Color-factor symmetry of the amplitudes of Yang-Mills and biadjoint scalar theory using perturbiner methods

Color-factor symmetry is a property of tree-level gauge-theory amplitudes containing at least one gluon. BCJ relations among color-ordered amplitudes follow directly from this symmetry. Color-factor symmetry is also a feature of biadjoint scalar theory amplitudes as well as of their equations of motion. In this paper, we present a new proof of color-factor symmetry using a recursive method derived from the perturbiner expansion of the classical equations of motion.


Introduction
The discovery of color-kinematic duality in the amplitudes of Yang-Mills theory, and subsequently in the amplitudes of a much broader class of field theories (see ref. [1] for a review), has unleashed a tool of great power, particularly in the calculation of gravitational amplitudes through the double-copy procedure [2][3][4]. In 2008, Bern, Carrasco, and Johansson (BCJ) showed that the assumption of color-kinematic duality in tree-level amplitudes of Yang-Mills theory implies a set of linear relations among the color-ordered amplitudes. The subsequent proof of these BCJ relations using string-theory techniques [5,6] and BCFW on-shell recursion [7,8] provided evidence for the conjecture of tree-level color-kinematic duality. Bern et al. also conjectured that color-kinematic duality applies to integrands of loop-level amplitudes [2,3]; while not proven, this conjecture has been tested for amplitudes of various multiplicities and loop levels in supersymmetric Yang-Mills theories, which have been used to construct supergravity amplitudes [1].
In 2016, R. W. Brown and the current author observed that tree-level gauge-theory amplitudes possess a color-factor symmetry, which acts as a momentum-dependent shift on the color factors of an amplitude, leaving the full amplitude invariant [9][10][11]. This symmetry was proved for both Yang-Mills theory and for gauge theories with massive particles of various spins using the radiation vertex expansion [12]. The BCJ relations follow as an immediate consequence of color-factor symmetry [9,10].
Color-kinematic duality and color-factor symmetry are closely related features of gauge theories: the former implies the latter (as proved using the cubic vertex expansion), but the latter implies a less stringent (but gauge-invariant) constraint than the former on the kinematic numerators of a tree-level amplitude [9]. Similarly color-kinematic duality implies color-factor symmetry for loop-level amplitudes, but no independent proof of the latter has yet been developed.
Color-factor symmetry is also a property of tree-level amplitudes of the biadjoint scalar (BAS) theory [13], whose fields transform in the adjoint representation of U(N) × U(Ñ ), as was proved using the cubic vertex expansion [9]. Cheung and Mangan [14] observed that the classical equations of motion of the BAS theory also possess color-factor symmetry, and that this implies the invariance of the tree-level amplitudes. They demonstrated a relation between the U(N) colorfactor symmetry of the equations of motion and the conservation of current associated with the dual U(Ñ ) symmetry. In ref. [15], these results were generalized to curved symmetric spacetime.
In this paper, we offer a new proof of color-factor symmetry based on a recursive approach. In 1987, Berends and Giele [16] introduced a method for computing tree-level QCD amplitudes using a set of partially off-shell amplitudes (subsequently known as Berends-Giele currents), which were then computed recursively. Rosly and Selivanov [17][18][19] later showed that a perturbative solution of the classical equations of motion (dubbed the perturbiner expansion) acts as a generating function for Berends-Giele currents. Mafra, Schlotterer, et al. [20][21][22][23] also used classical equations of motion to generate Berends-Giele currents in various theories. Mizera and Skrzypek [24] introduced the color-dressed perturbiner expansion, which, as we will see in this paper, is well adapted for the demonstration of color-factor symmetry of tree-level amplitudes.
Further developments in this subject include the work of Lopez Arcos, Quintero Vélez, et. al., who related the L ∞ -algebra that appears in Batalin-Vilkovisky quantization [25] to the perturbiner expansion for biadjoint scalar and Yang-Mills theories [26] as well as in gauge theories with matter [27]. Berends-Giele currents in BCJ gauge were constructed using Bern-Kosower rules [28], with this work extended to gravity using the double-copy procedure [29]. Gomez and Jusinskas have applied perturbiner methods to gravity coupled to matter [30], and in ref. [31], perturbiner methods were used to compute tree-level boundary correlators in anti-de Sitter space. The perturbiner approach has also been found effective for computing one-loop integrands [32]. The connection between tree-level Berends-Giele recursion relations and the L ∞ -algebra uncovered in ref. [25] was extended to loop-level recursion relations and the quantum homotopy algebra A ∞ in ref. [33]. For connections between the homotopy algebra and the double copy, see refs. [34][35][36].
We present this alternative proof of color-factor symmetry because the recursive methods employed may be more familiar to modern readers than the radiation vertex expansion used in ref. [9] to prove color-factor symmetry. Moreover, this recursive approach may be easier to generalize to the exploration of color-factor symmetry in other theories.
The outline of this paper is as follows. In sec. 2, we recall how color-factor symmetry acts on amplitudes, and how the BCJ relations follow as a consequence. In sec. 3, we show how the color-dressed perturbiner expansion is used to compute tree-level amplitudes, first in the biadjoint scalar theory, and then in Yang-Mills theory. In sec. 4, we then use the color-dressed perturbiner expansion to prove color-factor symmetry for tree-level amplitudes of the biadjoint scalar theory and of Yang-Mills theory. Sec. 5 contains our conclusions.

Color-factor symmetry and BCJ relations
Tree-level scattering amplitudes of a gauge theory are given by a sum of Feynman diagrams, and can be expressed as [37] where c i are color factors, consisting of the contraction of various color tensors f abc and (T a ) i j appearing in the Feynman diagrams, and a i depends on kinematic and spin factors. Each color factor can itself be represented as a Feynman diagram [38,39], one that contains only trivalent vertices. (If the full Feynman diagram contains only trivalent vertices, then it contributes to the color factor with the same Feynman diagram. If the full Feynman diagram also contains quartic vertices, then its contribution is parcelled out among different color factors by expressing the quartic vertex as products of trivalent vertices.) Note that, due to the group theory identities there exist (Jacobi) relations among the various color factors c i . Since the c i are not independent, there is some choice about how the coefficients a i are defined.

Color-factor symmetry
There exists a color-factor symmetry associated with each external gluon a contributing to the amplitude [9,10]. This symmetry acts on each color factor c i appearing in eq. (2.1) by a momentum-dependent shift δ a c i . For each color factor c i , the gluon leg a divides the associated tree-level diagram in two at its point of attachment. Let S a,i denote the subset of the remaining legs on one side of this point; it does not matter which side we choose. The shift of the color factor c i associated with gluon a then satisfies 1 where k µ a is the outgoing momentum of gluon a (satisfying k 2 a = 0), and k µ d are the outgoing momenta of the legs belonging to S a,i . The color-factor shift also respects the group theory identities, as we will see below.
We may regard the color-factor symmetry as acting directly on the color tensors appearing in c i . If gluon a (with color a) is attached to a gluon line, so that the color factor contains f bac , then the color-factor symmetry acts as [14] δ a f bac = α a δ bc (k 2 where k µ b and k µ c are the momenta flowing out of the vertex associated with f bac and α a is a constant parameter. If gluon a is attached to a line corresponding to a particle in some other representation, so that the color factor contains (T a ) i j , then the color-factor symmetry acts as where k µ i and k µ j are the momenta flowing out of the vertex associated with (T a ) i j . Using momentum conservation at each vertex, we may express these shifts as The relations (2.7) guarantee that the color-factor shifts satisfy eq. (2.4). We must also check that the color-factor shifts leave the group theory identities invariant. Using eq. (2.7) in eqs. (2.2) and (2.3), we find using momentum conservation and the masslessness of the gluon.
In ref. [9], the n-point amplitude eq. (2.1) was proved to be invariant under the color-factor shift associated with any of the external gluons it contains by rewriting the amplitude using the radiation vertex expansion. In sec. 4 we give an alternative proof of this fact using the recursive perturbiner approach.

BCJ relations
In the remainder of this section, we recall the demonstration [9] that color-factor symmetry of the amplitude implies the fundamental BCJ relation [5,7,40] among the color-ordered amplitudes.
For tree-level amplitudes containing fields in other representations (e.g. quarks) in addition to gluons, an independent basis of color factors is given by the Melia basis [43][44][45]. The independent amplitudes corresponding to this basis also satisfy BCJ relations that follow from the assumption of color-kinematic duality [46,47]. Color-factor symmetry can also be used to derive BCJ relations for these amplitudes [10].

Color-dressed perturbiner expansion
In this section, we review the color-dressed perturbiner expansion [24] of the solutions to the classical equations of motion for the biadjoint scalar theory and Yang-Mills theory, and how its coefficients (Berends-Giele currents) are used to obtain tree-level n-point amplitudes in those theories.

Biadjoint scalar theory
The biadjoint scalar theory is a theory of a massless scalar field φ aa ′ transforming in the adjoint where f abc andf a ′ b ′ c ′ are the structure constants 2 of U(N) and U(Ñ ) respectively. This Lagrangian yields the equation of motion Rosly and Selivanov [17][18][19] introduced the perturbiner ansatz, which is a solution to the nonlinear classical equation of motion obtained by first solving the free equation of motion ∂ 2 φ aa ′ = 0 with an arbitrary linear combination of plane waves 3 and then using this as a seed in eq. (3.2) to generate corrections higher order in λ. The coefficients of this expansion are used to compute tree-level amplitudes of the theory.
For the purpose of proving the color-factor symmetry of tree-level amplitudes in sec. 4, we find it convenient to use a version of the perturbiner ansatz developed by Mizera and Skrzypek [24], called the color-dressed perturbiner expansion (in distinction from the color-stripped perturbiner expansion). For the BAS theory, the ansatz can be written 4 summing over all non-empty ordered words P = p 1 p 2 · · · p m with 1 ≤ p 1 < p 2 < · · · < p m ≤ M, where k P = m j=1 k p j . By inserting eq. (3.5) into eq. (3.2) one obtains [24] where P = Q ∪ R denotes all possible divisions of P into two non-empty ordered words Q and R. The coefficients φ aa ′ P are Berends-Giele currents 5 of the BAS theory, computed recursively using eq. (3.6). One sees that φ aa ′ P has a pole at k 2 P = 0. To obtain the tree-level n-point amplitude A n , one first computes φ aa ′ P for P = 12 · · · (n − 1). Since momentum conservation for the n-point amplitude implies k P = −k n , and an on-shell amplitude has k 2 n = 0, one extracts the residue of the k 2 P pole of φ aa ′ P and contracts with φ aa ′ n to get [24] A n = lim To illustrate this procedure for the four-point amplitude we first use eqs. (3.3) and (3.6) to compute the rank-2 perturbiner coefficient 3.8) and from this the rank-3 coefficient where . This result agrees with four-point amplitude found in ref. [13].

Yang-Mills theory
We now describe the color-dressed perturbiner expansion for Yang-Mills theory [24]. The Yang-Mills Lagrangian implies the equation of motion 6 where the Yang-Mills field strength is given by Choosing Lorenz gauge ∂ ν A ν a = 0 (3.14) we can write eq. (3.12) as For convenience we define G νµa ≡ −i (∂ ν A µ a + F νµ a ), which becomes so that eq. (3.15) is expressed as The advantage of using two fields, A µa and G νµa , rather than just A µa is that eqs. (3.16) and (3.17) contain only quadratic (not cubic) terms, simplifying the recursion relations derived below [20][21][22][23][24].
We now solve these equations with the perturbiner ansatz. As in the previous subsection, we begin by solving the free equation with an arbitrary linear combination of plane waves Also to this order we have As before, the lowest-order solution is the first term of the color-dressed perturbiner expansion 7 The coefficients A µ a P are the (color-dressed) Berends-Giele currents of the Yang-Mills theory [16]. To obtain the tree-level n-gluon amplitude, one first computes A µa P for P = 12 · · · (n − 1), then extracts the residue of the k 2 P pole, and finally contracts 8 with the Berends-Giele current A µ a n of the last gluon [16] A n = lim A µ a n k 2 P A µ a P . (3.24) The recursion relations for the Berends-Giele currents A µa P are obtained by plugging eq. (3.23) into eq. (3.17) to obtain Similarly, plugging eq. (3.23) into eq. (3.16) we find We combine the three previous equations to obtain (3.28) Eqs. (3.25) and (3.28) play a key role in the proof of color-factor symmetry in the next section.
In the remainder of this section, we illustrate how they are used recursively to compute the four-gluon amplitude. We first use eqs. (3.25) and (3.28) together with eqs. (3.20) and (3.22) to obtain the rank-2 coefficients These are then used in eq. (3.25) to determine the rank-3 coefficient where c a 123 = f a 1 a 2 c f ca 3 a , (3.32) We may obtain a more explicit form for n µ 123 by using eq. (3.22) in eq. (3.33) and simplifying We observe that color factors appearing in the Berends-Giele current (3.31) satisfy the Jacobi identity c a 123 + c a 231 + c a 312 = 0, but the kinematic numerators do not because k 1 + k 2 + k 3 = −k 4 and ε 4 · k 4 = 0.

Recursive proof of color-factor symmetry
In this section, we present proofs that the tree-level n-point amplitudes of the BAS theory and Yang-Mills theory are invariant under the color-factor shifts described in sec. 2 using the recursion relations derived from the color-dressed perturbiner expansions in sec. 3.

Biadjoint scalar theory
We begin by combining eq. (3.7) with eqs. (3.3) and (3.6) to obtain the following expression for the tree-level n-point amplitude of the BAS theory We now determine how this amplitude transforms under the color-factor symmetry associated with scalar n. The color-factor symmetry acts only on the U(N) structure constants f abc , with the U(Ñ ) structure constantsf a ′ b ′ c ′ behaving as spectators. 9 From eq. (2.5) we have Since the sum over divisions of P into words B and C is symmetric under B ↔ C, we may relabel Now we again use eq. (3.6) to obtain (4.5) Using the invariance of the sum over P = B ∪ D ∪ E under any permutation of the words B, D, and E, we observe that the term in parentheses is invariant under cyclic permutations (4.6) 9 Naturally, one could alternatively define color-factor shifts that act onf a ′ b ′ c ′ .
Since the term in the left parenthesis vanishes by the Jacobi identity, the n-point amplitude is invariant under the color-factor symmetry associated with scalar n. Since the amplitude is Bose symmetric, it is invariant under the color-factor symmetry associated with any of the external fields δ a A n = 0 (4.7) as was previously established using the cubic vertex expansion [9].

Yang-Mills theory
To prove that the tree-level n-gluon amplitude is invariant under color-factor shifts, we begin by combining eq. (3.24) with eqs. (3.20) and (3.25) to obtain The color-factor symmetry associated with gluon n acts only on the explicit factor f anbc in the equation above, giving Our goal is to show that the right hand side of this equation vanishes, so we must first compute which unfortunately is a bit more complicated than the biadjoint scalar case. First we use eqs. (3.25) and (3.28) to find where k µ BC = k µ B + k µ C . We use eq. (3.26) to reexpress this as Equation (4.12) can be split into two contributions S µ P = S µ P,1 + S µ P,2 , (4.13) where S µ P,1 contains the terms in which the free index µ labels a momentum k and S µ P,2 contains those in which it labels a field A or H.
First we examine S µ P,1 . Relabelling B, b ↔ C, c in the last line (the k µ B term) of eq. (4.14) and using f acb = −f abc , we obtain two terms S µ P,1 = S µ P,1a + S µ P,1b , (4.16) For S µ P,1a , we relabel A, a ↔ B, b in the first two terms of eq. (4.17) and use f bac = −f abc to obtain Then we cyclically relabel the last two terms of eq. (4. 19) and use f bca = f cab = f abc to obtain S µ P,1a = gf abc where we have used k µ A + k µ B + k µ C = k µ P . Next, we turn to S µ P,1b , observing that the first two terms in eq. (4.18) cancel (since H νλ b Now we turn to S µ P,2 . The two terms on the first line of eq. (4.15) vanish using f abc = −f bac , leaving S µ P,2 = S µ P,2a + S µ P,2b , (4.25) For S µ P,2b , we use eq. (3.27) with J = D ∪ C in eq. (4.27) to obtain Letting A, a ↔ D, d in eq. (4.29) and using f dbj f jac = f cai f ibd , we obtain Recombining eqs. (4.28) and (4.30) and symmetrizing on B, b ↔ D, d we find S µ P,2 = S µ P,2a + S µ P,2b = S µ P,2c + S µ P,2d , (4.31) For S µ P,2c , we cyclically relabel ABD in four of the six terms in eq. (4.32) to obtain which vanishes by the Jacobi identity.
For S µ P,2d , we use eq. (3.27) with J = D ∪ E in eq. (4.33) to obtain we may replace which vanishes identically. 10 Hence we also have that S µ P,2d = 0.
In sum, we have shown that S µ P,2 = 0, leaving S µ P = S µ P,1 as given in eq. (4.24) Note that for P = 123 the first term is given by gf abc (n µ 123 + n µ 231 + n µ 312 ) as written in eq. (3.35), and the second term is absent.
Combining eq. (4.9) with eq. (4.37) we therefore have that the change in the amplitude under the color-factor shift associated with gluon n is δ n A n = g 2 α n ε µ n k µ P f abc where P = 12 · · · (n − 1). Momentum conservation n i=1 p µ i = 0 implies k µ P = −k µ n . Since ε n · k n = 0, we have established that the n-gluon amplitude is invariant under the color-factor shift associated with gluon n. Since the n-gluon amplitude is Bose symmetric, it is therefore invariant under a color-factor shift associated with any of the external gluons δ a A n = 0 (4.39) which is what we set out to prove.

Conclusions
We began by reviewing the color-factor symmetry of tree-level amplitudes of the BAS and Yang-Mills theories. This symmetry acts as a momentum-dependent shift on the color factors, leaving the amplitude invariant. The BCJ relations follow as a direct consequence of this symmetry.
Tree-level amplitudes can be obtained from Berends-Giele currents, which are computed recursively. The recursions relation for the currents can be derived from the classical equations of motion of the theory using the color-dressed perturbiner formalism. We used these recursion relations, together with a variety of group theory relations, to prove the invariance of tree-level amplitudes under a color-factor shift. This proof is a (somewhat) easier alternative to the proof of color-factor symmetry using the radiation vertex expansion given in ref. [9], and is amenable to generalization to other theories.
Cheung and Mangan [14] have shown that the color-factor symmetry of the BAS theory, with scalars transforming in the adjoint of U(N) × U(Ñ ), also applies to the equations of motion of the theory, and that the color-factor invariance of the equations of motion associated with U(N) is related to the conservation of current of the global symmetry of the Lagrangian under the dual group U(Ñ ). It would be interesting to find a similar relation for Yang-Mills and other theories possessing color-kinematic duality.