One-loop superstring six-point amplitudes and anomalies in pure spinor superspace

We present the massless six-point one-loop amplitudes in the open and closed superstring using BRST cohomology arguments from the pure spinor formalism. The hexagon gauge anomaly is traced back to a class of kinematic factors in pure spinor superspace which were recently introduced as BRST pseudo-invariants. This complements previous work where BRST invariance arguments were used to derive the non-anomalous part of the amplitude. The associated worldsheet functions are non-singular and demonstrated to yield total derivatives on moduli space upon gauge variation. These cohomology considerations yield an efficient organizing principle for closed-string amplitudes that match expectations from S-duality in the low-energy limit.


Introduction
Over the past decade, several superstring [1][2][3][4][5][6] and field-theory scattering amplitudes [7][8][9] have been computed in manifestly supersymmetric form using the pure spinor formalism [10]. Computations in the minimal pure spinor formalism relied extensively on the BRST invariance of the amplitude prescription as a way to organize the intermediate steps

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and to simplify the answers. At tree level, this method led to a general solution in closed form for the n-point integrand for both the open superstring [4] as well as its field-theory limit [7]. At higher loops -apart from the four-point one-and two-loop amplitudes of [1][2][3]11] -the superstring computations of [5,6] so far were restricted to the lowenergy limit of the integrand. This limit only receives contributions from a subset of the zero-modes of the pure spinor b-ghost and leads to a simpler analysis of OPE singularities among external vertex operators.
In 2012 [12], the one-loop open superstring n-point integrand restricted to the above zero-mode contributions of the b-ghost was computed in closed form in terms of scalar BRST invariants denoted by C i|A,B,C . These BRST invariants were later given a recursive construction in terms of ten-dimensional SYM superfields including a general expansion in terms of field-theory tree amplitudes [13]. Although the permutation-invariant integrands in [12] yield the desired low-energy behavior, they fail to reproduce the hexagon gauge anomaly on the boundary of moduli space.
The long-term goal of this project is to lift the restriction of b-ghost zero-modes from the one-loop analysis of [12] in order to obtain the complete and supersymmetric n-point one-loop amplitudes of the open superstring. In this paper we take the first step and write the complete six-point one-loop integrand for open and closed superstrings in pure spinor superspace. These results reproduce the pure spinor analysis of the gauge anomaly in [14] and match previous computations done with the RNS formalism. But unlike the RNS answer which is restricted to gluon amplitudes (see [15] for the parity-even and [16,17] for the parity-odd part), the result of this paper is fully supersymmetric and naturally unifies the contributions from both the even and the odd spin structures. Moreover, the worldsheet integrals for both open and closed strings are cast into a basis. For closed strings, a new factorized representation of the five-point kinematics paves the way for an efficient organization of the six-point result.
Since the gauge anomaly probes non-standard contributions from the b-ghost beyond the zero-mode analysis of [12], the six-point one-loop result of this paper harbors important insights about a difficult corner of the pure spinor formalism which currently inhibits further progress in multiloop computations.
2 Review: the hexagon anomaly and its cancellation

The pure spinor description of the anomalous gauge variation
The gauge variation of the six-point open-superstring amplitude at one loop using the pure spinor formalism was computed in [14]. This subsection briefly reviews that derivation.
The non-minimal pure spinor prescription to compute a one-loop amplitude in the type-I superstring with a SO(N ) gauge group is given by [18] (2.1) The sum is over the three worldsheet topologies at one-loop with G top and ∆ top denoting their corresponding Chan-Paton factors and integration domains for z j . Denoting the gen-JHEP04(2016)148 erators of SO(N ) in the fundamental representation by t a i , the Chan-Paton factors for the cylinder with all particles attached to one boundary and the Möbius strip are given by G P = N tr(t a 1 t a 2 t a 3 t a 4 t a 5 t a 6 ) and G N = −tr(t a 1 t a 2 t a 3 t a 4 t a 5 t a 6 ). When particles are attached to both boundaries of the cylinder one has, for example, G N P = tr(t a 1 t a 2 )tr(t a 3 t a 4 t a 5 t a 6 ). The integration domains will be elaborated in section 3.3. Furthermore, t is the one-loop Teichmüller parameter and µ the Beltrami differential, b is the b-ghost (see [18] for the expression in the non-minimal formalism and [1,19] for its schematic form in the minimal formalism), and (b, µ) ≡ d 2 w b(w)µ. The massless vertices are [10] with pure spinor λ α subject to (λγ m λ) = 0, linearized superfields [A α , A m , W α , F mn ] of ten-dimensional SYM [20] and worldsheet fields [∂θ α , Π m , d α , N mn ] of conformal weight h = 1 whose OPEs can be found in [10]. Finally, N regulates the integration over the non-compact space of pure spinors [18]. As in the original derivation of [21,22], the gauge variation of the amplitude can be computed directly by replacing the vertex operators by their gauge variation where Ω j are scalar superfields, and the BRST charge is defined by Since the total derivatives ∂Ω 2 ≡ ∂Ω 2 ∂z 2 from the integrated vertex operators are suppressed by the boundary contribution z i → z j of the integrand, the gauge variation of the six-point amplitude becomes To arrive at the second line the BRST charge was integrated by parts. The only nonvanishing contribution comes from the energy momentum tensor {Q, b} = T and gives rise to a factor of (T, µ) which in turn leads to a total derivative ∂ ∂t on moduli space [23]. The correlator in the second line of (2.5) can be easily evaluated by considering the saturation of fermionic zero-modes of the fermionic field d α . It is well known [18] that at one loop the regulator N provides eleven zero-modes of d α , so the vertices contribute the remaining five in order for the variation (2.5) to be non-vanishing, (dW 2 )(dW 3 )(dW 4 )(dW 5 ) (dW 6 ). Integrating the pure spinor zero-modes has the effect of replacing [24], and (2.5) becomes

Multiparticle kinematic building blocks
The zero-mode structure of the six-point one-loop amplitude in the pure spinor formalism (2.1) allows for two OPEs among massless vertex operators. Such OPEs can be recursively addressed using non-local multiparticle superfields K P ∈ {A P α , A m P , W α P , F mn P } of ten dimensional SYM [13]. They are referred to as Berends-Giele currents and defined by where the multiparticle label P = 12 . . . p encompasses p external legs. The sum in (2.9) instructs to deconcatenate P into non-empty words X = 12 . . . j and Y = j + 1 . . . p with j = 1, 2, . . . , p−1. The shorthand K [X,Y ] is used to represent all the four types of superfields simultaneously. More explicitly [25], Multiparticle momenta for P = 12 . . . p and their associated Mandelstam invariants are given by (2.14) Furthermore, we define the multiparticle version of the vertex operator V in (2.2) as such that M i = V i . The zero-mode saturation in the pure spinor one-loop amplitude prescription selects certain superfields from the integrated vertex operators U in (2.2), such JHEP04(2016)148 in the four-point amplitude [1]. Promoting the superfields to their Berends-Giele currents such as W α i → W α A suggests the following definitions [13,26], which automatically capture the results of iterated OPEs. In (2.17) and later places, the notation (a 1 , . . ., a p | a 1 , . . ., a n ) instructs to sum over all possible ways to choose p elements a 1 , a 2 , . . . , a p out of the set {a 1 , . . ., a n }, for a total of n p terms.

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Their gauge-invariant bosonic components C 1|A,B,C and C m 1|A,B,C,D determined from the zero-mode prescription (2.8) can be downloaded from [27]. As detailed in section 3.1, the scalars in (2.21) enter one-loop open-string amplitudes [12] but fail to explain the hexagon anomaly in view of their BRST invariance QC 1|A,B,C = 0. The vectors C m 1|A,B,C,D in turn are essential to efficiently represent the interactions between left-and right-movers in closed-string amplitudes, see section 4.

BRST pseudo-invariants
The hexagon gauge anomaly can be equivalently seen from a breakdown of BRST invariance, see appendix C for further details. Hence, the superfields in the anomaly kinematic factor (2.7) are required to appear in the BRST variation of the six-point open-string amplitude. We will refer to gauge and BRST anomalies interchangeably in the rest of the paper. The tensorial building block (2.19) selected by zero-mode arguments exhibits an anomalous BRST transformation of this type in its trace component [26], The same anomaly building block Y A,B,C,D,E appears in the context of a scalar anomaly current whose single-particle version reads [26] J 2|3,4,5,6 ≡ While the above definition suffices for the six-point amplitude, a general definition with multiparticle labels can be found in [26]. Instead of a BRST-invariant completion such as the scalars and vectors in (2.21) and (2.22), the recursions of [26]  these superfields are referred to as BRST pseudo-invariants. The motivation for this terminology stems from the purely parity-odd bosonic components which appear in the corresponding gauge variations such as (2.7) [26]. This ties in with the linearized gauge transformations e 1 → k 1 of the expressions for P 1|2|3,4,5,6 and C mn 1|2,3,4,5,6 on the webpage [27].

Worldsheet functions
String amplitudes augment kinematic factors with worldsheet integrals where the former conspire to BRST invariants or pseudo-invariants once the integrals are reduced to a basis. At one loop, the worldsheet integrand comprises doubly-periodic functions of the insertion points z i of the vertex operators such as the bosonic Green function on a genus-one surface with modular parameter τ , where z ij ≡ z i − z j and τ 2 ≡ Im(τ ). Derivatives w.r.t. the first argument of θ 1 (z|τ ) are interchangeably denoted by a tick and by ∂ ≡ ∂ ∂z . Exponentials of (2.29) give rise to the Koba-Nielsen factor from the plane-wave correlator seen for instance in (2.6): see (2.14) for the conventions for Mandelstam invariants s ij . As a main result of this paper, we give a representation for the six-point open-string integrand such that its BRST variation builds up the modular derivative of (2.30) required by the anomalous gauge variation (2.5). For this purpose, we recall a set of doubly-periodic functions ij with n = 0, 1, 2, . . . described in [28] which were identified as a convenient language for one-loop superstring amplitudes. In particular, it turns out that ji and Weierstraß function [29] ℘(z|τ ) ≡ −∂ 2 ln θ 1 (z|τ ) + θ ′′′ 1 (0|τ ) 3θ ′ 1 (0|τ ) (2.33) suffice to describe the six-point amplitude. They are related via Fay's identity as [28], and one can show that the short-distance singularities of (∂ ln θ 1 ) 2 and ℘ drop out from (2.32), rendering f ij non-singular as z ij → 0. The relation of f (2) ij with the τ derivative of the Green function (2.29) is explained and applied in section 3.3. The net result with t ′ ≡ 1/t connects the derivative in moduli space appearing in the gauge variation (2.6) with the function f (2) ij in the anomalous six-point correlator (3.7).

JHEP04(2016)148 3 The complete six-point amplitude of the open string
In applying the pure spinor one-loop prescription (2.1), the non-zero modes of the b-ghost lead to cumbersome CFT calculations. One way to address this difficulty is to use the BRST invariance of the pure spinor formalism as a guiding principle to write down the answers directly. This will be done in this section for the open-string six-point amplitude; the result contains two classes of kinematic factors: BRST invariants (K C ) and pseudoinvariants (K P ). Recalling the zero-mode prescription . . . in (2.8), our conventions are A separate analysis will be performed for each sector, and the pseudo-invariants K P will shortly be defined such as to make contact with the kinematic factor (2.7) of the anomalous gauge variation.

The non-anomalous part of the worldsheet correlator
A gauge-invariant subsector of one-loop open-string amplitudes which describes the lowenergy behavior has been analyzed to all multiplicities in [12]. Its kinematic factors are captured by the BRST-closed scalars C i|A,B,C in pure spinor superspace as exemplified in (2.21). Their derivation considers only the zero-mode contributions from the b-ghost leading to the scalar building blocks (2.16) and follows from integration by parts identities of the worldsheet functions associated to OPE singularities to reduce the integrals to a basis. More specifically, products of worldsheet propagators (2.29) and s ij in (2.14), can be conveniently manipulated by discarding 1 total derivatives acting on the Koba-Nielsen factor (2.30): A basis of worldsheet functions in open-and closed-string correlators can be attained by removing any explicit appearance of the fixed insertion point z 1 along with X 1j through the addition of total derivatives (3.3) with respect to z j . In terms of the worldsheet functions (3.2) and the BRST invariants C i|A,B,C , a permutation-invariant kinematic factor for the six-point amplitude (3.1) is given by [12] K C 6 = X 23 (X 24 + X 34 )C 1|234,5,6 + X 24 (X 23 + X 43 )C 1|243,5,6 + (2, 3, 4|2, 3, 4, 5, 6) (3.4) + X 23 X 45 C 1|23,45,6 + X 24 X 35 C 1|24,35,6 + X 25 X 34 C 1|25,34,6 + (6 ↔ 5, 4, 3, 2) .
1 Boundary terms in zi do not contribute since the exponential of α ′ sijGij vanishes as z α ′ s ij ij for zi → zj. This is obvious if sij has a positive real part, whereas the vanishing for generic momenta follows from analytic continuation [30].

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As initially observed in [12], the scalar BRST invariants C i|A,B,C can be re-expressed in terms of color-ordered SYM tree amplitudes. At six points, the identities [13] C 1|234,5,6 = s 56 s 45 A YM (1, 2, 3, 4, 5, 6) − s 35  allow to straightforwardly express all the polarization dependence of (3.4) in terms of A YM (. . .). However, the above BRST-invariant integrand K C 6 cannot be the complete answer for the six-point open string amplitude since it would imply manifest gauge invariance. 2 In the following, we will show how the anomalous part of the amplitude can be described using the BRST pseudo-invariants derived in [26] and reviewed in section 2.4.

The anomalous part of the worldsheet correlator
In order to correctly describe the anomalous part of one-loop amplitudes, the kinematic factor K P 6 in (3.1) cannot be BRST invariant. According to (2.5), its BRST variation must add up to a total derivative in moduli space and reflect a parity-odd gauge variation. For this purpose, the notion of a pseudo BRST cohomology was introduced in [26] along with recursive method to construct pseudo-invariants of arbitrary multiplicity and tensor rank. Its scalar six-point representative P 1|2|3,4,5,6 has been defined in (2.27), and its BRST variation −V 1 Y 2,3,4,5,6 in terms of the superfields (2.23) tie in with anomaly kinematic factor (2.7). That is why this superfield is suitable to describe the anomalous gauge variation of the six-point integrand.

Its BRST and gauge variations
will be identified as a boundary term in moduli space in section 3.3. Therefore the anomaly cancellation for gauge group SO(32) can be proven as in the RNS formalism and will not be repeated here [16,17,21,22,31].
In contrast to the BRST-invariant kinematic factors C i|A,B,C in (3.5) and (3.6), the pseudo-invariant P 1|2|3,4,5,6 cannot be expressed in terms of SYM tree-level subamplitudes. Two classes of tensor structures in its bosonic components [27] pose an obstruction: JHEP04 (2016)148 1. terms of the schematic form (e i · k j ) 6 where all the six gluon polarization vectors e i with i = 1, 2, . . . , 6 are contracted with an external momentum 2. parity-odd terms involving the ten-dimensional Levi-Civita tensor ǫ m 1 m 2 ...m 10 It is easy to see from Feynman rules and worldsheet supersymmetry that parity-even contractions (e i · k j ) 6 are absent in tree amplitudes of both SYM and the open superstring. 3
Moreover, we have checked that the field-theory limit of the above six-point amplitude reproduces the one-loop integrand of ten-dimensional SYM which has been derived in [8] from cohomology arguments. Upon dimensional reduction to D = 4, the pseudoinvariant P 1|2|3,4,5,6 and therefore the entire anomalous correlator (3.7) vanishes for MHV helicity configurations. Hence, the non-anomalous contribution (3.4) is sufficient to derive the BCJ representation of MHV amplitudes in [36] from the field-theory limit.

The BRST and gauge transformations as boundary terms
In this subsection, we discuss the scalar integrals accompanying the anomalous BRST and gauge variations (3.9) of the six-point amplitude. In particular, they are now demonstrated to describe boundary terms in the moduli space of open-string worldsheets.
In order to relate the BRST variation (3.9) of K P 6 to a total derivative with respect to the modular parameter, it is worthwhile to express the functions f (2) pq in terms of the τ derivative of the bosonic Green function (2.29). For generic complex arguments, the heat equation 4πi ∂θ 1 (z|τ ) ∂τ = ∂ 2 θ 1 (z|τ ) obeyed by the theta function in (2.33) implies that In a convenient parametrization of open-string worldsheets, the arguments z pq , τ of the Green function (2.29) have constant real parts and are integrated over their imaginary parts ν pq ≡ ν p − ν q and t: The integration domains ∆ top for vertex insertions in (2.1) and (3.1) are then given by where ∆ P and ∆ N are adapted to the single-traces over t a 1 t a 2 t a 3 t a 4 t a 5 t a 6 , and ∆ N P refers to the non-planar cylinder diagram with color factor G N P = tr(t a 1 t a 2 )tr(t a 3 t a 4 t a 5 t a 6 ). Hence, the functional dependence of G pq on the real parameters ν pq and t is given as follows in the three inequivalent configurations: Since the difference between planar and non-planar cylinders and the Möbius strip amounts to a constant shift of its arguments, G pq in (3.20) satisfies a universal differential equation, On the right hand side, the definition (2.32) of f (2) pq has been rewritten in terms of νderivatives of G pq . The function c(t) in (3.21) does not depend on ν p and will therefore JHEP04(2016)148 drop out from the later discussion. The differential operator on the left hand side can be recognized as a derivative 4 in the Jacobi transformed modular parameter: The original modular parameter t can be interpreted as the circumference of the cylinder or the worldline length in the field-theory limit. 5 Its Jacobi transform t ′ , on the other hand, describes the length of the cylinder or the proper time in the closed-string channel. Analogous statements hold for the Möbius strip. From (3.21) and (3.22), one can derive a universal relation for f (2) pq analogous to (3.17), This allows to rewrite the t ′ derivative of the Koba-Nielsen factor (2.30) in terms of f ij , ij on the right hand side reproduces the anomalous BRST transformation (3.9) of the six point correlator, Together with the Jacobi transformed integration measure dt = − dt ′ (t ′ ) 2 , one can finally identify the BRST anomaly of the six point amplitude in (3.1) as a boundary term in t ′ : where the transformation dz j = itdz ′ j has compensated for the factor of t −5 in (2.30). Note that modular invariance of the Koba-Nielsen factor allows to collectively replace G(iν pq |it) → G(ν ′ pq |it ′ ). By the universality of (3.23), this analysis is valid for all topologies of open-string worldsheets and the anomaly is canceled for the gauge group SO(32) [21,22].

The complete six-point amplitude of the closed string
This section is devoted to the six-point one-loop amplitude among massless closed-string states of type IIA/IIB superstring theories. Before presenting the six-point function we revisit the five-point amplitude result of [37] to rewrite its kinematics in a factorized form.

The one-loop five point function for closed strings
In [37] the pure spinor representation of the five-point closed-string amplitude in both type IIA/IIB was obtained 6 (in the type IIA the chirality of the right-movers is reversed)  The characteristic coefficient π τ 2 signals the mixing between left-and right-movers and arises from either the contraction Π m (z)Π n (z) or from left-moving derivatives acting on right-moving propagators in integration by parts identities, While the amplitude (4.1) is BRST invariant the kinematic factor L 5 is not manifestly BRST closed. However, by adding terms to L 5 that vanish in the cohomology one arrives at a manifestly BRST invariant expression (the vector C m where (note that Y 1,2,3,4,5 = 0 in the five-particle momentum phase space) [26], Therefore the five-point amplitude (4.3) becomes up to the Q-exact terms in (4.6) that do not contribute upon zero-mode integration. This representation is manifestly BRST invariant (since QC 1|A,B,C = QC m 1|A,B,C,D = 0) and organizes the kinematic dependence in a factorized form w.r.t. left-and right-movers. 6 The RNS and GS representations can be found in [38] and [30], respectively. 7 The shorthand C m 1|2,3,4,5C m 1|2,3,4,5 was assigned a different meaning in [37] and differs from the right-hand side of (4.6) by s23C 1|23,4,5C1|23,4,5 + (2, 3|2, 3, 4, 5).

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The compactness and manifest BRST invariance of (4.8) demonstrate the virtue of vectorial BRST invariants to describe closed-string amplitudes. From the five-point example, one can anticipate that BRST (pseudo-)invariants of rank r find a natural appearance in closed-string amplitudes at higher multiplicity r + 4, along with r powers of π τ 2 . In the subsequent, this expectation is confirmed for the six-point amplitude.

The six-point closed-string correlator
The six-point closed-string correlator M 6 combines the doubling of its open-string counterpart K 6 = K C 6 + K P 6 with an extended set of left-right interactions, where (see (3.4) and (3.7) for the expressions of K C 6 and K P 6 )  The pseudo-invariants in (4.11) obstruct a representation of L 6 as a tensor contraction of the form K mn 6K mn 6 but their presence compensates the anomalous BRST transformation (2.28) of the tensor C mn 1|2,3,4,5,6 such that QL 6 = 0. In addition, we will show in the next section that the form of (4.11) is fixed by the low-energy limit.

Low-energy limits and S-duality of type IIB
In this subsection, we discuss the low-energy limit of one-loop amplitudes among massless closed-string states and relate it to the S-duality implications in type IIB theory.
As explained in [30,37,39,40], the momentum dependence of torus integrals of the form (4.8) and (4.9) can be split into analytic and non-analytic contributions. 8 The leading analytic behavior α ′ → 0 follows unambiguously by setting I(s ij ) → 1 after taking the JHEP04(2016)148 kinematic poles due to integration over d 2 zI(s ij )∂G 23 ∂G 23 → 2πdr 23 r α ′ s 23 −1 23 with r 23 ≡ |z 23 | into account, see e.g. [30,37]. This gives rise to low-energy limits  4) and (4.10) for K C 6 and K m 6 have been inserted into (4.9), whereas the expression (4.11) for L 6 is treated as unknown at this point and will be derived in the subsequent. Note that the anomalous part K P 6 of the open-string correlator does not contribute to the low-energy limit due to the non-singular nature of the f (2) ij as z i → z j , see the discussion below (2.32).
At four and five points, the type IIB graviton components of (4.12) and (4.13) are proportional to the α ′ 3 order of tree-level amplitudes [37,41]. They originate from the R 4 operator in the type IIB low-energy effective action whose tensor structure is determined by supersymmetry and whose coefficient (determined by S-duality) is given by the nonholomorphic Eisenstein series E 3/2 [42][43][44]. The non-linear extension of R 4 equally affects multiparticle amplitudes at the α ′ 3 order at tree level and in the low-energy limit at one loop and leads to the following S-duality prediction, whose proportionality constant c q does not depend on the multiplicity N . The right-hand side borrows the notation of [45] for the low-energy expansion of tree-level amplitudes involving N closed-string states. The entries of the (N − 3)! × (N − 3)! matrices S 0 and M 3 are polynomials of degree N − 3 and 3 in the dimensionless Mandelstam invariants α ′ (k i · k j ), and S 0 is the momentum kernel [46] which appears in the field-theory limit of the KLT formula [47,48]. The matrix M 3 captures the α ′ 3 order in the low-energy expansion of genus-zero worldsheet integrals. 9 A slightly modified argument applies to the components of (4.15) which violate the U(1) R-symmetry of type IIB supergravity. This is indicated by the subscript q of the proportionality constant c q in (4.15). The simplest non-vanishing amplitude with U(1) violation occurs at multiplicity five and charge q = ±2, involving for instance four gravitons

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and one axio-dilaton, see [37,54,55] for its α ′ -expansion. It was argued via S-duality and confirmed through explicit calculation that the constants in (4.15) for charges q = 0, ±2 are related by c ±2 = − 1 3 c 0 [37]. An analogous discussion of the low-energy limit of two-loop five-point amplitudes and their dependence on R charges can be found in [6].
Since the coefficient L 6 of π 2 τ 2 2 in the six-graviton amplitude (4.9) contributes to the low-energy limit (4.14), it can be determined from the S-duality prediction (4.15). More precisely, the form of L 6 in (4.11) is inferred from the following reasoning.
The double contraction Π m Π n Π p Π q → 2δ m(p δ q)n π τ 2 ) 2 gives rise to a contribution of the form 1  Now the S-duality prediction relates the low-energy limit of the closed-string amplitude and the tree-level α ′ 3 terms via (4.15). Demanding the low-energy limit to be BRST invariant and permutation symmetric 10 uniquely fixes L 6 to the form (4.11). A component evaluation for six external gravitons confirms the matching with the tree-level amplitude at order α ′ 3 .

The BRST variation as a boundary term
It will be demonstrated in this section 11 that the BRST (or gauge) variation of the closedstring amplitude (4.9) gives rise to a total derivative in moduli space. BRST invariance of K m 6 and L 6 implies that ij can be expressed in terms of derivatives of the Koba-Nielsen factor with respect to τ and z j : The second step is based on translation invariance ∂ 1 I = − 6 j=2 ∂ j I. It turns out that the differential operator in (4.19) annihilates the right-moving correlatorK 6 since [28] ∂f (k) ij ≡ 1 imply that its constituents f (2) ij and f Hence, the BRST variation in (4.17) can be rewritten as In order to identify this as total derivatives, we have to commute the differential operators mutually cancel after summing p over 2, 3, . . . , 6, so we conclude 12 that the BRST variation of the six point function is a surface term in both τ and z p : The surface integral over the vertex insertions in the second line vanishes because the torus has no boundaries while the vanishing of the surface integral on moduli space follows from modular invariance [56][57][58].

Conclusion and outlook
In this work we combined the one-loop cohomology analysis of [26] with the worldsheet functions studied in [28] to write down the complete six-point one-loop amplitudes of the open and closed string. In doing so, we supplemented the BRST-invariant six-point correlator (3.4) that captures the worldsheet singularities among the external vertices with the non-singular pseudo-invariant correlator (3.7). The pseudo BRST invariance allows us to describe the hexagon gauge anomaly in pure spinor superspace while the non-singular worldsheet functions capture the regular parts of the correlator. Their composition given in (3.7) is such that its non-vanishing gauge variation (3.9) gives rise to a total derivative in moduli space (leading to the usual mechanism of anomaly cancellation [21,22,31]). This condition fixes the superspace form of the anomaly-containing part of the open-string correlator (3.7) and reproduces the bosonic results from earlier analyses within the RNS framework [15,16].

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The (pseudo-)invariant vector and tensor building blocks from the open string allow for elegant representations for closed-string one-loop amplitudes. As elaborated in section 4, any basis integral of the closed string is accompanied by a manifestly (pseudo-)invariant kinematic factor quadratic in the open-string (pseudo-)invariants. In order to arrive at the novel six-point result, in addition to an OPE-driven derivation of the singular part of the correlator, we also used S-duality considerations to completely fix its regular terms. This organizing principle for closed-string one-loop amplitudes has a natural extension beyond maximal supersymmetry, see [59] for examples in orbifold compactifications.
While the results of this paper demonstrate the value of the (pseudo-)cohomology framework of [26], it is imperative to derive them from first principles within the pure spinor formalism. This endeavor is expected to require a more in-depth understanding of how the non-zero modes of the b-ghost contribute to the final expressions in analogy to the RNS supercurrent in appendix B.2. These contributions are currently poorly understood and give rise to difficulties in extending the results for higher-loop amplitudes in [5,6] beyond their low-energy limits.
Furthermore, for gauge groups different than SO(32), additional boundary terms along the lines of [60] arise from regularizing the divergent modular integral. 13 These boundary terms give rise to worry about additional BRST anomalies and ambiguities associated with the choice of the regulator N for the non-compact space of pure spinors [18]. Subtleties of this type are not addressed in this work, and their treatment in a manifestly supersymmetric formalism is left as an interesting open problem.

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The key tool is the multiparticle version of the integrated vertex operator in (2.2), where the linearized superfields in U j are promoted to their multiparticle versions defined in section 2.2. As a consequence of their multiparticle equations of motion, the BRST variation QU = ∂V generalizes to [13] When inserting the left hand side into the amplitude prescription (2.1) in the place of V 1 U 2 , the analysis of {Q, b} around (2.5) can be repeated to show that by the definition (2.23) of the anomaly superfield. Using the Jacobi-transformed variables t ′ = 1 t and z ′ = z t in intermediate steps, the modular derivative of the Koba-Nielsen factor I(s ij ) in (2.30) has been evaluated via (3.24) and gives rise to the functions f (2) ij defined in (2.32). In the conventions of (3.1), one can read off the contribution of Y 12,3,4,5,6 6 i<j s ij f (2) ij to the antisymmetric part of the kinematic factor K 6 from (A.4). This needs to be compared with the amplitude prescription (2.1) involving the right hand side of (A.2): the total derivative ∂M 12 decouples by the suppression of boundary terms in z j via z α ′ s ij ij , and the leftover term V 1 U 2 − V 2 U 1 yields the desired difference between K 6 and its image under (1 ↔ 2). This completes the proof of (3.16).

B Comparison with the RNS computation
We have checked the six-point open-string amplitude (3.7) in pure spinor superspace to reproduce the gluon amplitude from the RNS formalism upon component expansion [27]. Since this comparison rests on the availability of both expressions in a basis of worldsheet integrals, we will sketch the underlying integral reduction on the RNS side in this appendix.

B.1 The parity-even part
The RNS prescription for the parity-even part of one-loop amplitudes is given by where V i denotes the vertex operator of the gluon in the superghost picture zero: The bracket . . . ν,τ instructs to evaluate the correlator in (B.1) on a genus-one Riemann surface with modular parameter τ , and ν = 1, 2, 3 encode the even spin structures of the worldsheet spinors ψ m associated with partition functions (−1) ν θ ν+1 (0|τ ) [15]. Correlators among x m and ψ m can be straightforwardly evaluated using Wick contractions x m (z i )x n (z j ) → −2α ′ δ mn G ij and ψ m (z i )ψ n (z j ) → δ mn S ν (z ij |τ ). The latter give rise to spin structure dependent Szegö kernels with ν = 1, 2, 3. Together with the partition function in the first line of (B.1), the summation over spin structures can be described by the following building block with x 1 + x 2 + . . . + x n = 0. As is well known, correlators with less than eight ψ m yield a vanishing spin sum, and G 4 is the first instance where Riemann identities yield a nonvanishing result, reflecting maximal spacetime supersymmetry. Representatives at multiplicity five and higher have been evaluated in [15,34,35] using Fay trisecant identities in slightly different guises. The results of these references are equivalent to [28] are defined by (2.31) and (2.32), respectively. In order to cast the RNS amplitude (B.1) into the same basis of integrals as seen in the expression (3.4) and (3.7) for K C 6 and K P 6 in pure spinor superspace, we organize the integral reduction into three steps:

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• (i) elimination of double derivatives: bilinears in ∂x m (z i ) from the vertex operator (B.2) contract to a double derivative ∂ 2 G ij of the Green function (2.29). Since there is always a partner term α ′ s ij (∂G ij ) 2 with the same tensor structure from the fermionic part ∼ ψ 2 of the vertex operators, the double pole of ∂ 2 G ij , (∂G ij ) 2 ∼ 1 z 2 ij turns out to be spurious. This can be seen from a total derivative relation involving the Koba-Nielsen factor I from (2.30): The residue of the double pole must be proportional to (1 − α ′ s ij ) since it would otherwise signal tachyon propagation.
• (ii) partial fraction relations: step (i) and (B.7) leave two topologies of bilinears in the propagator: ∂G ij ∂G ik with j = k and an overlapping leg i as well as the disconnected configuration ∂G ij ∂G pq with all of i, j, p, q distinct. The former requires an application of the Fay identity (2.34) before the pattern of functions X ij (X ik +X jk ) seen in (3.4) and suitable for step (iii) is manifest: • (iii) integration by parts: as explained below (3.3), the minimal set of worldsheet functions X ij is obtained by eliminating any instance of X 1j by discarding derivatives of the Koba-Nielsen factor (2.30) w.r.t. z j . This amounts to two equivalent manipulations after step (ii): X 12 (X 13 + X 23 ) = (X 23 + X 24 + X 25 + X 26 )(X 34 + X 35 + X 36 ) + ∂ 2 (. . .) + ∂ 3 (. . .) 10) The f (2) ij functions from (B.7) and step (ii) do not admit further simplification, in particular the five instances of f (2) 1j cannot be reduced to f (2) pq and X pq with p, q = 1. After performing the above steps, the agreement of bosonic components in the two formalisms, can be checked along with each instance of f (2) ij , X ij (X ik + X jk ) and X ij X pq , see (3.1) for the definition of the right hand side.

B.2 The parity-odd part
The parity-odd sector of the RNS six-point amplitude stems from the spin structure of ψ m with anti-periodic boundary conditions along both cycles of the Riemann surface. In JHEP04(2016)148 this case, zero modes of the β, γ ghosts as well as the ten components of ψ m have to be saturated in the path integral. This gives rise to the amplitude prescription where V 1 (e 1 , k 1 , z 1 ) denotes the gluon vertex operator in the picture of superghost charge −1, and the zero mode integration for the β, γ system has already been carried out [61] The worldsheet supercurrent ∂x p ψ p is a remnant of a picture changing operator whose position z 0 drops out from the correlator. In the evaluation of the correlator (B.12), the Wick contraction of ψ m is adjusted to the spin structure ψ m (z i )ψ n (z j ) → δ mn ∂G ij , and the zero mode integration amounts to absorbing ψ m 1 (w 1 )ψ m 2 (w 2 ) . . . ψ m 10 (w 10 ) → ǫ m 1 m 2 ...m 10 , (B.14) independently on w j . After simplifying the parity-odd kinematic factors and eliminating the double derivatives of G 0j in a way similar to (B.8), we arrive at the following expression for (B.12): The worldsheet functions contained in η ijk ≡ ∂G ij ∂G ik + cyc(i, j, k) can be rewritten as f (2) ij + f (2) jk + f (2) ki via (2.34), and the shorthand E pq encoding the polarization dependence is defined by (permutations of) E 23 ≡ 1 2 e m 1 (e 2 · k 3 )k n 2 − s 23 e n 2 ǫ mnr 3 s 3 ...r 6 s 6 k r 3 3 e s 3 3 . . . k r 6 6 e s 6 6 + (2 ↔ 3) . (B.17) The sum of the ten inequivalent E pq in (B.16) can be written as an antisymmetrization in eleven vector indices such that This is crucial to cancel the contributions of ∂G 2 01 and f (2) 01 in η 0pq − η 01p − η 01q − (∂G 01 ) 2 such that the position of the supercurrent in (B.12) drops out. We are left with

C.1 Gauge variation of multiparticle superfields
The response of linearized SYM superfields to a superspace gauge transformation δ i in particle i is given by

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The recursive construction of multiparticle superfields A P α , A m P , W α P , F mn P in (2.9) to (2.13) determines their linearized gauge variation from (C.1). As pioneered in appendix B of [26] and generalized in [25], multiparticle gauge transformations are conveniently captured by multiparticle gauge scalars Performing a linearized gauge transformation (C.1) in a single external leg (say leg i = 1) amounts to the initial condition G j = δ 1,j Ω 1 for the recursion (C.2), i.e. to having only one non-vanishing single-particle scalar G j . The induced gauge transformation of multiparticle superfields is given by [25] δ where the δ G operation reduces to the linearized variations (C.1) in any leg i for appropriate choices of initial conditions. As a consequence, the one-loop building blocks in (2.15) to ( The multiparticle response to gauge variations can be conveniently interpreted by assembling both the gauge scalars G P and the multiparticle superfields A P α , A P m , . . . in a generating series: while latter solve the non-linear equations of motion of ten-dimensional SYM [62], the resummation of the gauge scalars encodes their non-linear gauge transformations. The recursion (C.2) is obtained by demanding the non-linear gauge transformations to preserve the Lorentz-gauge condition for the generating series of A P m [25]. The benefits of certain different choices of multiparticle gauge scalars are discussed in the reference. The components δ G C mn 1|2,3,4,5,6 and δ G P 1|2|3,4,5,6 only depend on G 1 whereas any other G j =1 drops out. This reflects the initial observation in section 2.1 that gauge transformations of the integrated vertices U 2 , . . . , U 6 annihilate the six-point amplitude while the unintegrated vertex V 1 yields the anomaly (2.6) upon variation to QΩ 1 . Setting G j → δ j,1 Ω 1 e ik j x reproduces the anomaly kinematic factor K in (2.7): δ G C mn 1|2,3,4,5,6 → − 1 2 δ mn K , δ G P 1|2|3,4,5,6 → − 1 2 K .
(C. 16) The gauge anomaly in (C.14) and (C.15) obviously matches the anomalous BRST variations (2.28) upon adjusting G j → V j in the non-exact part. Hence, the mechanism of anomaly cancellation is completely analogous for gauge and BRST transformations, see [21,22] for open strings, and section 4.4 for the closed-string discussion.
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