Local BCJ numerators for ten-dimensional SYM at one loop

We obtain local numerators satisfying the BCJ color-kinematics duality at one loop for super-Yang-Mills theory in ten dimensions. This is done explicitly for six points via the field-theory limit of the genus-one open superstring correlators for different color orderings, in an analogous manner to an earlier derivation of local BCJ-satisfying numerators at tree level from disk correlators. These results solve an outstanding puzzle from a previous analysis where the six-point numerators did not satisfy the color-kinematics duality.

The multiparticle superfields and pure spinor one-loop building blocks lead to intuitive mappings between one-loop cubic graphs and pure spinor superspace expressions encoding the polarization dependence of ten-dimensional supersymmetric Yang-Mills states [1].

Description of the problem and its solution
This paper aims to answer a question left over from the pure spinor construction of oneloop integrands of super-Yang-Mills (SYM) using locality and BRST invariance [1]. Can one find a set of local and supersymmetric numerators for ten-dimensional SYM one-loop integrands at six points satisfying the Bern-Carrasco-Johansson (BCJ) 1 color-kinematics duality? We will see below that the answer is yes, and we will also outline the solution for seven-point integrands.
The one-loop integrands of SYM in ten dimensions for five and six points were constructed in [1], where it was shown that the numerators for the five-point amplitude satisfied the color-kinematics duality while those at six points did not. The proposal of [1] was based on two main ingredients: locality and BRST invariance. Using the multiparticle superfields in pure spinor superspace developed in [3], these requirements together with a basic understanding of the zero-mode saturation rules of the pure spinor formalism [4,5] led to intuitive rules mapping one-loop cubic graphs to superspace numerators, see fig. 1.
By assembling the numerators of the cubic graphs for all p-gons of a n-point amplitude such that their sum is in the pure spinor BRST cohomology (up to anomalous terms of the form discussed in [6,7]), the amplitudes of the color-ordered five and six-point amplitudes for the canonical color ordering were constructed. The six-point integrand was later successfully used in [8], passing some consistency checks. 1 A brief review of the BCJ color-kinematics duality sufficient for our purposes will be given below in section 3.1 but a much more in-depth review is contained in [2].

Genus-one open superstring correlators in pure spinor superspace
In this paper we will also use the same formalism of multiparticle superfields in pure spinor superspace to present local representations of the five-, six-and seven-point amplitudes that do obey the color-kinematics duality. Since we are using the same superfield language, it is therefore important to highlight the differences with respect to the previous analysis of [1].
The difference stems from the knowledge of the open-string one-loop correlators recently obtained in [9][10][11] up to seven points. They are given by The open string amplitudes for supersymmetric states are obtained from these correlators after integration over the vertex insertion points, over the loop momentum, and over the modulus τ of the genus-one Riemann surfaces where I n (ℓ) denotes the Koba-Nielsen factor, D top denotes an ordered region of integration over the insertion points z i , and C top denotes a group-theory factor which depends on the topology of the genus-one surface (cylinder, Möbius strip or non-planar cylinder) [12]. For simplicity we will consider only the planar cylinder topology in the following. For more details on this setup, see section 2 of [9].
To gain intuition why the one-loop open-string correlators lead to a representation of one-loop SYM numerators that satisfy the color-kinematics duality it will be illustrative to review the quest for local BCJ-satisfying ten-dimensional supersymmetric numerators at tree level, solved in pure spinor superspace in [13]. When the tree-level color-ordered amplitudes were first proposed in [14], the construction was based on the principles of locality and BRST invariance of pure spinor superspace expressions using multiparticle superfields. These same principles were later used when proposing SYM one-loop integrands in [1]. The difference between the expressions in [14] and [1] originates from the differences in the pure spinor amplitude prescriptions at tree level [5] and one loop [4]. The n-point tree-level numerators of [14] had to be built from three unintegrated (multiparticle) vertices V following the OPE contractions with (n − 3) integrated vertices U (z). For example, at tree level the five-point SYM amplitude in the canonical color ordering was obtained as 3 A SYM (1, 2, 3, 4, 5) = V [12,3] V 4 V 5 s 12 s 123 + V [1,23] V 4 V 5 s 23 s 123 + V [1,2] V [3,4]  3 For convenience we shall frequently omit from amplitudes such as (1.6) the pure spinor brackets . . . that extract the top element (λγ m θ)(λγ n θ)(λγ p θ)(θγ mnp θ) in the cohomology of the pure spinor BRST operator [5]. The component evaluation of ghost-number three expressions uses the identities from the appendix of [6].
where V [A,B] denotes the multiparticle unintegrated vertex operator in the BCJ gauge, see the review on multiparticle superfields in section 3 of [9] and section 4.3 of [15]. The expression (1.6) correctly reproduces the five-point tree amplitude of SYM in the canonical color ordering. The next task is to check whether this representation leads to numerators that satisfy the color-kinematics duality, this is where a subtle point arises.
A triplet of numerators participating in a kinematic Jacobi identity necessarily involves numerators from amplitudes with different color orderings, but the naive relabeling of the amplitude (1.6) does not lead to a representation satisfying the BCJ color-kinematics duality. Let us illustrate this point with an example. Using the parameterization of numerators from [16] where in order to check whether the numerators n 3 , n 5 and n 8 satisfy the kinematic Jacobi identity n 3 − n 5 + n 8 = 0 one needs to extract the numerator n 8 of the pole in 1/(s 25 s 34 ), n 3 of 1/(s 34 s 12 ) and n 5 of 1/(s 51 s 34 ). While n 3 and n 5 can be read off from the amplitude A (1,2,3,4,5) in (1.6), the numerator n 8 is found in the different color ordering A(1, 4, 3, 2, 5). If we assume that this color ordering is given by the relabeling of (1.6) the kinematic Jacobi relating these three numerators is not satisfied, where we used n 8 = V 1 V [43,2] V 5 obtained from n 4 = V 1 V [23,4] V 5 via the relabeling 2 ↔ 4.

Open superstring: Five-point tree numerators from the field-theory limit
The solution to the above problem was found in [13] by utilizing the n-point string disk correlator of [17] to generate different color orderings in its field-theory limit. These orderings follow from the various integration regions over the insertion points z i ordered along the boundary of a disk. For five points the superstring tree-level correlator is The string tree-level amplitudes with different color orderings are obtained by the different integration regions of the vertex insertion points relative to each other. The corresponding color-ordered SYM amplitudes follow from the field-theory limit α ′ → 0 of the disk integrals, encoded in the biadjoint scalar amplitudes [18] (see also [19]). More precisely, one can express the field-theory limit of the string correlator (1.9) as follows [20,21] where m(Σ|Ω) denotes the biadjoint tree amplitudes, m(P, n|Q, n) = s P φ P |Q (1.11) and φ P |Q are the Berends-Giele double currents [21]. They can be computed recursively in terms of generalized Mandelstam invariants s P = 1 2 k P · k P where k P is a multiparticle momentum defined by k P = k p 1 + k p 2 + · · · (for example k 123 = k 1 + k 2 + k 3 ).
Extracting the field-theory limit of the string disk integrals computed in the ordering z 1 ≤ z 4 ≤ z 3 ≤ z 2 ≤ z 5 -corresponding to Σ = 14325 in (1.10) -leads to the following color-ordered amplitude One can now read off the numerator n 8 = V 1 V 432 V 5 + V 12 V 43 V 5 and verify that the BCJ identity n 3 − n 5 + n 8 = 0 is identically 4 satisfied [13] 14) where the bracket notation reviewed in section 3.4.3 of [9] implies V 432 = V [23,4] and [3,4] .
The field-theory tree-level SYM numerators are extracted from the knowledge of the singular behavior of the correlator as vertex operators collide as encoded in the biadjoint  Fig. 2 The cubic graph associated to the pentagon N (5) 1|2,3,4,5 (ℓ) from equation (1.15). The convention for the loop momentum ℓ is to run from the last argument of the numerator to the first.
Berends-Giele currents. But we know that these limits constitute a local property of the Riemann surface and therefore must be independent of its genus. These results together with the analysis of [23] lead to the following expectation: The field-theory limit of the one-loop string correlators integrated along different vertex insertion orderings should give rise to a local representation for SYM oneloop integrands that satisfy the BCJ color-kinematics duality.
As an illustration of this method -to be fully developed in the next sections -let us apply it in the simplest case of the five-point SYM integrand/amplitude following from the string correlator (1.2). where the notation +(i, j|2, 3, 4, 5) denotes a sum over all possible ways to choose two elements i and j from the set {2, 3, 4, 5} while keeping the same order of i and j within the set. The cubic graph associated to this pentagon is displayed in fig. 2. Note the convention of assigning the loop momentum ℓ to the edge between 5 and 1.
Using BRST cohomology arguments the box and pentagon numerators following from relabelings (while respecting the loop momentum assignment convention and the constraint that leg 1 is contained in a multiparticle unintegrated vertex V ) were proposed in [1], For a cubic-graph parameterization of the five-point integrand to obey the BCJ colorkinematics duality the antisymmetric combination of two pentagons in the legs 1 and 2 must give rise to a box [24] From the figure above we see that the pentagon in the middle must come from the color ordering A(2, 1, 3, 4, 5) so as to keep the momenta in the common edges of the participating cubic graphs the same while respecting the loop momentum convention mentioned above.
However, the generic expression (1.17) has to ensure that the leg 1 appears in A, so the solution proposed in [1] satisfying both constraints was to assign the pentagon numerator N 1|3,4,5,2 (ℓ −k 2 ) to the middle diagram, with a shift in the loop momentum. Using that the 12-box numerator is V 12 T 3,4,5 , the expression (1.17) implies that the numerator translation of the diagrams above is given by The BCJ color-kinematic identity relating two pentagons with a box is satisfied, but only up to BRST-exact terms in pure spinor superspace that are annihilated by the pure spinor cohomology bracket . . . . The BRST exactness of the second line was shown in [22].

The BCJ pentagon from the field-theory limit of the string correlator
The five-point analysis of [1] was primarily based on the BRST cohomology properties of the integrands, and as we reviewed above this was enough to obtain a BCJ-satisfying parameterization up to BRST-exact terms. However, using the field-theory limit of the string correlator the resulting numerators for the pentagons improve the BCJ identity to be satisfied identically at the superspace level, requiring no cohomology manipulations.
To see this we consider the five-point correlator (1.2) written in terms of the Eisenstein- ij of [10], namely Z m 1,2,3,4,5 = ℓ m and Z 12,3,4,5 = g (1) 12 12 + (2 ↔ 3, 4, 5) + V 1 T 23,4,5 g 23 + (2, 3|2, 3,4,5) . (1.20) In the string-based formalism [25], the field-theory limit of the string propagator (in our case the g (1) ij functions) depends on the relative ordering of how the vertex insertion points are integrated by a term proportional to sgn ij . More precisely, if the color ordering of the resulting SYM integrand is P , the field-theory limit of g (1) ij contains a term 1 2 sgn P ij , where sgn P ij is defined in (2.16). Therefore the pentagons of the integrands in the A(1, 2, 3, 4, 5) and A(2, 1, 3, 4, 5) orderings differ by a sign in the term coming from g (1) 12 . This gives rise to the following pentagons: where we note that the constraint that leg 1 is within V is automatically satisfied because It is easy to see that the numerators (1.21) and (1.22) imply that the BCJ identity is identically satisfied at the superfield level, 23) where N 12|3,4,5 (ℓ) = V 12 T 3,4,5 . We thus see that the derivation of n-gon numerators from the field-theory limit of the open superstring correlator evaluated at different regions of integration implies that the associated BCJ identity is satisfied even before applying the pure spinor cohomology bracket to extract the polarization content of the superfields, unlike the case (1.18) obtained from relabeling. For five points this difference is immaterial as both approaches eventually satisfy the color-kinematics duality in the cohomology. However, we will see below that the field-theory limit technique leads to a six-point representation that satisfies the color-kinematics duality in contrast to the representation of [1].

SYM one-loop integrands from string correlators
The field-theory limit of the one-loop string correlators is obtained by shrinking the strings to points with α ′ → 0 while degenerating the genus-one surface with modular parameter τ to point-particle worldline diagrams with Im(τ ) → ∞ [26]. In principle this can be done using the tropical limit techniques of [27] or the string-based formalism [25], although the explicit form of the Kronecker-Eisenstein coefficient functions g (n) (z, τ ) lead to subtleties arising from the regular functions with n ≥ 2. Alternatively, one can combine the strengths of these approaches with the requirement that the field-theory integrands for different color orderings and loop-momentum parameterizations obtained from the string correlators are in the BRST cohomology of the pure spinor BRST charge. Some trial and error led to the combinatorial rules described below.

Kinematic poles and biadjoint Berends-Giele currents
The kinematic poles arise when the insertion points of the vertex operators approach each other z i → z j on the Riemann surface. The short-distance behavior of the Koba-Nielsen factor and the OPE propagator is independent of the genus of the Riemann surface. This means that the pole structure of the genus-one string correlators can be described by the same combinatorics of tree-level poles, given by the biadjoint scalar amplitudes (1.11).
These amplitudes are efficiently computed using the Berends-Giele double currents φ P |Q of explicit form given in (1.12) where the words P and Q encode the integration region and integrand.
In the one-loop case however, in addition to the tree-level kinematic poles in Mandelstam invariants the field-theory limit of the genus-one string correlators also yield Feynman loop momentum integrands to be integrated over a D-dimensional loop momentum ℓ with d D ℓ. Note the special role played by the label 1 in the above definition; this handling fixes the freedom to shift the loop momentum and is useful in obtaining BRST-closed SYM integrands [1].
In summary, the field-theory limit of genus-one open string correlators will be described by poles in Mandelstam invariants encoded in Berends-Giele double currents multiplied by Feynman loop momentum integrals.

Encoding different integration regions
In the same way as in the tree-level case, the color ordering of the resulting SYM integrand This map can be defined algebraically by This map will be used with the Berends-Giele double current to correctly generate kinematic poles for each integration region σ. It will be convenient to introduce the notation: for an amplitude with color ordering σ.

p-gon loop momentum integrands
Frequently we will need the Feynman loop momentum integrands (2.1) with a general shift in the loop momentum ℓ → ℓ + a i k i . This will be indicated by superscripts Explicitly we have where we defined for convenience In the event of an a i being zero, we will omit it from the notation. Note that the words characterizing the integrands (2.6) are totally symmetric e.g. I 1,342,5,6 = I 1,234,5,6 .
We will sometimes simplify the notation for the loop momentum integrands by dropping all indices which are single letters, and dropping the shifts in the loop momentum.
In a few instances, we may wish to use this notation when it is not immediately clear what the underlying color ordering is. In these circumstances we will include it as a superscript in the I. So, for example

Field-theory limit of Kronecker-Eisenstein coefficients
We are now ready to give the field theory limits. These are: These limits always have the same form; we take the subscripts of the g (p) ij , and sum over the possible ways to assign these to either a b (p) or a c (p) (to be defined below), and whenever we assign them to a c (p) they are also entered into the P function. In turn these are defined by where we used the notation (2.4). The cases provided above will be sufficient for our purposes.
Finally, the coefficients b (p) and c (p) for an integrand where B n denotes the n th Bernoulli number 5 and The function dist B a (i, j) measures the distance between i and j in the word B and returns +1 if it is larger than a and 0 otherwise, Note that when a i = 0 ∀ i, we must take 0 0 = 1 in the above. 5 The amplitudes up to seven points require up to B 3 :

A seven-point example
The field-theory limit of the term g 3,4 in the seven-point string correlator (1.4) for the SYM integrand with color ordering A(1, 2, 3, 4, 5, 6, 7; ℓ + 4k 4 − 6k 5 ) follows from (2.12) with a 4 = 4 and a 5 = −6, Many of these terms vanish. For instance using (2.13) the factor P (57) is proportional tô The non-zero terms are then given by The various b (1) ij and c (1) ij terms are given by (2.14) and (2.15). In the g 25 case, these are given by (recall that a 4 = 4, The others are given by Putting everything together, we see that the limit is given by

The one-loop SYM field-theory integrands
The one-loop correlators of the open superstring are integrated over the vertex insertions z i ordered along the boundary of a genus one surface. After taking the field-theory limit, the color ordering of the resulting SYM integrand corresponds to the ordering of the insertions z i . As alluded to in section 2.1, the field-theory limit of the open-string correlators will be written as a field-theory integrand depending on the loop momentum ℓ m . The parameterization of the one-loop graphs by Feynman loop integrals is notoriously plagued with the "labelling problem": arbitrary shifts of the loop momentum must not affect the integrated amplitude. This will be indicated by labelling a color-ordered SYM integrand with the explicit parameterization of the loop momentum as A(1, 2, ..., n; ℓ + a 1 k 1 + · · · + a n k n ) (2.24) This refers to the amplitude with color ordering 1, 2, ..., n, constructed such that the momentum going from the nth leg to the 1st leg is ℓ + a 1 k 1 + ... + a n k n . For example, the field-theory limit of the five-point correlator with insertion points ordered according to and loop momentum ℓ running between legs 4 and 1 is represented by the SYM integrand 6 A(1, 3, 5, 2, 4; ℓ). The statement of cyclicity -proven in the appendix B -in the color ordering becomes A(1, 2, ..., n; ℓ + a 1 k 1 + ... + a n k n ) = A(2, 3, ..., n, 1; ℓ + (a 1 − 1)k 1 + a 2 k 2 ... + a n k n ) (2.25) Using this, one can always choose to fix the color ordering of the SYM integrand to start with a leading 1.

The field-theory numerators
The field-theory limit of the open superstring n-point correlator for will be parameterized by a sum over p-gon cubic graphs ranging from p = 4 (boxes) to p = n: where N a 1 ,a 2 ,...,a n A 1 |A 2 ,...,A p (ℓ) denotes the kinematic Berends-Giele numerator of a p-gon constructed as described in the appendix A and I a 1 ,a 2 ,...,a n A 1 ,A 2 ,...,A p represents the p-gon integrand. We note that in extracting a local numerator N ... from (2.26) there will be a factor of 1/2 for each inverse Mandelstam invariant, see the definition (A.4).

Four points
The extraction of the field theory limit at four points is trivial as there is no propagator function [4]. The only limit to consider is the Koba-Nielsen factor and we get (2.27)

Five points
The five-point genus-one superstring correlator is given by [11] with the worldsheet functions [10] 12 . (2.29) This correlator gives rise to five terms with non-vanishing poles in the canonical color ordering, namely g 12 , g 23 , g 34 , g 45 , and g 51 . The parameterization of the integrand A(1, 2, 3, 4, 5; ℓ + a i k i ) from (2.26) is given by Since the field-theory limit rules behave differently for labels at the extremities of the color ordering, the 51-pentagon numerator is denoted N ′ 51|2,3,4 (ℓ). Using the field-theory limit (2.10) and comparing the outcome with (2.30) we can read off the box numerators. They are independent of the loop momentum and are uniformly described by (2.31) In particular, N ′ 51|2,3,4 = N 51|2,3,4 = V 51 T 2,3,4 . This result agrees with the analysis of [1].
The pentagon I a 1 ,...,a 5 1,2,3,4,5 (ℓ) arises from the pieces with no kinematic poles in (2.10) and collecting its associated superfields yields the numerator A straightforward but tedious calculation shows that with the f a 1 ...a 5 defined as in (2.7). It is then not hard to check that the above cancels the BRST variation of the box terms. For example, the terms proportional to (ℓ+f a 1 ...a 5 −k 123 ) 2 are given by can be used to show that where N 1|2,3,4,5 (ℓ) is given by (1.15) and I a 1 ,...,a 5 1,2,3,4,5 (ℓ) = I 1,2,3,4,5 (ℓ + a i k i ). This is an important consistency check on the field-theory rules spelled out in section 2.3.
All color ordering permutations of the five-point SYM integrand is available to download from [28].

Seven Points
At seven points, the numerators become far too complex to state here. One example can be found in the appendix D. The derivation of these numerators has one additional complication; as was discussed in [11] the refined worldsheet functions are given by Z 12|3,4,5,6,7 = ∂g 12 + s 12 g 12 g 12 − 3s 12 g 12 . (2.48) The derivative and the double pole are then removed by using partial integration with the Koba-Nielsen factor I 7 (ℓ) 12 )I 7 (ℓ) = ∂ 1 (g 12 I 7 (ℓ)) + g 12 which gives the reformulated expression for (2.48) Z 12|3,4,5,6,7 = −3s 12 g 12 + g 12 (ℓ · k 2 + s 23 g 23 + s 24 g This is the form of the refined worldsheet function we use to extract the numerators and the computation proceeds analogously as before. And we have verified the vanishing of the BRST variation of the resulting general expression.

Local BCJ-satisfying numerators
In this section we will obtain the kinematic numerators associated to various one-loop cubic graphs using the field-theory limit rules of section 2.3 applied to the superstring correlators for six external states as well as some seven-point numerators. The results of this section resolve a puzzle in the analysis of [1]. Namely, the representation in [1] of the six-point integrand did not satisfy the color-kinematics duality by terms which suspiciously were related to the gauge anomaly. We now show that the six-point integrand representation arising from the field-theory limit of the string correlator satisfies all the color-kinematic Jacobi dual relations of Bern-Carrasco-Johansson.

Color-kinematics duality
The color factors of amplitudes in gauge theory depend on the structure constants of some gauge group, f abc , that satisfy the Jacobi identity, The color-kinematics duality conjecture posed by Bern, Carrasco and Johansson (BCJ) states that the kinematic numerators of cubic-graph diagrams can be chosen to satisfy the same Jacobi identity relating their color factors [16]. That is, if a triplet of diagrams i, j, k whose color factors c i , c j , c k vanish due to the Jacobi identity (3.1), c i + c j + c k = 0, the corresponding numerators N i , N j , N k of the diagrams satisfy N i (ℓ) + N j (ℓ) + N k (ℓ) = 0 as well. Stated originally at tree-level [16] (and proven by the field-theory limit of string theory tree amplitudes [32,33]) the duality was conjectured at loop-level in [34], where the kinematic numerators also depend on loop momenta ℓ parameterizing various n-gon cubic graphs. Through this approach, properties of 4 ≤ N ≤ 8 supergravity up to four loops have been made manifest [35] (for the five-loop extension see [36,37]).
As part of the color-kinematics duality, once the gauge-theory amplitude is written down using kinematic numerators that satisfy all the kinematic Jacobi identities and automorphism symmetries of the cubic graphs, the gauge amplitude can be used to construct a gravity amplitude by replacing the color factors by a second copy of numerators c i →Ñ i (ℓ) [16,34]. For more details see the review [2].
We will now show that the numerators extracted from the one-loop string correlators using the field-theory rules of section 2.3 satisfy all the color-kinematics relations. However, starting at six points the numerators do not satisfy the required symmetries under shifts of the loop momentum required by the automorphism symmetries of the cubic graphs (see [24]), leading to subtleties in the construction of the gravity amplitudes which we defer to future work.
The one-loop five-point integrand of SYM in ten dimensions was already discussed in section 1.3.2 so we will focus on the six-point SYM integrand and briefly outline the discussion of the seven-point numerators.

Six points
The color-kinematics relations are manifestly satisfied within external tree graphs due to the BCJ gauge used in the multiparticle superfields [38,15]. Therefore we will discuss the kinematic Jacobi identities among p-gons with different values of p.

Kinematic Jacobi between pentagons and a box
The pure spinor superspace expressions of the numerators associated to the graphs in the following linear combination  fig. 2 this is the 23-pentagon N 23|1,4,5,6 (ℓ) from the integrand A(2, 3, 1, 4, 5, 6; ℓ) whose expression can be read off from the field-theory limit rules for this particular ordering.
However the assumption used in the parameterization of [1] was that this pentagon is obtained in a crossing symmetric way as N 1|4,5,6,23 (ℓ − k 23 ). As shown in [1], using these which is not in the cohomology of the BRST charge and therefore is not vanishing. In other words, the representation of the six-point integrand chosen in [1] does not satisfy the color-kinematics duality.
Taking the field theory limits and restricting ourselves to the s 23 single poles, we see that the numerator is given by This differs from the parameterization of this graph used in [1], namely N 1|4,5,6,23 (ℓ − k 23 ) with the expression for N And we note that the BCJ relation is identically satisfied at the superfield level (i.e. no BRST cohomology identity is needed). This trivial vanishing for the BCJ triplet at one loop parallels the superfield vanishing of the BCJ triplet of tree-level numerators obtained from the field-theory of the disk correlators as seen in (1.14).

Kinematic Jacobi between hexagons and a pentagon
In a given color ordering, all of the pentagons have a similar structure apart from the ijpentagon whose labels are cyclically split at the extremities A(i, . . . , j; ℓ). In this subsection we will demonstrate the validity of a BCJ relation involving such a numerator. The relation we will show is To find the hexagon numerators, we look at the piece of the field theory limits proportional to P = I. In the first case, this means making the substitution ij g . For the second hexagon, we consider the field-theory limit of the correlator with the color ordering A(1, 6, 2, 3, 4, 5; ℓ + k 1 ). The limits needed now have the form ij g kl → ij → ) .
Using these, the numerator is identified as N a 6 =1 1|6,2,3,4,5 (ℓ) = + 1 2 V 1 T mn 2,3,4,5,6 (ℓ m ℓ n + 2k m 1 k n 6 − 1 12 (k 1 m k 1 n + k 2 m k 2 n + · · · k 6 m k 6 n )) It is then simply a matter of plugging the numerators into the identity (3.12) to verify its validity. Fig. 3 The antisymmetry of the 61-pentagon from the integrand A(1, 2, 3, 4, 5, 6; ℓ). The momentum running into the 61 external tree in the graph on the right is ℓ + k 6 because in the color ordering 1, 2, 3, 4, 5, 6 a momentum ℓ must run between 6 and 1. Therefore in order to preserve the momentum assignment in the edges between the two cubic graphs, the pentagon on the left is part of the integrand A(1, 6, 2, 3, 4, 5; ℓ + k 6 ) with momentum ℓ + k 6 running between legs 5 and 1 as dictated by the convention (2.24). Therefore to extract this pentagon the field-theory rules of section 2.3 must be used with a 6 = 1. with the additional constraint that T ... ...,A1B,... = 0 (i.e., setting to zero all terms in which the label 1 is not assigned to a multiparticle vertex V P ). The exception arises for the ij-pentagon when the labels i, j are adjacent up to a cyclic rotation, e.g. the 61-pentagon in A(1, 2, 3, 4, 5, 6; ℓ) or the 12-pentagon in A(2, 3, 4, 5, 6, 1; ℓ) do not follow the general formula (3.19), as can be seen for example in (2.44). The reason this happens is due to a clash between the ij pentagon labels in A(j, P, i; ℓ) and the convention that the loop momentum ℓ runs between i and j. So to verify the antisymmetry of the 61-pentagon from A(1, 2, 3, 4, 5, 6; ℓ) one needs to compare it to the 16-pentagon from A(1, 6, 2, 3, 4, 5; ℓ + k 6 ) using the field-theory rules from section 2.3. The result is

Remaining BCJ triplets
There are then a number of relations between pentagons and boxes left to show in order to see that we have a BCJ representation of A(1, 2, 3, 4, 5, 6), and these can be seen in the cases a) to d) in the next figure. For each of these in turn we just follow the rules (2.10) for the following amplitudes with the following assignments of values for the a i a) These have been verified to give amplitudes which are BRST invariant and satisfy the relations a) to d) in the figure above. We will not detail their construction any further, as they can be obtained by analogous manipulations as discussed above.

Other parameterization of cubic graphs
Note that the choice of loop momentum to parameterize the cubic graphs plays an important role due to the inherent asymmetry of the numerators with respect to the label 1 (which must always be associated with V P ). The cases considered above are the ones Thus we conclude that the field-theory limit of the genus-one six-point string correlator

Seven points
At seven points, BCJ relations are analogously satisfied. Given their significantly more complex structure, we will not demonstrate these explicitly here and we will only outline their construction below.
As alluded to earlier, at seven points there is an extra complication that must be dealt with: the refined superfields. To find the field theory limits of the refined terms, we have to use an alternative method and partially integrate the worldsheet functions against the Koba-Nielsen factor. This then means that, when we want to verify BCJ relations, we must rearrange these refined terms. For relations in which the loop momentum structure is unchanged between terms (that is, BCJ relations in which there is always momentum ℓ going into leg 1), this amounts to canceling all (ℓ · k) terms. Take for instance the relation and consider the refined terms V 1 J 34|2,5,6,7 within it. In the standard ordering correlator, these terms are associated with the worldsheet function Z 34|1,2,5,6,7 and we would therefore expect the heptagon numerator N 1|2,3,4,5,6,7 (ℓ) to contain the terms Likewise, the other numerators we would expect to contain the terms N 1|2,4,3,5,6,7 (ℓ) ↔ − 1 12 The relation (3.28) is clearly not satisfied with these values.
Instead, we cancel the ℓ · k terms. For example, we rewrite (3.29) as We then cancel the (ℓ − k) 2 terms with the denominator of the Feynman loop integrand we have yet to identify a general algorithm for these situations. However, by explicitly rearranging amplitudes term by term, we have been able to structure them so that they satisfy all of the BCJ relations we have tested. Namely, we have been able to simultaneously satisfy the following more complex relations Though this is not an exhaustive test, we hope that it is sufficient to serve as a proof of concept that this method work, and that it should always be possible to rearrange the refined terms to satisfy the color-kinematics duality.

Supergravity amplitudes and the double copy
One of the goals in obtaining a parameterization of gauge theory 1-loop integrands that satisfies the color-kinematics duality is to construct corresponding supergravity integrands via the double-copy construction [2]. For five points this construction was carried out explicitly in four dimensions in [24] while the ten-dimensional analysis using pure spinor superspace was done in [1]. In the pure spinor superspace setup, the supergravity integrand obtained via the double copy must be checked to be BRST invariant, as that guarantees gauge and supersymmetry invariance of its component expression in terms of polarizations and momenta [5].
We will now repeat the five-point supergravity construction of [1] to highlight that it is BRST invariant but that it is so only because the numerators satisfy the dihedral symmetries of the cubic graphs in the cohomology of pure spinor superspace (see [24] for a discussion of these symmetries). While at five points our numerators satisfy these symmetries in addition to the color Jacobi identities, the corresponding symmetries at six points are not satisfied by our BCJ-satisfying six-point numerators and will prevent the double-copy construction of a BRST-closed supergravity integrand. Applying the doublecopy procedure at six points will be left for a future work.

The five-point supergravity integrand
Let us construct the five-point supergravity integrand using the double-copy procedure to highlight the existence of a subtlety: the consistency of the double-copy construction requires the five-point numerators not only to satisfy the kinematic Jacobi identities but also the dihedral symmetries of the cubic graphs. We will see that these symmetries, unlike the kinematic Jacobi identities, are satisfied in the cohomology rather than identically.
Starting with the color-dressed integrand (E.1) we replace the color factors by an extra copy of duality-satisfying kinematic numerators. This yields Note that the kinematic numerators on the left are written in terms of Berends-Giele numerators N of the appendix A while those on the right are the local numerators N .
After setting up the double-copy supergravity integrand (3.37) we must check its BRST variation. Since (3.37) is left/right symmetric 9 it is enough to consider the leftmoving BRST variation, which we will see vanishes only if the right-movers are in the cohomology of the right-moving pure spinor superspace. To see this surprising fact, consider the variation of the left-moving pentagon N 1|2,3,4,5 (ℓ) multiplied by the loop-momentum integrand I 1,2,3,4,5 : where we used identities such as (ℓ − k 1 ) 2 I 1,2,3,4,5 = I 12,3,4,5 that follow from (2.6).
These loop-momentum identities are trivial but one of them on the last line, namely ℓ 2 I 1,2,3,4,5 = I 1,2,3,4 , has a peculiar behavior: the right-hand side has no label 5. This seemingly innocuous fact will have a surprising implication in the double-copy construction of the five-point supergravity integrand when (3.38) appears multiplied by a right-moving factorÑ 1|2,3,4,5 (ℓ).
To summarize, the five-point supergravity integrand is BRST invariant. But there is a subtlety: the double-copy construction seems to require more than just the kinematic Jacobi identities, the numerators must also satisfy the dihedral symmetries of the cubic graphs 12 (which are satisfied in the cohomology of the right-movers).

Six-point double copy and automorphism symmetries
At six points a naive application of the double-copy procedure with BCJ-satisfying numerators obtained in the previous sections does not produce a consistent supergravity integrand: it fails to be BRST invariant in pure spinor superspace. This happens because the numerators, even though they satisfy the color-kinematics duality they do not satisfy the automorphism symmetries of their associated cubic graphs.
To see this it is enough to use the BCJ-satisfying six-point numerators in a tentative double-copy construction to obtain, among many others, the following terms under a leftmoving BRST variation QM 6 (ℓ), Therefore the naive application of the double-copy construction at six points is not consistent even though the numerators satisfy the color-kinematics duality.
It is interesting to observe that the automorphism symmetries of the graphs encoded
Since the issue with missing labels in the loop momentum integral as a result of a BRST variation will always be present for the BCJ-satisfying numerators obtained in this work, solving this problem seems to require a different approach to the double-copy construction in the pure spinor superspace context. Given that the failures are purely contact terms, the generalized double-copy prescription of [36] may be applicable 14 and it will be interesting to see how BRST invariance is restored. It is reasonable to speculate that the deformations of the right-moving BCJ triplets by contact terms as a result of loop momentum shifts due to canceled loop propagators in the left-moving BRST variation may be a generic feature of the double copy in pure spinor superspace. If true, the generalized double-copy formalism may be the norm by which gravity integrands are generated from gauge-theory integrands; a tree-level manifestation of this behavior was anticipated in [39].
We plan to investigate this problem in future work.
We note that supergravity integrands have been constructed using BCJ numerators in four dimensions for up to seven points in [40] and to all multiplicity in [20] using spinor helicity in the MHV sector. Supergravity amplitudes were also constructed in [41] but using a partial-fraction representation of the loop momentum integrands.

Conclusion
In this work we obtained a set of field-theory limit rules for the Kronecker-Eisenstein coefficient functions present in the genus-one superstring correlators derived in [9,10,11].
Using these rules we found local numerators for ten-dimensional SYM integrands at one loop for five, six and seven points that satisfy the BCJ color-kinematics duality. These 14 We thank Oliver Schlotterer for discussions on this point. results resolve the difficulties in an earlier analysis of the six-point SYM integrands which did not satisfy the color-kinematics duality [1].
These field-theory limits have an special affinity with the pure spinor superspace representation of the superstring correlators. They take into account arbitrary choices in the parameterization of the loop momentum integrands, shuffling terms among various numerators preserving BRST invariance of the SYM one-loop integrands while changing the BRST properties of individual numerators in a non-trivial way, see the discussion around (3.24). The prescription to find the field-theory limit of the correlator whose parameterization contains shifts of the loop momentum by arbitrary linear combinations of external particle momenta is crucial in demonstrating all the BCJ color-kinematic identities of our ten-dimensional SYM representation.
However, in attempting to use the BCJ-satisfying six-point numerators in a doublecopy construction of the supergravity integrand we learned that the numerators must satisfy, in addition to the kinematic Jacobi identities, also the various graph automorphism symmetries in order for the supergravity integrand to be BRST invariant. Unfortunately V [1,23] T [8,7],4,56 as the latter would violate the ordering constraint.
Another summation notation to note is A.0.1. Lie algebra notation and Berends-Giele currents We frequently use the notation of words and Lie brackets, especially when indexing SYM multiparticle superfields, see the discussion on section 3 of [9]. In any situation where a Lie bracket would be expected but a word A is found instead, this should be regarded as being the left-to-right Dynkin bracket ℓ(A) [42], A mapping from words to Lie brackets which will be particularly useful is the b-map defined by [43] 15 For example, b(12) = 1 2s 12 [1,2], and b(123) =  [2,3]]. Superfields are described in terms of two broad classes of objects. The first are local and denoted by V , T , J, and N . The composition of the first three of these objects can be found in more detail in [3,22]. The fourth will be used to refer to amplitude numerators and are detailed on a case by case basis. These objects have a number of slots for indices labelling their superfield contents, and all such indices will be Lie brackets. The second class of objects are Berends-Giele (BG) currents. These are related to the local objects previously described through the use of the b-map on each of their blocks of indices. The BG current of particular use to us is denoted by N , defined in terms of local objects N as For example, a seven-point box Berends-Giele numerator is expanded as [4,5],6],7 (ℓ) + 1 s 56 N 1| [2,3], [4,[5,6]],7 (ℓ) .
It should be noted that generalized Mandelstam invariants are defined with a 1 2 factor, Appendix B. Cyclic symmetry of the field-theory limit rules In this appendix we will show that the definitions for the field theory limits we have given yield the cyclic symmetry relations seen in (2.25) A(1, 2, ..., n; ℓ + Σ i a i k i ) = A(2, 3, ..., n, 1; We refer to terms from A(1, 2, ..., n; ℓ+Σ i a i k i ) with a (I), and A(2, 3, ..., n, 1; ℓ−k 1 +Σ i a i k i ) with a (II).
First, we compare their b ij terms. We restrict ourselves to the limit of a single Kronecker-Eisenstein coefficient function, as the limits of their products are the natural generalization of this and will follow accordingly. Referring to (2.14), and using the notation a ji := a j − a i , we see that they differ by Clearly in all cases where neither of i or j is 1 this vanishes. If we suppose i = 1, the first sgn function is 1, and the second is −1. Hence this difference becomes This can be shown to vanish. Taking for instance the p = 3 case, we have To show that (B.3) vanishes in general we expand the bracket (a j1 + 1) p−m , where we have separated out the terms of order (p − m) in the second line. In the right hand terms of the above, (1 − (−1) m ) vanishes when m is even, and B m vanishes when m is odd and not 1. Hence, this summation reduces to a single term, We then turn to the left hand terms of (B.5). Reordering the double summation, these have the form Then, we move onto the c piece. This difference is given by (1, j)) p−1 = a p−1 j1 j ≤ n − 2 (a j1 + 1) p−1 j > n − 2 .
When n = 4, 5, the only Kronecker-Eisenstein functions in amplitudes is g (1) ij , and we see that setting p = 1 in the above gives equivalence. When n = 6, these coincide in that n − 2 = 4. When n = 7 and p > 1, they differ when j = 5. However, this disagreement will not matter. At 7 points a term g (2+) 15 is multiplied by at most one other g (q) ij function, but we need at least two Kronecker-Eisenstein coefficient functions in order to make the corresponding P function non-zero. That is, for example, At 8 points, this will of course become an issue. However, the description of the dist function was chosen purely for simplicity. If we instead think of this function as asking whether the pole being approached crosses the boundary between particles n and 1, then consistency should be maintained to higher points.
Appendix C. The field-theory limit at higher points We anticipate that the field theory limit rules for an arbitrary product of g (n) ij functions should generalize in the natural way where P(12...n) denotes the power set of 12...n, A is an element of this, and A c its complement. We stress that the indices of the c (p) and those in the P function are identical.
The general P functions will be as in (2.13), with P (i 1 j 1 , ..., i n j n ) chaining together i m j m pairs as much as possible, and then using these as indices for φ and I functions. We can then extend this to the general a i case, though with less elegance. If instead of making the substitution (C.4) into (C.3), we instead use the general a i values of the b (1) terms, we find the relation (C.10) .
This cannot be as easily rearranged into a recursion relation. However, if we assume that ij is a polynomial in a j −a i up to order n, we may use the above to identify the polynomial coefficients. Doing this reveals the value of b (4) ij as would be expected from (2.10) as the unique solution. And then we have verified that the relation above is satisfied in a number of further cases if we assume this general form of b (n) .
We can perform a similar exercise for the c  We need not restrict ourselves to the a i = 0 ∀i case here, as the computation is simpler.
Looking at the s 1m single poles leads us to the relation Using that we know c 17 also. We can also repeat this calculation for poles of g Hence the form of c (n) ij presented in (2.10) is the natural generalization, and we expect (2.10) to hold to higher points.
We end this discussion though by stressing that this approach is highly speculative, and we have not tested these values produced in any way beyond the aforementioned discussion. They are however a strong candidate for what they are attempting to describe.

Appendix E. The five-point color-dressed integrand
In this appendix the five-point color-dressed integrand will be written down after the application of the color decomposition techniques of [46].  (2,3,4,5) where N denotes the Berends-Giele counterpart of the n-gon numerator as described in the appendix A while the color factors of the box and pentagon cubic graphs are The factor of 1 2 in (E.1) compensates the overcounting of graphs due to symmetries. Note that the box numerators do not depend on the loop momentum.