Double Copy in Higher Derivative Operators of Nambu-Goldstone Bosons

We investigate the existence of double copy structure, or the lack thereof, in higher derivative operators for Nambu-Goldstone bosons. At the leading ${\cal O}(p^2)$, tree amplitudes of Nambu-Goldstone bosons in the adjoint representation can be (trivially) expressed as the double copy of itself and the cubic bi-adjoint scalar theory, through the Kawai-Lewellen-Tye bilinear kernel. At the next-to-leading ${\cal O}(p^4)$ there exist four operators in general, among which we identify one operator whose amplitudes exhibit the flavor-kinematics duality and can be written as the double copy of ${\cal O}(p^2)$ Nambu-Goldstone amplitudes and the Yang-Mills+$\phi^3$ theory, involving both gluons and gauged cubic bi-adjoint scalars. The specific operator turns out to coincide with the scalar ${\cal O}(p^4)$ operator in the so-called extended Dirac-Born-Infeld theory, for which the aforementioned double copy relation holds more generally.


I. INTRODUCTION
The nonlinear sigma model (NLSM) [1][2][3] is an effective field theory (EFT) of Nambu-Goldstone bosons (NGB's) arising from spontaneously broken symmetries. Recent developments in the modern S-matrix program have led to renewed interest in the NLSM, which is frequently referenced. In particular, NLSM can be formulated in an entirely on-shell way, by imposing the consistency condition of the Adler's zero [4,5]. This powerful on-shell property, related to a shift symmetry at the Lagrangian level [6][7][8][9], leads to a wealth of constructions from totally different perspectives, including soft bootstrap [10][11][12] and single soft scaling or double soft theorems [13][14][15][16].
Furthermore, the NLSM is also a key element of the color-kinematics duality and the ensuing Bern-Carrasco-Johansson (BCJ) double copy [17], as well as the Cachazo-He-Yuan (CHY) formalism for S-matrix [18][19][20][21]. These formalisms have demonstrated a remarkable unity among naively distinct theories, by expressing for instance gravity as the double copy of Yang-Mills (YM), or Born-Infeld as the double copy between YM and NLSM [21]. At the leading O(p 2 ), the requirement of flavor-kinematics duality, together with locality and cyclic invariance of flavor-ordered amplitudes, can even uniquely constrain tree amplitudes in the NLSM [22].
More generally, in the space of consistent quantum theories, the NLSM can be related to YM through transmutation operators and dimensional reduction [23][24][25] or to bi-adjoint scalar through soft limits [9,[26][27][28]. These fascinating aspects are somewhat hidden in the traditional Lagrangian formulation. Finally, through a subset of higher derivative corrections starting from O(p 6 ), it also makes an appearance in string theory, as symmetrized sums over Z-theory amplitudes, which are objects carrying the α ′ dependence of the superstring theory [29,30]. The origin in string theory ensures the BCJ relation derived at O(p 2 ) is satisfied by all higher derivative operators in the Z-theory.
From the effective field theory perspective, a natural puzzle arises when one includes generic higher dimensional and higher derivative corrections to the leading renormalizable interactions: does any of the fascinating features, such as the double copy relation, continue to work in these cases? Some preliminary studies showed that for NLSM, direct applications of the BCJ relations from O(p 2 ) fail at O(p 4 ) [11,31].
However, recently new ingredients for constructing the color-kinematics duality are introduced at the level of 4-pt amplitudes [32][33][34], which involve new color (flavor) kinematic objects as linear combinations of color (flavor) structures with coefficients given by Mandelstam invariants. In this paper we extend the results on color-kinematics duality beyond the 4-pt amplitudes, to higher multiplicity, and investigate whether it is possible to construct double copy relations for NLSM at O(p 4 ).
It is also instructive to consider the double copy relation from the Kawai-Lewellen-Tye (KLT) bilinear form, which for NLSM at the leading O(p 2 ) can be expressed as [21,35]: where φ 3 denotes the cubic bi-adjoint scalar theory, and the universal KLT kernel matrix KLT ⊗ is the inverse of a matrix whose entries are doubly ordered amplitudes of φ 3 . How is the bilinear form modified after including higher derivative corrections in NLSM? Naively, when we include O(p 4 ) corrections to the left-hand side, it is conceivable that there is a version of higher derivative corrections to the cubic bi-adjoint scalar theory that could make the double copy relation work non-trivially. We will see that, for a particular choice of O(p 4 ) operator, there is indeed a theory that would form the double copy relation for NLSM (4) , where YM+φ 3 is a theory of biadjoint scalars with gauge interactions [35].
The paper is organized as follows. In Section II we begin with a discussion on the color structure of NLSM at O(p 2 ) and O(p 4 ). In Section III we review the color kinematics duality, including its recent modification, and how it applies to the O(p 4 ) single and double trace amplitudes at 4-pt. In Section IV we then extend to 6-pt, and find a flavor-kinematic solution that matches the O(p 4 ) double trace amplitude. In Section V we identify the double copy relation involving the NLSM amplitudes at O(p 4 ) and YM+φ 3 . We end with conclusions and future directions in Section VI.

II. THE NLSM UP TO O(p 4 )
The NLSM effective Lagrangian can be parameterized as the following: where π a are the NGB fields with flavor indices a, and Λ and f are constants of mass dimension 1, with f /Λ < 1. The low energy effective Lagrangian is a perturbative expansion of ∂/Λ, which is predictive when the energy scale of interest is much smaller than Λ. Because of Lorentz invariance, there are only even powers of ∂/Λ in the series expansion when we work in 4 spacetime dimensions. The leading order, O(p 2 ) Lagrangian, L NLSM , contains all the terms in Eq. (1) with two derivatives, and the subleading, O(p 4 ) Lagrangian L (4) NLSM contains all the terms of four derivatives, and so on. In other words, with L . At each order in the derivative expansion, the Lagrangian also admits an expansion of π/f , to all orders in 1/f . An n-pt tree amplitude has the low energy expansion: and at tree level, we have M NLSM up to n-pt will enter M (m) n . We review the Lagrangian and amplitudes of NLSM up to O(p 4 ) in the following.

A. The Lagrangian
Let us consider a general NLSM where the NGB fields π a furnish some representation R of a Lie group H. Using the bra-ket notation, |π a = π a , The O(p 2 ) Lagrangian is [6,36] where with T i being the generators of H in the representation R, written in a purely imaginary and anti-symmetric basis: The form of Eq. (4) is fixed by the requirement that the on-shell amplitudes vanish in the single soft limit. This implies a shift symmetry in the Lagrangian [6,36], where (|ε ) a = ε a represents an infinitesimal constant "shift" in π a , as well as a "closure condition" that the generators T i need to satisfy: Such a condition means that the NLSM can be embedded into a symmetric coset G/H.
In other words, it can be generated by the spontaneous symmetry breaking of some group G, with the coset G/H being symmetric. The generators of G include the "unbroken generators" T i associated with the group H, and "broken generators" X a associated with the coset G/H.

Then we can identify
as H is a subgroup of G, and f abc = 0 because we require G/H to be symmetric. Then the Lagrangian can be rewritten as [2,3] The interactions given by Eq. (10) are even powers of π a contracted with a single trace of generators X a .
The Lagrangian at the subleading order of O(p 4 ) in general contains four independent Parity-even operators: where and d µ = d a µ X a . In 4 spacetime dimensions, there can also be a Wess-Zumino-Witten term [37,38], which we will not consider for now.

B. Flavor ordering of the amplitudes
The on-shell method to construct the NLSM interactions for a general symmetric coset is soft bootstrap [10][11][12], where we consider flavor-ordered partial amplitudes. For the NLSM at O(p 2 ), which we will denote as NLSM (2) , the partial amplitudes are similar to the color-ordered amplitudes of the Yang-Mills (YM) theory [39], where the interactions involve the structure constant f ijk , which for NLSM can be identified with (T i ) ab as in Eq. (4). From the perspective of the unbroken group H, (T i ) ab is a group generator in some general representation; however, from the perspective of broken group G and coset Therefore, the color-decomposition of YM theories can be directly applied to general NLSM (2) . The flavor structure of the full amplitude can be expanded in the trace basis as M (2),a 1 ···an n (p 1 , · · · , p n ) = α∈S n−1 tr (X a 1 X a α(1) · · · X a α(n−1) ) M (2) n (1, α), where α is a permutation of {2, 3, · · · , n} and M n (1, α) is the single-trace flavor-ordered amplitude. The RHS of Eq. (13) is a sum of (n − 1)! terms.

III. FLAVOR-KINEMATICS DUALITY AT 4-PT
The color-kinematics duality of scattering amplitudes was first discovered for YM theories, the n-pt tree amplitudes of which can be written in the following form [17,42]: where the sum is over all distinct n−pt cubic graphs {g n }, while c g , n g and d g are the color numerators, kinematic numerators and denominators of each cubic graph. The denominators d g are given by the propagators associated with the cubic graphs, n g only contains kinematic information (Mandelstam invariants and polarization vectors), while the color structure are isolated in c g . The gauge fields are in the adjoint representation, and c g are constructed using structure constants, thus they satisfy anti-symmetry and the Jacobi identity. The duality for color and kinematics manifests in the fact that it is possible to find a representation for n g so that they satisfy anti-symmetry and the Jacobi identity as well.
It is known that such a duality also exists for the tree amplitudes of NLSM (2) [29,43,44].
However, as an EFT by construction, NLSM admits a derivative expansion, as shown in Eq. (3). A priori it is not clear whether the higher order contributions in the derivative expansion have the same property as well. For the next-to-leading order, i.e. O(p 4 ), previous works of directly applying the O(p 2 ) BCJ relations fail to hold [11,31]. It turns out that, for the flavor-kinematics duality to work at O(p 4 ), we need to generalize our definitions for the color/flavor numerators.

A. Building 4-pt numerators
We start with the lowest multiplicity, which is n = 4 for the NLSM amplitudes. Recently new ways to construct 4-pt numerators have been proposed [32,33], and we will discuss them systematically in the following.
At 4-pt, we can define a function with three indices j(1, 2, 3) and associate it with the 4-pt cubic graph in Fig. 3. If we impose anti-symmetry and the Jacobi identity, then it can be used as the numerator for 4-pt amplitudes. Specifically, we want It is also convenient to define with Let us first consider color/flavor numerators that do not contain any kinematic information. For YM, the color numerator is given by where f iab are structure constants of some Lie group H. More generally, for generators of H in some representation R, this can be generalized to assuming the closure condition given by Eq. (9). This is the flavor numerator for an NLSM (2) amplitude for a general group H. The color numerator c = f A is just a special case where the representation is the adjoint A.
Another valid flavor numerator is given by where the indices can be in any representation for any group: δ ab is always an invariant tensor. It is easy to check that f δ satisfies anti-symmetry and the Jacobi identity. One can also identify that when R is the fundamental representation of SO(N). The above is the consequence of the completeness relations of the generators in the fundamental representation of SO(N): and it is known that NLSM of fundamental SO(N) can be embedded to the symmetric coset SO(N + 1)/SO(N). In other words, for the SO(N) fundamental representation, f δ is not a new building block but is identical to f R . For other group representations, it is indeed new.
Next, let us consider numerators containing kinematic invariants. For simplicity we will restrict ourselves to numerators that are local. At the lowest mass dimension, we have the following numerator that only contains momenta: which is the kinematic numerator for single-flavor YM scalar theory.
We can use the simple building blocks discussed in the above to construct more complicated numerator j's. One way is to just multiply existing numerators with permutation invariant objects. There are two such objects that encode the internal symmetry: For non-adjoint representations, d 4 can be generalized to any rank-4 totally symmetric tensor d 4 , which may or may not exist. There are also two permutation invariant building blocks that only contain kinematic invariants: The other way to generate new numerators is to take two existing numerators j and j ′ , is a perfectly valid new numerator. Now let us build more numerators only containing momenta. We have which is the kinematic numerator for NLSM (2) [29,44]. We also have J(n ss , n nl ) = 1 6 Y n ss s , J(n nl , n nl ) = − 1 6 (Xn ss s + Y n nl s ).
This means that all numerators that only contain momenta can be written as a linear combination of n ss and n nl , each dressed with powers of permutation invariant objects X and Y [22].

B. 4-pt soft blocks at O(p 4 ) for NLSM
The full 4-pt amplitude for NLSM (2) is [17,42] M where we have suppressed the flavor indices. Here f R,s/t/u is the flavor factor defined in Eq. (23) for the s/t/u channel, while n nl is defined in Eq. (30). We can rearrange the amplitude to the DDM basis [40] using where the two terms correspond to the flavor-ordered partial amplitudes: Double-trace: S The full 4-pt amplitude can be written as 2 (σ, 4) where the dimensionless constants c i and d i are related to the Wilson coefficients in Eq. (11) by Now we want to construct a 4-pt amplitude at O(p 4 ) that is local and exhibits flavorkinematics duality, using numerators that satisfy the anti-symmetry and Jacobi identity in Eq. (19). One natural possibility is to replace the O(p 2 ) kinematic numerator n nl in Eq. (32) with O(p 4 ) kinematic invariants while leaving the flavor factors f R intact. In this case the coefficients of f R in the DDM basis simply correspond to flavor-ordered partial amplitudes, cf. Eq. (33), and satisfy KK and BCJ relations. However, it was shown in Refs. [11,31] that such local kinematic numerators do not exist at O(p 4 ).
An alternative possibility is to leave the O(p 2 ) kinematic numerator n nl unchanged and modify the flavor factor f →f, wheref is now O(p 2 ) in order for the full amplitudes to be O(p 4 ). Assuming the flavorkinematics duality,f u = −f t −f s , the full amplitude can be written as where we have plugged in Eq. (34). We see the ansatz in Eq. (40) amounts to expanding the O(p 4 ) full amplitudes in terms of O(p 2 ) partial amplitudes. We will present four different possibilities forf.
The first possibility iŝ This gives us a local full amplitude, and we can rewrite it in the DDM basis: Therefore, we obtain a partial amplitude in the single-trace basis: which is the unique single-trace soft block at 4-pt that satisfies KK relations [41].
The second modified flavor numerator iŝ Again, the corresponding full amplitude is local, while this time we write it in the trace basis: Then the partial amplitude is This is the unique single-trace soft block at 4-pt that is permutation invariant.
To obtain flavor-ordered partial amplitudes corresponding to the two double trace soft blocks we just need to replace f R in Eq. (41)  Let us first replace f R in Eq. (41) with f δ : The corresponding full amplitude is 1 Λ 2 f 2 δ a 1 a 2 δ a 3 a 4 (s 2 + 2tu) + δ a 1 a 3 δ a 2 a 4 (u 2 + 2st) + δ a 1 a 4 δ a 2 a 3 (t 2 + 2su) , (49) which gives the partial amplitude On the other hand, replacing d 4 in Eq. (44) with d 2 leads tô so that the full amplitude becomes where Y is defined in Eq. (29). Then the partial amplitude is  In the end, the four different modified flavor factors give rise to flavor-ordered partial amplitudes corresponding to the four soft blocks.

IV. FLAVOR-KINEMATICS DUALITY AT HIGHER MULTIPLICITY
Inspired by the 4-pt results discussed in the last section, we assume the following ansatz for the n-pt full amplitude of NLSM at O(p 4 ), where . Again using the Jacobi relations amongf i,g the full amplitude in the DDM basis is an expansion in the NLSM (2) partial amplitudes M (2) n : where σ is a permutation of {2, 3, · · · , n − 1}, hl(1, σ, n) is the corresponding half-ladder graph with 1 and n at two ends, as shown in Fig. 2. On the other hand, we can also expand M f i ,n in the trace basis: where σ corresponds to all the distinct trace structures, f i,σ is the flavor factor which is either a single trace or a product of two traces, and M where c Therefore, to findf i for higher multiplicity amplitudes, all we need to do is solve the coefficients c Once this is done, all the otherf i,g can be uniquely determined using the Jacobi relations.
In the most general case, without imposing any constraints on c (i) σ , the number of solutions are infinite. Instead, we will assume that c where α is an arbitrary constant. The degree of freedom in the solution, which is characterized by α, is a consequence of the single BCJ relation of M where we have used the short-hand notation for the traces: tr(123 · · · ) ≡ tr (X a 1 X a 2 X a 3 · · · ) .
To arrive at Eq. (62) we have used the cyclic and reverse ordering invariance of tr (1234) and M Eqs. (63) and (64) give the most general modified flavor factors that works for M (4) f 2 ,4 . If we also want them to be relabeling symmetric, in this case exchanging 2 ↔ 3 resulting in f 2,s ↔ −f 2,u , the constant α must be set to 0: we havê which is exactly what we know from Eq. (44).
where P 2 ijk··· ≡ (p i + p j + p k + · · · ) 2 . From Eq. (38) we see that this contribution corresponds to the following values for the Wilson coefficients in the Lagrangian: We will denote such a theory as NLSM d 2 . At 4-pt, we learned from Eqs.
where the coefficients c d 2 σ corresponding to the 24 orderings of σ are given in Table I where σ are permutations of {1, 2, · · · , 6} \ {i, j} modulo cyclic permutations. From Eq.
(70) we know that the above can be expanded in the form of Eq. (55), where the modified flavor numeratorsf d 2 ,g 1 for the half-ladder graph given in Fig. 4a has the following relabeling symmetric form: where T i a k a l ≡ (T i ) a k a l is the matrix entry of the group generator T i in the anti-symmetric basis. Note that we have omitted factors of δ a i a j , which can easily be restored from the two missing flavor labels in each term. The modified flavor factor of the other kind of 6-pt cubic graph, as shown in Fig. 4b, can then be directly calculated using the Jacobi relations.   NLSM (2) amplitudes can be written as the following where f R,g are flavor factors corresponding to cubic graph g and expressed in terms of generators in the representation R, the 4-pt example of which is given in Eq. (23). For the specific case of R to be the adjoint representation A, f A,g is the same as the color factor in the YM amplitude c g , as in Eq. (18). Replacing the NLSM (2) kinematic numerator n nl k,g in Eq. (76) with another copy of color factorc g (of a different group), we arrive at (up to coupling constants) which is the tree amplitude for the cubic bi-adjoint scalar theory φ 3 , generated by the where each scalar field φ aã carries two labels, a for the adjoint of group G andã is for the adjoint of groupG; f abc andfãbc are the structure constants for G andG, respectively. It is well understood that there is an intimate connection between the double copy structure and the KLT relations [42], and Eqs. (76) and (77) leads to the (trivial) KLT relation for NLSM (2) [20,21]: where M φ 3 (σ 1 ||σ 2 ) is the doubly ordered amplitudes for φ 3 , and S n (α||β) = M φ 3 n (1, α, n − 1, n||1, β, n, n − 1) is the KLT kernel [20]. Eq. (79) is trivial in the sense that we are multiplying the NLSM (2) amplitudes by unity, as S n (α||β) is the inverse of the cubic bi-adjoint amplitudes. What is less trivial is the fact that there is a universal KLT kernel for theories of adjoint fields, which subsequently is identified with the inverse of the cubic bi-adjoint amplitudes [20]. Notice that in the partial amplitude M φ 3 we use the double line "||" to separate the orderings of two different groups, in contrast to the double trace structure of a single group, where we use a single line "|" to denote. Eq. (79) can be written more compactly as For the O(p 4 ) operator that exhibits flavor-kinematic duality the natural question to ask is what happens when we replace n nl g withc g in the above? 1 In other words, can the object which leads us to deduce that the three point vertex in this theory is where legs 1 and 2 are scalars φ aã carrying two adjoint indices, a of group G andã of group G, while leg 3 is the vector boson Aã µ carrying the adjoint representation ofG, as well as a Lorentz index µ 3 . Such a vertex naturally arises in the following gauged kinetic term of the scalars: where D µ = ∂ µ + igAã µXã is the gauge covariant derivative,X is the generator forG, g being the gauge coupling, and φ a ≡ φ aãXã . Therefore we will call G the flavor group andG the gauge group.
Assuming the 3-pt vertex given in Eq. (86), the 4-pt contact term in Eq. (84) can come from the following interaction: Including the propagators for the massless vector states, we arrive at the following Lagrangian by examining the 4-pt amplitude in Eq. (85), where F µν = Fã µνXã is the field strength tensor of the gauge bosons. This is the Lagrangian for the well-known Yang-Mills scalar (YMS) theory, which can be seen as a dimensional reduction of the YM theory. Consequently, at the 4-pt level we are led to the observation that the following KLT relation holds (up to coupling constants): cannot exist on its own in a consistent quantum field theory; it is part of the derivative expansion in the 2 → 2 scattering amplitude that starts at O(p 2 ), where NLSM d 2 is a quantum field theory containing O(p 2 ) NLSM amplitudes and the O(p 4 ) amplitudes soft-bootstrapped from the soft block S It turns out a theory with the Lagrangian L YMS + L φ 3 has been studied previously and dubbed the YM+φ 3 theory [35], also called "generalized Yang-Mills scalar theory" in [21], which is generated by the following Lagrangian: The above can be seen as a specific "higher derivative" extension of the cubic bi-adjoint scalar theory φ 3 , where the groupG is gauged. So the conjectured double copy structure has the following KLT bilinear relation On the other hand, YM+φ 3 scalar amplitudes of higher orders in the derivative expansion actually conside with the NLSM ⊕ φ 3 theory discussed in Refs. [26,27], when all external scalars are bi-adjoint. Using the 6-pt amplitudes provided in Ref.
From the KLT relation in Eq. (93) one may be attempted to conclude NLSM d 2 as a double copy of NLSM (2) and YM + φ 3 , However, this is clearly not true as YM + φ 3 is a theory which also contains gluons as external particles. There are also trace and flavor structures in YM + φ 3 that are not present in the where the orderings α and β both contain an even number of labels.
So, more precisely, the double copy relation should read What is the theory that is the double copy of NLSM (2) and YM + φ 3 then? It turns out the question has been studied using the CHY formalism in Ref. [21] and the theory is an EFT called the extended Dirac-Born-Infeld (eDBI) theory [21], which involves scalars π a with flavors, and also a U(1) gauge boson A µ . The Lagrangian of eDBI is given by where F µν = ∂ µ A ν − ∂ ν A µ , and |d µ is the same as that in the NLSM defined in Eq. (5).
On the other hand, W µν is an infinite sum of terms, each being a single trace of odd powers of π a X a /f with two derivatives ∂ µ and ∂ ν acting on it; W µν is also anti-symmetric in µ and ν. 2 It has been proved using the CHY formalism that Brief overviews of the CHY formalism and the above double copy relation are provided in Appendices.
For our purpose, it is instructive to expand L eDBI to O(p 4 ), where L d 2 NLSM is exactly the Lagrangian of NLSM d 2 : 2 The exact form of W µν is not relevant for this work. Ref. [21] contains an expression for W µν in the Cayley parameterization [13] of the scalars; this cannot be used here as we are working in the exponential parameterization as shown in Eq. (10). To our knowledge W µν in the exponential representation has not been written down.
At O(p 4 ), the difference between M (4) eDBI,n where all external particles are scalars, and M (4) d 2 ,n , is that the former also contains double trace flavor structures where each trace contains odd powers of X a : these are contributions of the term (F µν + W µν ) 2 , and there can exist internal photons in the Feynman diagrams of these amplitudes. They are naturally generated in the KLT relation from YM + φ 3 , but are absent in NLSM d 2 . However, if we restrict to the partial amplitudes with an even number of NGB's in each trace, then the contributions of L d 2 NLSM and L eDBI are exactly the same. Therefore, we can write down the CHY formulas for these amplitudes, as the CHY representation of eDBI is known [21]: up to coupling constants.

VI. CONCLUSION AND DISCUSSIONS
In this work we have explored the possibility of extending the flavor-kinematics duality to O(p 4 ) operators in NLSM, by using new modified flavor numerators that mix flavor and kinematic factors. While at 4-pt all four operators have such a flavor-dual representation, we find that at 6-pt this is true only for one particular operator, corresponding to doubletrace amplitudes. Furthermore, these specific amplitude are seen to coincide with a subset of amplitudes given by eDBI, which is the double-copy of NLSM and YM + φ 3 . g are modified flavor/color and kinematic numerators of mass dimension k (note that d g has mass dimension 6 at 6-pt). From a bootstrap perspective, such an ansatz is allowed, but would be very difficult to solve already even at 6-pt for O(p 4 ). A related question is then whether partial amplitudes constructed via modified color/flavor factors satisfy BCJ-like relations, as these are typically much simpler to solve and would indicate the existence of a color-kinematic duality. It would also be interesting to understand whether all trace structures present in YM + φ 3 amplitudes have color-dual representations.
Also notice that in our work we have always assumed the "cubic adjoint" properties for the flavor structures: they correspond to cubic graphs that satisfy anti-symmetry and Jacobi relations. This is true when all interactions are dressed with structure constants f abc , as in the YM theory. For NLSM (2) , this is guaranteed by the closure condition Eq. (9) for the generators T i ab , while more general flavor structures appear at O(p 4 ). It is then natural to ask whether some version of flavor-kinematics duality can exist for these more general structures, perhaps involving theories beyond the cubic graphs.
Finally, these considerations can be explored beyond the O(p 4 ) operators investigated in this work. Composition rules such as those used in Eqs. (30) that generate color-kinematic solutions can be extended to higher multiplicity [34], and it would be fascinating to see how they can be used to bootstrap the infinite tower of corrections to the NLSM.
where the anti-symmetric matrix A n is given by and the reduced Pfaffian Pf ′ is defined as n being the matrix A n with rows and columns of labels a and b removed. Again, the choice of a and b does not affect the value of the CHY integral.
For the special case when m = 2 and v = 0, i.e. the double trace amplitudes of no external photons, the matrix Π in Eq. (A8) is reduced to the following 4 × 4 matrix: This is what appears in the CHY formula for NLSM d 2 .
When we restrict the ordering structure of I R Eq. (A24) to m = 2, with both α 1 and α 2 containing an even number of labels, the double copy relation is reduced to