Local BCJ numerators for ten-dimensional SYM at one loop
aa r X i v : . [ h e p - t h ] F e b Local BCJ numerators for ten-dimensional SYM at one loop
Elliot Bridges ⋆ , and Carlos R. Mafra α ′ Mathematical Sciences and STAG Research Centre, University of Southampton,Highfield, Southampton, SO17 1BJ, United Kingdom
We obtain local numerators satisfying the BCJ color-kinematics duality at one loop forsuper-Yang–Mills theory in ten dimensions. This is done explicitly for six points via thefield-theory limit of the genus-one open superstring correlators for different color orderings,in an analogous manner to an earlier derivation of local BCJ-satisfying numerators at treelevel from disk correlators. These results solve an outstanding puzzle from a previousanalysis where the six-point numerators did not satisfy the color-kinematics duality.February 2021 ⋆ email: [email protected] α ′ email: [email protected] ontents1 Description of the problem and its solution . . . . . . . . . . . . . . 21.1. Genus-one open superstring correlators in pure spinor superspace . . . . . 31.2. BCJ-satisfying local numerators at tree level from string disk correlators . . 41.3. BCJ-satisfying local numerators at one loop from string genus-one correlators 7 . . . . . . . . . . 102.1. Kinematic poles and biadjoint Berends-Giele currents . . . . . . . . . 102.2. p -gon loop momentum integrands . . . . . . . . . . . . . . . . . . 122.3. Field-theory limit of Kronecker-Eisenstein coefficients . . . . . . . . . 122.4. The one-loop SYM field-theory integrands . . . . . . . . . . . . . . 15 . . . . . . . . . . . . . . . . . . . 213.1. Color-kinematics duality . . . . . . . . . . . . . . . . . . . . . . 213.2. Six points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3. Seven points . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4. Supergravity amplitudes and the double copy . . . . . . . . . . . . . 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 A Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
B Cyclic symmetry of the field-theory limit rules . . . . . . . . . . . 41
C The field-theory limit at higher points . . . . . . . . . . . . . . . . 42
D The BRST analysis of a seven-point numerator . . . . . . . . . . . 45
E The five-point color-dressed integrand . . . . . . . . . . . . . . . . 48 ig. 1 The multiparticle superfields and pure spinor one-loop building blocks lead to intuitivemappings between one-loop cubic graphs and pure spinor superspace expressions encoding thepolarization dependence of ten-dimensional supersymmetric Yang–Mills states [1].
1. Description of the problem and its solution
This paper aims to answer a question left over from the pure spinor construction of one-loop integrands of super-Yang–Mills (SYM) using locality and BRST invariance [1]. Canone find a set of local and supersymmetric numerators for ten-dimensional SYM one-loopintegrands at six points satisfying the Bern-Carrasco-Johansson (BCJ) color-kinematicsduality? We will see below that the answer is yes , and we will also outline the solution forseven-point integrands.The one-loop integrands of SYM in ten dimensions for five and six points were con-structed in [1], where it was shown that the numerators for the five-point amplitude sat-isfied the color-kinematics duality while those at six points did not. The proposal of [1]was based on two main ingredients: locality and BRST invariance. Using the multiparticlesuperfields in pure spinor superspace developed in [3], these requirements together with abasic understanding of the zero-mode saturation rules of the pure spinor formalism [4,5]led to intuitive rules mapping one-loop cubic graphs to superspace numerators, see fig. 1.By assembling the numerators of the cubic graphs for all p -gons of a n -point amplitudesuch that their sum is in the pure spinor BRST cohomology (up to anomalous terms ofthe form discussed in [6,7]), the amplitudes of the color-ordered five and six-point ampli-tudes for the canonical color ordering were constructed. The six-point integrand was latersuccessfully used in [8], passing some consistency checks. A brief review of the BCJ color-kinematics duality sufficient for our purposes will be givenbelow in section 3.1 but a much more in-depth review is contained in [2]. .1. Genus-one open superstring correlators in pure spinor superspace In this paper we will also use the same formalism of multiparticle superfields in pure spinorsuperspace to present local representations of the five-, six- and seven-point amplitudes thatdo obey the color-kinematics duality. Since we are using the same superfield language, it istherefore important to highlight the differences with respect to the previous analysis of [1].The difference stems from the knowledge of the open-string one-loop correlators recentlyobtained in [9–11] up to seven points. They are given by K ( ℓ ) = V T , , Z , , , , (1 . K ( ℓ ) = V T m , , , Z m , , , , (1 . (cid:0) V A T B,C,D Z A,B,C,D + [
A, B, C, D | (cid:1) , K ( ℓ ) = 12 V T mn , , , , Z mn , , , , , (1 . (cid:0) V A T mB,C,D,E Z mA,B,C,D,E + [ A, B, C, D, E | (cid:1) + (cid:0) V A T B,C,D Z A,B,C,D + [
A, B, C, D | (cid:1) , K ( ℓ ) = 16 V T mnp , , , , , Z mnp , , , , , , (1 . (cid:0) V A T mnB,C,D,E,F Z mnA,B,C,D,E,F + [ A, B, C, D, E, F | (cid:1) + (cid:0) V A T mB,C,D,E Z mA,B,C,D,E + [ A, B, C, D, E | (cid:1) + (cid:0) V A T B,C,D Z A,B,C,D + [
A, B, C, D | (cid:1) − (cid:0) V J m | , , , , Z m | , , , , , + (2 ↔ , , , , (cid:1) − (cid:0)(cid:0) V A J B | C,D,E,F Z B | A,C,D,E,F + ( B ↔ C, D, E, F ) (cid:1) + [ A, B, C, D, E, F | (cid:1) − (cid:0) ∆ | | , , , , Z | , , , , + (2 ↔ , , , , (cid:1) , where the [ A , ..., A m | ...n ] notation is used to denote a sum over Stirling cycles [11], seethe appendix A for more details. The supersymmetric polarizations of ten-dimensionalgluons and gluinos are encoded in the pure spinor multiparticle building blocks V A T m...B,C,D... reviewed in [9]. The various Z m...A,B,C,... are worldsheet functions elaborated in [10] and theydepend on the insertion points of the vertices on the Riemann surface and on the loopmomentum ℓ m . These sums can also be described by all the ways in which 12 ...n can be completely decom-posed into m Lyndon words, with every letter appearing in precisely one such word. τ of the genus-one Riemann surfaces A n = X top C top Z D top dτ dz dz . . . dz n Z d D ℓ |I n ( ℓ ) | hK n ( ℓ ) i , (1 . I n ( ℓ ) denotes the Koba-Nielsen factor, D top denotes an ordered region of integrationover the insertion points z i , and C top denotes a group-theory factor which depends on thetopology of the genus-one surface (cylinder, M¨obius strip or non-planar cylinder) [12]. Forsimplicity we will consider only the planar cylinder topology in the following. For moredetails on this setup, see section 2 of [9].To gain intuition why the one-loop open-string correlators lead to a representation ofone-loop SYM numerators that satisfy the color-kinematics duality it will be illustrativeto review the quest for local BCJ-satisfying ten-dimensional supersymmetric numeratorsat tree level, solved in pure spinor superspace in [13]. When the tree-level color-ordered amplitudes were first proposed in [14], the constructionwas based on the principles of locality and BRST invariance of pure spinor superspaceexpressions using multiparticle superfields. These same principles were later used whenproposing SYM one-loop integrands in [1]. The difference between the expressions in [14]and [1] originates from the differences in the pure spinor amplitude prescriptions at treelevel [5] and one loop [4]. The n -point tree-level numerators of [14] had to be built fromthree unintegrated (multiparticle) vertices V following the OPE contractions with ( n − U ( z ). For example, at tree level the five-point SYM amplitude in thecanonical color ordering was obtained as A SYM (1 , , , ,
5) = V [12 , V V s s + V [1 , V V s s + V [1 , V [3 , V s s + V V [23 , V s s + V V [2 , V s s (1 . For convenience we shall frequently omit from amplitudes such as (1.6) the pure spinorbrackets h . . . i that extract the top element ( λγ m θ )( λγ n θ )( λγ p θ )( θγ mnp θ ) in the cohomology ofthe pure spinor BRST operator [5]. The component evaluation of ghost-number three expressionsuses the identities from the appendix of [6]. V [ A,B ] denotes the multiparticle unintegrated vertex operator in the BCJ gauge,see the review on multiparticle superfields in section 3 of [9] and section 4.3 of [15]. Theexpression (1.6) correctly reproduces the five-point tree amplitude of SYM in the canonicalcolor ordering. The next task is to check whether this representation leads to numeratorsthat satisfy the color-kinematics duality, this is where a subtle point arises.A triplet of numerators participating in a kinematic Jacobi identity necessarily involvesnumerators from amplitudes with different color orderings, but the naive relabeling of theamplitude (1.6) does not lead to a representation satisfying the BCJ color-kinematics dual-ity. Let us illustrate this point with an example. Using the parameterization of numeratorsfrom [16] where A SYM (1 , , , ,
5) = n s s + n s s + n s s + n s s + n s s (1 . A SYM (1 , , , ,
5) = n s s + n s s + n s s + n s s + n s s in order to check whether the numerators n , n and n satisfy the kinematic Jacobi iden-tity n − n + n = 0 one needs to extract the numerator n of the pole in 1 / ( s s ), n of 1 / ( s s ) and n of 1 / ( s s ). While n and n can be read off from the am-plitude A (1 , , , ,
5) in (1.6), the numerator n is found in the different color ordering A (1 , , , , n − n + n = V [1 , V [3 , V − V V [2 , V + V V [43 , V = 0 , (1 . n = V V [43 , V obtained from n = V V [23 , V via the relabeling 2 ↔ The solution to the above problem was found in [13] by utilizing the n -point string diskcorrelator of [17] to generate different color orderings in its field-theory limit. These order-ings follow from the various integration regions over the insertion points z i ordered alongthe boundary of a disk. For five points the superstring tree-level correlator is K ( z , . . . , z ) = V V V z z + V V V z z + V V V z z + (2 ↔ . (1 . α ′ → A SYM (Σ) = X XY =23 V X V ( n −
1) ˜ Y V n m (Σ | , X, n, Y, n − − | Y | +1 + (2 ↔ , (1 . m (Σ | Ω) denotes the biadjoint tree amplitudes, m ( P, n | Q, n ) = s P φ P | Q (1 . φ P | Q are the Berends-Giele double currents [21]. They can be computed recursively φ P | Q = 1 s P X XY = P X AB = Q (cid:0) φ X | A φ Y | B − ( X ↔ Y ) (cid:1) , φ P | Q = 0 if P \ Q = ∅ . (1 . s P = k P · k P where k P is a multiparticlemomentum defined by k P = k p + k p + · · · (for example k = k + k + k ).Extracting the field-theory limit of the string disk integrals computed in the ordering z ≤ z ≤ z ≤ z ≤ z – corresponding to Σ = 14325 in (1.10) – leads to the followingcolor-ordered amplitude A SYM (1 , , , ,
5) = 1 s s ( V V + V V + V V + V V ) V (1 . s s V V V − s s ( V V [4 , + V [1 , V ) V + 1 s s ( V V + V V ) V − s s V V [4 , V . One can now read off the numerator n = V V V + V V V and verify that the BCJidentity n − n + n = 0 is identically satisfied [13] n − n + n = V [1 , V [3 , V − V V [2 , V + ( V V V + V V V ) = 0 , (1 . V = V [23 , and V = − V [3 , .The field-theory tree-level SYM numerators are extracted from the knowledge of thesingular behavior of the correlator as vertex operators collide as encoded in the biadjoint Identically means that no BRST cohomology identity (of the type discussed in [22]) is requiredto verify the vanishing of the triplet of numerators; it vanishes at the superfield level. ℓ Fig. 2
The cubic graph associated to the pentagon N (5)1 | , , , ( ℓ ) from equation (1.15). The con-vention for the loop momentum ℓ is to run from the last argument of the numerator to the first. Berends-Giele currents. But we know that these limits constitute a local property of theRiemann surface and therefore must be independent of its genus. These results togetherwith the analysis of [23] lead to the following expectation:
The field-theory limit of the one-loop string correlators integrated along differentvertex insertion orderings should give rise to a local representation for SYM one-loop integrands that satisfy the BCJ color-kinematics duality.
As an illustration of this method – to be fully developed in the next sections – let us applyit in the simplest case of the five-point SYM integrand/amplitude following from the stringcorrelator (1.2).
As mentioned above, the five-point SYM integrand was proposed based on a few constraintssuch as locality and BRST invariance. The pentagon for the color order A (1 , , , ,
5) wasgiven as [1] N (5)1 | , , , ( ℓ ) = ℓ m V T m , , , + 12 (cid:2) V T , , + (2 ↔ , , (cid:3) + 12 (cid:2) V T , , + (2 , | , , , (cid:3) (1 . i, j | , , ,
5) denotes a sum over all possible ways to choose twoelements i and j from the set { , , , } while keeping the same order of i and j within theset. The cubic graph associated to this pentagon is displayed in fig. 2. Note the conventionof assigning the loop momentum ℓ to the edge between 5 and 1.7sing BRST cohomology arguments the box and pentagon numerators following fromrelabelings (while respecting the loop momentum assignment convention and the constraintthat leg 1 is contained in a multiparticle unintegrated vertex V ) were proposed in [1], N (4) A | B,C,D ( ℓ ) = V A T B,C,D (1 . N (5) A | B,C,D,E ( ℓ ) = ℓ m V A T mB,C,D,E + 12 (cid:2) V [ A,B ] T C,D,E + ( B ↔ C, D, E ) (cid:3) (1 . (cid:2) V A T [ B,C ] ,D,E + ( B, C | B, C, D, E ) (cid:3) . For a cubic-graph parameterization of the five-point integrand to obey the BCJ color-kinematics duality the antisymmetric combination of two pentagons in the legs 1 and 2must give rise to a box [24]
32 1 5 4 ℓ −
31 2 5 4 ℓℓ − k − ℓ
12 3 45 = 0From the figure above we see that the pentagon in the middle must come from the colorordering A (2 , , , ,
5) so as to keep the momenta in the common edges of the participatingcubic graphs the same while respecting the loop momentum convention mentioned above.However, the generic expression (1.17) has to ensure that the leg 1 appears in A , so thesolution proposed in [1] satisfying both constraints was to assign the pentagon numerator N | , , , ( ℓ − k ) to the middle diagram, with a shift in the loop momentum. Using that the12-box numerator is V T , , , the expression (1.17) implies that the numerator translationof the diagrams above is given by h N (5)1 | , , , ( ℓ ) − N (5)1 | , , , ( ℓ − k ) − N (4)12 | , , i = (1 . h k m V T m , , , + V T , , + V T , , + V T , , + V T , , i = 0 . (1 . h . . . i . The BRST exactness of the second line was shown in [22].8 .3.2. The BCJ pentagon from the field-theory limit of the string correlator The five-point analysis of [1] was primarily based on the BRST cohomology propertiesof the integrands, and as we reviewed above this was enough to obtain a BCJ-satisfyingparameterization up to BRST-exact terms. However, using the field-theory limit of thestring correlator the resulting numerators for the pentagons improve the BCJ identity tobe satisfied identically at the superspace level, requiring no cohomology manipulations.To see this we consider the five-point correlator (1.2) written in terms of the Eisenstein-Kronecker coefficient functions g (1) ij of [10], namely Z m , , , , = ℓ m and Z , , , = g (1)12 K ( ℓ ) = V T m , , , ℓ m + (cid:2) V T , , g (1)12 + (2 ↔ , , (cid:3) + (cid:2) V T , , g (1)23 + (2 , | , , , (cid:3) . (1 . g (1) ij functions) depends on the relative ordering of how the vertex insertion pointsare integrated by a term proportional to sgn ij . More precisely, if the color ordering of theresulting SYM integrand is P , the field-theory limit of g (1) ij contains a term sgn Pij , wheresgn
Pij is defined in (2.16). Therefore the pentagons of the integrands in the A (1 , , , , A (2 , , , ,
5) orderings differ by a sign in the term coming from g (1)12 . This gives riseto the following pentagons: N | , , , ( ℓ ) = ℓ m V T m , , , + 12 V T , , + 12 V T , , + 12 V T , , + 12 V T , , (1 . V T , , + 12 V T , , + 12 V T , , + 12 V T , , + 12 V T , , + 12 V T , , N | , , , ( ℓ ) = ℓ m V T m , , , − V T , , + 12 V T , , + 12 V T , , + 12 V T , , (1 . V T , , + 12 V T , , + 12 V T , , + 12 V T , , + 12 V T , , + 12 V T , , where we note that the constraint that leg 1 is within V is automatically satisfied becausethe correlator (1.20) is always the same, what changes is the relative ordering of integrationof the vertex positions.It is easy to see that the numerators (1.21) and (1.22) imply that the BCJ identity isidentically satisfied at the superfield level, N | , , , ( ℓ ) − N | , , , ( ℓ ) − N | , , ( ℓ ) = 0 , (1 . N | , , ( ℓ ) = V T , , . We thus see that the derivation of n -gon numerators fromthe field-theory limit of the open superstring correlator evaluated at different regions of9ntegration implies that the associated BCJ identity is satisfied even before applying thepure spinor cohomology bracket to extract the polarization content of the superfields, unlikethe case (1.18) obtained from relabeling. For five points this difference is immaterial as bothapproaches eventually satisfy the color-kinematics duality in the cohomology. However, wewill see below that the field-theory limit technique leads to a six-point representation thatsatisfies the color-kinematics duality in contrast to the representation of [1].
2. SYM one-loop integrands from string correlators
The field-theory limit of the one-loop string correlators is obtained by shrinking the stringsto points with α ′ → τ to point-particle worldline diagrams with Im( τ ) → ∞ [26]. In principle this can be doneusing the tropical limit techniques of [27] or the string-based formalism [25], although theexplicit form of the Kronecker-Eisenstein coefficient functions g ( n ) ( z, τ ) lead to subtletiesarising from the regular functions with n ≥
2. Alternatively, one can combine the strengthsof these approaches with the requirement that the field-theory integrands for different colororderings and loop-momentum parameterizations obtained from the string correlators arein the BRST cohomology of the pure spinor BRST charge. Some trial and error led to thecombinatorial rules described below.
The kinematic poles arise when the insertion points of the vertex operators approach eachother z i → z j on the Riemann surface. The short-distance behavior of the Koba-Nielsenfactor and the OPE propagator is independent of the genus of the Riemann surface. Thismeans that the pole structure of the genus-one string correlators can be described by thesame combinatorics of tree-level poles, given by the biadjoint scalar amplitudes (1.11).These amplitudes are efficiently computed using the Berends-Giele double currents φ P | Q of explicit form given in (1.12) where the words P and Q encode the integration regionand integrand.In the one-loop case however, in addition to the tree-level kinematic poles in Mandel-stam invariants the field-theory limit of the genus-one string correlators also yield Feynmanloop momentum integrands I A n +1 A ,A ,...,A n ( ℓ ) = 1( ℓ − k A ) ( ℓ − k A A ) · · · ( ℓ − k A A ...A n ) (2 . D -dimensional loop momentum ℓ with R d D ℓ . Note the special roleplayed by the label 1 in the above definition; this handling fixes the freedom to shift theloop momentum and is useful in obtaining BRST-closed SYM integrands [1].In summary, the field-theory limit of genus-one open string correlators will be de-scribed by poles in Mandelstam invariants encoded in Berends-Giele double currents mul-tiplied by Feynman loop momentum integrals. In the same way as in the tree-level case, the color ordering of the resulting SYM integrandfrom the field-theory limit of the genus-one open string correlator is associated to therelative ordering of the z i variables among each other on the boundary of the Riemannsurface. For example, the ordering z ≤ z ≤ z ≤ z ≤ z yields an integrand with colorordering σ = 13542.The presence or absence of kinematic poles depend crucially on the region of integra-tion relative to the ordering of the z ij variables being integrated. To encode this informationwe define a map Ord A ( B ) acting on two words A and B that crops the word A while main-taining the letters it shares with B . That is, we take the word B and return the smallestsequence of consecutive letters in the cyclic-symmetric object A containing every letter in B . For example,Ord (32) = 23 , Ord (13) = 123 , Ord (15) = 561 , (2 . (58) = 85 , Ord (465) = 4856 , Ord (78) = 7248 . This map can be defined algebraically byOrd A ( B ) = A i A i +1 ...A j − A j if A i , A j ∈ B, B ⊆ A i ...A j , j − i ≤ | A | A j A j +1 ...A | A | A A ...A i if A i , A j ∈ B, B ⊆ A i ...A j , j − i > | A | . σ . It will be convenient to introduce the notation:ˆ φ ( σ | A ) ≡ φ Ord σ ( A ) | A , (2 . σ . 11 .2. p -gon loop momentum integrands Frequently we will need the Feynman loop momentum integrands (2.1) with a general shiftin the loop momentum ℓ → ℓ + a i k i . This will be indicated by superscripts I a ,a ,...,a m A n +1 A ,A ,...,A n ( ℓ ) = I A n +1 A ,A ,...,A n ( ℓ + a k + a k + · · · + a m k m ) (2 . I a ,a ,...,a m A n +1 A ,A ,...,A n = 1( ℓ + f a ...a m − k A ) ... ( ℓ + f a ...a m − k A A ...A n ) , (2 . f a ,...,a m = a k + a k + ... + a m k m . (2 . a i being zero, we will omit it from the notation. Note that the wordscharacterizing the integrands (2.6) are totally symmetric e.g. I , , , = I , , , .We will sometimes simplify the notation for the loop momentum integrands by drop-ping all indices which are single letters, and dropping the shifts in the loop momentum.When this is done it should always be clear the color ordering of the amplitude. For ex-ample, in the canonical ordering A (1 , , ..., n ; ℓ ) we have I ∅ = I = I a ,...,a n , ,...,n , I = I a ,...,a n , , , ,...,n , (2 . I , = I a ,...,a n , , , , , ,...,n , I n , = I a ,...,a n n , , , , ,...,n − . In a few instances, we may wish to use this notation when it is not immediately clear whatthe underlying color ordering is. In these circumstances we will include it as a superscriptin the I . So, for example I ∅ = I = I , , , , , , I = I , , , , , I = I , , , . (2 . We are now ready to give the field theory limits. These are: g ( p ) ij → b ( p ) ij P + c ( p ) ij P ( ij ) (2 . g ( p ) ij g ( q ) kl → b ( p ) ij b ( q ) kl P + b ( p ) ij c ( q ) kl P ( kl ) + c ( p ) ij b ( p ) kl P ( ij ) + c ( p ) ij c ( q ) kl P ( ij, kl ) (2 . g ( p ) i j g ( p ) i j g ( p ) i j → b ( p ) i j b ( p ) i j b ( p ) i j P + b ( p ) i j b ( p ) i j c ( p ) i j P ( i j ) (2 . b ( p ) i j c ( p ) i j b ( p ) i j P ( i j ) + c ( p ) i j b ( p ) i j b ( p ) i j P ( i j )+ b ( p ) i j c ( p ) i j c ( p ) i j P ( i j , i j ) + c ( p ) i j b ( p ) i j c ( p ) i j P ( i j , i j )+ c ( p ) i j c ( p ) i j b ( p ) i j P ( i j , i j ) + c ( p ) i j c ( p ) i j c ( p ) i j P ( i j , i j , i j )12hese limits always have the same form; we take the subscripts of the g ( p ) ij , and sum overthe possible ways to assign these to either a b ( p ) or a c ( p ) (to be defined below), andwhenever we assign them to a c ( p ) they are also entered into the P function. In turn theseare defined by P = I (2 . P ( ij ) = ˆ φ ( σ | ij ) I ij P ( ij, kl ) = (cid:26) ˆ φ ( σ | ijl ) I ijl if j = k ˆ φ ( σ | ij ) ˆ φ ( σ | kl ) I ij,kl if all i unique P ( ij, kl, mn ) = ˆ φ ( σ | ijln ) I ijln if j = k , l = m ˆ φ ( σ | ijl ) ˆ φ ( σ | mn ) I ijl, mn if j = k , m, n / ∈ { i, j, k, l } ˆ φ ( σ | ijn ) ˆ φ ( σ | kl ) I ijn, kl if j = m , k, l / ∈ { i, j, m, n } ˆ φ ( σ | ij ) ˆ φ ( σ | kln ) I ij, kln if l = m , i, j / ∈ { k, l, m, n } ˆ φ ( σ | ij ) ˆ φ ( σ | kl ) ˆ φ ( σ | mn ) I ij, kl,mn if all i uniquewhere we used the notation (2.4). The cases provided above will be sufficient for ourpurposes.Finally, the coefficients b ( p ) and c ( p ) for an integrand A ( σ ; ℓ + P ni =1 a i k i ) are given by b ( p ) ij = p X m =0 (cid:0) sgn σij (cid:1) m B m ( a j − a i ) p − m m !( p − m )! (2 . c ( p ) ij = 12( p − (cid:0) ( a j − a i ) + sgn σij dist σ ( i, j ) (cid:1) p − (2 . B n denotes the n th Bernoulli number andsgn Bij = (cid:26) +1 : i is left of j in B − i is right of j in B (2 . Ba ( i, j ) measures the distance between i and j in the word B and returns+1 if it is larger than a and 0 otherwise,dist Ba ( i, j ) = (cid:26) +1 : if i is a or more letters to the left or right of j in B i is fewer than a letters to the left or right of j in B (2 . a i = 0 ∀ i , we must take 0 = 1 in the above. The amplitudes up to seven points require up to B : B = 1 , B = , B = , B = 0. .3.1. A seven-point example The field-theory limit of the term g (1)25 g (1)57 g (1)76 V T , , in the seven-point string correlator(1.4) for the SYM integrand with color ordering A (1 , , , , , , ℓ + 4 k − k ) followsfrom (2.12) with a = 4 and a = − g (1)25 g (1)57 g (1)76 → b (1)25 b (1)57 b (1)76 P + b (1)25 b (1)57 c (1)76 P (76)+ b (1)25 c (1)57 b (1)76 P (57) + c (1)25 b (1)57 b (1)76 P (25) (2 . b (1)25 c (1)57 c (1)76 P (57 ,
76) + c (1)25 b (1)57 c (1)76 P (25 , c (1)25 c (1)57 b (1)76 P (25 ,
57) + c (1)25 c (1)57 c (1)76 P (25 , , . Many of these terms vanish. For instance using (2.13) the factor P (57) is proportional toˆ φ (1234567 |
57) = φ | Ord (57) = φ | = 0. Similarly, we find P (25) = P (25 ,
76) = P (25 ,
57) = P (25 , ,
76) = 0 . (2 . P = I = I a ,a , , , , , , (2 . P (76) = ˆ φ (1234567 | I = φ | I = − s I a ,a , , , , , P (57 ,
76) = ˆ φ (1234567 | I = φ | I , , , , = − s s I a ,a , , , , The various b (1) ij and c (1) ij terms are given by (2.14) and (2.15). In the g (1)25 case, these aregiven by (recall that a = 4 , a = − b (1)25 = B ( −
0! 1! + B ( −
1! 0! = − −
112 (2 . c (1)25 = (cid:0) a − a + sgn dist (2 , (cid:1) − − (cid:0) − − × (cid:1) b (1)57 = 132 , c (1)57 = 12 , b (1)76 = − , c (1)76 = 12 . (2 . g (1)25 g (1)57 g (1)76 → I a ,a , , , , , , + 1438 1 s I a ,a , , , , , + 118 1 s s I a ,a , , , , (2 . .4. The one-loop SYM field-theory integrands The one-loop correlators of the open superstring are integrated over the vertex insertions z i ordered along the boundary of a genus one surface. After taking the field-theory limit,the color ordering of the resulting SYM integrand corresponds to the ordering of theinsertions z i . As alluded to in section 2.1, the field-theory limit of the open-string correlatorswill be written as a field-theory integrand depending on the loop momentum ℓ m . Theparameterization of the one-loop graphs by Feynman loop integrals is notoriously plaguedwith the “labelling problem”: arbitrary shifts of the loop momentum must not affect theintegrated amplitude. This will be indicated by labelling a color-ordered SYM integrandwith the explicit parameterization of the loop momentum as A (1 , , ..., n ; ℓ + a k + · · · + a n k n ) (2 . , , ..., n , constructed such that the mo-mentum going from the n th leg to the 1st leg is ℓ + a k + ... + a n k n . For example, thefield-theory limit of the five-point correlator with insertion points ordered according to z ≤ z ≤ z ≤ z ≤ z and loop momentum ℓ running between legs 4 and 1 is repre-sented by the SYM integrand A (1 , , , , ℓ ). The statement of cyclicity – proven in theappendix B – in the color ordering becomes A (1 , , ..., n ; ℓ + a k + ... + a n k n ) = A (2 , , ..., n, ℓ + ( a − k + a k ... + a n k n ) (2 . The field-theory limit of the open superstring n -point correlator for will be parameterizedby a sum over p -gon cubic graphs ranging from p = 4 (boxes) to p = n : A ( i i . . . i n ; ℓ + a j k j ) = n X p =4 X A ...A p +1 = i ...i n N a ,a ,...,a n A p +1 i A | A ,...,A p ( ℓ ) I a ,a ,...,a n i A ,A ,...,A p (2 . N a ,a ,...,a n A | A ,...,A p ( ℓ ) denotes the kinematic Berends-Giele numerator of a p -gon con-structed as described in the appendix A and I a ,a ,...,a n A ,A ,...,A p represents the p -gon integrand.We note that in extracting a local numerator N ... from (2.26) there will be a factor of 1 / For simplicity we will consider only the planar topology. .4.2. Four points The extraction of the field theory limit at four points is trivial as there is no propagatorfunction [4]. The only limit to consider is the Koba-Nielsen factor and we get A ( σ , σ , σ , σ | ℓ + a k σ + ... + a k σ ) = V T , , I , , , . (2 . The five-point genus-one superstring correlator is given by [11] K ( ℓ ) = V T m , , , Z m , , , , + (cid:2) V T , , Z , , , + (2 ↔ , , (cid:3) (2 . (cid:2) V T , , Z , , , + (2 , | , , , (cid:3) with the worldsheet functions [10] Z m , , , , = ℓ m , Z , , , = g (1)12 . (2 . g (1)12 , g (1)23 , g (1)34 , g (1)45 , and g (1)51 . The parameterization of the integrand A (1 , , , , ℓ + a i k i ) from (2.26) is given by A (1 , , , , ℓ + a i k i ) = N | , , , ( ℓ ) I a ,...,a , , , , ( ℓ ) (2 . s N | , , ( ℓ ) I a ,...,a , , , ( ℓ ) + 12 s N | , , ( ℓ ) I a ,...,a , , , ( ℓ )+ 12 s N | , , ( ℓ ) I a ,...,a , , , ( ℓ ) + 12 s N | , , ( ℓ ) I a ,...,a , , , ( ℓ )+ 12 s N ′ | , , ( ℓ ) I a ,...,a , , , ( ℓ ) . Since the field-theory limit rules behave differently for labels at the extremities of the colorordering, the 51-pentagon numerator is denoted N ′ | , , ( ℓ ). Using the field-theory limit(2.10) and comparing the outcome with (2.30) we can read off the box numerators. Theyare independent of the loop momentum and are uniformly described by N A | B,C,D = V A T B,C,D . (2 . N ′ | , , = N | , , = V T , , . This result agrees with the analysis of [1].16he pentagon I a ,...,a , , , , ( ℓ ) arises from the pieces with no kinematic poles in (2.10) andcollecting its associated superfields yields the numerator N a ,...,a | , , , ( ℓ ) = V T m , , , ℓ m + h V T , , (cid:0) a − a + 12 (cid:1) + (2 ↔ , , i (2 . h V T , , (cid:0) a − a + 12 (cid:1) + (2 , | , , , i . A straightforward but tedious calculation shows that QN a ,...,a | , , , ( ℓ ) = 12 V V T , , (( ℓ + f a ...a − k ) − ( ℓ + f a ...a − k ) ) (2 . V V T , , (( ℓ + f a ...a − k ) − ( ℓ + f a ...a − k ) )+ 12 V V T , , (( ℓ + f a ...a − k ) − ( ℓ + f a ...a − k ) )+ 12 V V T , , (( ℓ + f a ...a − k ) − ( ℓ + f a ...a − k ) )with the f a ...a defined as in (2.7). It is then not hard to check that the above cancels theBRST variation of the box terms. For example, the terms proportional to ( ℓ + f a ...a − k ) are given by 12 ( V V T , , − V V T , , ) = − s QV T , , (2 . ℓ + f a ...a − k ) I a ,...,a , , , , ( ℓ ) = I a ,...,a , , , ( ℓ ) . (2 . QN a ,...,a | , , , ( ℓ ) I a ,...,a , , , , = − QA box (1 , , , ,
5) and thereforethe five-point SYM integrand (2.30) is BRST invariant.The BRST cohomology identities [22] h V k m T m , , , i = h− V T , , + (2 ↔ , , i (2 . h V k m T m , , , i = h V T , , + (cid:2) − V T , , + (3 ↔ , (cid:3) i can be used to show that h N (5)1 | , , , ( ℓ + a i k i ) i = h N a ,...,a | , , , ( ℓ ) i (2 . N (5)1 | , , , ( ℓ ) is given by (1.15) and I a ,...,a , , , , ( ℓ ) = I , , , , ( ℓ + a i k i ). This is an im-portant consistency check on the field-theory rules spelled out in section 2.3.All color ordering permutations of the five-point SYM integrand is available to down-load from [28]. 17 .4.4. Six points The six-point genus-one superstring correlator is given by [11] K ( ℓ ) = 12 V A T mnA ,...,A Z mnA ,...,A + (cid:2) | A , . . . , A (cid:3) (2 . V A T mA ,...,A Z mA ,...,A + (cid:2) | A , . . . , A (cid:3) + V A T A ,...,A Z A ,...,A + (cid:2) | A , . . . , A (cid:3) , with the worldsheet functions [10], Z , , , = g (1)12 g (1)23 + g (2)12 + g (2)23 − g (2)13 , (2 . Z , , , = g (1)12 g (1)34 + g (2)13 + g (2)24 − g (2)14 − g (2)23 , Z m , , , , = ℓ m g (1)12 + ( k m − k m ) g (2)12 + (cid:2) k m ( g (2)13 − g (2)23 ) + (3 ↔ , , (cid:3) , Z mn , , , , , = ℓ m ℓ n + (cid:2) ( k m k n + k n k m ) g (2)12 + (1 , | , , , , , (cid:3) . To illustrate the field-theory rules of the previous section we will derive the SYM integrand A (2 , , , , , ℓ ) = A (1 , , , , , ℓ + k ). We begin with the field theory limit rules givenby (2.10) and (2.11) g (1) ij →
12 sgn ij I + 12 φ ij | Ord ( ij ) I ij (2 . g (2) ij → I + 12 s ( δ i δ j + δ j δ i ) I g (1) ij g (1) kl →
14 sgn ij sgn kl I + 14 sgn kl φ ij | Ord ( ij ) I ij + 14 sgn ij φ kl | Ord ( kl ) I kl + 14 P ( ij, kl )where P ( ij, kl ) = φ ijl | Ord ( ijl ) I ijl if j = k − φ ijk | Ord ( ijk ) I ijk if j = l − φ jil | Ord ( ijl ) I jil if i = kφ kij | Ord ( kij ) I kij if i = lφ ij | Ord ( ij ) φ kl | Ord ( kl ) I ij,kl else (2 . I = I , , , , , , we find the hexagon numerator N | , , , , ( ℓ ) = 12 V T mn , , , , (cid:0) ℓ m ℓ n −
112 [ k m k n + (1 ↔ , , , , (cid:1) (2 . V T m , , , ( ℓ m − k m + 16 k m ) + (2 , | , , , , −
12 ( V T m , , , ( ℓ m + 16 k m − k m ) + (2 ↔ , , , V ( T [[2 , , , , + T [2 , [3 , , , + (2 , , | , , , , V T , , + (2 , | , | , , , , −
14 ( V T , , + (2 | , | , , , , −
16 (( V − V ) T , , + (2 , | , , , , . We then identify the pentagon numerators, which in all but one case are given by a gen-eralization of the formulae from [1], N A | B,C,D,E = V E T mA,B,C,D ℓ m + 12 ( V [ A,E T B,C,D + ( A ↔ B, C, D )) (2 . V E T [ A,B ] ,C,D + ( A, B | A, B, C, D ))The exception to the above is the 12-pentagon, which differs as it has a contributionfrom the g (2) terms due to the color ordering 234561. It is given by the coefficient of1 / s I a =12 , , , , (note the absence of the label 1 from the ordering in I a =12 , , , , ) N ′ | , , , ( ℓ ) = − V T mn , , , , k m k n (2 . − ( V T m , , , k m + (3 ↔ , , V T m , , , ( ℓ m + k m − k m ) − ( V T m , , , k m + (3 ↔ , , V T , , + (3 , | , , , − ( V T , , + (3 | | , , , V − V ) T , , + (3 ↔ , , V superfield, and the other blocks of indices assigned to the TN A | B,C,D E = V D E T A,B,C , N E A | B,C,D = V E A T B,C,D . (2 . QA a =1 (1 , , , , ,
6) = 12 V Y , , , , ( I , , , , − ℓ I , , , , , ) (2 . a = ... = a = 0 result found in [1], and by ananalogous argument to the one presented there one finds the same result for the integratedanomaly Z d ℓQA a =1 (1 , , , , ,
6) = − π V Y , , , , . (2 . SO (32) is free of gauge anomalies[30], but this property does not survive the field-theory limit of its planar sector and thesix-point one-loop SYM amplitude in ten dimensions is anomalous [31]. The result (2.47)written in terms of the anomalous building block Y , , , , [22] is the pure spinor superspaceencoding of the field-theory anomaly [6,7]. At seven points, the numerators become far too complex to state here. One examplecan be found in the appendix D. The derivation of these numerators has one additionalcomplication; as was discussed in [11] the refined worldsheet functions are given by Z | , , , , = ∂g (2)12 + s g (1)12 g (2)12 − s g (3)12 . (2 . I ( ℓ )( ∂ g (2)12 ) I ( ℓ ) = ∂ ( g (2)12 I ( ℓ )) + g (2)12 ∂ I ( ℓ ) (2 . ∂ ( g (2)12 I ( ℓ )) + g (2)12 (cid:16) ( ℓ · k ) + s g (1)21 + s g (1)23 + ... + s g (1)27 (cid:17) I ( ℓ ) , which gives the reformulated expression for (2.48) Z | , , , , = − s g (3)12 + g (2)12 ( ℓ · k + s g (1)23 + s g (1)24 + ... + s g (1)27 ) (2 . See the discussion of [29] as summarized in section 4.5 of [1] to understand why (2.46) is nottrivially zero due to the cancellation of propagators in the integrand. . Local BCJ-satisfying numerators In this section we will obtain the kinematic numerators associated to various one-loop cubicgraphs using the field-theory limit rules of section 2.3 applied to the superstring correlatorsfor six external states as well as some seven-point numerators. The results of this sectionresolve a puzzle in the analysis of [1]. Namely, the representation in [1] of the six-pointintegrand did not satisfy the color-kinematics duality by terms which suspiciously wererelated to the gauge anomaly. We now show that the six-point integrand representationarising from the field-theory limit of the string correlator satisfies all the color-kinematicJacobi dual relations of Bern-Carrasco-Johansson.
The color factors of amplitudes in gauge theory depend on the structure constants of somegauge group, f abc , that satisfy the Jacobi identity, f abe f cde + f bce f ade + f cae f bde = 0 . (3 . i, j, k whose color factors c i , c j , c k vanish due to the Jacobi identity (3.1), c i + c j + c k = 0, thecorresponding numerators N i , N j , N k of the diagrams satisfy N i ( ℓ ) + N j ( ℓ ) + N k ( ℓ ) = 0as well. Stated originally at tree-level [16] (and proven by the field-theory limit of stringtheory tree amplitudes [32,33]) the duality was conjectured at loop-level in [34], where thekinematic numerators also depend on loop momenta ℓ parameterizing various n -gon cubicgraphs. Through this approach, properties of 4 ≤ N ≤ c i → ˜ N i ( ℓ )[16,34]. For more details see the review [2].We will now show that the numerators extracted from the one-loop string correlatorsusing the field-theory rules of section 2.3 satisfy all the color-kinematics relations. However,starting at six points the numerators do not satisfy the required symmetries under shifts21f the loop momentum required by the automorphism symmetries of the cubic graphs (see[24]), leading to subtleties in the construction of the gravity amplitudes which we defer tofuture work.The one-loop five-point integrand of SYM in ten dimensions was already discussedin section 1.3.2 so we will focus on the six-point SYM integrand and briefly outline thediscussion of the seven-point numerators. The color-kinematics relations are manifestly satisfied within external tree graphs due tothe BCJ gauge used in the multiparticle superfields [38,15]. Therefore we will discuss thekinematic Jacobi identities among p -gons with different values of p . The pure spinor superspace expressions of the numerators associated to the graphs in thefollowing linear combination ℓ
41 6 532 − ℓℓ − k
41 6 53 2 − ℓ
123 4 56 constitute a good example of how our methods give rise to a BCJ-satisfying parameteri-zation of the six-point integrand.Two of the above graphs are part of the integrand in the canonical color ordering A (1 , , , , , ℓ ). According to the color-kinematics identity that we are seeking to show,the loop momentum parameterization of the graphs must have the same momentum flowingin the edges that are kept the same for all graphs. Therefore the middle graph must haveloop momentum ℓ flowing from leg 6 to the fork 23. According to the convention shownin fig. 2 this is the 23-pentagon N | , , , ( ℓ ) from the integrand A (2 , , , , , ℓ ) whoseexpression can be read off from the field-theory limit rules for this particular ordering.However the assumption used in the parameterization of [1] was that this pentagon isobtained in a crossing symmetric way as N | , , , ( ℓ − k ). As shown in [1], using these22inematic numerators the algebraic translation of the BCJ triplet linear combination abovebecomes N | , , , ( ℓ ) − N | , , , ( ℓ − k ) − N [1 , | , , ( ℓ ) = (3 . k m V T m , , , + V T , , + (cid:2) V T , , + (4 ↔ , (cid:3) which is not in the cohomology of the BRST charge and therefore is not vanishing. Inother words, the representation of the six-point integrand chosen in [1] does not satisfy thecolor-kinematics duality.In contrast, using the field-theory limit rules of this work the cubic graphs abovecan be derived in their native color ordering and they satisfy the BCJ triplet numeratoridentity: N | , , , ( ℓ ) − N | , , , ( ℓ ) − N [1 , | , , ( ℓ ) = 0 . (3 . N [1 , | , , ( ℓ ), the coefficient of s s I , , , in the integrand A (1 , , , , , ℓ ). According to (2.11) and (2.13) theonly functions that can generate such a factor are g (1)12 g (1)23 and g (1)13 g (1)23 via P (12 ,
23) and P (13 , V T , , g (1)12 g (1)23 + V T , , g (1)13 g (1)32 . (3 . g (1)12 g (1)23 and g (1)13 g (1)23 gives rise to the box integrand through the P ( ij, jk ) terms in14 V T , , P (12 , V T , , P (13 ,
32) = 14 V T , , φ | I + 14 V T , , φ | I = 14 V T , , (cid:18) s s + 1 s s (cid:19) I + 14 V T , , (cid:18) − s s (cid:19) I . (3 . N [1 , | , , ( ℓ ) is given by the coefficient of
14 1 s s I , N [1 , | , , = V T , , − V T , , = V [1 , T , , (3 . N | , , , ( ℓ ) is given by the coefficient of s I in the field theory limit ofthe correlator K ( ℓ ) for the color ordering A (1 , , , , , ℓ ). Such factors arise from anyappearance of g (1)23 in (1.3), V T m , , , ℓ m g (1)23 + h V T , , g (1)12 g (1)23 + (2 ↔ i + h V T , , g (1)23 g (1)34 + (4 ↔ , i + h V T , , g (1)14 g (1)23 + (4 ↔ , i + h V T , , g (1)23 g (1)45 + (4 , | , , i (3 . s I we get N | , , , ( ℓ ) = V T m , , , ℓ m + 12 (cid:2) V [1 , T , , + (23 ↔ , , (cid:3) (3 . (cid:2) V T [23 , , , + (23 , | , , , (cid:3) . The numerators (3.6) and (3.8) agree with the numerators obtained in [1].The middle pentagon in the above figure is the 23-pentagon in the integrand of A (2 , , , , , ℓ ) since the loop momentum is running from leg 6 to 2. Alternatively, acyclic rotation as in (2.25) yields the integrand A (1 , , , , , ℓ − k ) whose field-theorylimit is computed using the rules of section 2.3 with with a = a = − a i = 0 for allother i . The calculation proceeds similarly to the above. The relevant terms are now V T m , , , ℓ m g (1)23 + h V T , , g (1)12 g (1)23 + (2 ↔ i (3 . h V T , , g (1)42 g (1)23 + V T , , g (1)43 g (1)32 + (4 ↔ , i + h V T , , g (1)14 g (1)23 + (4 ↔ , i + 12 h V T , , g (1)45 g (1)23 + (4 , | , , i . Taking the field theory limits and restricting ourselves to the s single poles, we see thatthe numerator is given by N | , , , ( ℓ ) = V T m , , , ℓ m − V [1 , T , , + 12 ( V [1 , T , , + (4 ↔ , V T [23 , , , + (23 , | , , , . N (5)1 | , , , ( ℓ − k )with the expression for N (5) A | B,C,D,E ( ℓ ) given in (1.17). While the representation of [1] failsto satisfy the color Jacobi identity, the new representation derived here obeys the color-kinematics duality. To see this we plug the superfield expressions of the new field-theoryrepresentations of the box (3.6) and pentagons (3.8), (3.10) in the kinematic Jacobi relation(3.3) to obtain N | , , , ( ℓ ) − N | , , , ( ℓ ) − N [1 , | , , ( ℓ ) = 0 . (3 . We exploit the total symmetry of the six-point correlator (1.3) in 2 , , , , .2.2. Kinematic Jacobi between hexagons and a pentagon In a given color ordering, all of the pentagons have a similar structure apart from the ij -pentagon whose labels are cyclically split at the extremities A ( i, . . . , j ; ℓ ). In this subsectionwe will demonstrate the validity of a BCJ relation involving such a numerator. The relationwe will show is ℓ
321 6 54 − ℓ + k
326 1 54 − ℓ + k
32 5 41 6 = 0which corresponds to N | , , , , ( ℓ ) − N a =11 | , , , , ( ℓ ) − N | , , , ( ℓ ) = 0 . (3 . P = I . In the first case, this means making the substitution g (1) ij →
12 sgn ij I , g (1) ij g (1) kl →
14 sgn ij sgn kl I , g (2) ij → I . (3 . N | , , , , ( ℓ ) = + 16 (( V [[1 , , + V [1 , [2 , ) T , , + (2 , | , , , , . V ( T [[2 , , , , + T [2 , [3 , , , + (2 , , | , , , , V [1 , T [3 , , , + (2 | , | , , , , V T [2 , , [4 , , + (2 , | , | , , , , V [1 , T m , , , ( ℓ m − k m + 16 k m ) + (2 ↔ , , , V T m [2 , , , , ( ℓ m − k m + 16 k m ) + (2 , | , , , , V T mn , , , , ( ℓ m ℓ n − k m k n − k m k n − · · · − k m k n ) . For the second hexagon, we consider the field-theory limit of the correlator with the colorordering A (1 , , , , , ℓ + k ). The limits needed now have the form g (1) ij →
12 sgn ij + δ j − δ i , (3 . g (1) ij g (1) kl → (cid:0)
12 sgn ij + δ j − δ i (cid:1)(cid:0)
12 sgn kl + δ l − δ k (cid:1) ,g (2) ij →
112 + 12 (cid:0) δ i (1 − sgn ij ) + δ j (1 + sgn ij ) (cid:1) . N a =11 | , , , , ( ℓ ) = + 12 V T mn , , , , ( ℓ m ℓ n + 2 k m k n −
112 ( k m k n + k m k n + · · · k m k n ))+ 12 ( V T m [2 , , , , ( ℓ m − k m + 16 k m ) + (2 , | , , , , − ( V T m [2 , , , , k m + (2 ↔ , , . V [1 , T m , , , ( ℓ m − k m + 16 k m + 2 k m ) + (2 ↔ , , V [1 , T m , , , ( 32 ℓ m − k m + 1312 k m )+ 16 V ( T [[2 , , , , + T [2 , [3 , , , + (2 , , | , , , , V [[1 , , + V [1 , [2 , ) T , , + (2 , | , , , −
13 (( V [[1 , , + V [1 , [2 , ) T , , + (2 ↔ , , V T [2 , , [4 , , + (2 , | , | , , , , V [1 , T [3 , , , + (2 | , | , , , −
34 ( V [1 , T [3 , , , + (2 | | , , , V [1 , T [2 , , , + (2 , | , , , s I , , , , from theintegrand A (1 , , , , , ℓ ). This can be found to be N ′ | , , , ( ℓ ) = + 12 (cid:2) ( V [[1 , , + V [1 , [2 , ) T , , + (2 ↔ , , (cid:3) (3 . (cid:2) V [1 , T [3 , , , + (2 | | , , , (cid:3) − (cid:2) V [1 , T [2 , , , + (2 , | , , , (cid:3) − (cid:2) V [1 , T m , , , k m + (2 ↔ , , (cid:3) + (cid:2) V T m [2 , , , , k m + (2 ↔ , , (cid:3) − V [1 , T m , , , ( ℓ m + k m − k m ) − V T mn , , , , k m k n It is then simply a matter of plugging the numerators into the identity (3.12) to verify itsvalidity. 26 ig. 3
The antisymmetry of the 61-pentagon from the integrand A (1 , , , , , ℓ ). The momen-tum running into the 61 external tree in the graph on the right is ℓ + k because in the colorordering 1 , , , , , ℓ must run between 6 and 1. Therefore in order to preservethe momentum assignment in the edges between the two cubic graphs, the pentagon on the left ispart of the integrand A (1 , , , , , ℓ + k ) with momentum ℓ + k running between legs 5 and1 as dictated by the convention (2.24). Therefore to extract this pentagon the field-theory rulesof section 2.3 must be used with a = 1. ij -pentagons from A ( i, P, j ; ℓ ) in i ↔ j As mentioned above, the color-kinematics duality relations within external tree diagramsis manifestly satisfied due to the usage of multiparticle superfields in the BCJ gauge. Forinstance, all the boxes and all but one pentagon for an integrand of arbitrary color ordering A ( P ; ℓ ) can be described by N A | B,C,D ( ℓ ) = V A T B,C,D ( ℓ ) + ( A ↔ B, C, D ) (3 . N A | B,C,D,E ( ℓ ) = (cid:2) V A T mB,C,D,E ℓ m + ( A ↔ B, C, D, E ) (cid:3) + 12 (cid:2) V A T [ B,C ] ,D,E + ( A | B, C | A, B, C, D, E ) (cid:3) + 12 (cid:2) V [ A,B ] T C,D,E + (
A, B | A, B, C, D, E ) (cid:3) (3 . T ......,A B,... = 0 (i.e., setting to zero all terms in whichthe label 1 is not assigned to a multiparticle vertex V P ). For example, using (3.19) werecover the 23-pentagon (3.10) N | , , , ( ℓ ) = V T m , , , ℓ m − V [1 , T , , + 12 ( V [1 , T , , + (4 ↔ , . V T [23 , , , + (23 , | , , , T ......,A B,... = 0. Since in the BCJ gauge [38,15] themultiparticle labels (words) in (3.18) and (3.19) satisfy generalized Jacobi identities, thecolor-kinematics duality are manifest within those words, with a notable exception.27he exception arises for the ij -pentagon when the labels i, j are adjacent up to a cyclicrotation, e.g. the 61-pentagon in A (1 , , , , , ℓ ) or the 12-pentagon in A (2 , , , , , ℓ )do not follow the general formula (3.19), as can be seen for example in (2.44). The reasonthis happens is due to a clash between the ij pentagon labels in A ( j, P, i ; ℓ ) and theconvention that the loop momentum ℓ runs between i and j . So to verify the antisymmetryof the 61-pentagon from A (1 , , , , , ℓ ) one needs to compare it to the 16-pentagon from A (1 , , , , , ℓ + k ) using the field-theory rules from section 2.3. The result is N a =116 | , , , ( ℓ ) = − (cid:2) ( V [[1 , , + V [1 , [2 , ) T , , + (2 ↔ , , (cid:3) (3 . − (cid:2) V [1 , T [3 , , , + (2 | | , , , (cid:3) + 12 (cid:2) V [1 , T [2 , , , + (2 , | , , , (cid:3) + (cid:2) V [1 , T m , , , k m − V T m [2 , , , , k m + (2 ↔ , , (cid:3) + V [1 , T m , , , ( ℓ m + k m − k m )+ V T mn , , , , k m k n . Comparing (3.21) with (3.17) one immediately verifies the color-kinematics identity de-picted in fig. 3 N a =116 | , , , ( ℓ ) + N | , , , ( ℓ ) = 0 . (3 . A (1 , , , , , ℓ ), namely N | , , , ( ℓ ) = V T m , , , ℓ m + 12 (cid:2) V T , , + (2 , | , , , V T , , + (2 ↔ , , (cid:3) . (3 . ℓ → ℓ + k in the 16-pentagon numerator (3.23) andcompare it with the 16-pentagon from the shifted amplitude A (1 , , , , , ℓ + k ) we findthat they are not BRST equivalent, Q (cid:0) N a =116 | , , , ( ℓ ) − N | , , , ( ℓ + k ) (cid:1) = Q ( s V J | , , , ) . (3 . A (1 , , , , , ℓ + k ) does notfollow from naively shifting ℓ → ℓ + k in A (1 , , , , , ℓ ).28 .2.4. Remaining BCJ triplets There are then a number of relations between pentagons and boxes left to show in orderto see that we have a BCJ representation of A (1 , , , , , a i a) ℓ ℓ + k
432 1 6 5 − ℓ + k
43 6 2 15 − ℓ + k
612 3 45 = 0b) ℓ ℓ + k
32 1 6 54 − ℓ + k
326 5 1 4 − ℓ + k
156 2 34 = 0c) ℓ + k ℓ − k
321 6 5 4 − ℓ + k ℓ + k − k
316 2 5 4 − ℓ + k
26 1 5 34 = 0d) ℓ + k ℓ − k
321 6 5 4 − ℓ + k − k ℓ − k ℓ + k
32 5 1 64 − ℓ + k
156 2 34 = 0 A (1 , , , , , ℓ + k ) ,A (1 , , , , , ℓ + k ) ,A (1 , , , , , ℓ − k ) ,A (1 , , , , , ℓ + k ) , a = a = a = a = a = 0 , a = 1 a = a = a = a = 0 , a = a = 1 a = a = a = a = a = 0 , a = − a = a = a = a = a = 0 , a = 1 (3 . Note that the choice of loop momentum to parameterize the cubic graphs plays an im-portant role due to the inherent asymmetry of the numerators with respect to the label1 (which must always be associated with V P ). The cases considered above are the oneswhich maximize the chances of failure. For instance, if we choose to position ℓ in the edgebetween 3 and 4 in the graphs depicted in a) in the previous figure the resulting triplet ofnumerators ℓ
432 1 6 5 − ℓ
43 6 2 15 − ℓ
612 3 45 is easily seen to satisfy the color-kinematics identity. In this case we get, N | , , , ( ℓ ) − N | , , , ( ℓ ) − N | , [6 , , ( ℓ ) = 0 . (3 . N | , , , ( ℓ ) = V T m , , , ℓ m − h V T , , + V T , , + V T , , (3 . − V T , , − V T , , − V T , , − V T , , + V T , , + V T , , + V T , , i N | , , , ( ℓ ) = V T m , , , ℓ m − h V T , , + V T , , + V T , , − V T , , − V T , , − V T , , − V T , , + V T , , + V T , , − V T , , i , from which we get N | , , , ( ℓ ) − N | , , , ( ℓ ) = − V T , , and (3.26) is satisfied since N | , [6 , , = V [6 , T , , = − V T , , .Thus we conclude that the field-theory limit of the genus-one six-point string correlator(1.3) for various color orderings as dictated by the ordering of vertex operators on theboundary of the Riemann surface satisfies all the color-kinematics identities.30 .3. Seven points At seven points, BCJ relations are analogously satisfied. Given their significantly morecomplex structure, we will not demonstrate these explicitly here and we will only outlinetheir construction below.As alluded to earlier, at seven points there is an extra complication that must bedealt with: the refined superfields. To find the field theory limits of the refined terms, wehave to use an alternative method and partially integrate the worldsheet functions againstthe Koba-Nielsen factor. This then means that, when we want to verify BCJ relations, wemust rearrange these refined terms. For relations in which the loop momentum structureis unchanged between terms (that is, BCJ relations in which there is always momentum ℓ going into leg 1), this amounts to canceling all ( ℓ · k ) terms. Take for instance the relation N | , , , , , ( ℓ ) − N | , , , , , ( ℓ ) − N | , , , , ( ℓ ) = 0 , (3 . V J | , , , within it. In the standard ordering correlator,these terms are associated with the worldsheet function Z | , , , , and we would thereforeexpect the heptagon numerator N | , , , , , ( ℓ ) to contain the terms − V J | , , , (cid:16) ℓ · k − k · k + 12 k · k (cid:17) . (3 . N | , , , , , ( ℓ ) ↔ − V J | , , , (cid:16) ℓ · k − k · k + 12 k · k (cid:17) (3 . N | , , , , ( ℓ ) ↔ . The relation (3.28) is clearly not satisfied with these values.Instead, we cancel the ℓ · k terms. For example, we rewrite (3.29) as − V J | , , , (cid:16)
12 ( ℓ − k ) −
12 ( ℓ − k ) + k · k − k · k + 12 k · k (cid:17) = − V J | , , , (cid:16)
12 ( ℓ − k ) −
12 ( ℓ − k ) + 12 k · k (cid:17) . (3 . ℓ − k ) terms with the denominator of the Feynman loop integrand I , , , , , , ( ℓ ) associated with this term, and so they contribute to hexagons instead. Hencethere is only one term of this form associated with the heptagon, N | , , , , , ( ℓ ) ↔ − s V J | , , , . (3 . N | , , , , , ( ℓ ) ↔ s V J | , , , . (3 . N | , , , , ( ℓ ) ↔ − s V J | , , , . (3 . s in its denominator compared with the heptagon. Now plugging(3.32), (3.33), (3.34) into the relation (3.28) we see it is now satisfied − s V J | , , , − s V J | , , , − (cid:16) − s V J | , , , (cid:17) = 0 . (3 . N | , , , , , ( ℓ ) − N a =11 | , , , , , ( ℓ ) − N [7 , | , , , , ( ℓ ) = 0 , (3 . N | , , , , , ( ℓ ) − N a =11 | , , , , , ( ℓ ) − N [1 , | , , , , ( ℓ ) = 0 ,N [7 , | , , , , ( ℓ ) − N a =1[7 , | , , , , ( ℓ ) − N [6 , [7 , | , , , = 0 ,N [6 , [7 , | , , , ( ℓ ) − N a =1[6 , [7 , | , , , ( ℓ ) − N [5 , [6 , [7 , | , , ( ℓ ) = 0 ,N a =1[1 , | , , , , ( ℓ ) + N [6 , | , , , , = 0 ,N [7 , | , , , , ( ℓ ) + N a =1[1 , | , , , , = 0 . Though this is not an exhaustive test, we hope that it is sufficient to serve as a proof ofconcept that this method work, and that it should always be possible to rearrange therefined terms to satisfy the color-kinematics duality.32 .4. Supergravity amplitudes and the double copy
One of the goals in obtaining a parameterization of gauge theory 1-loop integrands thatsatisfies the color-kinematics duality is to construct corresponding supergravity integrandsvia the double-copy construction [2]. For five points this construction was carried outexplicitly in four dimensions in [24] while the ten-dimensional analysis using pure spinorsuperspace was done in [1]. In the pure spinor superspace setup, the supergravity integrandobtained via the double copy must be checked to be BRST invariant, as that guaranteesgauge and supersymmetry invariance of its component expression in terms of polarizationsand momenta [5].We will now repeat the five-point supergravity construction of [1] to highlight thatit is BRST invariant but that it is so only because the numerators satisfy the dihedralsymmetries of the cubic graphs in the cohomology of pure spinor superspace (see [24]for a discussion of these symmetries). While at five points our numerators satisfy thesesymmetries in addition to the color Jacobi identities, the corresponding symmetries at sixpoints are not satisfied by our BCJ-satisfying six-point numerators and will prevent thedouble-copy construction of a BRST-closed supergravity integrand. Applying the double-copy procedure at six points will be left for a future work.
Let us construct the five-point supergravity integrand using the double-copy procedureto highlight the existence of a subtlety: the consistency of the double-copy constructionrequires the five-point numerators not only to satisfy the kinematic Jacobi identities butalso the dihedral symmetries of the cubic graphs. We will see that these symmetries, unlikethe kinematic Jacobi identities, are satisfied in the cohomology rather than identically.Starting with the color-dressed integrand (E.1) we replace the color factors by an extracopy of duality-satisfying kinematic numerators. This yields M ( ℓ ) = (cid:16) N | , , I , , , ˜ N | , , + 12 N | , , I , , , ˜ N | , , (3 . N | , , I , , , ˜ N | , , + 12 N | , , I , , , ˜ N | , , + 12 N | , , I , , , ˜ N | , , + N | , , , ( ℓ ) I , , , , ˜ N | , , , ( ℓ ) + perm(2 , , , (cid:17) Note that the kinematic numerators on the left are written in terms of Berends-Gielenumerators N of the appendix A while those on the right are the local numerators N .33fter setting up the double-copy supergravity integrand (3.37) we must check itsBRST variation. Since (3.37) is left/right symmetric it is enough to consider the left-moving BRST variation, which we will see vanishes only if the right-movers are in thecohomology of the right-moving pure spinor superspace. To see this surprising fact, considerthe variation of the left-moving pentagon N | , , , ( ℓ ) multiplied by the loop-momentumintegrand I , , , , : QN | , , , ( ℓ ) I , , , , = 12 V V T , , (cid:2) ( ℓ − k ) − ( ℓ − k ) (cid:3) I , , , , (3 . V V T , , (cid:2) ( ℓ − k ) − ( ℓ − k ) (cid:3) I , , , , + 12 V V T , , (cid:2) ( ℓ − k ) − ( ℓ − k ) (cid:3) I , , , , + 12 V V T , , (cid:2) ℓ − ( ℓ − k ) (cid:3) I , , , , = 12 V V T , , (cid:2) I , , , − I , , , (cid:3) + 12 V V T , , (cid:2) I , , , − I , , , (cid:3) + 12 V V T , , (cid:2) I , , , − I , , , (cid:3) + 12 V V T , , (cid:2) I , , , − I , , , (cid:3) where we used identities such as ( ℓ − k ) I , , , , = I , , , that follow from (2.6).These loop-momentum identities are trivial but one of them on the last line, namely ℓ I , , , , = I , , , , has a peculiar behavior: the right-hand side has no label 5. This seem-ingly innocuous fact will have a surprising implication in the double-copy construction ofthe five-point supergravity integrand when (3.38) appears multiplied by a right-movingfactor ˜ N | , , , ( ℓ ).The reason is that the right-moving pentagon ˜ N | , , , ( ℓ ) depends on the loop mo-mentum and picks up the shift ℓ → ℓ − k needed when rewriting I , , , → I , , , .More explicitly, one can show that the BRST variation of (3.37) contains QM ( ℓ ) = . . . + 12 V V T , , (cid:2) I , , , ˜ N | , , , ( ℓ ) + I , , , ( ˜ N | , , − ˜ N | , , , ( ℓ )) (cid:3) = . . . + 12 V V T , , I , , , (cid:2) ˜ N | , , + ˜ N | , , , ( ℓ − k ) − ˜ N | , , , ( ℓ ) (cid:3) . (3 . The left- or right-moving terminology refers to the two sides of the double-copy kinematicfactors, distinguished by the tildes. In the gauge-theory integrand the term V V T , , I , , , from the last line of (3.38) canbe trivially rewritten as V V T , , I , , , since its kinematic factor is invariant under the shift ℓ → ℓ − k .
34n the one hand we know from section 3 that the kinematic Jacobi identity˜ N | , , , ( ℓ ) − ˜ N | , , , ( ℓ ) + ˜ N | , , = 0 . (3 . . Therefore the vanishing of the left-moving BRST variation (3.39) hinges onthe dihedral symmetry of the pentagon ˜ N | , , , ( ℓ − k ) = ˜ N | , , , ( ℓ ). One can show thatthis symmetry is satisfied in the cohomology of the right-moving pure spinor superspacegiven by the pure spinor bracket h ˜ N | , , , ( ℓ − k ) i = h ˜ N | , , , ( ℓ ) i , (3 . h ˜ N | , , , ( ℓ ) − ˜ N | , , , ( ℓ − k ) i is given by h ˜ V ˜ T m , , , k m + ˜ V ˜ T , , + (cid:2) ˜ V ˜ T , , + (2 ↔ , , (cid:3) i = 0 , (3 . (which are satisfied in the cohomology of the right-movers). At six points a naive application of the double-copy procedure with BCJ-satisfying nu-merators obtained in the previous sections does not produce a consistent supergravityintegrand: it fails to be BRST invariant in pure spinor superspace. This happens becausethe numerators, even though they satisfy the color-kinematics duality they do not satisfythe automorphism symmetries of their associated cubic graphs.To see this it is enough to use the BCJ-satisfying six-point numerators in a tentativedouble-copy construction to obtain, among many others, the following terms under a left-moving BRST variation QM ( ℓ ), − s V V T , , (cid:16) I , , , ˜ N | , , , ( ℓ ) − I , , , ˜ N | , , , ( ℓ ) − I , , , ˜ N [1 , | , , (cid:17) = − s V V T , , I , , , (cid:16) ˜ N | , , , ( ℓ ) − ˜ N | , , , ( ℓ − k ) + ˜ N | , , (cid:17) . (3 . These numerators are readily available to download from [28]. At tree level for the double copy construction of supergravity amplitudes to be BRST invari-ant it is enough for the numerators to satisfy the kinematic Jacobi identities I , , , arise from loop-momentum cancellations in QN | , , , ( ℓ ) I , , , , . This is compensated by the shift ℓ → ℓ − k which is picked up by the right-moving pentagon in the second line. If the condition˜ N | , , , ( ℓ − k ) = ˜ N | , , , ( ℓ ) for the automorphism symmetry of the pentagon wassatisfied then the terms (3.43) would vanish identically since˜ N | , , , ( ℓ ) − ˜ N | , , , ( ℓ ) + ˜ N | , , = 0 , (3 . N | , , , ( ℓ − k ) = ˜ N | , , , ( ℓ ) and, unlike the case at five points, thisis not true even in the cohomology , h ˜ N | , , , ( ℓ − k ) i 6 = h ˜ N | , , , ( ℓ ) i . (3 . ℓ + a i k i are satisfied by the numerators from the integrandswith shifted loop momentum A ( σ ; ℓ + a i k i ). In the case of (3.45) we have the identity (validat the superfield level) N a = − ,a = − | , , , ( ℓ ) = N | , , , ( ℓ ) , (3 . A (1 , , , , , ℓ − k ). This integrand is computed with thefield-theory limits of section 2.3 with a = a = − ℓ − k − k . Unfortunately it is not clear how to use these numerators directlyas functions of ℓ rather than as functions of the shift parameters a i . Note that the last line of (3.43) is identical (apart from the left/right-moving nature of thenumerators) to the BCJ-triplet failure in the representation of [1], given in equation (6.12) ofthat reference. Unlike the representation of [1], the six-point integrand of gauge theory found heresatisfies all BCJ relations for the left- and right-moving numerators. However, once terms in theleft-moving BRST variation are collected we see that the BCJ failure of [1] in the left-movingsector appears here as a failure in the right-moving sector due to a shift of the loop momentum. .4.3. Comments on the double-copy construction in pure spinor superspace The failure of the automorphism symmetry (3.45) for the 23-pentagon is a contact termin s after its component expansion is evaluated through the pure spinor bracket, that is h ˜ N | , , , ( ℓ − k ) i − ˜ N | , , , ( ℓ ) i ∼ s ( . . . ). In pure spinor superspace we have N | , , , ( ℓ ) − N | , , , ( ℓ − k ) = k m V T m , , , + V T , , + (cid:2) V T , , + 4 ↔ , (cid:3) (3 . and it will be interestingto see how BRST invariance is restored. It is reasonable to speculate that the deformationsof the right-moving BCJ triplets by contact terms as a result of loop momentum shifts dueto canceled loop propagators in the left-moving BRST variation may be a generic feature ofthe double copy in pure spinor superspace. This characteristic may be especially importantat higher loops. We plan to investigate this problem in future work.We note that supergravity integrands have been constructed using BCJ numeratorsin four dimensions for up to seven points in [39] and to all multiplicity in [20] using spinorhelicity in the MHV sector. Supergravity amplitudes were also constructed in [40] but usinga partial-fraction representation of the loop momentum integrands.
4. Conclusion
In this work we obtained a set of field-theory limit rules for the Kronecker-Eisensteincoefficient functions present in the genus-one superstring correlators derived in [9,10,11].Using these rules we found local numerators for ten-dimensional SYM integrands at oneloop for five, six and seven points that satisfy the BCJ color-kinematics duality. Theseresults resolve the difficulties in an earlier analysis of the six-point SYM integrands whichdid not satisfy the color-kinematics duality [1]. We thank Oliver Schlotterer for discussions on this point.
Acknowledgements:
We thank Oliver Schlotterer for discussions and helpful commentson the draft. EB thanks Kostas Skenderis for useful discussions. CRM thanks Oliver Schlot-terer for collaboration on closely related topics. CRM is supported by a University ResearchFellowship from the Royal Society.
Appendix A. Conventions
In this appendix we briefly summarize some of the conventions used in the main text.Sums over deconcatenations are denoted by P A ...A n = a ...a m They represent the sumover all possible ways of generating n words from a ...a m , while maintaining the order.These words may be empty, but often when they are the terms being summed over will bezero. So, to give an example, the sum P ABC =12 denotes the sum over six cases; three ofthem are where two of A , B and C are empty and the third is 12, and the other three arewhere A = 1 , B = 2 , C = ∅ , A = 1 , B = ∅ , C = 2, and A = ∅ , B = 1 , C = 2.38nother notation commonly used is( terms ) + ( a , ..., a m | N , ..., N n ) , m ≤ n. (A.1)This notation works means a sum over all possible ways of replacing a , ..., a n in theterms with n terms from the ordered list N , ..., N n . Further generalizations of this fol-low naturally, with (terms) + ( a , ..., a m | b , ..., b m | N , ..., N n ) meaning sum over all waysof generating two ordered lists from N , ..., N n , one of length m , one of length m ,and substituting them in for a , ..., a m and b , ..., b m . For example, in V [1 , T [4 , , , +(23 | , | , , , ,
8) possible terms are V [1 , T [23 , , , and V [1 , T [7 , , , , but not V [1 , T [8 , , , as the latter would violate the ordering constraint.Another summation notation to note is( terms ) + [1 ...n | A , ..., A m ] , m ≤ n. (A.2)This denotes the sum over A , ..., A m all possible Stirling cycle permutations constructedfrom 1 , ..., n [11]. This means that you take the set of numbers 1, ..., n , and construct allpossible permutation cycles from it, select those involving m brackets, and canonicaliseby having the first term in each cycle be its lowest element, and the cycles ordered bytheir lowest elements. Each cycle is then substituted in for an A. For example, considerthe sum +[1234567 | A , ..., A ]. One possible permutation of 1 , . . . , A = 12, A = 3, A = 46, A = 57. These sums may bethought of as being A = 1 followed by any terms from 2 ...n in any order, then A is thenext lowest value left followed by any possible set of values in any order from the numbersleft, and so on. So in the above example, A = 15 would be a possible term, which wouldmean A starting with a 2 and so it could be A = 23, then A starts with a 4 and so wecould have A = 4, and then finally A follows the same rules and uses up all remainingletters, so A = 67. 39 .0.1. Lie algebra notation and Berends-Giele currents We frequently use the notation of words and Lie brackets, especially when indexing SYMmultiparticle superfields, see the discussion on section 3 of [9]. In any situation where aLie bracket would be expected but a word A is found instead, this should be regarded asbeing the left-to-right Dynkin bracket ℓ ( A ) [41], ℓ ( a ...a n ) ≡ [[ ... [[ a , a ] , a ] ... ] , a n ] . (A.3)For example, [[[1 , , , , [2 , , [4 , , [[6 , , b ( i ) = i, b ( P ) = 12 s P X XY = P [ b ( X ) , b ( Y )] . (A.4)For example, b (12) = s [1 , b (123) = s s [[1 , ,
3] + s s [1 , [2 , V , T , J , and N . The composition of the first three of these objects can befound in more detail in [3,22]. The fourth will be used to refer to amplitude numeratorsand are detailed on a case by case basis. These objects have a number of slots for indiceslabelling their superfield contents, and all such indices will be Lie brackets. The secondclass of objects are Berends-Giele (BG) currents. These are related to the local objectspreviously described through the use of the b-map on each of their blocks of indices. TheBG current of particular use to us is denoted by N , defined in terms of local objects N as N A | A ,...,A m ( ℓ ) ≡ N ( m ) b ( A ) | b ( A ) ,...,b ( A m ) ( ℓ ) (A.5)For example, a seven-point box Berends-Giele numerator is expanded as N | , , ( ℓ ) = N b (1) | b (23) ,b (456) ,b (7) ( ℓ )= 1 s s (cid:18) s N | [2 , , [[4 , , , ( ℓ ) + 1 s N | [2 , , [4 , [5 , , ( ℓ ) (cid:19) . It should be noted that generalized Mandelstam invariants are defined with a factor, s i ...i n ≡
12 ( k mi + ... + k mi n ) = X ≤ a
1. Hence this difference becomes b I ( p ) ij − b II ( p ) ij = p X m =0 B m m !( p − m )! (cid:16) a p − mj − ( − m ( a j + 1) p − m (cid:17) (B.3)This can be verified to vanish on a case by case basis with relative ease. Taking for instancethe p = 3 case, we have b I (3) ij − b II (3) ij = B (cid:16) a j − ( − ( a j + 1) (cid:17) + B (cid:16) a j − ( − ( a j + 1) (cid:17) + B (cid:16) a j − ( − ( a j + 1) (cid:17) + B (cid:16) a j − ( − ( a j + 1) (cid:17) = 16 (cid:16) a j − a j − a j − a j − (cid:17) + 14 (cid:16) a j + a j + 2 a j + 1 (cid:17) + 112 (cid:16) a j − a j − (cid:17) + 0 = 0 (B.4)With the aid of FORM [43] we have verified that this vanishes in at least the first 700cases. Similar will hold if we instead take j = 1. Hence, the b part of the field theory limitsmatches in both representations. 41hen, we move onto the c piece. This difference is given by c I ( p ) ij − c II ( p ) ij = 12( p − (cid:16)(cid:0) a ji + sgn ...nij dist ...n ( i, j ) (cid:1) p − (B.5) − (cid:0) a ji − δ j + δ i + sgn ...n ij dist ...n ( i, j ) (cid:1) p − (cid:17) Again, we need only consider the cases where one of i and j is 1. If we take i = 1 we get c I ( p ) ij − c II ( p ) ij = (cid:0) a j + dist ...n (1 , j ) (cid:1) p − − (cid:0) a j + 1 − dist ...n (1 , j ) (cid:1) p − p − a j + dist ...n (1 , j )) p − = (cid:26) a p − j j ≤ a j + 1) p − j > , (B.7)( a j + 1 − dist ...n (1 , j )) p − = (cid:26) a p − j j ≤ n − a j + 1) p − j > n − . When n = 4 ,
5, the only Kronecker-Eisenstein functions in amplitudes is g (1) ij , and we seethat setting p = 1 in the above gives equivalence. When n = 6, these coincide in that n − n = 7 and p >
1, they differ when j = 5. However, this disagreementwill not matter. At 7 points a term g (2+)15 is multiplied by at most one other g ( q ) ij function,but we need at least two Kronecker-Eisenstein coefficient functions in order to make thecorresponding P function non-zero. That is, for example, g (2)15 g (1)56 ⇒ P (15 ,
56) = φ | I = 0 , (B.8) g (2)15 g (1)56 g (1)67 ⇒ P (15 , ,
67) = φ | I = 0 . At 8 points, this will of course become an issue. However, the description of the dist function was chosen purely for simplicity. If we instead think of this function as askingwhether the pole being approached crosses the boundary between particles n and 1, thenconsistency should be maintained to higher points. Appendix C. The field-theory limit at higher points
We anticipate that the field theory limit rules for an arbitrary product of g ( n ) ij functionsshould generalize in the natural way n Y a =1 g ( p a ) i a j a → X A ∈P (12 ...n ) (cid:16) Y a ∈ A b ( p a ) i a j a (cid:17)(cid:16) Y b ∈ A c c ( p b ) i b j b (cid:17) P ( i B j B , ..., i B | B | j B | B | ) (cid:17)! (C.1)42here P (12 ...n ) denotes the power set of 12 ...n , A is an element of this, and A c its com-plement. We stress that the indices of the c ( p ) and those in the P function are identical.The general P functions will be as in (2.13), with P ( i j , ..., i n j n ) chaining together i m j m pairs as much as possible, and then using these as indices for φ and I functions. So,for instance, we would expect P (12 , , , , , ↔ ˆ φ ( σ | I (C.2) P (15 , , , ↔ ˆ φ ( σ | φ ( σ | I , As for the limits of b ( p ) and c ( p ) at higher points, these we expect will generalize from(2.10) in the natural way. As evidence of this, we look to the Fay identity for g ( n )12 g (1)23 g ( n )12 g (1)23 = − g ( n +1)13 + g (1)13 g ( n )12 − ng ( n +1)12 + n X j =0 ( − j g ( n − j )13 g (1+ j )23 . (C.3)We begin by looking at b ( n ) , and restrict ourselves to the case a i = 0 ∀ i initially. In thesecircumstances we know that b (1) ij = sgn ...nij , and we would expect the general order b ( n ) ij to depend only upon the order of i and j with respect to the color ordering. Hence, wesubstitute into (C.3) the values g (1)13 , g (1)23 → , g ( n )12 , g ( n )13 , g ( n )23 → b ( n ) . (C.4)Upon rearranging this gives us the recursion relation b ( n +1) = − n + 1 − ( − n n X j =1 ( − j b ( n − j +1) b ( j ) . (C.5)This can be seen to vanish for n even, n >
0, by virtue of the symmetry in the gg termsand the antisymmetry of the ( − j . For n odd, it simplifies to b (2 n ) = − n + 1 n − X j =1 ( − j b (2 n − j ) b ( j ) = − n + 1 n − X j =1 b (2 n − j ) b (2 j ) , (C.6)where the second equality follows from the vanishing of the b with odd indices. It may thenbe proved by induction that this is solved by b ( n ) = B n n ! , (C.7)43here B n is the n th Bernoulli number. Showing this requires an identity due to Euler [44], n − X k =1 (cid:18) n k (cid:19) B k B n − k = − (2 n + 1) B n , n ≥ . (C.8)Hence, we speculate that when a i = 0 ∀ i , the field theory limit of a general term fromthe Kronecker-Eisenstein series away from poles is given by (C.7). The first few (non-zero)values are b (0) = 1 , b (1) = 12 , b (2) = 112 , b (4) = − , b (6) = 130240 , (C.9) b (8) = − , b (10) = 147900160 , b (12) = − . We can then extend this to the general a i case, though with less elegance. If instead ofmaking the substitution (C.4) into (C.3), we instead use the general a i values of the b (1) terms, we find the relation (cid:18)
12 + a − a (cid:19) b ( n )12 = − b ( n +1)13 + (cid:18)
12 + a − a (cid:19) b ( n )12 − nb ( n +1)12 (C.10)+ (cid:18)
12 + a − a (cid:19) b ( n )13 + n X j =1 ( − j b ( n − j )13 b (1+ j )23 . This cannot be as easily rearranged into a recursion relation. However, if we assume that b ( n ) ij is a polynomial in a j − a i up to order n , we may use the above to identify the polynomialcoefficients. Doing this reveals the value of b (4) ij as would be expected from (2.10) as theunique solution. And then we have verified that the relation above is satisfied in a numberof further cases if we assume this general form of b ( n ) .We can perform a similar exercise for the c ( n ) ij pole terms. In its current form (C.3) isnot the most useful for this, as we would like the dist functions to be non-zero. Instead,we suppose the amplitude we are considering is A (1 , , ..., m ) for convenience, and look atan alternative Fay identity, g ( n )1 m g (1) m ( m − = − g ( n +1)1( m − + g (1)1( m − g ( n )1 m − ng ( n +1)1 m + n X j =0 ( − j g ( n − j )1( m − g (1+ j ) m ( m − . (C.11)We need not restrict ourselves to the a i = 0 ∀ i case here, as the computation is simpler.Looking at the s m single poles leads us to the relation c ( n )1 m (cid:18) −
12 + a m − − a m (cid:19) = (cid:18)
12 + a m − − a (cid:19) c ( n )1 m − nc ( n +1)1 m ⇒ c ( n +1)1 m = 1 n c ( n )1 m (1 + a m − a ) (C.12)44sing that we know c (1)1 m = , this becomes c ( n )1 m = 12( n − a m − a ) n − (C.13)This agrees with the known values of c (2)17 and c (3)17 also. We can also repeat this calculationfor poles of g ( n )12 to see what would happen if the dist function were not triggered, and findthe similar relation c ( n )12 = 12( n − a − a ) n − (C.14)Hence the form of c ( n ) ij presented in (2.10) is the natural generalization, and we expect(2.10) to hold to higher points.We end this discussion though by stressing that this approach is highly speculative,and we have not tested these values produced in any way beyond the aforementioneddiscussion. They are however a strong candidate for what they are attempting to describe. Appendix D. The BRST analysis of a seven-point numerator
In this appendix we identify the full expression for the [5 , [6 , A (1 , , , , , , ℓ + 4 k − k ), and confirm that its variation has the desired form. Webegin by finding the coefficient of one term contributing to the numerator in detail, namely V T , , . Within the string correlator this is associated with the worldsheet function Z , , , = g (1)25 g (1)57 g (1)76 + g (3)25 + g (3)57 + g (3)76 − g (3)62 + g (1)25 ( g (2)57 + g (2)76 − g (2)62 )+ g (1)57 ( g (2)25 + g (2)76 − g (2)62 ) + g (1)76 ( g (2)25 + g (2)57 − g (2)62 ) . (D.1)Only two of these terms contain the s s pole structure, g (1)25 g (1)57 g (1)76 and g (1)76 g (2)57 . Thecontribution of the former was identified in (2.23), and the latter follows from (2.11), c (1)76 c (2)57 = 12 ·
62 = 32 . (D.2)Summing these together, the V T , , contribution to the [5 , [6 , (cid:18) −
118 + 32 (cid:19) V T , , ˆ φ (1234567 | I = − s s V T , , I , , , , (D.3)45imilar calculations for all other terms in the correlator yield the numerator expression N a =4 ,a = − | , , , [5 , [6 , ( ℓ ) = 6 V T mn , , , , k m k n + V T m , , , [5 , ( ℓ m − k m + 6 k m ) − (cid:0) ( V T m , , , k m + (2 ↔ , V T m , , , k m + (5 ↔ [6 , (cid:1) + 12 (cid:0) V T , , [5 , + (2 ↔ , , [5 , (cid:1) + 12 (cid:0) V T , , [5 , + (2 , | , , , [5 , (cid:1) (D.4)+ 6 (cid:0) ( V T , [3 , , + (2 , | , , ↔ (cid:1) + 6 (cid:0) ( V T [2 , , , + (2 ↔ , ↔ [6 , (cid:1) + 6 (( V T , , + (2 ↔ , − (6 ↔ V T , , − (6 ↔ (cid:0) V T , , [5 , + (2 ↔ (cid:1) + 4 V T , , [5 , − V T , , [4 , [5 , + 6 V J m | , , , , ( k m − k m )+ 6 s (cid:0) ( V J | , , , + (2 ↔ , , V J | , , , , − (6 ↔ (cid:1) The
V J terms above are those which arise naively by looking to the s s poles in thecorrelator. As discussed previously it may be that they require some rearrangement to bein a BCJ representation, but for illustrating the field theory limit methods we give thenumerator in the above form. A lengthy calculation yields the variation QN a =4 ,a = − | , , , [5 , [6 , ( ℓ ) = 12 V V T , , [5 , (cid:0) ( ℓ − k + 4 k − k ) − ( ℓ − k + 4 k − k ) (cid:1) + 12 V V T , , [5 , (cid:0) ( ℓ − k + 4 k − k ) − ( ℓ − k + 4 k − k ) (cid:1) + 12 V V T , , [5 , (cid:0) ( ℓ − k + 4 k − k ) − ( ℓ − k + 4 k − k ) (cid:1) + 12 V V [5 , T , , (cid:0) ( ℓ − k + 4 k − k ) − ( ℓ − k + 4 k − k ) (cid:1) +( k · k ) (cid:18)(cid:0) V V T m , , , k m + (2 ↔ , , (cid:1) + V V T m , , , (cid:0) ℓ m + 6 k m (cid:1) + 6 V V T mn , , , , k m k n + 6 k m ( V V T m , , , + (2 ↔ , , (cid:0) ℓ m + 6 k m (cid:1) V V T m , , , + 6( V V T m , , , k m + (2 ↔ , V V T m , , , k m + 6 V V T m , , , k m + 6 V V T mn , , , , k m k n + V V T m , , , k m + (6 V V T m , , , k m + (2 ↔ , V V T m , , , k m + (2 ↔ , , V V [2 , T , , + (2 ↔ ,
12 ( V V T , , + (2 ↔ , −
12 ( V V T , , + (2 , | , , , V V T [2 , , , + (2 , | , , , V V T , , + (2 ↔ , V V T , , + (2 ↔ , , − V V T , , + 12 V V T , , + 6 (cid:0) ( V V T [3 , , , + (3 ↔ , ↔ , (cid:1) + 6( V V T [26 , , , + (2 ↔ , V V T , , + V V T , , + (2 , | , , V V T , , + (2 ↔ , V V T , , + (2 ↔ , V V T , , + (2 ↔ , (cid:0) ( V V T , , + (3 ↔ , V V T , , + (2 ↔ , (cid:1) + 6 V V T , , + 6( V ( V + V ) T , , + (2 ↔ V V T , , + 4( V V T , , + (2 ↔ , V V T , , + 4( V V T , , + (2 ↔ , V V T , , + 4 V V T , , + 2 V V T , , + 20 V V T , , + 6 V Y m , , , , , k m + 6( V Y , , , , + (2 ↔ , , , V Y , , , , − (6 ↔ (cid:19) +( k · k ) (cid:18)(cid:16)
12 ( V V [2 , T , , + V V T , , + (2 ↔ , V V T , , + 4 V V T , , + 12 ( V V T , , + (2 , | , , , V V T , , + (2 ↔ , − (5 ↔ (cid:17) −
12 ( V V T , , + (5 ↔ , (cid:0) ( V V T , , − (25 ↔ ↔ , (cid:1) − V V T , , − V Y , , , , (cid:19) +6( k · k )( k · k ) V V ( J | , , , + J | , , , ) − k · k )( k · k ) V V J | , , , This has intentionally been expressed with factors ( ℓ · k ) reformulated in terms of propa-gators. For an n -point amplitude in the canonical ordering with arbitrary loop momentumstructure, this is done with( ℓ · k i ( i +1) ...j ) = −
12 ( ℓ + n X m =1 a m k m − k ....j ) + 12 ( ℓ + n X m =1 a m k m − k .... ( i − ) − k i ( i +1) ...j · n X m =1 a m k m − k i ( i +1) ...j ! . (D.6)We may then be reassured of the validity of this numerator expression, as those terms inthe variation proportional to propagators cancel terms from other box numerators. For47xample, one such set of terms is V V T , , [5 , (cid:0) ( ℓ − k + 4 k − k ) − ( ℓ − k + 4 k − k ) (cid:1) I a =4 ,a = − , , , , = V V T , , [5 , (cid:16) I a =4 ,a = − , , , − I a =4 ,a = − , , , (cid:17) (D.7)This then cancels one term in the variation of the [3 , , [5 , [6 , , , [5 , [6 , Appendix E. The five-point color-dressed integrand
In this appendix the five-point color-dressed integrand will be written down after theapplication of the color decomposition techniques of [45].The five-point color-dressed one-loop integrand can be written as M ( ℓ ) = (cid:16) N | , , I , , , B , , , + 12 N | , , I , , , B , , , (E.1)+ 12 N | , , I , , , B , , , + 12 N | , , I , , , B , , , + 12 N | , , I , , , B , , , + N | , , , ( ℓ ) I , , , , P , , , , + perm(2 , , , (cid:17) where N denotes the Berends-Giele counterpart of the n -gon numerator as described inthe appendix A while the color factors of the box and pentagon cubic graphs are B , , , = f a f eab f b c f c d f d e , P , , , , = f a b f b c f c d f d e f e a . (E.2)The factor of in (E.1) compensates the overcounting of graphs due to symmetries. Notethat the box numerators do not depend on the loop momentum.The color-dressed integrand (E.1) is BRST closed. To see this we expand all colorfactors in terms of their pentagon constituents using the Jacobi identity as B , , , = P , , , , − P , , , , [45] and consider the terms proportional to P , , , , . Using the five-point numerators of section 2.4.3 these are M ( ℓ ) (cid:12)(cid:12)(cid:12) P , , , , = N | , , , ( ℓ ) I , , , , + 12 (cid:16) N | , , I , , , − N | , , I , , , (E.3)+ (cid:2) N | , , − N | , , (cid:3) I , , , + (cid:2) N | , , − N | , , (cid:3) I , , , + (cid:2) N | , , − N | , , (cid:3) I , , , + N | , , I , , , − N | , , I , , , (cid:17) . N ij | k,l,m = − N ji | k,l,m by (2.31) and performing the loop momentum shifts ℓ ′ = ℓ − k in I , , , and ℓ ′ = ℓ + k in I , , , these terms become the integrand A (1 , , , , ℓ )of (2.30), M ( ℓ ) (cid:12)(cid:12)(cid:12) P , , , , = N | , , , ( ℓ ) I , , , , + N | , , I , , , + N | , , I , , , + N | , , I , , , + N | , , I , , , + N | , , I , , , . (E.4)Hence, after considering all the permutations the color-dressed integrand (E.1) becomes M ( ℓ ) = A (1 , , , , ℓ ) P , , , , + perm(2 , , ,
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