Towards the n-point one-loop superstring amplitude I: Pure spinors and superfield kinematics

This is the first installment of a series of three papers in which we describe a method to determine higher-point correlation functions in one-loop open-superstring amplitudes from first principles. In this first part, we exploit the synergy between the cohomological features of pure-spinor superspace and the pure-spinor zero-mode integration rules of the one-loop amplitude prescription. This leads to the study of a rich variety of multiparticle superfields which are local, have covariant BRST variations, and are compatible with the particularities of the pure-spinor amplitude prescription. Several objects related to these superfields, such as their non-local counterparts and the so-called BRST pseudo-invariants, are thoroughly reviewed and put into new light. Their properties will turn out to be mysteriously connected to products of one-loop worldsheet functions in packages dubbed"generalized elliptic integrands", whose prominence will be seen in the later parts of this series of papers.

While tree-level correlators are completely determined by their singularity structure encoded in the OPEs of the vertex operators, the quest for genus-one correlators is guided by additional constraints: The homology cycles of the genus-one surface translate into a notion of double-periodicity in its complex coordinate. As we will see in part II, doubleperiodicity does not hold term-by-term in the genus-one correlators. Instead, the monodromies of individual terms cancel in similar patterns as the BRST variations of kinematic factors in pure-spinor superspace 2 . This will not only be a crucial guiding principle in constructing local representations of genus-one correlators in part III but also furnish a key incarnation of the duality between kinematics and worldsheet functions.
Apart from the parallels in their BRST-and monodromy variations, the kinematic building blocks and worldsheet functions in this work resonate in their symmetry properties under exchange of external legs. In a local form of the correlators, kinematic building blocks exhibit Lie-symmetries in several groups of labels, which translate into kinematic Jacobi relations in the tree-level subdiagrams of the field-theory limit [12,18,19]. The worldsheet functions of part II in turn are designed to vanish under shuffle products in several groups of labels which is reminiscent of the Kleiss-Kuijf relations of gauge-theory tree amplitudes [20]. Combinations of Lie-and shuffle symmetric objects are tailor-made to realize the permutation invariance of the correlators. At the same time, this symmetry structure is well known in the mathematics literature from a theorem by Ree [21] in the context of Lie polynomials [22]. Therefore we say that the local one-loop correlators of part III have a Lie-polynomial structure 3 .
We will also explore manifestly BRST-invariant but non-local representations of the correlators in part III, where the kinematic building blocks are dressed with the treelevel propagators. The resulting supersymmetric Berends-Giele currents [18,23] and their BRST-invariant combinations [24] realize shuffle symmetries on the kinematic side. Similarly, monodromy invariance of the correlators can be manifested by organizing the worldsheet functions into so-called generalized elliptic integrands (GEIs), see [17] and part II. In contrast to conventional elliptic functions, GEIs may involve loop momenta of the chiralsplitting formalism [25,26,27] that transform as well when punctures are taken along the homology cycle and cancel the monodromies of Jacobi theta functions. 2 BRST-invariance of a kinematic factor in pure-spinor superspace implies its components to be both gauge invariant and supersymmetric [2]. 3 Note that this same Lie-polynomial structure is already present in the calculation of the tree-level correlator in [9], but it remained unnoticed until now.
In part III, we will present expressions for (n ≤ 7)-point correlators in terms of BRSTinvariant superfields and GEIs such that both kinds of invariances are manifest. Given that BRST-invariant superfields and GEIs are shown in part II to obey the same kinds of relations, the role of kinematics and worldsheet functions can be freely interchanged. This generalizes the (n−3)!-term representations of tree-level correlators [9], where gauge-theory trees and Parke-Taylor factors enter on symmetric footing [15]. In analogy to these treelevel results, the one-loop amplitudes computed from such correlators are said exhibit a double-copy structure [17]. In the same way as the double-copy representations of gravitational loop integrands [28,29] hinges on the color-kinematics duality in gauge theories, the duality between kinematics and worldsheet function reveals a double-copy structure in one-loop open-superstring amplitudes.
A brief executive summary for the combined parts [1] of this series is as follows. In the first sections we will have self-contained discussions to set up preliminary notions regard- where V A and T m 1 ...m r A 1 ,...,A r+3 are local kinematic building blocks satisfying Lie symmetries while Z m 1 ...m r A 1 ,...,A r+4 are functions on the genus-one worldsheet satisfying shuffle symmetries (the unconventional notation for the permutations is explained in detail in the appendix III.A).
The need for "+ corrections" at n ≥ 7 points will be elaborated in detail invoking e.g., locality, BRST invariance, single-valuedness and several other related technical aspects introduced in the first sections. In section III.3, a multitude of representations for the correlators with n = 4, 5, 6, 7 is presented. The n = 8 correlator is proposed and, while it satisfies many non-trivial constraints, it fails to be BRST invariant by terms proportional to the holomorphic Eisenstein series G 4 . Unfortunately, this points to a certain weakness of our method since any Eisenstein series is a monodromy-invariant function with no dependence on the worldsheet punctures. ( Finally, one should not be overwhelmed by the total number of pages of this series; the wide areas of both mathematics and physics that it touches lead to several relationships and beautiful connections. The final results for the correlators are in fact quite compact.

Basic formalism
In this section we will review certain aspects of the one-loop amplitude prescription in the minimal 4 pure-spinor formalism [3]. In the later sections, this prescription will be used as a basis to formulate a general approach to assemble integrands for n-point open-superstring amplitudes at one loop from standard constraints such as single-valuedness and BRST invariance, among others.

Conventions
Throughout this work, we will use the shorthands 4 See [7] for the one-loop amplitude prescription in the non-minimal pure-spinor formalism.
Our convention for (anti)symmetrizing r vector indices does not include a factor of 1 r! and it always generates unit coefficients for each inequivalent term, even in the presence of symmetric tensors; for instance A (m B n) ≡ A m B n + A n B m as well as δ (mn k p) ≡ δ mn k p + δ mp k n + δ np k m . where the worldsheet fields on the right-hand side will be introduced below. The complicated expression (2.5) poses difficulties in a direct evaluation of (2.4) at multiplicities above four, especially when OPE contractions involving the b-ghost b(z 0 ) are considered.
For the five-point correlator, these contributions were shown to be total worldsheet derivatives with respect to z 0 and therefore could be dropped in the integrated amplitude [31]. 5 The ingredients of Z = Z J 10 P =2 Z B P 11 I=1 Y C I are explained in [3].
However, starting at six-points, the OPE contributions involving the b-ghost may introduce non-trivial functions of the punctures into the correlator although the dependence on z 0 ultimately drops out 6 . The expressions for one-loop correlators to be proposed in the later sections provide evidence that such OPE contractions can be reduced to zero-mode contributions.
When all the external states are massless, the unintegrated vertex operator V i (z i ) of conformal weight zero is given by 7 while the integrated vertices U i (z i ) are The vectorial and spinorial polarizations of the i th gluon and gluino as well as their light-like momenta k i enter through the ten-dimensional super-Yang-Mills superfields ] to be reviewed in section 3.
Integrated vertices (2.7) involve the Lorentz-current N mn of the pure-spinor ghost, and Π m = ∂x m + 1 2 (θγ m ∂θ) as well as d α are supersymmetric combinations of the matter variables in the pure-spinor worldsheet action [2]. 6 An explicit example of a related cancellation can be found in appendix B of [32] in the parity-odd sector of one-loop amplitudes in the RNS formalism. 7 For ease of notation, the dependence on z i via λ α (z), x m (z), θ α (z) as well as ∂θ α (z), Π m (z), d α (z), N mn (z) is left implicit on the right-hand side of (2.6), (2.7) and later equations.

Open-string integration domains
The vertex-operator locations or punctures need to be integrated over the torus or over the boundary components of the cylinder and the Möbius strip. For open strings, the integration over the boundaries has to match the cyclic orderings of the accompanying color factors: Each external state carries color degrees of freedom encoded in Lie-algebra 8 generators t a , and the color dependence of the amplitude enters via traces.
Given a torus of modular parameter τ in the parameterization of figure fig. 1, one can obtain the open-string worldsheets via suitable involutions [35,36], resulting in a purely imaginary modular parameter τ in case of the cylinder. The cylinder boundaries B 1,2 will be taken to be the B-cycle through the origin and the point z = 1 2 , respectively, i.e. they are separated by half the A-cycle. After using translation invariance to fix one puncture as z 1 = 0, the integration domain associated with the single trace tr( Additionally, the modular parameters need to be integrated, e.g. over τ ∈ iR + in case of the cylinder. As indicated in (2.4), the integrations over the different topologies along with their color traces are denoted by the subscript top .

Functional integration and OPEs
The worldsheet fields [∂θ α , Π m , d α , N mn ] in the integrated vertices (2.7) have conformal weight +1 and can be integrated after separating off the zero modes. Using d α (z) as an example, in a genus-g surface one gets where ω I (z) are g holomorphic one-forms normalized as A I dz ω J (z) = δ IJ when integrated around the A-cycles. The non-zero modesd α (z) in turn are characterized by A I dzd α (z) = 8 For the type-I superstring, the gauge group has to be chosen as SO (32) in order to guarantee cancellation of infinities [33] and gauge anomalies [34]. 0. In addition, when the holomorphic one-forms are integrated around the B-cycles one gets the period matrix Ω IJ = B I dz ω J (z). Note that ω 1 (z) = 1 at genus one, and the parameterization of the torus in fig. 1 involves the period matrix B dz = τ with τ ∈ C and Im τ > 0.
The non-zero modes are functionally integrated through OPE contractions, in particular at genus one we havê where the worldsheet singularities are captured by [3] (∂ ≡ ∂ ∂z ) 12) and the standard odd Jacobi theta function with q ≡ exp(2πiτ ) is given by (2.13) In the above OPEs, K(z) represents a generic superfield which depends on θ α (z) but not on any derivative ∂ n θ α (z) for n≥1, and whose x dependence is entirely contained in the plane-wave factor 9 e k·x . The functional integration of the variables x m (z, z) gives rise to the so-called Koba-Nielsen factor and will be reviewed in the next subsection. The above OPEs can be read off from their tree-level counterparts [37,2] via the substitution 1 z → g (1) (z, τ ).
It turns out that the four-point amplitude computed with the pure-spinor formalism does not involve any OPE contraction; its outcome is based purely on zero-mode integrations [3]. Therefore, the n-point amplitude admits at most n−4 OPE contractions, and it will be explained in the later sections II.7 and III.4 that this gives rise to modular forms of weight n−4 after integrating over the zero modes of Π m in (2.4).

Zero-mode integrations
The non-zero modes of the worldsheet fields are integrated out using OPE contractions (2.11), and a systematic procedure to capture the resulting tensor structures will be reviewed in section 3. Similarly, as in the tree-level correlator, we will assume that the OPE residues which feature double poles have been absorbed into single-pole residues using integration-by-parts identities [9]. The net effect of such manipulations can be accounted for via multiparticle superfields in the BCJ gauge [23] to be reviewed in subsection 3.4.
Once all non-zero modes are integrated out in that manner, the correlator (2.4) will depend only on the zero modes of the worldsheet variables [d α (z), N mn (z), Π m (z)] and [λ α (z), θ α (z)]. In this section we outline a practical procedure to integrate them using the pure-spinor zero-mode measures defined in [3]. Unlike the other worldsheet fields, the zero mode of Π m (z) is denoted with a different symbol ℓ m , called the loop momentum. Its integration will be discussed in the context of chiral splitting, see subsection 2.2.
As explained in [3], in performing the path integral of the prescription (2.4) the role of the picture-changing operators Z is to ensure that all bosonic and fermionic zero modes are absorbed correctly. In doing this it will be convenient to start integrating out d α and N mn while leaving the zero modes of Π m , λ α and θ α to be dealt with at later stages. The reason for this is that the integrations over d α and N mn can be performed, under mild assumptions, using group-theory arguments alone. This will lead to effective rules which will then be used as input on section 4 to define local kinematic building blocks based on the multiparticle SYM superfields of section 3.
To see how this comes about, the first thing to note is that in integrating out the zero modes d α and N mn using the pure-spinor measures of [3] one introduces two pure spinors λ α into the rest of the path integral. In addition, since the picture-changing operators contribute a fixed number of ten fermionic d α zero modes, an additional six zero modes must come from the b-ghost and the external vertices (2.7). We will consider the contributions from two classes of terms in the b-ghost (2.5), given by b (2) ≡ Πd 2 δ(N ) and b (4) ≡ d 4 δ ′ (N ).
When the b-ghost contributes b (4) ≡ d 4 δ ′ (N ), the external vertices must provide the remaining two d α zero modes. However, there must also be a N mn zero mode due to the factor of δ ′ (N ) [3]. Therefore, the non-vanishing configuration of zero modes from the external vertices must be proportional to d α d β N mn . Given the expression (2.7) for integrated vertex operators, this contribution is of the schematic form , see section 3.4 for multiparticle generalizations. Taking into account the introduction of two pure spinors from the pure-spinor measures, the integration from the external vertices can be summarized as where the integral sign represents the integration using the zero-mode measures of [3].
Up to an overall coefficient, this is the unique outcome because there is only one two-form irreducible representation in the tensor product of two pure spinors and two Weyl spinors 10 Hence, the net contribution from this sector to the correlator is given by a unique Lorentz- Similarly, when the b-ghost contribution comes from the term b (2) ≡ Πd 2 δ(N ), the external vertices must provide four d α zero modes, and this time there is no need for an additional N mn . Therefore, b (2) requires the external vertices to contribute d α d β d γ d δ W α 2 W β 3 W γ 4 W δ 5 , see section 3.4 for multiparticle generalizations. One can show that, up to an overall coefficient, the effective rule for integrating these zero modes is given by (2.17) The argument to see this is similar to (2.15): there is a single vector representation in the tensor product of λ 2 W 4 , [0, 0, 0, 0, 2] ⊗ [0, 0, 0, 0, 1] ∧4 = [1, 0, 0, 0, 0] + · · · .
(2.18) Therefore the unique Lorentz-invariant overall contribution from this zero-mode sector can be summarized by the following superfield combination (2,3,4,5) . (2.19) In section 4 we will see how (2.19) motivates the introduction of tensorial local building blocks that capture the kinematics of one-loop correlators.
The above rules are readily generalized for additional instances of zero modes of Π m , The analogues of (2.16) and (2.19) for the remaining terms of the b-ghost (2.5) besides b (2) and b (4) are currently out of reach. Instead, we will infer their contributions to oneloop correlators from first principles to be detailed in section III.2.4. Up to integrationby-parts equivalent terms, b (2) and b (4) provide the highest numbers of zero modes of d α , N mn and therefore start to contribute at the lowest multiplicities. Using these zeromode considerations it follows that the loop integrand for n-point open-string amplitudes (2.4) is a polynomial of degree n−4 in the loop momentum ℓ.

Pure-spinor superspace
The angle brackets . . . in the amplitude prescription (2.4) represent the complete path integral over all the worldsheet degrees of freedom. After integrating the zero modes of d α , N mn and all the other variables except for λ α and θ α , these . . . are replaced by . . .
which represent the remaining functional integration over zero modes of λ α and θ α . In integrating the variables in this order, the kinematic factors become expressions in purespinor superspace as defined in [39]. Pure-spinor superspace compactly encodes all states in the supermultiplet, and the components can be extracted using the prescription [2] (λγ m θ)(λγ n θ)(λγ p θ)(θγ mnp θ) = 2880 (2.22) for integration over zero modes of λ α and θ α . In fact, the amplitudes exhibit their most convenient form when written in pure-spinor superspace, i.e. without performing the integration in (2.22), and will be represented as such in this series of papers.

Chiral splitting of the Koba-Nielsen factor
The zero-mode integrations of the matter variables x m (z, z) or equivalently Π m (z) is performed employing the techniques of the chiral-splitting formalism of [25,26,27]. The idea is to defer the zero-mode integration for Π m within the path integral in (2.4) to the last step of the amplitude computation 11 and to interpret it as a string-theory antecedent of the loop momentum in Feynman integrals, to be denoted by In this setting, the contributions from the plane-wave factors e k·x can be reproduced from the Koba-Nielsen factor 25) and our notation I n (ℓ) for the Koba-Nielsen factor omits its dependence on the variables z j , k j , τ . Chiral splitting can be easily undone: In a closed-string context, the loop integration comprised by the path integral . . . closed over left-and right movers reproduces the more conventional and modular invariant form of the Koba-Nielsen factor, 11 More formally, chiral splitting is implemented by inserting the integrated delta function On the one hand, the combination log θ 1 (z, τ ) 2 − 2π Im τ (Im z) 2 in the exponent exhibits double periodicity under translations z → z + 1 and z → z + τ around the homology cycles of the Riemann surface. On the other hand, its second term ∼ (Im z ij ) 2 Im τ obstructs holomorphic factorization of the moduli-space integrand for closed-string amplitudes.
In an open-string context, the path integral . . . in (2.4) only comprises half the non-zero modes of x m as compared to its closed-string counterpart . . . closed in (2.26).
Accordingly, the plane-wave correlator of the open string yields half of the Koba-Nielsen Zero-mode integrations at multiplicities higher than four require generalizations of (2.26) and (2.27) and will be discussed in sections II.7 and III.4.

Definition of open-string correlators
The main challenge to be addressed in this work is to determine the dependence of the open-string amplitude (2.4) on the polarizations and momenta. The universal Koba-Nielsen factor (2.25) due to plane waves will be stripped off from The residual task is to identify kinematic factors K n (ℓ) in pure-spinor superspace that depend on the loop momentum as well as the zero modes of λ α , θ α and capture the superfield kinematics arising from the path integral. Given their origin from integrating out all the non-zero modes as well as the zero modes of d α and N mn , we will henceforth refer to these kinematic factors K n (ℓ) as correlators. They carry the key information on the amplitudes 29) and the computational methods and organizing principles for correlators to be developed in this work are tailored to reveal hidden double-copy structures.

Closed-string correlators and amplitudes
where F denotes the fundamental domain of the modular group SL 2 (Z) and the punctures z j are integrated over a torus of modular parameter τ . The reflection ℓ → −ℓ in the rightmoving correlator is due to our normalization conventions for external momenta. In situations where both K n (ℓ) andK n (−ℓ) depend on ℓ, we will see in section III.4 that quadratic and higher terms in the loop momentum introduces vector contractions between left-and right-moving superfields proportional to π/(Im τ ), see e.g. [41,42] and [43,44] in cases of maximal and reduced supersymmetry, respectively. This exemplifies how the double-copy structure of the closed-string integrand in (2.30) disappears after performing the loop integration [45]: While K n (ℓ) K n (−ℓ) is evidently a holomorphic square of open-string correlators, its loop integral over d D ℓ |I n (ℓ)| 2 no longer factorizes.
That is why chiral splitting is a convenient framework to study the double-copy properties of gravity amplitudes from a string-theory perspective.
By the appearance of open-string correlators K n in closed-string amplitudes (2.30), they need to be well-defined functions on the torus, at least after integration over ℓ. In particular -after stripping off a global factor of (Im τ ) −5 -the loop integral over |I n | 2 K n and |I n | 2 K nKn must have modular weight (n−4, 0) and (n−4, n−4), respectively.

Multiparticle SYM superfields
After introducing a convenient notation we review the recursive construction of multiparticle super-Yang-Mills superfields of [18]. A special emphasis will be given to their local representatives, as they will play an essential role in the construction of one-loop correlators in later sections.

Combinatorics on words
Let us first introduce a notation based on words and review a few associated results that will be used in the rest of this work. Good introductions to the combinatorics on words and related subjects can be found in [46,22].
In dealing with objects that contain multiple particle labels, one is faced with many permutations and associated operations acting on the labels of the participating particles.
A convenient framework to handle such things is to use the notion of words and linear maps acting on them. As such, permutations 12 of particle labels referring to the external legs are encoded in words composed from letters in the alphabet of natural numbers; {1, 2, 3, . . .}.
Words will be written in upper-case (e.g. P = 134256) and its letters in lower-case (e.g. i = 3). The length of the word P = p 1 p 2 . . . p n is the number n of its letters and is denoted by |P |. The reversal of the word P = p 1 p 2 . . . p n is the wordP = p n . . . p 2 p 1 .
The concatenation product of the words P = p 1 . . . p n and Q = q 1 . . . q m is the word P Q = p 1 . . . p n q 1 . . . q m . The empty word is denoted by ∅ and it is the unit with respect to the concatenation, i.e. P ∅ = ∅P = P . Unless otherwise noted, labeled objects are defined to be zero when their label is the empty word (such as the momentum k m ∅ ≡ 0). The deconcatenation of a word P into two words is denoted P = XY and is given by all pairs of words X, Y such that P = XY under concatenation (with obvious generalization for P = XY Z etc). For example, the deconcatenation of P = XY when P = 312 is given by the the words (X, Y ): (∅, 312), (3,12), (31,2) and (312, ∅). The deconcatenation map often occurs as a summation condition, e.g.
for arbitrary labeled objects T and F . The shuffle product of words of length n and m is defined recursively by A¡B ≡ a 1 (a 2 . . . a n ¡B) The deshuffle of P is denoted P = X¡Y and is the sum of all pairs of words X, Y such that P is a shuffle of X and Y . An efficient algorithm that generates X, Y in the deshuffle of P follows from the linear map δ(P ) = X ⊗ Y defined by For example, An alternative characterization is δ(P ) = X,Y P, X¡Y X ⊗ Y where ·, · denotes the scalar product on words defined by We will see in section 3.4.4 that the deshuffle coproduct describes the BRST variation of local multiparticle superfields just like the deconcatenation describes the BRST variation of their non-local counterparts.
As words are restricted to be permutations of the letters {1, 2, 3, . . .}, an explicit sum over permutations is often represented by a sum over words, e.g.
Furthermore, two common operations on words are given by the left-to-right bracketing map ℓ(A) and the rho map ρ(A). They are defined recursively as [22] ℓ(123 . . . n) ≡ ℓ(123 . . . n−1)n − nℓ(123 . . . n−1) , for example ℓ(123) = 123 − 132 − 231 + 321 and ρ(123) = 123 − 132 − 312 + 321. In sections 3.4 and 5.1 these maps will be used, among other applications, in the discussion of superfields in the BCJ gauge and as a practical prescription to convert non-local Berends-Giele currents into their local counterparts. There is a vast literature dealing with these and similar maps in the context of free Lie algebras, see for instance [22].
In addition, unless otherwise noted every labeled object considered in this series of papers is linear on words, e.g. T A+B ≡ T A + T B . This linearity will be frequently exploited to avoid unnecessary summation symbols, for instance To further illustrate the above points, the Kleiss-Kuijf amplitude relations [20] among Yang-Mills tree amplitudes become A P 1Qn = (−1) |P | A 1(P ¡Q)n , while the symmetry [47,23] obeyed by the Berends-Giele currents [48] is written as K A¡B = 0.
In this work we use the convention that whenever words of external-state labels in a subscript are separated through a comma (rather than a vertical bar), the parental object is understood to by symmetric under exchange of these words. For example, In addition to denoting a sum over permutations with standard notations such as more general permutations will be handled with the notation +(A, B|A, B, C, D); it instructs to sum over all ordered combinations of the words A, B taken from the set {A, B, C, D}, for example Generalizations of the form +(A 1 , . . . A n |A 1 , . . . A n+m ) for a total number n+m n of terms follow similarly.

Single-particle
A ten-dimensional covariant description of the SYM equations of motion makes use of four types of superfields seen in the vertex operators (2.6) and (2.7): the gluino and gluon po- and their field-strengths W α (x, θ), and F mn (x, θ). They satisfy the following linearized equations of motion [49,50] see (2.23) for the supersymmetric derivative D α in D = 10 superspace 13 .
We will use the collective notation for the four types of superfields describing the i th external leg in an open-string amplitude (2.29). The superfields K i will be referred to as single-particle superfields.

Two-particle
The vertex operators (2.6) and (2.7) for massless states in the pure-spinor formalism are expanded in terms of single-particle superfields. The computation of OPEs among the above vertex operators as required by the CFT amplitude prescription in the pure-spinor formalism leads to a natural definition of multiparticle superfields [18]. In contrast to the standard description of (3.14), these superfields encompass more than a single particle label. For example, after absorbing the double poles into total derivatives, the single pole in the OPE U 1 (z 1 )U 2 (z 2 ) can be written as [51] where the two-particle superfields are given by

17)
The last line involves the two-particle momentum k m 12 = k m 1 +k m 2 , see (2.2) for the definition of multiparticle momenta. By virtue of (3.14), one can check that the two-particle superfields (3.17) satisfy the following equations of motion: which augment the linearized equations of motion in (3.14) by contact terms ∼ k 1 · k 2 . Up to BRST exact terms [18], the two-particle version of the unintegrated vertex operator also appears in the OPE V 1 (z 1 )U 2 (z 2 ). Written in terms of the BRST charge Q = λ α D α , the equations of motion (3.18) become where the term (λγ m λ)A 12 m is absent by the pure-spinor constraint (2.8).

Multiparticle
Higher-point amplitudes can be elegantly described by multiparticle superfields that contain information on multiple particles at once. These superfields not only played a fundamental role in the derivation of the n-point disk amplitude in [9] but will also simplify the description of one-loop correlators.

Multiparticle vertex operators
As shown in [18], there is a multiparticle generalization of the the above superfields, that is suggested by iterated OPE calculations among vertex operators. Up to total derivatives and BRST-exact terms, the results of iterated OPEs boil down to the generalization see section 4 for the systematic construction of tensorial superfield building blocks.

Lie symmetries of multiparticle superfields
The construction of multiparticle superfields (3.21) is detailed in section 3 of [18]: Recursive equations following the structure of (3.17) are augmented by certain algorithmic redefinitions, which conspire to total derivatives or BRST-exact terms and were later identified as standard non-linear gauge transformations in [23]. More importantly, the symmetries resulting from these redefinitions are characterized by the generalized Jacobi identities or Lie symmetries (see section 8.6.7 of [22]), where ℓ(A) is the left-to-right bracketing (3.8). These are the same symmetries obtained by the following string of structure constants, and their simplest examples read By the correspondence (3.26) with contracted structure constants, the first two lines of  [18]. Accordingly, superfields that satisfy the symmetries (3.25) are said to be in the BCJ gauge [23].
Since the symmetries (3.25) are unchanged for any suffix word C, multiparticle superfields K P preserve the symmetries of their lower-multiplicity counterparts. For instance, the symmetry K 12 + K 21 = 0 (when C = ∅) carries over to K 123 + K 213 = 0 (when C = 3) and so forth for arbitrary C.

Nested bracket notation for superfields in BCJ gauge
Since the superfields K P in the BCJ gauge satisfy the same generalized Jacobi symmetries . .], p n ], it is convenient to use a notation where this is manifest. To this effect, a word P is understood as having a nested bracket structure P → ℓ(P ) and we define 14

28)
14 Note, however, that in the definition K P ≡ K ℓ(P ) one must not expand the Dynkin bracket as it would imply that K ℓ(P ) = |P |K P since K P satisfies the generalized Jacobi identities. So it is important to stress that (3.28) is a notational device. then follows from Baker's identity [22], that it is always possible to flatten brackets within local superfields, For example, Of course, one can check that the right-hand side of the last identity can also be written as −K 2341 . The above relations are equivalent to the Jacobi identities used in the context of kinematic numerators subject to the BCJ duality. They can be visualized as flattening out of the planar binary tree associated with the two branches A and B (see fig. 3). In the context of the pure-spinor superstring, the identities (3.31) have been firstly derived in [10,9] for the special case K P = V P as a consequence of the BRST algebra obeyed by V P to be reviewed below.

BRST variation of BCJ-gauge superfields
In evaluating the BRST variations of multiparticle superfields one is faced with an interesting pattern. Explicit calculations using the equations of motion of the single-particle superfields in the generalization of the definitions (3.17) to a multiparticle setup reveals the following behavior, for example [18] QV 1 = 0 , (3.32) It turns out that the deconcatenation and deshuffle maps defined in section 2 can be used to capture not only these identities for V P but also for the other multiparticle superfields in a precise manner 15 . That is, one can show that multiparticle superfields K P in the BCJ gauge satisfy the following BRST variations (k ∅ ≡ 0) [18] 15 In previous papers these BRST variations were formulated using sets and the powerset operation. Since sets are by definition unordered, this characterization was imprecise. This is rectified in (3.33) by using words together with the deshuffle map.
where P = XjY denotes the deconcatenation of the word P into the word X, a single letter j, and a word Y . Moreover, Y = R¡S denotes the deshuffle of the word Y into the words Note that when applying the formula (3.33) to QV 1 , the deconcatenation in 1 = XjY implies that at least two words among X, j and Y are empty. By defining k ∅ ≡ 0 the momentum contraction (k X · k j ) vanishes and we get the correct answer. The non-local counterparts of the subsequent building blocks in the form of supersymmetric Berends-Giele currents have been considered in [24], and we now complete that discussion by explicitly presenting their local versions. In the appendix D we display the BRST variations of every local building block relevant for correlators up to multiplicity eight. A subset of these local building blocks has been used in the construction of the four, five and six-point one-loop amplitudes of ten-dimensional SYM in [19] and the six-point string amplitudes in [32].

Scalars
The zero-mode integrations in the one-loop amplitude prescription (2.4) select certain superfields from the vertex operators according to their associated worldsheet variables.
For example, the b-ghost zero-mode contribution of the form b Using the BRST variation of the multiparticle superfields (3.33), it follows that the BRST variation of (4.1) is given by (k ∅ ≡ 0) where the notation for the sums is explained below (3.33). For example, the BRST variations of all T A,B,C up to multiplicity five are given by while the multiplicity-six and -seven BRST variations will be listed in the appendix D.
Since the right-hand side of the BRST variation (4.2) involves the same class of objects T B,C,D as seen on the left-hand side, the family of building blocks (4.1) is said to be BRST covariant. The appearance of V A on the right-hand side is inherited from the multiparticle equations of motion (3.33) and an integral part of our notion of BRST covariance.
Note that T A,B,C is symmetric in A, B, C by its definition (4.1), in agreement with the convention (3.11) adopted throughout this work.

Vectors
Vectorial building blocks can be defined from the zero-mode integrations of correlators According to the zero-mode integrations (2.16) and (2.19), the superfield expressions for the two cases above are given by where the effective rule (3.24) gives rise to The relative coefficient of these superfields is uniquely fixed as once we impose the covariant BRST transformation for example, the BRST variations of all T m A,B,C,D up to multiplicity six are given by while the examples at multiplicity seven are listed in the appendix D. In order to track the origin of BRST covariance, we first compute the BRST variations of the superfields in The linear combination in (4.6) is tailored to cancel the non-covariant term (λγ m W A )T B,C,D in which the vector index is carried by a gamma matrix and one arrives at (4.7). The remaining terms in (4.7) are compatible with the notion of BRST covariance: deshuffle sums where the vector index is carried by momenta. As firstly observed in [42,8], BRST covariant vector building blocks are crucial for BRST invariance of closed-string amplitudes that contain vector contractions between left-and right-movers.
The non-local counterparts of T A,B,C and T m A,B,C,D can be found in section 2.4 of [24].

Tensors
Local building blocks of higher tensor ranks can be defined from the zero-mode integrations of correlators that contain higher powers of loop momenta. For instance, with two loop or three loop momenta, and in general, Similarly as before, the terms in the first line originate from the Straightforward but tedious calculations using the BRST variations (3.33) of multiparticle superfields imply the rank-two variation, where the anomaly building block Y A,B,C,D,E and its generalizations will be introduced in the next subsection. Similarly, we find the following variation at rank three: In general, the BRST variation is given by

Anomalous building blocks
One-loop amplitudes of the open superstring at n ≥ 6 points are known to exhibit a gauge anomaly before combining the worldsheet topologies [34], also see [52].
The supersymmetric kinematic factor of the six-point anomaly derived with the purespinor formalism in [53] was given in terms of the pure-spinor superspace expression (λγ m W 2 )(λγ n W 3 )(λγ p W 4 )(W 5 γ mnp W 6 ). By promoting the W i to multiparticle superfields, one arrives at its higher-point extension as well as its tensorial generalization and their symmetry in B 1 , B 2 , . . . , B r+5 follows from the pure-spinor constraint and grouptheory arguments [24]. These definitions enter the BRST variations (4.14) to (4.16) of the higher-rank building blocks introduced above. By the arguments in appendix B.

Refined building blocks
In this section, we extend the system of T m 1 ...m r B 1 ,...,B r+3 by additional building blocks that preserve the key property of BRST covariance. This extension is initiated by the observation that the five-point linear combination k m 1 V 1 T m 2,3,4,5 + V 12 T 3,4,5 + (2 ↔ 3, 4, 5) is BRST closed [42]. Indeed, one can identify a local BRST generator, Although the emergence of the expression (4.22) for J 1|2,3,4,5 from the amplitude prescription is unclear, its independent study is motivated by the connection with the earlier building blocks via BRST covariance.
We emphasize that label 1 enters (4.22) on special footing, i.e it does not participate in the symmetrization of the other labels 2, 3, 4, 5. That is why the notation for this refined label 1 separates it from the rest by a vertical bar 18 . The refined building block J 1|2,3,4,5 can 18 Note that J A|B,C,D,E can be interpreted as the refinement of T m B,C,D,E and should be denoted by T A|B,C,D,E just like the other refined building blocks discussed below which share the parental notation, see e.g. (4.32). This inconsistency in the notation is a hysterical artifact. where

Higher-rank tensors
One can also define higher-rank tensors following the same logic, In doing so, the word A separated by a vertical bar is said to be refined. Straightforward but long and tedious calculations show that where the additional class of anomaly superfields Y m 3 ...m r A|B 1 ,...,B r+4 in the first line will be defined below. For example, the BRST variations of the above superfields up to multiplicity seven are given by 1|3,4,5,6,7 + (2 ↔ 3, 4, 5, 6, 7) .
The inclusion of J m 1 ...m r A|B 1 ,...,B r+4 and their generalizations into our system of ghost-number two building blocks is essential to rule out local cohomology objects: Up to and including multiplicity eight, they allow to identify a BRST generator for each local BRST-invariant at ghost number three which is constructed from the alphabet of building blocks introduced in this section, see the appendix III.B for more details.

Higher-refinement building blocks
It is possible to generalize the degree of refinement of multiparticle building blocks in a straightforward manner by contracting refined superfields with additional instances of A m B .
To define a general recursion for arbitrary tensor ranks and arbitrary degree d of refinement, it will be convenient to introduce In general, for scalars of refinement d = 2, one finds  and this will be the maximum degree of refinement present in the eight-point correlator.
For completeness, even higher degrees of refinement and tensor ranks are possible, where the objects Y m 3 ...m r A 1 ,...,A d |B 1 ,...,B d+r+3 in the first line will be defined next. The non-local counterparts of W

Anomalous building blocks
The higher-refinement generalization of the local superfields discussed above can also be applied to the anomalous building blocks. However, it turns out that already the simplest scalar building block with a double refinement can only appear starting at nine points, For example,

Trace relations
As an immediate consequence of their definition (4.24), refined scalar building blocks are related to traces of unrefined tensors [24],

Pure-spinor superspace: non-local superfields
The goal of this work is to assemble one-loop correlators in pure-spinor superspace from kinematic building blocks and their associated worldsheet functions. When comparing these two classes of ingredients, one discovers surprising parallels in their structures and relations which will be referred to as a duality between worldsheet functions and kinematics, see part II. One incarnation of this duality is based on the BRST pseudo-invariants discussed in [24] and has been pioneered in [17]. The purpose of this section is to review the BRST pseudoinvariants from the perspective of the above local building blocks. They will be related to their non-local Berends-Giele representations of [24] via the so-called Berends-Giele map.

The Berends-Giele map
Every local building block discussed in section 4 can be mapped to its non-local counterpart studied in [24]. This mapping is induced by a relation among the local superfields K P and their non-local Berends-Giele superfield K P given 19 by the Berends-Giele (BG) map: where S(A|B) i is the KLT matrix [54] (also known as the momentum kernel [55]) and Φ(A|B) i corresponds to its inverse [56], where δ A,B is equal to one if A = B and zero otherwise, see (3.6). Both matrices S and Φ are symmetric and subject to the conditions Φ(A|B) i = S(A|B) i ≡ 0 if A is not a permutation of B and they admit the following recursive forms [57,58] S(P, j|Q, j, where Φ(A|B) i ≡ φ(iA|iB). The first instances are given by, 19 For historic reasons the BG superfield associated with V P is denoted M P rather than V P .
Similarly, the BG image of the local building block T m 1 ... where S(32|32) 1 and Φ(32|32) 1 follow from S(23|23) 1 and Φ(23|23) 1 , respectively, by relabeling 2 ↔ 3. In (5.1) the notation B instructs to sum over all words B; the condition that S(A|B) i and Φ(A|B) i are zero if B is not a permutation of A leading to a finite sum.
The simplest applications of (5.1) to K P → V P are It is interesting to observe that the generalized Jacobi symmetries obeyed by the local superfields are translated to shuffle symmetries under the BG map. More explicitly, the BG superfields K P related to K P by (5.1) obey [47,23] Note that in writing (5.1) one needs to fix the first letter of the word P in both the local K P and non-local K P representatives to be the same. This can be done with The first relation follows from Baker's identity (3.29) while the second was proven in [59].
In general, applying the BG map to each individual slot in a local building block gives rise to its non-local Berends-Giele version. Therefore knowing one representation suffices to obtain the other, for example However, it is conceptually simpler (but equivalent to (5.8)) to define the BG counterpart of T A,B,C by directly using non-local multiparticle superfields in (5.1), for example [24] M A,B,C ≡ 1 3 (λγ m W A )(λγ n W B )F mn C + cyclic(A, B, C) . In addition to changing the symmetry properties within each word, the BG map also modifies the behavior under a BRST variation. The characteristic terms proportional to the momentum contraction (k X · k j ) in the BRST variation of the local superfields become a simpler deconcatenation sum, In the appendix of [18] an alternative relation between the local and the non-local superfields was given in terms of a diagrammatic map using planar binary trees. Yet one more relation between these objects will be given below in terms of the so-called S-map.
In summary, there is a multitude of perspectives on how these superfields are defined and the relations among them.

The S-map between local and non-local superfields
In this subsection we will describe the so-called S-map which relates local and non-local superfield representations. This map originally appeared in the appendix of [18] as a way to encode the BCJ relations among tree amplitudes and to rewrite scalar BRST cohomology objects in terms of super Yang-Mills trees.
After defining a weighted concatenation product ⊗ s of Berends-Giele superfields by the definition of the S-map can be written as A curious property of the S-map is that its iteration over all letters in a given word yields a translation between the Berends-Giele currents and its local counterparts in a way that preserves the bracketing structure. More precisely, see (3.31) for a discussion on the bracketing notation for local superfields in the BCJ gauge.
Note that the S-map plays a key role in deriving BCJ relations [11] of SYM tree amplitudes from the BRST cohomology [60], and that the definition (5.13) is equivalent 20 to

BRST pseudo-invariants
Following [24], let us now consider the non-local versions of the local building blocks discussed above. As mentioned in the previous section, they are denoted by M m... A,... or by the calligraphic letter of its local counterpart. The BRST variations for the simplest cases can be written as See [24] for more details and examples.

BRST invariants
The Berends-Giele currents were shown in [24] to be the natural building blocks in constructing recursion relations for BRST (pseudo-)invariants. For instance, it was shown using (5.18) that the following definitions are BRST invariant: These superfields are in the BRST cohomology and were dubbed BRST invariants in [18]. In general, a recursion relation was written down in [24] for these scalar and vector cohomology elements at arbitrary multiplicities. An alternative algorithm to generate the above combinations of Berends-Giele currents (and those of the subsequent tensorial generalizations C m 1 ...m r 1|A 1 ,...,A r+3 ) is described in appendix A.2.

BRST pseudo-invariants
The BRST variations of higher-rank tensors no longer vanish but they are proportional to superfields with an anomalous factor of Y m 1 ...m r A 1 ,...,A d |B 1 ,...,B d+r+5 in each term. Superspace expressions with a purely anomalous Q variation are referred to as BRST pseudo-invariant.
Similarly, a general recursion was written down in [24] and the first non-trivial instance is given by

Symmetries of pseudo-invariants
Following our convention for subscripts with words A 1 , A 2 , . . . separated by commas, the most general pseudo-invariant P m 1 ...m r 1|A 1 ,...,A d |B 1 ,...,B d+r+3 is separately symmetric in the refined slots A i and the unrefined slots B j but not under exchange of A i with B j . Also, the shufflesymmetries (5.6) of their leading terms propagate to the pseudo-invariants, e.g.

Trace relations
The trace relations (4.45) to (4.47) of the local building blocks straightforwardly generalize under the Berends-Giele map. Moreover, the (pseudo-)invariants inherit the trace relations of their leading term, e.g. 29) and at generic degree of refinement, These relations will play a key role for the modular properties of the correlators after integration over ℓ.

Anomaly counterparts of BRST invariants
We shall now review an interesting class of anomaly superfields ∆ ... As discussed in [24], it turns out that all the unrefined superfields ∆ 1|A,... are BRST exact after using momentum conservation, In contrast to the naive expectation from its slot structure, the component expansion of Since the left-hand side vanishes by (5.34) and each pole s −1 1j can occur in no term other than ∆ 1|j|... on the right-hand side, the respective residues must be zero.

Eight-point examples
Similarly, the eight-point topologies of ∆ 1|A|B,C,D,E,F are obtained by applying the rules

BRST cohomology identities
We have seen that the family of anomaly building blocks ∆ 1|A,... is obtained by redistributing the slots of M A M B,C,... in the Berends-Giele expansion of (pseudo-)invariants.

Conclusions
In this first part of the series of papers [1] towards the derivation of one-loop correlators in string theory, several aspects related to the description of the massless string states via superfield kinematics have been thoroughly discussed.
The whole setup starts with the standard superfields describing super-Yang-Mills states in ten-dimensions [49] contained in the massless vertex operators of the pure-spinor superstring. We then used a combination of OPEs, zero-mode integration rules and co- 22 Some of their properties were implicit in previous works (most notably in [24]) while in [19] it was realized that the one-loop amplitudes (up to six points) in field theory could be written using subsets of the definitions in this work.
We then reviewed their non-local representatives from [24] with special emphasis on the multitude of relations valid in the cohomology of the pure-spinor BRST charge. In this first part of the series, these relations represent cohomological identities among superfields.
In the sequel part II, it will be shown that these same identities are realized by a set of objects completely different in nature: functions on the genus-one worldsheet! The pursue of this unexpected connection dubbed "duality between worldsheet functions and kinematics" will lead us to a detailed study of so-called "generalized elliptic integrands" (GEIs), which were briefly introduced in [17].
Finally, in part III the numerous definitions as well as surprising relationships and identities uncovered in parts I and II will pave the way to the assembly of one-loop correlators in many different representations.

A.1.1. Scalar multi-word rhomap
Let us define a recursive two-word version of (3.9) by For example, The asymmetry ρ(A|B) = ρ(B|A) motivates the vertical bar notation (rather than a comma) for separating the two words, in accordance with the convention for building blocks. In addition, the definition (A.1) will be generalized for an arbitrary number of words using the following recursion (with ρ(A|B) ⊗ ρ ∅ ≡ ρ(A|B)) Since the words attached to ⊗ ρ end up becoming letters, the recursion eventually stops due to ρ(A|i) ≡ 0.

A.1.2. Tensorial multi-word rhomap
In order to upgrade the recursions above to a tensorial setting we modify the end point of the recursive definition (A.1) to All other definitions are kept unchanged except for having extra vector indices; e.g., and the generalization to multiple vector indices is straightforward.

A.1.3. Tensorial word-invariant maps
The multi-word generalization of the rhomap can be used to define the following word invariants, The reason for dubbing them "word invariants" will become clear shortly.

A.3. Change-of-basis identities
The word invariants also give rise to a simple algorithm to obtain various identities for the change of basis in BRST invariants. In summary, the word invariant map can be applied to any number of words (slots) and it unifies all C-like building block recursions from [24], as well as their change-of-basis identities.
As a final example of the unifying power of the word invariants, consider the mon-

B.1. BRST-closed expressions without momentum conservation
In the first series of checks, we treat all the momenta k 1 , k 2 , . . . , k n in an n-point superspace expression as independent, i.e. temporarily relax momentum conservation. An automated brute-force scan of all possible linear combinations of building blocks from section 4 using    can also be explicitly obtained using the alternative algorithm in (A.12). which manifests BRST invariance rather than locality. We can interpret (B.9) as the statement that the non-localities in the Berends-Giele expansion of k m 1 C m 1|2,3,4,5 in (5.21) are in fact spurious (which is easy to verify). Since the combinatorics of the Berends-Giele expansion of the building blocks L and C is the same (see (A.12)), the conclusion is that the ghost-number-two expression k 1 m L m 1|2,3,4,5,6 is also a local expression. Even though it was not guaranteed to be the case, computing its BRST variation leads to the manifestly local expression loc 6pt 1 . Similarly, the BRST-closed expression loc 6pt 1 can be rewritten in terms of non-local BRST pseudo-invariants Q k m 1 L m 1|2,3,4,5,6 = k m 1 ∆ m 1|2,3,4,5,6 + k m 1 k n 1 C mn 1|2,3,4,5,6 + s 12 P 1|2|3,4,5,6 + (2 ↔ 3, 4, 5, 6) , (B.10) where again the locality of the left-hand side (i.e., loc 6pt 1 ) is obscured by the representation with BRST (pseudo-)invariants on the right-hand side.
However, the same logic can be applied again: when promoting the BRST descendant on the right-hand side of (B.10) to a BRST generator via C mn...

1|A,B,C,D,E
and ∆ mn... 1|A,B,C,D,E → 0, locality follows from the equivalence of the respective expansions in terms of Berends-Giele currents. We therefore obtain the ghost-number-two terms k m 1 k n 1 L mn 1|2,3,4,5,6,7 + s 12 L 1|2|3,4,5,6,7 + (2 ↔ 3, 4, 5, 6, 7) that are guaranteed to generate a local ghost-number-three expression upon BRST variation. The fact that it exactly reproduces the unique expression for loc 7pt 1 obtained by a brute-force search demonstrates that the manifestly local BRST cohomology is empty at seven points (using the set of building blocks from section 4).
Similarly, we use the promotion C → L and ∆ → 0 in the BRST variation of the seven-point identity (see [24]  and it can be used to obtain the (tentatively unique) BRST-closed manifestly local combination at nine points.

B.2. BRST-closed expressions using momentum conservation
We shall now repeat the above analysis in presence of momentum conservation and count the number of manifestly local BRST invariants in an n-particle phase space.  At seven points, the BRST descendant in the third line of (B.14) can be promoted to a BRST generator whose locality is guaranteed by the last three lines of (B.14), 1|2|3,4,5,6,7 = −s 12 ∆ 1|2|3,4,5,6,7 in the cohomology. The six permutations of (B.16) under 2 ↔ 3, . . . , 7 are the only manifestly local BRST invariants at seven points which can be built from our alphabet of building blocks. By the availability of the BRST generator on the left-hand side of (B.16), all of them are excluded from the cohomology. The BRST generator inherits its locality from the seven-point analogue (B.15). Again, the anomalous terms on the right-hand side of (B.19) are not by themselves BRST invariant.
Hence, by the BRST generator on the left-hand side of (B.19), none of the seven local BRST invariants at eight points can belong to the cohomology. As long as we do not consider superspace combinations beyond the building blocks of section 4, the manifestly local BRST cohomology is therefore demonstrated to be empty at five to eight points.

Appendix C. Eight-point anomalous building blocks
This appendix is dedicated to the refined anomalous building blocks at eight points that enter the discussion of the eight-point correlator via (III. 3  Given that θ and fermionic wavefunctions are suppressed on the right-hand side, the superfields A m P and F mn P reduce to the bosonic Berends-Giele currents in the BCJ gauge [23]. The other topologies in (C.1) have similar expansions but were omitted and can be found as computer-readable files attached to the arXiv submission. Although it is not manifest in the form presented above, all four-channel s ijkl poles turn out to be absent 25