Multiparticle SYM equations of motion and pure spinor BRST blocks

In this paper a multiparticle generalization of linearized ten-dimensional super Yang--Mills superfields is proposed. Their equations of motions are shown to take the same form as in the single-particle case, supplemented by contact terms. A recursive construction of these superfields is inspired by the iterated OPEs among massless vertex operators in the pure spinor formalism. An enlarged set of BRST-covariant pure spinor blocks is then defined in a streamlined fashion and combined to multiparticle vertex operators. The latter can be used to universally describe tree-level subdiagrams in the perturbative open and closed superstring, regardless of the loop order. The inherent symmetries of the multiparticle superfields are reproduced by structure constants of the gauge group, hinting a natural appearance of the BCJ-duality between color and kinematics in the fundamentals of super Yang--Mills theory. We present one-loop applications where known scalar cohomology objects are systematically recomputed and a novel vector cohomology particularly relevant to the closed string is constructed for arbitrary multiplicity.


Introduction
Many examples have shown that String Theory inspires a deeper understanding of scattering amplitudes in field theories, see e.g. [1,2,3,4]. The world-sheet viewpoint on pointparticle interactions offers useful guiding principles through the multitude of Feynman diagrams. For example, tree-level subdiagrams of external particles arise when insertion points of string states on the world-sheet collide. This is captured by the operator product expansion (OPE) among vertex operators.
In this work, we study this mechanism in the context of ten-dimensional super Yang-Mills (SYM) theory. Its superspace description benefits from the use of pure spinors [5,6], and this formulation directly descends from the pure spinor superstring [7]. In previous work, a family of so-called BRST building blocks was identified in the pure spinor formalism [8,9] which encompasses the superfield degrees of freedom of several external particles.
These BRST blocks were argued to represent tree-level subdiagrams and led to an elegant and manifestly supersymmetric solution for multileg tree-level amplitudes in SYM theory [8] and the full-fledged open superstring 1 [9]. As initially suggested in [11], the driving forces in these constructions were: In step (ii), we benefit from the simple form of the BRST action on kinematic degrees of freedom, based on the SYM equations of motion for the superfields [12,13]. This appears to be special to the pure spinor formalism, at least we are not aware of an analogous implementation in the Ramond-Neveu-Schwarz (RNS) [14] or Green-Schwarz (GS) [15] framework.
The tree-level setup of [8,9] only made use of the mixed OPEs between one unintegrated vertex operator V and one integrated version U . In recent one-and three-loopcalculations [16,17,18], on the other hand, it became clear that pieces of the OPE among the amplitudes. In the following, we will complete the list of such BRST-covariant OPE ingredients and introduce multiparticle versions of the integrated vertex operator.
The multiparticle vertex operators are defined in terms of multiparticle superfields of ten-dimensional SYM theory. The latter in turn are constructed recursively where the rule for adding particles is extracted from the OPE among single particle vertex operators. The BRST transformation of these vertex operators is equivalent to equations of motion for the multiparticle superfields, which take the same form as their single-particle counterparts [12,13], but are enriched by contact terms. It points to very fundamental structures of SYM theory that these combinations of single-particle fields reproduce the "elementary" equations of motions.
In more mathematical terms [19,20,21], the recursion rule fusing two multiparticle superfields to a larger representative can be viewed as a Lie bracket operation which implements the algebraic structure of tree level graphs. In particular, the aforementioned contact terms present in multiparticle equations of motion directly realize the Lie symmetries of tree subdiagrams. This carries the flavour of a kinematic algebra which might shed further light on the duality between color and kinematics [22] in ten dimensions 2 . More specifically, the Lie symmetries of multiparticle BRST blocks imply kinematic Jacobi relations within the corresponding tree subdiagrams.
The multiparticle superfields and their BRST properties turn out to guide the construction of BRST-invariant kinematic factors. Together with the tight contraints from zero-mode saturation, this allows to anticipate the structure of scattering amplitudes in both field theory and string theory. As an example, we conclude this paper with an application to one-loop amplitudes of the open and closed (type II) superstring. The pure spinor formulation of the five graviton amplitude in [17] gave an example of how vector contractions between left-and right-moving superfields can be implemented in a BRST-invariant way. The backbone of this superspace construction is a vectorial BRST cohomology element which we recursively extend to higher multiplicity. From the field theory perspective, this amounts to identifying loop momentum dependent parts of the numerators, see [25,26].
The limit of infinite string tension α ′ → 0 leads to a worldline realization of the pure spinor setup [27] (see also [28] for the RNS equivalent). It has been shown in [29] that the worldline modifications of the worldsheet vertex operators and their OPEs give rise to the same SYM tree amplitudes as previously obtained from superstring methods [8,9].
Accordingly, it would be interesting to find the worldline equivalent of the present BRST block constructions.

Ten-dimensional SYM theory
Linearized super-Yang-Mills theory in ten dimensions can be described using the superfields 3 A α (x, θ), A m (x, θ), W α (x, θ) and F mn (x, θ) satisfying [12,13] (2.1) with gauge transformations δA α = D α Ω and δA m = k m Ω for any superfield Ω. The above equations of motion imply that the superfields A m , W α and F mn can be derived from the spinor superpotential A α , The notion that the superfield A α is enough to derive the others will be used in the next section to obtain a multiparticle generalization of the above equations of motion.

BRST building blocks from vertex operators
In the pure spinor formalism the massless sector of the open superstring (i.e. the gluon multiplet) is described by the vertex operators The superfields K ∈ {A α , A m , W α , F mn } and the pure spinor ghost λ α carry conformal weight zero whereas the worldsheet fields {∂θ α , Π m , d α , N mn } have conformal weight one.
When the superfields are on-shell and the pure spinor constraint (λγ m λ) = 0 is imposed, the vertices satisfy [7] QV = 0, where z ij = z i − z j are worldsheet positions. By K(x, θ), we collectively denote any superfield containing only zero-modes of θ α and whose x dependence is entirely given by the plane wave factor 4 e k·x .
Starting with the recursive definition of fermionic ghost-number one BRST building blocks were defined in [8,9] at rank two and three. More generally, where β j = {j + 1, . . . , p} and P (β j ) is the powerset of β j . Moreover, we identify T i ≡ V i for a single-particle label i and abbreviate multiparticle momenta by k 123...p To avoid factors of i in the formulae, we define ik m ≡ k m . Fig. 1 The correspondence of cubic graphs and BRST building blocks.

Lie symmetries of BRST building blocks
The symmetries (2.10) have been denoted "Lie" because a contraction of Lie algebra structure constants satisfies the same symmetries [16] 6 ,

Lie symmetries versus BRST variations
It is crucial to notice the interplay between the BRST variations (2.9) and the Lie symmetries (2.10) of cubic tree level subdiagrams: At rank two and three, we have 13) and the BRST variation (2.9) always has the precise form to make the sums in (2.10) BRST closed. This closure even holds before the redefinitions (2.8) are performed, e.g. Q(L 12 + L 21 ) = 0 for the direct outcome of the OPE (2.7). Any such BRST closed combination is also BRST exact since its conformal weight ∼ k 2 12...p is different from zero (unless p = 1) 7 . As detailed in [8,9], this implies that BRST exact terms (such as Q(A 1 · A 2 ) = L 21 + L 12 )) can be subtracted in the definition of T 12...p given in (2.8). Therefore the Lie symmetries obeyed by T 12...p are a consequence of the underlying BRST cohomology nature of the pure spinor superspace expressions which will ultimately describe the scattering amplitudes.
However, it was a matter of trial and error to find the BRST-"ancestors" of Q-closed L 21...p1 combinations, such as (A 1 · A 2 ) in the rank-two example and more lengthy expression at rank ≤ 5 given in [9]. In the following section, we develop a constructive method to generate these BRST completions in (2.8) without any guesswork. Moreover, our current approach based on integrated vertex operators U i delays the need for redefinitions (2.8) to rank three; all the rank-two BRST blocks will automatically be antisymmetric since they follow from the simple pole of the OPE between two integrated vertices.
The BRST building blocks play a key role in the recursive BRST cohomology method to compute SYM tree-level amplitudes [8,11] and in obtaining a manifestly local representation of BCJ-satisfying [22] tree-level numerators [32]. However, their explicit superspace expressions in [9] following from the more and more cumbersome OPE computations (2.7) become lengthy for higher rank and lack an organizing principle. We will describe a recursive method in the next section to find compact representations and to completely bypass the CFT calculations beyond rank two. 7 Recall that in a topological conformal field theory Qb 0 = L 0 implies that if Qφ h = 0 and [31].
This generalizes the mapping in fig. 1 from previous work [8,9] where only one representative T B at ghost number one was given.

Pure spinor BRST blocks
In this section we will show how to recursively define multiparticle superfields . As we will see, the recursion is driven by the OPE among two single-particle vertex operators and a system of multiparticle SYM equations of motion which generalize the standard description of (2.1). Throughout this paper, upper case letters from the beginning of the Latin alphabet will represent multiparticle labels, e.g.
with color factors, the BRST blocks reproduce symmetry properties of Lie algebraic structure constants. The BCJ compatibility of the explicit tree-level numerators in [32] are based on λ α A B α satisfying this symmetry matching. As described in the mathematics literature [19,20], the associated cubic graphs shown in fig. 2 (planar binary trees in mathematical jargon) can be mapped to iterated brackets and thereby give rise to a general construction of a Lie algebra basis. More details are given in Appendix A. 8 Throughout this paper, we will distinguish BRST building blocks T B as reviewed in section 2 from BRST blocks K B ∈ {A B α , A B m , W α B , F B mn } to be constructed in this section.
The BRST variation of the multiparticle unintegrated vertex operator defined by V B ≡ λ α A B α will be shown to have the same functional form as the BRST variation (2.9) of T B , thereby constituting a new representation of such objects. BRST-invariants built from T B do not change under a global redefinition T B → V B , hence the representations are equivalent. From now on, T B from [8,9] will not be used anymore and the new representation V B will take its place because it follows from simpler principles.

Rank two
The way towards multiparticle BRST blocks is suggested by the OPE between two integrated vertex operators. This is the largest and only CFT computation relevant for this work and has been firstly performed in [33], Using the relation ∂K = ∂θ α D α K + Π m k m K for superfields K independent on ∂θ α and λ α , we can absorb the most singular piece ∼ z −k 1 ·k 2 −2

12
into total z 1 , z 2 derivatives and rewrite Note that the last line can be viewed as a multiparticle generalization of the field-strength . In the prescription for computing string amplitudes the vertex operators are integrated over the worldsheet so the total derivatives can be dropped 9 and the composite superfields in (3.5) can be picked up via One can check using (2.1) that the above superfields satisfy which is a clear generalization of the standard equations of motion (2.1) with corrections proportional to the conformal weight ∼ 1 2 (k 1 + k 2 ) 2 = (k 1 · k 2 ) of the superfields. Furthermore, the single-particle relations k m A i m = 0 and k m (γ m W i ) α = 0 imply that, In other words, the (supersymmetrized) Dirac and YM equations k i m (γ m W i ) α = 0 and k m i F i mn = 0 for single-particle superfields are modified by the same kind of contact term ∼ (k 1 · k 2 ) as the field strength relation in (3.5) and the equations of motion (3.7) to (3.10).
Defining the rank-two unintegrated vertex operator as V 12 = λ α A 12 α (3.14) 9 In string calculations this cancellation involves a subtle interplay of BRST-exact terms and total derivatives on the worldsheet, see [11] and [34] for five-and six-point examples at tree level. One manifestation is the agreement of the superfields along with ∂ 1 , ∂ 2 in (3.4) with the At the level of diagrams, this can be interpreted as grafting the trees associated with K 12 and K 3 .
analogously to V i = λ α A i α , one can show that which generalizes (2.4) by contact terms and reproduces the BRST variation of the building block T 12 of [8]. It is interesting to note that (3.16) is compatible with the standard prescription relating integrated and unintegrated vertices, U 12 = b −1 V 12 [35].
Note that all rank-two BRST blocks are antisymmetric and therefore U 12 = −U 21 .

Rank three
Since the rank-two BRST blocks obey generalized SYM equations of motion one is tempted to define the rank-three BRST blocks following a similar approach. We know from (2.1) that the standard superfields A m , W α and F mn can be obtained from the spinor superpotential A α by recursively computing covariant derivatives. We will show that the a similar approach can be used to obtain their multiparticle generalizations starting from the following ansatz for the superpotential, This is a direct generalization of the expression for A 12 α in (3.5) as obtained from the OPE of U 1 (z 1 )U 2 (z 2 ). We have now inserted two-particle data represented by A 12 α , k 12 m , A 12 m and W α 12 into the OPE-inspired recursion. Once the BRST-trivial symmetry components are subtracted fromÂ 123 α (see section 3.2.1), the definition (3.17) can be interpreted in terms of a "grafting" procedure defined for example in [21]. As illustrated in fig. 3, (3.17) amounts to adjoining a further leg to the cubic graph associated with the BRST blocks K 12 at rank two, see Appendix A for more details.
A short computation shows that the action of the covariant derivative can be written in a form similar to (3.8) and therefore can be used to defineÂ m 123 , In turn, computing the covariant derivative of (3.19) and rewriting the result in a form analogous to the standard equation of motion for A m leads to the definition of W α 123 , Computing the covariant derivative of (3.21) leads to the definition of F 123 mn , where (3.12) has been used to arrive at, And finally, The above equations give rise to a natural rank-three definition of multiparticle SYM equations of motion: The non-contact terms in (3.18), (3.20), (3.22) and (3.24) perfectly tie in with those in the two-particle equations of motion (3.7) to (3.10). Note that the contact (3.20) and (3.22). The additional contact terms of the form mn have their two-particle analogues in the second line of (3.10).

Symmetry properties at rank three
The rank-three superfields defined above are manifestly antisymmetric in the first two labels, so they satisfy £ 2 from (3.1). However, one can show using the explicit expressions above that only a subset of the rank-three superfields satisfies £ 3 , This explains the non-hatted notation for W α 123 and F 123 mn ; they are BRST blocks already. To obtain BRST blocks for the other superfields they need to be redefined in order to satisfy the symmetry £ 3 . Fortunately, the underlying system of equations of motion greatly simplifies this task.
To see this, note that since And it turns out that k 123 m can be factored out in the cyclic sum ofÂ 123 m , where Therefore the redefinitions imply that A 123 α and A 123 m are BRST blocks since, This is a significant simplification compared to the redefinition (2.8). The latter required an "inversion" of the BRST charge on £ 3 •(L 2131 +. . .) whereas (3.27) extracts the rank-three redefinition H 123 from a straightforward £ 3 operation on the known expression (3.19) for It is easy to show that F 123 mn from (3.23) can now be rewritten as a field-strength using Thus (3.23) satisfying the symmetry £ 3 • F 123 = 0 can be understood as a property inherited from A m 123 since the contact term structure of (3.30) is the same as in the equation of motion D α A 123 m from which the BRST symmetry was derived in the first place. Defining rank-three vertex operators it follows that (2.4) as well as (3.15) and (3.16) have a rank-three counterpart, It is interesting to observe that £ 3 action translates to a total derivativê whereÛ 123 is related to U 123 in the obvious way A 123 α ↔Â 123 α and A 123 m ↔Â 123 m . The total worldsheet derivative suggests that the failure of the £ 3 symmetries in (3.34) decouples from string amplitudes and their SYM limit. In view of the diagrammatic interpretation of K 123 shown in fig. 3, the vanishing of U 123 + U 231 + U 312 can be viewed as the kinematic dual of the Jacobi identity f 12a f a3b + f 23a f a1b + f 31a f a2b = 0 among color factors. This indicates that the rank three superfields K 123 of SYM carry the fingerprints of the BCJ duality between color and kinematics [22].

Rank four
The patterns from the discussions above suggest how to proceed. The following superfieldŝ manifestly satisfy the £ 2 and £ 3 symmetries of (3.1). In general, by using the fully redefined for someF 1234 mn whose form is not important at this point. Note that the rank-three superfields in the terms proportional to (k 123 · k 4 ) are the true BRST blocks and not their hatted versions.

Symmetry properties at rank four
The hatted superfields appearing in the right-hand side of (3.38) to (3.40) can be rewritten in terms of BRST blocks by using the rank three redefinitionsÂ 123 The terms containing H ijk can be manipulated to the left-hand side in order to redefine the rank-four superfields. The outcome is, After the redefinitions of (3.41) it turns out that the superfield W ′α 1234 satisfies all the Lie symmetries (3.1) up to rank four, and therefore W α 1234 ≡ W ′α 1234 is a BRST block. Since W α 1234 satisfies (3.43), it immediately follows from the contact term structure of (3.39) that (3.26) has the following rank-four analogue (3.44) Furthermore, a straightforward calculation shows that k 1234 45) and the explicit expression for H 1234 is displayed in Appendix C.
Hence, the redefined superfields obey the required BRST symmetries: and therefore define rank-four BRST blocks.
Once the expression for A m 1234 is known the superfield F 1234 mn can be written down immediately in field-strength form, A straightforward but tedious calculation then shows that its expected equation of motion indeed holds, That is why the explicit form ofF mn 1234 was not strictly needed, one can directly write its BRST-block expression at the end of the redefinition procedure.
Defining rank-four vertex operators it follows that And similarly as at rank three, it is interesting that the failure of the £ 4 symmetry to hold for the primed superfields is equivalent to a total derivative in the integrated vertex U ′1234 ). Due to the general expectation for worldsheet derivatives to decouple from string amplitudes, this is another example for the fundamental role played by Lie symmetries. More specifically, £ 4 compatibility of U 1234 is a kinematic equivalent of Jacobi identities among permutations of f 12a f a3b f b4c . Hence, also the rank four BRST blocks satisfying £ 4 • K 1234 = 0 point towards the BCJ-duality [22].

Recursive construction at general rank
Suppose that all the BRST blocks up to rank p − 1 are known together with the superfields H 12...k for 3 ≤ k ≤ p − 1 used in their construction. The following steps can be used to obtain the explicit expressions for the rank-p BRST blocks: where the set γ j = {j + 1, . . . , p − 1} contains the p − j − 1 labels between j and p and P (γ j ) is its power set. Note that they manifestly obey all the £ k symmetries up to rank k = p − 1, but not (yet) £ p .
One can check that the superfieldsK 12...p satisfy equations of motion of the form (3.60) whose right-hand side contains not only lower-rank BRST blocks but also their hatted versions, for example, However, they can be redefinedK 12...p → K ′ 12...p such that equations of motion for K ′ 12...p are written entirely in terms of BRST blocks with rank less than p. This leads to the second step: 2 . Redefine the superfields according to with the constraints H i = H ij = 0. For example, As a consequence of (3.57), the following equations will hold, , where the set β j = {j + 1, j + 2, . . ., p} contains the p − j labels to the right of j and P (β j ) denotes its power set. Note that they satisfy all the Lie symmetries up to rank p.
It is conjectured that the BRST blocks defined in the three-step procedure above will satisfy the multiparticle equations of motion, Furthermore, defining the multiparticle vertex operators as one can show using the equations of motion (3.60) that they satisfy It is interesting to note that there is an alternative definition 10 of the rank-p BRST . This is convenient since it allows to get the complete set of rank p BRST blocks using H 12...k with k ≤ p − 1.
We have explicitly constructed BRST blocks up to rank four using the steps above.
Furthermore, preliminary checks also indicate that this construction works for rank five.

Berends-Giele currents
In the 1980's, Berends and Giele introduced the concept of gluonic tree amplitudes with one off-shell leg and found a recursive construction for these so-called "currents" [36]. Physical amplitudes are easily recovered by removing the off-shell propagator (as represented by 10 In fact, this is the representation chosen in all the checks performed with a computer.  11 . As pointed out in [37], this is implemented through the inverse momentum kernel [38,39] Fig. 6 The Berends-Giele current K 1234 of (4.2) is given by the sum of the superspace expressions associated with the above five cubic graphs with one leg off-shell. The mapping between the cubic graphs and BRST blocks is introduced in section 3, fig. 2 and explained in more detail in appendix A.
where σ ∈ S p−1 , and the momentum kernel S[·|·] 1 is defined as We use the shorthands s ij = k i · k j and i ρ ≡ ρ(i), and the object θ(j ρ , k ρ ) equals 1 (zero) if the ordering of the legs j ρ , k ρ is the same (opposite) in the ordered sets ρ(2, . . . , p) and σ(2, . . . , p). In other words, it keeps track of labels which swap their relative positions in the two permutations ρ and σ. At rank r ≤ 4, for example,  [8,9] which correspond to the unintegrated multiparticle vertex as   and (4.5) compared to (3.60) suggests that the Berends-Giele basis of tree subdiagrams is particularly suitable for a systematic construction of BRST-invariants, see section 5.

Symmetries of Berends-Giele currents
Under the momentum kernel multiplication (4.1), the Lie-symmetries of the multiparticle superfields K 12...p are mapped to a different set of Berends-Giele symmetries of K 12...p , which leave the same number (p−1)! of independent components at rank p. Universality of the momentum kernel implies that any of the K 12...p shares the same symmetry properties as M 12...p discussed in [8,9], namely 13 The notation {β T } represents the set with the reversed ordering of its n β elements and ¡ denotes the shuffle product. Furthermore, the convention K ...α¡β... ≡ σ∈α¡β K ...{σ}...
has been used. The multiparticle label B in K B now carries Berends-Giele symmetries (4.6) rather than the Lie symmetries (3.1) of the associated K B .
The symmetry properties (4.6) of rank-p currents can be viewed as rank-(p+1) Kleiss-Kuijf relation [40] obeyed by Yang-Mills tree amplitudes where the last leg p + 1 is off-shell and not displayed, leaving (p − 1)! independent components. Note, however, that the offshell-ness of one leg in the diagrammatic interpretation of Berends-Giele currents obstructs an analogue of the BCJ relations [22] among Yang-Mills tree amplitudes.
On the other hand, an interesting perspective on BCJ relations is opened up when the recursions (3.54) for BRST blocks are rewritten in terms of Berends-Giele currents. This observation is presented in Appendix B, which leads to a simplified rewriting of one-loop kinematics in terms of SYM amplitudes as compared to [16].

Application to the one loop cohomology
In

Tree level SYM amplitudes
As shown in [8], tree amplitudes A YM of ten-dimensional SYM theory take an elegant form in pure spinor superspace, The pure spinor bracket . . . in (5.1) denotes a zero-mode integration prescription of schematic form λ 3 θ 5 = 1. It extracts the gluon and gluino components of the enclosed superfields [7] as has been automated in [41]. The explicit form of the SYM amplitudes in terms of polarization vectors and gaugino wavefunctions up to multiplicity eight can be downloaded from [42].
The BRST cohomology techniques that were used in [8] to cast the SYM scattering amplitudes into the form (5.1) also played a crucial role in obtaining the general solution of the n-point tree-level amplitude of massless open superstrings [9].

Scalar BRST blocks at one-loop
In (λγ m W i )(λγ n W j )F k mn in the four-point amplitude [43] and motivates the following higherpoint definitions 15 , Using the universal form of QW α B and QF mn B , one sees that the BRST variation of (5.   [16], and the representation of W A and F B given in the reference is different from the current setup.

Scalar BRST cohomology at one-loop
The definition (5.4) of building blocks M A,B,C was used in [16] to construct BRST invariants C 1|A,B,C with up to eight particles by trial and error. We will now present a recursive method to generate them for arbitrary ranks.
The results of [16] suggest that each term of the form M i M A,B,C , with i a singleparticle label, can be completed to a BRST-closed expression of the schematic form Nilpotency Q 2 = 0 implies that QM A,B,C is also BRST closed, and the form of (5.5) suggests that it can be expanded as As we will see in the following Lemma, there is a neat interplay between action of the BRST charge and the ⊗ j operation defined in (5.9).
In the first step, we have isolated the first term of QM ij{δ} = M i M j{δ} +. . . and the second step made use of F {σ} = 0 ∀ {σ} as argued above.
The following recursive definition can be checked to generate BRST closed expressions for arbitrary ranks  As detailed in Appendix B, the C 1|A,B,C boil down to linear combinations of SYM tree amplitudes [16]. Nevertheless, their component expansion up to multiplicity seven can be downloaded from [42].

Vector BRST blocks at one-loop
In the five-point closed string computation of [17]   (5.14) The notation (i 1 , i 2 | i 1 , . . ., i n ) means a sum over all possible ways of choosing two indices i 1 and i 2 out of i 1 , . . ., i n , for a total of n 2 terms. Furthermore, another type of left/rightmixing zero-mode saturation was possible which required taking Π m d α d β N np from the integrated vertex operators, leading to terms of the form A m 2 T 3,4,5 .
The key observation in [17] was that the vectorial superfield This fact played a crucial role in demonstrating BRST invariance of the closed-string fivepoint amplitude [17] because it allows the BRST variation of the terms contracting leftand right-movers to factorize and cancel the variation of the holomorphic squared terms.
To generalize this construction to higher multiplicity one defines The ghost-number-three objects F m {σ} built from k m · M · M ·,·,· and M · M m ·,·,·,· again vanish by independence of the M j{σ} such that BRST invariance follows from (5.24) and Lemma 2. In view of the four slots A, B, C, D, the bracket [. . .] on the right hand side of (5.26) contains 8−n terms where n is the number of single-particle slots.
The first non-trivial applications of (5.26) are easily checked to be BRST closed,

Conclusion and outlook
In this work, we have constructed multiparticle vertex operators U 12...p through a recursive prescription described in subsection 3.4. This generalizes and streamlines the earlier construction of BRST-covariant building blocks in [8,9]. The coefficients of conformal weight-one fields {∂θ α , Π m , d α , N mn } in U B are interpreted as multiparticle superfields , F B mn } of ten-dimensional SYM with shorthands B = 12 . . . p for external p-particle trees. Their equations of motions are shown to have the same structure as their single-particle relatives -see (3.60) versus (2.1). In addition, they are enriched by contact terms where the multiparticle label B is distributed into two smaller subsets.
These multiparticle SYM fields furnish a kinematic analogue of the structure constants f abc of the color sector, and their Lie symmetries (3.1) guarantee that the tree-level subgraphs described by K B are compatible with the BCJ duality between color and kinematics [22]. Since the BCJ duality has been observed to hold in various dimensions, it will be interesting to explore lower-dimensional setups for multiparticle equations of motion.
It is worth emphasizing that the Lie-algebraic nature of the BRST blocks is completely general and can be understood in terms of its basic SYM superfield constituents.
The particular combinations of single-particle superfields constituting their multiparticle generalizations defined in this paper are suggested by OPE computations among vertex operators in the pure spinor formalism. Moreover, they are in lines with the BRST cohomology organization of scattering amplitudes suggested in [11] and brought to fruition in [8,9,16]. Given the general Lie symmetries obeyed by the multiparticle SYM superfields and their appearance in the OPEs of vertex operators, it is therefore natural to suspect that the BCJ duality between color and kinematics might be valid at the level of external tree subdiagrams to all loop-orders [44].
In section 5, which is devoted to one-loop applications, the zero mode saturation of the minimal pure spinor formalism [43]  Since the number of left-right contractions is unbounded for multiparticle one-loop amplitudes, the need for BRST invariants extends to tensors of arbitrary rank. The construction of tensorial BRST-blocks generalizing M A,B,C and M m A,B,C,D as well as their BRST-invariant embedding into full-fledged closed string amplitudes is left for future work [45]. Moreover, it remains to clarify how these tensors are related to the gauge anomaly of open superstring amplitudes and its cancellation [46]. For all of the aforementioned building blocks, the superspace representation in terms of elementary SYM superfields is explicitly accessible from this work. So the zero mode integration prescription of the schematic form λ 3 θ 5 = 1 [7] as automated in [41] allows to derive supermultiplet components in terms of gluon polarization vectors and gaugino wave functions. The gluon components of all the scalar and vector cohomology elements up to multiplicity seven can be found on the website [42].
Finally, it is worthwhile to note that the (non-minimal) pure spinor formalism can be interpreted as a critical topological string [47]. As shown in [48], the BRST cohomology of a topological CFT is endowed with a Gerstenhaber algebra structure and it would therefore be interesting to investigate possible connections with the BRST covariance property of multiparticle vertex operators. As pointed out by in [49], the associated Gerstenhaber bracket among vertex operators is a promising starting point to relate string amplitudes of different particle content. These references motivate further study of multiparticle vertex operators in view of both mathematical structures and applications to scattering of massive string states.

Appendix A. Physics of BRST blocks versus Mathematics of cubic graphs
In this Appendix we connect the recursive construction of BRST blocks with mathematical operations on planar binary trees, see [19,20,21] and references therein. As explained in the references, a mapping between planar binary trees and iterated brackets gives rise to an explicit Lie algebra basis construction. This will be used to manifest the Lie symmetries (3.1) of the BRST blocks and emphasize their connection with cubic graphs which play a central role for the duality between color and kinematics [22].

A.1. Iterated bracket notation
The antisymmetry of a rank-two BRST block K a 1 a 2 can be made manifest with the notation K [a 1 ,a 2 ] ≡ K a 1 a 2 . In general, the defining property of a rank-p BRST block to satisfy all Lie symmetries £ k with k ≤ p motivates the following notation with iterated brackets, The virtue of this bracket structure for the duality between color and kinematics was already emphasized in [50].   Fig. 7 Examples of the mapping between cubic graphs with one leg off-shell and BRST blocks. Together with the conventions (A.1), the fact that the BRST blocks furnish an explicit representation of the "Jacobi identity of trees" of the type discussed in [22] becomes manifest.

A.2. Diagrammatic representation of BRST blocks and their recursion
In the mathematics literature, such as [19,20,21] and references therein, there is a wellknown mapping between planar binary trees 17 and iterated brackets which is used to construct an explicit Lie algebra basis [20]. Given the iterated bracket convention discussed above, this can be immediately borrowed to create a mapping between cubic graphs with one leg off-shell and BRST blocks 18 , see fig. 7

A.3. Diagrammatic construction of Berends-Giele currents
It is possible to find the explicit expressions of Berends-Giele currents K B in terms of BRST blocks K B with a diagrammatic prescription which uses the mapping discussed 17 The precise definitions can be found in [19,20]. But for our purposes, a planar binary tree is nothing more than a cubic graph with one leg off-shell. 18 This prescription was already hinted (up to an overall sign) in the diagrammatic derivation of the symmetries obeyed by the building block T B discussed in [9]. The mapping now extends to the whole class of multiparticle superfields Fig. 8 The grafting operation on trees and its corresponding mapping in terms of BRST blocks. above. This can be used as an alternative to the inverse momentum kernel formula given in (4.1).
The Berends-Giele current with multiplicity p is obtained by the sum of the expressions associated with all the p + 1 cubic graphs with one leg off-shell, whose total number is given by the Catalan number C p−1 . It is convenient to recall that the Catalan number and their corresponding mapping in terms of cubic graphs and BRST blocks leading to the expression K 1234 were depicted in fig. 6. Higher-multiplicity examples are similarly handled.

A.4. Different superfield representations versus Lie symmetries
The definition of the hatted BRST blocks at multiplicity p has an explicit antisymmetrization of the form 12 . . . p − 1 ↔ p, where p is a single-particle label. As discussed above, the resulting BRST block is represented by a iterated bracket where the second slot of the outer bracket is a single-particle label. This motivates to check the outcome of a more general hatted superfield definition featuring a multiparticle label instead of p. As the brief discussion below suggests, the result is compatible with a linear combination of the "standard" BRST blocks following from the iterated bracket notation.

Appendix B. BCJ relations and one-loop scalar cohomology elements
The scalar cohomology elements C 1|A,B,C constructed in section 5.3 were argued in [16] to be linear combinations of SYM tree-level amplitudes multiplied by quadratic polynomials of Mandelstam invariants. Momentum conservation as well as BCJ and KK relations among color ordered SYM amplitudes A YM (. . .) [22,40] lead to a multitude of different such representations for C 1|A,B,C . In the following, we provide convenient representations at all multiplicities 19 in the sense that the total number of terms is systematically reduced and inverse powers of Mandelstam invariants are avoided. As we shall see, these  19 The explicit representation given at multiplicity five in [16] fails to satisfy the above criterion of having local Mandelstam coefficients along with A YM (. . .). The six-point representation was given only indirectly as an expansion in terms of A F 4 , which represent the α ′2 corrections of the string tree-level amplitudes.
4. Keep only the terms containing Mandelstams with labels distributed as in s ab s ac , with single-particle labels a ∈ A, b ∈ B and c ∈ C. Delete terms of the form s ab s bc .
5. The result is − C 1|A,B,C .
We have explicitly checked with the data available from [42] that the algorithm above is correct for all scalar cohomology elements up to multiplicity |A| + |B| + |C| + 1 = 7.