From maximal to minimal supersymmetry in string loop amplitudes

We calculate one-loop string amplitudes of open and closed strings with N=1,2,4 supersymmetry in four and six dimensions, by compactification on Calabi-Yau and K3 orbifolds. In particular, we develop a method to combine contributions from all spin structures for an arbitrary number of legs at minimal supersymmetry. Each amplitude is cast into a compact form by reorganizing the kinematic building blocks and casting the worldsheet integrals in a basis. Infrared regularization plays an important role to exhibit the expected factorization limits. We comment on implications for the one-loop string effective action.

We will not focus on phenomenology in this paper, but clearly it is of great interest to develop the state of the art of string effective actions with minimal supersymmetry, as opposed to maximal. We will argue that even at 1-loop order in minimal supersymmetry, there is much left to be understood about string amplitudes. For fundamental problems like moduli stabilization, without which there can be no reliable phenomenology 1 , the string effective action should be calculated at least to 1-loop order, as some stabilization effects are quantum-mechanical. Half-maximal supersymmetry provides a useful step on the way to minimal supersymmetry.
In this paper we study type II string compactifications on K3 and Calabi-Yau (toroidal) orbifolds that break supersymmetry down to half-maximal or quarter-maximal. For closed strings, quarter-maximal amounts to 8 supercharges which is N = 2 supersymmetry in D = 4 terminology. The basic technology to compute all 1-loop amplitudes in type IIB Calabi-Yau orbifolds (and orientifolds) has in principle been available for decades, but various technical obstacles have prevented progress.
Impressive progress on the gauge boson 1-loop 4-point amplitude in quarter-and half-maximal supersymmetry was made in 2006 [7], but in a form that was difficult to process further, for example to check supersymmetry Ward identities. Last year, this calculation was simplified [8] by specializing the external polarizations to spinor-helicity variables at an early stage of the calculation.
Recently, work on the graviton 1-loop 4-point amplitude for half-maximal K3×T 2 was presented in [9,10]. In contrast to those papers, we first perform the sum over the spin structures of the Ramond-Neveu-Schwarz (RNS) formalism and will later perform the field-theory limit. Also, in addition to K3 compactification to half-maximal supersymmetry, we also consider closed strings in Calabi-Yau compactification to quarter-maximal supersymmetry.
In this paper, we will approach the problems in the aforementioned papers from a new angle and  (6.27) and (6.32). The precise improvements on previous work will be clarified in those sections. The closed-string expressions are valid for generic massless NSNS external states (graviton, dilaton, and antisymmetric tensor).
One key aspect of these results is that connections between 1-loop amplitudes with different amounts of supersymmetry are revealed. First, the parity-even kinematic factors of open-string n-point amplitudes for half-maximal and quarter-maximal supersymmetry are identical. The amplitudes are only distinguished by the explicit functions of worldsheet moduli, see (4.43). Second, as will be detailed in section 5, the structure of half-maximal open-string amplitudes at multiplicity n is very similar to that of their maximally supersymmetric counterparts at multiplicity n+2. Finally, the progress we made on open-string amplitudes reverberates in our closed-string amplitudes in section 6, where the simplified expression (6.32) for the 4-point function closely resembles the maximally supersymmetric 6-point amplitude of [11].

Superstring effective action
We will not extract details of the effective action in this paper, but here is a short review of expectations and motivations.
The closed-string sector is somewhat more universal than the open-string sector, so let us begin there, but most of our comments extend to open strings. In D = 10 the leading correction to the type IIB string effective action appears at order α 3 . Even this leading correction is not completely known (especially for the RR sector), but many pieces are well understood. The gravitational part of the type IIB action in Einstein frame is (see e.g. [12,13,14,15] for some original references, or [16,17,18] for contemporary work) where R 4 is schematic for index contractions with the well-known tensor structure t 8 t 8 + 10 10 (see e.g. [19]). There is a simple way to include also the other massless NSNS fields: the Kalb-Ramond B-field and dilaton. As discussed e.g. by [20] in 1986, and more recently in e.g. [16,17,18], the idea is to shift the Riemann tensor by ∇H and ∇∇φ as where κ is the gravitational coupling, H is the NSNS 3-form field strength and φ is the dilaton. The geometric interpretation of this shift as torsion is discussed for example in [17]. (In eq. (2.2) and throughout this work, vector indices m = 0, 1, . . . , D−1 are taken as m, n, p, q, . . . from the middle of the latin alphabet, where the number D of dimensions will be clear from context.) We have not discussed terms that depend on RR fields, that we comment on in the outlook.
The best-understood type IIB coefficient in D = 10 is the one above, of the R 4 term [15], where Im (τ ) = g s and E s is the nonholomorphic Eisenstein series with series expansion 2 After compactification to e.g. D = 4 on some nontrivial (non-toroidal) space, much less is known than in D = 10, since as discussed above, the requisite amplitudes for the less-than-maximal type II superstring 2 For a recent review of the systematics of such expansions, also with toroidal compactification, consult [21].
have not been studied systematically until recently. (There is substantial literature on related issues in the heterotic string, some of which we review below.) As an illustration of the great simplifications of maximal supersymmetry, in (2.4) we see that there are no perturbative corrections beyond one loop.
This non-renormalization theorem does not extend to minimal supersymmetry. More relevant for our purposes is that in maximal supersymmetry, the α 3 R 4 correction is the leading-order α correction in a flat background. With minimal supersymmetry one generically expects all the lower-order terms to appear: R, α R 2 and possibly (see below) α 2 R 3 , both at tree level and at loop orders. So with less supersymmetry, the R 4 correction above that was leading in maximal supersymmetry becomes sub-subsub-leading in the α -expansion: where the subscript 1/4 means "quarter-supersymmetric". The loop-corrected coefficients ∆ i in general depend on the moduli, and one can extract aspects of this dependence from the 1-loop string amplitudes in this paper. Until recently this would not have been feasible.
Again, similar comments hold for open strings and string corrections to gauge-field effective actions.

Tree-level: review
As summarized above, in the type II string effective action in D = 10 there is the famous α 3 R 4 term that first appears at string tree level (sphere diagram). The lower powers α R 2 and α 2 R 3 are forbidden by 32 supercharges. By contrast, the heterotic string in D = 10 with 16 supercharges is known to have a tree-level R 2 term [20], unlike in type II. This can be explained by "double copy" (see e.g. [23]): there is a tree-level α F 3 term on the purely bosonic side of the heterotic string, and no α F 3 term on the supersymmetric side. "Multiplying out" two vectors to give a graviton in the sense of double-copy, one obtains a tree-level subleading term α R 2 in the heterotic string in D = 10. So an α R 2 term is allowed by 16 supercharges, but it is not required. For type II compactified on K3, Antoniadis et al. [19] explain (p.4) that there is no tree-level R 2 term.
The cubic curvature R 3 terms are absent in all these theories. The first evidence for this was by explicit calculation, but now it is understood more generally, see the next section.

Tree-level interactions of open and closed superstrings involve multiple zeta values (MZVs) upon
α -expansion, a first hint being the above single-zeta value in α 3 ζ(3)R 4 . In the closed-string sector, D m R n couplings in [24,25] can be traced back to the all-multiplicity results for open-string trees in [26,27] through the KLT relations [28] which imply identical graviton interactions in type IIB and type IIA theory. The patterns of MZVs and covariant derivatives can then be generated from the Drinfeld associator [29]. The study of D m R n interactions is important to assess the UV behavior 3 of N = 8 supergravity in four dimensions by testing their compatibility with its E 7(7) duality symmetry [30,32,33,31].
As initially observed in [24], systematic cancellations obscured by the KLT relations occur when assembling D m R n interactions from open-string amplitudes [25], leaving for instance only one tree-level interaction of type D m R n at the mass dimensions of D 2m R 4 with m = 0, 2, 3, 4, 5. The selection rules for the accompanying MZVs were identified with the single-valued projection [34]. Also beyond tree level, there is evidence that the single-valued MZVs and polylogarithms govern the closed-string α -expansion [35,36].
In the heterotic theory, an interesting tree-level connection between single-trace interactions in the gauge-sector and the type I superstring was found in [37], again based on the single-valued projection of MZVs. In the gravitational sector of the heterotic string, half-maximal supersymmetry allows for additional D m R n interactions absent for the superstring whose implications for counterterms of N = 4 supergravity were studied in [38]. At a given mass dimension, the D m R n interactions accompanied by MZVs of highest transcendental weight are universal to the heterotic and type II theories [39], and this universality of the leading-transcendental part in fact carries over to the bosonic open and closed string.
Accordingly, the MZVs along with non-universal D m R n interactions of the heterotic string starting with R 2 have lower transcendental weight as compared to their universal counterparts [39], suggesting a classification by weight at each mass dimension.

Factorization and ambiguities: tree level and 1 loop
The Gross-Sloan paper from 1986 mentioned above [20] contains a detailed discussion of field-theory pole subtractions, a key piece in the machinery of extracting an effective field theory from string amplitudes.
For less than maximal supersymmetry, the 4-point string amplitudes at 1-loop order factorizes onto 3point vertices, as drawn in fig. 1. Reducible field-theory diagrams with the gravitational 3-point vertices 3 Based on a symmetry analysis of D m R n matrix elements initiated in [30], any counterterm with the mass dimension of D 6 R 4 and below was ruled out, guaranteeing UV-finiteness of four-dimensional N = 8 supergravity up to and including 6 loops [31].
need to be subtracted from the low-energy limit of the string amplitude to isolate the irreducible fieldtheory 4-point coupling corresponding to D 2m R 4 terms in the effective action. This is a laborious procedure. As emphasized in [41], if we are interested in fewer powers of the Riemann tensor like R 2 and R 3 , we could in principle extract them from 2-point or 3-point functions, where one could expect there to be no reducible contributions at all. This can be taken as a general argument that for efficient computation one should strive to compute the lowest number of external legs that can probe the term of interest in the effective action.
However, 2-point and 3-point functions of massless states vanish on-shell unless they are infraredregularized, as we review in appendix C. This regularization is a key point in this paper and we will discuss it in more detail in section 4.2. Somewhat surprisingly, we will see that the same regularization procedure should also be applied to n-point functions for any n, to exhibit the expected factorizations in the spirit of this section.
A related issue is that since our string amplitudes are on-shell, there are ambiguities coming from field redefinitions, a typical example being a shift of the graviton h mn → h mn +R mn that can shuffle coefficients between the three terms R mnpq R mnpq , R mn R mn and R 2 in the string effective action, as explained for example in [42]. The coefficients are moduli-and background-dependent, so a more general background could lift some of these degeneracies. We discuss this issue a little further in section 6.5.1 and appendix C, but the main focus of this paper is the underlying string amplitudes and not the explicit construction of a string effective action.
So let us return to the factorization of the closed-string 4-point 1-loop amplitude as illustrated in fig.   1. (The following discussion will be general, but a few explicit expressions corresponding to the figures drawn here are given in appendix B.) The moduli space of string loop amplitudes is interesting already in this fairly simple example. As a first question, in the factorization limit in fig. 1, which of the two more specific diagrams in fig. 2 is actually realized? By conformal invariance of the worldsheet theory, we can always factor off a sphere from a bulk point on the worldsheet. When we draw this sphere explicitly as in fig. 2, we mean that the two external states on the left part of each diagram are closer to each other than to any other vertex operator 4 . In the worldsheet computation, this arises from a delta function of two punctures along with a propagator, caused by the collision of vertex operators. Now, we will encounter situations where an inverse propagator is generated from the contractions among vertex operators. If this cancels the propagator that arose with the delta function, we draw the diagram in the left panel of fig. 2. If the propagator is uncancelled, we draw it explicitly as in the right panel of fig. 2. 4 The moduli space of superstring amplitudes in figures like in this section is discussed more systematically in for example [43], for open strings in for example [44] and more recent discussions include Witten's extensive notes [45]. We admit that the qualification "closer to each other" restricts us to some class of worldsheet metrics.
In both diagrams in fig. 2, the delta function has reduced the number of integrations over punctures by • Figure 2: The distinction between "delta function" and "delta function and propagator".
one, so the moduli space of the remaining 1-loop integral is that of a 3-point 1-loop torus diagram, but where one of the external momenta is the sum k 1 + k 2 of momenta of the two states on the left side of the 4-point diagram. Unlike individual momenta of massless states, this sum is not constrained to be lightlike: (k 1 + k 2 ) 2 = 0. In field theory, the right side is then called a 1-mass triangle, but we emphasize that we have not taken a field-theory limit yet.
Analogously, we can ask whether there are any 1-mass bubbles, i.e. whether there is further factorization of the subdiagram on the right of fig limit would have generated a 3-particle propagator (k 1 + k 2 + k 3 ) −2 = (−k 4 ) −2 which is in fact infrared divergent in the 4-particle momentum phase space. The two spheres in the diagram on the right in fig.   3 each represent a delta function from a particular region in moduli space, so this leaves one integration over a puncture, as appropriate for a torus 2-point function with one fixed vertex operator. In the field-theory limit, this string amplitude will indeed generate a 1-mass bubble.
Factorization in the field-theory limit is an interesting topic in its own right, and will be discussed in a companion paper [46].
The above discussion was quite detailed, so let us make one broader statement, that we will explain in more detail in later sections. In maximal supersymmetry, it is well-known that the factorization of the 4-point function as in fig. 1 does not occur (i.e. has zero residue). In fact, the number of successive factorizations of an n-point function in maximal supersymmetry is n−4. We will find that for halfmaximal supersymmetry as well as parity-even contributions to 1-loop amplitudes in quarter-maximal supersymmetry, this number is n−2, as we have illustrated in the figures in this section. Parity-odd terms in quarter-maximal require a refined analysis, and preliminary arguments in later sections suggest  in the heterotic string in D = 10 (or on T 4 ), one would expect that supersymmetry would allow R 2 .
The details are interesting: it turns that the 1-loop correction to R 2 vanishes in IIB on K3 but does not vanish in IIA on K3. See for example [41,19] and especially [47] as well as section 6.3 for a review of this string amplitude computation. From the supergravity point of view, [19] explains the vanishing of 1-loop R 2 corrections in the D = 6 IIB string theory on K3 from reduction of the ten-dimensional 1-loop term (t 8 t 8 ± 10 10 )R 4 , where the relative sign gives cancellation in IIB but not IIA. There is also a duality argument: for IIA on K3 there should be a 1-loop R 2 correction but no tree-level R 2 , because in heterotic on T 4 there is a tree-level R 2 (as discussed above) and no 1-loop R 2 , and they should be exchanged by heterotic-IIA duality [19]. 5 These three arguments illustrate the variety of techniques that have been developed for half-maximal supersymmetry.
The previous discussion concerned D = 6. Compactification of type II on K3×T 2 to D = 4 is discussed in [51,47,19], where the authors calculate moduli-dependent couplings like where U is the complex structure and T is the Kähler modulus of the 2-torus, and they are exchanged by T-duality. Note that despite having the same amount of supersymmetry as in IIB on K3 above, compactification to D = 4 on this 2-torus allows an R 2 term in IIB. The authors of [47] argue that in the decompactification limit of the 2-torus, the coefficient would need to contain some power of the Kähler modulus T of the 2-torus to survive the large-torus limit, and ∆(U ) does not. This recovers the vanishing of the R 2 term in D = 6 for IIB and the non-vanishing for IIA.
Finally, there is a fairly detailed discussion of the heterotic 1-loop R 2 correction in [42], where the non-renormalization of the Einstein-Hilbert action is also discussed. See e.g. [52] for previous work and [53] for a useful summary of some of the older literature.
Let us move on to R 3 corrections. Reduction of R 4 from D = 10 on K3 or Calabi-Yau produces contractions of the schematic form (R external ) 3 R internal , where R internal leaves no room for anything else than the Ricci scalar of the compactification manifold, which vanishes by Ricci-flatness. In effective field theory, there is a general superspace argument that no superinvariant containing R 3 as bosonic component can be constructed (see for example [54]). Original explicit calculations showing the absence of R 3 terms go back to the 1970s, see for example [55,56,57,58,59]. Some of these explicit calculations are being revisited using modern techniques, see e.g. [60].
However, eq. (2.1) together with eq. (2. 2) indicate that there should be R 3 terms in nontrivial backgrounds, like flux backgrounds or internal dilaton gradients. Constructing such terms from string amplitudes in nontrivial backgrounds is challenging, see the conclusions for comments on this.
Finally, string loop corrections to the Einstein-Hilbert action in quarter-maximal supersymmetry were studied using the background field method in [61,62]. Amplitude calculations of this correction was discussed recently in [63] 6 which builds on, corrects and extends results from [62,64,65]. These results are all extracted from infrared-regularized low-point functions, in the strong sense that we discuss in detail in section 4.2. It would be desirable to compare with our results, though we do not do so in detail in this paper. The general conclusion from these papers is as expected from the effective supergravity discussion in [19], section 5: there is a 1-loop correction to R in type II on Calabi-Yau. It descends from the 10 10 term in (2.1), and the relative sign of the tree-level and 1-loop correction to R in type IIA is the opposite to that of type IIB. For completeness, let us also mention that there is a correction to R in type IIB orientifolds on K3 [66], which is also quarter-maximal due to the orientifolding.
Covariant derivatives D m R n of Riemann tensors have also been studied at loop level. In tendimensional type IIB theory, S-duality was exploited to determine the full moduli-dependent coefficients of the D 4 R 4 [67] and D 6 R 4 [68] interactions, including their non-perturbative completions. S-duality based predictions for the 2-loop and 3-loop coefficient of D 6 R 4 [69] were confirmed by the amplitude computations of [70] and [2], also see [71] for D 2 R 5 at two loops.
The amplitude calculations of this paper culminate in the compact expression (6.32) for the halfmaximal 1-loop amplitude involving four NSNS sector states in type IIB and type IIA. We lay some foundations for a systematic investigation of 1-loop D 2n R m≤4 couplings in half-maximal type II compactifications by identifying the gauge-invariant "seeds" in their matrix elements.
Here we have focused mostly on gravitational corrections. Other NSNS corrections involving B fields and dilatons have been studied somewhat less, but were discussed for example in [17,18], and our results here are equally relevant for those loop corrections, see e.g. section 6.5.2. We comment on RR fields in the conclusions.

Amplitude prescriptions and spin sums
In this section we define the computations that will occupy us for the remainder of the paper, including efficient techniques to sum over spin structures of the worldsheet spinors. We study compactifications of type I and type II superstrings on certain Z N orbifolds illustrated in fig. 4 that yield half-maximal and quarter-maximal supersymmetry. This is textbook material (see e.g. [40,72,73,74]), but before launching into the detailed prescriptions, we give a quick review.

Orbifolds
We will consider supersymmetric orbifolds of the form The "orbifold group" Z N is a discrete subset of the rotation group and one identifies points in spacetime that are related by the Z N action. With complexified string coordinates Z j = X 2j+2 + U j X 2j+3 , where j = 1, 2, 3 and U j is the complex structure of the j th 2-torus, the discrete orbifold rotation is diagonal: The rational numbers v j are such that Θ N = 1 (or occasionally one allows −1), and they satisfy v 1 + v 2 + v 3 = 0 to preserve some supersymmetry 8 , see table 2 below for examples. An orbifold theory is obtained from a "parent" theory in D = 10 by inserting the projector N −1 k=0 Θ k /N in amplitude trace computations. The power k in the trace is called the sector of the orbifold. The identification of points in spacetime that are related by the Z N action can create conical singularities at fixed points of the orbifold action, as in fig. 4. One can also mix in the worldsheet parity operation in the above orbifold group to make an orientifold, in the same sense that type I (open+closed strings) in D = 10 is an orientifold of type IIB. The D-branes on which open strings can end are added to cancel the negative D-brane charge of the orientifold plane. In noncompact models, one can get away without orientifolding, but in compact models, there is some additional work to compute the Möbius strip and Klein bottle 7 We assume factorizable tori, i.e. T 4 = (T 2 ) 2 and T 6 = (T 2 ) 3 . 8 A simple way to see this is to use the oscillator notation for gamma matrices, see e.g. Appendix B of [40].  The orbifold twist kv (that we will call γ, see eq. (3.37)) will occur in all our amplitudes.
amplitudes that might be needed for specific consistent string models. We will only consider annulus and torus amplitudes in this paper, but the key simplifications of the integrands should carry over straightforwardly to the remaining topologies. (We note that for closed strings, our torus amplitudes will be consistent by themselves, but for model-building one might want to orientifold also for closed strings, to allow moduli stabilization in minimal supergravity.) Table 2: Examples of (v 1 , v 2 , v 3 ) for supersymmetric orbifolds/orientifolds, see e.g. [74].

Open-string prescriptions
One-loop scattering amplitudes among unoriented open-string states receive contributions from cylinder and Möbius-strip diagrams. In this work, we will discuss the planar cylinder (annulus) with modular parameter τ 2 as a representative diagram where all external states are inserted on the same boundary component, and the corresponding color factor is a single-trace of gauge-group generators. In a parametrization of the non-empty cylinder boundary via purely imaginary coordinates z i with 0 ≤ Im (z i ) ≤ τ 2 , the universal n-point open-string integration measure will be denoted by We have 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 of eq. (4.5) below, which arises from the plane-wave factors of the vertex operators, see section 4.1. The measure (3.2) with modular parameter τ 2 can straightforwardly be adjusted to the remaining worldsheet topologies, and the delta-function δ(z 1 ) fixes the translation invariance of genus-one surfaces, by fixing one puncture to the origin. The number D of uncompactified spacetime dimensions is denoted as a superscript of dµ D 12...n , and the subscript 12 . . . n refers to the cyclic ordering of the open-string states along the boundary as well as the trace-ordering of the accompanying color factor.

Half-maximal supersymmetry
If one of the twist vector entries vanishes but the other two are nonzero, say v 3 = 0 and therefore v 1 = −v 2 as in table 2, the orbifold only breaks half of the supersymmetries. These orbifolds can be characterized by a single rational real number v that enters the partition functions through the vector v k ≡ k(v, −v). For brevity we will mostly discuss half-maximally supersymmetric orbifolds in their maximal spacetime dimension D = 6, i.e. arising from compactification from D = 10 on T 4 /Z N , which are special points in the moduli space of K3 manifolds. The 1-loop amplitude of n gauge bosons in this setting is given by (for textbook examples, see e.g. [74]) where the subscript "1/2" means "half-maximal", Γ (n) C denotes lattice sums over n-dimensional internal momenta, c 0 and c k are model-dependent constants determined by the action of the orbifold group on the Chan-Paton factors 9 , and the generalities of the constantsχ k = −[sin(πkv)/π] 2 are explained in appendix A. The external-state information is encoded in the integrands I ... whose dependence on the integration variables τ 2 and z i of the measure (3.2) will usually be suppressed. The subscripts "max" or "1/2" distinguish orbifold sectors that preserve all or half the supersymmetries, respectively. While the maximally supersymmetric integrand is parity-even 10 , the half-maximal integrand receives both parity-even and parity-odd contributions labelled by superscripts e and o. We write where v k highlights the dependence of the parity-even contribution on non-trivial orbifold sectors, i.e. on the internal partition function. The dependence of parity-odd integrands on orbifold twists v k cancels between the contributions to the partition function due to worldsheet bosons and worldsheet fermions 9 In toy models with just one gauge group, c k = ( tr γ k ) 2 , cf. appendix A. In models with more than one gauge group the traces are over sub-blocks of the matrices γ k . Explicit expressions are given in the companion paper [46]. 10 In general, for amplitudes of solely external states, like amplitudes of gauge bosons in D = 6, there is never a parityodd contribution to the maximally supersymmetric integrand. With only external excitation it is impossible to saturate the fermionic zero modes along the internal directions.
in the odd spin structure. Explicitly, we have where the second argument of the ϑ-functions is the purely imaginary τ = iτ 2 for the planar cylinder see (4.26) and (4.27) for the t 8 tensor.
The parity-odd part of the integrand contributions from the odd spin structure ν = 1, and the path integral over worldsheet spinors requires D zero-mode components to be saturated according to [40,78] with the D-dimensional Levi-Civita symbol on the right-hand side. The dependence on the position z 0 of P (+1) drops out on kinematic grounds, as expected from general arguments (see e.g. [79,40]), which we check in detail in appendix D. Note that the expression  Similarly to the maximally supersymmetric integrand, when there are internal directions that are unaffected by the orbifold rotation, the parity-odd contribution vanishes for external excitations. To save writing, we will mainly give D = 6 expressions, but the point here was to illustrate that the extrapolation is trivial, before performing τ integrals.

Quarter-maximal supersymmetry
A similar prescription applies to orbifolds with quarter-maximal supersymmetry which we will discuss in their maximal spacetime dimension D = 4. The quarter-maximal counterpart of (3.3), contains two kinds of lattice sums Γ (6) C , Γ C , and the twist vector is v k = k(v 1 , v 2 , v 3 ) with v 1 +v 2 +v 3 = 0. Half-maximal contributions arise in orbifold models where one of the three internal tori is fixed under the action of some orbifold sectors, e.g. if Θ k Z 3 = Z 3 for some k (i.e. kv 3 ∈ Z). In this case kv 1 = kv 2 , and I e n,1/2 ( v k ) is determined by (3.6) with v → v 1 . The quarter-maximal integrand contains parity-even and parity-odd contributions, where I o n,D=4 is a special case of (3.8), and the parity-even part is understood to depend on kv j / ∈ Z for all j = 1, 2, 3.

Spin sums
A major challenge in the evaluation of string amplitudes with half-and quarter-maximal supersymmetry is to perform the spin sums in the parity-even integrands (3.6) and (3.13). As elaborated in section 4.1, the worldsheet spinors in vertex operators V (0) i cause the correlators to depend on the spin structure ν through their two-point function, the Szegö kernel (3.14) Individual sectors with ν = 2, 3, 4 contain spurious worldsheet singularities that cancel upon summation, as a consequence of supersymmetry. Such spurious singularities are an inconvenient feature of the RNS formalism, and their cancellation in maximally supersymmetric cases can be manifested through the techniques of [75,80]. In this section, we will demonstrate that the method of the references can be adapted to address situations with reduced supersymmetry as well.

Worldsheet functions
We follow the notation of [80] where a doubly-periodic function f (n) for each non-negative integer n is defined by a non-holomorphic Kronecker-Eisenstein series starting with Note that f (1) is the only singular term of (3.15) with a simple pole at the origin as well as its translations z = n + mτ with m, n ∈ Z. For ease of notation, the dependence on the modular parameter τ will be suppressed in the following.
We note in passing that Ω(z, α, τ ) is closely related to the twisted fermion Green's function, which is in turn a nonholomorphic Eisenstein-Kronecker function E (k) s (w, z, τ ), as discussed for example in [81]. We note that there, w has direct interpretation as a twist of external orbifold-charged states, while here α is a formal expansion parameter.

Maximal supersymmetry
After pairwise contractions of the worldsheet spinors to Szegö kernels (3.14) via Wick's theorem, RNS amplitudes with maximal supersymmetry give rise to the spin sum An efficient method to evaluate (3.18) and to make its pole structure manifest was introduced in [75] (also see [50] for a variation). The functions f (n) in (3.15) allow to streamline the results as [80] G n (x 1 , x 2 , . . . , using the shorthand f The appearance of the holomorphic Eisenstein series G 4 as an extra constant in (3.24) generalizes in a pattern described in [75,80], and see also an alternative method in [82]. The associated x j -dependence in G N ≥9 can be cast into a convenient form through the notation A general definition can be compactly given in terms of the generating series Ω(z, α) in (3.15), The virtue of the functions V w to express G n at higher multiplicity is exemplified by [80] We see that without resorting to specific Riemann identities for large numbers of theta functions, these results let us write relatively compact expressions for integrands up to at least 10 external states without too much effort, incorporating the cancellations mentioned above.

Reduced supersymmetry
The results in the maximally supersymmetric sector that we reviewed above will now be extended to the most general spin sum in half-maximal and quarter-maximal amplitudes (3.3) and (3.11). The key idea is to rewrite the orbifold-twisted partition functions (which reflect reduced supersymmetry) in terms of fermion Green's functions with the twist as an insertion (which "uses up" additional external states).
To this end, we rewrite (3.6) and (3.13) by pulling out a factor like that of the maximal case (3.18) by hand: using the definition (3.14) of the Szegö kernel. The correlators of V (0) i yield the same cycles of two-point x i = 0 as seen in the maximal case. Hence, the most general spin sum resulting from (3.33) and (3.34), respectively, is given by In order to avoid proliferation of factors k, we introduce the shorthands for the orbifold twists. The expressions can be identified with the prototype spin sum (3.18) from the maximal case by viewing γ, −γ as x n+1 , x n+2 and γ 1 , γ 2 , γ 3 as x n+1 , x n+2 , x n+3 , respectively. They preserve the requirement on the x j to sum to zero, and they additionally imply that subsets of the arguments in the enlarged G n+2 and G n+3 add up to zero. As a convenient way to explore the resulting cancellations, we rewrite the expressions in (3.35) and (3.36) such as to manifest the symmetries As a result, the γ-dependence in the half-maximal (3.38) conspires to functions of even modular weight, In fact, all the F (k) 1/2 (γ) past k = 2 will be identified below as independent of γ, but we will keep the generic notation F (k) 1/2 (γ) to emphasize similarities to the quarter-maximal case. The analogous manipulations in the quarter-maximal case (3.39) only admit odd modular weight for the dependence on γ j , More generally, the γ-dependence in the results (3.38) and (3.39) is organized in terms of V n (. . .) from (3.28) above: with appropriate parity for n. With these definitions and the functions V n (x 1 , . . . , x n ) of worldsheet positions in (3.25) to (3.28), the spin sums for reduced supersymmetry can be evaluated as which suffices for eight-point amplitudes in half-maximal compactifications. Comparing to results derived by standard methods, the first three are well-known: G 2+2 comes from two fermion bilinears after so-called "spin sum collapse", using a standard theta function identity whose proof is outlined for example in [83], eqs. (120) to (132). To obtain G 2+3 from three fermion bilinears, one adapts a calculation from [7], in particular their eq.
While similar methods were used in [50,7] to determine G 2+4 , we are not aware of explicit results for G 2+n with n ≥ 5 in the literature.
The quarter-maximal analogues of (3.45) to (3.51) sufficient for seven-point amplitudes are given by (3.57) With standard methods, G 3+2 has been computed in the spin sum for two fermion bilinears, and the proof of the required spin sum identity is outlined for example in [83], eq. (130). Note that F 1/4 in G 3+2 is reminiscient of V 1 above but is independent of x i , just like for the two-fermion-bilinear piece in the half-maximal case. With three and four fermion bilinears, computations in [50,7] can be adapted to yield G 3+3 and G 3+4 above, but starting from G 3+5 we believe the results are new.
In addition to new explicit results, we emphasize the general applicability of this method. As an example, the following observation would be difficult to make without our strategy. For n ≥ 2, the structure of V k (x 1 , . . . , x n ) is obviously identical in the above expressions for G 2+n and G 3+n . If 1 = F (0) 1/2 is inserted in each term of G 2+n without an extra factor of F (k =0) 1/2 , the correspondence between (3.45) to (3.50) and (3.52) to (3.57) can be summarized by . (3.58) Hence, the resulting scattering amplitudes in half-maximal and quarter-maximal compactifications have the same structure in the parity-even sector, i.e. their integrands can be straightforwardly mapped into each other upon replacing F 1/4 (γ j ). However, the parity-odd contributions to half-maximal and quarter-maximal cases will exhibit differences as we will comment on in sections 4.7, 4.8 and 6.6.
As noted above, all the F (k) 1/2 (γ) past k = 2 turn out to be independent of γ. In fact they are given by holomorphic Eisenstein series: and then (3.49) to (3.51) can be further simplified to It would be interesting to find analogous simplifications in the quarter-maximal case.

Closed-string prescriptions
In this section, we recall the starting point for 1-loop closed-string amplitudes with non-maximal supersymmetry, specifically we consider half-maximal and quarter-maximal compactifications of type IIA and type IIB theories. Similar to the open-string integration measure (3.2), we capture the integration over inequivalent worldsheets of torus topology by the closed-string measure As before, the regularized external volume V D , the order N of the orbifold group Z N , and the Koba-Nielsen factor Π n in (4.5) are incorporated for later convenience. By modular invariance, the torus modulus τ is integrated over the fundamental domain F defined by |Re (τ )| ≤ 1 2 and |τ | ≥ 1. Externalstate insertions z i are integrated over the torus T (τ ) parametrized by the parallelogram in C that is bounded by 0, 1, τ +1, τ .

Half-maximal supersymmetry
The half-maximal 1-loop amplitude for n external states in D = 6 can be written as denotes n-dimensional closed-strings lattice sums, andχ k,k are constant coefficients that encode the degeneracies of orbifold-charged ("twisted") states, see appendix A and e.g. [121].
Similarly as for open strings, the maximally supersymmetric integrand J n,max can only receive contributions from the even-even sector. By contrast, the half-maximal integrand in general receives non-trivial contributions from all parity sectors, we write gives the dependence of the corresponding integrands on the internal partition function. At genus one, the total picture number of the vertex operators in the (e,ẽ) sector must be (0, 0) [40], and as is customary we choose all of them in the (0, 0) picture. In shorthand notation where, analogously as for the open-string integrand (3.33), we have expressed parts of the partition function in terms of Szegö kernels (3.14). Similarly, the inverse of the Koba-Nielsen factor Π n compensates for its inclusion into the measure (3.61), and the vertex operators V (0,0) j (whose arguments z j are suppressed for ease of notation) are defined in (6.1).
Note that the contributionsθν(0,τ ) andSν(γ k,k ,τ ) from the right-moving sector in (3.64) and (3.65) are understood to be the complex conjugate of ϑν(0, τ ) and Sν(γ k,k , τ ), respectively. The same sort of notation will appear in later equations on closed-string amplitudes, and will drop the obvious dependence of the above functions on τ andτ .
In the (e,õ) sector the super-moduli structure of the torus requires the total picture number of the vertex operators to be (0, −1) and the inclusion of the picture changing operator P (0,+1) . We have where the GSO projection of the type IIB and type IIA theories yields a + sign and a − sign, respectively, see appendix A. The expression for J o,ẽ n,1/2 ( v k,k ) in the (o,ẽ) sector obviously follows from (3.66) upon exchange of left-and right-movers except for a uniform sign ± → + in both type IIA and type IIB. Note that the spin sums in (3.64), (3.65) and (3.66) can be addressed through the methods of section 3.3.
In the (o,õ) sector we have In close analogy with (3.10) for open strings, the half-maximal amplitude (3.62) in D = 6 easily generalizes for half-maximal models in D = 4, where now also for the half-maximal integrand the only non-vanishing contribution is from the (e,ẽ) sector with J e,ẽ n,1/2 given by eq. (3.65).

Quarter-maximal supersymmetry
For compactifications down to four dimensions leading to quarter-maximal supersymmetry, the amplitude for n external NSNS states reads 11 The quarter-maximal integrand in general receives non-vanishing contributions from all parity sectors, and ± in the last two equations is a + sign for type IIB and − for type IIA .

Vertex operators and CFT basics
Gauge bosons as massless excitations of the open superstring are represented by vertex operators 12 in the zero superghost picture. The BRST invariance of these vertex operators is ensured by having lightlike momenta and transverse polarization vectors, For the parity-odd sector (3.8) we also need the vertex in the −1 superghost picture as well as the picture changing operator, where the fields e ±φ from bosonizing the β-γ superghost system [84,85] only enter through their zero modes in this work.
Correlation functions of the free conformal fields ∂X m (z) and ψ m (z) of weight h = 1 and h = 1 2 are determined by their two-point contractions on genus-one worldsheets, where f (1) and S ν are defined in (3.16) and (3.14), respectively, and the modular parameter τ is suppressed. The plane waves e k·X in the vertex operators yield the ubiquitous Koba-Nielsen factor, which is absorbed into the integration measure (3.2) by our conventions for the integrands I ... ... in (3.5), (3.6), (3.8) and (3.13). The boson Green's function G ij is and satisfies where s ij are Mandelstam variables see (4.4), or interact with the exponentials to yield Contractions of the fermions lead to the spin sums evaluated in section 3.3. The associated kinematic factors are gauge invariant Lorentz-traces over linearized field strengths e [m k n] which will be denoted by They are convenient to track intermediate steps of the subsequent computations, but an alternative system of kinematic building blocks will be introduced in section 5 to obtain simpler and more compact representations of the correlators and to highlight parallels with maximally supersymmetric cases.

Infrared regularization by minahaning
Any 3-point function of any massless external states naively vanishes by "3-point special kinematics".
This means that all 3-point would-be Mandelstam invariants (4.8) vanish identically 13 , as implied by momentum conservation and k 2 j = 0. This infrared zero can lead to 0/0 issues in presence of certain propagators. We will regularize by relaxing momentum conservation in intermediate steps: The three Mandelstam invariants s 12 , s 23 , s 13 then become nonzero, but subject to the single condition This is needed to ensure that exponentials of boson propagators in the Koba-Nielsen factor (4.5) of the string integrand are modular invariant. Other conditions on the deformed Mandelstam variables, for example the more symmetric but stronger s 12 = s 23 = s 13 , would violate modular invariance, as explained by Minahan in 1987 [43]. To see directly how the "deformation" momentum p m allows for nonzero Mandelstam invariants in the 3-point function, take scalar products with for example k 1 : i.e. the s ij in the 3-point function are only nonzero due to the deformation p m . We give some more details on this in appendix C. In general, we will refer to the procedure of relaxing momentum conservation subject to the constraint n i<j s ij = 0 as "minahaning" an n-point function. For the 3-point amplitude the need for some kind of infrared regularization is clear because of the infrared zero of 3-point special kinematics, but we will argue that there is a sense in which this should be done for any n-point amplitude.
As a first step in the subsequent calculations, we will combine the regularized (i.e. nonzero) Mandelstam invariants as in (4.14) with vanishing propagator denominators from string theory such that all indeterminate 0/0 expressions are taken care of. Then, for the purposes of this paper, we can safely set the deformation p m to zero in our final expressions for amplitudes. 14 13 We keep the kinematic identities covariant and dimension-agnostic in this work, i.e. factorization of s 12 = 1 2 (k 2 3 − k 2 2 − k 2 1 ) = 0 into four-dimensional spinor brackets 12 and [12] (one of which is often taken to be non-zero for complex momenta, see [33]) will not enter the discussion. 14 We note that the original procedure in [43] was a slightly stronger form of regularization, when terms in the effective action are computed without setting the deformation to zero at the end. This allows for the extraction of effective couplings from two-point functions, more recently used for example in [81] and references therein, which will not be discussed in this work. To distinguish the stronger form of regularization from the weaker "minahaning" used in this paper, one might be tempted to call the stronger procedure "maxahaning". We will resist this temptation.

Half-maximal parity-even 3-point amplitude
For three external states, the well-known half-maximal integrand in (3.6) is given by 12 (e 1 · e 2 )(e 3 · Q 3 ) + (3 ↔ 2, 1) − (e 1 · Q 1 )(e 2 · Q 2 )(e 3 · Q 3 ) (4.15) recalling that the Koba-Nielsen factor is absorbed into the measure (3.2) and the definition (4.9) of Q m i . The spin sum G 2 in the first line evaluates to zero whereas G 4 and G 5 in the second line are given by (3.45) and (3.46), respectively. Hence, the correlator (4.15) is homogeneous in f (1) ij , 12 K 12|3 + (12 ↔ 13, 23) , (4.16) whose antisymmetric kinematic factors K 12|3 = −K 21|3 can be simplified using relaxed momentum conservation (4.13) and order-p transversality (4.2) via (e 1 · k 3 ) = −(e 1 · k 2 ). We find The singular function f 12 ∼ (z 1 − z 2 ) −1 integrates to a kinematic pole in presence of the Koba-Nielsen factor Π 3 , i.e. z 1 dz 2 f (1) 12 e s 12 G 12 ∼ 1/s 12 , such that 3-particle momentum conservation for massless states would make this 1/0. However, the minahaning procedure explained in section 4.2 yields a finite integral for the function i.e. the prefactor s 12 in the kinematic factors (4.17) can be used to identify appropriate building blocks (4.18) which make the finiteness of the z-integral manifest: Again, minahaning means to first use only 3 i<j s ij = 0 and to only after performing the z-integrals at nonvanishing values of s ij finally impose momentum conservation again, s ij = 0. A similar procedure will be applied for the 4-point function.
In the low-energy limit α → 0, the analytic part 15 of the integrals over all of X 12 , X 23 and X 31 yield a constant, and (4.19) reduces to the 3-point tree-level amplitude, (1, 2, 3) . (4.20) 15 In addition to a power-series expansion in α , loop amplitudes in string theories give rise to logarithmic, non-analytic momentum dependence. As will be elaborated in a companion paper [46], the integration region of large τ 2 yields Feynman integrals of Yang-Mills along with their threshold singularities in s ij , see also [86,87,88,89]. Following the discussion of closed-string 1-loop amplitudes in [90,91,92,93], the analytic parts of the amplitude can be isolated in a well-defined manner.

Half-maximal parity-even 4-point amplitude
Also the half-maximal 4-point amplitude is well-known not to receive any contributions with less than two fermion bilinears,  All the spin sums are readily evaluated using (3.45) to (3.47) and give rise to functions f (1) f (1) or f (2) with various combinations of arguments.

Minahaning the 4-point function
The second class (ii) of functions ∼ f (2) ij is non-singular as z i → z j and therefore does not contribute to any factorization channel or the low-energy limit. However, the last two classes (iii) and (iv) of functions yield up to two simultaneous kinematic poles from the integration region where z i → z j . Worse, the 3- 13 Π n ∼ (s 12 s 123 ) −1 due to (iv) involves divergent propagators. This requires minahaning, as discussed in section 4.2.
We shall repeat the procedure of the 3-point amplitude and transform the integrals to a basis that is manifestly free of kinematic poles. The replacement f  We have used the shorthand 12 (s 13 f 13 for the combination of functions (iv) that does not integrate to any divergent propagators ∼ s −1 123 . Once the replacement (4.29) is coherently applied to the correlator I e 4,1/2 , the kinematic prefactors accompanying any f  [94]. The kinematic numerators of so-called "snail graphs" (see fig. 3 in section II.D of the reference) are found to be proportional to k 2 4 = 2s 123 such as to cancel the vanishing denominators (k 1 + k 2 + k 3 ) 2 from the external propagators. In the same way as these finite contributions are essential for the 4-loop UV divergence of N = 4 SYM, our way of minahaning the 4-point 1-loop amplitude for open strings will crucially impact its low-energy limit.

Comparison with [8]
Even though our result in  (4.29). In [8], on the other hand, spinor-helicity variables are introduced at an early stage, which implicitly drops contributions proportional to s 123 irrespective of the accompanying worldsheet functions. It will be interesting to check whether infrared-safe observables in field theory computed from (4.40) and the analogous expression in [8] might match in spite of the above differences in the string correlator.
Just like the result in [8], the D-dimensional expression (4.40) obeys the D = 4 corollary of supersymmetric Ward identities that amplitudes with 3 or 4 particles of alike helicity vanish [95,96]. All the kinematic factors L ij|kl , A tree (i, j, k, l) and t 8 (1, 2, 3, 4) have been tested for this property after dimensional reduction to D = 4 and conversion to spinor-helicity variables. As will be demonstrated in a companion paper [46], we are under the impression that the f (1) f (1) contributions in the second line of

Half-maximal parity-even amplitudes of higher multiplicity
Half-maximal amplitudes of higher multiplicity can be evaluated using the same principles. The required spin sums for up to eight external states are available in (3.45) to (3.51) and can be easily extended using the techniques of [80]. In the same way as the final form (4.40) of the 4-point correlator augments the simplest function F 1/2 (γ) of the orbifold twist γ = kv with the maximally supersymmetric kinematic factor t 8 (1, 2, 3, 4), the coefficient of F 1/2 (γ) in higher-multiplicity amplitudes will reproduce the maximally supersymmetric correlators in their dimensional reduction to D = 6. Starting from six points, new combinations of the f (i) will emerge where the γ-dependence F    (1) 34 ), the kinematic poles describe an (n−1)particle factorization channel which is plagued by a divergent propagator such as s −1 12...n−1 . Once the complete contribution to this channel is assembled from the correlator, the kinematic numerator is expected to yield compensating Mandelstam invariants (using no other relation than n i<j s ij = 0), see section 4.2. Generalizations of the functions X ij and X ij,k in (4.18) and (4.30) which remain smooth after integration over z j can be found in the context of maximally supersymmetric 1-loop correlators [77].

Quarter-maximal generalizations in the parity-even sector
In the parity-even sector, the quarter-maximal counterparts of the above correlators I e 3,1/2 and I e

Parity-odd integrands at lowest multiplicity
In the parity-odd sector, the zero-mode saturation rule (3.9)  Here and in later equations, we use the shorthand notation for vectors v m 1 , v n 2 , . . . , v p D , to avoid proliferation of indices. By antisymmetry of the -tensor, contributions from the sum in (4.44) with j = 2, 3 drop out immediately, and momentum conservation k 1 = −k 2 − k 3 leads to the same conclusion for the term with j = 1. Hence, the dependence on the position z 0 of the picture changing operator via f

Parity-odd integrands at next-to-lowest multiplicity
The simplest non-vanishing parity-odd contribution to half-maximal open-string amplitudes in D = 6 dimensions occurs at the 4-point level. A subtle chain of integral manipulations and kinematic rearrangements detailed in appendix D.1 confirms independence on the position z 0 of the picture changing operator, and the only z i -dependence turns out to enter through the non-singular function f (2)  Note that (4.48) and its generalizations to higher multiplicity vanish for external gauge bosons upon dimensional reduction to D < 6. That is why parity-odd contributions are excluded for amplitudes where the permutation sum in (4.52) along with f 23 includes any pair i, j subject to 2 ≤ i < j ≤ N . These expressions are derived in appendix D.1, where the ten-dimensional six-point analysis [11] is carried out in a dimension-agnostic manner.

Berends-Giele organization of open-string amplitudes
In this section, the kinematic organizing principles of the above open-string results are explored. They rely on bosonic Berends-Giele currents e m 12...p which recursively resum Feynman diagrams with p external on-shell states and an additional off-shell leg. While Berends-Giele currents were first used in the 1980's to elegantly address gluonic tree amplitudes [97] in YM theories, the value of this concept for superstring theories became apparent in [98,26,27]. In these references, tree-level amplitudes for any number of massless open-superstring states were computed in the pure spinor formalism [3]. The underlying supersymmetric Berends-Giele currents have been generalized and streamlined in [77,99,100], connected with the component currents from the 80's in [100,101] and exploited to compute and compactly represent loop amplitudes of the pure spinor superstring in [77,93,2,71,11].
The Berends-Giele representation of maximally supersymmetric string amplitudes led to a variety of insights on ten-dimensional SYM amplitudes in pure spinor superspace. In addition to the field-theory limit α → 0 of superstring amplitudes, ten-dimensional SYM amplitudes have been obtained from first principles -locality and BRST invariance. Locality amounts to imposing the Feynman-diagram content in the Berends-Giele constituents of the desired amplitude, and BRST invariance powerfully embodies both maximal supersymmetry and gauge invariance of bosonic components [3]. This program has been successfully applied at tree level [102,103], one loop [104,105] and two loops [106].
It will now be demonstrated that the Berends-Giele approach to string amplitudes can be extended to half-and quarter maximal supersymmetry. The structure of the above half-maximal 3-and 4-point amplitudes will be clarified using the bosonic components of supersymmetric Berends-Giele currents [97,100,101]. Apart from the conceptual benefit of extending the pure spinor methods, this will pave the way for a compact and enlightening representation of the closed-string computations in section 6.
Moreover, a first-principles approach to half-maximal SYM 1-loop amplitudes obtained in the fieldtheory limit will be discussed in a companion paper [46].

Definition of bosonic Berends-Giele currents
We will only define the minimal set of Berends-Giele currents that appear in half-maximal amplitudes with no more than four external legs 17 . Bosonic currents with a maximum of three on-shell legs are defined recursively via [97,100,101]   The cubic diagrams associated with the 2-particle and 3-particle currents e m 12 , f mn 12 and e m 123 , f mn 123 are depicted in fig. 6. Appropriate choices of e m ... versus f mn ... as suggested by string theory guarantee that quartic Feynman vertices of YM theories are absorbed into these cubic diagrams [101], in line with the BCJ duality between color and kinematics [107].   Once the Berends-Giele currents f mn A are resummed to yield a solution F mn of the non-linear YM field equations, the expressions in (5.7) and (5.8) can be generated from the Lagrangian ∼ F mn F mn , evaluated on this perturbative solution [108,100,101]. Note that the scalar building block in (5.4) is reminiscient of the maximally supersymmetric 1-loop building blocks defined in section 5.2 of [99] (see [77] for pioneering work) which were later identified as local box numerators in ten-dimensional SYM [105].

Vector & tensor building blocks for half-maximal loop amplitudes
While the scalar building block in (5.4) completely captures the kinematic coefficient of f (1) in halfmaximal open-string amplitudes at multiplicity n ≤ 4, the f (2) terms as well as the closed string will require various extensions. We will design vectorial and tensorial building blocks such that parityeven and parity-odd contributions to half-maximal string integrands are unified. For this purpose, the following basic building block for parity-odd kinematics is introduced, where the vertical-bar notation A|B, C is a reminder of the special role of the first slot, E m A|B,C = E m B|A,C , and E m 1|2,3 = m (e 1 , k 2 , e 2 , k 3 , e 3 ) is recovered in the single-particle case. We define the following frequently occurring composition of parity-even and parity-odd kinematics,  [99], see [93,11] and [105] for their role in closed-string amplitudes and pentagon numerators in SYM amplitudes, respectively.
In the same way as the maximally supersymmetric vectors were recursively extended to tensors of arbitrary rank [104], we define a two-tensor counterpart to the bosonic vector in (5.10): It will play an essential role for the closed-string 4-point function in section 6.4 and the loop-momentum dependent part of Feynman-diagram numerators in the field-theory limit [46].
Note that the combination of parity-even and parity-odd parts in (5.10) and (

Gauge-(pseudo-)invariant kinematic factors
Gauge transformations of the above building blocks yield a rewarding web of relations involving lowermultiplicity counterparts. These gauge variations resemble the BRST variations in pure spinor superspace [99,104] and will be thoroughly discussion in the companion paper [46]. For our present purposes, we simply state the gauge invariant combinations of the scalar, vectorial and tensorial building blocks  Following the terminology of [104], we will refer to quantities whose gauge variations can be exclusively expressed in terms of mnpqrs f mn B f pq C f rs D as "pseudo-invariant". Apart from the tensor (5.16), the following scalar is pseudo-invariant,
The virtue of organizing the kinematic factors of half-maximal string amplitudes in terms of the building blocks M A,B and their tensorial generalizations will become particularly obvious from the closed-string amplitudes discussed in the following section.  [11]. We will give some parity-even examples of novel effective couplings and check some known results, but we will not address field redefinitions, rescalings or frame-changing (see section 2.2). Our focus here is the string amplitudes, and we leave a detailed study of the loop-corrected string effective action to the future.

Vertex operators and left-right interactions
Massless NSNS-excitations of the closed superstring are represented by vertex operators where the delta-function on the right-hand side does not contribute in the presence of the Koba-Nielsen factor Π n in (4.5) and will therefore be suppressed 18  Integration by parts relations introduce additional interactions 19 between left-and right-movers since the worldsheet functions defined by (3.15) are no longer holomorphic at non-zero genus [80],  = e m 1 (e 2 · k 3 )(e 3 · k 2 ) + e m 2 (e 1 · k 3 )(e 3 · k 1 ) + e m 3 (e 2 · k 1 )(e 1 · k 2 ) .

Low-energy prescriptions
To study the implications of closed-string amplitudes for the low-energy effective action, the α → 0 behavior of the worldsheet integrals has to be extracted. Since a discussion of the Feynman diagrams in the supergravity limit along the lines of [1] is relegated to the companion paper [46], we will follow the procedure of [90,91,92,93] to truncate the integrals to their analytic momentum-dependence.
The leading low-energy behavior of closed-string integrals is determined by the piece with the highest number of kinematic poles. They originate from a "diagonal" pair of worldsheet singularities for functions g(z) that are regular at the origin. By repeated use of (6.19), only diagonal combinations of X ij andX kl affect the low-energy limit, e.g.
where the '→'-notation is understood to only keep track of the leading order of α occurring in the amplitude under discussion.
For the nested product X ij,k defined in (4.30), the analogous rules are determined by whereas different triplets of arguments do not yield any low-energy contribution at leading order, e.g.
Factors of π Im τ from the interactions (6.4) or (6.6) between left-and right-movers are of the same order in the low-energy expansion as a diagonal pair f (1)f (1) , e.g.
These schematic rules will be used in the following to extract matrix elements of the R 2 interaction from the low-energy limit of 3-point and 4-point closed-string amplitudes. Note that integrals involving non-singular worldsheet functions f (2) ij and F 1/2 on either the left-moving or the right-moving side do not contribute to the 4-point low-energy limit.
Subleading terms in the analytic low-energy expansions exhibit a gap at the mass dimension of R 3 such that the first non-vanishing interaction beyond the low-energy limit occurs at the order of R 4 . This follows from the low-energy behavior of torus integrals over z j in presence of f (1)f (1) [92,93] where any tentative contribution at subleading order in α is found to integrate to zero. The results of [93] for 5and 6-point integrals in the maximally supersymmetric case directly carry over to the subsequent 3-and 4-point integrals in the half-maximal case.

Half-maximal 3-point amplitude
The treatment of left-right interactions outlined in section 6.1 is easily applied to the 3-point amplitude.
The calculation can be found in the literature (see [47] and references therein), and we recalculate it using our methods and the notation of the previous sections to prepare for the 4-point generalization.
With the open-string kinematic factors in (4.19) and (4.44) as well as the chiral halves (6.10) and (6.11) of left-right contractions, the half-maximal closed-string correlator is given by By comparison with the vector building block in (5.10), parity-even and parity-odd terms combine into ij . Moreover, the sign of the right-moving parity-odd part e.g. inM m 1|2,3 differs between type IIB and type IIA due to the different GSO projections in the RR sector, as is clear from the partition function in appendix A. This The last equality involving the vector invariant C m 1|2,3 in (5.14) follows from 3-particle kinematics such as k m j M m 1|2,3 = 0 or s ij = 0 and manifests the structural similarity with the maximally supersymmetric 5-point amplitude in section 4.1 of [11].
In absence of worldsheet singularities, the Koba-Nielsen factor along with (6.27) can be replaced by its Taylor expansion which trivializes to Π 3 = 1 by 3-particle kinematics. Hence, the low-energy limit obtained from (6.23) does not receive any corrections at higher order in α , and its type IIB and IIA components will be discussed further in section 6.5. We remind the reader that we will not perform any integrals over the worldsheet modulus τ in this paper. It is of course important to do so to extract the moduli-dependence of the string effective action, and we would like to return to this issue in the future.
The slightly abusive notation M R 2 (1, 2, . . . , n) for the low-energy limit refers to matrix elements involving any combination of n NSNS sector states at the same order in α as the gravitational R 2 correction. The n-graviton component due to the R 2 interaction can be straightforwardly extracted by settingẽ m i → e m i and (e i · e i ) → 0. The vector and tensor integrands I m 4,1/2 and I mn 4,1/2 can be reconstructed from (6.12), (6.15) and (6.13), (6.16), respectively. After converting the kinematic factors into the building blocks of section 5 via Note that the last line of (6.31) will conspire with left-right interacting integrations by part and eventually contribute to the first three terms of the pseudo-invariant P 1|2|3,4 in (5.18).

Half-maximal 4-point amplitude
In view of the discussion in section 6.1.2 and appendix E, it is crucial to use the expressions for the left-right factorizing kinematic factors prior to any integrations by parts. More specifically, (6.29) requires the representation (4.22) for the parity-even part I e 4,1/2 and (D.5) for the parity-odd part I o 4,D=6 . We reduce the integrals in (6.29) to a basis by eliminating any instance of the first leg in f (1) 1j andf (1) 1j through the integration-by-parts rules of section 6.1.2 and appendix E. In this process, various corrections ∼ π Im τ and π Im τ 2 to the square of the simplified open-string correlator in (5.21) arise. Also, spurious dependences on z 0 as seen in (6.31) and the derivatives within (D.5) will cancel in this process.
It turns out that the vector invariant C m 1|23,4 in (5.15) as well as the pseudo-invariants C mn 1|2,3,4 and P 1|2|3,4 in (5.16) and (5.18) are tailor-made to express the closed-string 4-point correlator in a minimal form: They combine all the parity-even and parity-odd open-string constituents and capture the kinematic factors along with the basis integrals: By the modular weight (n, 0) of the functions f (n) [80], every term in (6.32) exhibits uniform modular weight (2,2), where factors of F 1/2 additionally mix different orbifold sectors k, k in (3.62). Together with the six-dimensional closed-string measure in (3.61), the weights of d 2 τ, τ −D/2 and 4 j=2 d 2 z j are compensated. Hence, (6.32) manifests modular invariance of the closed-string amplitude.
In the last line of (6.32), one can understand the presence of the "extra" P 1|2|3,4P1|2|3,4 + (2 ↔ 3, 4) pieces as follows. They compensate for the anomalous gauge transformation of the tensor contraction Note that the bilinears in pseudo-invariants seen in (6.32) mimic the patterns in the maximally supersymmetric 6-point amplitude, see section 4.2 of [11].
The anomalous gauge variations along with factors of f (2) ij in the first two lines of (6.32) conspire to total derivatives in τ and the z j . This follows from the same arguments as given for the maximally supersymmetric 6-point torus amplitude discussed in section 4.4 of [11].
The low-energy limit of (6.32) can be easily performed by means of the rules in section 6.2 and takes a very compact form: which vanishes by the BCJ relations s 12 C 1|234 = s 13 C 1|324 of C 1|234 = 2A tree (1,2,3,4) [107]. We also note that the parity-odd/odd part of (6.34) can be simplified to yield 1|3,4 +(2 ↔ 3, 4), see appendix B.2 for its factorization properties. We pause to contrast the expression above with the half-maximal closed-string 4-point amplitude discussed in [9,10]. That discussion was specialized early on to the field-theory limit and spinor-helicity expressions. After the manipulations performed here, we believe the present string amplitude clearly exhibits several interesting features that were not manifest in [9,10]. Apart from its applicability to arbitrary dimensions D ≤ 6, one important aspect is the presence and limitations of double-copy structure in this string amplitude. More precisely, the PP structure in the last line of (6.32) obstructs the naive expectation to find a pure tensor contraction T mnT mn along with ( π Im τ ) 2 . We expect this to be the source of the tension between worldsheet correlators and double copies of gauge-theory BCJ numerators observed in [10]. We hope to say more about the implications of (6.32) for the BCJ-duality between color and kinematics in the future.

The low-energy limit in type IIB and type IIA
This section is devoted to the type IIB and IIA components of the low-energy limits M R 2 (1, 2, . . . , n) in (6.28) and (6.34). The 3-point case has already been investigated in [47] where the parity-even IIB components were found to vanish for any combination of gravitons, B-fields and dilatons. The IIB cancellation relies on the interplay between the even/even and odd/odd spin structures and does not occur for type IIA because of the different GSO projections [47]: The contraction of tensors can be converted to the dot products seen in (6.10) via Gram determinants, Note that the parity-even type IIA result in (6.37) vanishes for an odd number of B-fields.
In the parity-odd sector, on the other hand, the GSO projections of type IIB and IIA yield [47] M

Comparison with the heterotic string
Matrix elements of the R 2 interaction also appear in tree-level amplitudes of the heterotic string [20] and the bosonic string [56] upon expanding to the linear order in α . This yields a KLT-like double copy of YM amplitudes and F 3 matrix elements known from the (α ) 1 -order of the bosonic open string [23], which also matches the bosonic-string result. The F 3 -constituents are given by [39] A F 3 (1, 2, 3) = (e 1 · k 2 )(e 2 · k 3 )(e 3 · k 1 ) (6.42) where the right-hand side of (6.43) manifests gauge invariance at the expense of manifest locality. Note that the structure of A F 3 (1, 2, 3, 4) = s 13 × {totally symmetric quantity} guarantees that the BCJrelations of A tree (. . .) [107] are also obeyed by A F 3 (. . .) [23] and that (6.41) is permutation invariant.
This discussion connects to that about field redefinitions in section 2.2: in D = 4, any tensor structure for the R 2 interaction is on-shell equivalent to the Gauss-Bonnet combination, that is topological if there is no moduli-dependent coefficient, cf. (2.6). The on-shell vanishing of (6.40) and (6.41) in D = 4 can be seen from the fact that there is no combination of graviton helicities where both A tree (. . .) and A F 3 (. . .) are non-zero [23].
In presence of B-fields, to be denoted by 1 B , 2 B , . . . in the following, the non-vanishing amplitudes are  In the type IIA low-energy limit, we have checked agreement of the 4-graviton component with the  1/2 (γ k,k ) and F (k+1) 1/4 (γ j k,k ) of orbifold twists γ k,k ≡ (k + k τ )v and γ j k,k ≡ (k + k τ )v j . Hence, the parity-even/even parts of n-point closed-string correlators are related by J e,e n,1/4 = J e,e n,1/2 .

(6.53)
In presence of parity-odd admixtures from either left-or right-movers, the universality breaks down by the discussion in section 4.8. From (4.52), for instance, parity-odd/odd contributions to quartermaximal 3-point amplitudes involve worldsheet functions of the type f pq and π Im τ 2 .
This departs from the factors of F 1/4 (γ j k,k ) π Im τ in the parity-even quarter-maximal terms (6.53) as well as their half-maximal counterparts ∼ π Im τ in (6.27). These structural differences in parity-odd contributions to half-maximal and quarter-maximal amplitudes also affect the low-energy behavior. For example, up to n−1 left-right contractions are compatible with the four-dimensional version of the n-point parity-odd/odd prescription (3.67), leading to tensorial 3-point kinematic factor ∼ e (m 2 n) (e 1 , k 3 , e 3 ) + (2 ↔ 3). This ties in with the counting of loop momenta in quarter-maximal SYM amplitudes [109].
We see that just as for half-maximal above, the parity-even sector of the low-energy limit of the closed-string 4-point function on Calabi-Yau orbifolds has the mass dimension of R 2 , so it does not produce a loop correction to the Einstein-Hilbert action, as expected from general arguments, see section 2.3. Only the parity odd/odd part of Calabi-Yau amplitudes has the right mass dimension to produce a loop correction to the Einstein-Hilbert action. However, this is delicate to see since it might require further minahaning; in the calculations above, we used strict momentum conservation in the odd/odd sector.

Conclusions and outlook
We made progress on calculating 1-loop string amplitudes with reduced supersymmetry through three key methods: • modular functions f (n) that let us generalize spin sums from the maximally supersymmetric case • the minahaning procedure of relaxing momentum conservation as an infrared regularization • building blocks of Berends-Giele type to capture gauge (pseudo-)invariant kinematic factors A companion paper [46] on the field-theory limit will elaborate on the value of the Berends-Giele organization of kinematic factors for 1-loop amplitudes of half-maximal SYM in six and lower dimensions.
Another domain of application of the current results that we have not pursued in detail is the string effective action. We have set the stage for a systematic α -expansion by expressing integrands in terms of useful modular objects, but we did not discuss their integration over τ here. 20 One issue with this is that we have not been too specific about string-theory models; for compact open-string models, we should include orientifolds for tadpole cancellation. As in for example [7], we believe that this can be done straightforwardly from our results.
We have not touched on RR fields at all in this paper. One interesting class of calculations concerns the completion of the dilaton and the NSNS field strength H 3 to the NSNS+RR axio-dilaton and selfdual field strength G 3 . As an example, the action at order α 3 contains for example |G 3 | 2 R 3 (see e.g. [112,113,114] as well as [93] for S-duality properties of higher-derivative corrections).
As emphasized earlier, it is important to remember that these calculations are performed at the orbifold point, and generalizations to smooth Calabi-Yau manifolds (including smooth K3) with the same amount of supersymmetry may vary from straightforward to highly nontrivial [115,116]. Whether or not these results are representative of generic points in moduli space, experience shows that explicit results at specific points will provide useful and highly needed guidance for generalizations.
It would be very interesting to revisit our amplitudes in a manifestly supersymmetric formalismeither by using the hybrid formalism [117,118] or by deforming the pure spinor formalism [3] to preserve half-maximal supersymmetry in D = 6 dimensions.

A Orbifold Partition Functions
In this appendix, we give further details on the vacuum amplitudes associated with the prescriptions in section 3. In compactifications of type I to D dimensions on orbifold limits of Calabi-Yau threefolds or K3, the cylinder vacuum amplitude (partition function) for open strings stretching between D9-branes can be written as 21 [120] Analogously, in orbifold compactifications of type IIA and type IIB, the torus vacuum amplitude (partition function) reads [121] In the main text we discuss gauge boson and graviton amplitudes for various orbifold compactifications, To write general expressions that cover all these cases and to account for the possible presence of half-maximal sub-sectors in D = 4, which depends on the rank N , we introduce the following slightly non-standard notation: by d k we denote the number of internal dimensions where for the given k the orbifold has a fixed direction, and we set For orbifold compactifications preserving some supersymmetry, which we always assume, the open-string partition function integrands can be expressed as 22 (−1) ν+ν+µδν ,1 ϑ ν (0, τ ) ϑ 1 (0, τ )θν (0,τ ) ϑ 1 (0,τ ) 4 (A.5) 21 In this paper we consider only D9-branes with no background fluxes. 22 In the literature, orbifold partition functions are often expressed in terms of ϑ functions with characteristics. These can easily be related to the above expressions using the basic definitions [122,123] and the supersymmetry constraint 23 This is schematic, but standard [119,120]. Explicit expressions for the matrices γ k are also given in the companion paper [46].
where µ takes the value 0 or 1 in type IIB or IIA, respectively. Eq. (A.6) applies to all supersymmetric orbifolds of the kind R 1,5 × T 4 /Z N and R 1,3 × T 2 × T 4 /Z N , but for Calabi-Yau limits it is only valid for R 1,3 × T 6 /Z N with no fixed direction, i.e. for N prime, and requires a slight generalization if not.
We have introduced coefficientsχ k,k = χ k,k /(2π) 10−D k , where χ k,k denotes the number of simultaneous fixed points under the Θ k and Θ k orbifold actions. The textbook way to generate χ k,k [74,121,73] is by starting with k = 0 and acting with modular transformations, for example the T transformation takes k → k +k . Individual orbifold sectors mix under modular transformations, but the full amplitude is of course invariant by construction. See also the comment below (6.32).

B.2 Closed string
As a sample of factorization of closed-string amplitudes, we consider the parity-odd/odd contribution to the 4-point low-energy limit in (6.36  Note that this check is again valid for any combination of gravitons, B-fields and dilatons.

C Kinematics of massless 3-point functions
In this appendix we give a few reminders about basic on-shell kinematics and connect the discussion with an interpretation of the minahaning procedure in section 4.2.

C.1 Scalar 3-particle special kinematics
Massless 3-point functions of scalars vanish on-shell by momentum conservation. Here is a quick reminder why this is the case. Momentum conservation with all momenta ingoing is Take the scalar product of this with k 1 and use on-shell masslessness k 2 1 = 0 to obtain k 1 · k 2 = −k 1 · k 3 . But this leads to 0 = (k 1 + k 2 + k 3 ) 2 = 2k 1 · k 2 + 2k 1 · k 3 =0 +2k 2 · k 3 = 2k 2 · k 3 (C.2) so k 2 · k 3 = 0, and similarly for the remaining two Mandelstam variables. We see that all Lorentz scalars k 2 i = k i · k j = 0, using momentum conservation and on-shell-ness.
C.2 Vector 3-particle special kinematics "Vectors" here mainly refer to non-Abelian gauge bosons. With vector polarizations e i we can make nonzero Lorentz scalars. A priori there are 6 independent e i · k j for each i = j, but e 1 · (k 1 + k 2 + k 3 ) = 0 (C.3) so by transversality e i · k i = 0 (no sum), we have e 1 · (k 2 + k 3 ) = 0 and cyclic. In other words, the only nonzero scalars are polarizations contracted with momentum differences k i − k j , which leaves three: This is enough to write the tree-level 3-point amplitude. However, at least in D = 4, even these three Lorentz scalars vanish due to 3-particle special kinematics. One way to think about this is that the momenta need to be collinear, so one can always reduce any e i · k j to e i · k i = 0.

C.3 Interpretation of the minahaning procedure
In quantum field theory, the fact that 3-point amplitudes of massless particles vanish on-shell is no problem: just go off-shell, k 2 i = 0. In (first-quantized) string theory there is no obvious self-consistent way to go off-shell. In the amplitude literature [33], one routinely uses 3-point functions as building blocks, but with complex momenta. As detailed in section 4.2, we use the minahaning procedure: we keep real momenta but relax momentum conservation, and maintain on-shell conditions k 2 i = 0. Then we have nonzero Lorentz scalars in the 3-point function, at least as an intermediate step. The basic idea is that the physical state conditions are not violated by relaxing momentum conservation.
But what does it mean to relax momentum conservation? One operational way to think of it is that the 3-point function is "embedded" in the 4-point function (so the 4 th momentum supplies the deformation), and the 4-point in the 5-point (as embodied in the notation s 123 for the deformation), and so on. This sounds surprising: why would we need to regularize the 4-point function, where there is no issue with "special kinematics" as above? A more physical way to relax momentum conservation is to use an external background field, for example a gravitational background, such as AdS or a sphere [61]. 24 In an orbifold, there is delta-function curvature at the fixed points, so the orbifold twist γ insertion mimics a background gravitation field insertion, as in fig. 7. (However, we emphasize that this is different from the insertion of an ordinary vertex operator, since the "position" of this insertion is the twist γ, which is not integrated over.) Considering a background field may make it clearer why the 4-point function 24 Of course, spheres may not be suitable as regulators if they break supersymmetry [41]. is affected by the infrared regularization: with a background field, there is potentially a background insertion in every n-point function.
For completeness, we also mention that Dp-branes for p < 9 provide another setting where momentum conservation in the naive sense is "naturally relaxed": momentum is not conserved transverse to the D-brane, since the D-brane is very massive in perturbation theory (see for example [44]).

E Integral reduction in the 4-point closed-string amplitude
In this appendix, we augment the general discussion in section 6.1.2 with further samples of corrections ∼ π Im (τ ) when reducing the closed-string integrals to a basis without any appearance of f (1) 1j and f (n) 0j . When both left-and right-movers contribute with two factors of f (1) as in (6.18) . Similarly, in presence off (2) ij on the right-moving side, left-moving integration by parts introduces corrections such as X 12f (2) 23 = − π Im τf (1) 23 + (X 23 + X 24 )f (2) 23 . (E.5) Cases of the form f pq do not admit any reduction via integration by parts and can be taken as basis elements regardless on i, j, p and q. For ease of notation, we have suppressed the Koba-Nielsen factor Π n in (E.1) to (E.5), i.e. relations of this type are understood to hold upon integration over the z j .
Moreover, the integration-by-parts removal of spurious double poles in the 4-point function, see Im (τ ) .
Finally, in presence of parity-odd contributions, integration by parts as seen in (D.5) is required to remove the spurious dependence on the position z 0 of the picture changing operator, ∂f 02X 12X34 = f (1) 02 (X 21 + X 23 + X 24 )X 12X34 + π Im (τ ) f where the derivatives are understood as ∂f (1) 02 ≡ ∂ 0 f 02 and∂f (1) 0j ≡∂ 0f (1) 0j . Iterating manipulations of the above type yields the final result (6.32) for the 4-point closed-string correlator in a basis of integrals.