Expansion of Einstein-Yang-Mills theory by Differential Operators

The factorization form of the integrands in the Cachazo-He-Yuan (CHY) formalism makes the generalized Kawai-Lewellen-Tye (KLT) relations manifest, thus amplitudes of one theory can be expanded in terms of the amplitudes of another theory. Although this claim seems a rather natural consequence of the above structure, finding the exact expansion coefficients to express an amplitude in terms of another amplitudes is, nonetheless, a nontrivial task despite many efforts devoted to it in the literature. In this paper, we propose a new strategy based in using the differential operators introduced by Cheung, Shen and Wen, and taking advantage of the fact these operators already relate the amplitudes of different theories. Using this new method, expansion coefficients can be found effectively.


Introduction
Witten's famous work [1] has opened a door to study scattering amplitudes from angles differed from the standard Feynman diagrams. One of such new angles is the Cachazo-He-Yuan (CHY) formalism [2,3,4,5,6]. The CHY formalism has made many beautiful properties manifest, such as the gauge invariance, the soft behavior [7,8,9,10,11,12,13,14] and the double copy structure [15,16], etc. Among those important results, two of them are related to our current work. The first one is that for all theories known now in the CHY formalism, the weight-4 CHY-integrands can be factorized as the product of two weight-2 ingredients, formally written as (1.1) For each weight-2 ingredient, when multiplying by another particular weight-2 ingredient, i.e., the color ordered Parke-Taylor factor PT(α) := 1 (z α 1 − z α 2 ) · · · (z α n−1 − z αn )(z αn − z α 1 ) , (1.2) they define corresponding color ordered amplitudes A L (α), A R (β). Thus (1.1) naturally gives the well known generalized Kawai-Lewellen-Tye (KLT) [17] relations (or the double copy form) 1 , A = σ, σ∈S n−3 A L (n − 1, n, σ, 1)S[σ| σ]A R (1, σ, n − 1, n) , (1.3) As pointed out in [19], the double copy form implies immediately that the original amplitude A can be expanded in terms of either the amplitudes A L or the amplitudes A R . For example, with a fixed σ ordering, we sum σ over all permutations of S n−3 and denote the result as where C( σ)'s serve as the expansion coefficients. Above short deduction seems to indicate that expanding amplitudes of one theory in terms of amplitudes of another theory is a trivial consequence of the double The plan of the paper is following. In section §2, we present some backgrounds needed for later discussions. They include the expansion of arbitrary color ordered YM amplitude in its BCJ-basis, the recursive expansion of sEYM amplitudes in the KK-basis of YM amplitudes, and the differential operators used heavily in this paper. In section §3, we re-derive the recursive expansion of sEYM amplitudes in the KK-basis of YM amplitudes using the proposed new method, i.e., using differential operators. As emphasized in previous paragraphes, results in this section show that differential operator method is an independent systematical new method to determine the expansion. In section §4, we apply the same method to Einstein gravity theory and Born-Infeld theory. In section §5, starting from the expansion of sEYM amplitudes in the KK-basis of YM amplitudes, with careful algebraic manipulations, we translate the KK-basis to the BCJ-basis, thus get the expansion in the BCJ-basis. In section §6, following the similar idea of §3, but with a different choice of building blocks, we derive the expansion of sEYM amplitudes in the BCJ-basis with one, two and three gravitons by applying differential operators directly. In section §7, some discussions have been given. Some technical calculations are collected in the Appendix. In the appendix A, we show terms with index cycle structure will drop out from the expansion. In the appendix B, we discuss the structure of functions when imposing manifestly gauge invariant conditions. These structures provide the building blocks used in the expansion in the BCJ-basis. In the appendix C, some omitted calculation details in §6 have been given here.

Some backgrounds
In this section, we will review some known results. They will serve as the backgrounds of our discussions in the whole paper. This section contains three parts. In the first part, we review the expansion of arbitrary color ordering YM amplitudes to its BCJ-basis. Especially we have rewritten the expansion coefficients in a compact form comparing to these given in [15,40]. In the second part, we review the recursive expansion of sEYM amplitudes in the KK-basis of YM amplitudes given in [19,31,33]. The result will be reproduced by using differential operators proposed in this paper. In the third part, we review differential operators introduced in [34]. They are our main tools. As a simple application of these tools, three simple but nontrivial new relations for EYM amplitudes have been derived in (2.31), (2.37) and (2.38).

(2.2)
Using the KK-relations (2.1), one can expand arbitrary color ordered Yang-Mills amplitudes in the basis, where two legs are fixed at the two ends (hence we will call such a basis the "KK basis").
The second kind is the Bern-Carrasco-Johansson (BCJ) relations [15]. To present these relations compactly, let us define some notations first. Given two ordered sets Ξ = {ξ 1 , ξ 2 , ..., ξ n } and β = {β 1 , ..., β r } where the set β is the subset of Ξ, for a given element p ∈ Ξ with its position K in Ξ (i.e., ξ K = p), we define: • (1) The momentum X p : It is given by , i.e., the sum of momenta of these elements, which are on the left hand side of p in the set Ξ.
• (2) The momentum Y p : It is given by , i.e., the sum of momenta of these elements, which are on the left hand side of p in the set Ξ, but do not belong to the set β. : To define them, we require p ∈ β, thus p split the set β into two subsets β L p and β R p , i.e., the collections of elements on the left hand side of p and on the right hand side of p respectively. Now we define , i.e., W (L,L) p is the sum of momenta of these elements, which are on the left hand side of p in the set Ξ, but do not belong to the subset β R p . In other words, W (L,L) p is the sum of momenta of these elements at the left hand side of p in the set Ξ satisfying either the condition that they do not belong to the set β or the condition that they belong to the subset β L p . This is the reason we call it W (L,L) p , where the first L is for these elements belonging to Ξ\β, and the second L is for these elements belonging to β L p . Similarly we define After above preparations, let us begin with the (generalized) fundamental BCJ relation [40]. If we divide the set {2, ..., n − 1} into two ordered subsets α = {a 1 , ..., α m } and β = {β 1 , ..., β t } (so m + t = n − 2), the relations are given by = (k p · k 1 + k q · (Y q + k p ))(k p · (Y p + k q )) K 1pq K 1p A YM n+1 (1, 2, {3, . . . , n − 1} ¡ {q, p}, n) for any set α. As shown in [40], by recursively using the relations (2.9), one can expand arbitrary color ordered Yang-Mills amplitudes in the BCJ-basis, i.e., those color ordered Yang-Mills amplitudes with three legs fixed at some particular positions (for example, at the first, second and the last positions), as A n (1, β 1 , ..., β r , 2, α 1 , ..., α n−r−3 , n) = (2.13) The expansion coefficients (we will call them the "BCJ coefficients") are first conjectured in [15] and then proved in [40]. Using our new defined notations, they are given by where with W 's defined in (2.8) and (2.6) and θ(x) = 1 when x > 0 and θ(x) = 0, otherwise 4 . Results (2.13), (2.14) and (2.15) are a little bit complicated and we give some explanations: • In (2.13), we have defined a new notation ¡ P , the "partial ordered shuffle", which is the sum over all permutations that maintain only the relative order of the subset {α} (i.e., the ordering of elements of the subset {β} in {ξ} can be arbitrary).
• The expression (2.14) tells us that for each element β k ∈ β, there is an associate kinematic factor.
The BCJ coefficients are the product of these factors.
Having explained the meanings of (2.13), (2.14) and (2.15), let us apply them to one example: where the sum is over all permutations ρ of three legs {p, q, r}. Coefficients in (2.16) involve two ordered lists: the first one {p, q, r} is the remembering of the original ordering on the left hand side, while the second one ρ{p, q, r} is the ordering in the expansion on the right hand side. For these six coefficients with orderings ρ{p, q, r}, we have: • The order {p, q, r}: For p, ξ 0 = +∞ > ξ p and ξ p < ξ q , so we have For q, ξ p < ξ q and ξ q < ξ r , so we have where momentum conservation has been used. For r, ξ q < ξ r and ξ r > 0, so we have This ordering is a special case since it is the same ordering comparing to the original ordering. This case is easy to generalize to a set with arbitrary length m and we have following results. For β r is not the first or the last elements of β, the corresponding kinematic factor is For the first element, the corresponding kinematic factor is For the last element, the corresponding kinematic factor is • The order {p, r, q}: For p, ξ 0 = +∞ > ξ p and ξ p < ξ q , so we have −k p ·(W For q, ξ p < ξ q and ξ q > ξ r , so we have k q · W (R,R) q = k q · Y q = −k q · X q . For r, ξ q > ξ r and ξ r > 0, so • The order {q, p, r}: For p, ξ 0 = +∞ > ξ p and ξ p > ξ q , so we have −k p ·(W For q, ξ p > ξ q but ξ q < ξ r , so we have −k q · (W (L,R) q − k 1 ) = −k q · (Y q − k 1 ). For r, ξ q < ξ r and ξ r > 0, • The order {q, r, p}: For p, ξ 0 = +∞ > ξ p and ξ p > ξ q , so we have −k p ·(W For q, ξ p > ξ q and ξ q < ξ r , so we have −k q · (W • The order {r, p, q}: For p, we have ξ 0 = +∞ > ξ p and ξ p < ξ q , so we will have −k p · (W • The order {r, q, p}: This case is also special, since the ordering {r, q, p} is the reversing of the original one {p, q, r}. In general, for the ordering β T , each element β r has the corresponding kinematic factor (2.20) For later convenience, we collect these six coefficients as following: Before ending this subsection, let us make an observation. For color ordered YM amplitudes, we have two basis: one is the KK-basis with (n−2)! elements and another one is the BCJ-basis with (n−3)! elements. As it has been seen in many examples, when we expand in the KK-basis, the expansion coefficients can be properly chosen to be polynomial functions, while when we expand in the BCJ-basis, the expansion coefficients will be rational functions in general. It is a simple but very useful observation. The reason for the polynomial functions rather than rational functions is clear: we must have some poles in these coefficients to match up pole structures at the both sides of the expansion.

Expansion of sEYM amplitudes by YM amplitudes
As explained in the introduction, in this paper we will focus on the expansion of sEYM amplitudes in terms of YM amplitudes. As mentioned before, the expansion in the KK-basis has been conjectured first by gauge symmetry consideration [19] and then proved using the CHY integrands in [31]. In this subsection, we recall the recursive expansion of sEYM amplitudes A |H| (1, 2 . . . r H) given in [19,31,33]   The expression (2.22) is manifestly invariant under all permutations and gauge transformations of gravitons in the set H\h a . The h a is special in the expansion and will be called as the fiducial graviton. The full permutation and gauge invariances for all gravitons, although not explicitly, are guaranteed by the generalized BCJ relations [40] (see explicit proof given in [42] ). When we use (2.22) recursively, we can expand any sEYM amplitude in the KK-basis of YM amplitudes with polynomial coefficients. In this paper, we will only use differential operators to reproduce the recursive expansion formula (2.22), thus provide a new strategy to find the expansion.
Before ending this subsection, let us give an important remark. Because of the generalized BCJrelations (2.9), even imposing the polynomial conditions, the coefficients C ha (h h h)'s in (2.23) can not be unique, since one can always add these zero combinations. Such a freedom can easily be fixed by making some choices for expansion coefficients of some elements in KK-basis.

Differential operators
In [34], to study connections of amplitudes of different theories, some kinds of differential operators have been defined. The first kind is the trace operator T ij where ǫ i ǫ j ≡ ǫ i · ǫ j is the Lorentz invariant contraction (similar understanding holds for all operators in this paper). The second kind is the insertion operator defined by As pointed out in [34], T ikj itself is not a gauge invariant operator, but when it acts on those objects obtained after acting a trace operator, it is effectively gauge invariant. Insertion operators will be extensively used in our paper. When it acts on the sEYM amplitudes, the operator T ikj has the physical meaning of inserting the graviton k between i and j when i, j are nearby in a trace. If i, j are not adjacent in a trace, for example, T ik(i+2) , we can write it as thus the physical meaning is clear. If i, j are not in same trace, the physical meaning is not clear. The third kind of useful operators is the gauge invariant differential operator where the sum is over all Lorentz contractions with ǫ i . When it is applied, ǫ i is effectively replaced by k i in any expression. Thus if an expression is gauge invariant, the action of W i will give zero result. The fourth kind of useful operators is longitudinal operators defined via 28) and the related operators The roles of L i and L ij are to add derivative interactions in vertexes while getting rid of polarization vectors. Using them, we can define the operators L and L as [34] L ≡ At the algebraic level, the actions of L and L are different. However, if we consider the combination L · T ab Pf ′ Ψ, and let subscripts of L i 's and L ij 's run through all legs in {1, 2, · · · , n} \ {a, b}, the effects of L and L are same and give a result which has a meaningful explanation. We will see it later.
As the simple applications of these kinds of differential operators, we derive three generalized relations for general EYM amplitudes. For simplicity we will use the sEYM amplitudes to demonstrate the idea. The first relation is the generalized color ordered reversed relation of EYM amplitudes: which is reduced to the familiar color ordered reversed relation for pure YM amplitudes when {h g } = ∅.
The second relation is the generalized photon decoupling relation of EYM amplitudes. By definition, we have (2.35) Putting them together gives Applying this identity on A EYM (1, · · · , n; h ∪ {h i }) leads to which is the wanted generalized photon decoupling relation for EYM amplitudes.
The third relation is the generalized KK relation, which is given by (2.38) First we want to point out that when α is the empty subset, it reduces to the generalized color-ordered reserved relation (2.31). The proof of the relation (2.38) is extremely similar to the proof of (2.31). Suppose In the third step, the relation T ikj = −T jki and the cyclic symmetry have been used. To understanding the meaning of last line in (2.39), recalling that all insertation operators are commutative and by the explanation of (2.26), the operator T 1b 1 n inserts b 1 at any position between 1 and n, thus turns Continuing the action until the last one, the last line in (2.39) gives (−) n β A EYM (1, α ¡ β T , n; {h g }), thus the generalized KK relation (2.38) is obtained.

Expansion in the KK-basis by differential operators
In this section, we will propose a new strategy to derive the expansion of sEYM amplitudes in the KK-basis of YM amplitudes by differential operators given in [34]. Before doing so, we make some remarks: • (1) Although it is well known that sEYM amplitudes can be expanded in terms of YM amplitudes, we don't need to assume this in the beginning. From the presentation in this section, we will see that assuming the factorization of polarization tensors ǫ µν i of gravitons into two polarization vectors ǫ and ǫ of gluons (i.e., ǫ µν i = ǫ µ iǫ ν i ), the web connections given in [34] will naturally lead to the expansion of sEYM amplitudes in terms of YM amplitudes.
• (2) Although in general, the expansion coefficients in the KK-basis can be rational functions as in [25,26], one can arrange them properly using the generalized BCJ relations (2.9) to get polynomial coefficients as written in (2.22). If we impose the condition that the expansion coefficients are polynomial, the KK-basis can be treated as independent up to the generalized BCJ-relations (2.9) 5 . Under this understanding, we see that the expansion coefficients in the KK-basis have the freedom caused by the generalized BCJ-relations.
• (3) Since we have assumed the factorization of polarization tensors of gravitons, we consider building blocks for the polarization vectors ǫ. For a given graviton, the expansion coefficients must be the function of Lorentz invariant combinations (k i · k j ), (ǫ i · k j ), as well as (ǫ i · ǫ j ) 6 . Among them, (ǫ i · k j ) and (ǫ i · ǫ j ) are the building blocks for the expansion when applying the differential operators. One reason using these building blocks is that we can transform the differential equations of expansion to linear algebraic equations of unknown coefficients of building blocks. One will see that such a choice of building blocks will naturally lead to the expansion in the KK-basis.
• (4) The mass dimension of expansion coefficients is |H|, the number of gravitons.
• (5) For the expansion of sEYM amplitudes A EYM n,m (1, · · · , n; {h 1 , · · · , h m }), the KK-basis is not general since the ordering of {1, 2, · · · , n} will be kept for each amplitude of the basis with nonzero coefficient. This fact can easily be understood from the pole structures of both sides and be proved using the insertion operators. Keeping only these KK-basis with the ordering {1, 2, · · · , n} has also partially fixed the freedoms caused by the generalized BCJ relations (2.9).
Above five points come from general considerations. Based on them, the structure of this section is following. In the first subsection, we consider the sEYM amplitudes with only one graviton. The calculations in this part are straightforward. In the second subsection, we consider the sEYM amplitudes with two gravitons. This part is very important because it contains the recursive strategy used in the third subsection for general situations. Based on the idea in the second part, recursive expansion for sEYM amplitudes with arbitrary number of gravitons has been presented in the third subsection. Thus we have shown that using differential operators, we could indeed derive the expansion of sEYM amplitudes in the KK-basis of YM amplitudes as given in (2.22).

The case with one graviton
According to the principles of Lorentz invariance and momentum conservation, the sEYM amplitude with 5 If we use the language of algebraic geometry, generalized BCJ relations (2.9) will generate an ideal in the polynomial ring of (n − 2)! variables. KK-basis will be the independent basis in the quotient ring. 6 Here we have excluded the contraction ǫi · ǫi. In other words, we have assumed that two polarization vectors coming from the same graviton should not contract with each other. This is a very natural assumption and could be seen from the traditional Feynman rules of gravitons.
one graviton A EYM n,1 (1, · · · , n; h 1 ) can always be written as where (ǫ h 1 · k i )'s 7 constitute a linearly independent building blocks with coefficients B i being functions of ǫ h 1 and (k i · k j ). By our experiences, to simplify the calculation we can choose a new basis of building blocks and the expansion is with K i = i j=1 k j , i = 1, ..., n − 1. Now we use differential operators to determine B i 's. To do so, we apply insertion operators T ah 1 (a+1) with a = 1, · · · , n − 1 to the expansion (3.2). Since T ah 1 Above manipulations show that after applying differential operators to the building blocks, we obtain linear algebraic equations for these B i 's. With the smart choice of building blocks, these linear equations are very easy to solve Here is a subtlety for the solution (3.4). As we have remarked, the KK-basis is independent only up to the generalized BCJ relations (2.9). Thus in principle, one can add these terms at the right hand side of (3.4). But with the polynomial condition as well as the mass dimension condition of coefficients, one can see that (3.4) is the only allowed solution. Similar phenomena will appear in later subsections and we will not discuss them further. Putting the solutions back, we get where in the second line we have used the definition of ¡ and Y h 1 (see (2.4)). 7 The ǫ h 1 · kn is not independent by the momentum conservation.
Although this example is trivial, it reveals the essence of our strategy: (1) The expansion in the KKbasis of YM amplitudes is the natural consequence of web connections established in [34] without relying other information such as the double copy form (1.3); (2) The differential operators act on building blocks only, and we get linear algebraic equations for unknown coefficients, which are much easier to solve than differential equations. This transmutation reflects the important role of building blocks.

The case with two gravitons
Now we move to the less trivial case A EYM n,2 . This example will show how to generalize our strategy to arbitrary amplitudes A EYM n,m . As in previous subsection, we expand A EYM n,2 according to the building blocks of ǫ h 1 as To determine B i , we use insertion operators T ah 1 (a+1) with a = 1, · · · , n − 1. After acting on both sides, we arrive Putting the solutions back, (3.6) becomes To determine the remaining two variables D h 2 , E h 2 , we use the gauge invariance of the graviton h 2 8 .There are two approachs. The first approach is that since the first term at the RHS of (3.8) is h 2 gauge invariant already, after ǫ h 2 → k h 2 in the remaining two terms, we get The second approach is to consider the commutation relation of insertion operators and gauge invariant operators (see (2.25) and (2.27)) 9 where T jk is the trace operator (2.24). For current case, we apply the commutation relation [T h 2 h 1 n , W h 2 ] = T h 2 h 1 to both sides of (3.8) and get The gauge invariance of graviton hi means the Ward's identity, i.e., the amplitude vanishes under the replacement ǫ h i → k h i . 9 Although for this simple example, the first approach is simpler, but when trying to generalize to general cases, the second approach is more suitable.
No matter which approach is used, now (3.8) becomes where the gauge invariance of h 2 is manifest. To determine the last variable D h 2 , we expand it according to the building blocks of ǫ h 2 as To determine the H i 's, we apply the combinations T ah 2 (a+1) T h 2 h 1 (a+1) , a = 1, ..., n − 1 of two insertion operators, and get (3.14) From it we can solve so the expansion becomes Now only the H h 1 is unknown. This one is manifestly gauge invariant and symmetric because ( To fix it, we use the special operator T jh 1 h 2 T jh 2 h 1 . While acting on the right hand side of (3.16) it gives when acting on the left hand side, the physical meaning of "insertion" is not clear at the level of amplitudes. However, as shown in [35,36], its action at the level of CHY-integrands is still clear. Doing it leads to To continue, we use the Pfaffian expansion given in [43,44] and keep only terms contributing under these operators. The related terms 10 are ( Using them, we find Since the left-hand side is zero, we find H h 1 = 0. Assembling all pieces together, we obtain the expansion of A EYM It is a recursive expansion of A EYM n,2 by A EYM n+1,1 and A YM n+2 . We can get the complete expansion of A EYM n,2 in terms of purely YM amplitudes by putting back the known expansion of A EYM n+1,1 in (3.5). From above calculations we see again that the expansion of sEYM amplitudes to the KK-basis of YM amplitudes is the natural consequence of web connections established in [34] by differential operators. Furthermore, we see that to find the expansion, we could do it step by step. At the first step, we organize expansion according to the building blocks of ǫ h 1 and then using insertion and gauge invariant differential operators to determine some coefficients. At the second step, we organize the remaining unknown variables according to the building blocks of the polarization vector of next gravitons, and then solve some variables again. Repeating the procedure, we will finally determine the expansion. In the next subsection, we will see more details about the iterative operations.
A nontrivial thing in above derivation is the possible appearance of the term It satisfies all general conditions, such as the multi-linearity of ǫ h i , gauge invariance and polynomial, etc. To show its vanishing, we need to consider the action of differential operators at the level of CHY-integrands instead of at the level of amplitudes directly. The vanishing result has been carefully proved in the Appendix A, which may imply some interesting things at the level of sEYM amplitudes.

The case with m gravitons
Having outlined our derivation strategy in the previous subsection, we consider the expansion of arbitrary sEYM amplitudes A EYM n,m . Let us begin with the expansion according to the building blocks of graviton h 1 At the first step we apply the insertion operators T ah 1 (a+1) with a = 1, · · · , n − 1 to (3.22) and get where we use h to denote the set of gravitons At the second step, using the gauge invariance conditions, i.e., the commutation relations [T hah 1 h , W ha ] = T hah 1 with a = 2, · · · , m to (3.24), we get One needs to notice that D h j 's are required to be gauge invariant for gravitons h = h 1 , h j 11 .
To determine the D h j , we need to expand (ǫ h j · D h j ) in (3.24) further according to h j 's building blocks Because of the gauge invariance of D h j , we can apply the commutation relations [T hah j n , W ha ] = T hah j (with a = 2, · · · , m, but a = j) to (3.27) and and (3.26) becomes To determine B h j i 's we apply the products of two insertion operators, i.e., where D h j h j 2 's are also gauge invariant for remaining gravitons. Now we compare (3.26) and (3.31). We see that to get (3.31) from (3.26), we need just do the replacement This replacement is derived by using proper products of insertions operators, which act either at the amplitude level or at the CHY-integrand level as shown in above. Now the pattern is clear. To expand (ǫ h j 2 · D h j h j 2 ), we act on with products of three insertion operators and find the result is equivalent to the replacement Continuing the procedure until the coefficient D does not contain any polarization vector ǫ (so the iteration stops), we obtain the complete recursive expansion of A EYM n,m as (3.34) In above derivation, an important thing is that at each step, we need to show building blocks like ( These building blocks belong to the "index circle structure" and in the Appendix A we will prove their coefficients are zero. Now we have reached our first goal, i.e., to expand sEYM amplitudes in the KK-basis of YM amplitudes, by using differential operators. Compared to other methods appearing in the literatures, our method provides a different angle. In the whole procedure, the use of building blocks is very crucial. By using them, we have translated the problem to the solving of linear algebraic equations. The solutions are naturally the KK-basis of YM amplitudes. In other words, instead of assuming it, we have derived the fact that sEYM amplitudes can be expanded in terms of YM amplitudes.

Further applications
As explained in the introduction, the expansion of sEYM amplitudes in the KK-basis of YM amplitudes is the primary example. In this section, we will demonstrate this claim by two new examples. In the first subsection, we will try to derive the expansion of gravity amplitudes using differential operators. Using the same strategy and results from the previous section, the expansion can be written down straightforwardly. In the second subsection, by using the observation that all other amplitudes can be obtained from acting on gravity amplitudes with proper differential operators [6,34], we derive the expansion of amplitudes of the Born-Infeld theory.

Expansion of gravity amplitudes in the KK basis of YM amplitudes
Let us consider the gravity amplitude A G with m gravitons. Expanding it according to the building blocks of h 1 , we write where k hm is eliminated using the momentum conservation. To determine D g , one can use the trace operator T h 1 hq , which turns the gravity amplitude to the sEYM amplitude with just two gluons. Applying it on the LHS of (4.1), it gives while acting on the RHS it gives Comparing two sides, we solve The vector B µ f can be determined from D f via the gauge invariance of graviton h f . Let us consider the commutation relation Applying this relation on (4.1) gives (4.7) 12 We want to point out that the expansion form (4.7) has also appeared in [19] based on the Pfaffian expansion given in [43].
To determine B µ f 's from the relations (4.6), we just need to find the expanded formula of sEYM amplitudes , which have been solved in previous section using the differential operators and are given in (2.22). A special feature of current situation is that there are no gluons {2, ..., r − 1} in (2.22) and we are left with only {h h h, h a }. Because of this fact, coefficient given in (2.23) will always have Y µ is = k µ hq in our case, which is exactly the combination −k hq · B q in (4.6). In other words, using (2.22) and (2.23), after removing the Y µ is , we get B µ f immediately. It seems that the set of amplitudes appeared in above expansion is larger than the KK-basis, since the appearance of the ordering (h q , h h h, h a , h 1 ) in (2.22). However, when we expand the sEYM amplitudes, the fiducial graviton h a can be chosen arbitrarily. Thus we can always choose h a = h m in (2.22). Then one can use the cyclic symmetry to re-write any ordering Thus, every amplitude appearing in the expansion has the ordering (h 1 , · · · , h m ) and belongs to the chosen KK-basis.

Expansion of Born-Infeld amplitudes in the KK basis of YM amplitudes
As pointed out in [34], starting from the expansion of gravity amplitudes expansions of amplitudes of different theories can be obtained by applying different combinations of differential operators on both sides of (4.9) simultaneously. The action of operators can be divided to two types.
, m) will be transformed to amplitudes of other theories, while expansion coefficients are not modified. For example, one can use this procedure to expand YM amplitudes in terms of bi-adjoint scalar amplitudes, with the same coefficients. The second type is more interesting, where the operators O ǫ are defined via the polarization vectors ǫ g . While it turns the gravity amplitudes to amplitudes of other theories on the LHS, it modifies only expansion coefficients on the RHS. Using this observation we will expand amplitudes of Born-Infeld(BI) theory in the KK-basis of YM amplitudes in this subsection.
The n-point BI amplitudes can be created from the gravity ones by applying differential operators as we have Without loss of generality, let us choose a = 1 and b = n. From results in the previous subsection, we know how to expand the amplitude A EYM 2,n−2 (1, n; {g}/{1, n}) in terms of YM ones as (4.13) Applying the operator L on the LHS of (4.13) gives the BI amplitude A BI n ({g}). For the RHS, L acts only on coefficients C(¡) since YM amplitudes do not carry the polarization vectors ǫ. The operator L turns all (ǫ i · k j )'s to (k i · k j )'s, therefore only terms with the form i (ǫ i · K i ) can survive under the action of L with K i 's being the sum of some external momenta. In C n−2,2 (¡), such a part is given by i (ǫ i · X i ).
Carrying it out, the RHS gives (4.14) Comparing the LHS with the RHS, the expansion of BI amplitudes is obtained as with the very compact expansion coefficients.
The expansion of BI amplitudes can also be derived via another definition of the operator L, where L ij ≡ −(k i · k j )∂ ǫ i ǫ j and the pairs {i, j} run over the set {g}/{1, n}. Applying this definition on the RHS of (4.13), the survived terms are these in which each polarization vector ǫ i is contracted with another polarization vector ǫ j . Using the results (2.22) and (2.23), such a part is found to be where Or 1 ∪Or 2 ∪· · · Or t = {2, · · · , n−1}, and the summation |Or k |even is over all possible ordered splitting that the lengths of all subsets are even. Furthermore, M k (¡) for length-r subset Or k = {γ 1 , γ 2 , · · · γ r } is defined as 13 Then we get  .15) gives while (4.20) gives with the chosen order 2 ≺ 3. These two results are equal to each other only after using BCJ relations (2.13). The example shows that although (4.20) looks more complicated, it gives shorter expressions.
As an interesting remark, if we reverse the logic, i.e., assuming the equivalence between two differential operators, we will arrive the equivalence of results (4.15) and (4.20). From them, we can derive the generalized fundamental BCJ relations (2.9).

Expansion of sEYM amplitudes in the BCJ-basis of YM amplitudes
As pointed in the introduction, the expansion of sEYM amplitudes in terms of YM amplitudes can be divided to two types: the expansion in the KK-basis and the expansion in the BCJ-basis. Up to now, expansion in the literatures are in the KK-basis, where the expansion coefficients are much simpler. In this paper, we will initiate the investigation of expanding in the BCJ-basis. There are two different approaches we could think of. In the first approach, we start from the expansion in the KK-basis (2.22) and then transform the KK-basis to the BCJ-basis using (2.13). This approach is straightforward, but as shown in this section, the algebraic manipulations are not so trivial. In the second approach, encouraged by results in the previous sections, we will use differential operators. Both approaches will be explored in this paper. In this section, we will take the first approach, while in the next section, we will take the second approach. As emphasized in the introduction, we are looking for a more efficient method to find the expansion coefficients, which now will be a more complicated rational functions. Results obtained in this section will be compared with results found in the next section.

Case with one graviton
We start with the sEYM amplitudes with just a graviton. The expansion in the KK-basis is given by To expand in the BCJ-basis, we use a trick, i.e., the gauge invariance of p, so we have Solving the equation and putting it back, we have Using the definition of field strength, we arrive A EYM n,1 (1, 2, . . . , n; p) = (1, 2, {3, . . . , n − 1} ¡ {p}, n) .

(5.4)
The expansion coefficient's are rational functions with pole (k p · k 1 ), which is crucial to match up the structure of physical poles at both sides. The observation implies that the expansion coefficients in the BCJ-basis will be, in general, the rational functions with proper pole structures.

Case with two gravitons
Using the recursive expansion (2.23), we could write down The gauge invariance of p leads to the following equation Solving the A EYM n+1,1 (1, p, {2, . . . , n − 1}, n; q) and putting it back, we get A EYM n,2 (1, 2, . . . , n; p, q) = Although we have not reached the expansion in the BCJ-basis, the gauge invariance of all gravitons is manifest because the only appearance of f p , f q . For the first term of (5.6), we use the result (5.4) to reach where it is important to notice that since for q, the p is just a gluon, thus we must use the X q instead of the Y q . Expanding further to different orderings, we get For the second term in (5.6), we split it to For the first term of T 2 , we use the result (2.11) to get For the second term of T 2 , we use (2.10) to get (1, 2, {3, . . . , n − 1} ¡ {p, q}, n) .
Putting T 2,1 , T 2,2 back, we get (1, 2, {3, . . . , n − 1} ¡ {q, p}, n) . with coefficients The first coefficient can be rewritten as When comparing the factor (kp·(Yp−k 1 ))(kq ·Xq) K 1pq K 1p in the second term of C({p, q}, ¡) with the second term of (2.11), we see they are the same. It is not a coincidence and we will see its reasoning in the next section.
The second coefficient C({q, p}, ¡) is not relate to the first coefficient by permutation at the current form, thus we need to do some manipulations. It is easy to see that the first two terms can be rewritten as where we have used the identity (5.14) 14) The identity (5.14) can be easily remembered as following 14 : The two elements B, k 1 in (B · f p · k 1 ) are exchanged with the element k q in (k q · k p ). Putting it back, the second coefficient becomes The formula is also like the Schouten identity ab cd = ac bd + ad cb . Now the permutation relation with C({p, q}, ¡) is manifest.
Before ending this subsection, we give some remarks for above results: • (1) First, the two coefficients (5.13) and (5.15) are related to each other by a permutation as it should be. This relation can also be used as a consistent check of our calculations.
• (2) Each coefficient contains two terms. The first term is the product of two factors The same factor has also appeared in (5.4) for the case having just one graviton, and can be interpreted as "turning a graviton to a gluon in the BCJ-basis" in a loose sense. The second term has the factor (k 1 f p f q k 1 ), which contains the contraction ǫ p · ǫ q as a singal of the mutual interaction. Thus the physical picture of these two terms is clear.
• (3) One of important observations is that the factor (k 1 f p f q k 1 ) appears naturally in C({p, q}, ¡), but not in C({q, p}, ¡). Only with some manipulations, we can transfer the form (k 1 f p f q Y q ) to (k 1 f p f q k 1 ). As we will see for the case with three gravitons, the same pattern will appear again. In other words, the contraction of ǫ · ǫ can always be included inside the form k 1 f...f k 1 . Observations given in the second and third points provide information of the building blocks for expansion in the BCJ-basis when using differential operators.
• (4) Although it is straightforward, the whole calculation is nontrivial. An indication is that in middle steps, the BCJ coefficients (2.13) have been used many times, thus expressions in middle steps are lengthy and complicated. However, when summing them up, cancelations happen and the final result (5.13) is very simple. Similar phenomena have been met many times in the history of scattering amplitudes. Thus we believe that there should be a more efficient method where cancelations in middle steps are automatically avoided.
The first term: For the first term, we can use the result (5.11) directly only paying the attention to the meaning of Y q , Y r since for q, r, the p likes a gluon. Thus we have Now we can find contributions to six coefficients in (5.18). Since we expect they are related to each other -30 -by permutations, we will only write down contributions to the coefficient C({p, q, r}) for simplicity 15 The second term: The second term can be split to 1, 2, {3, . . . , n − 1} ¡ {q, p}, n; r) Among these two terms, using the result (5.4), the first term can be expanded to It is important to notice that this term does not contribute to the coefficient C({p, q, r}), thus we can forget it at this moment 16 . The second term can be expanded to where to distinguish different situations, it is crucial to expand X r to Y r . Among these two terms, the first one needs to be expanded further as q, p, r, {2, 3, . . . , n − 1}, n) . (5.20) 15 As pointed out in the previous subsection, for a given choice of fiducial graviton in the KK-basis expansion, some orderings will have simpler expressions for coefficients. This is the reason we consider this ordering. 16 This is also the reason we consider the coefficient C({p, q, r}) for simplicity. Although other coefficients will be similar, but to lead to the pattern observed from this coefficient, nontrivial algebraic manipulations will be needed.
Using the (2.10), we can read out the contribution to the coefficient C({p, q, r}) from the first term as Using the (2.11), we can read out the contribution to the coefficient C({p, q, r}) from the second term as Using the (2.21), we can read out the contribution to the coefficient C 3 ({p, q, r}) from the third term as 17 Now we sum up above three coefficients. The sum of the first two is Adding up the third one we get For the second term of T 2,2 , we can expand to another three terms (1, q, r, p, {2, 3, . . . , n − 1}, n) .
The first term does not contribute to the coefficient C({p, q, r}). Using (2.11) and (2.21), the contributions from the second and third terms are 17 Now it is clear that why in (5.20) we write Xr out explicitly, since when we use (2.21) to get (5.23), there is another Xr appears. The meaning of these two Xr's is different and we must make clear distinction.

(5.29)
Now we treat term by the term. For the first term, we expand it further as r, p, q, {2, . . . , n − 1}, n) .

(5.30)
Reading out contributions to the coefficient C({p, q, r}) we get For the second term we expand to Reading out contributions to the coefficient C({p, q, r}) we get (the first term does not contribute) Putting two parts together we get The fourth term: The fourth term can be expanded to four terms: (1, 2, {3, . . . , n − 1} ¡ {r, q, p}, n) (1, r, 2, {3, . . . , n − 1} ¡ {q, p}, n) The first and the second terms do not contribute to the coefficient C({p, q, r}). The remaining two terms give The fifth term: We expand it to four terms (1, 2, {3, . . . , n − 1} ¡ {q, r, p}, n) Again the first and the second terms do not contribute to coefficient C({p, q, r}). The remaining two terms give In total: Now we sum up these contributions and simplify them to We can simplify further, by using

43)
This is a much nicer expression than the one given by our direct calculation (5.39), since each A i has more manifest physical pattern.

Expansion in the BCJ-basis by differential operators
In the previous section we have illustrated how to expand sEYM amplitudes in the BCJ-basis of YM amplitudes starting from their expansion in the KK-basis. Although the whole procedure is systematical, the algebraic manipulation is not so easy. As we have emphasized, the strategy of using differential operators is itself an independent method, thus in this section, we will show how to do the calculation.
Before going to calculation details, let us give some general considerations. A sEYM amplitude A EYM n,m (1, 2, · · · , n; {h 1 , h 2 , · · · , h m }) with n ≥ 3 gluons 18 and m gravitons can be expanded in the BCJ basis of YM amplitudes as following where the color order of n gluons is kept and ρ represents a permutation of m gravitons, ρ{h 1 , h 2 , · · · , h m } = {ρ 1 , ρ 2 , · · · , ρ m }. The coefficients C(¡, ρ) depend on the permutation ρ and the shuffle ¡, because gravitons don't have color order and can occur in all possible orderings.

(6.2)
A very important fact of the expansion (6.1) is that by the gauge invariance of EYM amplitudes, when ǫ i is replaced by k i , A EYM n,m (ǫ i → k i ) = 0 at the left hand side, thus C ¡,ρ (ǫ i → k i ) must be zero at the right hand side because each amplitude of the BCJ-basis is independent to each other. In other words, these coefficients are also gauge invariant like amplitudes. From experiences in the previous section we see that these coefficients are functions of field strength f µν only, thus are manifestly gauge invariant.
Another general feature is that coefficients are multi-linear functions of polarization vectors ǫ i . Thus Lorentz invariance means coefficients can only be polynomial functions of Lorentz invariant contractions (ǫ i · ǫ j )'s and (ǫ i · k j )'s, but can be rational functions of (k i · k j )'s. We can separate terms according to contraction types sketchily as Above two general considerations are very important since if we impose such conditions, i.e., the gauge invariance and multi-linearity of polarization vectors ǫ i , we could derive the expansion of sEYM 18 The case n = 2 is special and we will not discuss it in this paper. amplitudes in the BCJ-basis naturally as will be shown in this section. Comparing to expansion in the KK-basis, building blocks used in this section are much more complicated. In the appendix B, we will give a more systematical discussion about the building blocks for the expansion in the BCJ-basis, although complete understanding is still not clear for us. Now we present several examples to show how to derive the expansion of sEYM amplitudes in the BCJ-basis by differential operators.
Putting it back, we get the wanted expansion in the BCJ-basis where K a is exactly the Y p defined before. Y p has implicit dependence on the shuffle. We want to emphasize that using insertion operators and gauge invariance for the building blocks only, we have found the expansion of sEYM amplitudes with single graviton in the BCJ-basis of YM amplitudes without any other preassumption, such as the KLT relations.

The case with two gravitons
Using the results given in (B.13) in the Appendix B, we can expand the sEYM amplitude with two gravitons according to the building blocks A EYM n,2 (1, 2, · · · , n; p, q) = Now we determine these coefficients one by one: • For E: Since the building block contains the index circle structure (k 1 · f q · f p · k q ), i.e., according to the general argument given in Appendix A, E = 0.
Before ending this part, let us give another remark. To solve D we have used the differential operator T 1qp T 1p2 and the result (2.11). However, we can use another operator T 1q2 T qp2 to find D. If we act it on both sides of (6.22), we will get the same L and different R in (6.23). Using these two different expressions of R, we can solve D. Putting the D back, we get the L, which is nothing, but the expression (2.11). In other words, we have derived the expansion of YM amplitudes to its BCJ-basis for this special case, just like the derivation of the fundamental BCJ-relations in the previous subsection (6.10). From the new angle, the BCJ relations are consistent conditions of these differential operators. It is similar to the observation that consistent conditions for different KLT relations [17,45,46,47,48,49,50] will imply these BCJ relations.

Another derivation
The starting point of the previous derivation is the expansion (6.11) of A EYM n,2 using building blocks of two gravitons. Now we consider another derivation, which has the recursive structure like the expansion in the KK-basis. We can first view A EYM n,2 as the polynomial functions of ǫ p and write its manifestly gauge invariant expansion by building blocks of ǫ p as A EYM n,2 (1, · · · , n; p, q) = n−1 with B a and B q being polynomials of ǫ q . Now we use differential operators to determine these unknown variables one by one: • (1) First, we use insertion operators T ip(i+1) with 2 ≤ i ≤ n − 1 to determine B a . After applying these operators, we get • (2) For the remaining two variables, there are no differential operators involving only graviton p to select them one by one, and we need to expand it further according to ǫ q as Because of the index circle structure, we have immediately B qp = 0.
From it, we can solved

Thus we can write
A EYM n,2 (1, · · · , n; p, q) = • (4) The determination of E pq will be exactly the same as the determination of D in (6.25) and we will not repeat the calculation.
In this paper, we have proposed a new method to efficiently find expansion coefficients by using the differential operators introduced in [34]. Moreover, we have actually achieved more than our initial goal. Using the proper building blocks, we have shown that the differential operators, together with the web connections established in [34], naturally lead to the expansion of sEYM amplitude in the KK-basis or BCJ-basis without any extra assumptions. Furthermore, with the use of these building blocks, we have translated the problem into a set of linear equations, thus greatly reducing its complexity.
For the expansion in the KK-basis, the expansion coefficients are polynomial and the natural building blocks are ǫ · ǫ and ǫ · k. With this simplification, the recursive expansion of sEYM amplitudes to YM amplitudes (2.22) has been reproduced by the new method. As a further demonstration of the efficacy of the new techniques, we have also discussed the expansion of gravity theory and Born-Infeld theory in the KK-basis.
For the expansion in the BCJ-basis, which has not been dealt with much in previous works, finding the expansion coefficients is a much more difficult task. The technical challenge lies in that a proper understanding of the building blocks, for arbitrary number of gravitons, is still missing. As explained above, the building blocks ǫ · ǫ and ǫ · k in the KK-basis expansion are too large. Even if we constrain ourselves to the gauge invariant combinations, such as (k 1 f X) and (kf...f k), they are still too large. As we have argued, the index cycle structure should not appear. Furthermore, as given in (5.40), not all allowed gauge invariant building blocks are independent of each other. Thus a proper understanding of these building blocks becomes one of the most important problems.
Both expansions are worth studying. For the expansion in the KK-basis, since all physical poles are included in the basis, expansion coefficients can be arranged to be polynomials. Thus such an expansion is more suitable when considering the analytical structure of amplitudes. However, in this manner, the manifest gauge invariance of gravitons is lost. On the other hand, for the expansion in the BCJ-basis, the gauge invariance of all gravitons is manifest. The price to pay is that some physical poles are moved to the coefficients.
Due to the difficulty related to gauge invariant building blocks, as pointed out before, we could not give the complete solution for the expansion of sEYM amplitudes in the BCJ-basis of YM amplitudes at this moment. Thus a clear understanding of the building blocks will be an important future problem. Our examples of the previous section, especially the recursive construction, may provide us with some guidance in this issue. Our results here show that differential operators introduced in [34] provide a much wider set of applications (such as the three generalized relations presented in (2.31), (2.37) and (2.38)) and have a deeper meaning (such as the relation between expansion coefficients and BCJ expansion coefficients given in (6.44)) than previously thought. Therefore, further studies are needed in order to exploit their full potential, such as, for instance, their applications to soft and collinear limits.
In (A.4), Ψ (2··· ) represents any cycle containing h 2 and other gravitons than h 1 , and similar understanding for Ψ (1··· ) and Ψ (1···2··· ) . All of them are annihilated by at least one of insertion operators. Thus among all terms in the sum of PfΨ, only the first two give nonzero contributions. Carrying it out explicitly, we get The next simple case is the one with three gravitons, for example, the string (h 1 , h 2 , h 3 ). By similar reason, the action of T ah 1 h 2 T ah 2 h 3 T ah 3 h 1 is with PfΨ H m−2 the Pfaffian of other gravitons. Carrying it out explicitly, first we note that For the remaining three terms, we have From the previous two simple examples, we can see how to generalize to strings of s gravitons, for example, the (h 1 , h 2 , ..., h s ). With the action of insertion operators, we have T ah 1 h 2 T ah 2 h 3 · · · T ahsh 1 PfΨ Hm =T ah 1 h 2 T ah 2 h 3 · · · T ahsh 1 1≤i 1 ≤i 2 ≤···≤im≤n i 1 +i 2 +···+im=n (−1) n−m P i 1 i 2 ···im =T ah 1 h 2 T ah 2 h 3 · · · T ahsh 1 1≤j 1 ≤j 2 ≤···≤j l ≤s j 1 +j 2 +···+j l =s where PfΨ H m−s is the Pfaffian of other gravitons. Because of the particular circle order (h 1 , h 2 , · · · , h s ), the action of insertion operator T ah j h j+1 is not zero when and only when there is the factor ǫ h j · k a or ǫ h j · k h j+1 . Thus only terms, which are either the multiplication of s's length one cycles or the length s cycles having the same or reversing order, will contribute. With the simplification, we have So finally we have proved that any term with the building block having index circle structure will vanish.

B. General discussions of manifestly gauge invariant functions
In the appendix, we make some general discussions of manifestly gauge invariant functions, which are important for the construction of building blocks when expanding in the BCJ basis. For amplitudes involving gauge particles, they must satisfy some basic requirements: Lorentz invariance, on-shell condition, momentum conservation, transversality 19 , gauge invariance, etc. With those requirements, properties of amplitudes have been thoroughly discussed in [51,52,53,54,55]. Especially, in [52] starting from the fact that amplitudes are multi-linear functions of ǫ's the authors are able to write down a set of linear equations coming from gauge invariance condition. After solving them, they can derive some important consequence. Going further in [54] the authors prove that using only gauge invariance, locality and minimal power counting, one can determine amplitudes uniquely.
Since gauge invariance is a so important and strong constraint, it is better to construct functions which are manifestly gauge invariant by sacrificing other properties [55]. By Feynman rules amplitudes are multilinear functions of polarization vectors. Gauge invariant condition means that when replacing ǫ µ i by k µ i for each i, amplitudes vanish. A simple and naive combination satisfying above condition is (ǫ µ i − k µ i ), which is wrong since the dimension of two terms does not match. Correcting with dimension, one leads to the form (ǫ µ i k ν j − ǫ ν r k µ i ). Imposing gauge invariant condition for each i, we end up the combination ǫ µ i k ν i − k µ i ǫ ν i , which is nothing, but the familiar field strength f µν i = k µ i ǫ ν i − ǫ µ i k ν i . Above argument seems to indicate that any manifestly gauge invariant function could be rewritten as the function of f µν . Although we could not give complete proof about this statement, for some special cases, such as the sEYM amplitudes studied in this paper, we will show it now. Actually, we can go further. Lorentz invariance requires that f µν must be contracted by a momentum or another f µν , thus eventually we will reach two types of contractions: Type-I with the form (k a ·f b · · · f c ·k d ) and Type II with the form tr(f a · f b · · · f c · f d ) ≡ (f µ aν f ν bρ · · · f α cβ f β dµ ). Lorentz and gauge invariant building blocks should be functions of both types, but in this paper we only encounter the Type-I contractions 20 , so we just discuss some properties of them.
For the Type-I contraction, we use the number of f 's appearing in the term to characterize them (called the f -degree), thus we have d f = m − 2, where d f is the f -degree while m is the mass dimension. Some examples are (k a f b k c ) of degree one and (k i f j f l k s ) of degree two. Two properties of Type-I contractions can be easily proved. The first is The second is which is just the identity (5.14). The formula (B.2) tells us that any Type-I contraction with higher degrees can always be decomposed to the sum of terms as the product of Type-I contractions with lower degrees. This decomposition terminates at Type-I contractions of degree one and degree two, which will be called fundamental. Relations (B.2) give the first indication why the building blocks for BCJ-basis are much more complicated.
Having above general discussions, let us move to the case related to sEYM amplitudes.

B.1 Having only one polarization vector
Now we consider the simplest example, i.e., a general function of (n+1) momenta and only one polarization vector. As the linear functions of ǫ, the functions can be generally written as F = F (k 1 , · · · , k n , k p , ǫ p ) = where α i 's are unknown functions of (k j · k l ). Here we have used momentum conservation to eliminate the momentum k n , so all remaining (ǫ p · k i )'s are independent. Gauge invariant condition leads to α i (k p · k i ) (k p · k 1 ) , (B.5) from the equation and putting it back, we get with (k 1 f p k i ) = k 1µ (k µ p ǫ ν p − ǫ µ p k ν p )k iν and K 1p = (k p · k 1 ). Thus we have shown the appearance of Type-I contractions in this case 21 . Result (B.6) tells us that the basis of gauge invariant building blocks of single polarization vector is given by (k 1 f p k i ) (or (k 1 fpk i ) K 1p ) with i = 2, ..., n − 1. A equivalent, but more convenient basis can be taken as (k 1 f p K i ) (or (k 1 fpK i ) K 1p ) with i = 2, ..., n − 1 and K i = i t=1 k t .
Now we solve these equations. Using the first set of equations we get β ij (k p · k i ) (k p · k 1 ) for j = 1, 2, · · · , n − 1, (B.9) 21 We should note that the number of momenta should be greater than 3, since when n + 1 = 3, the function will vanish because of the special kinematics of three particles.
For simplicity, in the sum we just indicate the unallowed indexes, for example, i = n, n + 1 means the sum range is i = 1, 2, · · · , n − 1, n + 2, n + 3. From these three sets of equations we can solve some β variables and then put them back to get the manifestly gauge invariant form of F as F = i =1,n,n+1 j =1,n,n+2 l =1,n,n+3 As before, the special role of k 1 comes from our particular choice when solving equations. Above form is applicable when n + 3 > 4. When n + 3 = 4, F is simplified to with i = 3, 4, j = 2, 4 and l = 2, 3. Using gauge invariant conditions to get three sets of equations and solving them, we find Like the case in previous subsection, result (B.18) gives us also the basis of gauge invariant building blocks for three polarization vectors. Since it is straightforward, we will not write down them explicitly.
Although in this paper we will only use results up to three polarization vectors, above calculations show that the manifestly gauge invariant form of F does have some patterns, and we conjecture that these patterns will also appear even with more than three polarization vectors. But there must be some new things happen when the number of momenta is equal to that of polarization vectors. For example, when there are only four gluons involved, the manifestly gauge invariant form of F has been given in [55] consisting of the Type-II contractions, while the Type-I contractions don't appear. We will discuss these in the future work.