One-loop Correlators and BCJ Numerators from Forward Limits

We present new formulas for one-loop ambitwistor-string correlators for gauge theories in any even dimension with arbitrary combinations of gauge bosons, fermions and scalars running in the loop. Our results are driven by new all-multiplicity expressions for tree-level two-fermion correlators in the RNS formalism that closely resemble the purely bosonic ones. After taking forward limits of tree-level correlators with an additional pair of fermions/bosons, one-loop correlators become combinations of Lorentz traces in vector and spinor representations. Identities between these two types of traces manifest all supersymmetry cancellations and the power counting of loop momentum. We also obtain parity-odd contributions from forward limits with chiral fermions. One-loop numerators satisfying the Bern-Carrasco-Johansson (BCJ) duality for diagrams with linearized propagators can be extracted from such correlators using the well-established tree-level techniques in Yang-Mills theory coupled to biadjoint scalars. Finally, we obtain streamlined expressions for BCJ numerators up to seven points using multiparticle fields.

As a result, we will present new expressions for BCJ numerators, not only for tendimensional SYM but also for lower-dimensional gauge theories with reduced or without spacetime supersymmetry. The numerators of this work manifest the power counting of loop momenta by representation-theoretic identities between Lorentz traces in vector and spinor representations. Moreover, our construction preserves locality, i.e. the BCJ numerators do not involve any poles in momentum invariants.
Furthermore, we will also present two new results on one-loop correlators and BCJ numerators. First, we will compute parity-odd contributions to the correlators by taking forward limits with chiral fermions, both in D = 10 SYM and in the D = 6 case with a chiral spectrum. In addition, we will simplify the BCJ numerators using the so-called multiparticle fields [61,73], which can be viewed as numerators of Berends-Giele currents [74] that respect color-kinematics duality, derived in the BCJ gauge [60,75].

Conventions
In the conventions of this paper, the CHY representation of tree-level amplitudes with a double-copy structure is given by where the theory-dependent normalization factor N for instance specializes to −2(− g √ 2 ) n−2 for gauge-theory amplitudes with YM coupling g. 6 Inside the CHY measure dµ tree n , the prime along with the product instructs to only impose the n−3 independent scattering equations, see [72] for additional details. Depending on the choice of the half-integrands I tree L,R , (1.1) can be specialized to yield tree amplitudes in gauge theories, (super-)gravity and a variety of further theories [33,76]. Color-ordered gauge-theory amplitudes are obtained from a Parke-Taylor factor I tree L → (σ 12 σ 23 . . . σ n1 ) −1 and taking I tree R to be the reduced Pfaffian given in (2.5). The one-loop analogue of the amplitude prescription (1.1) is reviewed in appendix A. 6 The combination g/ √ 2 in the normalization factor N of gauge-theory amplitudes can be understood as rescaling the color factors.

Summary
The main results of the paper can be summarized as follows.
• We present new expressions for tree-level correlators with two and four fermions and any number of bosons. By taking forward limits in a pair of bosons/fermions, we obtain a new formula (3.15) for one-loop correlators in D = 10 SYM.
• By combining building blocks with vector bosons, fermions or scalars circulating the loop, we obtain a similar formula (4.4) for one-loop correlators in general, possibly non-supersymmetric gauge theories in D < 10.
• Since the worldsheet dependence is identical to that of single-trace correlators for (YM+φ 3 ) tree amplitudes, we can recycle tree-level results to extract one-loop BCJ numerators in these theories.
• We will derive parity-odd contributions (5.3) to one-loop correlators from forward limits with chiral fermions.
• We present various BCJ numerators at n ≤ 7 points in a compact form by using the multiparticle fields.
The paper is organized as follows. We start in sec. 2 by collecting some results which will be used in the subsequent: First we spell out tree-level correlators with n bosons and those with n−2 bosons and 2 fermions in the RNS formalism for ambitwistor string theory. Then we review how the tree-level input can be used to construct one-loop correlators by taking the forward limit of a pair of bosons or fermions with momenta in higher dimensions.
Next, we study one-loop correlators and BCJ numerators in D = 10 SYM in sec. 3. After expressing them as combinations of vector traces and spinor traces of linearized field strengths, we propose a key formula (3.11) for converting spinor traces to vector traces, which allows us to simplify the one-loop correlators of D = 10 SYM. In particular, the power counting in loop momentum follows from representation-theoretic identities between vector and spinor traces. Once the correlator is in this form, it is straightforward produce BCJ numerators as the problem is equivalent to that for tree-level amplitudes in YM+φ 3 .
We move to general gauge theories in even dimensions D < 10 in sec. 4. By also including one-loop correlator from forward limit of two scalars, we obtain a general formula for the case with n v vectors, n f Weyl fermions and n s scalars. In particular, we apply the general formula to obtain explicit results for specific theories in D = 6 and D = 4.
In sec. 5, we derive parity-odd contributions to one-loop correlators from forward limits in chiral fermions, which are parity-odd completions of correlators in D = 10 SYM and those in lower dimensions. Finally, in sec. 6, by using multiparticle fields, we provide particularly compact expressions for the BCJ numerators in various theories, which combine contributions from the Pfaffian and the field-strength traces in the correlators.
The discussion in the main text is complemented by three appendices: Our representation of one-loop integrands will be reviewed in appendix A; we review CFT basics and give the derivation for tree-level correlators with zero, two and four fermions in appendix B; we also prove the identity for reducing spinor traces to vector traces in appendix C.1.

Basics
In this section, we use the RNS formulation of the ambitwistor string in D = 10 dimensions [28,30] (see [77,78] for the RNS superstring) to review tree-level correlators with n gluons (bosons). The latter evaluate to the well-known Pfaffian in the CHY formulation [25], and we will present new representations for correlators with 2 gluinos (fermions) and n−2 gluons, also see appendix B.4 for four-fermion correlators. On the support of scattering equations, the Pfaffian can be expanded into smaller ones dressed by Lorentz contraction of field strengths with two polarizations. As we will see, the correlator with 2 gluinos and n−2 gluons can be simplified to a similar form, which are smaller Pfaffians dressed by gamma-matrix contracted field strengths, with wave functions for the two fermions. We will see that these representations of correlators are most suitable for combining the forward limits of two gluons/gluinos and studying the resulting supersymmetry cancellations.

Tree-level correlator for external bosons
Given gluon vertex operators, one can compute the tree-level correlator for n bosons The 2n × 2n antisymmetric matrix Ψ was first introduced in [25], with columns and rows labelled by the n momenta k i and polarizations i for i = 1, 2, · · · , n, and it also depends on the punctures σ i . The entries of Ψ are reviewed in appendix B.2 to fix our conventions. The reduced Pfaffian Pf | . . . | 1,n in (2.5) is defined by deleting two rows and columns 1, n of the matrix Ψ with a prefactor 1/σ 1,n . More generally, one can define it by deleting any two columns and rows 1 ≤ i < j ≤ n with a prefactor (−1) i+j+n−1 /σ i,j : this amounts to having the gluons i, j the −1 picture, and while the correlator is manifestly symmetric in the remaining n−2 particles, on the support of scattering equations it becomes independent of i, j thus completely symmetric as required by Bose symmetry.
( 1 · f 2 · n ) = µ 1 (f 2 ) µν ν n , so we reproduce the well-known three-point example respectively. It has been known since [26] that using scattering equations, one can expand the correlator as a linear combination of Parke-Taylor factors, say in a Kleiss-Kuijf basis, and the coefficients are BCJ master numerators for the corresponding (n−2)! half-ladder diagrams. One way for doing so is to start from (2.6), and the challenge is identical to extracting BCJ numerators for single-trace YM + φ 3 amplitudes. See [35,68,[70][71][72] for more details.

Tree-level correlator for two external fermions
In the subsequent, we will cast two-fermion correlators involving two spin fields S α [81,82] into simple forms by virtue of the current algebra generated by ψ µ ψ ν along the lines of [84]. Note that such simplifications are partly motivated by (2.6) since such a correlator with external fermions can also be expanded in a similar form.
In the first representation, we have the two fermions, say, leg 1 and n−1, both in the − 1 2 ghost picture, and one of the gluons, say, leg n, in the −1 picture. Throughout this work, we will use the subscript "f" to denote fermions (gluinos) and suppress any subscript for the vector bosons (gluons). On the support of scattering equations, one can show that the reduced tree-level correlator reads (see appendix B.3 for details) where we sum over all the splittings of the set {2, 3, · · · , n−2} into disjoint sets A, B and C, with again Pf(Ψ A ) times a sum over permutations ρ and τ for labels in B and C respectively. Similar to (2.6), we have a Parke-Taylor factor PT(1, ρ(B), n, τ (C), n−1) defined by (2.7) for each term. The main difference is that instead of the vector-index contraction, the linearized field strengths (2.8) are now contracted into gamma matrices. More specifically, with the conventions the last line of (2.11) features gamma-matrix products with the gluons in ρ(B), τ (C) entering via / f j , gluon n entering via / n , and the fermion wavefunctions χ 1 , χ n−1 contracting the free spinor induces, e.g. ( In view of their contractions with Weyl spinors χ 1 , χ n−1 , the gamma matrices in (2.11) are 16 × 16 Weyl-blocks within the Dirac matrices in 10 dimensions. Our conventions for their Clifford algebra and antisymmetric products are A variant of (2.11) with / n moved adjacent to χ n−1 has been studied by Frost [85] along with its implication to the forward limit in the fermions. At n = 3 points, the two-fermion correlator (2.11) specializes to 14) and the sum over {2} = A ∪ B ∪ C in its (n = 4)-point instance gives rise to the following three terms instead of the four terms in the bosonic correlator (2.10) (also see [30]): The formula (2.11) for the two-fermion correlator is manifestly symmetric in most of the gluons 2, 3, · · · , n−2 except for the last one n which is earmarked through the hat notation in I tree 2f (. . . ,n). On the support of scattering equations and the kinematic phase space of n massless particles, one can show that (2.11) is also symmetric in all ofn and 2, 3, . . . , n−2. But this no longer the case in the forward-limit situation of sec. 5, where we extract parity-odd contributions to one-loop correlators from (2.11).
Note that the expression (2.11) for the two-fermion correlator can be straightforwardly generalized to any even spacetime dimension since the structure of the underlying spin-field correlators is universal (see appendix B.3). However, only D = 2 mod 4 admit χ 1 and χ n−1 of the same chirality since the charge-conjugation matrix in these dimensions is off-diagonal in its 2 D/2−1 × 2 D/2−1 Weyl blocks. In order to extend (2.11) to D = 0 mod 4 dimensions, χ 1 and χ n−1 need to be promoted to Weyl spinors of opposite chirality.

Alternative representation of the two-fermion correlator
In this section, we present an alternative representation of the two-fermion correlator which is manifestly symmetric in all its n−2 gluons. To do that, we put the two fermions, say leg 1 and n in − 1 2 and − 3 2 picture, respectively, and on the support of scattering equations we find (see appendix B.3 for details) which takes a form even closer to (2.6) since we also sum over partitions {2, 3, · · · , n−1} = A ∪ B with disjoint A, B. All the (gamma-matrix contracted) field strengths (2.12) in ρ(B) are sandwiched between χ 1 and ξ n . At n = 3, the sum over {2} = A ∪ B in (2.16) involves two terms: In order to relate this to the earlier result (2.14) for the fermionic three-point correlator, we have rewritten Pf Ψ {2} = ( 2 ·k 1 )σ 1,3 σ 2,1 σ 2,3 and χ 1 / f 2 ξ 3 = − 1 2 χ 1 / k 3 / 2 ξ 3 in passing to the second line. These identities are based on both momentum conservation and the physical-state conditions 2 · k 2 = χ 1 / k 1 = 0. Finally, the Clifford algebra (2.13) gives rise to χ 1 / k 3 / 2 ξ 3 = 2( 2 · k 3 )χ 1 ξ 3 − χ 1 / 2 / k 3 ξ 3 , and one can identify the wavefunction χ 3 = / k 3 ξ 3 by (2.3). In this way, we reproduce the permutation of the earlier three-point result (2.14). Even though this may appear to be a detour in the computation of the three-point correlator, the similarity of (2.16) with the bosonic correlator (2.6) will be a crucial benefit for the computation of forward limits. At n = 4 we have the four contributions similar to (2.10): , We remark that again we can further expand the Pf Ψ A in both cases, and on the support of scattering equations eventually one can expand the correlator as a linear combination of (length-n) Parke-Taylor factors. Their coefficients can be identified with BCJ numerators [26,68,70,71], now involving two external fermions on top of n−2 bosons. In the following, we will mostly work with the second representation (2.16) of the two-fermion correlator when we take the forward limit in the two fermions and combine it with the bosonic forward limit of (2.6). The parity-odd part of one-loop numerators in chiral theories in turn will be derived from the first representation (2.11) of the two-fermion correlator, see sec. 5. Similar to the results of the previous section, the two-fermion correlator (2.16) generalizes to any even spacetime dimension. The chiralities of χ 1 and ξ n remain opposite in any D = 2 mod 4, whereas dimensions D = 0 mod 4 require a chirality flip in one of χ 1 or ξ n .
As detailed in appendix B.4, four-fermion correlators with any number of bosons can be brought into a very similar form. Six or more fermions, however, necessitate vertex operators in the +1/2 superghost picture that feature excited spin fields and give rise to more complicated n-point correlators [86][87][88]. Still, the results are available from the manifestly supersymmetric pure-spinor formalism [89], where n-point correlators in Parke-Taylor form are available in superspace [13,14]. Its components for arbitrary combinations of bosons and fermions can be conveniently extracted through the techniques of [60,61].

Forward limits and gluing operators
Finally, we review the prescription for taking forward limit of a pair of legs, which can be both bosons or both fermions. The momenta of the two legs are + and − respectively, which should be taken off shell, i.e. 2 = 0. 7 Moreover, we need to sum over the polarization states and other quantum number of the two legs. For example, we consider all particles (both gluons and gluinos) to be in the adjoint representation of e.g. U (N ) color group, then we have to sum over the U (N ) degrees of freedom of the pair of legs. In this way, the oneloop color-stripped amplitude can be obtained by summing over tree-level ones with the two adjacent legs inserted in all possible positions. This is the origin of the one-loop Parke-Taylor factors (A.4), also see [38,45] for more details.
We shall now define the kinematic prescription for forward limit of two bosonic or fermionic legs. For that on bosonic legs i and j, we define with auxiliary vector¯ µ subject to ·¯ = 1. Note that we have used the completeness relation of polarization vectors. For the forward limit on fermionic legs i and j, we define where we have used the completeness relation for fermion wave functions. When applied to a pair of vertex operators with total superghost charge −2, the prescriptions (2.20) and (2.21) implement the gluing operators of Roehrig and Skinner [65]. Before proceeding, we remark that after taking forward limit of a pair of gluons/gluinos in the tree-level correlator, (2.6) and (2.16), the only explicit dependence on loop momentum is in Pf Ψ A through diagonal entries of the submatrix C A ; there is no loop momentum in other parts of Pf Ψ A or factors involving particles in B. We will see in the subsequent that this observation immediately yields the power counting of loop momentum for BCJ numerators in various gauge theories.

One loop correlators and numerators of ten-dimensional SYM
In this section, we study one-loop correlators with external bosons for ten-dimensional SYM, which in turn give explicit BCJ numerators at one-loop level. We begin by taking the forward limit of tree-level correlators with two additional bosons and fermions, (2.6) and (2.16), respectively; in order to combine them, we present a key result of the section, namely a formula to express a spinor trace with any number of particles in terms of vector traces. Moreover, the relative coefficient is fixed by maximal supersymmetry, thus we can write a formula for the one-loop correlator with all the supersymmetry cancellations manifest at any multiplicity.
Even though this section is dedicated to ten-dimensional SYM, we will retain a variable number D of spacetime dimensions in various intermediate steps. This is done in preparation for the analogous discussion of lower-dimensional gauge theories in section 4 and justified by the universality of the form (2.16) of two-fermion correlators.

The forward limit of two bosons/fermions
Implementing the forward limits (2.20) and (2.21) via gluing operators [65] sends the presentation (2.6) and (2.16) of the tree-level correlators to FWL 1,n I tree bos (1, 2, . . . , n) = The contribution of B = ∅ stems from contractions η µν (η µν − µ¯ ν − ν¯ µ ) = D −2 and δ α β δ β α = 2 D/2−1 in (2.20) and (2.21), the latter being the dimension of a chiral spinor representation in even spacetime dimensions D. In spelling out the contributions of B = ∅ to the bosonic forward limit, we have exploited that the terms ∼ µ¯ ν + ν¯ µ in (2.20) do not contribute upon contraction of with vectors different from i , j [65]. We shall introduce some notation for the frequently reoccurring traces over vector and spinor indices, delaying the discussion of parity-odd pieces to sec. 5: We remark that the spinor trace in (3.2) would in principle contain parity-odd terms, but here we define tr S (1, 2, . . . , p) to be the parity-even part by manually discarding parity-odd terms. 8 Note that the B = ∅ contribution to (3.2) formally arises from tr S (∅) = 2 D/2−1 and non-empty traces exhibit the parity properties In order to study the supersymmetry cancellations in one-loop correlators, we will be interested in linear combinations of bosonic and fermionic forward limits with theory-dependent relative weights. The main results of this work are driven by the observation that most of the structure in (3.1) and (3.2) is preserved in combining bosons and fermions such that the linear combinations are taken at the level of the field-strength traces: with an a priori undetermined weight factor α ∈ Q, we have bos,α (1, 2, . . . , n) = FWL +,− I tree bos (+, 1, 2, . . . , n, −) + α · I tree 2f (+ f , 1, 2, . . . , n, .n} ) will be proportional to at least one power of loop momentum since a plain Pfaffian in a tree-level context is known to vanish on the support of the scattering equations. The diagonal entries C jj in the expansion of Pf(Ψ A ) within (3.6) still involve terms µ j ( µ σ j,+ − µ σ j,− ) which would be absent in the naive tree-level incarnation of Pf(Ψ {12...n} ) without any reference to extra legs +, −.

From spinor traces to vector ones
In this subsection we propose the identities which allow us to convert any spinor trace to vector ones. Our result will be useful in the subsequent sections when we study the one-loop correlator and BCJ numerators for various gauge theories.
Our starting point is the well-known formula for traces of chiral gamma matrices We will review a recursion for such traces in appendix C.1, and based on that it is easy to show that tr S (∅) = 2 D/2−1 generalizes to where we have used the parity properties (3.5): for single-trace terms we have 6 (cyclically inequivalent) permutations but only 3 of them are independent under parity.
Moving to the n = 5 case, we find that tr S (1, 2, 3, 4, 5) is again given by combinations of single and double traces, where only 4!/2 = 12 single-trace terms, and 5 2 = 10 double-trace terms are independent under parity.
As we will show recursively in Appendix C.1, in general the n-point spinor trace can be written as a sum of terms with j = 1, 2, · · · , n/2 vector traces with suitable prefactors, where for each j, we sum over partitions of {1, 2, . . . , n} into j disjoint subsets A 1 , A 2 , . . . , A j , and the factor 1 j! compensates for the overcounting of partitions due to permutations of A 1 , A 2 , . . . , A j ; for each subset A i we sum over all cyclically inequivalent permutations σ ∈ S |A i | /Z |A i | , e.g. by fixing the first element in tr V to be the smallest one in A i ; finally the sign ord id σ counts the number of descents in σ (compared to the identity permutation). For example, ord id 132 = −1, ord id 1243 = ord id 1324 = −1, and ord id 1432 = 1. An alternative representation of the parity-even spinor trace (3.4) in terms of a Pfaffian can be found in (4.35a) of [65].
More generally, if the spinor trace has an ordering ρ, one can choose the first element σ 1 to be the smallest in ρ, and the sign ord ρ σ can be factorized as where the sgn ρ ij factors are defined to be ±1 according to the conventions of [44] sgn For example, ord 132 132 = 1 instead of ord id 132 = −1 and ord 1243 1432 = −1. Let's end the discussion with an example for triple-trace contribution (j = 3) of tr S (1, 2, 3, 4, 5, 6), which reads (3.14)

Ten-dimensional SYM
Since we have not been careful about the normalization of the fermionic tree-level correlator (2.11), the normalization constant α in (3.6) for a single Weyl fermion will be fixed by the example of ten-dimensional SYM. The supersymmetry cancellations are well-known to yield vanishing (n ≤ 3)-point one-loop integrands in D = 10 SYM [90]. Accordingly, there exists a choice α = − 1 2 in (3.6) such that both the |B| = ∅ contributions D−2 + α · 2 D/2−1 D=10 and those with |B| = 2, 3 vanish: Recall that the term tr S is defined to contain the parity-even part only. We have used the relation (3.8) between vector and spinor two-and three-traces in D = 10 dimensions, tr S (1, 2) = 2tr V (1, 2) and tr S (1, 2, 3) = 2tr V (1, 2, 3). Throughout this work, the external states of the one-loop correlators are gauge bosons. Thus we will no longer specify bos in the subscripts of I (1) . The first contribution to (3.15) from the field strengths at |B| = 4 turns out to not depend on the permutation ρ and reproduces the famous t 8 -tensor, cf. (3.9), which is known from one-loop four-point amplitudes of the superstring [90] and defined by Hence, the four-point instance of (3.15) is the well-known permutation symmetric combination of Parke-Taylor factors, Let us already emphasize here that (3.15) after rewriting tr S (. . .) in terms of tr V (. . .) applies to any dimensional reduction of ten-dimensional SYM, for instance N = 4 SYM in D = 4 (cf. section 4).
By analogy with (3.16), one may define higher-rank tensors beyond t 8 in (3.17). We can use the difference of vector and spinor traces to define higher-point extensions of (3.17) that will capture the kinematic factors besides the Pf(Ψ A ) in the correlators (3.15) D = 10 SYM. As exemplified by the five-point case (3.19), higher-point tr V (. . .) − 1 2 tr S (. . .) will involve t 8 tensors with nested commutators of f j w.r.t. Lorentz indices in its entries. The only new tensor structures that are not expressible in terms of t 8 with commutators arise from the permutation symmetric combination 9 involving an even number n of field strengths. The permutation sum vanishes for odd n by the parity properties (3.5). Rewriting correlators of D = 10 SYM in terms of (3.20) is the kinematic analogue of decomposing color traces in gauge-theory amplitudes into contracted structure constants and symmetrized traces, where only the latter can furnish independent color tensors [91].
The simplest instance of (3.20) beyond t 8 is a rank-twelve tensor t 12 occurring at n = 6. As detailed in appendix C.2, the case of t 12 admits an exceptional simplification that is not possible for t 16 and any higher-rank tensor (3.20): One can reduce t 12 to products, where the four-traces and products tr V (i 1 , i 2 )tr V (i 3 , i 4 )tr V (i 5 , i 6 ) conspire to t 8 . Here and throughout the rest of this work, the notation +(1, 2|1, 2, . . . , k) instructs to add all permutations of the preceding expression where the ordered pair of labels 1, 2 is exchanged by any other pair i, j ∈ {1, 2, . . . , k} with i < j. A similar notation +(1, 2, . . . , j|1, 2, . . . , k) with j < k will be used to sum over all possibilities to pick j elements from a sequence of k, for a total of k j terms. The exceptional simplification of t 12 in (3.21) can be anticipated from the fact that sixtraces tr V (1,2, . . . ,6) cancel from the combination (3.20) after rewriting the spinor traces via (3.11). For any higher-rank t 2n at n ≥ 8 in turn, the coefficient of tr V (1, 2, . . . , n) is non-zero when expressing the spinor traces of (3.20) in terms of tr V (. . .). These coefficients are worked out in terms of Eulerian numbers in appendix C.2.
In summary, the tensor structure of the n-point correlators (3.15) in D = 10 SYM is captured by Pf(Ψ A ) and even-rank tensors t 2n in (3.20) including t 8 in (3.17) contracting nested commutators of field strengths.

BCJ numerators versus single-trace YM+φ 3 at tree level
Given the general formula (3.15) for the one-loop correlator in ten-dimensional SYM, one can read off the BCJ master numerators N (1) of an n-gon diagram as soon as all the σ jdependences of the Parke-Taylor factors and the Pf(Ψ A ) are lined up with where we need to use scattering equations at (n+2) points. 10 More specifically, the numerator N (1) (+, ω(1, 2, . . . , n), −) refers to one of the (n+2)-point half-ladder diagram in the right panel of figure 1 that arises from the partial-fraction decomposition of the n-gon propagators reviewed in appendix A. For a given partition {1, 2, . . . , n} = A ∪ B in (3.15), the leftover task is to absorb the σ j -dependence of the Pfaffian into the (|B|+2)-point Parke-Taylor factors, such as to form (n+2)-point Parke-Taylor factors. The kinematic factors K A (ω, ρ(B)) are multilinear in the polarization vectors of the set A that enter via Pf(Ψ A ). The identical challenge arises at tree level when computing the BCJ master numerators of single-trace (YM+φ 3 )-amplitudes. Recall that both gluons and scalars in (YM+φ 3 ) amplitudes are in the adjoint representation of a color group, and the scalars are additionally in the adjoint representation of a flavor group. A color-stripped amplitude has all the n particles in an ordering, thus CHY half-integrand is given by a (length-n) Parke-Taylor factor. In addition, by "single-trace" we mean the scalars are also in an ordering after stripping off the flavors, and the other CHY half-integrand is given by a Parke-Taylor factor for scalars in legs 1, . . . , k and a Pfaffian for gluons in legs k+1, . . . , n [92]. BCJ master numerators are obtained by reducing to Parke-Taylor factors using scattering equations [26]: Here without loss of generality, we have chosen the ordering for scalars to be 1, τ (2, . . . , k−1), k with τ ∈ S k−2 , and the second line of (3.24) can be attained by the techniques of [35,68,71]: The Parke-Taylor coefficients N tree YM+φ 3 (. . .) are BCJ master numerator associated with a halfladder diagram, with 1 and k on opposite ends and the permutation π ∈ S n−2 acting on the remaining particles (the second ordering 1, τ (2, . . . , k−1), k is for the scalars w.r.t. the flavor group).
By matching (3.23) with (3.24), one can identify the kinematic factors K A (ω, ρ(B)) in a one-loop context with a (YM+φ 3 )-master numerator at tree level. One needs to pick the scalars to be in +, −, B and the gluons to be in A, and choose the two orderings to match the permutations ω, ρ: Two of the current authors present an improved method of computing the necessary N tree in Ref. [72].
As an illustration, let us consider the simplest case with one gluon, i.e. k = n−1, then Pf Ψ {n} = C n,n and partial-fraction manipulations are sufficient to show that [68], where the ordering for the scalars has been chosen as 1, 2, · · · , n−1 for simplicity. By . . , a, n, a+1, . . . , n−1) (for a = 1, · · · , n−2), which read K(ω, ρ a (B)) = a i=1 n · k i . One can proceed similarly in case of more gluons: for k = n−2, by expanding PfΨ {n−1,n} and using scattering equation of leg n, after some algebra we obtain [68] PfΨ where we have Parke-Taylor factors with label n and n−1 inserted at various positions. In this way, one can continue with more and more gluons and obtain BCJ master numerators for single-trace amplitudes in YM+φ 3 [35]. Similar techniques have been used in e.g. [68,70,71], and more recently in [72,93,94].

General gauge theories
In this section, we move to more general gauge theories in even dimensions whose spectrum may involve an arbitrary combination of adjoint scalars, fermions and gauge bosons. Accordingly, their one-loop correlators are built from forward limits of not only vectors and Weyl fermions but also scalars. As we will review, the tree-level correlator with 2 scalars and n−2 gluons can be obtained from dimension reduction of the n-gluon one [38]. By combining all the building blocks from forward limits, we then have a formula for one-loop correlators with n v vectors, n f Weyl fermions and n s scalars in D dimensions 11 . We will present examples of such correlators in various theories in D = 6 and D = 4.

Forward limits in general gauge theories
Before we present a formula for general one-loop correlators in even dimension, let us first review the tree-level correlator involving two scalars. In fact, the bosonic tree-level correlators (2.6) can be straightforwardly adapted to two external scalars in legs 1, n by taking their polarizations 1 , n to satisfy which can also be realized from dimensional reduction. The resulting scalar correlator solely features the B = ∅ term of (2.6), where we have used the subscript "s" to denote scalars (recall that gluons have no subscript). The scalar forward limit analogous to (2.20) simply amounts to FWL i,j (k i , k j ) = (+ , − ),   11 We denote these numbers of different species by boldface n, to avoid confusion with the n th external leg.
Note that n s only appears in the term with B = ∅ and A = {1, 2, . . . , n}, and we again have the loop-momentum dependence C jj = µ j ( µ σ j,+ − µ σ j,− ) + . . . in Pf(Ψ A ) for any choice of A. Moreover, the coefficient n v (D−2) + n s − n f 2 D/2−2 of Pf(Ψ {12...n} ) can be recognized as the difference of bosonic and fermionic on-shell degrees of freedom: D-dimensional vector bosons and Weyl fermions have D−2 and 2 D/2−2 physical degrees of freedom, respectively. Hence, the B = ∅ contribution to (4.4) is absent in supersymmetric theories.
Given that tr V (. . .) and tr S (. . .) vanish at |B| = 1, supersymmetric theories admit at most |A| = n−2 particles in Pf(Ψ A ). As a consequence, the maximum power of loop momenta in the parity-even part of supersymmetric correlators is n−2 , reproducing the power counting of [44] (such power-counting has been studied since the early days of unitarity methods [95][96][97]). As will be detailed below, the parity-odd contributions to D = 4 correlators with four supercharges may exceed this bound and involve up to n−1 powers of .
(ii) A six-dimensional hypermultiplet w.r.t. 8 supercharges contains a single Weyl fermion n f = 1 and two scalars n s = 2 with a total of 2 + 2 on-shell degrees of freedom, (4.7) The simplest contributions at |B| = 2, 3, 4 are The expressions in (4.6) and (4.8) confirm the decomposition of a ten-dimensional gauge multiplet into one vector multiplet and two hypermultiplets in D = 6: By adding two copies of (4.8) to (4.6), the two-and three-traces drop out, and one recovers the four-trace of D = 10 SYM in (3.16). In sec. 6.3, we will spell out simplified expressions for (n ≤ 5)-point BCJ numerators resulting from (4.7) in terms of multiparticle fields.
(iii) Reducing all the way to D = 4, we can examine a gauge multiplet N = 1 SYM, which has two fermionic degrees of freedom, so with n v = 1 and n f = 2 (1,2,0,D=4) (1, 2, . . . , n) = FWL +,− I tree bos (+, 1, 2, . . . , n, −) − I tree 2f (+ f , 1, 2, . . . , n, − f ) The first three contributions in the |B| ≥ 2 sector can be easily read off from (3.8) and (3.9), as in the previous examples, Examinations of extended N = 4, 2 supersymmetry in D=4 are redundant since the respective correlators are equivalent to the D=10 example in (3.15) and the D=6 example in (4.4). In absence of supersymmetry, the four-point instance of (4.4) has been used in [40] to reproduce the BCJ numerators of [98] with up to four powers of loop momentum for the box diagram. Finally, we remark that the BCJ numerators in these general gauge theories can be extracted from the same worldsheet techniques as for D = 10 SYM: In all cases, their σdependence exclusively enters in the form of PT(+, 1, 2, . . . , j, −) Pf Ψ {j+1...n} whose rewriting in terms of n+2-point Parke-Taylor factors can be reduced to a solved tree-level problem as discussed in section 3.4. We will present some examples for such BCJ numerators in sec. 6 and simplify them using multiparticle fields.

Parity-odd contributions
In this section, we derive parity-odd contributions to one-loop correlators from forward limits in chiral fermions. More specifically, this amounts to a parity-odd completion of the correlators (3.15) for D = 10 SYM and those instances of (4.4) with a chiral spectrum.

General prescription and low-multiplicity validation
The worldsheet prescription for the parity-odd sector of one-loop amplitudes has been discussed in [99][100][101] for conventional strings and in [30] for ambitwistor strings. Both approaches have in common that one of the bosonic vertex operators needs to be inserted in the ghost picture −1. This insertion of V (−1) in (2.1) is essential for zero-mode saturation in the ghost sector and gauge anomalies such as the hexagon anomaly of D = 10 SYM [102][103][104]. Accordingly, the forward-limit implementation of the parity-odd sector should start from a tree-level correlator that also has an insertion of V (−1) . That is why the forward limit is performed in the representation (2.11) of the two-fermion correlator at tree level, where both two fermions are in the −1/2 ghost picture. The forward-limit prescription follows from (2.3) & (2.21). To ensure the correct relative normalization between the parityodd and parity-even sectors, we repeat the exercise from sections 3.1 & 3.3 of fixing the relative factor β ∈ Q between the bosonic and fermionic forward limits using known properties, With judicious application of scattering equations, the choice β = −1 reproduces the n = 4 result calculated in (3.15) & (3.16). 12 The forward limit (5.2) has also been studied by Frost [85], where the singularities in σ +,− were demonstrated to cancel between the bosonic and fermionic contribution. Also, the fermionic forward limit was related to the τ → i∞ limit of the Ramond-sector contribution to bosonic one-loop correlators which generalizes the analysis of [65] to ghost pictures (− 1 2 , − 1 2 , −1). The parity-odd forward limit inherits this choice, converting the tree-level correlator (2.11) into The notation tr odd (. . .) instructs to only keep the parity-odd part of the chiral trace 13 proportional to the Levi-Civita symbol ε µ 1 µ 2 ...µ D , Higher points for the parity-even sector could in principle be done using this forward limit. However, the presence of / and / 1 (without an accompanying / k1) obscure the supersymmetry cancelations, requiring increasingly complicated application of scattering equations. As we will see shortly, the choice for β is also reinforced by matching the expected relative factor between parity-odd and parity-even results. 13 When contracting µ , f µν i and µ 1 with 2 D/2 ×2 D/2 Dirac gamma matrices Γ µ instead of the 2 D/2−1 ×2 D/2−1 Weyl blocks γ µ , one can obtain tr odd (. . .) by inserting the D-dimensional chirality matrix ΓD+1 into the trace. Accordingly, tr odd (. . .) with less than D gamma matrices in the ellipsis automatically vanish, tr odd (γ µ 1 γ µ 2 . . . γ µp ) = 0 ∀ p < D . (5.5) Hence, the partitions of {2, 3, . . . , n} into A, B, C must have at least |B| + |C| ≥ D 2 − 1 to allow for a non-vanishing trace, starting with This implies a minimum multiplicity n = D 2 to obtain non-zero parity-odd correlators ( in lines with the analysis of fermionic zero mode in one-loop worldsheet prescriptions [30,[99][100][101]. Moreover, the tensor structure of the ( D 2 )-point correlator (5.8) is entirely determined by the fermionic zero modes. Like this, the permutation-symmetric sum over Parke-Taylor factors in (5.8) is consistent with the worldsheet derivation. In order to avoid proliferation of indices, we employ shorthands for Levi-Civita contractions of D-dimensional vectors v j . In this notation, the permutationsymmetric BCJ-numerators following from (5.8) are given by after absorbing the leading factor of 1 2 following the definition of N (1) from (3.22). In D = 10 dimensions, this becomes a five-point numerator that reproduces the parityodd part of the pentagon numerator iε 10 ( , 1 , k 2 , 2 , k 3 , 3 , . . . , k 5 , 5 ) in ten-dimensional SYM [15,44]. With the normalization of (5.10) and (3.15), we arrive at the relative factor of parity-even and -odd terms known from [105] that plays an important role for S-duality of the five-point one-loop amplitude of type-IIB superstrings. Similarly, (5.10) in D = 6 yields the parity-odd term iε 6 ( , 1 , k 2 , 2 , k 3 , 3 ) in the triangle numerator of chiral six-dimensional SYM with eight supercharges [44,106].

Anomalies and their singled-out leg
In order to reproduce the expected gauge anomalies from our parity-odd correlators, we need to evaluate the forward-limit prescription (5.3) at multiplicities ≥ D 2 + 1. This requires chiral gamma traces beyond (5.5) and (5.6) such as 14 and its generalizations, details of which are provided in appendix C.3 (also see (4.35b) of [65] for an alternative form of the all-multiplicity result). We have checked for the six-point correlator of D = 10 SYM and for the four-point correlator of chiral SYM in D = 6 that the forward-limit prescription ( where the notation (2, 3|2, 3, 4, . . . , D 2 +1) is explained below (3.21). The ρ-dependent signs sgn ρ ij are defined in (3.13), and we have introduced the following shorthands for the tensor structures in the last two lines: Note that we have used the overantisymmetrization identity in deriving (5.12) from (5.3). As a major advantage of the forward-limit prescription (5.3), it bypasses the reference to the spurious position of the picture-changing operator in the one-loop worldsheet prescription [30]. Like this, the Parke-Taylor decomposition of (n ≥ D 2 + 1)-point correlators is greatly facilitated by the approach in this section.
On the support of the scattering equations, (5.12) vanishes under linearized gauge variations j → k j in all the legs j = 2, 3, . . . , n except for the first one. The variation 1 → k 1 in the leg which is singled out by the hat notation in (5.12) is proportional to 2 I (1) odd (1, 2, . . . , n) and therefore yields rational loop integrals, see section 5.5 of [44] for details in a CHY context 15 . Given the asymmetric gauge variations, the ( D 2 +1)-point correlator (5.12) cannot be permutation invariant, not even on the support of scattering equations. Indeed, the difference between singling out legs 1 and 2 through the ghost picture (−1) in (2.11) is given by [44] I see [62] for the analogous asymmetry of the one-loop six-point amplitude of the pure-spinor superstring.

BCJ numerators in terms of multiparticle fields
In this section, we provide alternative representations of the BCJ numerators, where the contributions from the Pfaffian and the field-strength traces in the correlators (3.15), (4.4) and (5.3) are combined. The driving force for particularly compact expressions are so-called multiparticle fields -essentially the numerators of Berends-Giele currents [74] in BCJ gauge, where the color-kinematics duality is manifest [60,75]. Multiparticle fields were initially constructed in pure-spinor superspace [73] (see [13,14] for tree-level precursors) and later on formulated in components for arbitrary combinations of bosons and fermions [61]. They became central ingredients of BCJ numerators [13,15,17,44] and correlators for multiparticle string amplitudes [58,[62][63][64]108].

Brief review
Multiparticle polarizations µ P and field strengths f µν P will be indexed by words P = 12 . . . p or multiparticle labels. This subsection simply collects the definitions relevant to later equations, and the reader is referred to [61,106,109] for further background.
Two-particle versions of polarization vectors and field strengths are defined by and obey µ 12 = − µ 21 as well as f µν 12 = −f µν 21 . Here and below, the notation for multiparticle momenta is k 12...p = k 1 + k 2 + . . . + k p . (6.3) Three-particle polarizations are defined in two steps: Promoting (6.1) to a recursion with labels (1, 2) → (12, 3) yields the intermediate expression Some of the later numerators involve the four-particle field strength that can be assembled from  [44], also see [15] for their supersymmetrization. In the same way as the t 8 -tensor (3.17) furnishes the four-point BCJ numerators in (3.18), higher-point numerators will boil down to its contraction with multiparticle field strengths such as (6.2), (6.6) and (6.7), The symmetries of f µν A in its multiparticle label A = 12 . . . propagate to (6.8) in the obvious manner, e.g. Note that the contribution f µ 1 λ f λν 2 − f µ 2 λ f λν 1 to the two-particle field strength in (6.2) stems from the commutators [f 1 , f 2 ] µν in the t 8 -representation of the five-traces (3.19). The remaining contributions to f µν 12 such as ( and its permutations in (3.15). The worldsheet origin of the five-point numerators (6.12) is also explained in appendix D of [44], using the one-loop ambitwistor-string prescription in the RNS formalism, and its supersymmetrization can be found in [15]. Additionally, antisymmetrizing (6.12) in 1,2, we find the numerator of a massive box diagram (with legs 1,2 in a dangling tree) to be t 8 (12,3,4,5).
In the seven-point generalization of (6.13), all -dependent terms can be anticipated by adjoining a vector index to the building blocks of the above N The -independent terms in the last five lines contain the new seven-point information 16 . We have introduced a vectorial and a two-particle version of the permutation symmetric hexagon building block (6.14), Additionally, we gather those terms which could not be lined up with multiparticle polarizations in the new ∆ building block, This object is antisymmetric in the two labels to the left of the vertical bar, and as such contributes to the seven point hexagon numerator where those two legs have been pulled out as the dangling tree. It would be interesting to relate (6.17) to a component version of the so-called refined building blocks J in pure-spinor superspace [63,110].
Its simplest instance t 4 (1, 2) = (k 1 · 2 )(k 2 · 1 ) − (k 1 · k 2 )( 1 · 2 ) vanishes in the momentum phase space of two massless particles, but we will find non-vanishing multiparticle examples. In particular, one can attain linearized gauge invariance at the level of loop integrands by relaxing momentum conservation: The numerators of this section are understood to rely on no Mandelstam identity other than s 12...n = 0 at n points. This proposal goes back to work of Minahan in 1987 [111] and will be referred to as Minahaning (also see [7,58,106] for four-point implementations).
At three points for instance, Minahaning amounts to keeping nonzero s ij while imposing s 12 +s 13 +s 23 = 0, and it introduces non-vanishing s ijk at four points. For dot products with polarization vectors, transversality and momentum conservation will be used as usual, i.e. ( 1 · k 12...n ) = ( 1 · k 2...n ) = 0. These choices lead to t 4 (12, 3) = (k 1 · k 2 )( 1 · 2 )(k 1 · 3 ), where the factor of (k 1 · k 2 ) cancels the formally divergent propagator (k 1 +k 2 ) −2 of a threepoint diagram with an external bubble. More generally, any potentially divergent propagator introduced by Parke-Taylor integrals (i.e. forward limits of doubly-partial amplitudes) will be cancelled by the corresponding Mandelstam invariant from the numerators of this section. However, this mechanism does not cure forward-limit divergences in the tree-level propagators that arise when integrating non-supersymmetric correlators (4.4) in terms of doubly-partial amplitudes.
Similar to (6.9) and (6.10), the subsequent numerators are built from vector and tensor generalizations of the scalar building block (6.19), where the quantity t 8 (1,2,3,4) in the last line generalizes (6.14) to half-maximal supersym-of this new five-point result has been greatly facilitated by the representation (4.4) of the correlator induced by forward limits.

Parity-odd examples
The forward-limit prescription (5.3) for parity-odd correlators can also be lined up with compact BCJ numerators in terms of multiparticle fields. On top of the simplest non-vanishing numerator (5.10) at multiplicity n = D 2 , the correlator (5.12) at D 2 +1 points leads to the BCJ numerators −iN

Summary and outlook
In this work, we have constructed streamlined representations of one-loop correlators in various gauge theories by taking forward limits of tree-level correlators. Our results are driven by new representations of two-fermion correlators at tree level which closely resemble their bosonic counterparts. The combination of their forward limits therefore manifests all supersymmetry cancellations, and the power counting of loop momenta follows from representationtheoretic identities between Lorentz traces over vector and spinor indices.
Our results apply to gauge-theory correlators in arbitrary even dimensions and with any combination of adjoint scalars, fermions and gauge bosons running in the loop. Also in the non-supersymmetric case, we expand the correlators in terms of Parke-Taylor factors in a subset of the external legs accompanied by Pfaffians. It is then straightforward to extract BCJ numerators w.r.t. linearized propagators by rearranging the Parke-Taylor factors according to well-established tree-level techniques in the YM+φ 3 theory.
A variety of interesting follow-up questions is left for the future, for instance: • The strategy of this work calls for an application to higher-loop correlators, starting from the two-loop case on a bi-nodal Riemann sphere [39,41]. It remains to identify suitable representations of tree-level correlators to perform multiple forward limits, and the four-fermion correlator in appendix B.4 could be a convenient starting point. The gluing operators of [65] and the discussion of double-forward limits in [42] will give crucial guidance in this endeavor.
• The Parke-Taylor decompositions of the one-loop correlators in this work lead to BCJ numerators w.r.t. linearized propagator in the loop momenta. Their algorithmic recombination to quadratic propagators is still an open problem (see [49][50][51][52][53] for recent progress along this direction) and has not yet been understood at the level of the (n+2)point tree-level building blocks. We hope that our representations of BCJ numerators in general gauge theories provide helpful case studies to (i) pinpoint the key mechanisms in the conversion to quadratic propagators (ii) offer a way to preserve the BCJ duality in this process.
• The description of our one-loop BCJ numerators in terms of multiparticle fields has not yet been generalized to arbitrary multiplicity. Even though the Berends-Giele currents for tree-level subdiagrams in BCJ gauge are available to all multiplicity [60,75], their composition rules in one-loop numerators involve additional structures. An allmultiplicity construction of one-loop BCJ numerators from multiparticle fields is likely to shed new light on the long-standing questions concerning a kinematic algebra. each term in the sum over i can be interpreted as one way of opening up the n-gon and is associated with an (n+2)-point tree diagram involving off-shell momenta ± [38,45]. Each of the cubic diagrams can have different kinematic numerators, leaving a total of n! inequivalent n-gon numerators. The manipulations in (A.1) also apply to one-loop integrals with massive momenta k A , k B , k C and k D such as k A = k a 1 + k a 2 + . . . + k ap for A = a 1 a 2 . . . a p . E.g. a massive box admits the following four-term representation: These rearrangements uniquely decompose the one-loop integrand for color-ordered singletrace amplitudes into n terms dubbed partial integrands [43], similar to the decomposition (A.1) of the n-gon. Each partial integrand can be interpreted as the forward limit of a colorordered (n+2)-point tree amplitude with off-shell momenta, where for instance the momenta of the two legs between n and 1 in figure 1 are identified as and − [38]. Although it is an open problem to perform loop integrals over linearized propagators, the above rearrangements of loop integrals have to yield the same result as integrating the quadratic propagators. Such integrals naturally arise from one-loop CHY formulas, which can be obtained by performing forward limits on tree-level CHY formulas, or by localizing the τ integral of ambitwistor string formula at genus one [30] at the cusp τ → i∞, where the torus degenerates to a nodal sphere [36,37]. A general formula for e.g. one-loop amplitudes of gravity and gauge theories in D spacetime dimensions reads (with the normalization factor N from the tree amplitude (1.1)) where E i is the i-th tree-level scattering equation of (n+2) points and we take forward limit by k ± → ± . As indicated by the prime, three of the equations are redundant due to the SL(2, C) symmetry. For gauge theories, one of the two half integrands I Note that these integrals with linearized propagators not only naturally appear from CHY formulas, but also enter the Q-cut representation of loop amplitudes [112]. Such representations provide a well-defined notion of "loop integrands" for non-planar diagrams and generic theories 19 and offer valuable perspectives on the structure of loop amplitudes. It also allows one to generalize KLT and BCJ relations to one loop [43,44].

B.1 CFT basics
In the worldsheet conformal field theory (CFT) of the RNS formalism in D = 10, the free-field OPEs relevant for the correlators of gluon vertex operators (2.1) read The spin field in the fermion vertex (2.2) interacts with worldsheet spinor ψ µ via As a result of the OPEs, we have two and three-point correlation functions (σ ij = σ i −σ j ) see [113] for higher-point spin-field correlators in various dimensions. 19 Also see [23] for the emergence of global loop integrands from the field-theory limit of string amplitudes.
The conformal fields ψ µ and S α are primary fields of a Kac-Moody current algebra at level k = 1 with generators ψ µ ψ ν . By Kac-Moody Ward identities, current insertions in a correlator can be removed by summing over all OPE singularities such as [80,82,84] with the normalization conventions 2η λ[ν η µ]ρ = η λν η µρ −η λµ η νρ for antisymmetrization brackets. Hence, current-algebra techniques can be used to straightforwardly compute spin-field correlators with any number of ψ µ ψ ν insertions. In this way, the contributions ∼ f µν ψ µ ψ ν (σ) of bosonic vertex operator (2.1) in the zero picture can be addressed in presence of spin fields.

B.2 Bosonic correlators and the Pfaffian
When the bosonic correlator (2.4) is evaluated as the reduced Pfaffian in (2.5), the antisymmetric 2n × 2n matrix Ψ {12...n} is organized into n × n blocks A, B and C [26] with C T denoting the transpose of C. The entries of the n × n matrices A, B, C are given by 20 . (B.9) We define the Pfaffian of a 2n × 2n anti-symmetric matrix as As a consequence of momentum conservation and scattering equations, the matrix Ψ has two null vectors such that Pf Ψ = 0. The reduced Pfaffian in (2.5), by contrast, yields a non-vanishing bosonic correlator on the support of momentum conservation and scattering equations.
The diagonal terms of the C-matrix in (B.9) arise when the first term ∼ µ P µ (σ) in V (0) contracts the plane waves of the remaining vertex operators, 2) for the underlying OPEs. Accordingly, when multiple V (0) contribute through the conformal field P µ , the plane-wave correlators relevant to any number of bosons and fermions evaluate to This is the CFT origin of those term in the correlators (2.6), (2.11) and (2.16), where the Pfaffian Pf Ψ A contributes via products of the C jj for all the labels in the set A. The admixtures of the Aand B-blocks in (B.9) as well as the non-diagonal C ij at i = j will be discussed in the next subsections.

B.3 Two-fermion correlators
For the first representation (2.11) of the fermionic vertex operator, the three-point example spelt out in (2.14) is an immediate consequence of the spin-field correlator (B.5). We shall now derive the contributions from the additional insertions of V (0) j (σ j ) at n ≥ 4 points from the recursive techniques outlined above.
At four points, the first term V In passing to the last line, we have inserted the three-point correlator (B.5) and used the gamma-matrix identity γ µ αβ η νλ − γ ν αβ η µλ = 1 2 (γ µν γ λ ) αβ + 1 2 (γ µν γ λ ) βα . Upon contraction with Iterating these OPEs leads to products of gamma matrices, where the multiplication order is correlated with the labels of the accompanying σ −1 ij . Partial-fraction manipulations and the commutators of γ µν can be used to arrive at the same number of gamma matrices and at a chain-structure (. . . σ ij σ jk σ kl . . .) −1 in each term. By analyzing the combinatorics of this algorithm and keeping in mind that the correlator does not depend on the order in which the ψ µ ψ ν are eliminated via Ward identities, one arrives at the n-point expression in (2.11). The same logic has been used in deriving the n-point tree-level correlator in the purespinor formalism [14], where the double-pole contributions have been absorbed to redefine the kinematic factors of the simple poles and to eventually obtain multiparticle superfields.
The same way of applying Kac-Moody Ward identities gives rise to the alternative form (2.16) of the two-fermion correlator. For instance, the three-point correlator in (2.17) follows from the same use of Ward identities that eliminated a single Lorentz current in (B.13). On the one hand, the three-point correlator involving fermionic ghost pictures V (−1/2) V (−3/2) shares certain intermediate steps with the four-point correlator from V (−1/2) V (−1/2) . On the other hand, we can give the same kind of all-multiplicity results (2.11) and (2.16) for both ghost-picture assignments. The discussion in sec. 3.1 illustrates that (2.16) due to n−2 insertions of V (0) instead of n−3 is more suitable to manifest the interplay with the bosonic correlator (2.6) upon forward limits.
These techniques to successively remove insertions of ψ µ ψ ν from the correlator are universal to the SO(D) Kac-Moody symmetry of the RNS model [80,82,84] in any number of spacetime dimensions D. Since the Clifford algebra (2.13) also takes the same form in any number of dimensions, the structure of the gamma-matrix product in the two-fermion correlators (2.11) and (2.16) is universal to any even value of D. The only D-dependent aspect of these correlators is the relative chirality of the fermion wavefunctions χ j which can be understood from the three-point correlator (B.5) for lower-dimensional spin fields that initiates the recursion based on Ward identities. The D-dimensional three-point correlator is nonzero in case of alike chiralities in D = 2 mod 4 and opposite chiralities in D = 4 mod 4, see e.g. section 3 of [113]. Hence, the two-fermion correlators in (2.11) and (2.16) can be used in any even D ≤ 10 provided that one of the chiralities is flipped in D = 4 mod 4.

B.4 Four-fermion correlators
The recursive computation of two-fermion correlators can be straightforwardly extended to the four-fermion case. In this case, Ward identities reduce correlators with Lorentz-current insertions to the basic spin-field correlator Note that this result is specific to D = 10 dimensions, see [84,113] for the tensor structure of lower dimensional four-spin-field correlators. Permutation invariance under exchange of (α, σ 1 ) ↔ (β, σ 2 ) is obscured on the right-hand side of (B.19) but can be checked using the gamma matrix identity γ µ(αβ γ µ γ)δ = 0 in ten dimensions. It can be manifested by rewriting the correlator as a reduced determinant with entries γ µ αβ /σ 12 . As an immediate consequence of (B.19), the four-fermion correlator is given by (B.20) Additional bosonic vertex operators yield the same contributions of C jj from V as detailed in the two-fermion case. For instance, the five-point correlator is obtained in the following form Note that the exchange of 2 and 3 acts on both the χ j and on the punctures in the four-point Parke-Taylor factor as well as the σ −1 ij inside the square brackets. One may eliminate one of the field-strength contractions via to manifest the quadratic falloff as σ 5 → ∞, but we chose to display (B.21) in the more symmetric form, where the generalization to higher multiplicity is more apparent. Similar to the two-fermion case, the general formula is then given by a sum over all subsets A of the bosons {5, 6, . . . , n} along with Pf Ψ A . For a fixed choice of A, it remains to sum over all possibilities to insert gamma-matrix contracted field strengths / f j of the bosons in the complement of A adjacent to the four fermion wavefunctions.
To simplify the notation, let us define a "field-strength-inserted" fermion wave function, X i,B i for a fermion i and a set of bosons where we sum over permutations of B i . In (B.21), we have one of the simplest examples With this definition, the numerator (χ 1 γ µ χ 2 )(χ 3 γ µ χ 4 ) is generalized to (X 1,B 1 γ µ X 2,B 2 ) ×(X 3,B 3 γ µ X 4,B 4 ), which has four sets of field-strength insertions B 1 , B 2 , B 3 , B 4 , associated with fermions 1, 2, 3, 4, respectively. This is symmetric for bosons in each set B i (i = 1, 2, 3, 4), and by the gamma-matrix identity underlying (B.22) has the correct SL 2 weights for the σ j of all the bosons involved. Now it becomes clear how to write down the general form of the n-point correlator with four fermions: It would be interesting to apply double-forward limits of this result to supersymmetric twoloop amplitudes [42].
C Details of gamma-matrix traces Here we present a derivation of the decomposition of tr S in terms of tr V given in (3.11). First, we remind the reader of a well-known recursive formula for calculating γ traces. Using that formula, we'll find relative signs and the overall factor for the length n tr V within the length n tr S . Then we show how the multitrace terms arise from the recursive calculation of the γ traces. For notational simplicity, we will focus on the sequential ordering of labels tr S (1, 2, . . . , n) in (3.4), with the understanding that that other orderings can be reached by application of suitable permutations. The parity even piece of a generic-length γ trace, in arbitrary dimension, can be computed using tr [γ µ 1 γ µ 2 . . . γ µn ] = n j=2 (−1) j η µ 1 µ j tr γ µ 2 . . .γ µ j . . . γ µn , (C.1) where tr γ µ 2 . . .γ µ j . . . γ µn is the tr of n − 2 γs with γ µ j removed. The recursion ends with tr [id CA ] which depends on the representation of the Clifford algebra, and therefore carries the D dependence of the traces. We can use this formula to evaluate the tr S , using (2.12) to rewrite tr S (1, 2, . . . , n) = 2 −n f µ 1 ν 1

1
. . . f µnνn n η ν 1 µ σ (2) . . . η ν σ(n) µ 1 (C.3) with σ ∈ S n−1 , and the explicit reversal contribution is included to demonstrate the factor of 2 −j in (2.12). The η ν 1 µ σ(2) term can be directly sourced out of (C.1) by rotating the tr in (C.2) using cyclicity so that γ ν 1 is the first in the string. Then, since γ µ σ(2) will always occupy the even slots in the trace, this term always carries a +. The final η ν σ(n) µ 1 can always be chosen as the last step of the (C.1), and thus also always carries a +. However, γ µ 1 remaining in the tr until the end is vitally important, as it is what breaks the symmetry between the two intermediate cases: σ(i) coming before σ(j) in 1, 2, . . . , n, or coming after. If σ(i) comes first, then the tr can be cycled such that γ ν σ(i) is at the front of the trace, and this cycling will never put γ µ 1 between γ ν σ(i) and γ µ σ(j) . Since each pair of γ µ γ ν can be removed in adjacent steps of this recursion, there will always be an even number of γ between γ ν σ(i) and γ µ σ(j) , and thus (C.1) will provide a + contribution. On the other hand, when σ(i) comes after σ(j), the process of cycling γ ν σ(i) to the front will always leave γ µ 1 between γ ν σ(i) and γ µ σ(j) . As in the previous case, there will always be an even number of γs removed between σ(i) and σ(j), but now γ µ 1 shifts the counting by 1, so (C.1) will introduce a − sign. We collect all of the resulting signs into the ord function introduced in (3.12) to get which is provides the leading trace term from (3.11). Notably, the reversed tr V from (C.3) is included as one of the elements of ρ. The recursive realization of the tr[γ . . . ] in (C.1) also naturally generates the multi-Lorentz-trace terms. Each of the subtraces can be resolved, one at a time, in the same method as above. The γ not participating in the targeted subtrace always cycle together, and thus only shift the counting between targeted γ by an even number, never changing the sign. Each tr V picks up a factor of 1 2 as in (C.3) to account for the reversal overcount, leading to the factor of 2 −j in (3.11).

C.2 Higher t 2n tensors from D = 10 SYM
This appendix gives more details on the permutation symmetric tensors t 2n defined in (3.20). More specifically, we will determine the coefficients of tr V (1, 2, . . . , n) once the spinor traces are rewritten in terms of vectorial ones via (3.11). This will allow to verify the cancellation of the six-trace from the exceptionally simple expression (3.21) for t 12 .
Using (3.11), we can count the + and − contributions of the longest tr V (1, 2, . . . , n) to the permutation sum (3.20) defining t 2n . Since t 2n is fully permutation symmetric, it suffices to count the number of permutations in S n−1 that generate a positive coefficient for tr V (1, 2, . . . , n) vs those that generate a negative one. These counts can be expressed directly in terms of the Eulerian numbers which count the number of permutations of length i that have k permutation ascents; adjacent labels in the permutation ρ that have ρ j < ρ j+1 are a permutation ascent. This is exactly the information needed by the ord ρ σ sign (3.12), and as such those terms with k even will carry a + sign, while k odd will carry a −. 21 The symmetric tensor t 2n (f 1 , f 2 , . . . , f n ) will contain the term tr V (1, 2, . . . , n) with a coefficient given by The additional overall factor of 2 is due to the parity properties (3.5). As a necessary condition for the simplification (3.21) of t 12 , the case with n so there is no contribution of tr V (1, 2, . . . , 6) to correlators (3.15) of D = 10 SYM up to and including seven points. However, all other even n admit length-n Lorentz traces.
will fully contract the ε ν... first, and then leave behind tr(γ µ i γ µ j ) that are not contracted into the ε. Thus, running the recursive evaluation until the ε is completely contracted, we find tr odd (γ µ 1 . . . γ µ D γ µ D+1 γ µ D+2 ) = A (D) ∪B (2) =(1,2,...,D+2) i(−1) skip A B ε µ A 1 ...µ A D tr(γ µ B 1 γ µ B 2 ) , (C. 10) where the summation range A (D) ∪ B (2) = (1, 2, . . . , D+2) follows our convention of A and B being disjoint ordered subsets of (1, 2, . . . , D+2), with the additional constraint that A is length D, and B length 2. The sign (−1) skip A B compensates for skipping over the γ µ B i as the γ µ A j are paired with γ ν k , ensuring that all of the terms in the remaining B trace eventually have the correct relative signs. For the simple case in (C.10), skip A B is the number of A i between the two elements of B, which can in turn be reduced to the representation given in (5.11).
To generalize this computation to D+2j γs, we need to more carefully account for skip. As mentioned, it needs to restore the signs required in (C.1) that were dropped when separating the indices into the A and B set. A convenient definition for skip A B that accomplishes this is number of elements of B before A i . (C.11) Note that this definition exactly captures the behavior described by (5.11): an even separation between the B i will have skip A B = 0 + · · · + 1 + · · · + 1 even +2 + · · · → (−1) skip A B = 1 , (C. 12) whereas an odd separation will give skip A B = 0 + · · · + 1 + · · · + 1 odd +2 + · · · → (−1) skip A B = −1 . i(−1) skip A B ε µ A 1 ...µ A D tr(γ µ B 1 . . . γ γ B j ) . (C.14) Notably, this construction specifically includes (5.6) as the j = 0 case, since we are using the convention tr(id CA ) = 2 D/2−1 .