String-motivated one-loop amplitudes in gauge theories with half-maximal supersymmetry

We compute one-loop amplitudes in six-dimensional Yang-Mills theory with half-maximal supersymmetry from first principles: imposing gauge invariance and locality on an ansatz made from string-theory inspired kinematic building blocks yields unique expressions for the 3- and 4-point amplitudes. We check that the results are reproduced in the field-theory limit $\alpha' \rightarrow 0$ of string amplitudes in K3 orbifolds, using simplifications made in a companion string-theory paper 1603.05262.


Introduction
The last few years have seen significant progress on massless scattering amplitudes of string and gauge theories with less than maximal supersymmetry. As an example, open-string 4-point 1-loop amplitudes with minimal supersymmetry were discussed in [2,3], and the closed-string counterparts for half-maximal supergravity amplitudes can be found in [4,5]. In a companion paper [1] we simplified and generalized these results, using an infrared regularization procedure due to Minahan [6] to maintain manifest gauge invariance.
In this paper, we will present novel representations for 1-loop 3-and 4-point amplitudes in 6dimensional gauge theories with 8 supercharges, inspired by their string-theory ancestors from the companion paper [1]. In contrast to the 4-dimensional string-theory expressions in other work [4,5,3], we maintain 6-dimensional Lorentz-covariance (the maximum allowed by half-maximal supersymmetry) as we did in [1]. We use 4-dimensional spinor helicity variables only for specific checks.
The general philosophy of our calculational strategy will be: • String theory motivates a generic "alphabet" of kinematic building blocks for field-theory amplitudes. As we will see in examples, imposing locality and gauge invariance on a suitable ansatz drawn from this alphabet fixes the amplitudes we consider. Building blocks with up to two loop momenta and the systematics of their gauge variations will be discussed at general multiplicity.
• In general, there is a tension between manifest locality and manifest gauge invariance. We begin from a local representation, with crucial input from the cancellation of gauge variations of different diagrams. Then, by manipulating integrands, we rearrange kinematic factors into gauge-invariants of the same form as in the string amplitudes from the companion paper [1].
• In the pure-spinor description of 10-dimensional super Yang-Mills (SYM) [7,8,9], gauge invariance and supersymmetry are unified to BRST invariance [10]. Along with locality, this has been used to determine multiparticle amplitudes in pure-spinor superspace up to and including two loops [11,12,13]. To extend this approach below maximal supersymmetry, we consider half-maximal SYM in the maximal spacetime dimension D = 6 where 8 supercharges can be realized.
• A useful check is provided by comparison with [12]: the n-point 1-loop amplitudes in our halfmaximal setup follow the same structure as the corresponding (n+2)-point amplitudes with maximal supersymmetry.
This last feature is inherited from the structure of the string integrands, where comparison of the pure-spinor superspace results in [14] with the orbifold amplitudes in [1] reveals the same +2 offset in multiplicity. The counting is uniform with the relevant string compactifications: at complex dimension 2, we find the first nontrivial Calabi-Yau manifold (K3), that breaks half the supersymmetry. At complex dimension 3, supersymmetry is broken to a quarter (N = 1 in 4-dimensional counting), and the multiplicity offset in 1-loop amplitudes is +2 in the parity-even and +3 in the parity-odd sector, respectively. This means the parity-even part of our results, written in dimension-agnostic variables, applies universally to gauge-theory amplitudes with N = 2 and N = 1 supersymmetry. The parity-odd contributions to 4-dimensional N = 1 amplitudes, on the other hand, are quite different from the present 6-dimensional results with half-maximal supersymmetry. The methods of this work could be applied there too, but we postpone this to the future.
This paper is mostly about gauge-theory amplitudes, but it is of great interest to pursue the analogous calculations for supergravity. In particular, it is interesting to test to what extent the Bern-Carrasco-Johansson (BCJ) duality [15] holds in our calculations, and whether the requisite supergravity (1-loop) amplitudes with 16 supercharges can be obtained from the double-copy construction [16]. We report our results on this in section 4, where we find that the 3-point function in half-maximal gauge theory satisfies the BCJ duality, but -in contrast to the 4-dimensional expressions in [17,18] -our representation of the 4-point function does not naturally lend itself to the duality.
For comparison with the literature, we consider compactification on T 2 from 6 to 4 dimensions, and specialize to a 4-dimensional helicity basis, finding a perfect match with known results. The match involves a single free numerical factor that depends on the field content of the specific model. Many consistent string models contain exotic matter, whereas much of the work on field-theory amplitudes does not, so a completely general match goes beyond the scope of this work. Instead, in appendix E we describe simplified string models that are by themselves inconsistent (in particular when restoring couplings to a gravitational sector), but can usefully be compared to existing work on amplitudes.
2 Kinematic building blocks for 1-loop In this section, we introduce a system of kinematic building blocks for 1-loop amplitudes of half-maximal SYM. The overall guiding principle is invariance under linearized gauge transformations, that can be compactly implemented through the Grassmann (and thereby nilpotent) operator We will follow the ideas of Berends and Giele [19] to construct multiparticle generalizations of the polarization vector e m and its gauge-invariant linearized field strength, 3) The multiparticle variables of this section are designed to represent tree-level subdiagrams with an offshell leg and therefore transform covariantly under (2.1). They provide a suitable starting point to obtain both local and gauge-invariant expressions for the 1-loop amplitudes under investigation.

Local multiparticle polarizations
In order to attach the tree-level subdiagrams in figure 1 to a graph of arbitrary loop order, we define local 2-and 3-particle generalizations of polarizations and field strengths. In the conventions for multiparticle momenta and Mandelstam invariants where can be used to relate the factorization limit of n-point amplitudes on a 2-particle channel ∼ (k i +k j ) −2 to an (n−1)-point amplitude with one gluon polarization replaced by e m ij . Their 3-particle counterparts read , (2.8) and the combinations of e m ijl and f mn ijl that we will encounter in the next section capture the polarization dependence of 3-particle factorization channels ∼ (k i +k j +k l ) −2 . These definitions can be motivated by a resummation of Feynman diagrams [19] or through OPEs of vertex operators in string theory -see [20] for a supersymmetric derivation in the pure-spinor formalism − ω 2 f mn 13 + ω 13 f mn 2 ) , which will play a central role in this work. Given that the right-hand side is entirely furnished by multiparticle polarizations and multiparticle gauge scalars ω 12 ≡ 1 2 ω 2 (k 2 · e 1 ) − ω 1 (k 1 · e 2 ) , ω 123 ≡ 1 2 ω 3 (k 3 · e 12 ) − ω 12 (k 12 · e 3 ) , (2.10) the gauge algebra of the e m 12...p , f mn 12...p and ω 12...p is said to be covariant. Nilpotency of the gauge variation (2.1) can be checked from the covariant transformation of the fermionic gauge scalars ω 12...p , δω 12 = s 12 ω 1 ω 2 , δω 123 = (s 13 + s 23 )ω 12 ω 3 + s 12 (ω 1 ω 23 − ω 2 ω 13 ) . (2.11) Note that the gauge algebra (2.11) of multiparticle gauge scalars resembles the BRST variation 1 of multiparticle vertex operators V 12...p in the pure-spinor superstring [20]. 1 BRST invariance in pure-spinor superspace powerfully combines the supersymmetry of 10-dimensional SYM with gauge invariance of the bosonic superfield components [10].

Berends-Giele currents
In order to simplify the recursive definition and the gauge algebra of the above multiparticle polarizations As one can see from the 3-particle instances e m 123 and f mn 123 , the cubic graphs in figure 1 are combined according to a color-ordered 4-point amplitude with one off-shell leg, see figure 2.  ..p and f mn 12...p combine multiparticle polarizations with appropriate propagators so as to reproduce the cubic-vertex subdiagrams in a colorordered (p+1)-point tree amplitude with an off-shell leg · · · . The 2-and 3-particle instances (2.12) can be reproduced from the compact recursion [19,21]  (2.11) to the Berends-Giele framework, We have introduced Berends-Giele currents Ω P associated with the multiparticle gauge scalars (2.10), 18) which are reproduced from the recursion

Tree-level building blocks
Based on arguments in pure-spinor superspace [23], tree-level amplitudes of YM theories in arbitrary dimension have been expressed in terms of the kinematic structure [21] M   which only require currents of multiplicity ≤ n 2 as anticipated in [24]. It is easy to check gauge invariance of (2.25) via (2.23), and manifestly cyclic expressions at higher multiplicity can be found in [11] in pure-spinor superspace.

Parity-even 1-loop building blocks
The Berends-Giele organization also applies to loop amplitudes: The uplifts of e m P and f mn P to pure-spinor superspace [23] have been used to construct BRST invariant and local expressions for 5-and 6-point 1-loop amplitudes [12] as well as 2-loop 5-point amplitudes [13] in 10-dimensional SYM. To extend the method beyond maximal supersymmetry, we shall now introduce kinematic building blocks for 1-loop amplitudes in 6 dimensions with half-maximal supersymmetry. In absence of the no-triangle property of maximal SYM [25], we expect loop integrals of bubble and triangle topology in the half-maximal setup.
Since we will be interested in both local and gauge-invariant amplitude representations, we start by introducing local 1-loop building blocks before giving their Berends-Giele counterparts based on e m P and f mn P in section 2.6. As motivated by the string-theory discussion of [1], suitable kinematic numerators for bubble diagrams as in figure 3 are given by The multiparticle labels A and B in the subscripts of the multiparticle field strengths f mn 12...p , see (2.6) and (2.8), refer to the tree-level subdiagrams seen in figure 3. Their gauge variations in (2.9) imply covariant transformation for T A,B in (2.26) such as δT 123,4 = (s 13 + s 23 )(ω 12 T 3,4 − ω 3 T 12,4 ) + s 12 (ω 1 T 23,4 − ω 23 T 1,4 − ω 2 T 13,4 + ω 13 T 2,4 ) .
As will become clearer from the examples in sections 3.1 and 3.4, loop momenta m in the numerators of triangle diagrams and higher n-gons require vectorial and tensorial kinematic building blocks to contract with. This leads us to define generalizations of (2.26), B = e m A e n B + e n A e m B . Again, the covariant gauge variations (2.9) of local multiparticle polarizations propagate to the transformation of (2.28), e.g.
In section 3, we will identify combinations of scalar, vector and tensor building blocks whose gauge variation cancels -dependent propagators ( −k 12...p ) 2 , i.e. which qualify as triangle-and box numerators.

Parity-odd 1-loop building blocks
The running of chiral fermions in the loop introduces Levi-Civita tensors into the integrands of multiplicity ≥ 3. This requires a parity-odd completion of the above building blocks whose form is inspired by the contribution of worldsheet fermions with odd spin structures in the RNS superstring [26,27] The lack of symmetry under A ↔ B or A ↔ C (represented by the vertical bar A| . . . in the subscript) is an artifact of the asymmetric superghost pictures in the string computation [1]. Throughout this work, we will choose reference leg 1 to be part of A in each term.
In contrast to the variations (2.26) and (2.29) of the parity-even building blocks, the gauge algebra of (2.30) now relies on momentum conservation: Only by imposing k m A +k m B +k m C = 0, one can show that In analogy to the parity-even building blocks (2.28), the parity-odd vector (2.30) allows for tensorial generalizations such as Once the gauge variation of this tensor building block is simplified via 2η mn abcdef = η ma nbcdef + η na mbcdef − η mb nacdef − η nb macdef + (ab ↔ cd, ef ) (2.33) based on "overantisymmetrization" [abcdef η g]n = 0 in 6 dimensions, the trace component contributes to an anomalous gauge variation: The scalar building block represents the chiral box anomaly specific to 6 dimensions [28], that we will encounter in the 4-point 1-loop amplitude.

Berends-Giele building blocks at 1-loop
Recombining the local multiparticle polarizations into Berends-Giele currents e m P , f mn P in (2.13) and (2.14) leads to simplified gauge variations (2.15) and (2.16). Accordingly, the Berends-Giele versions , with the obvious Berends-Giele version of the anomaly building block (2.35): We recall that the gauge algebra (2.40) relies on momentum conservation. Note that the above gauge variations take a similar form as seen in the BRST algebra of maximally supersymmetric 1-loop building blocks in [20] and [29]. In particular, the generalization of the BRST covariant building blocks to tensors of arbitrary rank [29] can be easily adapted to the half-maximal framework, and the resulting definition and gauge algebra of T m 1 m 2 ...mr A 1 ,A 2 ,...,A r+2 or E m 1 m 2 ...mr A 1 |A 2 ,...,A r+2 will be explored in the future.

Gauge-invariant and pseudo-invariant 1-loop building blocks
The turn out to exhibit the same pattern of combining different multiparticle labels as seen in the parity-odd sector: While the parity-even gauge algebra (2.37) along with momentum conservation allows for invariants with additional free vector indices, the anomalous term η mn Ω A Y B,C,D in the variation (2.40) of E mn A|B,C,D complicates the construction of tensor invariants in the parity-odd sector. Hence, we follow the terminology of [29] to relax the requirement of gauge invariance such that anomaly kinematics (2.41) is admitted: Kinematic factors whose gauge variation can be expressed in terms of Ω A and Y B,C,D ∼ mnpqrs f mn B f pq C f rs D will be referred to as pseudo-invariant. Then, one can view the combination of different tensor ranks in that the construction of gauge-(pseudo-)invariants follows the same rules. Accordingly, we introduce a unifying notation for combinations of parity-even and parity-odd building blocks and vector invariants are constructed from the same combinations of multiparticle labels as the maximally supersymmetric BRST invariants C 1|2,3,4 , C 1|23,4,5 , C 1|234,5,6 as well as C m 1|2,3,4,5 , C m 1|23,4,5,6 defined in section 5 of [20]. The tensor pseudo-invariant on the other hand, resembles the 6-point tensor C mn 1|2,3,4,5,6 in pure-spinor superspace defined in (3.14) of [29] -see [14] for its appearance in closed-string amplitudes. From the gauge algebras (2.37) and (2.40), it is straightforward to check that δC 1|A = 0 , δC m 1|A,B = 0 , δC mn 1|2,3,4 = 2iω 1 η mn (k 2 , e 2 , k 3 , e 3 , k 4 , e 4 ) = ω 1 η mn Y 2,3,4 , (2.53) using momentum conservation for the vectors and the tensor. In addition to the tensor (2.52), one can construct scalar pseudo-invariants from the additional building block is pseudo-invariant as well: δP 1|2|3,4 = 2iω 1 (k 2 , e 2 , k 3 , e 3 , k 4 , e 4 ) = ω 1 Y 2,3,4 . (2.57) Its composition from M A,B and M m A|B,C follows the patterns of the BRST pseudo-invariant P 1|2|3,4,5,6 in (5.22) of [29]. More generally, the recursion introduced in [20,29] allows to construct BRST (pseudo-)invariants at arbitrary multiplicity and tensor rank, including comparable generalizations of P 1|2|3,4 in (2.56). The master recursion for an arbitrary number of multiparticle slots in section 8 of [29] can be used to obtain (pseudo-)invariants of higher multiplicity n ≥ 5 in the half-maximal setup.
The summation range Γ 12...n selects cubic 1-loop graphs i that are compatible with the cyclic ordering 1, 2, . . . , n of the color-stripped single-trace amplitude in (3.2). Any graph i is associated with (possibly loop-momentum dependent) internal momenta p 1,i ( ), p 2,i ( ), . . . , p n,i ( ) from its edges α, and the design 6 Tadpole subgraphs are incompatible with the string prescription that motivates our choice of building blocks: The tadpoles have n−1 propagators with external momenta only, that cannot arise from the maximum number of n−2 kinematic poles s −1 i...j admitted by the singularity structure of the string-theory integrands in [1]. We note that this is not directly related to the general consistency relations known as tadpole cancellation in string theory (see e.g. the textbooks [26,30]) -we have not (yet) demanded tadpole cancellation in the models of appendix E. of their kinematic numerators N i ( ) is guided by the gauge variation (2.1): Each term of δN i ( ) must cancel a propagator, i.e. contain a factor of p 2 α,i ( ) , α = 1, 2, . . . , n . (3.3) By locality of the numerators, this is a necessary condition for gauge invariance of the overall integrand.
To ensure it is also sufficient, it remains to check that all the contributions from δN i ( ) with fewer propagators cancel between diagrams. The global delta function imposing momentum conservation

The local form of the 3-point amplitude
At the 3-point level, our ansatz for a cubic-graph expansion for half-maximal 1-loop amplitudes without tadpoles involves three bubble-diagrams and one triangle: = ..p fixes its gauge variation to be using the vanishing of Mandelstam invariants (2.4) in 3-particle momentum phase space 7 In view of the gauge algebra (2.29), the minimal solution to (3.6) is 2 m T m 1,2,3 . However, we are led to the nonminimal solution 8) 7 We keep kinematic identities covariant and dimension-agnostic (except in section 5, for checks in D = 4). Hence, we will not use the common strategy of factorizing s 12 = 1 2 (k 2 3 − k 2 2 − k 2 1 ) = 0 into 4-dimensional spinor brackets 12 and [12], one of which is taken to be non-zero for complex momenta [31]. Instead, in the next section 3.2 we will introduce a D-dimensional infrared regularization to track the cancellation of the vanishing 3-particle s ij in intermediate steps.
which will appear in 4-point gauge variations and play out with the BCJ duality. Moreover, (3.8) resembles the structure of the maximally supersymmetric pentagon numerator in (4.5) of [12]. We have allowed for the parity-odd term m E m 1|2,3 ∼ ( , e 1 , k 2 , e 2 , k 3 , e 3 ) defined by (2.30), for chiral fermions to run in the triangular loop.
The freedom to choose nonminimal solutions might seem to be at odds with the "fixing" of the amplitudes claimed in the introduction. In fact, the parity-even extensions of the triangle numerator by T ij,k in (3.8) vanish as detailed in section 3.2 below, but their inclusion parallels certain non-vanishing contributions to the triangle numerators in the 4-point amplitude, see eqs. (3.24) to (3.27). The parityodd term m E m 1|2,3 is local and gauge-invariant by itself, but the string-motivated combination m M m A|B,C , defined in (2.46) at generic multiplicity, tells us that m T m 1,2,3 should appear only in the combination . As a check, this term can be directly calculated from string theory as in [1].

Infrared regularization
In (3.7), we see the usual vanishing s ij = 0 of 3-particle Mandelstam invariants for massless external states. This threatens to introduce singular propagators of the form "1/0" in the bubble terms in (3.4).
The problem of singularities ∼ s −1 12...n−1 in the phase space of n massless particles also arises in string amplitudes in orbifolds with half-maximal supersymmetry. In [1], where we constructed the string-theory input for the SYM amplitudes in this paper, we used the following proposal by Minahan in 1987 [6], that we referred to as minahaning. Infrared singularities are regularized by a lightlike "deformation" momentum p m perpendicular to all the polarization vectors, that deforms 3-particle momentum conservation to so s 12 is linear in the deformation p. (In string theory, the virtue of this particular regularization procedure is that it preserves modular invariance of 1-loop amplitudes while keeping the external states on-shell.) Then, by (3.9), the dependence on p m automatically drops out from the bubble contributions, and the singular propagator is cancelled as visualized in figure 4, This casts the 3-point amplitude (3.4) into the following form, (3.13) in the one-mass bubble diagram is compensated by the formally vanishing numerator T 12,3 = s 12 (e 1 · e 2 )(k 1 · e 3 ). If the diagram on the left is a "snail", then the diagram on the right is a "shy snail". All our snails are shy.
One might worry whether the deformation (3.10) of the kinematic phase space interferes with the gauge algebra, since we used momentum conservation in section 2.6 and 2.7 to identify gauge-invariants.
This worry is unfounded -indeed we used momentum conservation for our vectorial and tensorial building blocks, but only scalar building blocks are minahaned. For instance, we use n-particle momentum conservation to cast δE m 1|23,4 in (2.31) into covariant form and to rewrite δT m 1,2,3 in (2.44) as where the gauge-invariant completion is more evident. The gauge algebra for the scalars M A,B ∼ f mn A f mn B , where minahaning is required in case of single-particle slots A or B, is not tied to any phase-space constraints.
In more general terms, the vector and tensor building blocks seen in section 2 are built from products of at least 3 Berends-Giele currents, see (2.36) and (2.38). Hence, the associated external propagators contain at most n−2 massless momenta, and the naively singular propagators of the scalars M 12...n−1,n are bypassed. To summarize: in the sector of the gauge algebra that relies on momentum conservation, we are in fact free to set the deformation vector p in (3.10) to zero from the outset.
As expected, the representation (3.13) of the 3-point amplitude integrates to zero in dimensional regularization: the scale-free bubble integral vanishes by cancellation between infrared and ultraviolet divergences, and the triangle contributions with tensor structure m → k m j vanish upon integration, on kinematic grounds. While the main emphasis of this work is on integrands and their systematic construction via gauge invariance and locality, we will also study the integrated expressions as consistency checks.

The gauge-invariant form of the 3-point amplitude
to the integrand, setting s 12 = 0 in the brackets [. . .]. With the definitions (2.48) and (2.50) of the scalar and vectorial gauge-invariants C 1|23 and C m 1|2,3 , we arrive at the alternative representation with manifest gauge invariance. (We also included the vanishing scalar triangle s 23 C 1|23 to make contact with the maximally supersymmetric pentagon in (5.5) of [12].) To compare (3.15) with the manifestly local expression (3.13), we write out polarizations and momenta: (3.16) .
We see that the gauge-invariant form (3.16) of the 3-point amplitude happens to also have manifest locality, but as emphasized earlier, amplitudes at higher multiplicity generically exhibit a tension between locality and gauge invariance. At 4 points, for instance, the gauge-invariant triangle "numerators" such as m C m 1|23,4 that will appear in section 3.5 involve kinematic poles (say s −1 12 ) that do not match the propagator structure of the triangle diagram under discussion (say (s 23 . When all diagrams of the amplitude (3.2) are assembled, those superficially non-local contributions will collapse to local expressions, as is guaranteed from the manifestly local starting point of our construction.
The gauge-invariant bubble coefficient in (3.16) can be recognized as the 3-point tree, It arises as the leading UV-divergence of (3.15) when performing the d D -integral in (3.16) in D ≥ 4 dimensions, for which we introduce the shorthand | UV , as in

The local form of the 4-point amplitude
The 4-point analogue of the ansatz (3.4) in terms of cubic diagrams without tadpoles reads The propagators in the first line are associated with bubble diagrams, and the numerators N A|B,C ( ) and N box 1|2,3,4 ( ) of the triangles and the box, respectively, will be inferred from gauge invariance.

Bubbles
Our ansatz for the bubbles in the 4-point amplitude (3.19) is again based on the scalar building block Similarly to our previous discussion around (3.12), the propagators of an external bubble include a 3-particle Mandelstam invariant s 123 that naively vanishes in 4-particle phase space, but this is compensated by a zero of the numerators T 123,4 and T 321, 4 . In more detail, we extend the minahaning procedure from section 3.2 to 4 points: a lightlike deformation momentum 4 j=1 k m j = p m amounts to using no Upon insertion into (3.19), this pinpoints the bubble contributions to the 4-point amplitude in its local , is compatible with (3.3) and takes the right form to conspire with the triangle diagrams.

Parity-even triangles
By analogy with the 3-point expression (3.8), the numerators of the 4 triangle diagrams in (3.19) will have both parity-even and parity-odd contributions, to be denoted by N even 12|3,4 ( ) and N odd 12|3,4 ( ), respectively: extending the pattern of (3.8), provide the right interplay between scalar and vector contributions to produce differences of the inverse propagators 2 12...j and s ij in their gauge variation: The shorthand N even i,j,k ( ) on the right-hand sides refers to the parity-even triangle numerator in (3.8): The exceptional terms 2k m 4 T m 41,2,3 − s 14 e m 4 T m 1,2,3 − s 14 (e 1 · e 4 )T 2,3 in the fourth triangle numerator (3.27) without any counterparts in N even 12|3,4 ( ), N even 1|23,4 ( ) and N even 1|2,34 ( ) can be justified as follows: The gauge variation of the naive ansatz 2 m T m 41,2,3 + T 412,3 + T 413,2 + T 41,23 for N even 41|2,3 ( ), does not satisfy the necessary condition is easily seen to complete the first line of (3.30) to be expressible via differences of 2 12...j and can be motivated by a diagrammatic argument: In our conventions for the shifts of the integration variable, is the momentum in the n-gon edge between the external legs 1 and n. For the "special" triangle graph with legs 1 and 4 forming a tree-level subdiagram, the momentum in the analogous adjacent edge is + k 4 rather than , see figure 6. Hence, it is not surprising that our analysis driven by gauge invariance points towards a numerator of the form N even 41|2,3 ( ) = 2( m + k 4 m )T m 41,2,3 + . . .. The remaining terms of the ellipsis -specifically the subtraction of s 14 e m 4 T m 1,2,3 + (e 1 · e 4 )T 2,3 -can be inferred by demanding the variation to follow the structure of (3.28), The appearance of 2 12...j in the triangles' gauge variation cancels the contribution (3.22) from the bubbles. The remaining terms ∼ s ij in the above δN even A|B,C need to conspire with the box graph. In particular, the sign change of ω 24 T 1,3 + ω 34 T 1,2 and the conversion N even 4,2,3 ( ) → N even 2,3,4 ( ) going from (3.30) to (3.31) is essential to render the desired gauge variation of the box numerator linear in : Only the relative minus sign in 2 12...j − 2 12...j−1 makes the quadratic piece ∼ 2 disappear.

Parity-odd triangles
The parity-odd part of the triangles can be reconstructed by demanding all the N A|B,C ( ) to be expressible in terms of the string-theory motivated "parity-(odd+even)" building blocks M m A|B,C and J 1|i|j,k defined in (2.46) and (2.54), respectively, where the last term ∼ −2 2 ω 1 k m 4 E m 4|2,3 will be seen to play an important role for the 6-dimensional gauge anomaly.

The box numerator
The -dependent part of the box numerator can be readily written down by promoting the constituents of the triangle in (3.8)   it is easy to verify that (3.36) naively cancels the entire parity-odd variation of the triangles in (3.34), but in the next section we will see the expected box anomaly. It remains to find a scalar completion N scal 1|2,3,4 of the -dependent parity-even building blocks m T m A,B,C and m n T mn 1,2,3,4 in (3.35). The expression for N scal 1|2,3,4 will be designed to cancel the gauge variations (3.28) and (3.31) of the triangles which are not yet accounted for by the bubbles (3.22): One can check that this is accomplished by the following local expression for the scalar box in (3.35):  in terms of the "parity-(odd+even)" building blocks (2.46).

The box anomaly
The anomaly kinematics Y 2,3,4 = 2i (k 2 , e 2 , k 3 , e 3 , k 4 , e 4 ) from both δN 41|2,3 ( ) and δN box 1|2,3,4 ( ) deserves particular attention since this is where the tensor trace δ( m n M mn 1|2,3,4 ) = 2 ω 1 Y 2,3,4 + . . . conspires with the special triangle with k 41 in a massive corner (i.e. where the −2 propagator is absent): Naively, one would be tempted to set (3.40) to zero since the integrand appears to vanish. As is well known, dimensional regularization reveals a logarithmic divergence that requires a refined analysis. One 9 Note that the gauge variation of the second line of (3.39) is given by can show with conventional (see e.g. [35] or section 5.1 of [28]) or worldline techniques (see e.g. section 4.5 of [12]) that tensor n-gon integrals in D = 2n − 2 dimensions give rise to the following rational terms when combined with an appropriate scalar (n−1)-gon. Hence, we identify the following anomalous gauge variation in the above 4-point amplitude, (3.42) As can be "discovered" from the field-theory perspective (see e.g. [28]), this anomaly can be cancelled by contributions due to additional fields in the gravitational sector, once the relations between couplings in the gauge and gravitational sectors are suitably tuned. From the string-theory point of view, this is the Green-Schwarz mechanism generalized to D = 6 (see e.g. [36]). The additional states are p-form fields, possibly on collapsed cycles of the K3 orbifold, but in string theory no couplings need to be adjusted: the coupling relations suitable for anomaly cancellation arise from the same open-string loop diagrams as those that gave rise to the anomaly (e.g. diagrams discussed in [1], but in the long-cylinder limit instead of the field-theory limit discussed later in this paper).

The gauge-invariant form of the 4-point amplitude
To make pseudo-invariance of the 4-point amplitude (3.19) manifest, one can repeat the procedure of section 3.3 and perform algebraic rearrangements of the integrand similar to (3.14). The guiding principle is to eliminate those cubic diagrams where the reference leg 1 is involved in a non-trivial tree-level subdiagram 10 , i.e. where either 2 or 2 1 is absent. However, the above discussion of the box anomaly suggests that the special triangle with propagators 2 1 2 12 2 123 is an exception. In the process of these rearrangements, the kinematic building blocks T A,B , M m A|B,C , M mn A|B,C,D and J 1|2|3,4 are assembled into the gauge (pseudo-)invariants C 1|A , C m 1|A,B , C mn 1|A,B,C and P 1|2|3,4 introduced in section 2.7. By tedious but straightforward manipulations, one can show that the integrand of (3.19) agrees with upon insertion of all the numerators, with the following shorthand for the gauge-invariant scalar box: This representation generalizes the form (3.15) of the 3-point amplitude and confines the leading UV contribution to the single bubble in the first term such that, in analogy with (3.18),

An empirical invariantization map
Following the same steps as for the maximally supersymmetric setup explained in section 5.1 of [12], there is a systematic and intuitive mapping from the above local representations to the manifestly gauge (pseudo-)invariant expressions in (3.15), (3.43) and (3.44). Whenever the reference leg 1 enters a kinematic building block through a single-particle slot A = 1, it signals a (pseudo-)invariant according to

A simplified representation
In contrast to the maximally supersymmetric 6-point amplitude in [12], the present 4-point context turns out to admit additional simplifications. The BCJ relations [15] among different permutations of C 1|234 ∼ A tree (1, 2, 3, 4) imply the vanishing of the scalar triangle numerators, and additional on-shell relations detailed in appendix C cast the scalar box numerator into a compact form: Upon insertion into (3.43), we arrive at the following simplified and manifestly pseudo-invariant form of the 4-point amplitude with half-maximal supersymmetry, The diversity of gauge theories increases rapidly when reducing from maximal to half-maximal supersymmetry (running of gauge couplings, variety of supermultiplets and representations). We would like to highlight that all this additional complexity of the 4-point 1-loop amplitude in general dimensions is compactly captured by the kinematic building blocks in (3.48). As we discuss further in the conclusions, upon dimensional reduction of this theory to D = 4, the parity-even part of minimally supersymmetric gauge theory is also given by this expression.

Supergravity from the duality between color and kinematics
A major virtue of the cubic-graph organization of gauge-theory amplitudes is that it often admits the construction of supergravity amplitudes at various loop orders by double-copy [16]. For this to work, the kinematic constituents must mirror all the properties of the color factors [15], in other words they should satisfy the BCJ duality between color and kinematics 11 . In this section, it will be demonstrated 11 See [37] for a recent pedagogical account. that the 3-point gauge-theory amplitude presented in sections 3.1 and 3.3 obeys the BCJ duality, so we can infer the related half-maximal supergravity amplitude. However, in the formulation of the 4-point amplitude we gave in sections 3.4 and 3.5, the duality is not manifest and further work is needed.

Review of the BCJ duality and double-copy construction
The BCJ duality between color and kinematics is based on the dictionary between cubic graphs and structure constants f abc of an arbitrary gauge group. The color representative c i of a graph is obtained by dressing each cubic vertex with a factor of f abc and by contracting the color indices a, b, c across the internal lines. Then, the Jacobi identity any supersymmetry such as [40,41,42,43,18,44], and the 4-dimensional version of the half-maximal 1-loop amplitudes under investigation has been cast into BCJ form in [17,18]. 13 Ambiguities for local cubic-graph representations of gauge-theory amplitudes arise from the freedom to assign the contributions from quartic gluon vertices to different cubic diagrams. Redistributions as required by the BCJ duality are often referred to as "generalized gauge freedom" [15,45,16], and a concrete non-linear gauge transformation to implement such rearrangements of tree-level diagrams was identified in [23]. of the supergravity amplitude M g-loop N + N . As for the state dependence of N i andÑ i , the supergravity spectrum emerges as the tensor product of the gauge-theory states, e.g. graviton polarization tensors follow from the traceless parts of the gluon polarizations e (mẽn) , and the N = 8 supergravity multiplet arises as a double copy of the N = 4 SYM multiplet.
The standard double-copy realization of pure N = 4 supergravity is as (N = 0) × (N = 4) SYM, an asymmetric ("heterotic") realization (see e.g. [46]). In this work, we have in mind the symmetric double copy (N = 2) × (N = 2) SYM, that gives N = 4 supergravity coupled to N = 4 matter multiplets with maximum spin 1 and 3 2 , respectively. String constructions of these matter-coupled supergravities leave some freedom to tune the matter content, see e.g. [4] and references therein.

BCJ duality and double copy of the 3-point amplitude
The local representation (3.4) of the 3-point amplitude will now be shown to obey the BCJ duality.
There are 3 inequivalent classes of kinematic Jacobi relations N i + N j + N k = 0 to check: • Symmetry of the bubbles versus absence of tadpoles: As depicted in figure 8, the symmetry of bubble numerators T A,B = T B,A and the absence of tadpoles is consistent with the BCJ duality. • Antisymmetrizing a triangle numerator in legs 1,2 is also is consistent with the bubble numerators under the BCJ duality. Since our shift conventions for the loop momentum fix to reside in the edge next to leg 1, the momentum routing in figure 10 requires particular care, and the numerator of the second triangle takes − k 2 instead of as its argument. Given that the representation (3.4) of the gauge-theory amplitude satisfies all these Jacobi identities, it is qualified to enter the double-copy construction. By converting the color factors of the bubble-and triangle diagrams to additional kinematics, see (4.2), one obtains after combining both orientations of the triangle. However, by the formally vanishing bubble numerator T 12,3 = s 12 (e 1 · e 2 )(e 3 · k 1 ) derived in (3.9), the -independent part of the bubble contributions yields .

(4.5)
Given that all of T m 1,2,3 , E m 1|2,3 ,T n 1,2,3 ,Ẽ n 1|2,3 are perpendicular to the external momenta, the only contribution to the integrated expression has tensor structure m n → η mn and leaves a no-scale integral.
Of course, the same discussion applies to the manifestly gauge-invariant representation of the SYM numerators given in section 3.3, i.e. to the image of the numerators under the invariantization map (3.46). In the alternative form the UV-divergence due to the trace component m n → η mn of the tensor integral manifestly agrees with the low-energy limit of the corresponding string computation [1], see [47,1] for a discussion of the components with different numbers of gravitons, B-fields and dilatons. For 3 gravitons, the counterterm associated with (4.7) is an operator R 2 which is on-shell equivalent to the Gauss-Bonnet term, and therefore yields vanishing amplitudes in strictly 4 dimensions. Likewise, the parity-odd contribution to (4.7) obviously drops out in D = 4 for any state configuration.

Deviations from the duality in the 4-point amplitude
Since the 4-point gauge-theory amplitude (3.48) has been constructed from the same principles as its BCJ-satisfying 3-point counterpart, it is tempting to hope for the duality to hold also at 4 points.
However, in our representation one can identify a simple counterexample among the kinematic Jacobi identities that renders the 4-point supergravity amplitude inaccessible to naive 14 double copy of the present building blocks. Just like the 3-point BCJ discussion was identical for the manifestly local and the manifestly gauge-invariant representation, for our 4-point arguments we could choose either (3.19) or (3.48). For convenience we pick the latter.
As depicted in figure 11, the antisymmetrization of two triangles with massive momentum k 2 + k 3 in one corner should reproduce a bubble numerator that vanishes in the parametrization of (3.48). 14 We note that two new approaches to double-copy constructions of gravity amplitudes have been developed since the first preprint version of this article [48,49] which do not require a BCJ representation of the gauge-theory input.
However, we obtain instead of the vanishing bubble numerator from the analogous antisymmetrization in figure 11. This is an obstacle to BCJ duality and prevents us from immediately writing down a double-copy expression for the supergravity amplitude. This obstacle was anticipated from the analogous closed-string amplitude in our companion paper [1]. It closely parallels the obstacle observed in the representation of the maximally supersymmetric 6point amplitude in section 6.3 of [12]. Both here and there, the mismatch in kinematic Jacobi identities boils down to pseudo-invariant kinematic factors as seen in (4.8). It is tempting to speculate that there is a tension between anomalies and the BCJ duality [12], but ongoing work [50] and the recent results of [51] indicate that there is no direct connection.

Comparison with 4-dimensional results
So far, we have discussed half-maximal SYM amplitudes in the highest dimensions D = 6 where 8 supercharges can be consistently realized. This section is devoted to their dimensional reduction to D = 4, where the parity-odd sector of gluon amplitudes including their box anomaly drops out and their dimension-agnostic parity-even integrands 15 are expressed in terms of (D = 4) spinor-helicity variables.
In the manifestly gauge-invariant framework of section 3, the initially 6-dimensional kinematic factors will be specialized as with explicit spinor-helicity expressions for the 4-dimensional objects C ... ... and P ... . The resulting 4dimensional 4-point amplitudes with N = 2 supersymmetry will be shown to match the expressions in the literature [52,35,17,18].

Spinor-helicity expressions versus polarization vectors
In this section, we collect spinor-helicity expressions for the parity-even gauge-invariant building blocks (5.1) in D = 4. Conventions for the relevant momentum spinors and σ-matrices are summarized in appendix B. As expected from supersymmetric Ward identities [53,54], only the MHV components of the scalar, vectorial and tensorial kinematic factors in (5.1) are non-zero, e.g.
In the MHV sector, one obtains 3-point helicity components such as , [23] , as well as At 4 points, we have the following inequivalent cases for scalar kinematic factors: for vector kinematic factors we have: identify s ij P ij|kl with the gluon components of the kinematic variables κ ij + κ kl in [18].
It is worth stressing the crucial difference between the helicity-agnostic expressions underlying the above components of C 1|234 and those underlying P 12|34 . Using Lorentz traces over linearized field it was found in the string-theory companion paper [1] that where only C 1|234 is expressible in terms of the famous t 8 -tensor from maximal supersymmetry, By contrast, the object P 12|34 of (5.12) only preserves 8 supercharges. 16 In (5.12), the dependence of P 12|34 on the polarization vectors involves tensor structures ∼ 4 i=1 (e i · k j i ), i.e. terms without any contraction of the type 17 (e i · e j ). An important difference between P 12|34 and the t 8 -tensor (5.13) is that expressions of the form 4 i=1 (e i · k j i ) in P 12|34 cannot be eliminated via momentum conservation and transversality (e i · k i ) = 0. Note also that the symmetries P 12|34 = P 21|34 = P 12|43 = P 34|12 (5.14) are manifest in (5.12). Certain helicity configurations lead to identical expressions for s 23 C 1 − |2 − 3 + 4 + and P 1 − 2 − |3 + 4 + -see (5.5) and (5.6) -but the vanishing of the MHV component P 1 − 2 + |3 − 4 + shows that there can be no helicity-independent proportionality between C 1|234 in (5.11) and P 12|34 in (5.12).

Disentangling the supermultiplets in the loop
As emphasized below (3.48), in section 3.4 we constructed a particular 4-point solution to locality and gauge invariance in the presence of bubble and triangle diagrams, but this particular solution admits the freedom of adding the maximally supersymmetric 1-loop amplitude A 1-loop N =4 with arbitrary coefficient. With this in mind, we want to compare our results to the well-known N = 2 supersymmetric 1-loop 16 This can be seen by relating P 12|34 to the κ ij -variables in (4.8) and (4.9) of [18], for example. 17 The absence of tensor structures n i=1 (e i · k ji ), i.e. the omnipresence of at least one factor of (e i · e j ) in n-point tree-level amplitudes of the open superstring has been stressed and exploited in [56,57]. An investigation of (possibly non-supersymmetric) gauge-invariant scalar kinematic factors including n i=1 (e i · k ji ) can be found in [58]. amplitudes in D = 4 field theory (see for example [52,35]). We focus on the recent work [18], that discusses the contribution of the hypermultiplet running in the loop. In general, the single-trace part of the color-ordered 1-loop amplitude in an N = 2 theory can be written schematically as translates into the following amplitude contributions from these multiplets in the loop: which is meaningful in both D = 6 and D = 4. Hence, one can write the generic one-loop amplitude (5.15) asÃ and normalizing the hypermultiplet contribution to unity, we have It is now evident that if we can match A hyp N =2 to [18] with unit normalization, the detailed modeldependence of the original (5.15) (for example, on the field content) is confined to the coefficient of the scalar-box contribution A 1-loop N =4 . We need to compare the generic form (5.19) with the 4-point amplitude (3.48) shifted by any multiple c of A 1-loop N =4 (1, 2, 3, 4), , 18 A hypermultiplet has two complex scalars. Following the references we are using in this section, we write 2(N = 2) hyp , where the factor of 2 means that what we call "hypermultiplet" here is actually a "half-hypermultiplet" with two real scalars. See also the conclusions for comments on supersymmetry decomposition with other multiplets.
where we used the explicit form of the maximally supersymmetric contribution , (5.21) and the representation of the scalar box as s 12 s 23 C 1|234 = 2t 8 (1, 2, 3, 4). As we will see in section 6, the field-theory limit of string amplitudes in 4-and 6-dimensional orbifold compactifications identifies the hypermultiplet contribution with (5.20) at c = − 1 3 where t 8 (1, 2, 3, 4) drops out, In the next section, we will check that this recreates the D = 4 expressions of [18].

Matching 4-dimensional spinor-helicity expressions
Now we use the 4-dimensional MHV components of section 5.1 to simplify the hypermultiplet contribution to the 4-point amplitude, and match our result to [18]. In D = 4 − 2 dimensions 19 , performing the integrals in dimensional regularization, from eq. (5.22) and (5.20) we have Up to and including order 0 , we arrive at the simple result The amplitudes (5.25) and (5.26) perfectly match known results [52,35,18]. 20 As is well-known, the maximally supersymmetric 4-point amplitude (5.21) is proportional to the corresponding tree amplitude, by a supersymmetry Ward identity. Here in half-maximal supersymmetry, we have the additional 8-supercharge tensor structure P 12|34 in (5.12). This leads to the situation that even though the two helicity components (  with half-maximal supersymmetry in 6 and 4 dimensions, respectively (see e.g. the textbooks [59,30] or the review [60]). It is convenient to use complexified coordinates on the two twisted 2-tori, i.e. z j = x 2j+4 + U j x 2j+5 , with j = 1, 2 and U j denoting the complex structure of the j th 2-torus. The generator Θ of the cyclic group Z N acts on the tori (and on the string spectrum) as a discrete rotation, when the string spectrum is quotiented out by worldsheet parity, to create an orientifold (see e.g. [59,30]), where one also computes unoriented string diagrams like the Möbius strip. Although doing so is straightforward, it is possible to extract partial amplitudes for a large class of gauge theories by performing the field-theory limit of just the oriented-string worldsheet topologies. (We should warn the reader that such truncated theories may contain spurious divergences that are cancelled in the complete theory, but we have not seen any indication that this should affect the discussion here.) We will now reproduce the 1-loop color-ordered single-trace amplitudes computed in the previous sections from the field-theory limit of the planar cylinder diagram.

1-loop open-string amplitudes with half-maximal supersymmetry
Extending previous work in [2,4,5,3], 1-loop open-string amplitudes with external gluons in the context of orbifold compactifications were studied and simplified in [1]. In particular, toroidal orbifolds preserving half-maximal supersymmetry give rise to the following decomposition of the planar cylinder diagram for n-point 1-loop amplitudes: depend solely on the orbifold sector labelled by k = 1, 2, . . . , N −1, and the integrands I n,max and I n,1/2 ( v k ) are determined by the external states and the partition function. The subscripts "max" or "1/2" distinguish orbifold sectors that preserve all or half the supersymmetries, respectively. While the maximally supersymmetric integrand I n,max is parity-even and independent of the dimension D, the half-maximal integrand I n,1/2 ( v k ) has both parity-even and parity-odd parts. The parity-odd part vanishes in D < 6. Finally, the integration measure in eq. (6.2) is given by where τ 2 is the modular parameter of the cylinder worldsheet whose non-empty boundary is parametrized by purely imaginary coordinates z i with 0 ≤ Im (z i ) ≤ τ 2 . In the measure (6.4) we have also incorporated the regularized external volume V D , the order N of the orbifold Z N , as well as the ubiquitous Koba-Nielsen factor Π n which arises from the plane-wave factors of the vertex operators; with the conventions of reference [1] we have with z ij ≡ z i − z j , the Jacobi theta function ϑ 1 , and G(z, τ ) denotes the bosonic Green's function on a genus-one worldsheet with modular parameter τ .

The worldsheet integrands
The maximally supersymmetric integrands in the 3-and 4-point amplitudes (6.2) are well-known [62], 23 C 1|23 , (6.7)  10) which are evaluated at purely imaginary coordinates z j and purely imaginary modular parameters of the cylinder, τ ≡ iτ 2 with τ 2 real. Note that f (1) (z, τ ) has a simple pole at the origin z → 0, and the combination is the Weierstrass ℘-function, that encodes the entire dependence of (6.8) on the orbifold twists (6.1).
(See also [2] for the role of ℘ in string-theory setups with half-and quarter maximal supersymmetry.) 22 A similar reduction to a basis of integrals was performed in [3]. The 4-dimensional spinor-helicity setup of this reference departs from the infrared regularization of this work (section 3.2), leading to different final expressions for the simplified integrands. In [3], the field-theory limit was performed for the integrands of particular orientifold string models, and it would be interesting to compare those results in greater detail to the results in this section. 23 See appendix A for an intuitive motivation.

The field-theory limit of the open-string amplitudes
It is well-known how to perform the field-theory limit of 1-loop string amplitudes to obtain the corresponding Feynman diagrams in Schwinger parametrization, see e.g. [62,64,65,66] and references therein. The limit is α → 0 while simultaneously degenerating the genus-one surface to a worldline by sending τ 2 → ∞. The limits are taken such that α τ 2 is kept finite and reduces to the worldline length α τ 2 → t in the Schwinger parametrization of the corresponding field-theory diagrams. The finite parts ν of the cylinder coordinates Im(z) = τ 2 ν are then identified with proper times on the worldline. In this limit, the bosonic Green's function in eq. (6.5) reduces to Then, the limit of the Koba-Nielsen factor is 13) and the measure in eq. (6.2) reduces to 24 , (6.14) see [67] for an extension of these techniques to higher loops. Note that the definition of Q n in (6.13) admits massive external legs with composite momenta k A = k a 1 + k a 2 + . . . + k am for A = a 1 a 2 . . . a m .
In the above limit, the worldsheet functions (6.9) and (6.10) involved in the half-maximal integrands of the 3-and 4-point functions reduce to while the function F 1/2 (kv) of the orbifold twists in the 4-point integrand degenerates as follows: .
We would like to highlight the following two additional aspects in performing the field-theory limit of string amplitudes: • The quantized momenta in the internal dimensions need to decouple, this is obtained by shrinking the size of the internal dimensions at the same rate as the string-length goes to zero [62,66] so 24 On the right-hand side of eq. (6.14) we have suppressed the overall constant factor V D 8N (8π 2 ) D/2 from eq. (6.4). In this section, we will suppress all such overall prefactors that do not depend on the kinematic variables. that lattice sums become unity 25 Γ (...) C → 1 . (6.17) • The worldsheet functions f (1) ij have simple poles when vertex operators collide, f ij ∼ 1/z ij as z i → z j . These regions of the integration domain require separate treatment and yield kinematic poles through interplay of the singular functions f (1) ij with the Koba-Nielsen factor (6.5): (6.18) For example, the regime of z 2 → z 3 in the 3-point integrand (6.7) gives rise to The ubiquitous combinations s ij f (1) ij which accompany the gauge-invariant kinematic factors in the half-maximal correlators of section 6.1 cancel these kinematic poles. For example, the factor of s −1 23 on the right-hand side of (6.19) (due to the integration region where z 2 → z 3 ) cancels in . (6.20) To make contact with the momentum-space construction in section 3, it remains to translate the resulting Schwinger parametrization of field-theory amplitudes into Feynman integrals. The following map will be sufficient for the 3-and 4-point amplitudes to trade the integration over worldlines with length t and proper times ν j for the integration over the loop momentum : , 25 For toroidal compactifications, given the Kähler moduli T i 2 , the internal momenta decouple when T i 2 → 0, such that T i 2 /α stays finite. In this limit, with the conventions adopted in [1], we have Often in the literature on maximal supersymmetry, e.g. in [66], the Γ (2m) C differ from ours by a factor of (α τ 2 ) m . This is because in these papers Γ where α, β m and γ mp are arbitrary scalars, vectors and tensors, respectively, and In the following, we will apply these techniques to compute the field-theory limit of the above 3-and 4-point functions, taking the manifestly gauge-invariant integrands in (6.7) and (6.8) as a starting point.

The 3-point amplitude
At 3 points, the maximally supersymmetric integrand vanishes, and the half-maximal open-string amplitude reads Starting from the half-maximal integrand in (6.7) and formally equating the vanishing expressions By virtue of (6.21), this is easily seen to reproduce (3.15).

The 4-point amplitude
The 4-point string amplitude reads with the gauge-invariant form of the integrand I 4,1/2 ( v k ) given in eq. (6.8). It is convenient to organize the field-theory limit according to the configurations of colliding vertex operators: • Simultaneous contact of 3 vertex operators occurs for products of two f (1) ij , reproducing the bubble integral in (3.48). We take the compactification dependent coefficients N −1 k=1 c kχk into account to combine the resulting contributions with the maximally supersymmetric sector of (6.25).
• From worldsheet regions where only one pair of z i and z i+1 collides, we instead have The vanishing kinematic combinations in the second line translate into the spurious scalar triangles in eq. (3.43), and the vector triangles in (3.48) will be traced back to the irreducible part of the worldsheet integrand, that we consider next.

Streamlining the model dependence in the 4-point amplitude
Collecting the above results, and using the expression (6.16) forF 1/2 (kv) as well asχ k = −(sin(πkv)/π) 2 , we can finally write the field-theory limit of the 4-point amplitude as with A 1-loop N =4 and A hyp N =2 given by (5.21) and (5.22), respectively. In eq. (6.32), the unambiguous identification of the hypermultiplet contribution (and of its normalization relative to the N = 4 multiplet) is possible because the coefficients arising from Chan-Paton factors, c 0 and c k , are related to the numbers of multiplets, c vec , c hyp , as

Conclusions
In this work, we have presented a method to construct 1-loop amplitudes of 6-dimensional gauge theories with half-maximal supersymmetry from first principles: starting from a string-theory inspired ansatz of kinematic building blocks, imposing locality and gauge invariance led us to unique answers for 3 and 4 external gauge bosons, respectively, see (3.16) and (3.48) for the manifestly gauge-invariant results. We have checked that, when dimensionally reducing the integrands to D = 4, these expressions integrate to results known from the field-theory literature, and also emerge from the field-theory limit of the corresponding open-string amplitudes [1] (also see [2,4,5,3] for earlier work). Most of the building blocks are introduced in generality for any number of legs, so it appears feasible to construct highermultiplicity amplitudes along the same lines.
A similar strategy has been applied to tree and loop amplitudes of 10-dimensional SYM [11,12,13] where supersymmetry is imposed along with gauge invariance and the kinematic ansatz is inspired by the pure-spinor superstring [10]. Accordingly, the 1-loop amplitudes of this work inherit their structure from their maximally supersymmetric counterparts in pure-spinor superspace [12] with two additional legs.
It would be desirable to reexpress and supersymmetrize the present results in a comparable superspace setup and to derive their superstring ancestors from the hybrid formalism [68,69].
The structural similarities between loop amplitudes with n legs and 16 supercharges and n+2 legs and 8 supercharges naturally lead to questions about further supersymmetry breaking to 4 supercharges.
For N = 1 supersymmetric amplitudes in 4 dimensions, the parity-even part is a straightforward dimensional reduction of the parity-even N = 2 contributions in our results. This is due to the enhanced supersymmetry when summing over the fundamental and antifundamental chiral multiplets in the loop [18]. However, it remains an open challenge to construct parity-odd parts of SYM amplitudes with quarter-maximal supersymmetry and pure Yang-Mills amplitudes from the principles we used here. It would be equally interesting to extend this to minimal supergravity; as reviewed in [1], only minimal supersymmetry allows for 1-loop corrections to the Einstein-Hilbert term in the low-energy effective action. There can also be interesting couplings of open-and closed-string sectors, for example one could investigate loop amplitudes of gauge bosons coupled to 6-dimensional tensor multiplets, that we did not include in our decomposition (5.17) in the context of that section.
While the 3-point amplitude of this work obeys the BCJ duality between color and kinematics and thereby yields the half-maximal supergravity amplitude (4.6) as a byproduct, we encounter an obstacle at the 4-point level to satisfy kinematic Jacobi identities. This ties in with the findings of [12] on maximally supersymmetric 5-and 6-point amplitudes. It is a particularly burning question how to reconcile the present approach to D-dimensional 1-loop n-point amplitudes with the BCJ duality and the double-copy construction 27 . As a rewarding first step forward, it would be helpful to directly compute the 6-dimensional 1-loop supergravity amplitudes beyond the current reach of the double copy, based on the field-theory limit of the closed-string amplitudes in [14,1] and comparison with the 4-dimensional BCJ analysis of [5].

Acknowledgements
We program. 27 We are grateful to Yu-tin Huang and Henrik Johansson for informing us about their unpublished BCJ form of the 4-point 1-loop amplitude in D = 6 half-maximal SYM, including parity-odd terms [50]. We recall that 4-dimensional BCJ representations are available in [17,18].

A RNS approach to Berends-Giele currents
In units 2α = 1, the color-stripped vertex operator for the gluon in the RNS formalism is given by V (e, k, z) = (e m ∂X m (z) + 1 2 f mn ψ m (z)ψ n (z))e k·x(z) . (A.1) the 2-particle currents e m 12 and f mn 12 in (A.5) identifies Berends-Giele currents as a suitable language to describe the kinematic dependence of string amplitudes. In the same way as multiparticle correlators give rise to nested OPEs with the same 2-point contractions among X m and ψ m at each step, the iteration of the Berends-Giele recursion (2.13) and (2.14) yields currents e m P and f mn P of arbitrary multiplicity.

B Spinor-helicity conventions
In this appendix we summarize the spinor-helicity conventions used in this paper. Given solutions to the Weyl equations we use the following conventions and notation The computations in section 5.1 are performed by choosing the following expressions for the polarization vectors e m ± of gluons with helicity ±1, where q i are arbitrary massless reference momenta that cancel in gauge-invariant expressions.
With the above choices, by use of completeness these are used to derive the spinor-helicity expressions in section 5.1.

C Relations for the kinematic building blocks
In this appendix we summarize a series of on-shell identities relating the scalar blocks to the vector and tensor blocks introduced in section 2. These identities are used in the main text, in particular to make contact between the Schwinger parametrization of the field-theory amplitudes, which naturally comes in terms of just scalar blocks, and the corresponding Feynman-integral representations. While contractions with external momenta give rise to In addition, the scalar pseudo-invariants P 1|2|3,4 satisfy s 12 P 1|2|3,4 + s 13 P 1|3|2,4 + s 14 P 1|4|2,3 = s 23 s 34 C 1|234 , (C.3) that is, this combination is gauge-invariant because the parity-odd part cancels.
The following corollaries of (C.1) and (C.

D Feynman integral "basis"
We use the following "basis" of Feynman integrals for the computations in D = 4 − 2 dimensions in section 5.3 (though the result will ultimately be expressed only in terms of I 2 (s ij ) and I D=6

4
): We express our generic integrals in terms of these basis integrals as follows.

E Explicit 4-dimensional models from oriented-string orbifolds
In this appendix we show that eq. (6.33), which relates Chan-Paton traces in string theory to the gauge group and supermultiplet content of the effective theory, is satisfied in explicit string models. As emphasized in the main text, the oriented-string orbifold models we consider here are not full-fledged string models, but if taken by themselves suffer from inconsistencies, especially when coupled to gravity.
It would be straightforward to generalize them to consistent string models, e.g. as orientifolds, but they suffice as they are for our purposes here: to compare specific coefficients in specific amplitudes.
The spectrum of effective gauge theories with half-maximal supersymmetry arising from Z N orbifold compactifications of the RNS string can be obtained by projecting the spectrum of the maximally supersymmetric open string as follows 29 We denote color-stripped string states by |w and Chan-Paton matrices by λ. The rotation operators J 1 , J 2 act on the twisted tori T 2 1 and T 2 2 , and the matrices γ k for k = 0, 1, . . . , N −1 represent the orbifold group Z N on Chan-Paton factors. If we wanted to build fully consistent string models, e.g. free of closed-string tadpoles and gauge anomalies, certain constraints would arise on the number of Chan-Paton factors and the allowed representations spanned by the γ k matrices (see e.g. [72,61]). 29 As in the main text, we consider the class of Z A N models in the language of [61] but without imposing the worldsheet parity projection and considering only D9-branes.

E.1 Simplest models without hypermultiplets
More generally, the spectrum of the D-dimensional effective theory of these simple string models consists of a half-maximal vector multiplet with U (M ) gauge symmetry, and no hypermultiplets. Referring to eq. (5.19), in these models we then have c vec = M and c hyp = 0. We now wish to verify eq. (6.33), i.e.
we want to show that in this class of models (where v = 1/N ), The coefficients c 0 and c k are determined by traces of the γ k matrices acting on the Chan-Paton factors.
In models with just one U (M ) gauge group, we have i.e. c k clearly does not depend on k when γ k are in the trivial representation, and we obtain as desired, where we used the elementary finite sum N −1 k=1 sin(πkv) 2 = N −1 k=1 sin(πk/N ) 2 = N/2.

E.2 Simplest models with hypermultiplets
To obtain models with hypermultiplets, the matrices γ k should form a non-trivial representation of Z N . with m 0 + m 1 + · · · + m N −1 = M , the total Chan-Paton degeneracy at each string endpoint. In the Z 2 case, eq. (E.6) specializes to γ 1 ≡ diag(1 m 0 ×m 0 , −1 m 1 ×m 1 ) , (E. 7) and the massless NS states that survive the Z 2 projection have the following block structure is equal to (E.9) and confirms eq. (6.33) that we wanted to check.