F mn [P,Q] superfields in the BCJ gauge are found. These are most easily described by constructing them in terms of the above redefined BCJ gauge superfields and using the contact-term map (3.5), whereȧ andḃ denote the letterifications of A and B as defined in the appendix A and We have explicitly tested that the superfields up to and including multiplicity nine following from the formulas (4.9) and (4.14) satisfy the generalized Jacobi identities 8 . Since new corrections cubic in H [A,B] could be present at multiplicity nine, the fact that these formulas lead to superfields satisfying the BCJ symmetries suggest that (4.14) is correct for arbitrary multiplicity.

From Lorenz gauge to BCJ gauge
Alternatively, one can generate superfields in the BCJ gauge by starting from the superfields in the Lorenz gauge obtained through the recursions (4.2). The redefinitions are more involved in this case and one can show that to obtain their BCJ gauge counterparts requires the following iterated redefinition, where the operator L j is defined by To simplify the algebra we tested the bosonic components. Since the backbone of the recursion (4.14) is given by the supersymmetric H A,B,C we believe that (4.14) also leads to correct fermionic components.
where the H [A,B] are defined as they were in (4.14) -(4.16), andĤ m A ≡ k m AĤ A . Furthermore, the mapsC andC ′ are the variants of the contact-term map C defined in the section 3.2.1.
To demonstrate the meaning of these maps we will now provide examples. First of all note that theC andC ′ maps in (4.25) are both associated with pairs ofĤ superfields, each of which requires three indices, and so these terms will only be non-zero when |A|+|B| ≥ 6.
Thus at lower multiplicities these relations reduce to equation (3.15) of [2], as theC and C ′ terms only start contributing at multiplicity 6+. An example of the relations in this case is as follows:Ĥ  [4,5] ) .
We will now outline an example of (4.25) for the multiplicity six redefinition term Unfortunately to see an example where theC ′ map in the definition ofĤ ′ comes into affect requires going to multiplicity seven, which considerably increases the number of terms involved and makes any such example less easy to follow. The process is not terribly different from the one just outlined though, there are just more terms involved.
It might raise some concerns that (4.25) and (4.14) -(4.16) are in some places defined in terms of BCJ gauge superfields, and so this might not represent a true gauge transformation. This is however not an issue, as a purely Lorenz gauge version of (4.25) can be found by just replacing the BCJ superfields with their Lorenz gauge expansions (4.18). Some difficulty may arise doing this for H A,B,C due to the presence of F mn P terms.
However, we do the same thing, and plug the Lorenz gauge expansions into (4.13) to get The notation of (4.25) has just been chosen for its compactness and clarity.

BCJ symmetries and standard gauge transformations
In this section we will briefly review the result of [2] that the redefinitions of a local superfield K from the Lorenz to the BCJ gauge amount to a standard gauge transformation of the corresponding non-linear superfield K introduced in section 2.2. However, the discussion of [2] was based on examples up to multiplicity five and consequently missed an infinite number of correction terms. As a result, the gauge transformations were identified only in infinitesimal form. We will prove that the iterative redefinitions (3.5) lead to a finite gauge transformation instead.
To show this one uses the perturbiner series expansion K as given in (2.8) in terms of its Berends-Giele currents. Before proceeding, we review the definition of the Berends-Giele currents using a formulation based on the b map (5.2).

Berends-Giele currents and contact terms from maps on words
We will define the notion of a Berends-Giele current from a purely combinatorial point of view based on the map b(P ) acting on words. In order to do this for arbitrary labelled objects such as multiparticle superfields, let us define the a replacement of words by arbitrary superfields as In turn, this definition can be used to define the Berends-Giele currents and related concepts through the b and C maps.
Definition 4 (Berends-Giele map). If K P ∈ {A P α , A m P , W α P , F mn P } is a local multiparticle superfield, its associated Berends-Giele current is represented by a calligraphic letter K P ∈ {A P α , A m P , W α P , F mn P } and is given by

BCJ symmetries of local superfields as a gauge transformation
It was already pointed out in [2] that the redefinitions of the local multiparticle superfields in the Lorenz gauge correspond to a gauge transformation of the corresponding Berends-Giele current.
Indeed, if we define the Berends-Giele currents using (5.2) However, the identification of (5.7) as the gauge transformation relating the superfields in the different gauges is not complete. This is because the analysis of [2] was restricted to multiplicity five, whereas we know from (4.14) and (4.15)  Therefore, at multiplicity six the transformation between Lorenz and BCJ gauge follows We will now demonstrate that there is an infinite series of non-linear corrections to (5.9) which generate a finite gauge variation.

BCJ symmetries from finite gauge transformations
If H represents a generating series of Berends-Giele superfields H P (5.6), one can show that the series representation of the recursive iterations (4.19) for the gauge superpotential A m is given by Iterating the series representation of the transformation A BCJ Unsurprisingly, the expression (5.11) is nothing more than the series expansion of the finite gauge transformation given by

Conclusions and outlook
One of the main achievements of this paper is the recursive solution to the redefinition superfields H [A,B] given in (4.14). These superfields encode the non-linear gauge variations required to obtain local multiparticle superfields in the BCJ gauge. The pursuit of this formula led to improvements to and clarifications of earlier discussions given in [1,2].
In particular, in going beyond the multiplicity-five examples of [2], we found an infinite set of higher-order corrections leading to the perturbiner representation of a finite gauge transformation (5.11).
We also introduced new combinatorial maps on words and rigorously proved key statements that address some natural although not crucial questions previously left unanswered.
For instance, we found closed formulas for the gauge redefinition of K We will end this paper with some observations that could lead to further investigations.

Tree-level amplitudes using redefinition superfields
The gauge transformations responsible for the BCJ gauge require redefinitions by superfields of ghost-number zero H [A,B] determined recursively through (4.14). Customarily, after performing the redefinitions using the redefining superfields one writes down the tree amplitudes of SYM using the newly obtained superfields [7]. For example, using the compact language of the pure spinor superspace [22] one gets where V P ≡ λ α A P α is a BCJ-satisfying superfield whose explicit expression contains the redefinition superfields H ′ A,B,C in various combinations. So, in the usual formulation, we see that the superfields in the BCJ gauge are used to write down the local numerators of tree-level SYM amplitudes. These numerators have ghost number three [6] and, if one wishes to produce expressions written in terms of particle polarizations and momenta, require the standard pure spinor zero-mode rule (λγ m θ)(λγ n θ)(λγ p θ)(θγ mnp θ) = 1 [6] to integrate out the pure spinors. Somewhat surprisingly, it turns out that the redefinition superfields themselves give rise to numerators of the tree amplitudes of SYM.

Tree-level amplitudes as a map on planar binary trees
The observation above can be made more intuitive and intriguing if we frame it in terms of the b map (3.1). The SYM tree amplitudes can be viewed as a map A SYM • acting on the Lie polynomials in the expansion of (3.1). More precisely, where the map A SYM • admits two formulations In hindsight, the statement that tree-level amplitudes can be written using the definition of H A,B,C could be made when putting together the results of [7] and [2]. But now we have explicitly checked up to multiplicity nine that all the new corrections introduced in (4.15) that lead to the definition of H ′ A,B,C do not affect the final results of the amplitudes. These observations give rise to the speculation that the new prescription to compute tree level amplitudes from [23] naturally gives rise to the amplitudes written in terms of H ′ A,B,C . After all the prescription in [23] does not involve unintegrated vertices (so no pure spinors) and the end result will have to involve the double poles in the OPEs among integrated vertices. This agrees with the mechanism in the usual formulation [1] where the double poles are distributed among the simple poles using integration by parts, and it is after this step that the superfields in the numerators satisfy BCJ symmetries. This may give rise to a systematic derivation of the H ′ A,B,C redefinitions via OPE calculations and it is an interesting question left to the future. BCJ numerators were constructed for gauge theories deformed by α ′ F 3 and α ′ 2 F 4 interactions by finding appropriate α ′ corrections to the H P fields [24]. Since low-multiplicity examples show that these corrections can also be written in terms of α ′ -corrected H A,B,C in a similar manner as discussed in this paper, one may wonder whether the all-multiplicity formulas found here can be applied with minimal changes to the setup of [24].
The color-kinematics duality has given reasons to speculate about the existence of a "kinematic algebra" [25] in the same way as the color factors are related to standard Lie algebras. It will be interesting to connect this line of thought with the gauge variation approach pursued here. See [26] for a recent account on the quest for the kinematic algebra.
Finally, the Berends-Giele recursion relations have been recently derived using the technology of an L ∞ -algebra in [27]. It would be interesting to find a new derivation of the recursions for the gauge parameter H [A,B] using the methods of [27].
Acknowledgements: EB thanks Kostas Skenderis for useful discussions. CRM thanks Oliver Schlotterer for collaboration on closely related topics and for comments on the draft. CRM is supported by a University Research Fellowship from the Royal Society.

Appendix A. Some common operations on words
In this appendix we list some of the operations on words used in this paper. With the exception of the letterification introduced below, the following definitions are standard and can be found in [17].
In certain formulas such as (4.14) it is necessary to handle a word as if it were a single letter to avoid it being split by other maps. To deal with these situations we introduce a letterfication operation whereby a word Q is mapped to a letterq, Since a letter can not be deconcatenated this freezes the individual letters within Q. In the endq is restored by its original word Q. For example, suppose that the word Q = 12 has been letterified toq = 12 -as may be the case in a formula such as (4.14)-and that P = 3. Then deconcatenating QP is different than deconcatenatingqP . For example, one gets only one term Appendix B. Equations of motion for localK [P,Q] In this appendix we will write down the equations of motion satisfied by the multiparticle α is the local counterpart of ∇ α ≡ D α − A α and is defined by where C[[·, ·]] is the contact-term coproduct map on words defined in (3.5) and (C.5). To illustrate the above equations, consider ∇ (L) αÂ m [1,2] = (γ mŴ [1,2] ) α + k m 12Âα [1,2] where  where we used the first example in (C.7). Therefore the equation of motion ofÂ m [1,2] reads

C.1. Symmetries of Berends-Giele currents
We have seen on section 3.1 that b(P ) is a Lie polynomial. A standard result in the theory of free Lie algebras states that any Lie polynomial is orthogonal to non-trivial shuffles [17].
This implies that is annihilated by proper shuffles, i.e. (note K A¡B ≡ σ∈A¡B K σ ) Note that the original currents J m P defined by Berends and Giele in [8] were argued to satisfy J m A¡B = 0 in [28]. One can show that, in our conventions, J m P = A m P [7].

C.2. Deconcatenation terms in the equations of motion
The equations of motion of local multiparticle superfields (see the appendix B) contain contact-term corrections with respect to their single-particle counterparts. When expressed in terms of Berends-Giele currents, these contact terms corrections are translated to a deconcatenation structure. For example, the Berends-Giele counterpart of the local equation of motion D αÂ m [1,2] = (γ mŴ [1,2] ) α + k m 12Âα [1,2] is given by These observations can now be given a universal justification as follows. If one assigns the superfields K and S to the contact terms of a Lie polynomial [P, Q] as which demonstrates several deconcatenation formulas of this kind from a local superfield perspective. Using the contact-term map C displayed in (D.1), the simplest example applications of (C.5) read In addition, the contact terms generated with the formula (C.5) can be used to write down the BRST variations of the multiparticle unintegrated V P for arbitrary nested Lie bracketings. This generalizes the previous formula valid for the left-to-right nesting [1].
More precisely, the BRST variation can be written as For example, using (C.8) one can write down the BRST variation of V [1, [2,3]] directly, (C.10) It is worth mentioning that (3.15) shows the equivalence between (C.8) and (C.10).

Appendix D. Example applications of the C andC maps
In this appendix we display some example applications of the C andC maps acting over some simple Lie polynomials. These examples help to elucidate how the algorithms are used, and can be used to verify that the redefinition formulas arising from the general formulas match the formulas for the simplest cases that were previously known.

D.1.1. Examples of the C map
To demonstrate the (3.5) algorithm, the first few expansions generated from it are  [4,5]) .
This will be of particular use in the example discussed in section 4.3.1.

Appendix E. Freedom in defining Hs
There is considerable freedom in defining the Hs, arising from the symmetries within the  , with n ≤ |A|. One half of (4.14) will disappear under the action of the £, as where in the second sum X is not constrained to be non-empty. The final equality then just comes from the fact that H ′ A,B,C is constructed so as to satisfy generalized Jacobi identities in each of A, B, and C. Using this and (4.14) it then just follows that, if £ n • H We emphasize the typographical convention of representing a left-to-right nested bracket by its composing letters, e.g.Ĥ [[ [1,2],3],4] ≡Ĥ 1234 , even though the parent superfields do where the redefinition termsĤ P are defined so as to enforce generalized Jacobi identities upon superfields, and can be identified most easily with repeated use of (4.25) and (4.14) -(4.16).
The rank six Berends-Giele currents are found using the Berends-Giele map (5.2), and for a general superfield K is Verifying that the redefinitions (F.1) -(F.6) amount to a gauge transformation in the Berends-Giele currents means plugging them into the above, and checking that in the resulting expression the Mandelstams cancel perfectly and the formula (5.8), which has the form of a gauge transformation, is produced.
Clearly this calculation requires considerable effort, but it has been performed and the result works as it should. A more efficient alternative approach based on (3.7) of Propo-