Berends-Giele currents in Bern-Carrasco-Johansson gauge for $F^3$- and $F^4$-deformed Yang-Mills amplitudes

We construct new representations of tree-level amplitudes in D-dimensional gauge theories with deformations via higher-mass-dimension operators $\alpha' F^3$ and $\alpha'^{2} F^4$. Based on Berends-Giele recursions, the tensor structure of these amplitudes is compactly organized via off-shell currents. On the one hand, we present manifestly cyclic representations, where the complexity of the currents is systematically reduced. On the other hand, the duality between color and kinematics due to Bern, Carrasco and Johansson is manifested by means of non-linear gauge transformations of the currents. We exploit the resulting notion of Bern-Carrasco-Johansson gauge to provide explicit and manifestly local double-copy representations for gravitational amplitudes involving $\alpha' R^2$ and $\alpha'^2 R^3$ operators.


Introduction
Recent investigations of scattering amplitudes in gauge theories and gravity revealed a wealth of mathematical structures and surprising connections between different theories. For gravitational theories in D spacetime dimensions, traditional methods for tree amplitudes and loop integrands naively give rise to an exasperating proliferation of terms. Still, the final answers for these quantities across various loop-and leg orders take a strikingly simple form: The dependence on the spin-two polarizations can often be reduced to squares of suitably chosen gauge-theory quantities.
The double-copy structure of perturbative gravity originates from string theory where Kawai Lewellen and Tye (KLT) identified universal relations between open-and closed-string tree-level amplitudes [1]. The KLT relations have been later on reformulated in a field-theory framework by Bern, Carrasco and Johansson (BCJ) [2][3][4] such as to flexibly address multiloop integrands. In this way, numerous long-standing questions on the ultraviolet properties of supergravity theories have been resolved [5][6][7][8][9][10], bypassing the spurious explosion of terms in intermediate steps.
This double-copy approach to gravitational amplitudes takes a particularly elegant form once a hidden symmetry of gauge-theory amplitudes is manifested -the duality between color and kinematics due to BCJ [2]. At tree level, the BCJ duality in gauge theories has not only been explained and manifested in string theories [11][12][13][14][15][16] but also extends to various constituents of string-theory amplitudes [17][18][19][20][21][22]. In particular, the following terms in the gauge-field effective action of the open bosonic string 1 in D spacetime dimensions preserve the BCJ duality to the order of α 2 [18], where F µν and α denote the non-abelian field strength and the inverse string tension, respectively. In presence of the effective action (1.1), KLT formulae and BCJ double-copy representations known from Einstein gravity extend 2 to gravitational tree amplitudes 3 from α R 2 + α 2 R 3 operators [18] involving higher powers in the Riemann curvature R. The schematic notation R 2 and R 3 for operators in the gravitational effective action is understood to comprise additional couplings of a B-field and a dilaton ϕ (such as e −2ϕ R 2 ) known from the low-energy regime of the closed bosonic string [25]. The interplay of higher-mass-dimension operators D 2m F n and D 2m R n in string theories with the BCJ duality and double copy is well understood from the worldsheet description of tree-level amplitudes [18,[20][21][22]. Also, D-dimensional amplitudes of the F 3 operator and their double copy have been studied in the CHY formalism [26]. The purpose of this work is to explore a complementary approach and to manifest the BCJ duality of the α F 3 + α 2 F 4 operators directly from the Feynman rules of the action (1.1). We will follow some of the ideas in earlier work on ten-dimensional super-Yang-Mills (SYM) [14,15,27,28] and realize the BCJ duality at the level of Berends-Giele currents [29] -up to the order of α 2 .
We will reorganize the Feynman-diagrammatics of (α F 3 + α 2 F 4 )-deformed Yang-Mills (YM) theory such as to find an explicit off-shell realization of the BCJ duality. The key idea is to remove the deviations from the BCJ duality by applying a concrete non-linear gauge transformation to the generating series of Berends-Giele currents. Our starting point for the currents is Lorenz gauge, and their transformed versions which obey the color-kinematics duality are said to implement BCJ gauge in (α F 3 + α 2 F 4 )-deformed YM theory 4 . Particular emphasis will be put on the locality properties of our construction, i.e. the absence of spurious kinematic poles in the gauge-theory constituents. Like this, the gravitational amplitudes from α R 2 +α 2 R 3 operators obtained via double copy reflect the propagator structure of cubic-vertex diagrams and facilitate loop-level applications based on the unitarity method [30][31][32][33][34]. Moreover, locality of the gauge-theory building blocks will be crucial for one of our main results: a kinematic derivation of the BCJ relations [2] among colorordered amplitudes of (YM+F 3 +F 4 ) [18], a manifestly gauge invariant formulation of the BCJ duality.
Finally, the complexity of the Berends-Giele currents of (YM+F 3 +F 4 ) will be systematically shortened by adapting techniques [14,15,35,36] from ten-dimensional SYM. Our manipulations resemble BRST integration by parts of the pure-spinor superstring [37] and allow for manifestly cyclic amplitude representations as well as streamlined expressions for the gauge parameter towards BCJ gauge.
The results of this work on the currents and amplitudes of (YM+F 3 +F 4 ) are valid up to and including the order of α 2 . At higher orders in α , effective operators including α 3 D 2 F 4 as provided by the bosonic string are required to maintain the BCJ duality [18,21]. Moreover, our results hold in any number D of spacetime dimensions: Apart from the critical dimension D = 26 of the bosonic string and the phenomenologically interesting situation with D = 4, this allows for a flexible unitarity-based investigation of loop integrands in various dimensions and dimensional regularization, see e.g. [38,39].
By its close contact with Lagrangians, the construction in this work resonates with recent developments in scalar theories with color-kinematics duality and double-copy structures [40,41]: For the color-kinematics duality of the non-linear sigma model (NLSM) of Goldstone bosons [40], a Lagrangian origin along with the structure constants of a kinematic algebra has been identified in [42]. This new formulation of the NLSM can be derived from higher dimensional YM theory [43], and a string-inspired higher-derivative extension of the NLSM 5 [44] has been recently obtained from the analogous dimensional reduction of α F 3 in a companion paper [45]. In view of these connections, we hope that the notion of BCJ gauge inspires a reformulation of the (YM+F 3 +F 4 )-Lagrangian (1.1) where -similar to [42] -the D-dimensional kinematic algebra is manifest 6 .
Another source of motivation for this work stems from the renewed interest in the gravitational α R 2 + α 2 R 3 interactions in D = 4 dimensions. While R 3 is well-known to be the first (non-evanescent) two-loop counterterm for pure gravity [50,51], the evanescent one-loop counterterm R 2 was recently found to contaminate dimensional regularization at two loops [52,53]. Moreover, evanescent matrix elements of R 2 are closely related to certain anomalous amplitudes of N = 4 supergravity [54] through double copy [55]. Finally, when viewed as ambiguities in defining quantum theories, matrix elements of higher dimensional operators can be crucial to restore symmetries when using a non-ideal regulator for loop amplitudes [56]. We hope that our D-dimensional double-copy representations for tree-level amplitudes of (α R 2 + α 2 R 3 )-deformed gravity shed further light into these loop-level topics: either by unitarity or by using the BCJ-gauge currents as building blocks for loop amplitudes that universally represent tree-level subdiagrams 7 .

Outline
This work is organized as follows: In section 2, we review the basics of Berends-Giele recursions, the BCJ duality as well as the double copy and establish the associated elements of notation. Section 3 is dedicated to amplitudes of (YM+F 3 +F 4 ) in different types of Berends-Giele representations including a systematic reduction of the rank of the currents. In section 4, an explicit off-shell realization of the BCJ duality is obtained from the Berends-Giele setup. Finally, section 5 relates this realization of the BCJ duality to non-linear gauge freedom and combines the off-shell ingredients from the previous section to manifestly local amplitude representations of (YM+F 3 +F 4 ) and gravity with α R 2 + α 2 R 3 operators. A derivation of the BCJ relations to the order of α 2 from purely kinematic arguments is given in section 5.2.

Review and notation
In this section, we set up notation and review the key ideas and applications of Berends-Giele recursions for tree-level amplitudes in YM theory, in particular • the resummation of Berends-Giele currents to obtain perturbiner solutions to the nonlinear field equations • manifestly cyclic Berends-Giele representations of YM amplitudes involving currents of smaller rank than naively expected.
We will also review the BCJ duality and the double copy from a perspective which later on facilitates the implementation of these features in tree amplitudes and Berends-Giele currents of (YM+F 3 +F 4 ) as well as gravity with α R 2 + α 2 R 3 operators.

Berends-Giele recursions
An efficient approach to determine the tensor structure of D-dimensional tree amplitudes in pure YM theory has been introduced by Berends and Giele in 1987 [29]. The key idea of the reference is to recursively combine all color-ordered Feynman diagrams involving multiple external on-shell legs and a single off-shell leg. This recursion is implemented via currents J µ 12...p that depend on the polarization vectors e µ i and lightlike momenta k µ i of the external particles i = 1, 2, . . . , p subject to the following on-shell constraints e i · k i = k i · k i = 0 ∀ i = 1, 2, . . . . Currents of arbitrary multiplicity can be efficiently computed from the Berends-Giele recursion [29] J µ i = e µ i , The external states have been grouped into multiparticle labels or words P = 12 . . . p. We will represent multiparticle labels by capital letters P, Q, X, Y, . . . and denote their length, i.e. the number of labels in P = 12 . . . p, by |P | = p. The summation over XY = P on the right-hand side of (2.2) instructs to deconcatenate P into non-empty words X = 12 . . . j and Y = j+1 . . . p with j = 1, 2, . . . , p−1 and therefore generates |P |−1 terms 8 . Similarly, XY Z = P encodes 1 2 (|P |−1)(|P |−2) deconcatenations into non-empty words X = 12 . . . j, Y = j+1 . . . l and Z = l+1 . . . p with 1 ≤ j < l ≤ p−1.
and XY Z = P in (2.2) is to connect lower-rank currents J µ X , J ν Y and J λ Z via Feynman vertices in all possible ways that preserve the color order of the on-shell legs in the word P = 12 . . . p.
Accordingly, color-ordered on-shell amplitudes at n = p+1 points are recovered by taking the off-shell leg in the rank-p current J µ P on shell: This on-shell limit is implemented by contraction with the polarization vector J µ n = e µ n of the last leg and removing the propagator s −1 12...p in the p-particle channel of J µ P which would diverge by n-particle momentum conservation For instance, the rank-two current due to (2.2) with X = 1 and Y = 2 yields the following 8 For instance, the summation over XY = P with P = 1234 of length four incorporates the pairs (X, Y ) = (123, 4), (12,34) and (1,234). 9 Here and in later equations of this work, we keep both instances of a contracted Lorentz index in the uppercase position to avoid interference with the multiparticle labels of the currents. The signature of the metric is still taken to be Minkowskian, regardless of the position of the indices.
representation of the three-point amplitude where cyclicity may be manifested via e 3 · k 2 = −e 3 · k 1 by means of on-shell constraints and momentum conservation. Note that Berends-Giele formulae similar to (2.6) have been given for tree amplitudes in ten-dimensional SYM [36], doubly-ordered amplitudes of bi-adjoint scalars [61] and worldsheet integrals for tree-level scattering of open strings [62].

Perturbiners as generating series of Berends-Giele currents
The Berends-Giele construction of the previous section can be related to solutions of the non-linear field equations: Generating series of Berends-Giele currents turn out to solve the equations of motion from the action S YM of pure YM theory We use the following conventions in deriving the Lie-algebra valued gluon field A µ and its non-linear field strength F µν from a connection ∇ µ , (2.12) 10 The shuffle product P ¡Q of words P = p1p2 . . . p |P | and Q = q1q2 . . . q |Q| is recursively defined by P ¡∅ = ∅¡P = P , P ¡Q = p1(p2 . . . p |P | ¡Q) + q1(q2 . . . q |Q| ¡P).
The relation of tree-level amplitudes with solutions of the field equations via generating series goes back to the "perturbiner" formalism [66][67][68][69][70]. In these references, generating series of MHV amplitudes are derived from self-dual YM theory, see [71] for supersymmetric extensions. The connection between perturbiner solutions and the dimension-agnostic Berends-Giele currents of [29] was established in [27,28] and will now be reviewed. Lorenz gauge ∂ µ A µ = 0 simplifies the equations of motion [∇ µ , F λµ ] = 0 to the wave equation with the notation = ∂ µ ∂ µ for the d'Alembertian, One can derive formal solutions to (2.13) by means of the perturbiner ansatz (2.14) The summation variables i, j, l, . . . = 1, 2, 3, . . . refer to external-particle labels in an unbounded range, and we have introduced a compact notation P =∅ for sums over nonempty words P = 12 . . . p in passing to the second line. The dependence on the spacetime coordinates x µ enters through plane waves 11 e k P ·x , see (2.5) for the multiparticle momenta k P . The color degrees of freedom in (2.14) are represented through matrix products of the Lie-algebra generators t a i whose adjoint indices a 1 , a 2 , . . . are associated with an unspecified gauge group. Upon insertion into the second line of (2.13), the perturbiner ansatz (2.14) can be verified to solve the non-linear field equations [∇ µ , F λµ ] = 0 if its coefficients J µ P obey the Berends-Giele recursion (2.2). Hence, generating series of Berends-Giele currents are formal solutions to the field equations 12 . By the shuffle symmetry (2.8) of the currents J µ P , the matrix products t a i t a j of the Lie-algebra generators on the right-hand side of (2.14) conspire to nested commutators, and the perturbiner solution is guaranteed to be Lie-algebra valued [72].
As a convenient reorganization of the Berends-Giele recursion (2.2), one can write the field equations as in the first line of (2.13) and insert a separate perturbiner expansion for the non-linear field strength, The conventional form of plane waves e ik·x with an imaginary unit in the exponent can be recovered by redefining the momenta in this work as k → ik. The equations in the main text follow the conventions where external momenta are purely imaginary in order to keep factors of i from proliferating. 12 Strictly speaking, contributions with several factors of t a j referring to the same external leg j need to be manually suppressed by adding nilpotent symbols to the perturbiner ansatz [66]. For ease of notation, we do not include these symbols into the equations in the main text, and all terms with repeated appearance of a given external leg are understood to be suppressed.
The expressions for the field-strength currents B µν P in terms of J λ Q are determined by the definition (2.12) of F µν , and their non-linear terms P =XY J Y have already been studied in [73]. Then, inserting (2.14) and (2.15) into (2.13) yields a simpler but equivalent form of the recursion (2.2) [28] The trilinear term {J X , J Y , J Z } in (2.4) which represents the quartic vertex of the YM Lagrangian has been absorbed into the non-linear part of the field-strength current B µν P in (2.15). The leftover deconcatenations P = XY in (2.16) can be interpreted as describing cubic diagrams, see   ..p , only cubic-vertex diagrams have to be considered in their recursive construction from lower-rank currents. In the depicted example at rank p = 3 with an additional off-shell leg · · · , only two cubic diagrams of s-channel and t-channel type contribute to the four-point amplitude obtained from s 123 J µ 123 J µ 4 .
Note that the Lorenz-gauge condition and the field equations imply the relations including transversality of the gluon polarizations for single-particle labels P = i. Moreover, the non-linear gauge symmetry of the action (2.11) under , Ω] acts on the currents via The scalar currents Ω P are defined by the perturbiner expansion Ω(x) = P =∅ Ω P t P e k P ·x of the gauge scalar in δ Ω . We will later on spell out a choice of gauge-scalar currents Ω P which manifests the BCJ duality at the level of Berends-Giele currents.
Another specific choice of Ω P → Ω lin P allows to track the effect of linearized gauge transformations e µ i → k µ i on the i th leg of the Berends-Giele currents in (2.18): One can line up the replacement e µ i → k µ i with a set of gauge transformations that preserves Lorenz gauge. The condition δ Ω lin (∂ µ A µ ) = ∂ µ (δ Ω lin A µ ) = 0 then translates into the recursion [27] which needs to be supplemented with the initial conditions Ω lin j → δ i,j if the linearized gauge transformations e µ i → k µ i only applies to the i th leg. Precursors of the formula (2.19) for linearized gauge transformations of Berends-Giele currents can be found in [63].

Manifestly cyclic reformulation
Given that the Berends-Giele formula (2.6) for color-ordered amplitudes A YM (1, 2, . . . , n) singles out the last leg n which is excluded from the current J µ 12...n−1 , cyclic invariance in the external legs is obscured. We shall now review a reorganization of the Berends-Giele currents for YM tree amplitudes such that the n th leg enters on completely symmetric footing. Moreover, the subsequent rewritings reduce n-point amplitudes to shorter Berends-Giele currents of rank ≤ n 2 instead of the rank-(n−2) currents in the recursion (2.2) for J µ 12...n−1 . The backbone of the manifestly cyclic Berends-Giele formulae is the building block [28] composed of three currents with multiparticle labels X, Y, Z each of which represents treelevel subdiagrams. The resulting diagrammatic interpretation of M X,Y,Z is depicted in figure  3, and the definition (2.20) along with B µν X = −B νµ X implies permutation antisymmetry M X,Y,Z = −M Y,X,Z and M X,Y,Z = M Y,Z,X expected from the cubic vertex in the figure. Figure 3: Diagrammatic interpretation of the building block M X,Y,Z in (2.20) with multiparticle labels X = x 1 x 2 . . . x p , Y = y 1 y 2 . . . y q and Z = z 1 z 2 . . . z r .
Using k P · J P = 0 and k X + k Y + k Z = 0, it was shown in [28] that the n-point amplitude (2.6) can be rewritten as As demonstrated in appendix A.2, momentum conservation k P + k Q = 0 and (2.17) imply the following identity which will be referred to as "integration by parts" 13  .
Note in particular that the rank of the currents in the manifestly cyclic n-point amplitudes (2.24) is bounded by 14 n 2 rather than n−2 as expected from the recursions (2.2) or (2.16) for J µ 12...n−1 . In section 3, similar expressions with manifest cyclicity and Berends-Giele currents of maximum rank n 2 will be given for the deformed (YM+F 3 +F 4 ) theory.

BCJ duality
The organization of the Berends-Giele recursion (2.16) in terms of cubic-vertex diagrams as exemplified in figure 2 resonates with the BCJ duality between color and kinematics [2]: According to the BCJ duality, scattering amplitudes in non-abelian gauge theories can be represented in a manner such that color degrees of freedom can be freely interchanged with the kinematic variables. While "color" refers to contractions of structure constants f a i a i a k , polarizations and momenta are referred to as "kinematics", and the notion of "freely interchanging" will be shortly made precise. The three-index structure of the contracted structure constants can be visualized via cubic-vertex diagrams with a factor of f a i a i a k for each vertex and contractions of the adjoint indices along the internal edges. Similarly, the kinematic dependence on e µ i , k µ i should also be organized in terms of cubic diagrams to manifest the BCJ duality.
The non-linear Y of the field-strength current B µν P in (2.15) absorbs the contributions from the quartic vertex Tr[A µ , A ν ][A µ , A ν ] in the YM action (2.11). This can be seen from that fact that the non-linear terms have fewer propagators than the rest of (2.16). Hence, the use of field-strength currents amounts to inserting 1 = such that a quartic vertex is "pulled apart" into two cubic vertices connected by the "fake" propagator k 2 P . The choice of the channel P in 1 = In figure 4, this situation is visualized in a four-point tree-level context, but there is no limitation to cubic-diagram parametrizations of n-point tree amplitudes as well as multiloop integrands [3,4]. Although the BCJ duality conjecturally applies to loop integrands [3,4], we shall focus on its well-established tree-level incarnation.
Of course, contributions from the higher-order vertices of (Tr F 3 )-and (Tr F 4 )-type can also be cast into a cubic-graph form by repeated insertions of 1 = . For the action (1.1) of (YM+F 3 +F 4 ), the color structure of the F 3 and F 4 operators also boils down to contracted structure constants [18], and the ambiguities due to Jacobi identities (2.25) arise in this situation as well. In the subsequent review of the BCJ duality, the color-dressed tree-level amplitudes where Γ n denotes the set of cubic tree-level graphs with n external legs. The color factors C I represent the contracted structure constants that arise from the traces in (2.26). The kinematic numerators N I are combinations of e µ i and k µ i that can be assembled from the Berends-Giele currents of the theory. Finally, the propagators s −1 e comprise Mandelstam variables (2.5) for the multiparticle momenta in the internal edges e of the graph I.
The parametrization (2.27) is said to manifest the BCJ duality if all the symmetries of the color factors C I carry over to the kinematic numerators N I . More specifically [2]: • If two graphs I and I are related by a single flip of a cubic vertex, antisymmetry f a i a j a k = f [a i a j a k ] implies the color factors to have a relative minus sign. In a dualitysatisfying representation (2.27), the kinematic numerators exhibit the same antisymmetry properties under flips: • For each triplet of graphs I, J, K where the Jacobi identities (2.25) lead to the vanishing of triplets C I +C J +C K , the BCJ duality requires the corresponding triplet of kinematic numerators to vanish as well  Figure 5: Triplets of cubic graphs I, J, K whose color factors C · and kinematic factors N · are both related by a Jacobi identity if the duality between color and kinematics is manifest. The dotted lines at the corners represent arbitrary tree-level subdiagrams and are understood to be the same for all of the three cubic graphs.
In later sections, we will construct local representatives of the kinematic numerators N I in (2.27) of (YM+F 3 +F 4 ) which do not exhibit any poles in s P and obey the BCJ duality up to and including the order of α 2 . By the Jacobi identities (2.25) of the color factors, the numerators are still far from unique after imposing locality, and generic choices at n ≥ 5 points will fail to obey some of the kinematic Jacobi relations (2.29). Hence, finding a manifestly color-kinematics dual parametrization (2.27) requires some systematics in addressing quartic and higher-order vertices via 1 = k 2 P k 2 P . The additional requirement of locality is particularly restrictive, and we will see that suitable gauge transformations (2.18) of the Berends-Giele currents in (YM+F 3 +F 4 ) give rise to local solutions, generalizing the construction in tendimensional SYM [28].
Still, the very existence of duality satisfying kinematic numerators is sufficient to derive BCJ relations among color-ordered amplitudes [2] n−1 j=2 (k 23...j · k 1 )A(2, 3, . . . , j, 1, j+1, . . . , n) = 0 . (2.30) By combining different relabellings of (2.30), any color-ordered amplitude can be expanded in a basis of size (n−3)!. BCJ relations were shown to apply to A → A YM+F 3 +F 4 up to and including the order of α 2 [18] by isolating suitable terms in the monodromy relations of openstring tree-level amplitudes [11,12]. For a variety of four-dimensional helicity configurations, kinematic numerators of YM + F 3 subject to Jacobi relations (2.29) can be found in [18]. We will derive generalizations to helicity-agnostic expressions in D dimensions and include the α 2 order of (YM+F 3 +F 4 ). Kinematic antisymmetry relations (2.28) and Jacobi identities (2.29) leave (n−2)! independent instances of N I . A basis of kinematic numerators under these relations can be assembled from the "half-ladder" diagrams depicted in figure 6 which are characterized by a fixed choice of endpoints 1 and n as well as permutations ρ ∈ S n−2 of the remaining legs 2, 3, . . . , n−1. We will denote the basis numerators of the half-ladder diagrams in figure 6 by N 1|ρ(2,3,...,n−1)|n and refer to them as "master numerators".

Double copy
The BCJ duality allows to convert cubic-graph parametrizations (2.27) of gauge-theory amplitudes into gravitational ones: Once the gauge-theory numerators N I satisfy the same symmetry properties as the color factors C I (i.e. flip antisymmetry (2.28) and kinematic Jacobi identities (2.29)), then the double-copy formula enjoys linearized-diffeomorphism invariance. In case of undeformed YM theory, (2.31) yields tree-level amplitudes of Einstein-gravity including B-fields, dilatons and tentative supersymmetry partners [2,46]. The polarizations of external gravitons, B-fields or dilatons in the j th leg are obtained by projecting the tensor products e µ jẽ ν j in (2.31) to the suitable irreducible representation of the Lorentz group.
In case of (YM+F 3 +F 4 )-numerators, the gravitational amplitudes descend from a deformation of the Einstein-Hilbert action by higher-curvature operators of α R 2 + α 2 R 3 [18] as seen in the low-energy effective action of the closed bosonic string [25], see section 5.4 for details. The tilde along with the second copyÑ I of the gauge-theory numerator N I indicates that the i th external gravitational state may arise from the tensor product of different polarization vectors e µ i andẽ µ i . In the same way as (2.31) is obtained from gauge-theory amplitudes (2.27) by trading color for kinematics, C I →Ñ I , one can investigate the converse replacement N I →C I : This double copy of color factors (withC I comprising structure constantsfb ibjbk of possibly different Lie algebra generatorstb) describes tree amplitudes of biadjoint scalars φ = φ a|b t a ⊗tb with a cubic interaction f a 1 a 2 a 3fb1b2b3 φ a 1 |b 1 φ a 2 |b 2 φ a 3 |b 3 [75]. The two species t a andtb admit a two-fold color decomposition (2.26), and we define its doubly-partial amplitudes m(·|·) by peeling off two traces with possibly different orderings ρ, τ ∈ S n−1 [75], .
We will later on use the matrices S(P |Q) 1 and Φ(P |Q) 1 to relate shuffle independent Berends-Giele currents to kinematic numerators subject to Jacobi identities.

Perturbiners and Berends-Giele representations for F and F 4
In this section, we apply the Berends-Giele methods of sections 2.1 to 2.3 to the deformed (YM+F 3 +F 4 ) theory known from the low-energy regime of open bosonic strings. The treelevel amplitudes following from the action reproduce the leading orders α 0 , α 1 in the low-energy expansion of bosonic-string amplitudes [78], and a well-defined sector of the α 2 order: The effective action of both bosonic and supersymmetric open strings comprises an operator α 2 ζ 2 Tr F 4 which can be cleanly distinguished from (3.1) by its transcendental prefactor ζ 2 = π 2 6 [79 -81]. Up to and including the order of α 2 , the amplitudes computed from (3.1) obey BCJ relations (2.30) while the α 2 ζ 2 Tr F 4 -operator excluded from (3.1) is incompatible with the BCJ duality [18]. As explained in the reference, these BCJ relations to the order of α 2 rely on the interplay between single-insertions of the α 2 Tr F 4 -operator in (3.1) and double-insertions of α Tr F 3 .
More generally, the accompanying multiple zeta values are instrumental to identify the D 2m F n -operators in string effective actions that admit color-kinematics dual representations. For instance, the entire single-trace gauge sector of the heterotic string obeys BCJ relations [20]. The subsector of open-bosonic-string amplitudes compatible with the BCJ duality was identified in [21], and the amplitude contributions without any zeta-value coefficient were derived from a field-theory Lagrangian [22].
The subsequent Berends-Giele recursions for the amplitudes of (3.1) follow a two-fold purpose: on the one hand, they will be used to generate economic and manifestly cyclic amplitude representations along the lines of section 2.3. On the other hand, they set the stage for • an off-shell realization of the BCJ duality in section 4 • a kinematic proof of the BCJ relations in section 5.2 • a construction of manifestly local gauge-theory numerators subject to kinematic Jacobi relations in section 5.3.
All of these results hold to the order of α 2 and are based on a non-linear gauge transformation of the generating series of Berends-Giele currents similar to (2.18).

Berends-Giele recursions for F 3 and F 4
Our Berends-Giele approach to (YM+F 3 +F 4 ) follows the lines of section 2.2 to derive recursions for the currents from the non-linear equations of motion. The field variation of the action (3.1) is given by 15 and augments (2.11) by α -corrections. In Lorenz gauge ∂ µ A µ = 0, setting (3.2) to zero amounts to a wave equation analogous to (2.13), Since we will only be interested in the amplitude contributions up to the order of α 2 , we can simplify (3.3) by dropping terms of order α 3 and higher. At the first order in α , this allows This form of the field equations gives rise to an efficient Berends-Giele recursion: We will study formal solutions of (3.4) modulo α 3 that descend from a perturbiner ansatz where Lorenz gauge and the definition F µν = −[∇ µ , ∇ ν ] of the field strength imply In comparison to the perturbiners (2.14) and (2.15) of undeformed YM theory, the Berends-Giele currents have been renamed as J µ P → A µ P and B µν P → F µν P in order to distinguish these α -dependent quantities from the YM currents in (2.16) and (2.15) , . The extensions of this lemma to Fµν -dependent quantities X follows straightforwardly from the Leibniz rule.
In the same way as the Berends-Giele recursion of undeformed YM theory benefits from field-strength currents B µν P , the perturbiner solutions to (3.4) are conveniently expressed in terms of the additional auxiliary currents By their definition in (3.8), the auxiliary currents are determined by A µ P and F µν P , in the same way as the F µν P in (3.6) boil down to the elementary currents A µ P . All the above currents can be shown to obey shuffle symmetry by repeating the arguments for the currents J µ P and F µν P of pure YM theory [27,63]. With the above definitions, the Berends-Giele recursion induced by the field equation (3.4

) takes the simple form
In analogy to (2.6), the leading orders α ≤2 of the tree amplitudes resulting from the action (3.1) are then given by For instance, the rank-two current due to (3.11) with X = 1 and Y = 2 and the resulting three-point amplitude read The last term of the three-point function illustrates a fundamental difference between the tensor structure of YM amplitudes and their F 3 corrections: Contractions of the type (k · e) n do not occur in n-point amplitudes of YM [82][83][84]. Hence, the expressions for A YM+F 3 +F 4 in this work do not belong to the class of D 2m F n amplitudes that are accessible from the open superstring through a combination of color-ordered SYM trees [17].
By the shuffle symmetry (3.10) of the currents, the amplitude representation (3.12) can be used to demonstrate A YM+F 3 +F 4 (1, 2, . . . , n) to also obey the Kleiss-Kuijf relations (2.9). The α 1 -order of our results up to and including n = 6 points has been checked to match the D-dimensional CHY formulae of 16 [26]. The D = 4 helicity components of the CHY expressions 17 in turn have been verified to agree with the results of [86][87][88].
We emphasize that the form of the recursion in (3.11) only involves deconcatenations P = XY into two words X, Y rather than three-word expressions with P = XY Z as seen in (2.2). Hence, the amplitudes (3.12) naturally arise in a cubic-graph parametrization (2.27) as visualized in figure 7. The cubic-graph organization extends to the order of α 2 , although each term of the Tr F 4 vertex in (3.1) involves at least four powers of the A µ field. Still, the quartic-vertex origin of the terms G (3.11) can only contribute at minimum length |X| + |Y | = 3, i.e. to amplitudes (3.12) at multiplicity n ≥ 4. ..p of rank p combine the diagrams and propagators expected in a color-ordered (p+1)-point tree amplitude of (YM+F 3 +F 4 ) with an off-shell leg · · · . Vertices marked with a white and black dot represent the first and second order in α on the right hand side of (3.11), i.e. the cubic-graph parametrization of α F 3 and α 2 F 4 insertions.
Note that the equation of motion (3.2) translates into the following expression for the 16 We are grateful to Song He and Yong Zhang for providing the analytic expressions. tensor divergence of F µν P :

Manifestly cyclic Berends-Giele representations
This section is dedicated to a reformulation of the Berends-Giele formula (3.12) for A YM+F 3 +F 4 such as to manifest cyclicity and to reduce the maximum rank of the Berends-Giele constituents (3.11) on the right-hand side. This amounts to identifying a deformation of the cyclic M X,Y,Z building block in (2.20) that preserves the structure of the economic and manifestly cyclic amplitude representations (2.24) up to and including the order of α 2 . The desired cyclic building block analogous to M X,Y,Z reads  .15), the central vertex can either be of YM type (first term), of (α F 3 )-type (white dot) or of (α 2 F 4 )type (black dot). The blobs labelled by X, Y, Z represent currents of (YM+F 3 +F 4 ).
In analogy with (2.21), the amplitude formula (3.12) can be rewritten as To first order in α , the equivalence with (3.12) is proven to all multiplicities in appendix A.1. At second order in α 2 , (3.16) has been checked analytically to multiplicity n = 6 and numerically up to and including n = 8. In the same way as we are only interested in the orders α ≤2 of the amplitudes (3.16), we will consistently drop terms at the orders α ≥3 in later equations of this work and skip the disclaimer O(α 3 ) for ease of notation. Amplitude representations with manifest cyclicity and lower-rank Berends-Giele currents can be obtained by an integration-by-parts property that takes the same form as (2.22), also see [28,35,36] for the antecedents of these representations in ten-dimensional SYM.

Gauge algebra of F 3 + F 4 building blocks
The action of non-linear gauge transformations δ Ω A µ = ∂ µ Ω − [A µ , Ω] is not altered by the higher-mass-dimension operators in the action (3.1). Hence, given perturbiner components Ω P for the gauge scalars Ω, the α -deformed currents of the previous subsection follow the transformations of the YM currents (2.18), One can therefore verify non-linear gauge invariance of the amplitude formula (3.12) by repeating the arguments of the undeformed gauge theory: Among the three terms in the gauge variation the first one vanishes by the Lorenz-gauge condition k 12...n−1 · A 12...n−1 = 0 and the second one due to transversality k n · A n = 0 (using momentum conservation k µ 12...n−1 = −k µ n in both cases). The currents in the second line of (3.23) have multiplicity |X|, |Y | ≤ n−2 and are therefore regular as s 12...n−1 → 0, so multiplication with s 12...n−1 causes this term to vanish as well. This rests on the reasonable assumption that the gauge scalars Ω P descend from a perturbiner and can only have poles in s Q for subsets Q ⊆ P .
By the Leibniz property of δ Ω , momentum conservation and the expression (3.14) for contractions of the form k µ P F µν P , one can infer the non-linear gauge transformation of the building blocks (3.15). The result is most conveniently expressed in terms of a scalar quantity Ω X,Y,Z,W = Ω [X,Y,Z,W ] which is totally antisymmetric in four multiparticle labels X, Y, Z, W , While the α → 0 limit of (3.24) descends from BRST variations of superspace building blocks in ten-dimensional SYM theory [28], a proof to the first order in α is given in appendix A.3. At the order of α 2 , we have tested (3.24) to the order of |X| + |Y | + |Z| = 6.
Based on the gauge algebra (3.24) and permutation antisymmetry Ω X,Y,Z,W = Ω [X,Y,Z,W ] , one can check the non-linear gauge invariance of the manifestly cyclic amplitude representations in (3.18), (3.19) and (3.21). In particular, this will be exploited in later sections to evaluate the M X,Y,Z in BCJ gauge which is tailored to manifest the BCJ duality via local numerators.

Kinematic Jacobi identities in off-shell diagrams
The purpose of this section is to manifest the BCJ duality between color and kinematics in off-shell diagrams of (YM+F 3 +F 4 ). We will construct local solutions to the kinematic Jacobi relations (2.29) in the subdiagram with an off-shell leg drawn in figure 9. This amounts to assigning kinematic numerators to the cubic-vertex diagram in the figure which share the symmetries of the associated color factors The adjoint indices a 1 , a 2 , . . . , a p refer to p on-shell legs, and an off-shell leg is associated with a free adjoint index b carried by the rightmost factor in (4.1). When identifying the dotted off-shell line in figure 9 with an external on-shell leg, we recover the half-ladder diagrams of figure 6 that define the master numerators at n = p+1 points. Accordingly, permutations of figure 9 in 2, 3, . . . , p will be associated with the master numerators in an off-shell setup: Cubic diagrams which are not of half-ladder form or do not have leg 1 and the off-shell leg · · · at their endpoints can be reached from (p−1)! permutations of figure 9 through a sequence of Jacobi identities. In specific examples of (4.1) at rank p ≤ 5, antisymmetry f a i a j a k = f [a i a j a k ] and Jacobi identities (2.25) imply the so-called Lie symmetries for the color factors, The ellipsis in the subscript of each term indicates that lower-rank symmetries in the first labels extend to higher rank. For instance, can be shown to persist at any rank p > 2 by contraction with f ba 3 c f ca 4 d . . . f xa p−1 y f yapz which yields C z 123...p = −C z 213...p . The generalization of the Lie symmetries (4.2) to higher rank will be spelt out in (4.28) and can be checked to leave (p−1)! independent permutations of C b 123...p at rank p.
We will now describe the construction of local kinematic factors for (YM+F 3 +F 4 ) with the same Lie symmetries of (4.2) which imply kinematic Jacobi relations. The recursive procedure presented in this section closely follows the steps of [15], where local superspace building blocks with Lie symmetries have been constructed for ten-dimensional SYM.

Local multiparticle polarizations up to rank three
As we already saw for the Berends-Giele currents of the previous section, each cubic vertex of (YM+F 3 +F 4 ) may introduce powers of α 0 , α 1 or α 2 into the kinematic factors. We no longer distinguish these contributions from the individual vertices (as done by the white and black circles in figure 7 and 8) and collectively refer to all contributions at orders α ≤2 through the off-shell diagram in figure 9. We will start from the numerators in the Berends-Giele recursion (3.11) to construct solutions to the kinematic Jacobi identities -i.e. realizations of the Lie symmetries in (4.2) -up to the order of α 2 .
Kinematic representatives for the diagram in figure 9 with Lie symmetries will be referred to as multiparticle polarizations and denoted by lowercase parental letters a µ 12...p , f µν 12...p , f ..p . This notation will help to distinguish the local multiparticle polarizations from the Berends-Giele currents A µ P , F µν P , F µ|νλ P and G µν|λρ P with kinematic poles. In the same way as all the four species of Berends-Giele currents enter the cyclic building block M X,Y,Z in (3.15), we will later on see that the analogous four species of multiparticle polarizations can be combined to Jacobi-satisfying kinematic numerators N I in the sense of section 2.4.
At rank one, the local multiparticle polarizations are defined to match their Berends-Giele counterparts which include the transverse polarization vectors e µ i and do not exhibit any kinematic poles, The simplest multiparticle polarization a µ 12 at rank two is defined by isolating the numerator in the Berends-Giele current (3.13), where the absence of contributions at order α 2 is plausible by the valence of the Feynman vertices from α 2 F 4 . The alternative presentation of (4.4) as a µ 12 = s 12 A µ 12 generalizes to the -25 -following two-particle polarizations at higher mass dimension, The local multiparticle polarizations are still proportional to their Berends-Giele counterparts since the latter only describe a single cubic diagram, see figure 10. By the shuffle symmetry 123 . Indeed, antisymmetry a µ 123 = − a µ 213 in the first two indices is inherited from the property (4.6) of the rank-two input. However, the first non-trivial kinematic Jacobi identity for the triplet of cubic diagrams in figure 11 requires a µ 123 + a µ 231 + a µ 312 to vanish, which is not the case. Still, the obstruction takes a special form, where one can factor out the overall momentum k µ 123 and isolate a scalar quantity h 123 that captures the deviation from the Lie symmetries a µ 123 + a µ 231 + a µ 312 = 3k µ 123 h 123 . Amusingly, the explicit form of can be reproduced from the cyclic building block of (3.15). Already the left-hand side of (4. ..p at higher mass dimension involves contact terms ∼ s ij that preserve the Lie symmetries. For instance, the rank-three generalizations of (4.5) are easily checked to reproduce the symmetries (4.11) of a µ 123 . These contact terms are the local equivalents of the deconcatenation terms in the Berends-Giele currents F µν P , F µ|νλ P and G µν|λρ P in (3.6) and (3.9), see section 5.1 for more details and [15] for superspace analogues. While the difference between a µ 123 and a µ 123 in (4.10) drops out from the definition (4.12) of f µν 123 , it will be crucial at higher rank to always build f µν 12...p , f µ|νλ 12...p and g µν|λρ 12...p from the redefined fields subject to Lie symmetries.

Local multiparticle polarizations at rank four and five
The numerators of the Berends-Giele recursion (3.11) serve as a starting point to construct higher-rank multiparticle polarizations for the diagrams in figure 9 that satisfy the Lie symmetries (4.2) of the dual color factors. The higher-rank systematics of the redefinition in (4.10) is most conveniently illustrated via examples at p = 4, 5.
Given the multiparticle polarizations at rank three in (4.10) and (4.12), their Lie symmetries imply that the rank-four object obeys a µ 1234 = − a µ 2134 and a µ 1234 + a µ 2314 + a µ 3124 = 0. However, the Lie symmetry at rank four is not yet satisfied by (4.13), and in contrast to (4.8), it is not possible to factorize k µ 1234 from a µ 1234 − a µ 1243 + a µ 3412 − a µ 3421 . Instead, we will need redefinitions a µ 1234 → a µ 1234 → a µ 1234 in two steps, where an intermediate object a µ 1234 is built from permutations of the scalar h ijk in the rank-three redefinition, see (4.9), (4.14) The pattern of subtractions in (4.14) has been inferred by mimicking BRST transformations in ten-dimensional pure-spinor superspace [15], and it should be possible to give a similar motivation from a study of linearized gauge variations. The key benefit of the redefinition in (4.14) is that the deviation from the rank-four Lie symmetry now takes a factorized form This final form of the multiparticle polarization a µ 1234 can be used to construct its counterparts at higher mass dimensions, where the Lie-symmetry preserving contact terms in (4.12) generalize to 18 At higher rank, analogous redefinitions in two steps a µ 123...p → a µ 123...p → a µ 123...p will be sufficient to attain Lie symmetries, i.e. there are no additional intermediate steps at p > 4. For instance, the Lie-symmetry satisfying multiparticle polarizations (4.16) and (4.18) at rank four can be used to recursively construct a rank-five quantity subject to the symmetries (4.17) in its first four labels. The rank-five Lie symmetry can be enforced by first performing subtractions analogous to (4.14) and [15], and then identifying a rank-five scalar h 12345 along the lines of (4.15),  The remaining multiparticle polarizations are given by

Local multiparticle polarizations at higher rank
The recursive construction of multiparticle polarizations will now be summarized in terms of all-rank formulae that closely follow their superspace antecedents [15] but incorporate αcorrections. The Lie symmetries of a µ 12...q , f µν 12...q , f with h i = h ij = 0 as well as h ijk and h ijkl defined in (4.9) and (4.25), respectively. These subtractions vanish at rank p ≤ 3, and their instances at p = 4, 5 are spelt out in (4.14) and (4.20), respectively. These equations might be helpful to see an example of the summation prescription of the form a 1 a 2 . . . a k = X¡Y in (4.27): For a given k-particle label a 1 a 2 . . . a k , the sum runs over all the 2 k pairs of words X and Y whose shuffle product contains a 1 a 2 . . . a k , for instance 20 (X, Y ) = (∅, ∅) in case of k = 0 as well as (X, Y ) = (a 1 , ∅) and (X, Y ) = (∅, a 1 ) in case of k = 1. The outcome of (4.27) still obeys the Lie symmetries in the first p−1 labels and is claimed to factorize k µ 12...p when probing the rank-p Lie symmetry: : p = 2n even .
see the explanation above for the summation prescription j+1, j+2 . . . p = X¡Y . The pattern of contact terms on the right-hand side preserves the Lie symmetries in all the p labels and will be connected with Berends-Giele currents in section 5.1. In the next section, these four Jacobi-satisfying kinematic representatives (4.29) and (4.31) of the off-shell diagram in figure  9 will be combined to on-shell numerators of (YM+F 3 +F 4 ).

Berends-Giele currents in BCJ gauge
In this section, we relate the Jacobi-satisfying numerators for cubic off-shell diagrams as constructed in the previous section to gauge-transformed Berends-Giele currents. The idea is to compare the Lorenz-gauge currents A µ P , F µν P , F      More generally, each cubic diagram contributing to A µ, BCJ 12...p can be derived from the halfladder topology via kinematic Jacobi relations. The half-ladder numerators at rank p in turn can be expanded in the (p−1)!-element basis {a µ 1ρ(23...p) , ρ ∈ S p−1 } by their Lie symmetries (4.30). As already noted in a superspace context [15,19], currents and master numerators are related by the inverse of the (p−1)! × (p−1)! KLT matrix in (2.42), . Hence, the former also descend from Lie-algebra valued perturbiners [72] such as with gauge parameters Ω P → H P . These transformations are checked to apply to all currents up to and including rank five, see appendix B.3 for the explicit form of H 12345 , and the existence of suitable H 12...p is conjectural at higher rank p ≥ 6. As the punchline of (5.7), the local Jacobi-satisfying numerators for off-shell diagrams are related to Lorenz-gauge currents through a non-linear gauge transformation generated by H P t P e k P ·x . (5.8) In the next sections, the local multiparticle polarizations will be used to manifest the BCJ duality between color and kinematics in tree-level amplitudes of (YM+F 3 +F 4 ). Hence, the transformed currents A µ BCJ P , . . . , G µν|λρ, BCJ P related by (5.2) are said to be in BCJ gauge [27,28]. Note that the first perturbiner solutions to the field equations of ten-dimensional SYM were actually constructed in BCJ gauge [89].

Kinematic derivation of the BCJ relations
As a first application of BCJ-gauge currents, we derive the BCJ relations (2.30) of A YM+F 3 +F 4 amplitudes by inverting their correspondence (5.2) with multiparticle polarizations. The same sequence of arguments has been applied to derive BCJ relations for tree amplitudes of tendimensional SYM from superspace currents in BCJ gauge [28], and we adapt the reasoning of the reference to the bosonic amplitudes up to the α 2 -order.
At rank p, inversion of (5.2) relates multiparticle polarizations to BCJ-gauge currents via where S(σ|ρ) 1 denotes the KLT matrix defined in (2.39). The Lie symmetries of the numerators of the BCJ-gauge currents ensure that the Mandelstam invariants from S(σ|ρ) 1 cancel all of their kinematic poles on the right-hand sides of (5.9). However, when repeating these matrix multiplications with Lorenz-gauge currents A µ P , some of the kinematic poles in (m ≥ 3)-particle channels persist 22 in τ ∈S p−1 S(σ|τ ) 1 A µ 1τ (23...p) with p ≥ 3. Hence, it is a peculiarity of BCJ-gauge currents that local objects are obtained from matrix multiplication with S(σ|ρ) 1 . Similarly, the pole s −1 12...p in the p-particle channel drops out from the following rank-p combinations of BCJ-gauge currents, Similarly, the rank-(p≤5) combinations in (5.10) and (5.11) with regular s 12...p → 0 limit imply the following five-and six-point BCJ relations after multiplication with the vanishing 22 For instance, the product τ ∈S 2 S(2, 3|τ )1A µ 1τ (23) = s12(s23A µ 123 − s13A µ 213 ) can be thought of as a Lorenzgauge analogue of a µ 123 and exhibits a pole in s123 with residue ∼ s12k µ 123 h123. This calls for an all-multiplicity formula for analogous combinations with regular behaviour as s 12...p → 0: The right-hand sides of (5.10) can be generated through the S-map [15,28] involving words P = p 1 p 2 . . . p |P | and Q = q 1 q 2 . . . q |Q| . BCJ gauge of the currents implies that the S-map defined in (5.15) removes the pole in s P Q [15] and therefore paves the way for the following form of the BCJ relations [28]  which is equivalent to (2.30) by the KK relations A YM+F 3 +F 4 ((X¡Y ), n) = 0 ∀ X, Y = ∅.

Local Jacobi-satisfying numerators
In this section, we will exploit the multiparticle polarizations of (YM+F 3 +F 4 ) to construct local and Jacobi-satisfying cubic-diagram numerators. The most direct approach is to expand the BCJ-gauge current in the amplitude representation (5.12) via (5.2), propagators in Φ(τ |ρ) 1 , the expressions a µ 1ρ(23...n−1) e µ n will be shown below to take the role of master numerators of the half-ladder diagrams depicted in figure 6. The (n−2)! KKindependent permutations of A YM+F 3 +F 4 in (5.18) incorporate each cubic diagram at least once and therefore define all of the numerators.
In order to demonstrate that the numerators in (5.18) obey kinematic Jacobi identities, we bring it into the form of the general amplitude representation (2.36) with manifest colorkinematics duality, Figure 13: Diagrammatic interpretation of the local building block N X,Y,Z with multiparticle labels X = x 1 x 2 . . . x p , Y = y 1 y 2 . . . y q and Z = z 1 z 2 . . . z r referring to three off-shell halfladder diagrams that are connected by the central vertex. Note that we no longer distinguish between the orders of α carried by the individual cubic vertices and therefore suppress the white and black dots of figure 8.
There is an ambiguity in relating cubic diagrams to the combinations N X,Y,Z in (5.22): Each of the n−2 cubic vertices may be associated with the central vertex in figure 13, e.g. all of N 123,4,5 , N 12,3,45 and N 1,2,543 describe the same cubic diagram. A valid (n−2)!-set of N X,Y,Z to serve as the master numerators for half-ladder diagrams is given by As a defining property of these master numerators N ... , the central vertex of figure 13 is always chosen to be adjacent to leg n which therefore enters in a single-particle slot. As depicted in figure 14, the numerators in (5.23) describe half-ladder diagrams with endpoints 1 and n−1, where the location of leg n decides about the partition into the three subdiagrams associated with the slots of N X,Y,Z . The remaining labels a 1 , a 2 , . . . , a p , b 1 , b 2 , . . . , b q are a permutation of 2, 3, . . . , n−2 with p+q = n−3. Together with the n−2 different choices of p = 0, 1, . . . , n−3, this exhausts the total of (n−2)! permutations of the larger set 2, 3, . . . , n−2, n. The collection of N 1a 1 a 2 ...ap|n|bq...b 2 b 1 n−1 in (5.23) and figure 14 can be used as an alternative to the master numerators a µ 1ρ(23...n−1) e µ n in (5.19). As a practical advantage of the N ... , their constituents in (5.22) only require multiparticle polarizations of maximal rank n−2 instead of the rank-(n−1) quantities a µ 1ρ(23...n−1) . As demonstrated in appendix C 23 , they yield Jacobi-satisfying amplitude representations of the form (2.36), 1, 2, . . . , n)) = Note that the α → 0 order of (5.24) follows from the field-theory limit of the pure-spinor superstring based on the amplitude representations of [14,61] and the superspace gauge described in [27]. Moreover, one can further reduce the maximum rank of the multiparticle polarizations by a generalization of the integration-by-parts relation (3.17). The latter still holds when the M A,B,C are constructed from BCJ gauge currents 24 and multiplied by KLT matrices for the slots A, B, C, so the rank-(n−2) cases in (5.24) can be reduced as follows

(5.26)
The right-hand sides of (5.25) and (5.26) at n ≥ 5 can be assembled from multiparticle polarizations of maximum rank n−3, and the spurious poles in s n−1,n cancel after combining all the terms. The same strategy applies to permutations of N 12...n−2|n|n−1 and N 1|n|23...(n−2)(n−1) in 2, 3, . . . , n−2. Like this, the n-point amplitude representation (5.24) with manifest BCJ duality and local numerators is completely determined by multiparticle polarizations of rank n−3. For instance, the explicit construction of multiparticle polarizations up to rank five in section 4 is sufficient to pinpoint all the eight-point numerators in (5.24). Note that the special footing of legs 1, n−1 and n in (5.24) breaks the crossing symmetry even more heavily than the numerators in (5.19). Still, one can restore crossing symmetry by averaging over all choices of singling out legs i, j, k ∈ {1, 2, . . . , n} instead of 1, n−1 and n.

Relation to string-theory and gravity amplitudes
A major motivation for the construction of (YM+F 3 +F 4 ) numerators with manifest locality stems from their connection with gravitational quantities through the double copy. Following the lines of [18], the double copy of A YM+F 3 +F 4 to amplitudes from higher-curvature operators α R 2 + α 2 R 3 can be extracted from the string-theory KLT relations [1] (also see [23,24] comprise our results for A YM+F 3 +F 4 (and the aforementioned contribution from the supersymmetrizable F 4 -operator which is incompatible with the BCJ duality and can be distinguished by its coefficient ζ 2 [18]). By the interplay with the trigonometric factors in the KLT formula, both copies of A super−F 4 (σ) drop out from relevant orders of the closed bosonic string [21], where S(ρ|τ ) 1 is the field-theory KLT matrix defined in (2.39) and the permutations ρ, τ act on 2, 3, . . . , n−2. Hence, to the orders considered, the right-hand side of (5.28) describes amplitudes from the low-energy effective action of the closed bosonic string [25] S closed where ϕ denotes the dilaton and H = dB is the field strength of the B-field. In spite of the dilaton admixtures via e −2ϕ , e −4ϕ , the operators along with the first and second order of α are collectively referred to as R 2 and R 3 . While the R 2 operator can be reconciled with up to sixteen supercharges [91], the R 3 operator is not supersymmetrizable [92]. Given the multitude of propagators in KLT formulae of the form (5.28), the locality properties of gravity amplitudes are more transparent in representations involving Jacobisatisfying numerators as in (2.34). For instance, our master numerators for (YM+F 3 +F 4 ) in where each term has the propagator structure of cubic diagrams. The subscript GR+R 2 +R 3 is just a schematic shorthand for the amplitudes generated by the action (5.29) to the orders of α 2 . As emphasized in [18], the α 2 -order of (5.30) receives contribution from both singleinsertions of R 3 operators and double-insertions of R 2 operators. In D = 4 spacetime dimensions, the R 2 contribution to (5.29) is the topological Gauss-Bonnet term. The components at the first order in α of (5.30) with graviton helicities are therefore guaranteed to vanish. Still, the double insertions of R 2 contribute to the α 2 -order of graviton components in four dimensions since the prefactor of e −2ϕ in (5.29) allows for dilaton exchange [18].
On the right-hand side of (5.28) or (5.30), the α 2 order receives both symmetric and asymmetric contributions: Terms of the form A YM+F 3 +F 4 α Ã YM+F 3 +F 4 α where both gaugetheory halves contribute a factor of α have been carefully analyzed in D = 4 helicity components [18]. Our results on the α 2 -order of A YM+F 3 +F 4 and its master numerators additionally deliver the contributions to D-dimensional amplitudes of GR + R 2 + R 3 , where both powers of α stem from the same gauge-theory factor. These contributions involving A YM+F 3 +F 4 α 2 explain 25 the departure of M GR+R 2 +R 3 α 2 from the double copy of the first α -order A YM+F 3 +F 4 α which has already been observed in certain D = 4 helicity components [18]. Hence, there is no need to consider higher orders from the α -expansion of the sin( πα 2 k i · k j ) terms in the string-theory KLT relations as speculated in the reference. At the order of α , one may extract new representations 26 for supersymmetrized matrix elements of R 2 from (5.30) by trading A YM+F 3 +F 4 for color-ordered amplitudes of tendimensional SYM and their dimensional reductions. These supersymmetrizations play a key role in recent studies of divergences and duality anomalies of N = 4 supergravity [55,56].

Conclusions and outlook
In this work, we have studied various representations for tree-level amplitudes of D-dimensional gauge theories with α F 3 + α 2 F 4 deformations. Our results are independent on the choice of gauge group and hold to the order of α 2 , where the interplay of the F 3 and F 4 -operators is known to result in the BCJ duality between color and kinematics [18]. While the BCJ duality has originally been explained by the realization of the α F 3 + α 2 F 4 operators from the open bosonic string, our work takes a different approach by identifying the seeds of the duality in the Berends-Giele currents of (YM+F 3 +F 4 ).
We study the Berends-Giele currents of (YM+F 3 +F 4 ) in the perturbiner formalism [66][67][68][69][70], where non-linear gauge transformations can be mapped to reparametrizations of the scattering amplitudes [27]. We pinpoint a specific non-linear gauge transformation up to the order of five on-shell legs which rearranges the naive Feynman-diagram output of the action such as to manifest the BCJ duality. Like this, we derive the BCJ relations among colorordered amplitudes to the order of α 2 from purely kinematic considerations. Furthermore, two kinds of explicit cubic-diagram parametrizations are given for (YM+F 3 +F 4 )-amplitudes where the manifestly local numerators obey kinematic Jacobi relations.
Our construction is inspired by superspace kinematic factors of ten-dimensional SYM [27,28] whose properties were inferred from the conformal-field-theory description of the purespinor superstring [14,15,35]. We identify extensions of these superspace-inspired structures to higher orders in α and to operators F 3 , F 4 that do not admit any supersymmetrization. It would be interesting to find a conformal-field-theory derivation of the local multiparticle polarizations that drive our BCJ-duality-satisfying amplitude representations. One possible starting point is to combine the worldsheet description of the bosonic string with the off-shell techniques of [16]. Alternatively, it might be helpful to identify a vertex-operator origin of the CHY formulae for F 3 amplitudes [26] 27 along with generalizations to higher orders in α .
A complementary approach to the α F 3 + α 2 F 4 operators of the bosonic string is suggested by the recent double-copy description of bosonic-string amplitudes [22]: After peeling off the worldsheet integrals that are common with superstring amplitudes, an all-order family of α -corrections of the bosonic string can be traced back to a massive gauge theory dubbed (DF ) 2 + YM. The latter has been constructed in [95] by imposing the BCJ duality on a collection of dimension-six interactions between gauge bosons and massive scalars, and it should reproduce the (YM+F 3 +F 4 )-amplitudes in this work upon low-energy expansion. It would be interesting to study our results from the (DF ) 2 + YM-perspective and to generalize them to arbitrary orders in α by integrating out its massive modes.

A Properties of the cyclic building blocks
In this appendix, we prove several properties of the building block M A,B,C defined in (3.15) to first order in α . These proofs are based on transversality k P · A P = 0 and the truncation of (3.14) to the first order in α , Moreover, we will use the generalization of (A.1) to higher mass dimension, which follows from the definition [∇ µ , ∇ λ ] = −F µλ and a corollary of the equations of motion, This can be inserted into (A.2) to find By virtue of (A.1), (A.2) and (A.4), one can rewrite any contraction of F µν P and F µ|νλ P with the corresponding momentum k P in terms of deconcatenations. We will always work to first order in α , but we will split the proofs into different orders for the convenience of the reader.

A.1 Appearance in the amplitudes
The first property we wish to prove is where, as a special case of (3.15), ) .

(A.7)
We first focus on the zeroth order of α . Notice how the last two terms on the first line of (A.7), when contracted with A µ n , exactly match the first and third terms on the first line of (A.6). To see that the remaining terms are equal to each other, we notice that where we have used momentum conservation k µ n = −k µ X −k µ Y and transversality. This matches with the missing terms at the α 0 order of (A.7), upon contraction with A µ n . We now show that the same matching occurs between the terms of order α in both expressions. Equating them leads to Now we have to make sure that the remaining terms C, D, E, F and G on the left-hand side of (A.9) cancel each other. The first two can be rewritten as XY =12...n−1 where the second line follows from (A.2). In passing to the third line, we have again used momentum conservation and exploited antisymmetry F νλ n = F [νλ] n . Hence, the statement (A.5) is proved to first order in α . which translates into the following claim at the zeroth order in α , (A.14) We can rewrite the second term ∼ A X A Y in the first line using the antisymmetry of F µν B and the definition (3.6) of F µν A : The analogous sum in the second line of (A.14) has a similar term related by A ↔ B, 16) where F µν A F µν B cancels against (A.15). For the remaining terms F µν Inserting this into (A.14) and ignoring the O(α )-term in the last line, we conclude that the property (A.13) is indeed satisfied to zeroth order in α . We now show that the property is still valid at the first order in α . In doing that, we have to combine the last term of (A.17) with the O(α )-terms in the M A,B,C of (A.13). Hence, the leftover task is to prove that Let us start by using the Jacobi identity (A.3) to rewrite the following terms: The right-hand sides cancel the first two terms inside the sums in (A.18) over A = XY and B = XY . Then, using the recursive definition of F µ|νλ P in (3.9), we can write ) . (A.21) Since the contribution from G + H takes the same form with A ↔ B, the terms F ν|λµ B F µ|νλ A cancel from the combination Also, using Jacobi identity (A.3) as well as (A.2), we have and using the same manipulations The terms of the form k µ · F νλ · F ν|µλ · cancel between (A.22), (A.23) and (A.24) by momentum conservation k A + k B = 0. Finally, the contributions of the form A=XY F νλ B F µλ X F µν Y cancel between (A.23), (A.24) and the leftover terms of (A.18). This concludes our proof of the integration-by-parts identity (A.13) to the order of α .

A.3 Gauge algebra
Finally, we want to see how a non-linear gauge transformation (3.22) acts on M X,Y,Z . To zeroth order in α , we get (A. 25) Let us look at the terms which are not inside a sum. We can start by grouping all the ones with the same Ω · coefficient, and use momentum conservation k µ X + k µ Y + k µ Z = 0: We then rewrite these four terms via and the same identities with (Y ↔ Z). In the first line, we rewrote k µ Y F µν Y using the zeroth order in α of (A.1). The last two lines of (A.27) are based on the antisymmetry of F µν Y and the definition (3.6) of F µν Z . Notice that F µν Y F µν Z cancels in the antisymmetrization w.r.t. Y ↔ Z in (A.26). Hence, all the leftover terms in (A.25) involve a sum over deconcatenations, either X=AB or one of (X ↔ Y, Z). We collect all the expressions from the cyclic permutations in (A.25) where the sum is X=AB (A. 28) It turns out that the coefficient of each of the Ω's inside the above sum can be identified as some M P,Q,R with various combinations of the three words: The object inside the sum over X = AB is totally antisymmetric in A, B, Y, Z and can be identified as Ω A,B,C,D as defined in (3.24). Hence, the zeroth order of the gauge transformation (A.25) can be written as We now want to extend the proof of (A.30) to the first order in α . First of all, terms of O(α ) have been neglected when inserting (A.1) into the first term of (A.27). Therefore, we carry forward the following terms in δ Ω M X,Y,Z , where we have spelt out all the terms of the same form X=AB as in (A.28). This needs to be combined with the gauge variation of the O(α ) terms in the definition (3.15) of M X,Y,Z : The only term in (A.32) which is not yet in the form of a deconcatenation sum will now be rewritten via momentum conservation k µ X + k µ Y + k µ Z = 0: (A.33) The first term calls for the relation (A.2),  such that all the terms in the quantity G have been expressed via deconcatenation sums: F νλ Z ± perm(X, Y, Z) . δ Ω M X,Y,Z α 1 = 1 2 In this appendix, we spell out the rank-five generalization of the gauge scalars H P in (5.