The Structure of n-Point One-Loop Open Superstring Amplitudes

In this article we present the worldsheet integrand for one-loop amplitudes in maximally supersymmetric superstring theory involving any number n of massless open string states. The polarization dependence is organized into the same BRST invariant kinematic combinations which also govern the leading string correction to tree level amplitudes. The dimensions of the bases for both the kinematics and the associated worldsheet integrals is found to be the unsigned Stirling number S_3^{n-1} of first kind. We explain why the same combinatorial structures govern on the one hand finite one-loop amplitudes of equal helicity states in pure Yang Mills theory and on the other hand the color tensors at quadratic alpha prime order of the color dressed tree amplitude.

With the advent of the pure spinor formalism this rewarding description was put into the context of the full superstring theory with a underlying BRST symmetry and a new kind of superspace [20].
Using the ideas of [21] for the field theory amplitudes, it was suggested in [13] and proven in [12,16] that BRST invariance together with the propagator structure of cubic diagrams are sufficient to determine tree-level amplitudes of D = 10 SYM to any multiplicity. The recursive BRST cohomology method obtained in [16] leads to compact and elegant supersymmetric answers and makes use of so-called BRST building blocks which can be regarded as superspace representatives of cubic diagrams. The field-theory techniques of [16] were subsequentely exploited to also calculate the general color-ordered open superstring tree amplitudes in [12,22]. The punchline is that the n-point string amplitudes are written as a sum of (n − 3)! field theory subamplitudes dressed by hypergeometric integrals [22].
The problem of computing one-loop amplitudes in open superstring theory has been dealt with since the 1980's, the first successful result at four-points being [23] in the NS sector and [24] in the R sector. In spite of the technical difficulties caused by the spin structure sums required by the RNS model, [25] provides progress towards higher multiplicity up to seven-points. In the context of heterotic theories, five-and six gluon amplitudes as well as their implications for effective actions were analyzed in [26]. Pure spinor techniques have been applied to one-loop scattering in [2,27,28,8], superspace results up to five-points are available from these references. As for two-loop amplitudes, after an amazing effort by D'Hoker and Phong the four-point amplitude was computed within the RNS formalism in [29] (see also [30]). Two-loop calculations using the pure spinor formalism can be found in [4,6,9].
Can this BRST line of reasoning within the pure spinor formalism be extended to loop amplitudes? With this intention in mind, in this paper we apply the technique of BRSTcovariant building blocks to address one-loop amplitudes in superstring theory. For any number of massless SYM states, we determine the BRST invariant part of their worldsheet integrand which is unaffected by the hexagon anomaly [31]. The complete kinematic factor turns out to be organized in terms of color-ordered tree-level amplitudes at order α ′2 that are dressed with worldsheet functions in a minimal basis. A beautiful harmony in the combinatorics of both ingredients arises. However, evaluating the (worldsheet-and modular) integrals is left for future work, in particular the extraction of field theory loop integrals as α ′ → 0 along the lines of [32].
Superstring theory has proven to be a fruitful laboratory to learn about hidden structures in the S matrix of its low energy field theories. The open superstring did not only inspire the color organization of gauge theory amplitudes but also provided an elegant proof for Bern-Carrasco-Johansson (BCJ) relations among color-ordered tree amplitudes [33,34], based on monodromy properties on the worldsheet. Another difficult field theory problem which found a string-inspired answer is the explicit construction of local kinematic numerators for gauge theory tree amplitudes which satisfy all the dual Jacobi identities, see [35]. After these tree-level examples of cross-fertilization between superstring and field theory amplitudes, we hope that this work helps to provide further guidelines to organize multileg one-loop amplitudes in maximally supersymmetric SYM in both ten and four dimensions 3 . Even though the low energy behaviour of the worldsheet integrals is not addressed, our result for the kinematic factor heavily constrains the form of these field theory amplitudes. In particular, the gauge invariant kinematic building blocks C 1,... to be defined later on appear to be a promising starting point to construct kinematic numerators for higher multiplicity. They could potentially generalize the crossing symmetric factor s 12 s 23 A YM (1,2,3,4)  This paper is organized as follows. In section two, we review the construction of the n-point SYM tree amplitude from first principles. We start with the massless vertex operators in terms of SYM superfields and sketch how their singularity structure give rise to BRST building blocks representing cubic subdiagrams. As we will argue, BRST invariance forces them to pair up such that color-ordered SYM amplitudes emerge. Section three sets the formal foundation for the computation of one-loop amplitudes using the minimal pure spinor formalism. It motivates the construction of a further family of BRST 3 The four-dimensional N = 4 SYM theory can be obtained by standard dimensional reduction from its ten-dimensional version with N = 1 supersymmetry [36].
building blocks which is carried out in section four. The fourth section follows a line of reasoning similar to the tree-level review -the BRST variation of the one-loop specific building blocks allows to a priori determine any BRST invariant to be expected in a one-loop computation. Then in section five, these BRST invariants are derived from an explicit conformal field theory (CFT) computation, in particular the associated worldsheet functions are determined. Section six connects the BRST invariants with α ′ corrections to tree-level amplitudes and explains why their symmetry properties agree with those of finite one-loop amplitudes in pure Yang-Mills theory. Finally, in the last section, we point out that also the color factors present at the α ′2 order of tree amplitudes align into the same combinatorial patterns. This leads to a duality between the worldsheet integrand of one-loop amplitudes and color-dressed tree amplitudes at O(α ′2 ).
To give a brief reference to the main results of this work -the final form for the n-point kinematic factor can be found in equation (5.31) whose notation is explained in subsection 5.4. Subsection 6.2 contains the general conversion rule (6.18) between the BRST invariants C 1,... and color-stripped O(α ′2 ) trees A F 4 as well as low multiplicity examples thereof. According to subsection 7.3, the representation (7.21) of the colordressed O(α ′2 ) tree manifests a duality to the one-loop kinematic factor (5.31).

Review of tree-level cohomology building blocks
In this section, we shall review the construction of tree-level amplitudes in ten-dimensional SYM, based on BRST building blocks in pure spinor superspace [13,14,16]. Although the problem at hand is of purely field theoretic nature, we shall use the vertex operators and the BRST charge of the pure spinor superstring [1] as the starting point. These ingredients suggest a pure spinor superspace representation for color-ordered tree subdiagrams with one off-shell leg. BRST invariance and the pole structure in the kinematic invariants s 12...p ≡ 1 2 (k 1 + k 2 + · · · + k p ) 2 (2.1) turn out to be sufficient in order to determine the tree-level SYM amplitude A YM n with any number n of external legs [12,16]. The compactness of the final expression At the level of the full-fledged superstring theory, the main virtue of the BRST building block representation for A YM is the possibility to identify these SYM constituents within the CFT computation of the superstring disk amplitude. The supersymmetric n-point tree amplitude in superstring theory was shown in [12,22] to decompose into a sum of (n − 3)! color-ordered field theory amplitudes, each one of them being weighted by a separate function of α ′ . The main result of the current work is a similar decomposition of one-loop supersymmetric amplitudes, based on a new family of BRST building blocks.
The equations of motion (2.5) imply that the vertex operators in (2.4) obey QV i = 0 and QU j = ∂V j . Since their ingredients V i and U j are BRST closed, superstring amplitudes (and in particular their field theory limit) should inherit this property.
The correlator (2.3) can be computed by integrating out the conformal worldsheet fields of unit weight within the U j vertex operator. This amounts to summing over all worldsheet singularities in z i → z j which the fields in question can produce. In any CFT, this information is carried by operator product expansions (OPEs), the first example being This defines a composite superfield L 21 associated with the degrees of freedom of the states with labels 1 and 2, respectively. By iterating this OPE fusion, we define a family of superfields of arbitrary rank [16] which will be referred to as OPE residues 5 . After the fields with conformal weight one have been integrated out using their OPEs, the zero modes of the pure spinor λ α and θ α are integrated using the (λ 3 θ 5 ) = 1 prescription reviewed in [12]. 5 It turns out that even if OPE contractions are firstly carried out among U i (z i )U j (z j ) and then merged with V 1 , the result is still a combination of L 2131...m1 permutations. In other words, at tree-level the OPE U i (z i )U j (z j ) does not introduce any independent composite superfields.

From OPE residues to BRST building blocks
A major shortcoming of the OPE residues L 2131...m1 is their lack of symmetry under exchange of labels 1, 2, 3, . . . , m. However, the obstructions to well-defined symmetry properties can be shown to conspire to BRST-exact terms. As a simple example, the symmetric rank-two combination is where Q = λ α D α denotes the BRST operator of the pure spinor formalism [1] and A m i is the vectorial superfield of D = 10 SYM. Using the BRST transformation properties of L 2131... , these BRST-exact admixtures have been identified in [12,16] up to rank five, and their removal leads to a redefinition of the OPE residues 6 The outcome of (2.10) is an improved family of superfields T 123...m which we call BRST building blocks. They are covariant under the action of the BRST charge, e.g.
where V 1 ≡ T 1 . The set β j = {j + 1, j + 2, . . ., k} encompasses the k − j labels to the right of j, and P (β j ) denotes its power set. In other words, Q acting on a BRST building block of higher rank yields products of two lower rank analogues together with a Mandelstam variable.
As discussed in [12], at each rank the BRST building blocks obey one new symmetry in its labels while still respecting all the lower-rank symmetries. For example, since the ranktwo building block satisfies T (12) = 0 all higher-order building blocks also obey T (12)34... = 6 We define (anti-)symmetrization of p indices to include 1 p! , e.g. L [21] = 1 2 (L 21 − L 12 ). and leaves (m − 1)! independent components at rank m. It turns out that the above symmetries are shared by color factors of nonabelian gauge theories formed by contracting structure constants f ijk of the gauge group. At lowest ranks, we have which states their total antisymmetry and Lie algebraic Jacobi identities, and similarly In general, the symmetries of a rank m building block are the same as those of a string of structure constants with m + 1 labels, where the free color index a m reflects an off-shell leg m+1 in the associated cubic diagram.
Therefore the basis of rank m building blocks being (m − 1)!-dimensional is equivalent to the well-known fact that the basis of contractions of structure constants with p free adjoint indices has dimension (p − 2)! after Jacobi identities.   1 The correspondence of tree graphs with cubic vertices and BRST building blocks.

From BRST building blocks to Berends-Giele currents
Given the dictionary between cubic tree subdiagrams and BRST building blocks, the next challenge is to combine different diagrams in order to arrive at BRST-invariant SYM am- involves all partitions of the m on-shell legs on two Berends-Giele currents which are compatible with the color ordering. The situation is depicted in Fig. 2b.

The D = 10 SYM amplitude as a pure spinor cohomology problem
Using the Berends-Giele currents reviewed in the previous subsection, a method to recursively compute the ten-dimensional SYM tree-level scattering amplitudes was developed in [16]. It was later shown in [12] that the expressions found in [16] also follow from the field theory limit of tree-level superstring amplitudes computed with the pure spinor formalism.
The method relies on finding an expression in the cohomology of the pure spinor BRST charge, i.e. which is BRST-closed but non-exact, QA YM (1, 2, . . . , n) = 0, A YM (1, 2, . . . , n) = QX n . If we additionally require this cohomology element to reproduce the kinematic poles of a color-ordered SYM subamplitude, the result is uniquely determined to be (2.20) In order to show that the right-hand side is in the BRST cohomology first note that The number of cubic diagrams in the color-ordered n-point tree amplitude is given by the Catalan number C n−2 , see [42], which satisfies the recurrence relation C p+1 = p i=0 C i C p−i with C 0 = 1. By its diagrammatic construction, M 12...j gathers C j−1 pole channels, so the number of poles in the expression (2.20) for the n-point subamplitude is given by n−3 i=0 C i C n−3−i , which is precisely the recursive definition of C n−2 . The expression (2.20) therefore contains the same number of cubic diagrams as the color-ordered n-point amplitude, and the fact that Berends-Giele currents have a notion of color ordering guarantees that the pole channels in (2.20) are precisely those of A YM (1, 2, . . . , n). The factorization properties of the expression (2.20) are depicted in fig. 3, and the reader is referred to [12] for more details.

One-loop amplitudes with the minimal pure spinor formalism
This section sketches the prescription towards one-loop amplitudes within the minimal pure spinor formalism. The main goal is to make the one-loop zero mode saturation rule (3.8) for the correlator V 1 n j=2 U j plausible instead of giving an exhaustive review. The reader is referred to [2] for the details omitted in the following discussion.
The prescription to compute n-point one-loop amplitudes for open superstrings is [2] where µ is the Beltrami differential, t is the Teichmüller parameter and b is the b-ghost whose contribution will be discussed below. The sum runs over all one-loop open string worldsheet topologies, i.e. over planar and non-planar cylinder diagrams as well as the Moebius strip, see [43]. The integration domain D top for t has to be adjusted accordingly, and the associated color factors C top are single-or double traces over Chan-Paton generators associated with the external states. Both the Chan-Paton traces and the integration region for the z j must reflect the cyclic ordering of the vertex operators on the boundaries of the genus one worldsheet. The main focus of this work is the simplification of the t integrand in (3.1), so we do not specify further details of D top and C top or comment on the interplay between the topologies.
In order to introduce the remaining elements appearing in (3.1), note that the computation of the CFT correlator at one-loop starts by separating off the zero mode of the conformal weight one variables. The role of the picture-changing operators Z B , Z J and Y C is to ensure that the zero modes of bosonic and fermionic variables are absorbed correctly, see [2]. The angle brackets . . . in (3.1) initially denote the path integral over all the worldsheet variables in the pure spinor formalism. The non-zero modes are integrated out using their OPEs as described below and we will follow a procedure where the zero modes of d α , N mn and the ghost current J are integrated out first, leaving those of λ α and θ α for a last step in the computation, e.g. after the superfield expansions of (2.6) are substituted in the expressions of various building blocks. And since general group theory arguments will be used to determine the integrals over zero modes of d α , N mn and J the precise details of the zero-mode measures of [2] will not be needed.
So unless otherwise stated, every appearance of the pure spinor angle brackets . . .
in this paper denotes the zero-mode integration of λ α and θ α only and will be taken as the definition of pure spinor superspace [20]. This integration can be performed using symmetry arguments alone and follows from the tree-level prescription (λ 3 θ 5 ) = 1 of [1].
Since this tedious process has been mostly automated in [44] we will restrict ourselves to presenting our one-loop results in compact pure spinor superspace form as in the tree-level approach of [12]. Furthermore, the correlation function of the matter variables x m (z, z) and Π m (z) is computed as in [45,46] and will receive no special treatment in the following.
The non-zero-modes are integrated out using their OPEs [45] Singularities in colliding worldsheet positions enter through the function η ij which is defined on a given Riemann surface as the derivative of the bosonic Green's function It behaves as z −1 ij as the positions approach each other but respects the periodicity properties required by a higher-genus Riemann surface. The representation in terms of Jacobi theta functions will not be needed in the following discussions, only its antisymmetry η ij = −η ji will play a fundamental role.
In the amplitude prescription (3.1), the b-ghost is a composite operator whose form is given schematically by [2,47], b = (Πd + N ∂θ + J∂θ) d δ(N ) + (w∂λ + J∂N + N ∂J + N ∂N )δ(N ) where δ ′ (x) = ∂ ∂x δ(x) is defined through integration by parts and the precise index contractions are being omitted. It will be argued in the appendix A that the zero-mode contribution from the b-ghost is unique and given by an expression of the form d 4 δ ′ (N ).
Furthermore, the result of the zero-mode integrations in this case is fixed by group theory up to an overall constant, and this is the contribution which will concern us in this paper.
We do not have a constructive proof that the b-ghost does not contribute via OPE contractions (i.e. via nonzero modes), but an indirect argument based on total symmetry of the kinematic factor will follow in subsection 7.4.
In general, the evaluation of the one-loop amplitude (3.1) involves two separate challenges summarized by the formula 7 Firstly, the computation of the kinematic factor K n in pure spinor superspace whose generic form is given by is unique to the pure spinor formalism and will be dealt with in the following sections. Its BRST invariant ingredients will be identified and related to the α ′2 terms in the expansion of the corresponding tree-level amplitudes. The expressions for the integrals over the Riemann surface are exactly like in RNS or Green-Schwarz [48] formalisms and will not play a role in this article. Extracting information on the integrals -in particular their field theory limits -will be left for future work.

The one-loop prescription for d α , N mn zero mode saturation
When the number of external states is four, the saturation of d α zero modes in is unique and determines the amplitude up to an overall coefficient [2,27]. The picture changing operators, the b-ghost and the external vertices provide ten, four and two d α zero-modes, respectively, thereby saturating all the sixteen zero modes of d α . Furthermore, as mentioned after (3.3), the terms with four d α zero modes from the b-ghost also contain factors which absorb extra zero modes of N mn , either 1, 2 or 3. For the four-point amplitude the only possibility is the absorption of one zero mode of N mn through an 7 Since the Koba Nielsen factor KN= n i=1 e ik i ·x(z i ,z i ) due to the plane wave correlator is a universal prefactor, we define the kinematic factor K n not to contain KN. Nevertheless, its presence is relevant for integration by parts relating different worldsheet functions, see subsection 5.3. overall factor of δ ′ (N ). Summing it all up, the contribution from the external vertices is proportional to 1 2 V 1 (dW 2 )(dW 3 )F 4 mn N mn + cyclic(234) (3.6) and the remaining zero mode integration is given schematically by As one can check in the expressions given in [2], Consequently, (3.6) leads to the following kinematic factor for the four-point one-loop amplitude whose BRST invariance one can easily check using the pure spinor constraint (λγ m λ) = 0 and elementary corollaries (λγ m ) α (λγ m ) β = 0 and (λγ m γ pq λ) = 0.
According to the arguments in appendix A, the replacement rule (3.8) still applies to one-loop amplitudes with n ≥ 5 legs. It passes the superspace kinematic factor built from one unintegrated and n − 1 integrated vertex operators to the tree-level zero mode prescription λ 3 θ 5 = 1: Studying the interplay of (3.8) with the non-zero modes of the conformal fields in U j is the subject of the next section. Integrating out all but three weight one fields d α d β N mn obviously requires n − 4 OPEs, and we will see that they give rise to new families of BRST building blocks.

BRST building blocks for loop amplitudes
As reviewed in section 2, tree-level BRST building blocks T 12...k are defined by a two step procedure. Its starting point have been the residues of the single poles in iterated OPEs of integrated vertex operators U (z j ) with the unintegrated one V (z 1 ). As a second step, the BRST trivial components of these residues had to be subtracted to obtain symmetry properties suitable for a diagrammatic interpretation. On the genus zero worldsheet governing tree-level amplitudes, conformal fields of weight +1 have no zero modes, so all of d α and N mn are completely integrated out in generating the residues entering BRST building blocks. However, this is no longer the case at one-loop.
As seen in the previous section, the kinematic factor at one-loop comes from the terms in the external vertices which contain two zero modes of d α and one of N mn . Hence, we have to integrate out weight one fields from the n − 1 integrated vertex operators until we are left with the combination (d) 2 N which requires a total of n − 4 OPE contractions. In doing so, one is naturally led to define the composite superfieldsJ mn 12 ,K α 12 and higher rank generalizationsJ mn 12...k ,K α 12...k as the remaining single-pole terms ∼ d α or ∼ N mn in nested OPEs of multiple integrated vertex operators: The ellipsis · · · indicates terms with Π m and ∂θ α as well as double poles in individual z ij , they do not contribute to the end result for one-loop amplitudes. Given the prescription As a rank k = 2 example, let us consider the OPE of two integrated vertices. It contains single and double poles and one can read of from the superfields contracted with d α and N mn , respectively.
The definitions in (4.1) lead to the following rank ≤ 3 expressions Expressions for the rank four building blocksK m 1234 andJ mn 1234 are available from the authors upon request. Similar to their tree-level counterparts T 12...k [12], the new composite superfields have two essential virtues: On the one hand, they have symmetry properties which reduce the independent rank k components to (k − 1)! and thereby suggest an interpretation in terms of tree-level subdiagrams with one off-shell leg. On the other hand, they possess covariant BRST variations, However, the appearance of the OPE residue L 21 in the right-hand side of QK m 123 instead of the BRST building block T 12 signals the need for a redefinition ofK m 123 analogous to the redefinitions of L 2131... toT 123... at tree-level, see subsection 2.2. In order to justify this, let us recall the following general lesson from the tree-level analysis: Quantities whose Q variation contains BRST exact constituents such as L (21) = − 1 2 Q(A 1 · A 2 ) combine to BRST trivial parts of the amplitude. It is economic to remove these terms in an early step of the computation, i.e. to study the BRST building block from now on whose BRST transformation gives rise to T 12 rather than L 21 : Also the higher rank cases K m 12...k =K m 12...k + . . . and J mn 12...k =J mn 12...k + . . . at k ≥ 4 require modification to ensure BRST building blocks T 12...k rather than the OPE residues L 21...k1 (with BRST exact components) in their Q transformation. However, in contrast to the tree-level redefinitions T 12...k = L 21...k1 + . . . , the symmetry properties of loop-specific building blocks are already present in OPE residuesK m andJ mn . For instance, we already have an antisymmetric residueK m 12 =K m [12] at rank two whereas the OPE residue L 21 has to be projected on its antisymmetric part T 12 = L 21 − L (21) . In summary, the Q variations of the BRST building blocks which will appear in loop amplitudes are given by (2.11) and or J mn ... enter with a relative minus sign (where the tree-level building block T ... is always understood to be placed on the left of K m ... and J mn ... ). The above variations generalizes as follows to rank k:

Rank three is the first instance where modifications
. ., k} encompasses the k − j labels to the right of j, and P (β j ) denotes its power set.

Unified notation for one-loop BRST building blocks
For each contraction pattern among integrated vertex OPEs, there are three kinematic factors associated with the same z i → z j singularity structure. This corresponds to the three ways of extracting the worldsheet fields d α d β N mn from three nested U j OPEs a la Note that (4.22) is completely symmetric in i, j, k and under moving the T i , T j and T k ..a p . As can be seen from the K m ... K n ... ∼ (λγ m ) α (λγ n ) β in the definition (4.22), the combination T i T j T k has ghost-number two. In combination with the unintegrated vertex V 1 (or OPE contractions thereof with U j ), we arrive at the total ghost number three, as required by the λ 3 θ 5 = 1 prescription.
In the notation (4.22), the BRST variations QK m 12...k and QJ mn 12...k can be written in a unified way as Of course, it has to be kept in mind that only expressions containing a full triplet of loop building blocks are well defined. Recall that the set β j = j + 1, j + 2, ..., n encompasses n − j labels to the right of j, and P (β j ) denotes its power set.
..c r as the kinematic factor of a box diagram. The four tree subdiagrams at the corners are identified with building blocks T and T i .

Diagrammatic interpretation of the loop building blocks
According to our discussion above, the T i ... share the symmetry properties and the structure of their Q variation (in particular the Mandelstam variables therein) with the tree-level building blocks T ... . So we also think of T i 12...k together with the s −1 12 , s −1 123 , . . . , s −1
Since the conformal weight-one fields from U i can also be contracted with the V 1 vertex, the correlator of (3.1) additionally involves tree-level building blocks T ..c r that it does not take the asymmetric role of the treelevel BRST building block T d 1 ...d s into account, i.e. the lack of (a 1 . . . a p ) ↔ (d 1 . . . d s ) symmetry. Moving the one-loop building blocks (i.e. the i, j, k superscripts) to different positions amounts to reshuffling contact terms due to the quartic gluon vertex in the SYM action between cubic graphs. For instance, the difference ( is proportional to s 12 when evaluated in components and therefore cancels the propagator present in the common diagram 8 . 8 In order to see this, consider the two terms on the right hand side of A particular motivation for the suggestive box notation comes from the low energy limit of superstring amplitudes. After dimensional reduction to four dimensions, they are supposed to reproduce amplitudes of N = 4 SYM -see e.g. [32] for a derivation of the four-point box integral in field theory from a D-dimensional superstring computation in the α ′ → 0 limit. The fact that only quadruple T ... and no triple T ... enter the superspace kinematics in the string computation reminds of the "no triangle" property of the underlying field theory [49]. In view of these matching structures in loop diagrams of SYM and kinematic constituents of string amplitudes, we found it natural to represent the central .. as a box. However, this does not claim a one-to-one correspondence between a particular superspace kinematic factor and a box coefficient in field theory. The systematic reproduction of N = 4 SYM amplitudes via α ′ → 0 limits of the present results is not addressed in this paper and left for future work instead.

Berends-Giele currents for loop amplitudes
As the next hierarchy level of building blocks we define loop-level Berends-Giele currents, covariance under Q with no additional Mandelstam factors The combinatorics of zero mode saturation implies that the end result for amplitudes always involves a sum of all the three terms on the right hand side. That is why we will always use the notation on the left hand side of (4.27) in the rest of this work.

BRST-invariant kinematics for loop amplitudes
Amplitudes computed with the pure spinor formalism give rise to superspace kinematic factors in the cohomology of the BRST operator. We have motivated K and J building blocks from their appearance in the iterated OPEs of integrated vertex operators (along with the d α and N mn worldsheet fields) and argued that their have covariant BRST variations (4.26) connecting different pole channels. Given the strong constraints which BRST invariance imposes on tree-level SYM amplitudes -see subsection 2.4 -it is natural to explore the Q cohomology using the one-loop building blocks. In this as four Berends-Giele currents (i.e. collections of tree subdiagrams guided by color-ordered tree-level amplitudes), glued together by a central quartic "box"-vertex.
subsection we will write down BRST invariants constructed from the above elements dictated by the minimal formalism. This amounts to anticipating the admissible kinematic structure in the result of the CFT computation of one-loop scattering amplitudes.
As mentioned in subsection 4.2, the one-loop prescription (3.1) containing one unintegrated vertex operator V 1 implies that one tree-level building block T 1... (combined to a Berends-Giele current M ...1... ) has to appear in these BRST invariants, in addition to three The diagrammatic interpretation of such a term follows from the fact that Berends-Giele currents represent color-ordered tree amplitudes with one off-shell leg, see fig. 5.
The difference between the two (M ... , M i ... ) assignments corresponds to a reshuffling of contact terms in the cubic subdiagrams at the corners of the box.
In the following, we shall give a list of BRST invariants built from M ..

Symmetry properties of the BRST invariants
In this subsection, we examine the symmetry properties of the BRST invariants of the previous subsection and determine the number of independent permutations (at least under linear relations with constant coefficients). In particular, we will argue that the C 1,... with label "1" in the first entry form a suitable basis. This ties in with the one-loop prescription (3.1) for string amplitudes: The special role of the unintegrated vertex V 1 implies that only C 1,... can appear in the CFT computation, and these ingredients must be able to capture any permutation C i =1,... via linear combination.
In order to see that the reduction to C 1,... is possible, first note that the invariants ..c r inherit the symmetry properties of the Berends-Giele currents for each of individual three sets of labels (a 1 , . . . , a p ), (b 1 , . . . , b q ) and (c 1 , . . . , c r ), i.e.
directly carry over to The notation for the sets α, β, σ is the usual one appearing in the Kleiss-Kuijf relation [50]. The latter implies the subcyclic property (or photon decoupling identity) However, the above symmetries do not relate C i,... to C j =i,... (with different labels i, j in the first slot). Equations of that type follow from the BRST cohomology of pure spinor superspace, i.e. from the vanishing of BRST exact terms at ghost number three, The left hand side is always organized into linear combinations of C's, let us illustrate this by examples: The four-point BRST invariant turns out to be totally symmetric, and five-point invariants can be reduced to C 1,ij,k,l = C [1,ij],k,l by means of and (c 1 . . . c r ). This is the defining property of the unsigned Stirling numbers of first kind

One-loop amplitudes in pure spinor superspace
The pure spinor BRST cohomology of building blocks will now be used to deduce the form of the n-point one-loop open superstring amplitudes. Apart from the four-and fivepoint amplitudes which were previously computed without explicit use of building blocks [2,27,28], the results for higher-points are strongly guided by their cohomology properties.
From the discussion of section 3, the n-point kinematic factor for one-loop amplitudes is given, up to OPE terms with the b-ghost, by the following correlator where the subscript ddN is a reminder that the substitution rule (3.8) must be applied.
It is easy to see that n − 4 OPE contractions among the vertex operators will have to be performed before the zero-mode combination d α d β N mn can be extracted. Throughout this section, we will immediately trade all the OPE residues L 2131...ℓ1 andK m ℓ+1...p ,J mn ℓ+1...p for the corresponding BRST building blocks T 12...ℓ and K m ℓ+1...p , J mn ℓ+1...p . Experience with the tree-level computation [12,13,14] shows that their difference can only contribute to BRST trivial kinematics and drops out through total worldsheet derivatives.
The calculation of the kinematic factor will be divided into three steps: 1. Express the correlator (5.1) in terms of BRST building blocks 2. Group these building blocks into Berends-Giele currents 3. Use integration by parts to combine different currents to BRST invariants C 1,...
Starting from six-points, we will use BRST invariance as an extra input in steps 1 and 2 to fix certain parts of the correlator: This concerns the failure of η ij η jk products to obey the partial fraction identity (z ij z jk ) −1 + cyc(ijk) = 0 from tree-level. This relation plays an important role for the basis reduction of worldsheet integrals at tree-level, see [22].
After these steps are performed the correlator (5.1) becomes a linear combination of the BRST invariants C 1,... constructed in subsection 4.4, which we can regard as the one-loop analogue of the tree-level subamplitudes A YM . Hence, up to the aforementioned partial fraction subtlety, the one-loop strategy follows the same logical step as the calculation of the n-point tree amplitude in [12]. Imposing BRST invariance from the beginning makes us blind to the hexagon anomaly in D = 10 dimensions arising from the boundary of the t integration [31], so in our method we are not able to reproduce the superspace anomaly computed in [51]. In other words, we compute the non-anomalous or BRST-invariant part of the amplitude.
Although our final result for K n won't include leg one on the same footing as all the others, we will prove its hidden total symmetry in subsection 7.4. The basis choice C 1,... for the kinematic constituents reflects the special role played by leg one entering the computation through the unintegrated V 1 vertex. New cross-connections to color structures at tree-level will be pointed out in section 7 which trivialize the outstanding symmetry proof.

Step 1: CFT correlator in terms of building blocks
Using the definitions of the building blocks, the CFT correlator (5.1) will encompass all possible combinations of building blocks allowed by its total permutation symmetry in (234 . . . n). As mentioned before, n − 4 OPE contractions must be performed before the ddN zero-modes can be extracted and leave a triplet of building blocks T i ... T j ... T k ... behind. As a trivial starting example, the four-point kinematic factor does not require any OPE and can be written down immediately using the definitions (4.6), (4.9) and (4.22) The ten possible OPEs in the five-point kinematic factor give rise to two classes of terms, depending on whether the contraction involves the unintegrated vertex or not: The resulting BRST building blocks are 5) and the validity of the replacement L 21 → T 12 follows from BRST-closedness of T i 3 T j 4 T k 5 .
Applying this kind of analysis to the six-point correlator leads to an ambiguity: We firstly find those contractions which closely resemble the tree-level procedure (up to But in addition to that, the correlator could contain terms with worldsheet functions which are invisible in the z i → z j limit since (z ij z ik ) −1 + cyc(ijk) = 0. These parts of the CFT correlator cannot be fixed on the basis of the leading OPE singularity and symmetry arguments in (23 . . . n). Instead, we will keep them undetermined for the moment and use BRST invariance in the following subsections to argue their absence in the end result. The precise way to combine permutations will be discussed in the next subsection.
Similarly, the seven-point kinematic factor receives contributions from where the seven different types of OPEs yield These six-and seven-point cases give an idea of the general pattern for the n-point correlator: The kinematic factor K n encompasses all tree-level building blocks involving the unintegrated vertex η 12 η 23 . . . η ℓ−1,ℓ T 12...ℓ , multiplied with all the possible topologies of ..n ) of the remaining n − ℓ legs where zero modes of d α d β N mn are extracted: × (η p+1,p+2 . . . η q−1,q T j p+1...q ) (η q+1,q+2 . . . η n−1,n T k q+1...n ) + permutations + η ijk (. . .) . (5.11) The next tasks to be addressed in the following subsections are to trade the BRST building blocks for Berends-Giele currents and to resolve the ambiguity about the η ijk terms.

Step 2: Berends-Giele currents
In the n-point tree amplitude computations of [12]  It has already been proven at tree-level [16] that the Berends-Giele currents are the natural objects to describe the SYM amplitudes. The identity (5.12) was the key step in identifying the n-point superstring amplitude as sum of (n − 3)! SYM amplitudes [12] dressed by hypergeometric worldsheet integrals [22]. To what extent can the tree-level identity (5.12) and its corollaries be generalized to one-loop?
In order to answer this question note that the tree-level proof of (5.12) required two assumptions: the symmetries of the building blocks and the partial fraction identities In order to prevent overcrowding in the formulae below the following shorthand notation will be used The five-point correlator (5.3) is also rather trivially converted to Berends-Giele currents M 12 = T 12 /s 12 and M i 23 = T i 23 /s 23 . The permutations generated by (5.4) and (5.5) combine to ten terms The six-point amplitude is the first instance where the identity (5.13) finds non-trivial application. Dropping the terms proportional to η ijk in lines with the BRST reasoning, the six-point topologies (5.7a) -(5.7d) give rise to At this point, we shall be more explicit about the permutations within the correlator.
As mentioned before, the correlator must be symmetric in all the legs (23 . . . n) of integrated vertices, but the last term in K 6 only contains 2 × 10 out of the 60 possible terms M i pqr X pq (X pr + X qr ) with p, q, r ∈ {2, 3, 4, 5, 6}. It turns out that by the symmetry properties of Berends-Giele currents (e.g. For these reasons, the following expression for the seven-point kinematic factor, is totally symmetric even though only those six M i σ(2345) permutations σ ∈ S 4 with fixed point σ(2) = 2 occur.
The n-point generalization of the above patterns is given by

Step 3: Integration by parts
In this step the number of one-loop worldsheet integrals will be reduced using partial integration identities. These manipulations have been crucial in the computation of the n-point disk amplitude [12,22] and had already found appearance in the string inspired rules towards field theory amplitudes [54,55]. As emphasized in the references, integration by parts allows to eliminate double derivatives of the bosonic Green function.
After this reduction is performed the kinematic factor for the one-loop amplitude becomes a sum over manifestly BRST invariant objects multiplied by n − 4 powers of X ij ; schematically, this means K n = X n−4 C 1,... .
In order to see how these partial integrations can be performed note that the worldsheet integrands at any loop order contain a universal factor proportional to the correlation function of the plane wave exponential factors, the so-called Koba-Nielsen factor The precise form of the bosonic Green's function x(z i , z i )x(z j , z j ) in terms of Jacobi theta functions is irrelevant for the analysis in the following. What matters is its appearance in the Koba-Nielsen factor and the antisymmetry of its derivative η ij = This identity still holds in presence of further η pq factors in the integrand as long as none of the p, q labels coincides with the differentiation leg i, for instance KN X 12 (X 13 + X 23 ) = KN (X 34 + X 35 + · · · + X 3n ) (X 23 + X 24 + · · · + X 2n ) The ubiquitous p k=2 k−1 m=1 X mk products in equation (5.20) for K n turn out to be maximally partial-integration-friendly. This has already been exploited in tree-level computations [12].
Once we have removed any appearance of z 1 from X ij via integration by parts (5.23), the remaining terms in the correlator will build up various BRST invariants C 1,... . This is a trivial statement in the four-point correlator (4.28), whereas the five-point kinematic factor requires X 12 = X 23 + X 24 + X 25 and (2345) permutations thereof (which is valid under integration against KN only). After eliminating the X 1j at j = 2, 3, 4, 5 in (5.16), we find the manifestly BRST-invariant expression which agrees with the expression from [28] when its component expansion is evaluated [44]. Once we eliminate z 1 from every X rs in the integrand, the remaining X n−4 polynomials are guaranteed to be minimal under (5.22) and the superfield prefactors must be BRST closed ..q,q+1...n . Let us consider the six-point amplitude to see these mechanisms in action. The first two lines in (5.17) require integration by parts in the form X 12 X 34 = X 34 (X 23 + X 24 + X 25 + X 26 ) and X 12 (X 13 + X 23 ) = (X 23 + X 24 + X 25 + X 26 )(X 34 + X 35 + X 36 ) in order to eliminate all the X 1j . The remaining two lines already involve integrands in the z 1 independent basis, and the associated kinematics receive corrections In order to make the permutations in (5.28) more precise and to compactly write down its n-point generalization, we shall introduce some notation that facilitates the bookkeeping of the S n−1 3 terms in K n .
The map Ξ p,q,r cuts a given S k permutation of (12 . . . k) into three tuples (12 . . . p), (p + 1 . . . p + q) and (p + q + 1 . . . k) of cardinality, p, q and r, respectively. Each of them is modded out by the corresponding cyclic group Z p , Z q , Z r , and in case of coinciding cardinalities (p = q or q = r or both), we divide by permutations S ν(p,q,r) of these tuples of equal size. Indeed, we can check that the number of elements in the individual (p, q, r) contributions to (5.29), reproduce the entries of table 1.
The structure of the n-point kinematic factor is described by The variables p, q are related to the cardinality of the permutations σ, π, ρ via p = |σ| + 1 and q − p = |π| and should not be confused with the summation variables in (5.29).
We shall conclude this section with a comment on the rigid s ij η ij = X ij combinations in the worldsheet integrand (5.31). The z i → z j singularities from η ij = z −1 ij + O(z ij ) in connection with the Koba Nielsen factor (5.21) give rise to kinematic poles in the corresponding Mandelstam variable, at least for some choices of the integration region. The connection between worldsheet poles and massless propagators was thoroughly explored at tree-level [22], and since the z i → z j singularities are local effects on the worldsheet regardless of its global properties, we expect the pole analysis to carry over to higher genus.
The fact that short distance singularities on the worldsheet always occur in the combination X ij = s ij η ij , any potential kinematic pole is immediately smoothed out by the Mandelstam numerator s ij . That is why the z i integrals do not introduce any poles in kinematic invariants 11 , i.e. that all massless open string propagators enter through the BRST invariants C 1,... . However, this does not rule out branch cut singularities in s ij as they are expected from the polylogarithms in field theory loop amplitudes. Systematic study of the non-analytic momentum dependence is a rewarding challenge which we leave for future work.

One-loop kinematic factors built from tree-level data
In this section, we will show that the BRST invariant constituents C 1,... of the one-loop kinematic factor K n can be expanded in terms of SYM tree amplitudes. More precisely, these kinematic building blocks for one-loop amplitudes are local linear combinations of the α ′2 correction A F 4 to color-ordered superstring tree amplitudes, defined by (6.1) The notation A F 4 is motivated by the fact that the first string correction to (6.1) at order 12 α ′2 can be attributed to a supersymmetrized F 4 operator in the low energy effective action [56], see later remarks. Comparing with the central result of [12,22] A tree (1, 2, . . . , n; α ′ ) = σ∈S n−3 for the disk amplitude, one can identify the O(α ′2 ) power of the functions F σ as the expansion coefficients of A F 4 in terms of (n − 3)! field theory subamplitudes: The first examples up to multiplicity n = 6 read This does not exclude massless poles from the modular t integration due to closed string exchange in non-planar cylinder diagrams [31]. 12 Higher dimensional operators such as D 2n F 4 and F 4+n with n ≥ 1 contribute to (6.1) at orders α ′2+n and are not reflected in C 1,... which carries the same mass dimension as A F 4 . and a O(α ′3 ) momentum expansion for the n = 7 functions F σ -i.e. the defining data for A F 4 (1, 2, 3, 4, 5, 6, 7) -can be found in the appendix of [22].
Since Cubic operators of type ∼ Tr{D 2k F 3 } can be ruled out since none of them is supersymmetrizable. That is why one-loop kinematics in maximally supersymmetric theories have no other choice than reproducing the A F 4 which have firstly been observed at tree-level.
The organization of this section proceeds as follows: We will first develop a pure spinor superspace representation for A F 4 in terms of quadruple Berends-Giele currents   6 The building block prescription for the four-and five-point A F 4 diagrams. The rule is that the Berends-Giele current with leg one is always to the left, carries no i, j, k labels and the combination of superfields must contain the same kinematic poles of the graph.

Diagrammatic expansion of tree-level α ′2 corrections
Following the ideas of [21], a method which associates pure spinor building blocks to cubic tree diagrams of SYM amplitudes in D = 10 was reviewed in section 2 on the basis of [13,14,16]. The pure spinor superfield method of [16] rests on two basic assumptions: 1. the kinematic numerator of a cubic graph can only contain BRST building blocks whose Q variation cancels one of the kinematic poles 2. the sum of the expressions associated to all cubic graphs must be in the pure spinor BRST cohomology. Now we are interested in an analogous diagrammatic method for constructing the tree-level α ′2 corrections and relating them to one-loop kinematic structures.
At n-points, A F 4 (1, 2, . . . , n) has n − 4 simultaneous poles corresponding to diagrams with n − 4 cubic vertices and one quartic vertex. Since we are using the same superspace ..c r present in one-loop BRST invariants, the box notation introduced in subsections 4.2 and 4.4 will be kept and can be identified with the tree-level quartic vertex 13 ∼ Tr{F 4 }. The unified diagrammatic language for both α ′2 13 Even though the diagrammatic rules in this subsection might suggest an association of the kinematic factors A F 4 with box integrals in the field theory limit, they will also find appearance along with pentagons and higher n-gons. The α ′ → 0 limit of the worldsheet integrals (to be analyzed in later work) will determine the kinematic coefficient of higher n-gons in terms of A F 4 -at least up to anomalous terms. corrected trees and loop-level kinematic factors emphasizes that they can be represented by the same class of subamplitudes A F 4 . As mentioned before, this can be traced back to the uniqueness of N = 1 supersymmetric dimension eight operators in D = 10.
The four-and five-point diagrams associated with the tree-level α ′2 correction are depicted in Fig. 6, together with their pure spinor superfield mapping. The expression correctly reflects the absence of poles in A F 4 (1, 2, 3, 4) and is BRST closed.
However, we could not find a superspace proof for (6.6) to agree with the A YM combination  amplitudes. Moreover, as a sufficient condition, we have explicitly checked their agreement up to n = 6 by computing the bosonic component expansions [44] and comparing with (6.3). It is highly plausible that the (well-tested) experimental rule BRST-closed objects with the same kinematic pole structure are proportional persists for n ≥ 7 legs.
The above expressions for A F 4 (1, 2, . . . , n) are not manifestly cyclic invariant in (1, 2, . . . , n) because the leg number one is treated differently. This is an artifact of the one-loop prescription from section 3 which associates only leg number one with the unintegrated vertex operator V 1 . But it can be shown that the difference to another choice of V i =1 is BRST-exact and therefore zero, . . , n, 1) = Q X n = 0, (6.12) for example 14 , ). (6.17)

KK-like identities for A F 4 and finite QCD amplitudes
We have argued in subsection 4.5 that the symmetries (4.36) of the C 1,... align them into a is invertible and we can solve it for C 1,... in terms of A F 4 permutations. We will now give an indirect argument that this is indeed the case.
Relations of type (6.27) and (6.28) have already been observed in [57] for finite one- ..c r . Reference [60] puts the idea to derive relations between box coefficients from quartic expressions in Berends-Giele currents into a more general context.

3
-dimensional basis found 15 The expression (6.31) for pure Yang Mills amplitudes A (1) n;1 was observed in [59] to agree with dimension-shifted one-loop amplitudes of N = 4 SYM in D → D + 4 dimensions. 16 The A YM representation (6.2) of A F 4 is dimension-agnostic -the functional dependence of SYM trees on gluon polarization vectors is the same in any number of dimensions, and one can use spinor helicity variables and the Parke Taylor formula [63] in the four-dimensional MHV situation.
be extra relations with Mandelstam coefficients between A F 4 that are independent under KK-like relations.
At five-points, extra identities with bilinear coefficients in Mandelstam variables reduce the A F 4 or pure Yang-Mills amplitudes A (1) n;1 to two independent ones (in agreement with the (n − 3)! basis of A YM ). Examples on the A (1) n;1 side are shown in equation (5.2) of [57], we have checked that they are also satisfied by A F 4 . However, the most compact relations we could find between five-point BRST invariants involve the C 1,... rather than with constants x ij , then the ansatz s 23 P 1 C 1,23,4,5 + s 24 P 2 C 1,24,3,5 + s 25 P 3 C 1,25,3,4 +s 34 P 4 C 1,34,2,5 + s 35 P 5 C 1,35,2,4 + s 45 P 6 C 1,45,2,3 = 0 (6.32) is sufficient to find a two-dimensional basis of BRST invariants. The ansatz (6.32) is motivated by the fact that the 1 s 23 pole in C 1,23,4,5 does not appear in any other C 1,... , so it must be cancelled by a s 23 prefactor for C 1,23,4,5 . Plugging in the polynomials P i = 5 j=1 x ij s j and solving the system of equations which follow from a component evaluation of (6.32) using [44] lead to four independent quadratic relations between C 1,... . As a result,

Harmony between color, kinematics and worldsheet integrands
In this section, we will explore the common combinatorial structures that govern on the one hand the kinematic building blocks C 1,... of one-loop amplitudes and the corresponding worldsheet integrands X ij = s ij η ij , on the other hand also the color factors from the α ′2 In the open string sector, the color-dressed tree amplitude is given by Tr T a σ(1) T a σ(2) · · · T a σ(n−1) T a n A tree (σ(1, 2, . . . , n−1), n; α ′ ) (7.1) where the summation includes all cyclically inequivalent permutations of the labels modded out by the (−1) n parity of color-stripped n-point amplitudes. The T a i denote the Chan-Paton factors 18 in the fundamental representation of the gauge group, and parity weighting is represented as ↔ Tr T a 1 T a 2 · · · T a n := Tr T a 1 T a 2 . . . T a n + (−1) n T a n T a n−1 · · · T a 1 .

(7.2)
A convenient basis for these parity weighted traces involves structure constants f abc and symmetrized traces d a 1 a 2 a 3 ...a 2n of even rank only, the latter being defined as [64], We will use shorthands f 123 ≡ f a 1 a 2 a 3 and d 12...k ≡ d a 1 a 2 ...a k for the (adjoint) color degrees of freedom.
As mentioned in [64], the explicit computation of symmetrized traces is tedious to perform by hand but it is also well-suited for a computer implementation. The first nontrivial relations are relatively compact [65,66] ↔ Tr ( but the lenghty relations for n = 6 and 7 were computed using the color package of FORM Once the color-dressed disk amplitude (7.1) is rewritten in this color basis, the subamplitude relations at various orders in α ′ impose selection rules on what kinds of tensor contribute to M tree n (α ′ ) at the order in question. Keeping the first two terms in (6.1) ∼ α ′0 , α ′2 , the KK identities [50] between A YM select those color tensors with n − 2 powers of structure constants and project out any symmetrized trace ( see [65] for the color structure at higher order in α ′ . The notation for higher multiplicity versions of (7.1) shall be lightened using (7.10) and the α ′2 correction M F 4 n will be the object of main interest in this section where we show its tight connection to the one-loop integrand (5.31).
Before looking at the color tensor structure at order α ′2 and their interplay with A F 4 symmetries, let us review the color organization at the SYM level α ′0 . At five-points, the KK relations for the field theory subamplitudes yield in agreement with the color-decomposition proposed by [53]. More generally, this reference suggests the following (n − 2)! element Kleiss-Kuijf bases f 1σ(2)a f aσ(3)b · · · f zσ(n−1)n , σ ∈ S n−2 , A YM 1, σ(2, 3, . . . , n − 1), n , σ ∈ S n−2 (7.11) for the color factors (f bcd ) n−2 and the SYM subamplitudes (using Jacobi identities for the former and KK relations for the latter). In this setting, one can reproduce the (n − 2)! color-decomposition proven in [53] M YM (3)b · · · f zσ(n−1)n A YM 1, σ(2, 3, . . . , n − 1), n , (7.12) starting from (7.1), and the cancellation of d 12...2k contributions at order α ′0 becomes manifest due to KK relations. In the remainder of this section, we will find remnants of (7.12) in M F 4 n , in particular the basis choice (7.11) for (f bcd ) n−2 color factors is pathbreaking for the organization of the color tensors d a 1 a 2 a 3 a 4 (f bcd ) n−4 relevant at α ′2 order. drop out, see the first two lines of (7.6). Secondly, the expansion A F 4 (1, 2, 3, 4, 5) = In the next subsection, we shall put these observations into a more general context. Note that d 123456 and d 12345a f a67 tensors (or more generally d a 1 ...a 6 (f bcd ) n−6 and d a 1 ...a 2k (f bcd ) n−2k at k ≥ 3) from the rank ≥ 6 traces do not contribute at O(α ′2 ) because of the KK-like amplitude relations between A F 4 .

Dual bases in color and kinematic space
We conclude from the calculations above that the BRST invariants C 1,... are natural objects to appear not only in the one-loop integrand but also in color-dressed tree-level amplitudes.
According to (6.18), they are related to subamplitudes A F 4 at order α ′2 by a change of (S n−1 In fact, arriving at the simple results (7.13), (7.14) and (7.15) for the α ′2 correction to the color-dressed amplitude crucially relies on the fact that the dimension of the basis for color factors and the kinematics matches. This fact has been exploited to choose "compatible" bases of color structures and corresponding kinematics, generalizing the treelevel correspondence (7.11) between color factors (f bcd ) n−2 and A YM in their (n − 2)! KK bases. In the SYM case, the reduction to (n − 2)! bases makes use of Jacobi identities on the color side and the KK relations for the subamplitudes.
We shall now explain why also the d a 1 a 2 a 3 a 4 (f bcd ) n−4 color factors align into a basis of S n−1 3 elements. The reduction algorithm consists of two steps: The diagram associated with the leading term of is accompanied by a f p−1 color factor which is adapted to the permutation σ ∈ S p−1 according to the tree-level rule (7.12): The three chains of f corresponding to the M i ... , M j ... and M k ... are then contracted with the x p , y q , z r indices of d 1x p y q z r , i.e. glued to the three corners of the box where leg one is not attached to. This amounts to the following rule how the dual S n−1 3 element bases for color-and kinematic factors enter M F 4 n : Permutations of C 1,a 1 ...a p ,b 1 ...b q ,c 1 ...c r for fixed sets {a 1 , a 2 , . . . , a p }, {b 1 , b 2 , . . . , b q } and {c 1 , c 2 , . . . , c r } always appear in the combination in agreement with our results (7.13), (7.14) and (7.15) for M F 4 n≤7 . This can be recognized as sum over the S n−1 3 partitions of legs 23 . . . n into three cycles, see subsection 5.4 for the associated set S n−1 3 . Using the latter notation defined in (5.29), we can compactly write the n-point color-dressed amplitude as ,y p+2 f y p+2 ,p+3,y p+3 · · · f y q−1 ,q,y q ρ f q+1,q+2,z q+2 f z q+2 ,q+3,z q+3 · · · f z n−1 ,n,z n . (7.21) As in (5.31), the numbers p and q are defined through the cardinality of the permutations to be p = |σ| + 1 and q − p = |π|.

Duality between one-loop integrands and M F 4 n
This subsection is devoted to the close relationship between M F 4 n and the one-loop kinematic factor K n . Our final expressions (5.25), (5.27), (5.28) and (5.31) for K 5 , K 6 , K 7 and K n can be obtained from the corresponding M F 4 n using a well-defined one-to-one map between d 1pqr (f bcd ) n−4 colors factors and the (X rs ) n−4 polynomials in the worldsheet integrand. The color basis choice of having leg one attached to d 1... corresponds to integrating by parts on the worldsheet such that only X rs with r, s = 1 enter the minimal form of K n . The two six-point topologies C 1,234,5,6 , C 1,23,45,6 in K 6 and M F 4 6 (given by (5.27) and (7.14), respectively) are accompanied by 24) and the C 1,2345,6,7 , C 1,234,56,7 and C 1,23,45,67 at seven-points are dressed by  (7.25) have the same symmetries in the labels 23 . . . n -the left hand side because of Jacobi identities, the right hand side due to algebraic identities such as X 23 (X 24 + X 34 ) + cyc (234) BRST symmetry T 23 [45] + T 45 [23] = 0 under the map (2.15)).
More generally, the three independent cubic subdiagrams contracted with the x p , y q , z r indices of d 1x p y q z r each correspond to a separate nested product of worldsheet functions like p k=3 Combining the three subdiagrams with the central quartic vertex, we arrive at the following dictionary between d 1pqr (f bcd ) n−4 color tensors and (X rs ) n−4 worldsheet integrands: Given the most general definition (7.26) of the double-arrow notation, the final forms (5.31) and (7.21) for K n and M F 4 n , respectively, are related by M F 4 n ←→ K n . (7.27) This map allows to construct the one-loop kinematic factor by knowledge of the corresponding color-dressed tree amplitude at order α ′2 .

Proving total symmetry of K n
In this subsection, we use the M F 4 n ↔ K n duality (7.26) to carry out the outstanding proof that K n as given by (5.31) is completely symmetric in all labels (12 . . . n).
Representing K n and M from r, s = 1 to r, s = 2. We will instead apply an indirect argument.
The mapping (7.26) between color factors and (X rs ) n−4 integrands respects not only the standard Jacobi identities (7.7) but also those relations which are required for the aforementioned change of basis: The generalized Jacobi relations (7.8) are dual to integration by parts. The simplest non-trivial example can be found at five-points, where the validity of the X 1j relation rests on integration against the Koba Nielsen factor, see subsection 5.3. At higher multiplicity, the form p k=3 k−1 m=2 X mk of the worldsheet functions is sufficiently integration-by-parts-friendly such that they still obey four term identities of type (7.8), e.g. f 12a f a3b d b456 = f 12a (d a45b f b36 +cyc(456)) ←→ X 12 (X 13 +X 23 ) = X 12 (X 34 +X 35 +X 36 ) as well as f 23a f a1b d b456 + sym(1456) = 0 ←→ X 23 (X 21 + X 24 + X 25 + X 26 + 2 ↔ 3) = 0 at six-points. Generalizations to higher multiplicity are straightforward.
Since the mapping (7.26) preserves the generalized Jacobi relations (7.8), the hidden total symmetry of M F 4 n implies that of K n . Our computation of M F 4 n started with the manifestly 1 ↔ 2 symmetric expression (7.1) summing over all cyclically inequivalent permutations, so we can be sure that the representation (7.21) is totally symmetric. Our derivation of the final result (5.31) for K n , on the other hand, started with the V 1 ↔ U 2 asymmetric prescription (3.1) and involved incomplete arguments about the absence of additional b-ghost contributions. It is quite assuring to see that (5.31) must be totally symmetric as well -if the b-ghost contributed to K n via OPE contractions, then this would probably modify its symmetry properties due to the asymmetric response of V 1 and U j≥2 , suggesting their absence.

Correspondence between color and kinematics in M F 4 n
It was argued in [21] that the symmetric role of kinematic numerators and color factors in SYM amplitudes suggests to impose dual Jacobi identities in the kinematic sector. They have been successfully applied to simplify the calculation of multiloop amplitudes in both SYM and gravity [68,69]. The BRST building blocks technique can be used to obtain local BCJ numerators at tree-level for any number of external legs [35] through the low energy limit of string amplitudes. Therefore, it seems worthwhile to search for possible BCJ generalizations at the next order in the momentum expansion of the superstring. for color-dressed SYM tree amplitudes. The sum over i encompasses all cubic diagrams with n − 3 propagators α i s −1 α i , and c i , n i denote the associated color-and kinematic structures. One rewarding property of (7.28) is the fact that gravity tree amplitudes can be immediately obtained by replacing color factors c i →ñ i by another copyñ i of the kinematic numerators n i , provided that the latter satisfy Jacobi identities dual to the color factors c i . This encouraged us to build the M F 4 n analogue (7.32) of (7.28), we regard it as the first step towards a double copy construction that could ultimately yield a gravity analogue of A F 4 amplitudes. Instead of the cubic diagrams in (7.28), the diagrams in M F 4 n are built from one totally symmetry quartic vertex and n − 4 cubic vertices.
The expansion of M F 4 n in terms of BRST invariants C 1,... takes a very compact form, but since each C 1,... encompasses several kinematic poles (i.e. diagrams of the form Fig. 4), it is not immediately obvious from (7.21) how the kinematic numerators associated to these poles combine with color factors. In section 4, we have constructed these numerators in pure spinor superspace, they are quartic expressions T d 1 ...d s T i a 1 ...a p T j b 1 ...b q T k c 1 ...c r in tree subdiagrams T ... and T i,j,k ... attached to a totally symmetric quartic vertex. As an artifact of inserting leg one via unintegrated vertex operator V 1 , each numerator obeys 1 ∈ {d 1 , d 2 . . . d s }. multiplicity + 60 terms f 12a f a3b d b456

Conclusions
In this article, we have derived BRST invariant worldsheet integrands K n for one-loop open superstring amplitudes involving any number n of massless gauge multiplets. Our main result (5.31) is expressed in terms of kinematic building blocks C 1,... which are implicitly given in terms of O(α ′2 ) tree subamplitudes via (6.18). Since we have used BRST invariance in determining the associated worldsheet functions, our setup is by construction blind to the hexagon anomaly [31]. A superspace treatment of anomalous amplitude ingredients along the lines of [51] is left for future work.
Both the superspace kinematics C 1,... and the associated worldsheet functions fall into a basis of dimension S n−1

3
, an unsigned Stirling number of first kind. The same kind of symmetries also govern the color-dressed tree amplitude M F 4 n at order α ′2 , so we point out a duality between its minimal form (7.21) in a color basis and the one-loop integrand K n given by (5.31). The link is a one-to-one dictionary (7.26) between color factors d ijkl (f bcd ) n−4 (encompassing one symmetrized four-trace and structure constants otherwise) and worldsheet functions X n−4 ij ≡ (s ij η ij ) n−4 (built from η ij = ∂ i x(z i , z i )x(z j , z j ) ) present in K n .
A detailed analysis of the S n−1 3 worldsheet integrals is left for future work. The only comment we want to make at this point is that the integrand structure closely parallels the tree-level result from [12,22]: Each z i → z j singularity in both the tree-level and the oneloop integrand is always accompanied by a corresponding Mandelstam numerator s ij , i.e.
we have s ij η ij = s ij /z ij + O(z ij ). This guarantees that the integration does not introduce any poles in kinematic invariants, i.e. that the full propagator structure due to open string exchange is captured by the C 1,... . On the other hand, loop amplitudes additionally involve non-analytic momentum dependencies, so the main challenge in further studying the worldsheet integrals is to identify the polylogarithms that arise in both leading and subleading orders in α ′ . However, the following argument supports that the zero-mode b-ghost contribution at one-loop is unique and given by d 4 δ ′ (N ). In order to see this note that the zero-mode contribution of the picture changing operators is fixed and given by (d) 10 (λ) 10 δ 10 (N )δ(J)(θ) 11 δ 11 (λ), which is responsible among other things for absorbing all 11 bosonic zero-modes of w α [2]. Now assume that the b-ghost zero-mode contribution contains (d) n δ m (N ) and note that performing the zero-mode integral of uniform type under integration by parts. Therefore the zero-mode contribution from the b-ghost is unique and given by (d) 4 δ ′ (N ). In this paper we studied the cohomology properties of precisely this class of terms in order to anticipate its appearance in the final expression for the superspace kinematic factors.
When the b-ghost is allowed to contribute non-zero modes the number of possibilities increases, but only those which also contain either 0 or 4 zero modes of d α can have a nonzero impact on the amplitude. As argued in [28], terms involving only one OPE contraction of the b-ghost vanish because they are proportional to a derivative with respect to the position z 0 of the b-ghost insertion. Since z 0 appears nowhere else in the correlation function, those terms are total derivatives which integrate to zero due to the doubling trick.
Having excluded single OPEs with the b ghost, it follows that the five-point amplitude gets no contribution at all from b ghost OPEs [28], but from six-points onwards these terms are not excluded. For example, the b-ghost term (d) 4  Tr(T 1 T 2 T 3 T 4 T 5 T 6 ) + Tr(T 6 T 5 T 4 T 3 T 2 T 1 ) = 2d 123456