Berends-Giele recursion for double-color-ordered amplitudes

Tree-level double-color-ordered amplitudes are computed using Berends-Giele recursion relations applied to the bi-adjoint cubic scalar theory. The standard notion of Berends-Giele currents is generalized to double-currents and their recursions are derived from a perturbiner expansion of linearized fields that solve the non-linear field equations. Two applications are given. Firstly, we prove that the entries of the inverse KLT matrix are equal to Berends-Giele double-currents (and are therefore easy to compute). And secondly, a simple formula to generate tree-level BCJ-satisfying numerators for arbitrary multiplicity is proposed by evaluating the field-theory limit of tree-level string amplitudes for various color orderings using double-color-ordered amplitudes.


Introduction
As discussed in [1], the bi-adjoint cubic scalar theory with the Lagrangian 1 gives rise to double-color-ordered tree amplitudes m(A|B), Tr(t a 1 t a 2 . . . t an )Tr(t b 1t b 2 . . .t bn )m(a 1 , . . . , a n |b 1 , . . . , b n ), (1.2) and a diagrammatic algorithm to compute them was described. It was also demonstrated that these double-color-ordered amplitudes are related to the entries of the field-theory inverse KLT matrix [2][3][4] as well as the field-theory limit of string tree-level integrals [5][6][7]; thus providing an alternative method for their calculation which does not involve inverting a matrix nor evaluating any integrals [6].

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The algorithm to compute m(A|B) described in [1] involves drawing polygons and collecting the products of propagators associated to cubic graphs which are compatible with both color orderings. Their overall sign, however, requires keeping track of the polygons orientation in a process that can be challenging to automate. The connection of these double-color-ordered amplitudes with the Cachazo-He-Yuan approach [8] led to other recent proposals for their evaluation [9][10][11] (see also [12]).
Given the importance of the double-color-ordered tree amplitudes for the evaluation of the field-theory limit of string disk integrals, a fully recursive and algebraic algorithm to compute them will be given in this paper. This will be done using the perturbiner approach of [13] (recently emphasized in [14]) to derive recursion relations for Berends-Giele double-currents from a solution to the non-linear field equation of the action (1.1). The double-color-ordered tree amplitudes are then computed in the same manner as in the Berends-Giele recursive method [15].
Two immediate applications of this new method are given. In section 4, the relation between the inverse field-theory KLT matrix and double-color-ordered amplitudes observed in [1] is shown to greatly simplify when the amplitudes are written in terms of Berends-Giele double-currents. And in section 6, the efficient evaluation of the field-theory limit of string tree-level integrals for various color orderings will lead to a closed formula for BCJsatisfying tree-level numerators [16] at arbitrary multiplicity, tremendously simplifying the case-by-case analysis of [17].

On notation
Multiparticle labels correspond to words in the alphabet {1, 2, 3, 4, . . .} and are denoted by capital letters (e.g., A = 1243) while single-particle labels are represented by lower case letters (e.g., i = 4). A word of length |P | is given by P ≡ p 1 p 2 . . . p |P| while its transpose isP = p |P| p |P|−1 . . . p 2 p 1 . The notation XY =P means a sum over all possible ways to deconcatenate the word P in two non-empty words X and Y . For example, The shuffle product ¡ between two words A and B is defined recursively by [18] and ∅ denotes the empty word. To lighten the notation and avoid summation symbols, labeled objects are considered to be linear in words; e.g., T 1¡23 = T 123 +T 213 +T 231 . Finally, the Mandelstam invariants are defined by

Review of Berends-Giele recursions for Yang-Mills theory
In this section we derive the Berends-Giele currents for Yang-Mills theory [15] from a solution to the non-linear field equations. This approach has been recently emphasized in [14] and resembles the perturbiner formalism of [13]. The same procedure will be applied in the next section to the bi-adjoint cubic scalar theory (1.1).

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The Lagrangian of Yang-Mills theory is given by t a is a Lie algebra-valued field with t a the generators of a Lie group satisfying [t a , t b ] = f abc t c . The non-linear field equation [∇ m , F mn ] = 0 following from (2.1) can be rewritten in the Lorenz gauge ∂ m A m = 0 as To find a solution to the equation (2.2) one writes an ansatz of the form [14,19] where the sum is over all words P restricted to permutations. One can check using a planewave expansion A m P (x) = A m P e k P ·x that the ansatz (2.3) yields the following recursion, and A m i with a single-particle label satisfies the linearized field equation A m i = 0. It can be shown [20] that the recursion (2.4) is equivalent to the recursive definition for the Berends-Giele current J m P derived in [15] using Feynman rules for the cubic and quartic vertices of the Lagrangian (2.1). Note however that (2.4) contains only "cubic" vertices; the quartic interactions are naturally absorbed by the non-linear terms of the field-strength. This is conceptually simpler than previous attempts for absorbing those quartic terms [21].
One can also show using either group-theory methods [22] or combinatorics of words [14] that the currents A m P satisfy which guarantees [23] that the ansatz (2.3) is a Lie algebra-valued field (the equivalence between the two statements in (2.5) follows from the theorems proved in [23] and [24]). Finally, the color-ordered tree-level n-point amplitude is given by [15] A YM (1, 2, . . . , n) = s 12..

Berends-Giele recursions for the bi-adjoint cubic scalar theory
In this section we derive recursion relations for Berends-Giele double-currents using a perturbiner expansion for the solution of the non-linear field equations obtained from the bi-adjoint cubic scalar Lagrangian. These double-currents will then be used to compute the tree-level double-color-ordered amplitudes.

Berends-Giele double-currents
The field equation following from the Lagrangian (1.1) can be written as Following [14,19], a solution to the field equation (3.1) can be constructed perturbatively in terms of Berends-Giele double-currents φ P |Q with the ansatz, Since the ansatz (3.2) contains the plane-wave factor e k P ·x (as opposed to e k Q ·x ), in order to have a well-defined multiparticle interpretation φ P |Q must vanish unless P is a permutation of Q, i.e. φ P |Q ≡ 0 if P \ Q = ∅. Plugging the ansatz (3.2) into the field equation (3.1) leads to the following recursion where s P is the multiparticle Mandelstam invariant (1.4) and the single-particle double-

) is antisymmetric in both [XY ] and [AB]
, the combinatorial proof of the Berends-Giele symmetry (2.5) given in the appendix of [14] also applies to both words in the double-currents φ P |Q , and, in particular, φ Ai|Q = (−1) |A| φ iÃ|Q (with similar expressions for the symmetries w.r.t. the word Q in φ P |Q ). The symmetries (3.4) generalize the standard Berends-Giele symmetry (2.5) to both sets of color generators and guarantee that the ansatz (3.2) is a (double) Lie series [23], thereby preserving the Lie algebra-valued nature of Φ(x) in (3.1). Using φ i|j = δ ij a few example applications of the recursion (3.3) are given by as well as In the appendix B, the Berends-Giele double-current φ P |Q is given an alternative representation in terms of planar binary trees and products of epsilon tensors.

Double-color-ordered amplitudes from Berends-Giele double-currents
Without loss of generality, one can use that m(R|S) is cyclically symmetric in both words R and S to rewrite an arbitrary n-point amplitude as m(P, n|Q, n), where |P | = |Q| = n − 1. Therefore, a straightforward generalization of the gluonic amplitude (2.6) using the Berends-Giele double-currents yields a formula for the double-color-ordered amplitudes 3 (recall that φ n|n = 1), It is easy to see using the symmetries (3.4) obeyed by the double-currents that the Kleiss-Kuijf relations are satisfied independently by both sets of color orderings. Since the doublecurrents φ P |Q obey the recursion relation (3.3), the computation of double-color-ordered amplitudes is easy to automate and their overall sign requires no additional bookkeeping. 4

The field-theory KLT matrix and its inverse
In this section we demonstrate that the entries of the inverse field-theory KLT matrix [2,3] (also called the momentum kernel matrix [4]) are equal to the Berends-Giele double currents and therefore are easy to compute. This computational simplicity is important because, apart from applications related to gauge and gravity amplitudes, the field-theory KLT matrix and its inverse relate [7] the local and non-local versions of multiparticle super Yang-Mills superfields 5 with manifold applications in recent developments within the pure spinor formalism applied to the computation of scattering amplitudes in both field-and string theory [5,[31][32][33].

The field-theory KLT matrix
The symmetric matrix S[P |Q] defined by gives rise to the KLT matrix S[A|B] i when the first letters on both words coincide The convention for the sign of the Mandelstam invariants here is such that m here (P, n|Q, n) = (−1) |P | m there (P, n|Q, n) in comparison with the normalization of [1]. 4 An implementation using FORM [28,29] is attached to the arXiv submission. 5 The relations (4.1) apply for all types of SYM superfields (Aα, Am, W α , . . .) [30]. The restriction to VP in (4.1) was chosen for simplicity.

The inverse KLT matrix
The inverse KLT matrix S −1 [A|B] i can be computed from the entries (4.3) using standard matrix algebra. However, this task quickly becomes tedious in practice and the direct outcome of the matrix inversion usually requires further manipulations to be simplified.
To see this one uses the Berends-Giele amplitude formula (3.7) in (4.4) to obtain In the first line the label (n − 1) has been moved to the front using (3.4) and in the second line the condition φ P |Q = 0 unless P is a permutation of Q implies that s iA(n−1) φ (n−1)Ãi|(n−1)iB = φ (n−1)|(n−1) φÃ i|iB . For example, which agrees with the results of [7]. Higher-multiplicity examples follow similarly. Using the Berends-Giele representation of the inverse KLT matrix (4.5), the first relation in (4.4) simplifies to and therefore provides an efficient algebraic alternative to the diagrammatic method to compute M P described in the appendix of [30].

JHEP07(2016)080 5 The field-theory limit of tree-level string integrals
The n-point open-string amplitude computed using pure spinor methods in [5] can be written in terms of (local) multiparticle vertex operators V P [30] as where the deconcatenation in XY includes empty words and Z Σ (N ) is given by [7], The factor 1/vol(SL(2, R)) compensates the overcounting due to the conformal Killing group of the disk 7 and the region of integration Σ is such that z σ i < z σ i+1 for all i = 1 to i = |M| − 1. The pure spinor bracket . . . is defined in [34] but will play no role in the subsequent discussion.
As pointed out in [1], the field-theory limit of the string disk integrals (5.2) is given by the double-color-ordered amplitudes, which agree with (C.1) and (C.5) of [7]. Higher-multiplicity examples follow similarly. So the SYM tree amplitudes with color ordering Σ obtained from the field-theory limit of the string amplitude (5.1) are given by It was shown in [17] that a set of BCJ-satisfying numerators for SYM tree amplitudes can always be obtained from the field-theory limit of the string tree-level amplitude (5.1), and explicit expressions for numerators up to 7-points were given in that reference. Since the Berends-Giele algorithm to evaluate the double-color-ordered amplitudes is easy to automate, one can quickly obtain higher-point BCJ numerators this way. Studying their patterns leads to a proposal for a general formula giving BCJ-satisfying tree-level numerators for arbitrary multiplicities. This will be done in the next section. 7 It amounts to fixing three coordinates zi, zj and z k and inserting a Jacobian factor |zijz jk z ki |.
However, for general color orderings (5.5) and (6.1) no longer manifestly coincide. For example, the field-theory limit of the string amplitude (5.5) with ordering 12435 is while the field-theory formula (6.1) yields One can see from (6.5) and (6.4) that the numerators generated by the SYM amplitude formula (6.1) are mapped to the following BCJ-satisfying numerators in the string theory amplitude,

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Comparing the field-theory limit of the string amplitude (5.5) for various orderings with the outcomes of the SYM amplitude (6.1), one can check that the BCJ-satisfying numerators following from the string tree amplitude can be obtained by a mapping • ij defined by acting 8 on the field-theory numerators given by the SYM amplitude (6.1). In (6.7), P (γ) denotes the powerset of γ, (C) is the left-to-right Dynkin bracket [18], Defining M X • ij M Y by its action on the products of V A • ij V B from the expansion of M X and M Y given by (4.9) one can check a few cases explicitly that the following superfield is BRST closed (Q is the pure spinor BRST charge [34]) Assuming that E (ij) P is BRST invariant to all multiplicities, one is free to use this "gaugefixed" version of E P in the SYM amplitude formula (6.1) to obtain By construction, the SYM amplitudes generated by the formula (6.11) manifestly coincide with the field-theory limit of the string tree amplitude and therefore give rise to BCJsatisfying numerators for all n-point tree amplitudes. Incidentally, the powerset appearing in the definition (6.7) naturally explains why the number of terms in BCJ-satisfying numerators is always a power of two, as firstly observed in [17].
In the appendix A the mapping (6.7) is shown to be the kinematic equivalent of the color Jacobi identity which expresses any cubic color graph in a basis where labels i and j are at the opposite ends.

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Acknowledgments I thank Ellis Yuan for discussions about the algorithm of [1] and Oliver Schlotterer for an enlightening discussion about the perturbiner expansion of Berends-Giele double-currents as well as for collaboration on related topics. I also acknowledge support from NSF grant number PHY 1314311 and the Paul Dirac Fund.

A Proof of manifest BCJ numerators
In this appendix we prove that the rewriting of field-theory numerators given by (6.7) corresponds to the Jacobi identity obeyed by structure constants.
In a BCJ gauge of super Yang-Mills superfields, the multiparticle vertex operator V P satisfies generalized Jacobi identities (see e.g. [18]) and therefore its symmetries correspond to a string of structure constants [30] where (A) denotes the Dynkin bracket (6.8). Similarly, the symmetries of three vertices are mapped to where F (A) is the multi-peripheral color factor [26] F (1, 2, 3, . . . , (n − 1), n) ≡ f 12a 3 f a 3 3a 4 f a 4 4a 5 · · · f a (n−1) (n−1)n . where˜ (P ) = (P ). One can also derive a closed formula to arrive at the DDM basis while keeping track of the relative positions of three labels (say i, j and n), where P (γ) is the powerset of γ and (C) is to be considered a single letter in P (γ). To arrive at the second line one uses the identity 9 (see [35])

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Finally, combining the results above one gets where in the penultimate line we transposed the set δ (while considering (C) a single letter) and used˜ (C) = (−1) |C|+1 (C) when (C) is part of a multiparticle label. Therefore the expression (6.7) for the product V iAjB • ij V C V n is the kinematic counterpart of the color identity (A.5).

B Berends-Giele double-currents from scalar φ 3 theory
In this appendix an alternative derivation of the Berends-Giele double-currents is given which resembles the algorithm of [1].
The field equation φ = φ 2 of the standard scalar φ 3 theory can be solved in a perturbiner expansion as φ(x) = P φ P e k·x ξ P , where ξ P = ξ p 1 ξ p 2 . . . ξ p |P| is an auxiliary parameter and the coefficients φ P obey the recursion relations of planar binary trees, It is straightforward to check that (B.1) gives rise to the recurrence relation for the Catalan numbers, C 0 = 1, C n+1 = n i=0 C i C n−i , where C n refers to the number of terms in the pole expansion of φ 12...n+1 . Examples of φ 123...n up to n = 4 are given by, Note that the above binary trees naturally capture the kinematic pole expansion of "compatible channels" in a color-ordered tree amplitude. The restriction of φ P by an ordering given by a word A is denoted φ P A and is defined by suppressing a term from φ P if it contains any factor of s abcd... whose letters are not adjacent in the word A. where "i < j inside A" is true if the letter i appears before j in A. For example, (1324|1, 4) = +1 but (1324|4, 1) = −1. If P = 123 . . . p is the canonical ordering, the sign factor simplifies to (P |Q) = q 1 q 2 q 2 q 3 . . . q |Q|−1 q |Q| where ij is the standard anti-symmetric tensor; ij = +1 if i < j and ij = −1 if i > j.
One can check that the Berends-Giele double-currents (3.3) can be written as Comparing (B.6) with the algorithm of [1] one concludes that the cumbersome factor of (−1) n flip of [1] admits a simpler representation in terms of epsilon tensors (this observation was made en passant in [11]).
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