The seeds of EFT double copy

We explore the double copy of effective field theories (EFTs), in the recently proposed generalized color-kinematics and Kawai-Lewellen-Tye (KLT) approaches. In the former, we systematically construct scalar numerators satisfying the Jacobi identities from simpler numerator"seeds"with trace-like permutation properties. This construction has the advantage of being easily applicable to any multiplicity, which we exemplify up to 6-point. It employs the linear map between color factors formed by single traces of generators and by products of the structure constants, which also relates the generalized KLT and color-kinematics formalisms, allowing to produce KLT kernels at arbitrary order in the EFT expansion. At 4-point, we show that all EFT kernels are generated and that they only yield double-copy amplitudes which can also be obtained from the traditional KLT kernel. We perform initial checks suggesting that the same conclusions also hold at 5-point. We focus on single-trace massless scalar EFTs which however also control the higher-derivative corrections to gauge and gravity theories.

that may simultaneously depend on both color and kinematics, while still satisfying Jacobilike identities. A set of composition rules is defined to systematically build such numerators at a given order in the EFT expansion. This approach can be extended from 4-to 5-point amplitudes [35].
In the approach taken by [36], the KLT kernel is generalized. The initial observation is that the map between open-and closed-string amplitudes involves a kernel with higherderivative corrections, which is the inverse matrix of the color-ordered amplitudes in a bi-adjoint EFT with operator coefficients correlated in a specific way [37]. This notion is generalized by allowing for more free parameters in the KLT kernel, arising from a more general bi-adjoint EFT. A set of bootstrap equations on the KLT kernel and input theories are proposed to guarantee a healthy analytical structure in the resulting doublecopy amplitudes. Solving the bootstrap then yields a systematic study of the operators possibly involved in the double copy. Correct factorization properties for a theory with fixed particle content need to be imposed as an additional constraint on both the kernel and the input amplitudes.
In this paper, we aim to shed light on these two approaches to the double copy of EFTs and on their relation, by introducing a new method to construct generalized numerators at any multiplicity. The traditional color numerators, consisting of products of Lie-group structure constants, can be written as linear combinations of the single traces of products of group generators. In the same way, we show that all generalized numerators can be constructed from simple numerator seeds, which satisfy the same permutation properties as the single traces of generators.
The construction of numerators has previously been investigated from different perspectives. They have for instance been extracted from known amplitudes and the KLT kernel [38,39]. Dual trace factors, analogous to our numerator seeds but involving momenta and polarization vectors, have also previously been identified in Yang-Mills amplitudes [40][41][42][43][44]. In contrast, we use numerators to construct EFT amplitudes, and seeds built out of color and momenta to construct generalized scalar numerators. Based on the kinematic algebra, vector numerators for Yang-Mills and heavy-quark effective theories have also recently been constructed from simpler "pre-numerators" [32,45].
A further advantage of numerator seeds is that they can be directly related to KLT kernels. This enables the study of the operators involved in the KLT double copy, through a method that is alternative to the bootstrap of [36]. Numerator seeds thus provide further insight into the relation between the double copy approaches of [34,35] and [36], and into the structure of the generalized KLT kernel. In particular, at 4-point, we show that the double-copy amplitudes obtained with a generalized KLT kernel can equivalently be achieved by the traditional kernel, multiplying healthy local input amplitudes including higher-derivative corrections. As emphasized in [36], the generalized kernel does however allow for more general EFT inputs to the double copy. We also report on various results which indicate that this observation extends to higher multiplicities.
The structure of this paper is as follows. To be self-contained and set the notation, we first provide in Sec. 2 a detailed review of the two aforementioned approaches to the double copy of EFTs. The construction of generalized numerators from seeds at any multiplicity is then presented in Sec. 3. In this section, we also show how this construction facilitates the reorganization of CK-dual representations of amplitudes in terms of color-ordered amplitudes, which are the building blocks of the KLT formalism. Moreover, for any input amplitude that can be double-copied with a generalized kernel, we identify new objects which yield the same double copy with the traditional kernel. This holds provided the generalized kernel can be constructed from numerator seeds, and provided one can ensure locality of the new objects in order to call them amplitudes. We discuss these two caveats at 4-and 5-point in the subsequent sections. Restricting to 4-point amplitudes, Sec. 4 and Sec. 5 illustrate our method and show that it generates all solutions to the KLT bootstrap. We also analyse the double-copy structure in the KLT formalism, and the factorization properties of the amplitudes involved. The two caveats above are successfully addressed in this 4-point case. Moving on to 5-point amplitudes in Sec. 6, we demonstrate that the lowest-order bootstrap solutions provided in [36], can be reproduced from our numerator construction. We also present partial results suggesting that no new double copies are generated by the generalized kernel at 5-point either.

The systematic double copy of EFTs
We start with a review on the generalized KLT method of [36] and the generalized numerator method of [34,35].

The generalized KLT approach
The KLT formula for an amplitude with n external particles in the adjoint representation of SU(N ) (or U(N )) symmetry groups is given by Here, A L n and A R n are the color-ordered amplitudes of potentially different theories, called single copies, and M n is the double-copy amplitude. Due to the Kleiss-Kuijf (KK) [46] and BCJ [2] relations, the number of independent color-ordered amplitudes forming a BCJ basis is (n − 3)!. The indices α, β in Eq. (2.1) refer to the color-orderings of the single-copy amplitudes and the sums run over the elements of any two BCJ bases, while the KK and BCJ relations ensure that the double-copy amplitude does not depend on the chosen bases.
The multiplication of the single copies is governed by S n , the KLT kernel, which is a scalar function of Lorentz invariants. Its form depends on the BCJ bases considered in the sum. The kernel plays a crucial role in ensuring that the resulting double-copy amplitude has a healthy analytical structure. It cancels poles that are present in both ordered amplitudes, to prevent double poles in M n , and provides missing poles so that all physical factorization channels are generated. This requires the KLT kernel to have a precise structure, which was found to be closely related to the amplitudes of the bi-adjoint scalar theory (BAS) [6].
The BAS Lagrangian is given by where the scalar field has two adjoint color-group indices. 1 We normalize the adjoint generators and structure constants such that [T a , T b ] = f abc T c and Tr T a T b = δ ab . Using the decomposition of the structure constants, the full bi-adjoint n-point amplitude can be written in terms of linearly independent traces of the generators, A bas n = α,β∈S n−1 Tr(T aα 1 T aα 2 · · · T aα n ) m n [α|β] Tr(Tã β 1Tã β 2 · · ·Tã βn ) . (2. 3) The objects m n [α|β] are called doubly color-ordered amplitudes. They can be computed by summing over the trivalent graphs that contribute to the color orderings of both arguments, with appropriate relative signs. where (1234) ≡ Tr(T a 1 T a 2 T a 3 T a 4 ), etc. We definec 0 similarly as c 0 for the color factors with tilded indices. The full BAS amplitude is then written compactly in matrix form as, 2 where we use the conventions s = s 12 , t = s 13 and u = s 14 with s ab = (p a + p b ) 2 and all momenta incoming. Remarkably, the KLT kernel can be identified as the inverse matrix of doubly colorordered amplitudes [6], where the indices α and β should be restricted to any two BCJ bases (in which colorordered amplitudes are independent), such that the m n [α|β] sub-matrix is of full rank and can be inverted. The rows and columns of m n all satisfy the KK and BCJ relations, and the double copies of BAS amplitudes are trivial: where the uncontracted α, δ indices correspond to the color orderings that are left untouched in this relation. The BAS theory can also be double-copied with another single-copy theory, in which case the KLT product encodes the KK and BCJ relations. This is simple to illustrate at 4-point, where m 4 [α|β] has rank 1 and the kernel is simply the inverse of a number for fixed α, β. In this case, Eq. (2.6) implies that there are choices of BCJ bases that render the KLT relation trivial, such as in when α = γ. However, because of the KK and BCJ relations, the left-hand side does not depend on the β, γ bases chosen in the product. For example, (note α = γ, but arbitrary β) is exactly a BCJ relation, while one of the KK relations is given by It was emphasized in [36] that the BAS behaves like an identity element in the KLT product, which is why it is also referred to as the zeroth copy.
The identification of the BAS as the identity element of the KLT product leads to generalizations of this product associated to modifications of the BAS theory. This is exactly the case for the field theory form of the KLT relation between the open-and closed-string amplitudes [37]. However, not all modifications of the BAS theory result in acceptable KLT kernels. It was found that these corrections should preserve the rank of the matrix of doubly color-ordered amplitudes, which is (n − 3)! [36]. This is called the minimal rank condition. In this paper, we will focus on higher-derivative (h.d.) corrections suppressed by powers of an EFT cutoff scale Λ, i.e. m h.d. n = m n +O(1/Λ). In the decoupling limit, Λ → ∞, one therefore recovers the traditional KLT product.
In and allow to bootstrap the single-copy amplitudes A n,l/r that can take part in the double copy, Depending on the form of m h.d.
n and S h.d. n , the generalized KKBCJ relations for A n,r and A n,l may be different. We emphasize that both A n,l/r and A n,l/r may in principle contain higher-derivative corrections. The prime indicates that the amplitudes satisfy generalized KKBCJ relations, which allow for more operators.
That Eq. (2.13) produces a healthy amplitude when m h.d.
n has minimal rank is a nontrivial empirical result [36]. The generalized KLT formalism allows for a systematic study of the space of theories that can appear as input and output of the double-copy procedure. Although the bootstrap equations strongly constrain the higher-derivative corrections that are allowed in the input (single-copy) amplitudes, the generalized KLT formalism allows for more independent operators in the single copies than its traditional version. However, up to the orders checked explicitly in [36], it was found that the space of generalized output (double-copy) amplitudes M n is the same as M n . At 4-point, where the doublecopy relation contains a single term, 'similarity transformations' were proposed in [36] to explain this fact (see also [47]). We aim to shed further light on this observation in Sec. 3 and Sec. 5.
As an example, the 4-point amplitude of the BAS theory with higher-derivative corrections is (2.14) Solving the minimal rank condition, the m h.d.

4
matrix of doubly color-ordered amplitudes corresponding to BAS+h.d. can be written as [36] m h.d.
and f 2 satisfies the bootstrap equation Furthermore, the aforementioned assumption m h.d. , withm h.d.
This structure also exists if we turn off the higher-derivative corrections, with Restricting to the smallerm h.d.

4
andm 4 matrices will prove useful in the following sections. We will actually only use these hereafter and drop the tildes for convenience. However, such a block matrix structure generally only exists at lowest derivative order for n > 4 particles. The 4-point case is special because the kinematics is invariant under reversal of the particle labels:f (1, 2, 3, 4) ≡ f (s 12 , s 13 ) = f (s 43 , s 42 ) ≡f (4, 3, 2, 1), for any function f of the Mandelstam invariants.

The generalized numerators approach
Another approach to the double copy is based on the color-kinematics (CK) duality [2,3]. The basic idea is to use the decomposition of an on-shell n-point amplitude A n on trivalent graphs g, where d g is the product of the (inverse) propagators it involves; c g traditionally correspond to color factors associated to that same graph (i.e. combinations of generators of the gauge algebra); while n g are the kinematic numerators that depend on Lorentz invariants and possibly on polarization vectors. Given an amplitude A n , the numerators n g are not unique. CK duality is then a property of amplitudes for which there exists a choice of numerators n g which verify the same algebraic relations as those of the color factors c g , inherited from the gauge algebra. In certain theories, such as Yang-Mills, all tree-level amplitudes satisfy the CK duality. In particular, the SU(N) color factors of 4-point amplitudes obey Jacobi identities of the form 3 and antisymmetry relations upon interchanging two legs on one vertex in the g a , g b , g c graphs. Any numerator satisfying these adjoint algebraic relations will be called an adjoint numerator.
In gauge theories, the color relations ensure the gauge invariance of the amplitude under shifts of the polarization vectors contained in n g , A| i →p i = 0, for any particle label i. This implies, in turn, that c g can be replaced by any expression which satisfies the same relations without spoiling gauge invariance, and the latter applies in particular to another copy of a CK-dual n g . The BCJ double-copy procedure [2,3] is thus schematically where M n is the double-copy amplitude. The numeratorñ g is not necessarily the same as n g , but both are CK-dual to the same c g color factors. 4 For two copies of Yang-Mills, the resulting amplitude describes the scattering of gravitons (as well as scalars). Let us illustrate this approach at 4-point with the well-known case of Yang-Mills theories. There are three trivalent diagrams associated to the s, t, u channels, and the amplitude reads where we defined with an arbitrary function α. Square and angle brackets denote spinor-helicity variables (for pedagogical introductions see e.g. [48,49]). The Jacobi identity reads c s + c t + c u = 0 and the explicit formulae above allow to check that n s + n t + n u = 0 for any α. The color factors also satisfy antisymmetry relations such as c s | 1↔2 = c s | a↔b = −c s . For α = 1/3, the numerators are antisymmetric too: e.g. n s | 1↔2 = n s | t↔u = −n s . This shows the CK duality of the 4-point YM amplitude. The double-copy method then leads to a diffeomorphism-invariant four-gravitons amplitude, The bi-adjoint scalar theory considered in the previous subsection also plays the role of a zeroth copy in the CK approach. It has a (color-color) dual structure, where the two color groups lead to c g andc g . Being both color factors in the adjoint representation, they verify Jacobi and antisymmetry relations and any of the two can be treated as a CK-dual numerator n g . Moreover, replacing c g in Eq. (2.22) by one of the BAS color factors has a trivial effect.
Higher-derivative effects in the BAS theory can be included, while preserving the dual CK structure, by building generalized numerators, c h.d. (c, s ab ) [34,35] (see also [31]). These verify the same adjoint algebraic relations as the c g , but may depend on both the Mandelstam invariants s ab and color factors c. The color factors are themselves not necessarily of adjoint type, but more generally built from products of traces of group generators. Only their combination with Mandelstams is required to satisfy the adjoint algebraic relations. In this paper, we focus on single traces and do for instance not consider generalizations in the form of double traces (see e.g. [31]).
Choosing c h.d. = c g + O(s ab /Λ 2 ), the generalized numerators result in EFT amplitudes for the bi-adjoint scalar theory that retains the CK-duality, (2.27) Similarly, higher-order corrections in a gauge theory that preserve the CK-duality can be obtained by replacing the color factors by generalized numerators, (2.28) for unmodified n g . This can also be interpreted as the double copy between A ym n and A bas+h.d. n amplitudes. The gauge invariance of the amplitude is maintained because the c h.d. g satisfy the same relations as the c g . This is the approach taken in [34,35] to extend the CK double-copy method to higher-derivative EFT amplitudes.
To find the most general allowed numerators, one could construct an ansatz for c h.d. at each order in the Mandelstams and impose the adjoint algebraic relations. Exploiting the underlying structure instead, [34] demonstrated that all purely kinematic adjoint 4-point numerators can be built using a simple composition rule acting on existing lower-order adjoint kinematic numerators j and k (vectors), This requires only one building block made out of kinematic invariants, Furthermore, all adjoint 4-point numerators involving one factor of color can be generated using Eq. (2.29) with one of the two input numerators containing color, and the additional rule where d is the color factor that is fully symmetric under external particle permutations, The only other color structures required as primary building blocks are the adjoint ones, c s = f abe f ecd , etc. encountered in Eq. (2.24). All possible adjoint structures at 4-point can then be obtained by successive applications of these composition rules, and linear combinations of these yield the most general c h.d. [34]. At 5-point, the situation is complicated by the presence of more composition rules and algebraic structures [35].
In the next section, instead of constructing adjoint numerators from lower-order adjoint numerators and applying composition rules, we will build all of them from simpler non-adjoint objects. This alternative construction procedure can be extended to higher multiplicities without much complication.

Numerator construction from seeds at any multiplicity
We now propose an alternative method to construct generalized adjoint numerators of the SU(N ) (or U(N )) unitary group. At this stage, we are only interested in the adjoint numerators themselves, without regard to the factorization properties and particle content of amplitudes in which they enter. Section 5 discusses which extra constraints are imposed by such considerations. The current section applies to any number of particles, while the construction is repeated explicitly in the next section at 4-point.

Numerator seeds
The main observation is that the adjoint color factor, c adj , consisting of products of the structure constants, can be written in terms of linear combinations of single traces of the group generators (see also [44]), The matrix J contains only {±1, 0} entries and will play a central role in our construction. Its entries are determined by decomposing the structure constants in terms of traces through Products of traces can then be combined using the SU(N ) completeness relation, and one can show that the 1/N terms cancel for products of structure constants. In this way, the matrix J relates the simple algebraic structure of single traces (contained in the c 0 vector) to the more involved adjoint algebraic properties. It encodes the Jacobi identities, and the antisymmetry relations which follow from the permutation properties of the single traces. The matrix J can be decomposed as follows: where B is the (n − 2)! × (n − 1)! matrix of rank (n − 2)! that relates c 0 to the color factors in a Del Duca-Dixon-Maltoni (DDM) basis [50], and A is the (2n − 5)!! × (n − 2)! matrix of rank (n − 2)! that relates the DDM basis to c adj . Conventions can also be chosen such that both A and B are sub-matrices of J, see [44]. Therefore, any object n 0 that satisfies the same algebraic properties as the single traces will be mapped to an adjoint numerator under multiplication by J. The vector of single traces has linearly independent entries that are given in terms of permutations of one functional form, which is cyclically invariant in its arguments, (3.6) In matrix notation, the algebraic properties can be summarized as follows. Under a relabeling σ of the particles, c 0 transforms as where M c 0 ,σ is a permutation matrix. We shall call objects that obey the algebraic properties of the single traces, numerator seeds, or seeds for short, as they can be used to generate adjoint numerators. We shall use both n 0 and c h.d.

0
to refer to numerator seeds. The latter notation emphasizes that the cyclically invariant functional form of the seeds depends on both Mandelstam invariants and color factors. Such seeds generate the generalized adjoint numerators discussed in Sec. 2.2.
In App. A, we prove that any adjoint numerator can be constructed from a numerator seed, i.e. n adj = J · n 0 . (3.9) Since the seeds are straightforwardly constructed, this provides an efficient way to explore the space of possible (generalized) adjoint numerators. A similar result was proven for kinematic numerators in renormalizable Yang-Mills theory using a different method [42]. A set of linearly independent numerator seeds generally maps to a redundant set of adjoint numerators, namely J · n 0 = J · n 0 could happen even for n 0 = n 0 . Therefore, identifying a set of seeds which generates independent adjoint numerators requires an extra step of reduction. This can be done by directly inspecting the general expression of n 0 , or by relying on a construction which removes redundancies. We give explicit examples of the former below, while the latter can be achieved using the Moore-Penrose pseudoinverse of J, called J + (see App. A): it also follows from the argument of App. A that J + · J · n 0 = J + · J · n 0 is a valid numerator seed. 5 A complete and independent set of k 1 2 3 4 5  6  7  8  9  10  11  12  13 14 15   4-pt 1 1 1 2  2  2  3  3  3  4  4  4  5  5  5  5-pt 0 0 1 2  5  8  14  21  32  45  63  84 112 144 185 6-pt 1 3 9 23 54 120 243 469 861 1509 2546 4158 · · · · · · · · · Table 1. Number of independent scalar kinematic numerators at O(1/Λ 2k ) in the EFT expansion. The Gram determinant constraints relevant in 4 spacetime dimensions have been accounted for. The counting at 4-and 5-point was achieved in [34] while the 6-point one is provided here for the first time. Although straightforward in principle, the numerically intensive reduction of the overcomplete set of numerators was not pushed beyond k = 12 at 6-point.
numerator seeds can thus be obtained by projecting with J + · J on all cyclically invariant functions.
As an example of redundancies, permutation invariant functional forms result in valid numerator seeds, but they are mapped to zero and thus do not give rise to independent adjoint numerators. In addition, J always combines seed entries with reversed ordering of particle labels, since c adj → (−1) n c adj under reversal, at n-point. The entries in a numerator seed can thus be ordered such that J has a block matrix structure, schematically: J a×b = J a×(b/2) , (−1) n J a×(b/2) . Therefore, the general numerator seed n 0 = n 0 (1, 2, ..., n), ... , n 0 (n, ..., 2, 1), ... , (3.10) and the seed on which we impose (anti)symmetry under reversal on the functional form, n 0 = 1 2 n 0 (1, 2, ..., n) + (−1) n n 0 (n, ..., 2, 1), ... , n 0 (n, ..., 2, 1) + (−1) n n 0 (1, 2, ..., n), ... , (3.11) result in the same adjoint numerator. At 4-point, these are the only sources of redundancy in the construction of adjoint numerators. There are further redundancies in the construction of CK-dual amplitudes, called generalized gauge transformations that will be addressed in section Sec. 4.3. At higher multiplicity, redundancies can take a more complicated form, to be exemplified at 5-point in Sec. 6. Even without identifying the specific algebraic origin of the redundancies, it is straightforward to just build the overcomplete set of seeds and identify numerically a basis of independent adjoint numerators. We provide the counting of the latter in Table 1, for numerators taking the form of polynomials of Mandelstam invariants. This table can be compared with Table 2 in App. B, listing the number of independent seeds, which grows faster with n than the number of independent numerators. The construction of adjoint numerators at 4-point agrees with the observation made in [34], namely that higher-order adjoint numerators can be obtained from lower order ones by multiplication with a permutation invariant function. The results at 5-point also agree with the number of independent numerators listed in Table 2 of [35] and we have constructed all the adjoint kinematic scalar numerators up to 6-point and O(1/Λ 24 ). 6 We also provide explicit examples of numerator seeds and adjoint numerators of lowest orders in App. B.

Double copy and color-ordered amplitudes from numerators
The matrix J is also useful to obtain color-ordered amplitudes from adjoint numerators. This approach was previously taken in [7], with a similar definition for J. 7 For instance, writing the full bi-adjoint scalar amplitude of Eq. (2.26) in matrix form and using the fact that c adj and c 0 are related by J through Eq. (3.1) yields where P contains the propagators (and coupling constants) of the trivalent graphs on its diagonal. By definition, this produces the BAS matrix of doubly ordered amplitudes (see also [44]), Similarly, one can write the single-copy (color-ordered) amplitudes in terms of a numerator seed (replacingc adj by n adj = J · n 0,r ), These single-copy amplitudes satisfy the traditional KK and BCJ relations as a consequence of the explicit factor of m and of the relation in Eq. (2.7). The double-copy amplitude can be obtained in the CK way, M = n adj,l · P · n adj,r = n 0,l · m · n 0,r , (3.15) or equivalently through the KLT relations, This exposes the special role played by the BAS matrix of color-ordered amplitudes to ensure the correct propagator structure of the double-copy amplitude.
The same method can be applied to obtain color-ordered amplitudes from generalized numerators. Defining a matrix H h.d. , which depends only on Lorentz invariants, one can decompose the numerator seeds (which, for simplicity, we build using only single traces) as c h.d.
It follows that the higher-derivative color-ordered amplitudes can be constructed by leftand right-multiplication of the lowest order matrix m, Similarly, starting from Eq. (2.28), one can write the higher-derivative single-copy (full) amplitude as and color-ordered amplitudes as where we defined n 0,r such that n adj = J · n 0,r . Assuming that H h.d.
l is of full rank, which always holds for an EFT expansion of the form H h.d.
l · n 0,r , and we obtain Note that n 0,r also satisfies the properties of numerator seeds, and A r satisfies the generalized KKBCJ relations. The naive procedure for obtaining a double-copy amplitude in the generalized CK formalism is to replace c h.d.
where we distinguish sums over all (n − 1)! color factors and sums over BCJ bases.
We have thus derived that the double-copy amplitudes obtained by the generalized KLT relations can equivalently be obtained through the traditional KLT double copy. However, there are two caveats to this statement. First, we derived this statement assuming that m h.d. is constructed via generalized numerators. It is unclear whether it then reproduces all solutions to the KLT bootstrap. In Sec. 5, we prove that this is the case at 4-point, and we have performed initial checks at 5-point presented in Sec. 6. Second, while the double-copy amplitude obtained by A l/r and a generalized kernel is the same as the one obtained by A l/r = (H h.d. l/r ) −1 · A l/r and a traditional kernel, it is unclear whether A l/r are physical amplitudes and what is their particle content. In Sec. 5, we show at 4-point that the assumption that H h.d.
l/r does not affect the BAS particle content implies that this is also the case for A l/r .

Seeds and generalized numerators at 4-point
In this section, we will work out the 4-point construction of scalar adjoint numerators from their seeds. This serves as an illustration of the method, and prepares for a comparison with the generalized KLT formalism in the next section.
A product of structure constants can be written in terms of single traces as follows, As noted before, the traces appear together with their reversed ordering in this relation. The adjoint color numerator can thus be written compactly in terms of traces as This defines the matrix J 4 with {±1, 0} entries, which encodes the Jacobi identity as the vanishing sum of its rows. Notice that we redefined the vector of traces c 0 shown in Eq. (2.4), combining traces and their reversed orderings such that We have conventionally chosen the ordering in the arguments of c 0 entries such that the s, t, u Mandelstams are invariant under the cyclic permutation of the (1,3,2,4), (1,2,3,4), (1,2,4,3), respectively. For instance, s| 1→3→2→4 = s. The 4-point BAS amplitude can now be rewritten as It is clear that J 4 multiplying any numerator seed that satisfies the same algebraic relations 9 as c 0 results in an adjoint numerator. However, it is not immediately clear that all adjoint numerators can be constructed in this way. In the following, we prove that indeed the construction via numerators seeds leads to the complete set of adjoint numerators. For the general proof at any multiplicity, see App. A. First, notice that J T 4 /3 satisfies J 4 ·J T 4 /3·J 4 = J 4 , and thus (since J 4 encodes the Jacobi identities as the sum of its rows) J 4 ·J T 4 /3·n adj = n adj for any vector n adj that satisfies the Jacobi identities. Therefore, J T 4 · n adj /3 is the pre-image of any n adj . Importantly J T 4 · n adj /3 is also a numerator seed: is invariant under cyclic permutations of its arguments and the reversal of their order, thanks to the algebraic properties of n adj . This completes the proof. As an example, consider multiplying the adjoint color factor by J T 4 , The resulting seed is equivalent to the usual c 0 up to the addition of a fully permutationinvariant quantity (mapped to zero by J 4 ), as it should since they generate the same adjoint color numerator. It is illustrative to compare the construction via numerator seeds with the composition method to construct adjoint numerators [34], reviewed in Sec. 2.2. Both the composition rule of Eq. (2.29) and the basic kinematic building block of Eq. (2.30) can be rewritten as a numerator seed multiplied by J 4 , The composition rule of Eq. (2.31) can similarly be rewritten, where j 0 is in fact a numerator seed (vector). This last composition rule encodes the simple statement that a fully symmetric object, such as d, takes an adjoint numerator to another one. This means that, at 4-point, there is a direct correspondence between the construction of adjoint numerators from seeds and by composition. This is however not the case at 5point, where the composition rules [35] are not equivalent to a simple multiplication by J 5 .

Kinematic numerator seeds
Analogous to Eq. (4.3), purely kinematic numerator seeds for a scalar theory have the form n 0 = n 0 (1, 3, 2, 4), n 0 (1, 2, 3, 4), n 0 (1, 2, 4, 3) T , n 0 (a, b, c, d) = g(s ac , s ab ) , (4.10) where g is a function of the Mandelstam invariants s ab ≡ (p a + p b ) 2 . Invariance under reversal is automatic at 4-point, while cyclic invariance requires g(s, t) = g(s, −s − t) = g(s, u). A general polynomial expansion of g(s, t) can then be written as 10 (4.11) Consistent factorization on the poles and assumptions on the particle spectrum of a theory can impose further restrictions on the a i,j coefficients. From this numerator seed, a singlecopy scalar amplitude can be constructed following Eq. (3.14), which leads to where we note that the function g(s, t) only appears through a permutation-invariant overall factor (so that it does not affect the traditional KK and BCJ relations).

Generalized numerator seeds
At zeroth order in Mandelstam invariants, one can verify that there is only one linear combination of single traces (c adj defined in Eq. (4.2)) that satisfies the adjoint algebraic properties. Therefore, the only necessary numerator seed containing only color information is given by the c 0 vector defined in Eq. (4.3). At this order, any other seed is related to c 0 by the addition of a permutation invariant combination of traces. Such seeds also map to c adj because permutation invariant combination of traces map to zero under multiplication by J. At higher orders in the kinematics, the most general functional form that is cyclically and reversal invariant is 10 To see this, first express g(s, t) in term of t + u and t − u. The requirement that g(s, t) = g(s, −s − t) = g(s, u) imposes a symmetry under t ↔ u exchange which requires that t − u only arises in even powers. However, since (t − u) 2 = s 2 − 4 t u, one concludes that g(s, t) can be written as an expansion in powers of just s and t u, as claimed.
analogously to Eq. (4.3). Here g(s, t) = g(s, −s − t) = g(s, u) which is the same constraint as before (Eq. (4.10)), and h(s, t) is a priori a general function. Notice that it is not trivial that we can write the equation above in terms of the vector c 0 instead of the single traces separately. This is a feature of the 4-point kinematics, which is invariant under reversal of particle labels: f (1, 2, 3, 4) ≡ f (s 12 , s 13 ) = f (s 43 , s 42 ) ≡ f (4, 3, 2, 1), for any function f of the Mandelstam invariants. 11 Whilec h.d.

0
is the most general numerator seed, for the purpose of constructing independent adjoint numerators, the g(t, s) c 0 (1, 2, 3, 4) term is redundant. It can be canceled by adding a permutation invariant function and redefining the arbitrary h(s, t). This means that we can restrict to the numerator seed and still generate all possible adjoint numerators with color. We could have reached the same conclusion regarding the fact that the function g can be absorbed in h using the systematic algorithm which makes use of J + which we discussed in Sec. 3.

Generalized gauge transformations
Up to this point, we have considered numerators independently from the amplitudes they generate. There does exist a freedom to shift a numerator without affecting the amplitude, if the other numerators they multiply satisfy Jacobi identities. For instance, at 4-point, the redefinition n s → n s + s ∆, n t → n t + t ∆, n u → n u + u ∆, for any function ∆, results in which is just A 4 if the color vector c satisfies the Jacobi identity. In matrix notation, any shifts in the vector n proportional to (s, t, u) T leave the amplitude A = c · P 4 · n invariant. Here (s, t, u) T is the null vector of J T 4 · P 4 , where J T 4 arises if c satisfies the Jacobi identity. Such shifts are called generalized gauge transformations because an actual gauge transformation, i → i + p i for any particle label i, results in a similar vanishing shift of the amplitude. Nevertheless, generalized gauge transformations are also present in non-gauge theories.
At the level of the numerator seeds, the generalized gauge transformations allow for shifts proportional to the null-vectors ofm 4 = J T 4 · P 4 · J 4 , which are (u, 0, −s) T and (t, −s, 0) T . This includes the permutation invariant shift proportional to (1, 1, 1) T (the null-vector of J 4 ) that was used before and does not affect the constructed adjoint numerator. Other shifts are possible that change the permutation properties of the seed and, in turn, may correspond to a non-adjoint numerator. A particular generalized gauge 11 At higher multiplicity n, this is not generally true. While generalized numerator seeds can always be organized in terms of an (n − 1)!/2 dimensional vector, factorizing out the color from the kinematic dependence generally requires all (n − 1)! single traces separately.

transformation, given bȳ
has the property that the shift is itself a numerator seed and, therefore, maps into a valid seed c h.d.

0
. Hence, to capture all amplitudes in a CK-dual theory,c h.d.  (if g(s, t) is allowed to have simple poles). We stress that this is not the most general numerator seed, nor does it construct the most general adjoint numerator, but it constructs the most general amplitude.  has minimal rank, is that the diagonal matrices G h.d. l/r have full rank and therefore preserve the rank of the matrix m 4 . Conversely, the relation above can be inverted to express g l/r in terms of f 2 (not uniquely), which means that they encompass any EFT solution f 2 of the bootstrap equations. One can for instance take

Seeds and the generalized KLT bootstrap at 4-point
Thanks to the bootstrap condition for f 2 , these satisfy the constraint g(s, t) = g(s, −s − t) = g(s, u), as required for numerator seeds. This shows that the generalized numerator seed of Eq. (4.20) generates any matrix of doubly ordered amplitudes that appears in the generalized KLT formalism. The choice above is not unique since m h.d. 4 only depends on the product of g l and g r . To illustrate how Eq. (5.3) works in an EFT expansion, let us consider the lowest-order terms in the bootstrap solution for a pure scalar theory [36], These functions give rise to the numerator seeds and the associated adjoint numerators of the BAS+h.d. theory. The minimal-rank bootstrap equations also allow for solutions that are not of the BAS+h.d. form. However, modifications to the lowest-order 4-point kernel were found to increase the rank at higher multiplicities and lead to unhealthy double-copy structures [36].

Generalized single and double copies
which obeys the traditional KK and BCJ relations, and similarly for A l . Although the notation suggests otherwise, A r may still contain higher-derivative corrections, but they are such that they do not affect the form of the traditional KK and BCJ relations. If A r can be argued to be a valid amplitude, this could be an efficient method to construct generalized single copies. In addition, it would imply that the generalized KLT formalism does not lead to double copies with additional higher-derivative corrections besides the ones which can be obtained with the usual KLT kernel. Indeed, where we stress that both primed and unprimed amplitudes may contain higher-derivative corrections and S 4 stands for the traditional BAS KLT kernel. However, as it stands, A r [α] cannot be interpreted as a physical amplitude, since it does not necessarily factorize properly on all channels, whose associated residues can be affected by G h.d.
r . We will study the functional form of G h.d. l/r , under the assumption of a fixed BAS particle content in Sec. 5.2. This will lead to a slightly adapted but equivalent form for Eq. (5.10), such that A r [α] and A r [α] have the same residues on all poles.
Leaving momentarily aside the question of physical residues, Eq. (5.10) also provides a means to construct generalized single copies. They can be obtained by multiplying amplitudes that satisfy the traditional KK and BCJ relations with G h.d.
l/r . The same conclusion is reached in the numerator formalism (see Eq. (3.21)). Explicitly, comparing with Eq. (4.12), we have that where n 0 = g(s, t), g(t, s), g(u, s) T . All generalized amplitudes can be constructed in this way, showing that one can associate a generalized color numerator to any amplitude obtained from the generalized KKBCJ relations. The first equality in Eq. (5.12) also applies to gauge theories in which case n 0 contains polarization vectors. Imposing particular locality properties on this amplitude restricts the coefficients inside g(s, t) and g r (s, t). We note that such constraints may be less restrictive than the constraints on g l/r coming from imposing a fixed particle content on m h.d. 4 . We will get back to this point in the following.

Factorization properties and particle spectrum
So far, we have not been concerned with the particle content (and factorization properties) of the BAS+h.d. amplitudes constructed through Eq. (5.1). Following [36], we now impose that m h.d. 4 reduces to the BAS matrix m 4 at lowest order and that the particle content of the theory is fixed to one bi-adjoint scalar. This implies that f (s, t) = −s f 2 (s, t)/g 2 φ in Eq. with polynomial g l/r (s, t), the residues might be modified. Such modifications are either non-physical or can be interpreted as new particles appearing in the factorization channels. However, at 4point, only contact-term higher-derivative corrections are allowed with a fixed single scalar particle content (since the 3-point amplitudes are not modified in the solution to the KLT bootstrap [36]). Imposing such conditions yields the following constraints: which enforce in particular the consistency conditions g l/r (s, 0) = g l/r (−s, 0) , (5.14) when using g l/r (s, t) = g l/r (s, −s − t). This implies that, on the t and u poles, the first variable of g(s, t) necessarily appears in even powers. The lowest order terms in the solutions are then where we emphasize that the same coefficient a l2,0 appears in both expansions. These solutions are fully consistent with the expansions in Eq. (5.5), which were obtained from the bootstrap solution f 2 (s, t), assuming a fixed particle content [36]. 12 With these solutions, one of the generalized KKBCJ relations, showed in Eq. (5.9), is given by  (t, 0)). On the poles, these factors can also be obtained from the function g r ( (s 2 + t 2 + u 2 )/2, 0)| t→0 = g r (s, 0), where the square root always appears in even powers in the Taylor expansion over the first variable thanks to the consistency condition of Eq. (5.14). Since g r ( (s 2 + t 2 + u 2 )/2, 0) is permutation invariant, we can redefine the amplitude A r [α] as A r [α] (5.17) which still satisfies the traditional KK and BCJ relations while also having the same poles and residues as A r [α]. Defining simultaneously it follows from Eq. (5.13) that g l (s 2 + t 2 + u 2 )/2, 0 = 1/g r (s 2 + t 2 + u 2 )/2, 0 , which leads to the conclusion that the double-copy amplitude remains unchanged. In other words, there is an interplay between the left and right amplitudes and the generalized kernel that allows for the cancellation of any correction to the kernel, in a manner that does not affect the residues and poles of the amplitudes.
We therefore conclude that a double-copy amplitude obtained with a generalized kernel can equivalently be generated with the traditional BAS kernel (c.f. Eq. (5.11)). The singlecopy amplitudes may then still include higher-derivative corrections, but only those that do not spoil the usual KK and BCJ relations. To prove this statement, we assumed that no extra particles are added to the BAS spectrum. At 4-point, the generalization of the KLT formalism does thus not enlarge the space of possible double copies, but it does enlarge the space of single copies that can be used as input.
In our derivation, it is also clear that Eqs (5.17, 5.18) can be used to obtain amplitudes that satisfy generalized KKBCJ relations (A l/r ) from amplitudes satisfying the usual KK and BCJ relations (A l/r ), with the same particle content. This, indeed, has exactly the same form in the generalized numerators approach shown in Eq. (5.12).

Results at 5-point
Here, we use the numerator seeds described in Sec. 3 to construct adjoint numerators at 5-point. We also discuss the seed redundancies, previously described at 4-point in Sec. 4. The statement that the generalized KLT formalism does not enlarge the space of double-copy amplitudes had two caveats, see Sec. 3.2. While these were fully addressed at 4-point in the previous sections, we only provide partial results at 5-point. We check explicitly that the generalized numerators generate all the leading-order KLT kernels of [36]. Furthermore, we study the factorization properties and the particle spectrum of the objects A l/r = (H h.d. l/r ) −1 · A l/r for a restricted set of higher-derivative corrections. For the corresponding generalized kernels, we achieve the same conclusion as at 4-point, namely that the double-copy amplitudes it produces can equally be obtained with the traditional BAS kernel and physical single-copy amplitudes.

Numerator seeds
At 5-point, the kinematic numerator seeds have the functional form and all scalar adjoint numerators are easily built from n adj = J 5 · n 0 . The explicit form of J 5 is given in Sec. C.1. We chose a cyclic basis of Mandelstam invariants, which is possible at any multiplicity (see e.g. [52]) and simplifies the particle permutation properties. In addition, there are two independent adjoint numerators containing only color information. These are obtained from the following single-trace seeds, As before, it is not necessary to impose antisymmetry under reversal. Any other allowed single-trace color numerator maps to a linear combination of J 5 · c 0,1 and J 5 · c 0,2 . For instance, the two independent adjoint numerators c a,1,2 of [35] are 13 Besides numerator seeds featuring only kinematic and color information, we study generalized seeds that contain both, i.e. c h.d.

BAS+h.d. amplitudes
At any multiplicity, the BAS matrix of doubly color-ordered amplitudes that derives from the generalized numerators can be written as (see Eq. for those special values of g 1,2 is equal to an amplitude built withc h.d.

0
, because both seeds are related by a generalized gauge transformation (nevertheless, the adjoint numerators obtained via these seeds are linearly independent). Therefore any 5-point BAS matrix of doubly color-ordered amplitudes can be written as is non-diagonal, with only one non-zero entry on each row/column, as can be seen in Sec. C.1. The non-diagonal matrix is necessary because the 5-point BAS matrix m 5 contains zero entries, which may become non-zero when higher-derivative corrections are included.
It is non-trivial to verify whether Eq. (6.11) covers all solutions to the KLT bootstrap of [36]. As argued in general in Sec. 3.2, m h.d. 5 has minimal rank just as m 5 since G h.d.
2,r/l has full rank. We have reproduced the solution of the KLT bootstrap for all orders explicitly provided in [36] and the forms of the necessary functions are listed in Sec. C.2. Going beyond this, we also checked that the numerator seeds reproduce the lowest-order 5-point contact terms, which are cubic in the Mandelstam invariants. These are captured by Eq. (6.11) with is of third order in the Mandelstam invariants. This exposes a simple structure of higher-order corrections.

Factorization properties and particle spectrum
As discussed in Sec. 3.2, given a single-copy amplitude A that satisfies the generalized KKBCJ relation, the object A = (H h.d. ) −1 · A satisfies the usual KK and BCJ relations (where we momentarily omitted the subscript L/R, for simplicity). Moreover, the same double copy can be constructed using either of these two single copies. However, it is not immediately clear that A and A share the same analytic properties. At 4-point, we showed that this is indeed the case. At 5-point, achieving a fully general proof seems far more challenging. Therefore, we start the exploration of this question with simplifying assumptions.
Since one can write H h.d. = (G 1 + G 2 ) at 5-point, it follows that A = (G 1 + G 2 ) · A. Restricting to the two independent 12345 and 13524 color orderings, we have that   (1, 2, 3, 4, 5)g 1 (1, 3, 5, 2, 4) + g 2 (1, 2, 3, 4, 5)g 2 (1,3,5,2,4) . (6.15) Studying the analytic properties of the amplitude A is challenging for two main reasons. First, as discussed in Sec. 6.2 and in contrast to the 4-point case, the functions g 1 and g 2 may contain poles. So a general parametrization has a complicated form and studying whether the poles of A and A agree is non-trivial. Second, even if we take analytic g 1 , g 2 so that A and A poles are identical, a suitable redefinition of A may be necessary to guarantee that its residues match those of A (similarly to the 4-point case presented in Eq. (5.17)).
Working out such a redefinition at 5-point, or even proving that one always exists, is beyond the scope of this paper. However, in the simplest setup, namely that of a kernel whose first EFT correction is a 5-point contact term constructed from a vanishing g 2 and a polynomial g 1 , we can identify a suitable redefinition of A l/r such that their residues agree with those of A l/r while leaving the double copy unchanged (we checked this property up to order O(1/Λ 10 )). Explicitly, we found that the locality properties of the kernel at 5-point impose that (for seeds up to order O(1/Λ 10 )) where c l/r are free constants and p l/r (1, 2, 3, 4, 5) = p l/r are permutation-invariant functions such that p l p r = 1 whenever a Mandelstam invariant vanishes. Consequently, have the same residues on the poles as A l and A r , respectively. Therefore, the doublecopy amplitudes associated to the kernel obtained from the seeds of Eq. (6.16) can also be obtained with the traditional KLT kernel and single-copy amplitudes which verify the usual KKBCJ relations. Whether such manipulations can be performed in full generality at 5-point is a question left for future investigation.

Conclusions and outlook
In this paper, we revisited recent proposals for the systematic double copy of effective field theories. Inspired by the decomposition of adjoint color factors into single traces of Lie algebra generators, we proposed a method to construct generalized adjoint numerators. These satisfy Jacobi-like and antisymmetry relations, while depending on both color and kinematics. Starting from numerator seeds satisfying the permutation properties of single traces, we proved that all adjoint numerators can be obtained through the linear map J between single-trace and adjoint color factors. While generalized numerators have previously been constructed up to 5-point in [34,35], the construction from numerator seeds is advantageous because the algebraic properties of single traces are simpler than the adjoint algebraic relations. We showed that this method works for any multiplicity and it is convenient to explore the higher-derivative corrections that allow for a color-dual representation.
The matrix J is also instrumental in relating the amplitudes represented with trivalent graphs involving (generalized) adjoint numerators to color-ordered ones. The construction of generalized adjoint numerators therefore facilitates the comparison between the generalized numerators construction of [34,35] and the generalized KLT formalism of [36]. At 4-point, we showed that the generalized adjoint numerators encode all the KLT bootstrap solutions, to any order in the EFT expansion. The two approaches therefore allow for exactly the same higher-derivative corrections to the bi-adjoint scalar amplitudes. The single-copy amplitudes are also the same in the two formalisms. This insight consequently exposed the structure of double-copy amplitudes. While the generalized KLT formalism does expand the range of operators in the single-copy amplitudes, we find (at 4-point) that any resulting double copy can also be obtained with the traditional KLT kernel. We provide partial 5-point results suggesting that these conclusions may extend to higher multiplicity. However, due to the more complicated KLT bootstrap and structures involved, further investigations on higher multiplicities are left to future work.
There are several directions that deserve further attention. For example, we have focused on the construction of scalar numerators, as opposed to gauge-theory numerators involving polarization vectors. The relevant purely kinematic numerator seeds have been considered for Yang-Mills theory in [40][41][42][43][44], but not in the EFT context or for generalized numerators. Gauge invariance must hold at the level of amplitudes (i.e. A| i →p i = 0), and individual entries in a numerator are typically not gauge invariant. The necessary extra constraint on numerator seeds may therefore take a complicated form, especially beyond 4point. Methods proposed to identify the possible gauge-invariant structures in Yang-Mills theories [52,53] could be useful. In particular, a basis of cyclically invariant structures which can be used as numerator seeds was provided in [53].
Furthermore, we have only considered color factors consisting of single traces of Lie algebra generators. However, the construction of adjoint numerators from numerator seeds does apply more generally. For instance, at 4-point there exist theories with double traces, which can be combined into the seed c 0 (1, 3, 2, 4) = Tr(T a 1 T a 2 )Tr(T a 3 T a 4 ) = δ 12 δ 34 . The resulting generalized adjoint numerator was previously identified in [31]. Constructing generalized color factors involving products of traces may lead to interesting new single or double copies, which would also be worth studying in the generalized KLT formalism.
Besides the assumption of single traces, we have not included the possibility of extra particles beyond the bi-adjoint scalar discussed in Sec. 5. For the double copy of a gauge theory with matter in the fundamental representation, the KLT kernel is for instance constructed from a bicolor theory containing two scalars [10,12]. Including new particles in factorization channels, or even externally, in the generalized KLT formalism would allow for a larger space of single-copy amplitudes. It would then be worthwhile to extend the numerator seeds to amplitudes with a more complicated particle spectrum.
Altogether, the simple construction of adjoint numerators from numerator seeds has been useful to explore the structures in the generalized CK and KLT double-copy formalisms. Still, the double copy of effective field theories retains various unexplored aspects which promise exciting new findings for the years to come. n 0 = J + · n is a numerator seed. In this subsection, we prove that the numerator seeds constructed through Eq. (A.5), transform according to the same rule as the color factor c 0 , under a permutation of the particle labels, σ. First, note that adjoint numerators transform analogously under σ, with the same M c adj ,σ ( = M c 0 ,σ ) for any adjoint numerator. Since the entries of c 0 are linearly independent, The numerator seeds transform as which we want to show is the same as J + · n adj  3 4 5  6  7  8  9  10  11  12  13  14  15   4-pt 1 2 2 3 3  4  4  5  5  6  6  7  7  8  8  5-pt 0 0 2 4 10 16 28 42  64  90 126 168  224  288  370  6-pt 2 8 22 58 133 298 600 1166 2132 3754 6324 10351 16368 25266 38004   Table 2. Counting of scalar kinematic numerator seeds up to 6-point and at O(1/Λ 2k ) for k ≤ 15 in the EFT expansion. The Gram determinant constraints relevant in 4 spacetime dimensions have been accounted for. Therefore (A.14) Thus, as we wanted to show, the numerator seed n 0 = J + ·n adj (which exists for any adjoint numerator) transforms as

B Examples of seeds and adjoint numerators
In this appendix, we present the construction of kinematic adjoint numerators from numerator seeds. In general, a numerator seed can be obtained from any function f (1, 2, ..., n) through n 0 (1, 2, ..., n) = f (1, 2, ..., n) + (−1) n f (n, ..., 2, 1) + cyclic . (B.1) The number of independent numerator seeds built using this equation, up to 6-point and dimension 30, is provided in Table 2. After multiplying them by J, one needs to explicitly verify the linear independence of the resulting adjoint numerators. To determine the counting provided in Table 1, we first construct vectors by evaluating the numerators numerically for different values of the momenta. The rank of the matrix formed with these vectors as columns is the number of independent adjoint numerators. A systematic correspondence can be established between the entries of Table 1 and  Table 2. At 5-point, the number of independent numerators is, for instance, exactly half of that of independent seeds. This can be understood from the algebraic properties of color factors decomposed into structure constants f abc and symmetric d abc... , as discussed in [35] up to 5-point. The 4-and 5-point cases are detailed below. The other valid numerator seed at second order, n 0 (1, 2, 3, 4) = t u maps to the same adjoint numerator, because the permutation invariant s 2 + t 2 + u 2 = 2(s 2 − t u) maps to zero. At third order (and any higher order), the adjoint numerators are permutation invariant functions multiplying the lowest two orders [34]: Upon comparing Table 1 and Table 2 (recall (12...5) ≡ Tr(T a 1 T a 2 ...T a 5 ), etc.). The comparison between Table 1 and Table 2 shows that the number of independent adjoint numerators is exactly half that of independent seeds. As in the 4-point case, this can be understood from the algebraic properties of the color factors which are generated by combinations of single traces of group generators with the same behavior under reversal symmetry as the seeds. At 5-point, the seeds are antisymmetric under reversal, while the color factors which can be decomposed onto antisymmetric combinations of single traces can be identified from the classification in [35]. In the language of this reference, they correspond to adjoint and hybrid structures, which correspond to combinations of color factors of the form f abx f xcy f yde and d abcx f xde . Therefore, antisymmetric seeds generate all expressions having the algebraic properties of these two kinds. In addition, there exists a bijection between adjoint and hybrid structures [35]. Namely, for each expression with adjoint properties, there exists one with hybrid properties, and reciprocally. It then follows that the number of independent seeds is exactly twice that of independent adjoint numerators. where s abc = (p a + p b + p c ) 2 . These map to one independent adjoint numerator after multiplication by J. At second order, there are three independent adjoint numerators. Out of the eight independent seeds, these can for instance be constructed from A block structure in J 5 has been made manifest by ordering c 0 into the schematic form: c 0 = (c 012 , reversed(c 012 )) T , where c 012 contains 12 entries of c 0 that are not related by reversing the order of the particle labels.
The matrix G 2 , which can be obtained from the numerator seed Eq. (6.7) by stripping of the single traces, forg 12345 ≡ g 2 (1, 2, 3, 4, 5), is  and e i are free parameters, which reproduce the results of [36] when e 1 = a 1,0 − a 1,1 , e 2 = a 2,0 , and e 3 = a 1,1 . Note that the e 2 parameter could have equivalently been part of g 1,r and g 2,r without changing the amplitudes.

C.3 Generalized KLT kernel from seeds
At 4-point, the generalized kernel for any choice of BCJ bases was obtained by multiplying the left and right side of the traditional KLT kernel by the diagonal matrices G h.d. l/r for the same BCJ bases, see Eq. (5.6). This simple structure extends to 5-point only for a particular BCJ bases, as the structure gets more involved due the presence of the nondiagonal matrix G h.d. 2 in Eq. (6.11). This is not necessarily a problem as one is always allowed to choose a particular basis to compute double-copy amplitudes.
For example, for the usual biadjoint scalar theory, the sub-matrix of ordered amplitudes for α, β ∈ {12345, 13524} reads