BCJ relations from a new symmetry of gauge-theory amplitudes

We introduce a new set of symmetries obeyed by tree-level gauge-theory amplitudes involving at least one gluon. The symmetry acts as a momentum-dependent shift on the color factors of the amplitude. Using the radiation vertex expansion, we prove the invariance under this color-factor shift of the $n$-gluon amplitude, as well as amplitudes involving massless or massive particles in an arbitrary representation of the gauge group with spin zero, one-half, or one. The Bern-Carrasco-Johansson relations are a direct consequence of this symmetry. We also introduce the cubic vertex expansion of an amplitude, and use it to derive a generalized-gauge-invariant constraint on the kinematic numerators of the amplitude. We show that the amplitudes of the bi-adjoint scalar theory are invariant under the color-factor symmetry, and use this to derive the null eigenvectors of the propagator matrix. We generalize the color-factor shift to loop level, and prove the invariance under this shift of one-loop $n$-gluon amplitudes in any theory that admits a color-kinematic-dual representation of numerators. We show that the one-loop color-factor symmetry implies known relations among the integrands of one-loop color-ordered amplitudes.


Introduction
In 2008, Bern, Carrasco, and Johansson discovered a novel set of linear relations satisfied by treelevel color-ordered amplitudes in gauge theories [1]. They arrived at these relations by writing the tree-level n-gluon amplitude as a sum over (2n − 5)!! diagrams assembled from cubic vertices where the color factor c i associated with the diagram is composed of group theory structure constants f abc , the denominator d i consists of the product of the inverse propagators associated with the diagram, and the kinematic numerator n i depends on the momenta and polarizations of the gluons. All contributions from diagrams with quartic vertices are redistributed among the cubic diagrams. By virtue of the Jacobi identity f abe f cde +f ace f dbe +f ade f bce = 0 satisfied by the structure constants, the color factors c i obey a set of Jacobi relations of the form Because of these linear dependences, the kinematic numerators n i are not uniquely defined, but can undergo generalized gauge transformations n i → n i + δn i which leave eq. (1.1) unchanged [2,3].
The authors of ref. [1] conjectured that there exists a generalized gauge in which the kinematic numerators satisfy the same algebraic relations as the color factors; in particular, they can be made to satisfy kinematic Jacobi relations From this assumption of color-kinematic duality, they demonstrated the existence of new relations (subsequently known as BCJ relations) satisfied by the color-ordered amplitudes A(1, · · · , n). These relations can be derived from the fundamental BCJ relation (and permutations thereof) [4][5][6] n b=3 b−1 c=1 k 2 · k c A(1, 3, · · · , b − 1, 2, b, · · · , n) = 0 (1.4) where k a are the (outgoing) momenta of the gluons. Besides color-kinematic duality, these relations rely on the properties of the propagator matrix [7], constructed from the inverse denominators 1/d i (see sec. 8 for a precise definition). Specifically, as a consequence of momentum conservation, this (n − 2)! × (n − 2)! matrix has rank (n − 3)!, and consequently possesses a set of (n − 3)(n − 3)! eigenvectors with eigenvalue zero.
Despite the fact that the BCJ relations for n-gluon amplitudes have been definitively established, interest in tree-level kinematic numerators continues, not least because the numerators that are naturally generated by Feynman rules 1 generally do not obey the relations (1.3) except in the case of four-point amplitudes [23,24]. Many approaches have been developed to obtain kinematic numerators that obey color-kinematic duality directly from a Lagrangian approach [3,[25][26][27][28][29][30][31][32][33][34].
In this paper, we introduce a new set of symmetries obeyed by tree-level gauge-theory amplitudes, associated with each external gluon in the amplitude. 2 These symmetries act on the color factors c i of the amplitude, shifting them by momentum-dependent quantities. Since color factors do not carry any momentum dependence, this is a purely formal operation; we prove, however, that the tree-level n-gluon amplitude is invariant under these shifts by writing it in an alternative form known as the radiation vertex expansion [35].
We then show that the BCJ relations (1.4) follow as an immediate consequence of the color-factor symmetry of the n-gluon amplitude. Although BCJ relations have been previously established, our results reveal a more direct connection to the symmetries of the Lagrangian formulation of gauge theory and its Feynman rules (i.e., gauge and Poincaré invariance) and provide a basis for generalizations.
Let us describe this symmetry in a bit more detail, reserving a full description for section 2. Given a tree-level n-gluon color factor c i , the choice of one of the external gluon legs a divides the 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 must satisfy δ a c i ∝ c∈S a,i k a · k c . (1.5) the diagram in clockwise order. Flipping a to the other side of a leg changes the sign of the color factor due to the antisymmetry of f abc . As a result of the Jacobi identity, the color factors obey 3 r=1 c (r) = 0. By eq. (1.5), the shifts of these color factors are δ a c (r) = α (r) k a · K (r) (1.6) where K (r) is the momentum flowing out of each leg. Requiring 3 r=1 δ a c (r) = 0 implies that α (r) is independent of r, as a result of momentum conservation and masslessness of the gluon. A more detailed description of the color-factor shifts is given in sec. 2.
The symmetry we have introduced has roots in the radiation symmetry [36] that underlies the general radiation zero theorem [35,37,38]. In theories with local gauge couplings and spins ≤ 1, all single-photon tree amplitudes vanish if the ratios Q c /k a ·k c are all equal, 3 where k a is the photon momentum, and c labels external particles with momentum k c and charge Q c . These spin-independent zeros have spin-dependent counterparts where Q c are replaced by numerators J c , closely related to the kinematic numerators n i in eq. (1.1). The underlying radiation symmetry refers to invariance under Q c → Q c + αk a · k c as well as J c → J c + βk a · k c for arbitrary α and β. The extension to nonabelian "charges" has also been considered and the details behind a nonabelian radiation vertex expansion discussed [35,36]. The general color-factor symmetry introduced here, however, incorporates crucial nonabelian constraints (Jacobi relations) on α, which lead to a complete set of BCJ relations, and have not heretofore been developed. Nevertheless, since the color-factor symmetry relies on the presence of massless gauge bosons, we may regard it as a generalized radiation symmetry.
We also introduce in this paper the cubic vertex expansion of an n-point amplitude A n with respect to one of the gluons a. Consider the set of cubic diagrams I that contribute to the (n − 1)-point amplitude of all the particles in A n except for gluon a. We show that, for any a, the amplitude A n can be written as a triple sum over the legs r of the vertices v of the cubic diagrams I: c (a,I,v,r) n (a,I,v,r) 2k a · K (a,I,v,r) . (1.7) Here d (a,I,v,r) is the product of inverse propagators that branch off from leg r of vertex v of diagram I, c (a,I,v,r) is the color factor of the n-point diagram obtained by attaching gluon a to leg r of vertex v of diagram I (exactly as in fig. 1), and n (a,I,v,r) is the associated n-point kinematic numerator. The shift of c (a,I,v,r) associated with gluon a is δ a c (a,I,v,r) = α (a,I,v) k a · K (a,I,v,r) , where, as explained above, α (a,I,v) is independent of r. Since the alternative radiation vertex expansion shows that the amplitude A n is invariant under the color-factor shift, we may conclude from the cubic vertex expansion of A n that Note that this constraint on the kinematic numerators, less stringent than the kinematic Jacobi relations (which state that 3 r=1 n (a,I,v,r) = 0 for each vertex), is nonetheless sufficient to imply the BCJ relations (1.4). Moreover, unlike the kinematic Jacobi relations, the condition (1.8) is invariant under generalized gauge transformations. A constraint of precisely the form (1.8) was derived in refs. [39,40] for the five-gluon amplitude using the monodromy properties of string theory amplitudes.
We show in this paper that more general gauge-theory amplitudes, with both gluons and massless or massive particles in an arbitrary representation of the gauge group and with arbitrary spin ≤ 1, are also invariant under the color-factor symmetry. Consequently, the kinematic numerators of these amplitudes obey a constraint analogous to eq. (1.8). We further show that color-factor symmetry implies BCJ relations for the color-ordered amplitudes of a class of n-point amplitudes involving n − 2 gluons and a pair of particles in an arbitrary representation of the gauge group and arbitrary spin, as previously conjectured in refs. [41,42].
BCJ relations for the primitive amplitudes of a more general class of amplitudes containing gluons and an arbitrary number of pairs of differently flavored fundamentals (based on a proper decomposition developed by Melia [43][44][45] and Johansson and Ochirov [42]) were conjectured by Johansson and Ochirov [42], and subsequently proven using BCFW on-shell recursion by de la Cruz, Kniss, and Weinzierl [46]. In a sequel to this paper [47], we prove that these BCJ relations also follow as a direct consequence of the color-factor symmetry.
The amplitudes of the theory of massless bi-adjoint scalars with cubic interactions [48] also exhibit invariance under color-factor symmetry, as we show using the cubic vertex expansion. In this case, the color-factor shifts are associated with each massless adjoint scalar in the amplitude. As a consequence, we demonstrate the reduced rank of the propagator matrix for the n-gluon gauge-theory amplitude by deriving the set of its null eigenvectors.
Finally, we generalize the cubic vertex expansion and color-factor symmetry to loop-level amplitudes containing at least one external gluon. We exhibit an independent set of shifts that act on the color factors of one-loop n-gluon amplitudes and which depend on the loop momentum as well as external momenta. These one-loop amplitudes are invariant under color-factor shifts in theories that admit a color-kinematic-dual representation of numerators. The color-factor symmetry also implies certain relations among the integrands of one-loop color-ordered amplitudes that were previously uncovered in refs. [49][50][51].
The contents of this paper are as follows. In sec. 2 we define the color-factor shift for the ngluon amplitude and derive the BCJ relations as a consequence of the invariance of the amplitude under this shift. We also introduce the cubic vertex expansion, and use it to derive a generalizedgauge-invariant constraint on the kinematic numerators of the n-gluon amplitude. We introduce an analogous set of shifts of the kinematic numerators, and show that they correspond to a generalized gauge transformation. In sec. 3, we prove the invariance of the four-gluon amplitude under the colorfactor symmetry, and in sec. 4, we extend this to the n-gluon amplitude by using the radiation vertex expansion. In sec. 5, we define the color-factor shift for more general amplitudes, and derive the BCJ relations for the class of amplitudes containing n − 2 gluons and a pair of particles in an arbitrary representation R. In sec. 6, we prove the invariance of the four-point amplitude with two gluons and a pair of massive particles of arbitrary spin ≤ 1 and representation R under the color-factor symmetry, and in sec. 7, we extend this to a general n-point amplitude containing gluons and other particles. In sec. 8, we prove the invariance of the amplitudes of the bi-adjoint scalar theory under the color factor symmetry, and derive the null eigenvectors of the propagator matrix. In sec. 9, we generalize the cubic vertex expansion and color-factor symmetry to loop-level amplitudes, and derive a constraint on the integrands of one-loop color-ordered amplitudes. Section 10 contains a discussion and conclusions. In appendix A, we write the shifts for all the color factors of the five-gluon amplitude, and derive the explicit constraint on the kinematic numerators that follow from the color-factor symmetry.
2 Color-factor symmetry and its consequences We begin this section by introducing the color-factor symmetry in the simplest context, the treelevel four-gluon amplitude where and s, t, and u are Mandelstam variables. We define the four-point color-factor shift to act as 4 where α is arbitrary. Equation (2.3) preserves the Jacobi relation c s + c t + c u = 0 by virtue of momentum conservation s + t + u = 0.
The statement that eq. (2.1) is invariant under eq. (2.3) implies the kinematic Jacobi relation It is well-known [23,24] that eq. (2.4) is satisfied in the case of the four-gluon amplitude, and we will show this explicitly in sec. 3. This serves as proof of the invariance of the four-gluon amplitude under the color-factor shift.
Recall that the kinematic numerators n i are not uniquely defined by eq. (2.1) because a generalized gauge transformation n s → n s + β s, n t → n t + β t, n u → n u + β u (2.5) (with β arbitrary) leaves eq. (2.1) unchanged by virtue of the Jacobi identity c s + c t + c u = 0. In the case of the four-gluon amplitude, however, the sum n s + n t + n u is well-defined: it is invariant under the generalized gauge transformation (2.5) due to momentum conservation.
The four-gluon amplitude can be written in terms of color-ordered amplitudes as which is the four-gluon BCJ relation [1].

Color-factor shift for n-gluon amplitudes
Next we turn to tree-level n-gluon amplitudes with n > 4, which may be written as a sum over diagrams composed of cubic vertices (referred to as cubic diagrams) [1] Associated with each cubic diagram i is a color factor c i obtained by sewing together structure constants f abc . Among these color factors c i we may identify the subset of half-ladder color factors c α defined by (see fig. 2) The color factors c i are not independent but obey a set of Jacobi relations. Using the procedure outlined in ref. [52], the Jacobi identity f abe f cde + f ace f dbe + f ade f bce = 0 may be repeatedly applied to reduce each c i to a linear combination of half-ladder color factors where γ denotes a permutation of {2, · · · , n−1}. The (n−2)! half-ladders c 1γn form an independent set. Alternatively, M i,1γn may be computed by rewriting c i using f abc = Tr([T a , T b ]T c ), reducing the resulting expression to a linear combination of single traces, and then identifying the coefficient of Tr(T a 1 T a γ(2) · · · T a γ(n−1) T an ) (see e.g., ref. [53]).
We now define a set of momentum-dependent shifts, associated with each external gluon a in the amplitude, that act on the color factors c i . The action of the shift δ a c i associated with gluon a is constrained by two requirements: (I) that it preserve all the Jacobi relations satisfied by c i , and (II) that it satisfy where S a,i denotes the subset of the external particles on one side 5 of the point at which a is attached to c i . In particular, if c i is a color factor in which gluon a is attached to an external leg b, the shift is proportional to k a · k b , which is an inverse propagator in the associated Feynman diagram. More generally, eq. (2.11) is related (see eq. (2.20)) to the propagators in the Feynman Figure 3: Diagrams with color factors c 1···σ(b−1)[aσ(b)]σ(b+1)···n and c 1···σ(b−1)aσ(b)···n diagram associated with c i .
Consider the subset of n-point color factors obtained from a given (n − 1)-point cubic diagram I by attaching gluon a to it in all possible ways. One of these color factors has gluon a attached to external leg 1 of the (n − 1)-point diagram 6 ; define its shift to be α I k a · k 1 . One may easily verify (using the argument in the introduction) that the conditions (I) and (II) above uniquely fix the coefficients of the shifts of all the other color factors in this subset. The coefficients α I for different (n − 1)-point diagrams are then constrained by Jacobi relations among their color factors.
We now demonstrate that there is an (n−3)!-parameter family of color-factor shifts associated with each gluon a in the n-gluon amplitude. First choose a ∈ {2, · · · , n − 1}, and consider the subset of half-ladder color factors c 1aσn , where σ ∈ S n−3 denotes a permutation of {2, · · · , n − 1} \ {a}. We define the color-factor shift associated with gluon a to act on these half ladders as where α a,σ are a set of (n − 3)! arbitrary, independent constants (or functions) for each a. Let c 1···σ(b−1)[aσ(b)]σ(b+1)···n denote the color factor shown in fig. 3; its shift is proportional to k a · k σ(b) . This together with eq. (2.12) and the Jacobi relation implies that δ a acts on the independent half-ladder color factors c 1γn as 7 (2.14) consistent with eq. (2.11). The action on the remaining color factors is given by which is also consistent with eq. (2.11), as may be shown using the procedure described in ref. [52].
The color-factor shifts associated with the gluons {2, · · · , n − 1} are not all independent. In particular, the (n − 3)!-parameter family of shifts associated with gluon n − 1 are linear combinations of shifts associated with a ∈ {2, · · · , n−2} as a result of momentum conservation. 8 We may also define (n − 3)!-parameter families of shifts associated with gluons 1 and n. These are also not independent of the others. Thus the dimension of the (abelian) group of color-factor shifts is (n − 3)(n − 3)!.

Fundamental BCJ relations from the color-factor symmetry
By using eq. (2.10), the tree-level n-gluon amplitude (2.8) may be rewritten in the Del Duca-Dixon-Maltoni half-ladder decomposition [52,54] A n = γ∈S n−2 where the coefficients are color-ordered amplitudes belonging to the Kleiss-Kuijf basis [55]. In the previous subsection, we defined the action of the color-factor symmetry associated with a given gluon a. The variation of eq. (2.16) under the shift associated with a = 2 gives (2.18) In secs. 3 and 4, we prove that A n is invariant under this shift. Since α 2,σ are arbitrary and independent, this implies that n b=3 k 1 · k 2 + b−1 c=3 k 2 · k σ(c) A(1, σ(3), · · · , σ(b − 1), 2, σ(b), · · · , σ(n − 1), n) = 0 (2.19) which is the fundamental BCJ relation (1.4). All other permutations of this relation can be obtained using the invariance of the amplitude under the color-factor shifts associated with gluons 3 through n − 1.
It is known [1] that the BCJ relations reduce the number of independent color-ordered amplitudes from (n − 2)! to (n − 3)!. Not surprisingly, the difference between these, (n − 3)(n − 3)!, is the dimension of the group of color-factor shifts that leave the amplitude invariant.

Cubic vertex expansion
In order to examine the implications of the color-factor symmetry for the kinematic numerators n i appearing in the cubic decomposition (2.8), we introduce in this section the cubic vertex expansion of the amplitude with respect to one of the gluons. This expansion is similar to, but distinct from, the radiation vertex expansion [35] that will be used in secs. 4 and 7 to prove the invariance of n-point gauge-theory amplitudes under color-factor shifts.
The cubic decomposition (2.8) is a sum over the (2n − 5)!! cubic diagrams of an n-gluon amplitude, but for any a ∈ {1, · · · , n} it can be viewed as a sum over the (2n−7)!! cubic diagrams of an (n−1)point function with external legs {1, · · · , n} \ {a}, to each of which gluon a is attached in 2n − 5 different ways. Let us label these (n−1)-point cubic diagrams by I and their denominators by d (a,I) .
Each (n − 1)-point cubic diagram I has n − 3 vertices, the set of which we denote by V (a,I) . For each vertex v ∈ V (a,I) , we can break d (a,I) into three factors 3 r=1 d (a,I,v,r) , where d (a,I,v,r) is the product of propagators that branch off from leg r of the vertex. If leg r is an external leg of the diagram, then d (a,I,v,r) = 1.
We can attach gluon a either to one of the n − 1 external legs or to one of the n − 4 internal lines of I, yielding altogether 2n − 5 of the terms in the sum (2.8). Let K be the momentum running through one of the internal lines of I. Attaching gluon a to this line will replace the factor K 2 in d (a,I) with K 2 (K + k a ) 2 . We split the inverse denominator into two terms using the identity and we associate each of the terms on the right hand side of the equation with one of the two vertices to which the internal line is connected. Thus, with this doubling of internal line terms, we now have a total of (n − 1) + 2(n − 4) = 3(n − 3) terms for each I; namely, one term for each of the legs of each of the n − 3 vertices of I. We label this term by (a, I, v, r), and write the cubic vertex expansion of the n-gluon amplitude with respect to gluon a as  The ± freedom in the definition of c (a,I,v,r) is used to make the relative signs in the Jacobi relation positive: The denominators in each triple also sum to zero 3 r=1 k a · K (a,I,v,r) = 0 (2.23) by momentum conservation k a + 3 r=1 K (a,I,v,r) = 0 and the masslessness of the gluon k 2 a = 0. A priori, however, there is no reason for the kinematic numerators n (a,I,v,r) associated with each vertex to sum to zero. We will see in the next subsection, however, that the color-factor symmetry of the amplitude leads to a constraint on the sum of kinematic numerators.

Constraint on kinematic numerators from the color-factor symmetry
Having introduced the cubic vertex expansion of the n-gluon amplitude with respect to gluon a, we now consider the effect of a color-factor shift δ a on the amplitude. The shift associated with gluon a acts on the color factors appearing in the cubic vertex expansion (2.21) as δ a c (a,I,v,r) = α (a,I,v) k a · K (a,I,v,r) (2.24) where α (a,I,v) is a linear combination of α a,σ uniquely determined by the Jacobi relations. These shifts respect eq. We prove that δ a A n = 0 in secs. 3 and 4; hence the color-factor symmetry implies the following constraint on the kinematic numerators Now if the constants α (a,I,v) were all independent, then we could conclude from this argument that 3 r=1 n (a,I,v,r) = 0, i.e. that the kinematic numerators necessarily obey Jacobi relations. This would be in conflict with the well-known fact that the kinematic numerators obtained from Feynman rules in general do not satisfy color-kinematic duality 9 ; indeed this is not a valid inference from eq. (2.26) because the α (a,I,v) are not independent. The set of α (a,I,v) for all the vertices of a given diagram I are equal (up to signs) because any two adjacent vertices share a common color factor (see the example discussed in the appendix). In fact, with an appropriate choice of signs for c (a,I,v,r) , the α (a,I,v) may be made independent of v. The α (a,I) for different diagrams I are further constrained by the Jacobi relations among the color factors of I.
While eq. (2.26) does not imply that the numerators satisfy the kinematic Jacobi relations, it does impose a set of (generalized-gauge-invariant) conditions that the color-kinematic violations ∆ ijk = n i + n j + n k must satisfy. We wish to emphasize that, while the kinematic Jacobi relations n i + n j + n k = 0 are not invariant under generalized gauge transformations 10 (hence the actual claim of color-kinematic duality is that there exists a generalized gauge in which they hold true), the conditions (2.26) are invariant under generalized gauge transformations. The argument for this is simple. A generalized gauge transformation is a transformation n i → n i that leaves the amplitude (2.8) unchanged. Hence by starting with A n = i (c i n i /d i ) and following the steps above (since the condition δ a A n = 0 is also gauge invariant), we obtain the same result (2.26) except with n (a,I,v,r) replaced with n (a,I,v,r) .
To obtain a more explicit form of eq. (2.26), we would need to identify all the linear dependences among the α (a,I,v) required by the color Jacobi relations. Previously, we observed that the number of independent color-factor shifts was (n − 3)(n − 3)!, parametrized by constants α a,σ , where a = 2, · · · , n − 2 and σ ∈ S n−3 denotes a permutation of {2, · · · , n − 1} \ {a}. If we were to write α (a,I,v) in terms of these independent constants, eq. (2.26) would yield (n − 3)(n − 3)! independent constraints on the ∆ ijk . In appendix A, we carry out this procedure for the five-gluon amplitude.
While the BCJ relations (2.19) were originally derived as a consequence of the assumption of colorkinematic duality, it was known [39,40] from early on that they are equivalent to a set of weaker conditions on the numerators. The conditions for five-gluon numerators were derived in refs. [39,40] as a consequence of the monodromy properties of string-theory amplitudes. These conditions are equivalent to eq. (2.26), as we show in appendix A. In this section, we have demonstrated that both eq. (2.26) and the BCJ relations (2.19) are a consequence of the invariance of the amplitude under the color-factor symmetry.

Kinematic numerator shift symmetry
We have considered the effect on the amplitude of a shift of the color factors. One may ask what effect an analogous shift of the kinematic numerators would have. 11 We show in this section that such a shift is simply a generalized gauge transformation.
Let us define the kinematic shift associated with leg a, where a ∈ {2, · · · , n − 1}, on the half-ladder numerator n 1γn to be δ a n 1σ (2) (2.27) where σ ∈ S n−3 denotes a permutation of {2, · · · , n − 1} \ {a}, and β a,σ are a set of arbitrary constants (or functions). The action on all other numerators n i is then defined by δ a n i = γ∈S n−2 M i,1γn δ a n 1γn . (2.28) Note that we have not assumed that the n i obey the Jacobi relations (1.3). However, the numerator shifts defined by eqs. (2.27) and (2.28) will satisfy δ a (n i + n j + n k ) = 0 (2.29) so if the n i do satisfy kinematic Jacobi relations, the shifted numerators will continue to do so, and if they do not, then the neither will the shifted numerators.  c (a,I,v,r) n (a,I,v,r) 2k a · K (a,I,v,r) . (2.30) The kinematic shift with respect to gluon a acts on the numerators appearing in this expression as δ a n (a,I,v,r) = β (a,I,v) k a · K (a,I,v,r) (2.31) and therefore on the amplitude itself as 3 Proof of color-factor symmetry for four-gluon amplitudes In this section we prove that the tree-level four-gluon amplitude is invariant under the color-factor symmetry. In doing so, we develop some results that will be necessary for our more general proof of the invariance of the n-gluon amplitude in the next section.
The four-gluon amplitude can be constructed from a three-gluon vertex by attaching a fourth gluon to a propagator emanating from each of the legs of the vertex or to the vertex itself. This yields The kinematic numerator n (r) receives contributions from the diagram in which gluon 4 is attached to leg r as well as from the four-gluon vertex. The color-factor symmetry (associated with gluon 4) acts on the color factors (3.2) and the four-gluon amplitude (3.1) as Thus, by showing that 3 r=1 n (r) = 0, we will establish that δ 4 A 4 = 0. This we now proceed to do.

Attaching a gluon to a leg
The three-gluon vertex is 12 Our structure constants are normalized by This differs from the standard textbook convention by a factor of i √ 2. We use η00 = 1 in this paper.
and k a are outgoing momenta. In Feynman gauge, the gluon propagator is −iη µν δ ab /k 2 . Attaching gluon 4 to leg 1 yields the expression The contribution of this diagram to the four-gluon amplitude (3.1) is obtained by contracting with 4 a=1 ε aµa and dividing by i. The first and second terms in the square brackets vanish using ε a · k a = 0. The third term vanishes due to using k 2 2 = k 2 3 = 0. The contribution of the remaining terms of eq. (3.6) to the kinematic numerator n (1) can be written are the spin-one angular momentum matrices acting on gluon r. These satisfy the Lorentz algebra commutation relations (3.10)

Attaching a gluon to a vertex
Attaching gluon 4 directly to the three-gluon vertex, we obtain the four-gluon vertex Using eq. (3.5), this can be recast as The contribution of this vertex to the four-gluon amplitude is obtained by contracting with 4 a=1 ε aµa and dividing by i. This contribution is then parceled out among the three terms in eq. (3.1); the contribution to n (1) is The reader may be concerned about the use of eq. (3.12) for the following reason. The three-gluon vertex (3.5) can be rewritten using momentum conservation 3 a=1 k a = 0. For example, we can eliminate k 3 from eq. (3.5), writing it as (3.14) The partial derivatives (∂/∂k aµ 4 )V (k 1 , k 2 , k 3 ) obtained from eq. (3.14) differ from those obtained from eq. (3.5), so that eq. (3.12) gives Nonetheless eq. (3.15) is equal to eq. (3.11) courtesy of the Jacobi relation 3 r=1 c (r) = 0. Equation (3.15) simply corresponds to a different way of parceling the four-gluon vertex among the color factors c (r) , and the expressions n (r) will differ by a generalized gauge transformation (2.5). The amplitude (3.1) of course remains unchanged.

Kinematic numerators of the four-gluon amplitude
Using the fact that the three-gluon vertex (3.5) is linear in momenta, we rewrite eq. (3.8) using and then combine eqs. (3.8) and (3.13) to obtain where the orbital angular momentum operators are defined as These satisfy the Lorentz algebra commutation relations Similar expressions are obtained for n (2) and n (3) . Finally, defining the total angular momentum operator we can write all the kinematic numerators as where we have suppressed the polarization vectors ε 1 , ε 2 , and ε 3 . The subscripts on J r and S r indicate on which polarization indices these operators act.
Note that under a gauge transformation ε 4 → ε 4 + λk 4 of gluon 4, the last two terms in eq. (3.21) vanish by virtue of the antisymmetry of J αβ r and S αβ r , and the first term gives n (r) → n (r) +λg 2 k 4 ·k r in accord with eq. (2.5).
We observe that the operators L αβ r and S αβ r are the same as those appearing in the Burnett-Kroll form [56] of the subleading terms of the Low soft-photon theorem [57] as applied to gluons [58][59][60][61][62].
(See refs. [63][64][65] for recent derivations of the soft-gluon theorem from gauge invariance.) The first two terms in eq. (3.21) correspond precisely to the leading and subleading terms in the k 4 → 0 expansion of the four-gluon amplitude; the third term is higher order in the soft momentum. We emphasize, however, that eq. (3.21), and the vertex expansion that we will derive in the next section, are exact, and not dependent on taking a soft limit.

Kinematic Jacobi relation
We will now show that the sum of the four-gluon kinematic numerators (3.21) vanishes. This result, which has been known at least since 1980 [23,24] inspired the conjecture of color-kinematic duality [1]. We demonstrate it in a way that will facilitate the proof of color-factor symmetry of the n-gluon amplitude.
Using eq. (3.21), we see that The first sum on the right-hand side of eq. (3.22) vanishes by momentum conservation 4 r=1 k r = 0 together with ε 4 · k 4 = 0. The second sum on the right hand side of eq. (3.22), which may be written more explicitly as is the first-order Lorentz transformation of the three-gluon vertex. This vanishes, as may be verified by explicit computation, because V µ 1 µ 2 µ 3 (k 1 , k 2 , k 3 ) is a Lorentz tensor. Alternatively, we can define the spin-one angular momentum operator to act on polarization indices [63,64] S αβ is a Lorentz-invariant function of k r and ε r . The third sum in eq. (3.22) is proportional to This expression is antisymmetric in β and γ. Since it multiplies k 4β k 4γ in eq. (3.22), the whole expression vanishes. The cancellations that we have just exhibited were originally used in ref. [35] to prove the radiation zero theorem.
We have shown that the sum of kinematic numerators vanishes (the kinematic Jacobi relation), and thus have demonstrated the invariance of the four-gluon amplitude δ 4 A 4 = 0 under the color-factor shift symmetry. We would like to emphasize that in proving the vanishing of eqs. (3.23) and (3.27) we did not use that k r were on-shell, nor did we use ε r · k r = 0 for r = 1, 2, 3. Thus we will be able to use these results in sec. 4 for an off-shell three-gluon vertex V µ 1 µ 2 µ 3 (k 1 , k 2 , k 3 ).

Proof of color-factor symmetry for n-gluon amplitudes
We now turn to the proof that the tree-level n-gluon amplitude is invariant under a color-factor shift associated with any of the gluons. To do so, we employ a decomposition of the amplitude known as the radiation vertex expansion [35]. This is a recursive approach which constructs an n-point amplitude by attaching a massless vector boson to all possible (n − 1)-point diagrams.
Consider the set of all tree-level (n − 1)-gluon diagrams with external legs {1, · · · , n} \ {a} constructed using Feynman rules. Label these diagrams by I . Please note that this set differs from the set of (n − 1)-point diagrams appearing in the cubic vertex expansion (cf. sec. 2) because it includes not only cubic diagrams but also diagrams with four-gluon vertices. (In the cubic decomposition (2.8), all diagrams containing quartic vertices are redistributed among the cubic diagrams.) We will construct all possible n-gluon diagrams by attaching gluon a to each (n − 1)-gluon diagram I in all possible ways. This includes: (1) attaching gluon a to an external leg, (2) attaching gluon a to an internal line, or (3) attaching gluon a to one of the three-gluon vertices of I to make a four-gluon vertex. By rearranging terms and discarding pieces that vanish by Ward identities, we obtain simple expressions for the contribution to the n-gluon amplitude from each vertex of diagram I . We then show that each such contribution is invariant under the color-factor shift associated with gluon a.

Attaching a gluon to an external leg
First we single out one of the external legs, b, of I , denoting the resulting expression as where k b and ε b are the momentum and polarization vector of gluon b, and · · · denotes momenta belonging to gluons {1, · · · , n} \ {a, b}. Attaching gluon a to external leg b and using eq. (3.5) we obtain Contracting this with ε bµ b ε aµa eliminates the first two terms in the square brackets. The third term is proportional to This term does not vanish by itself as it did in the case of the four-gluon amplitude, but when we obtain the total n-gluon amplitude by including all (n − 1)-point diagrams I , the sum of such terms vanishes due to gauge invariance (Ward identity). Therefore we are left with the last two terms in square brackets, which yield the two terms in the following expression We set this expression aside for now.

Attaching a gluon to an internal line
Next we single out one of the internal lines of I , which divides the diagram into two subdiagrams B and C, and splits the external legs {1, · · · , n} \ {a} into two complementary sets S a,B and S a,C . The contribution of the diagram can thus be written as where K = d∈S a,B k d is the momentum running through the line, and the · · · in B and C denote momenta belonging to S a,B and S a,C respectively. Attaching gluon a to the line connecting the two subgraphs yields Writing the three-gluon vertex (3.5) as we see that the first term vanishes when contracted with ε aµa . The second and third terms give terms proportional to Again these terms do not vanish by themselves, but when we include all (n − 1)-gluon diagrams I , B bµ will be replaced by the sum over all diagrams containing external legs S a,B plus one additional off-shell line, and similarly for C cν , and these expressions will vanish by gauge invariance (Ward identity). We are thus left with Now we use the identity (2.20) to rewrite this as We associate each of the terms in this equation with one of the two vertices to which the line is attached.

Radiation vertex expansion
In the previous subsections, we showed that attaching gluon a to an (n − 1)-gluon diagram yields one term (4.4) for each external leg and two terms (4.10) for each internal line, or in other words, one term for each leg of each vertex of I . We can therefore reorganize these terms into a sum over the legs of the vertices of each of the (n − 1)-gluon diagrams I . This is the radiation vertex expansion [35].
First we choose one of the three-gluon vertices v of I (if it has any). Such a vertex divides the external legs into three non-overlapping subsets S (a,I ,v,r) , r = 1, 2, 3 such that 3 r=1 S (a,I ,v,r) = {1, · · · , n} \ {a}. The contribution of diagram I to the (n − 1)-gluon amplitude can be expressed where K r = d∈S (a,I ,v,r) k d is the momentum flowing out of each leg of the vertex, and · · · in A (r) denotes momenta belonging to S (a,I ,v,r) . If any of the legs is external, then A (r) We now attach gluon a to each of the legs of this three-gluon vertex, either to an external leg or to an internal line. From eqs. (4.4) and (4.10) this yields We can also attach gluon a directly to the three-gluon vertex itself. Using eq. (3.12), this yields We now use eq. (3.16) in eq. (4.12), and combine eqs. (4.12) and (4.13) as we did in sec. 3. Leaving the indices on V µ 1 µ 2 µ 3 (K 1 , K 2 , K 3 ) implicit, we obtain the contribution of the three-gluon vertex to the radiation vertex expansion where the subscripts on J r and S r indicate on which indices of V µ 1 µ 2 µ 3 (K 1 , K 2 , K 3 ) these operators act.
Next we choose one of the four-gluon vertices v of I (if it has any). Such a vertex divides the external legs into four non-overlapping subsets S (a,I ,v,r) , r = 1, · · · , 4 such that 4 r=1 S (a,I ,v,r) = {1, · · · , n} \ {a}. The contribution of the diagram I can be expressed as (4.15) We now attach gluon a to each of the legs of this four-gluon vertex. From eqs. (4.4) and (4.10) we obtain (4.16) One cannot attach gluon a to the four-gluon vertex itself since there are no five-gluon vertices.
Thus the contribution to the radiation vertex expansion from the four-gluon vertex is To summarize this section, we have expressed an n-gluon amplitude as a sum over the vertices of all of the (n − 1)-gluon diagrams I , comprising a term (4.14) for each three-gluon vertex of I and a term (4.17) for each four-gluon vertex of I .

Invariance of the radiation vertex expansion under color-factor symmetry
Computing the variation of the radiation vertex expansion of the amplitude under a color-factor shift is somewhat more delicate than calculating the variation of the cubic vertex expansion of the amplitude (as we did in sec. 2) because each factor A (r) crµr in eqs. (4.14) and (4.17) can contain more than one color factor c i due to the possible presence of four-gluon vertices.
First let us consider the contribution (4.14) of a three-gluon vertex to the radiation vertex expansion, and for the moment let us assume that the subdiagrams corresponding to A  crµr contain four-gluon vertices, we can use eq. (3.11) to expand these expressions and then use the argument above to show that each separate contribution will vanish under the color-factor shift. Therefore the contribution of the three-gluon vertices (4.14) to the radiation vertex expansion is invariant under the color-factor shift associated with gluon a.
Second let us consider the contribution (4.17) of a four-gluon vertex to the radiation vertex expansion. Again we begin by assuming that the diagrams corresponding to A (r) crµr contain only three-gluon vertices. We now expand the four-gluon vertices V µ 1 µ 2 µ 3 µ 4 c 1 c 2 c 3 c 4 in eq. (4.17) into several terms, one of which is The variation of c (a,I ,v,r) under the color-factor shift associated with gluon a is The variation of eq. (4.20) under the color-factor shift therefore contains two sums. The first vanishes by momentum conservation, k a + 4 r=1 K r = 0, together with ε a · k a = 0. The second is the first-order Lorentz transformation of the tensor η µ 1 µ 3 η µ 2 µ 4 − η µ 1 µ 4 η µ 2 µ 3 , which vanishes.
The variation under the color-factor shift of the other two terms from the expansion of V µ 1 µ 2 µ 3 µ 4 c 1 c 2 c 3 c 4 similarly vanishes. Furthermore, the same argument applies when the diagrams corresponding to A (r) crµr contain four-gluon vertices, by expanding these expressions using eq. (3.11). Therefore the contribution of the four-gluon vertices (4.17) to the radiation vertex expansion is invariant under the color-factor shift associated with gluon a.
In fine, we have shown that each contribution to the radiation vertex expansion is invariant under the color-factor shift associated with gluon a, and therefore the entire n-gluon amplitude is invariant under this shift. QED 5 Color-factor symmetry for more general amplitudes In secs. 3 and 4, we proved the color-factor symmetry of n-gluon amplitudes, from which follow the BCJ relations for color-ordered amplitudes. Color-factor symmetry is a property of a much larger class of tree-level gauge-theory amplitudes, namely those containing at least one gluon together with massless or massive particles in arbitrary representations of the gauge group with arbitrary spin ≤ 1, with the usual gauge-theory couplings. We will establish the invariance of this larger class of gauge-theory amplitudes under a color-factor shift in secs. 6 and 7.
Consider a tree-level n-point gauge-theory amplitude A n with gluons as well as particles ψ and ψ, either massless or massive, with spin zero, one-half, or one, in an arbitrary representation of the gauge group. For convenience, throughout the next three sections we refer to ψ (andψ) as fundamentals (and antifundamentals), but they can be in any representation. This amplitude has the cubic decomposition where we decorate the color factors, kinematic numerators, and denominators with primes to distinguish them from the analogous quantities for n-gluon amplitudes. The denominator d i now consists of the product of inverse propagators for both massless and massive particles. The color factor c i associated with each cubic diagram is obtained by sewing together ggg vertices f abc and ψgψ vertices (T a ) i j where T a denote the generators in the appropriate representation. Contributions from Feynman diagrams with quartic vertices (either gggg orψggψ in the case of a scalar or vector ψ) are parceled out among the cubic diagrams. The number of cubic diagrams in the sum (5.1) will generally be fewer than for the n-gluon amplitude, as some will be excluded for violating fermion number or flavor symmetry.
Just as in the case of the n-gluon amplitude, the amplitude A n can be written in a cubic vertex expansion with respect to gluon a: c (a,I,v,r) n (a,I,v,r) 2k a · K (a,I,v,r) .
The only difference between the derivation of this expression and that for the n-gluon amplitude given in sec. 2 is that we must use the modified identity when gluon a is attached to a propagator of a massive field.
As usual, the color-factor shift associated with gluon a is defined by two requirements: (I) that it satisfy all the algebraic symmetries (e.g., Jacobi relations) obeyed by the color factors c i , and (II) where S a,i denotes the subset of the external particles on one side of the point at which a is attached to c i . These together imply that δ a c (a,I,v,r) = α (a,I,v) k a · K (a,I,v,r) . on the sums of kinematic numerators appearing in the cubic decomposition.
As in the case of n-gluon amplitudes, color-factor symmetry can be used to derive BCJ relations among the color-ordered amplitudes associated with A n . BCJ relations for n-point amplitudes with gluons and a single pair of massive fundamentals were conjectured in ref. [41] and more generally for amplitudes containing an arbitrary number of pairs of fundamentals in ref. [42], based on the assumption of color-kinematic duality. A proof of these BCJ relations using BCFW on-shell recursion was given in ref. [46]. In order to derive these relations, however, it is necessary to write the amplitude in a proper decomposition [45], i.e., in terms of an independent set of color factors and generalized-gauge-invariant primitive amplitudes. For general amplitudes, this is a subtle problem, which was recently solved for the case of multiple pairs of distinct-flavor fundamentals by Melia [43][44][45] and Johansson and Ochirov [42]. In a sequel to this paper [47], we review their solution, and then derive the BCJ relations using the color-factor symmetry.
There is one class of amplitudes, however, for which the story is practically identical to the n-gluon case, namely, n-point amplitudes with n − 2 gluons and a single pair of fundamentals. For that case, an independent set of (n − 2)! color factors is given by the half-ladders All other color factors c i can be reduced to half ladders by repeatedly applying f abc (T c ) i j = T a , T b i j , similar to the case of n-gluon amplitudes [52]. The coefficients M i,1γn are precisely the same as in the n-gluon case. The n-point amplitude can then be written in a proper decomposition [66,67] A n (ψ 1 , g 2 , g 3 , · · · , g n−1 , ψ n ) = γ∈S n−2 c 1γn A (1, γ(2), · · · , γ(n − 1), n) (5.9) where the primitive amplitudes are given by We define an (n − 3)!-parameter family of shifts associated with each gluon a ∈ {2, · · · , n − 1} via where γ is a permutation of {2, · · · , n − 1}, σ is a permutation of {2, · · · , n − 1} \ {a}, and α a,σ is a set of (n − 3)! arbitrary constants for each a. As in the case of the n-gluon amplitude, the dimension of the (abelian) group of color-factor shifts is (n − 3)(n − 3)!. We show in secs. 6 and 7 that the amplitude A n (ψ 1 , g 2 , g 3 , · · · , g n−1 , ψ n ) is invariant under the color-factors shifts (5.11). As a consequence, the color-ordered amplitudes defined in eq. (5.9) obey BCJ relations that have the same form (when expressed in terms of invariants k a · k b , where k a is the momentum of a gluon) as those for the n-gluon amplitude, namely n b=3 together with all permutations of this equation with 2 replaced by a, as conjectured in refs. [41,42].
6 Proof of color-factor symmetry for the A 4 (ψ 1 , ψ 2 , g 3 , g 4 ) amplitude In this section, we prove that the tree-level four-point amplitude with two gluons and two massive particles in an arbitrary representation of the gauge group (which for convenience we refer to as fundamentals) with spin zero, one-half, or one is invariant under the color-factor symmetry. We will use these results for the proof of the invariance of the more general n-point amplitude in sec. 7.
The four-point amplitude A 4 (ψ 1 , ψ 2 , g 3 , g 4 ) can be constructed from theψψg vertex by attaching a gluon to a propagator emanating from each of the external legs of the vertex, or (in the case of a spin-zero or spin-one fundamental) to the vertex itself. This yields where the color factors The color-factor symmetry (associated with gluon 4) acts on the color factors (6.2) and the four-point amplitude (6.1) as We will establish that δ 4 A 4 = 0 by showing that 3 r=1 n (r) = 0, a result that has long been known [23,24]. We will do this separately for spin zero, spin one-half, and spin one fundamentals.

Kinematic numerators for spin-one-half fundamentals
We begin with the case of a spin-one-half fundamental, which is simpler due to the absence of ā ψψgg vertex. Theψψg vertex 13 and Dirac propagator are Attaching gluon 4 to (fermion) leg 1 yields the expression The contribution of this diagram to eq. (6.1) is obtained by sandwiching eq. (6.5) betweenū(k 1 ) and u(−k 2 ), contracting with ε 3µ 3 ε 4µ 4 , and dividing by i. The first term in the square brackets vanishes usingū(k 1 )(−/ k 1 + m) = 0, leaving where the spin-one-half angular momentum matrices Σ αβ satisfy the Lorentz algebra commutation relations Similarly, attaching gluon 4 to (fermion) leg 2 yields (6.8) Finally, attaching gluon 4 to (gluon) leg 3 yields The sum of the kinematic numerators is thus The first sum on the right-hand side of this equation vanishes by momentum conservation 4 r=1 k r = 0 together with ε 4 · k 4 = 0. The second sum vanishes because is the first-order Lorentz transformation of γ µ (acting on both spinor indices as well as the vector index) and hence vanishes. Thus the sum of kinematic numerators for the amplitude A 4 (ψ 1 , ψ 2 , g 3 , g 4 ) is zero for spin-one-half fundamentals.

Kinematic numerators for spin-zero fundamentals
Next we turn to the case of spin-zero fundamentals. Theψψg vertex is where k a are outgoing momenta. The scalar propagator is iδ i j /(k 2 − m 2 ). Thus attaching gluon 4 to (scalar) leg 1 yields the expression 14) The contribution of this diagram to eq. (6.1) is obtained by contracting with ε 3µ 3 ε 4µ 4 and dividing by i. The first term in the square brackets vanishes using ε 4 · k 4 = 0, leaving Similarly, attaching gluon 4 to (scalar) leg 2 yields Attaching gluon 4 to (gluon) leg 3 yields Attaching gluon 4 directly to theψψg vertex, we obtain theψψgg vertex which can be written as The rest of the story proceeds exactly as in sec. 3, allowing us to write the kinematic numerators as where we have suppressed the polarization vector ε 3 . Because two of the legs are scalars, we have S αβ 1 = S αβ 2 = 0, whereas S αβ 3 is given by eq. (3.9).

Now consider the sum of kinematic numerators
where the scalar-scalar-gluon vertex is V µ 3 (k 1 , k 2 , k 3 ) = (k 1 −k 2 +λ[k 1 +k 2 +k 3 ]) µ 3 with λ arbitrary due to momentum conservation. The first sum on the right-hand side of this equation vanishes as usual by momentum conservation. The second sum, which may be written more explicitly as is the first-order Lorentz transformation ofψψg vertex. This vanishes, as may be seen by explicit computation, because V µ 3 (k 1 , k 2 , k 3 ) is a Lorentz tensor. The third term on the right-hand side of eq. (6.21) is which automatically vanishes due to k 2 4 = 0 and ε 4 ·k 4 = 0. Again, we emphasize that in proving the vanishing of eqs. (6.22) and (6.23) we did not use that k r were on-shell, nor did we use ε r · k r = 0, for r = 1, 2, 3. Thus these results remain valid for an off-shell vertex V µ 3 (k 1 , k 2 , k 3 ).

Kinematic numerators for spin-one fundamentals
Finally we consider massive spin-one fundamentals withψgψ vertex of the form where V µ 1 µ 2 µ 3 (k 1 , k 2 , k 3 ) is given by eq. (3.5). We emphasize that although we refer to the vector particles as fundamentals, they could be in any representation, including the adjoint, in which case (T a 3 ) a 1 a 2 = f a 1 a 3 a 2 and eq. (6.24) is equal to eq. (3.4), except that now the vector boson is massive. The propagator for a massive spin-one particle is In the case in which the vector boson gets its mass from a spontaneously-broken symmetry, eq. (6.25) is the propagator in unitary gauge; this is most convenient for tree-level calculations as we need not compute contributions involving Goldstone bosons.
Attaching gluon 4 to leg 1 yields the expression The contribution of this diagram to eq. (6.1) is obtained by contracting with 4 a=1 ε aµa and dividing by i, giving where we have used 0 = k 2 4 = ε 4 · k 4 = ε 1 · k 1 and k 2 1 = m 2 1 but we did not use eq. (3.7), which is not valid in this case because k 2 2 = k 2 3 . Analogous expressions are obtained for n (2) and n (3) . Attaching gluon 4 directly to theψψg vertex, we obtain theψψgg vertex Again, the rest of the story proceeds exactly as in sec. 3, allowing us to write the kinematic numerators as where we have suppressed the polarization vectors ε r for r = 1, 2, 3. Note that the kinematic numerators n (r) have exactly the same form as the kinematic numerators for the four-gluon amplitude, even though the masses for particles 1 through 3 can be nonzero. The proof of the vanishing of the sum of numerators proceeds exactly as in sec. 3.
In this section, we have explicitly shown that the sum of kinematic numerators for four-point amplitudes A 4 (ψ 1 , ψ 2 , g 3 , g 4 ) vanishes (where ψ can have spin zero, one-half, or one) and thus have demonstrated the invariance of the four-point amplitude under the color-factor symmetry associated with gluon 4. As the results of the last subsection have shown, this result remains valid even when particles 1 through 3 are massive; only the gluon associated with the color-factor symmetry need be massless. The results we have derived will be used in the next section to prove a more general result.
7 Proof of color-factor symmetry for more general amplitudes In this section, we use the radiation vertex expansion to demonstrate the invariance under the color-factor symmetry of tree-level gauge-theory amplitudes containing at least one gluon together with massless or massive particles in arbitrary representations of the gauge group (but referred to as fundamentals for convenience) and with arbitrary spin ≤ 1. For concreteness, we focus on the n-point amplitude A n (ψ 1 , g 2 , g 3 , · · · , g n−1 , ψ n ) with n − 2 gluons and a pair of fundamentals ψ, but it will be clear that the proof applies to more general amplitudes. The proof is very similar to that given in sec. 4 for n-gluon amplitudes, and so we only highlight the differences.
The radiation vertex expansion constructs the n-point amplitude A n (ψ 1 , g 2 , g 3 , · · · , g n−1 , ψ n ) by attaching gluon a ∈ {2, · · · , n − 1} to all possible (n − 1)-point diagrams I , with two fundamentals and n − 3 gluons, in all possible ways, and reorganizing this as a sum over all the vertices of the (n − 1)-point diagram. We have already shown in sec. 4 that the contributions of the three-and four-gluon vertices are invariant under the color-factor symmetry, so we only need to demonstrate the same for vertices involving two fundamentals. We do this separately for fundamentals with spin zero, one-half, and one.

Vertices involving spin-one-half fundamentals
To derive the contribution of theψψg vertices to the radiation vertex expansion for spin-one-half fundamentals ψ, we examine the effect of attaching a gluon to a fermion leg, either external or internal.
First we single out (fermion) leg 1, denoting the contribution of an (n − 1)-point diagram I to the amplitude asū(k 1 )C i 1 (k 1 , · · · ) where · · · denotes momenta belonging to {2, · · · , n}\{a}. Attaching gluon a to external fermion leg 1, usingū(k 1 )(−/ k 1 + m) = 0, and contracting with ε aµa , we obtain Next, we single out (fermion) leg n, denoting the contribution of the diagram I to the amplitude as B in (k n , · · · )u(−k n ), where · · · denotes momenta belonging to {1, · · · , n − 1} \ {a}. Attach gluon a to external fermion leg n, use (/ k n + m)u(−k n ) = 0, and contract with ε aµa to obtain Now we single out one of the internal fermion lines of I , which divides the diagram into two subdiagrams B j and C j , and splits the external legs {1, · · · , n} \ {a} into two complementary sets S a,B and S a,C . The contribution of the diagram I can thus be written as where K = d∈S a,B k d is the momentum running through the line, and the · · · in B and C denote momenta belonging to S a,B and S a,C respectively. Attaching gluon a to the line connecting the two subgraphs and contracting with ε aµa , we have Now we use the identity [35] to rewrite this as Each term can be associated with one of the two vertices to which the line is attached.
We now choose one of theψψg vertices v of I . Such a vertex divides the external legs into three non-overlapping subsets S (a,I ,v,r) , r = 1, 2, 3 such that 3 r=1 S (a,I ,v,r) = {1, · · · , n} \ {a}. The contribution of the diagram I to the (n − 1)-point amplitude can be expressed as where K r = d∈S (a,I ,v,r) k d is the momentum flowing out of each leg of the vertex, and the · · · in B, C, and A denote momenta belonging to S (a,I ,v,1) , S (a,I ,v,2) , and S (a,I ,v, 3) respectively. If either fermion leg is external, then B j 1 =ū(k 1 )δ i 1 We now attach gluon a to each of the legs of thisψψg vertex, either to an external leg or to an internal line. Using the expressions above as well as those in sec. 4, we obtain which is the contribution of theψψg vertex to the radiation vertex expansion.
We now wish to show that eq. (7.8) is invariant under the color-factor symmetry. As in sec. 4, we first assume that the subdiagrams corresponding to A, B, and C contain no four-gluon vertices. Designate by c (a,I ,v,r) with r = 1, 2, 3 the color factors associated with each of the three terms in eq. (7.8), including factors of f abc and (T a ) j k in the subdiagrams. These color factors manifestly satisfy 3 r=1 c (a,I ,v,r) = 0, and the variation of c (a,I ,v,r) under the color-factor shift associated with gluon a is δ a c (a,I ,v,r) = α (a,I ,v) k a · K r (7.9) which preserves 3 r=1 c (a,I ,v,r) = 0. The variation of eq. (7.8) under eq. (7.9) is then proportional to The first term vanishes by momentum conservation, and the second by the transformation properties of γ ν , as we saw in sec. 6. If the subdiagrams A, B, and C do contain four-gluon vertices, we can expand eq. (7.8) into individual pieces, each of which is invariant under the color-factor symmetry. Together with the result from sec. 4 that the contributions to the radiation vertex expansion from the three-and four-gluon vertices are also invariant, we have thus shown that the amplitude A n (ψ 1 , g 2 , g 3 , · · · , g n−1 , ψ n ) with spin-one-half fundamentals is invariant under the colorfactor symmetry. In fact, this proof applies to an amplitude with an arbitrary number of pairs of fundamentals, and will be used in the sequel [47] to prove the BCJ relations [42] for that class of amplitudes.

Vertices involving spin-zero fundamentals
To derive the contribution of theψψg andψψgg vertices to the radiation vertex expansion for spin-zero fundamentals ψ, we examine the effect of attaching a gluon to a scalar leg, either external or internal.
First we single out (scalar) leg 1, denoting the contribution of an (n − 1)-point diagram I to the amplitude as C i 1 (k 1 , · · · ) where · · · denotes momenta belonging to {2, · · · , n} \ {a}. Attaching gluon a to external scalar leg 1 we obtain Contracting this with ε aµa eliminates the first term in the square brackets, leaving Similarly, attaching gluon a to external scalar leg n, we obtain Next we single out one of the internal scalar lines of I , which divides the diagram into two subdiagrams B j and C j , and splits the external legs into two complementary sets S a,B and S a,C . The contribution of the diagram I can thus be written as B j (−K, · · · ) iδ j k K 2 − m 2 C k (K, · · · ) (7.14) where K = d∈S a,B k d is the momentum running through the line, and the · · · in B and C denote momenta belonging to S a,B and S a,C respectively. Attaching gluon a to the line connecting the two subgraphs yields Contracting with ε aµa and using ε a · k a = 0, we have Now we use the identity (5.3) to rewrite this as (7.17) We associate each of the terms in this equation with one of the two vertices to which the line is attached.
Next we choose one of the scalar-scalar-gluon vertices v of I (if it has any). Such a vertex divides the external legs into three non-overlapping subsets S (a,I ,v,r) , r = 1, 2, 3 such that 3 r=1 S (a,I ,v,r) = {1, · · · , n} \ {a}. The contribution of the diagram I can be expressed as where K r = d∈S (a,I ,v,r) k d is the momentum flowing out of each leg of the vertex, and the · · · in B, C, and A denote momenta belonging to S (a,I ,v,1) , S (a,I ,v,2) , and S (a,I ,v,3) respectively. If either scalar leg is external, then B j 1 = δ i 1 in . If the gluon leg is external, then A We now attach gluon a to each of the legs of this scalar-scalar-gluon vertex, either to an external leg or to an internal line. This yields We can also attach gluon a directly to the scalar-scalar-gluon vertex itself. Using eq. (6.19), this yields We now use eq. (3.16) in eq. (7.19), and combine eqs. (7.19) and (7.20) as we did in sec. 6. Leaving the µ 3 index implicit, we obtain the contribution of the scalar-scalar-gluon vertex to the radiation vertex expansion where J 3 and S 3 act on the µ 3 index of V (K 1 , K 2 , K 3 ) .
At the end of the last subsection, we discussed the color-factor symmetry acting on theψψg vertex contribution to the radiation vertex expansion. Under eq. (7.9), the variation of eq. (7.21) is proportional to In sec. 6, we demonstrated that each of the three terms in eq. (7.22) vanishes. Therefore the contributions of theψψg vertices (7.21) to the radiation vertex expansion are invariant under the color-factor shift associated with gluon a.
Finally we choose one of the scalar-scalar-gluon-gluon vertices v of I (if it has any). Such a vertex divides the external legs into four non-overlapping subsets S (a,I ,v,r) , r = 1, · · · , 4 such that 4 r=1 S (a,I ,v,r) = {1, · · · , n} \ {a}. Using eq. (6.18), the contribution of diagram I can be expressed as We now attach gluon a to each of the legs of this scalar-scalar-gluon-gluon vertex, either to an external leg or to an internal line. This yields We now need to consider the variation under the color-factor symmetry of this contribution to the radiation vertex expansion. Designate by c (a,I ,v,r) with r = 1, · · · 4 the color factor associated with each of the terms in eq. (7.24), including the factors of f abc and (T a ) j k in the subdiagrams. These color factors satisfy 4 r=1 c (a,I ,v,r) = 0 by virtue of The variation of c (a,I ,v,r) under the color-factor shift associated with gluon a is δ a c (a,I ,v,r) = α (a,I ,v) k a · K r . (7.26) The variation of eq. (7.24) under eq. (7.26) is therefore proportional to 4 r=1 ε a · K r η µ 3 µ 4 − iε aα k aβ (S αβ 3 ) µ 3 ν η νµ 4 + (S αβ 4 ) µ 4 ν η µ 3 ν . (7.27) The first term vanishes by momentum conservation, k a + 4 r=1 K r = 0, and ε a · k a = 0. The second term is the first-order Lorentz transformation of the tensor η µ 3 µ 4 , which vanishes.
Thus each vertex involving scalars that contributes to the radiation vertex expansion is invariant under the color-factor shift associated with gluon a. Together with the result from sec. 4 that the contributions from the three-and four-gluon vertices to the radiation vertex expansion are also invariant, we have shown that the full amplitude A n (ψ 1 , g 2 , g 3 , · · · , g n−1 , ψ n ) with spin-zero fundamentals is invariant under the color-factor shift. In fact, any amplitude built withψψg and ψψgg vertices for scalar ψ will have the color-factor symmetry.

Vertices involving spin-one fundamentals
To derive the contribution of theψψg andψψgg vertices to the radiation vertex expansion for spin-one fundamentals ψ, we examine the effect of attaching a gluon to a massive vector leg, either external or internal.
First we single out (vector) leg 1, denoting the contribution of an (n − 1)-point diagram I to the amplitude as ε 1µ 1 C i 1 µ 1 (k 1 , · · · ) where · · · denotes momenta belonging to {2, · · · , n}\{a}. Attaching gluon a to external vector leg 1 we obtain Contracting with ε 1µ 1 ε aµa , we obtain where we used k 2 4 = ε 4 ·k 4 = ε 1 ·k 1 = 0 and k 2 1 = m 2 1 . We did not use the vanishing of eq. (4.3). Similar expressions result from attaching gluon a to the other legs. By comparing eqs. (4.4) and (7.29), we observe that the expression is the same for a massless or a massive vector particle.
Next we single out one of the internal lines of I , which divides the diagram into two subdiagrams B and C, and splits the external legs {1, · · · , n} \ {a} into two complementary sets S a,B and S a,C . The contribution of the diagram can thus be written as where K = d∈S a,B k d is the momentum running through the line, and the · · · in B and C denote momenta belonging to S a,B and S a,C respectively. Attaching gluon a to the line connecting the two subgraphs and contracting with ε aµa , we obtain Now we use the identity [35] P µλ (K)ε aµa V λκµa (K, −K − k a , k a )P κν (K + k a ) to rewrite this as We did not use the vanishing of eq. (4.8). We associate each of the terms in curly brackets with one of the two vertices to which the line is attached.
For the rest of the discussion, we can be brief. The expressions for the contributions to the radiation vertex expansion fromψψg andψψgg vertices are similar to those for spin-zero fundamentals, except that we must include J r and S r terms for r = 1 and 2. The proof that these vertex contributions are invariant under the color-factor symmetry relies on some of the results from sec. 4.
Thus we have shown that the full amplitude A n (ψ 1 , g 2 , g 3 , · · · , g n−1 , ψ n ) with massive spin-one fundamentals is invariant under the color-factor shift. In fact, any amplitude built withψψg and ψψgg vertices with a massive vector particle ψ will have the color-factor symmetry.
We can go even further and state that any amplitude built from ggg andψgψ vertices (with ψ having arbitrary spin ≤ 1) where not only ψ but also some of the gluons are massive (i.e., through spontaneous symmetry breaking) will be invariant under the color-factor shift. The only particle in the amplitude that must be massless is the gluon associated with the color-factor symmetry.

Null eigenvectors of the propagator matrix
The symmetry that we have introduced in this paper is possessed not only by gauge-theory amplitudes but also by the amplitudes of the much simpler theory [48]  are double-partial amplitudes of the bi-adjoint scalar theory [48]. The m(1γn|1δn) are also the entries of (n − 2)! × (n − 2)! propagator matrix defined in ref. [7]. Vaman and Yao argued that the propagator matrix has rank (n − 3)! by virtue of momentum conservation, using explicit low n examples. Cachazo, He, and Yuan confirmed this for general n by expressing the double-partial amplitudes as a sum over the (n − 3)! solutions of the scattering equations [48].
The cubic vertex expansion of the n-point bi-adjoint scalar amplitude with respect to external scalar a (see sec. 2) is given by Using the invariance of A scalar n , together with independence of α 2,σ andc 1δn , we obtain n b=3 , · · · , σ(n − 1), n|1δn) = 0 (8.10) i.e., we have derived a set of (n − 3)! null eigenvectors of the propagator matrix. Other sets of null eigenvectors are obtained from the color-factor shifts associated the other massless scalars in the amplitude.
Since the (n − 2)! × (n − 2)! propagator matrix is known [48] to have rank (n − 3)!, at most (n − 3)(n − 3)! of these null eigenvectors can be independent. Thus, the color-factor symmetry associated with n − 3 of the massless scalars suffices to guarantee the reduced rank of the propagator matrix. This nicely explains a result found by one of the present authors in ref. [68], viz., that the propagator matrix persists in having rank (n − 3)! even when up to three of the external particles in the amplitude are massive, but if more than three particles are massive, the rank of the propagator matrix is greater. We now see that for m ≥ 3 massive and n − m massless particles, the number of null eigenvectors generated by the color-factor symmetry will be (n − m)(n − 3)! , and therefore the rank of the propagator matrix will be (m − 2)(n − 3)! for m = 3 through m = n.
Returning now to Yang-Mills theory, if we assume that the numerators of the n-gluon amplitude obey color-kinematic duality, i.e. they obey the same Jacobi relations as c i , then they can similarly be expressed in terms of (n − 2)! half-ladder numerators n 1γn via Thus the null eigenvectors of the propagator matrix (8.10) imply that the color-ordered n-gluon amplitudes obey the BCJ relations (2.19).
As we saw in sec. 2, however, it is not necessary to require color-kinematic duality in order to prove the BCJ relations. The BCJ relations follow from the weaker constraint (2.26), and both eq. (2.26) and the BCJ relations are a consequence of the color-factor symmetry of the amplitude, which is established through the radiation vertex expansion.

Loop-level amplitudes
Given the connection established in this paper between the color-factor symmetry of tree-level gauge-theory amplitudes and color-kinematic duality/BCJ relations, it is naturally of great interest to see whether these ideas can be extended to loop level. In this section, we generalize the cubic vertex expansion introduced in sec. 2 to loop-level amplitudes. We then define a set of shifts of oneloop color factors that depend on the loop momentum as well as the momenta of external particles, and ask whether the cubic vertex expansion of the one-loop amplitude is invariant under these shifts.
It is obvious from the variation of the cubic vertex expansion (2.25) that the tree-level amplitude will be invariant under a color-factor shift if the numerators satisfy color-kinematic duality. The reader may have noticed, however, that up until now we have taken great pains not to invoke color-kinematic duality to prove color-factor symmetry. We have chosen rather to show that it follows directly from a Lagrangian approach. At loop level, we no longer have that luxury, at least at the current stage of development. Instead we will explicitly invoke loop-level color-kinematic duality (for the theories in which it has been shown to hold [1,2,[10][11][12][13][14][15][16][17][18][19][20]) in order to demonstrate the color-factor symmetry of the one-loop amplitude. We then show that the invariance of the amplitude under color-factor shifts implies a set of relations among the integrands of its colorordered amplitudes.

Cubic vertex expansion for loop-level amplitudes
To construct the cubic vertex expansion of an L-loop n-gluon amplitude with respect to gluon a, we begin with the set of L-loop (n − 1)-point cubic diagrams I with external legs {1, · · · , n} \ {a}.  where the denominators in eq. (9.1) are the products of inverse propagators associated with the diagrams. (There could be different sets of denominators depending on the mass of the particle circulating in the loop.) Explicit definitions of the color factors are given below in eq. (9.6). The terms obtained by attaching the gluon to an internal line -in this case, the last three terms of eq. (9.1) -are split into two by applying the identity (2.20) or, in the case of massive internal lines, eq. (5.3). The terms are then reorganized into a sum over the vertices of I. For example, the terms in eq. (9.1) are reorganized into 2k 4 · k 1 + c 1234 n 1234 2k 4 · − c 1423 n 1423 2k 4 · ( + k 1 ) [34] n 12 [34] 2k 4 · k 3 + c 1243 n 1243 where in the last term of the last line of eq. (9.2) differs from the label in fig. 5 by a shift. 14 If the particle circulating in the loop has mass m, the expressions ( + · · · ) 2 in eq. (9.2) (i.e., those outside the square brackets) are all replaced by ( + · · · ) 2 − m 2 . We hasten to remind the reader that A c ···b,n,b+1··· n ···b,n,b+1··· Recently, partial fraction identities similar to eq. (2.20) have been employed to recast one-loop amplitudes into a new form whose denominators contain factors linear in the loop momentum [69][70][71][72][73][74], somewhat analogous to eq. (9.5). These new expressions are those that naturally emerge from a scattering-equation approach to loop-level amplitudes.

Color-factor symmetry at one-loop level
Next we consider the behavior of one-loop amplitudes under momentum-dependent shifts of its color factors. First we must define a set of shifts consistent with the requirements elucidated earlier in the paper. For the one-loop four-gluon amplitude, the color-factor shift associated with gluon 4 must satisfy [34] = α k 4 · k 3 (9.9) since these diagrams have gluon 4 attached to an external leg. Requiring the color-factor shift to respect the Jacobi relations (9.4) implies that Unlike the tree-level case, however, these requirements alone are not sufficient to fix the values of all the shifts; one of them, δ 4 c 1234 , remains arbitrary. In analogy with eq. (9.3), we define the remaining shift to be δ 4 c 1234 = α k 4 · (9.11) where is the loop momentum. 15 The effect of this shift on eq. (9.2) is Similar expressions are obtained for the contributions to the cubic vertex expansion from the other three-point diagrams.
A goal consistent with the development in this paper would be to prove by some alternative means (such as the radiation vertex expansion) that δ 4 A 4 vanishes under the one-loop color-factor shift. That would imply the vanishing of the r. h. s. of eq. (9.12) plus the expressions obtained from the other three-point diagrams, imposing a generalized-gauge-invariant constraint on the one-loop kinematic numerators (namely, that the terms in the square brackets in eq. (9.12) add up to something that integrates to zero). But at this point in the development of the subject, we have no such proof, and therefore we will turn the argument around, and use the knowledge that there exist kinematic numerators for the one-loop four-gluon amplitude that obey color-kinematic duality 0 = n [14]23 + n 1234 − n 1423 = n 1 [24]3 + n 1423 − n 1243 = n 12 [34] + n 1243 − n 1234 (9.13) for pure Yang-Mills theory (with only gluons circulating in the loop) [19] as well as for theories with other particles circulating in the loop [1,20]. In these cases, the kinematic Jacobi identities (9.13) imply that eq. (9.12) vanishes, as do the other contributions to the cubic vertex expansion. Thus, the one-loop four-gluon amplitude in these theories is invariant under the color-factor shift specified by eqs. (9.9), (9.10), and (9.11).
The cubic vertex expansion of the one-loop four-point amplitude of the bi-adjoint scalar theory may be obtained by replacing the kinematic numerators n i with a second copy of the color factors c i . Since the latter obey the one-loop color Jacobi identities (9.4), the bi-adjoint scalar one-loop four-point amplitude is also invariant under the color-factor shift.
(9.16) Relations of this form were first uncovered in refs. [49][50][51] from the perspective of on-shell recursion relations, and revisited recently using monodromy relations in string theory [75]. (See also ref. [76] for BCJ-type relations among loop-level integrands.) Conversely, if the integrands of the one-loop amplitude of a theory can be shown to satisfy eq. (9.16), then it follows that the one-loop amplitude is invariant under the color-factor shift, avoiding the need to invoke color-kinematic duality.
It is straightforward to generalize these considerations to one-loop n-gluon amplitudes. We may define a color-factor shift associated with any external gluon a, but for simplicity of presentation we will focus on gluon n. Let σ denote a permutation of {2, · · · , n − 1}. The color factor c ···[σ(b)n]··· shown in fig. 6 undergoes a shift because gluon n is attached to gluon leg σ(b). Requiring the shifts to respect the Jacobi relation We must additionally define the shifts δ n c 1σ(2)···σ(n−1)n = α n,σ k n · (9.19) for a set of half 16 of the permutations σ, where α n,σ are a set of (n − 2)!/2 independent arbitrary constants. Together these conditions imply that the shifts of the ring color factors are given by δ n c 1σ(2)···σ(b−1)nσ(b)···σ(n) = α n,σ k n · + k n · k 1 + b−1 c=2 k n · k σ(c) , b ∈ {2, · · · , n − 1} (9.20) and the shifts of all other one-loop color factors are fixed by requiring that they respect the Jacobi relations. Thus there is an (n − 2)!/2-dimensional family of one-loop color-factor shifts associated with gluon n.

Discussion and conclusions
In this paper, we have introduced a new set of symmetries of gauge-theory amplitudes, which act as momentum-dependent shifts on the color factors appearing in the cubic decomposition of the amplitude. These symmetries are intimately linked to the presence of massless gauge bosons in the amplitude (or massless adjoint scalars in the case of the bi-adjoint scalar theory) and can be considered generalizations of the radiation symmetry of ref. [36]. We demonstrated that a wide class of tree-level gauge-theory amplitudes are invariant under these shifts, using a representation of the amplitude known as the radiation vertex expansion [35]. We also introduced a related but distinct cubic vertex expansion of the amplitude, and used this to derive a set of generalized-gauge-invariant constraints on the kinematic numerators appearing in the cubic decomposition of the amplitude. All known BCJ relations for tree-level gauge-theory amplitudes [1,4,5,41,42] follow as a direct consequence of the color-factor symmetry (this paper and ref. [47]). Finally, we generalized the cubic vertex expansion and color-factor symmetry to loop level. We showed that one-loop amplitudes that satisfy color-kinematic duality are invariant under the one-loop color-factor symmetry, and derived a set of relations among the integrands of one-loop color-ordered amplitudes.
Let us take a look at the connection between the color-factor symmetry and more fundamental symmetries of the Lagrangian, gauge and Poincaré invariance [35]. The color-factor symmetry follows as a result of the vanishing of certain expressions, namely eqs. (4.19), (7.10), and (7.22), associated with the cubic vertices of a gauge-theory amplitude, and eqs. (4.23), (4.24), and (7.27), associated with the quartic vertices. It is illustrative to examine the various contributions in a soft expansion in the gluon momentum k a , even though the color-factor symmetry is exact in k a .
The leading term in the soft expansion corresponds to the O(k 0 a ) term in each of eqs. (4.19), (4.23), (7.10), (7.22), and (7.27). These are all proportional to r ε a · K r where K r are the momenta flowing out of each leg of the vertex. This vanishes by ε a · k a = 0 together with momentum conservation k a + r K r = 0 -a result of symmetry under spacetime translations.
The subleading term in the soft expansion corresponds to the O(k 1 a ) term in each of eqs. (4.19), (7.10), (7.22), and (7.27), and to eq. (4.24). These expressions are all given by a sum of angular momentum generators J αβ r , which act as a first-order Lorentz transformation on the relevant vertex factors. They vanish by Lorentz invariance.
Thus the first two terms in the soft expansion vanish as a result of Poincaré invariance. It is a little more difficult to pin down the underlying symmetry responsible for the vanishing of the sub-subleading terms in eqs. (4.19) and (7.22), together with an analogous expression for spin-one particles. The O(k 2 a ) term in eq. (4.19) is proportional to = 2i −η αµ 1 η βµ 2 η γµ 3 + η αµ 1 η γµ 2 η βµ 3 + (cyclic permutations of 123) (10.1) and the O(k 2 a ) term in eq. (7.22) is proportional to Neither expression vanishes by itself but both do when contracted with ε aα k aβ k aγ for gluon a. These identities, which go beyond the first-order Poincaré cancellation and are connected to Yang-Mills gauge symmetry, are key ingredients contributing to the color-factor symmetry.
Returning to our discussion in the introduction of the connection of the color-factor symmetry with the photon radiation symmetry and radiation zeros (for a collection of early references, see refs. [23,24,[35][36][37][38][77][78][79][80][81]), we have uncovered some additional analogs. For example, we can write a factorized form for the four-gluon amplitude (2.1) which vanishes at c s /s = c t /t = c u /u = const. This is a non-abelian version of the original radiation zero studied almost forty years ago in qqW γ and eνW γ reactions, with a zero at Q c /k c · k a = constant. This original radiation factorization and its zero led to the prediction of a measurable experimental dip, which has now been confirmed [82,83]. The analogous zero in the four-gluon amplitude is washed out, however, by the color averaging that must be performed in observable quark-gluon processes.
In the generalization to tree-level n-point amplitudes, the abelian radiation symmetry and existence of zeros for Q c /k c · k a = constant (for photon momentum k a ) rest on having gauge-theory couplings, as noted earlier. The non-abelian color-factor symmetry uncovered in this paper can also lead to zeros in n-point amplitudes, but with an important complication. The invariance under c i → c i + α i c∈S a,i k c · k a for the attachment of a gluon with momentum k a cannot be used to systematically cancel out the complete n-gluon amplitude using an overall common value for α i . Because of the Jacobi relations, that overall common value must vanish. There are in principle zeros, however, for separate islands of α i values. Although they are again washed out by coloraveraging, the generalized factorization coming from the color-factor symmetry remains useful for theoretical analysis of tree amplitudes. In a different direction, note that the BCJ form of the gluon amplitudes has been utilized in the planar zeros studied recently in refs. [84,85].
The analogs described above are a bridge to a final overall remark. It has been helpful to think of gluon emission or absorption as effecting a (first-order) transformation in both color and kinematic space simultaneously on the graph to which it is attached. In particular, the attachments lead to transformations of the various legs and vertices of the "parent" diagram in either momentum or space-time representations. All the parent wave functions end up transformed, and identities derived from eq. (2.20) for the different spins, e.g., eqs. (7.5) and (7.32), yield exactly the two terms expected from the propagator with its bilinearity in the wave functions. The cancellations highlighted throughout this paper arise precisely because we consider those theories whose amplitudes transform covariantly under color and kinematic transformations. Adding all possible massless gluon attachments to a complete set of parent graphs leads to a sum of corresponding color shifts that vanishes because of invariance. Such a picture should help in finding directions in the diagrammatic analyses of a variety of extensions of the gauge theories considered in this paper, supersymmetric and otherwise.

A Five-gluon amplitudes
In this appendix, we use the five-gluon amplitude to provide an explicit example of the cubic vertex expansion (2.21) introduced in sec. 2, and the relations among kinematic numerators (2.26) resulting from the color-factor symmetry.
The cubic decomposition (2.8) of the five-gluon amplitude is given by where c α are half-ladder color factors defined in eq. (2.9). Let us recast this amplitude in a cubic vertex expansion with respect to gluon 2. We have already arranged the terms in eq. (A.1) so that each line corresponds to one of the four-gluon diagrams I obtained by omitting gluon 2. We rewrite the denominator of the third term of the first line as and similarly the denominators of the third terms of the other two lines to obtain According to eq. (2.14), the color-factor shifts associated with gluon 2 act as which is precisely of the form of eq. (2.26). The color-factor shift with respect to gluon 3 instead yields δ 3 A 5 = α 3,24 n 1 − n 4 − n 15 s 45 + n 3 − n 1 + n 12 s 12 + n 11 − n 2 + n 5 s 15 + n 10 − n 11 + n 13 s 24 (A.9) + α 3,42 n 6 − n 8 − n 9 s 25 + n 14 − n 6 + n 7 s 14 + −n 11 + n 2 − n 5 s 15 + −n 10 + n 11 − n 13 s 24 = 0 .
Since α 2,34 , α 2,43 , α 3,24 , and α 3,42 are independent arbitrary constants, each expression in parentheses vanishes, yielding four independent constraint equations on the kinematic numerators of the five-gluon amplitude. No additional independent constraints are obtained from the color-factor symmetries associated with the other three gluons. These "generalized Jacobi relations" for fivegluon amplitudes were previously derived in refs. [39,40] by using the properties of string theory amplitudes.