Expansion of tree amplitudes for EM and other theories

The expansions of tree-level amplitudes for one theory into amplitudes for another theory, which have been studied in various recent literatures, exhibit hidden connections between different theories that are invisible in traditional Lagrangian formulism of quantum field theory. In this paper, the general expansion of tree EM (Einstein-Maxwell) amplitudes into KK basis of tree YM (Yang-Mills) amplitudes have been derived by applying the method based on differential operators. The obtained coefficients are shared by the expansion of tree $\phi^4$ amplitudes into tree BS (bi-adjoint scalar) amplitudes, the expansion of tree sYMS (special Yang-Mills-scalar) amplitudes into tree BS amplitudes, as well the expansion of tree DBI (Dirac-Born-Infeld) amplitudes into tree special extended DBI amplitudes.


I. INTRODUCTION
The modern researches on S-matrix have exhibited remarkable properties of scattering amplitudes that are not evident up on inspecting traditional Feynman rules. The expansion of tree-level amplitudes of one theory into that of another theory, which hints the hidden connections between different theories, is a significant example. Such unexpected expansions was first implied by the well known Kawai-Lewellen-Tye (KLT) relation [1], which represents the tree-level amplitudes of gravity as the double copy of color-ordered tree amplitudes of Yang-Mills (YM) theory, where S n−3 denotes permutations on n−3 external gluons and S[σ σ] stands for the kinematic kernel. One can arrive the expansion of gravitational amplitudes into YM ones by summing over all S n−3 permutations σ (orσ) in the relation (1). The Cachazo-He-Yuan (CHY) formula proposed in 2013 indicates a much richer web of expansions among a wide range of theories [2][3][4][5][6]. In the CHY framework, the n-point tree-level amplitudes arise as a multi-dimensional contour integral over n auxiliary complex variables, formally written as where the auxiliary variables are fully localized by the scattering equations. The integrand I CHY depends on the theory under consideration, which can always be factorized as two ingredients Integrands for Gravity and YM take the formulae I L G = Pf ′ Ψ(k i , i , z i ), I R G = Pf ′ Ψ(k i ,̃ i , z i ), and I L YM = Pf ′ Ψ(k i , i , z i ), I R YM = PT(α) (or I L YM = PT(α), I R YM = Pf ′ Ψ(k i ,̃ i , z i )), respectively. Here Ψ(k i , i , z i ) is a 2n × 2n anti-symmetric matrix depends on external momenta k i , polarization vectors i , as well as auxiliary variables z i . Pf ′ Ψ(k i , i , z i ) denotes the reduced pfaffian of the matrix Ψ(k i , i , z i ). PT(α) is the so-called Parke-Taylor factor writes Notice that the polarization tensors of gravitons are expressed as the product of polarization vectors µ ĩ ν i . In the framework, the expansion can be understood as expanding the reduced pfaffian Pf ′ Ψ(k i ,̃ i , z i ) into the Parke-Taylor factors PT(α). One can go further since the CHY integrands reflect the double copy structure for a verity of other theories beyond gravity. For instance, integrands for Einstein-Yang-Mills theory (EYM), Einstein-Maxwell theory (EM), Born-Infeld theory (BI) also carry I L = Pf ′ Ψ(k i , i , z i ) [6], thus amplitudes of these theories can also be expanded into YM ones by expanding I R into Parke-Taylor factors.
Although conceptually simple and straightforward, the evaluation of the explicit coefficients of expansion is rather complicate. In order to overcome the inadequacy, various methods have been developed from different angles [7][8][9][10][11][12][13][14][15][16][17]. Among these investigations, the recently proposed approach based on differential operators is one of the efficient and systematic way [17]. The differential operators introduced by Cheung, Shen and Wen [18] link on-shell tree amplitudes for a series of theories together, and organize them into an elegant unified web 1 . In this unified web, tree amplitudes of other theories can be generated by acting proper differential operators on tree amplitudes of gravity, thus the connections between amplitudes of different theories can be represented by differential equations. With these relations, expansions of tree amplitudes can be arrived via two paths. One is solving the corresponding differential equations, together with considering some physical constraints such as gauge invariance condition. Through the manipulation, the expansion of tree amplitudes for EYM and gravity into tree pure YM amplitudes 2 have been derived efficiently [17]. Another way is getting the expansion of other amplitudes by applying corresponding differential operators on the given expansion of gravitational amplitudes since amplitudes of other theories arise from amplitudes of gravity via differential operators. Along this path, the expansion of BI amplitudes into YM ones have been obtained directly, and the expansion of YM amplitudes into bi-adjoint scalar (BS) ones has also been discussed [17]. It is worth to emphasize that the connections between amplitudes of different theories, which are reflected by differential operators previously, are now established by expansions.
The unified web based on differential operators includes much more theories than gravity, EYM, BI and YM, thus it is natural to expect that the same method can be applied to other theories in the web, and expansions for these theories could also be found. The current short paper devote to apply the method based on differential operators to tree amplitudes of Einstein-Maxwell theory (EM), φ 4 theory, special Yang-Mills-scalar theory (sYMS), as well as Dirac-Born-Infeld theory (DBI). By applying appropriate operators on the expansion of tree gravitational amplitudes, it is simple to obtain the expansion of tree EM amplitudes into KK basis. Then, the expansion of tree φ 4 amplitudes into tree BS amplitudes, the expansion of tree sYMS amplitudes into tree BS amplitudes, and the expansion of tree DBI amplitudes into tree special extended DBI (seDBI) amplitudes, can be readout straightforwardly by acting operators on the expansion of EM amplitudes. Thus the web of the expansions which reflects the deep connections between different theories have been generalized to include EM, 1 A similar web for CHY integrands of various theories has been provided earlier in [6], and the relation between two pictures was established in [19,20]. 2 More precisely, the expansion of amplitudes of EYM and gravity into KK basis. The KK basis will be explained in the next section.
This paper is organized as follows. In section II, we give a brief review of differential operators and the formula of the expansion of gravitational amplitudes into KK basis. These ingredients serve as backgrounds for the work in this paper. With these preparations, we derive the general expression of the expansion of tree EM amplitudes into KK basis in section III. As by products, the expansion of φ 4 amplitudes into BS ones, the expansion of sYMS amplitudes into BS ones, and the expansion of DBI amplitudes into special extended DBI ones, are also provided in this section. Some explicit examples are given in section IV. Finally, we end with a summary and discussions in section V.

II. BACK GROUNDS
In this section, we review some already known results, which are crucial for discussions in later sections. Firstly, we introduce differential operators which are main tools in this paper. Secondly, we review the KK basis of pure YM amplitudes, and the explicit formula of the expansion of tree gravitational amplitudes into KK basis.

A. Differential operators
The differential operators defined by cheung, Shen and Wen, act on variables constructed by Lorentz contractions of external momenta and polarization vectors, establish unifying relations for a variety of theories [18][19][20]. There are three kinds of basic operators • (1) Trace operator: where i is the polarization vector of ith external leg.
• (2) Insertion operator: where k i denotes the momentum of the ith external leg.
• (3) Longitudinal operator: Here the up index means the operators are defined through polarization vectors i . For gravitons with two copies of polarization vectors, operators can also be defined via another independent copỹ i . Four kinds of combinatory operators can be defined as products of these basic operators • (1) For a length-m ordered set α = {α 1 , α 2 , ⋯, α m } of external particles, the operator T [α] is given as which generates the color-ordering (α 1 , α 2 , ⋯, α m ).
• (2) For n external particles, the operator L is defined as Two definitions L andL are not equivalent to each other at the algebraic level. However, when acting on appropriate on-shell physical amplitudes, two combinations L ⋅ T ab andL ⋅ T ab , with subscripts of L i and L ij run through all nodes in {1, 2, ⋯, n} {a, b}, give the same effect and have meaningful physical interpretation.
• (3) For a length-2m set I, the operator T X 2m is defined by • (4) The operator T X 2m is defined as where δ I i k I j k forbids the interaction between particles with different flavors.
The explanation for the notation ∑ ρ∈pair ∏ i k ,j k ∈ρ is necessary. One can divide the length-2m set I into m pairs, then the set ρ of pairs for this partition can be written as with conditions i i < i 2 < ⋯ < i m and i t < j t , ∀t. Under such partitions, ∏ i k ,j k ∈ρ stands for the product of T i k j k for all pairs (i k , j k ) in ρ, and ∑ ρ∈pair denotes the summation over all possible partitions. The combinatory operators above translate the tree amplitudes of gravity into tree amplitudes of other theories. Here we only focus on the generated theories which will be considered later, and list them in Table I

B. KK basis and expansion of gravitational amplitudes
When expanding an amplitude into other amplitudes, a necessary requirement is that the basis is complete and linear independent. Such condition can not be satisfied by all color-ordered tree YM amplitudes with the same external particles, due to the well known Kleiss-Kuijf (KK) relation [21] The KK relation (13) indicates that an arbitrary color-ordered YM amplitude can be expanded into color-ordered YM amplitudes with two external legs are fixed at two ends. In this sense, (n − 2)! amplitudes with two legs fixed at two ends can be chose as the complete basis (so-called KK basis). Actually, the KK basis is not independent to each other due to the Bern-Carrasco-Johansson (BCJ) relations among them [22]. The BCJ relations suggest the truly independent basis which include (n − 3)! amplitudes with three legs are fixed. However, as discussed in [17], the coefficients contain poles when expanding to BCJ basis. If we hope the coefficients are rational functions of Lorentz invariance kinematic variables and all physical poles are included in the basis, the only candidate is the KK basis. In other words, for rational coefficients, the KK basis is independent. With the KK basis introduced above, now we discuss the expansion of tree gravitational amplitudes into such basis. The expansion of gravitational amplitudes into color-ordered YM ones in the framework of ordered splitting is given as [17] where elements in {h} n are labeled as {h} n = {1, 2, ⋯, n}. Some explanations are in order. The ordered splitting for n elements used here is a little different from that used for EYM amplitudes in [13,14,16], which can be defined as follows. First, a reference ordering for elements should be given, for instance n ≺ n − 1 ≺ ⋯ ≺ 1. For the current case, n must be fixed at the lowest position. Once the reference ordering is fixed, the ordered splitting is defined by the ordered set of subsets {Or 0 , Or 1 , ⋯, Or t } subsequently, satisfying following conditions: • Each subset Or l ⊂ {h} n is ordered.
• Or 0 = {1, ⋯, n}, i.e., 1 and n belong to Or 0 and are fixed at the first and last positions respectively. Notice that two gravitons 1 and n can be chosen arbitrary in the set • Denoting h l as the last element in Or l , then h 0 ≺ h 1 ≺ ⋯ ≺ h t according to the reference ordering (it fixes the ordering of subsets Or l in the set {Or 0 , Or 1 , ⋯, Or t }).
• In each subset Or l , h l must satisfy h l ≺ α for arbitrary α ∈ Or l , α ≠ h l , according to the reference ordering. On the other hand, there is no requirement for ordering of all other elements.
The subset Or ′ 0 occurs in (15) is defined by Or ′ 0 ≡ Or 0 {1, n}. With the ordered splitting is given, L l for a length-r subset Or l = {γ 1 , γ 2 , ⋯γ r } is defined as while F 0 for Or ′ 0 = {γ 1 , γ 2 , ⋯γ r } is given by Here f µν γ i stands for the strength tensor f µν i is the sum of momenta of external legs satisfy two conditions: (1) legs at the LHS of the leg i in the color-ordered YM amplitude, (2) legs at the LHS of i in the labeled chain defined by the ordered splitting. The labeled chain used here for a given ordered splitting is the ordered set {1, Or ′ 0 , Or 1 , ⋯, Or t , n}. The summation over shuffles ∑ ¡ was defined previously when introducing KK basis. The summation ∑ {Or l } means summing over all possible ordered splittings.
In the formula (15), the gravitational amplitude is expanded into color-ordered YM amplitudes with two legs are fixed at two ends, thus the coefficients of expansion into KK basis can be readout conveniently. The formula (15) serves as the starting point for computations in the next section.
Before ending this subsection, we emphasize that in the expansion (15), the basis only carry polarization vectors , and all polarization vectors̃ are included in the coefficients.

III. DERIVATION OF EXPANSIONS
In this section, we derive the general formulae of expansions of tree amplitudes. We first apply operators T̃ X 2m and T̃ X 2m on two sides of (15) simultaneously to obtain the ordered splitting formula of the expansion of tree EM amplitudes into YM ones. Then, we propose the rule of getting the coefficients of KK basis. Finally, we explain that coefficients of the expansion of gravitational amplitudes into KK basis are shared by the expansion of φ 4 amplitudes into BS ones, the expansion of sYMS amplitudes into BS ones, as well as the expansion of DBI amplitudes into special extended DBI ones.

A. Expansion of EM amplitudes into YM ones: ordered splitting formula
We start with tree amplitudes of general Einstein-Maxwell theory that photons do not carry any flavor. To generate such amplitudes, one can act the operator T̃ X 2m on gravitational amplitudes, which can be seen in Table I. Now we are going to apply this operator on two sides of (15). On the LHS, the operator T̃ X 2m transmute the gravitational amplitude into the EM one. On the RHS, since basis depend on polarization vectors i and all̃ i are included in coefficients, the operator T̃ X 2m only modifies coefficients. Thus, the action of operator provides the expansion of EM amplitudes into pure YM amplitudes, with the coefficients determined by acting the operator on coefficients in the expansion of gravitational amplitudes.
As a warm up, we first restrict ourselves on the special case that all external particles of the EM amplitude are photons, i.e., 2m = n. Under this condition, subscripts i and j of T̃ ij in T̃ X 2m run over all external particles. To study the effect of T̃ X 2m , let us consider the operator ∏ i k ,j k ∈ρ T̃ i k j k under a given partition ρ. If a term on the RHS of (15) contains all (̃ i k ⋅̃ j k ) with i k , j k ∈ ρ, the operator ∏ i k ,j k ∈ρ T̃ i k j k turns all these (̃ i k ⋅̃ j k ) into 1. Otherwise, the term will be annihilated by ∏ i k ,j k ∈ρ T̃ i k j k . Thus, in each term which can survive under the action of T̃ X 2m , every polarization vector must contract with another one. This requirement indicates that not all ordered splittings are allowed, each subsets in the ordered splitting must take even length according to the observation that each (̃ i k ⋅̃ j k ) only occurs in L l or F 0 . Acting T̃ X 2m on terms correspond to these selected splittings, one can get the non-vanishing contributions. For proper ordered splittings with even length, L l and F 0 contain and (−) respectively. After turning all (̃ i k ⋅̃ j k ) into 1, we get the expansion of the tree EM amplitude as where and If the set {Or ′ 0 } is empty, we have E 0 = 1. Then we turn to the general case that the n-point tree EM amplitude contains 2m photons and (n − 2m) gravitons as external particles. Since two special gravitons 1 and n in (15) can be chosen arbitrary, we assume that both 1 and n are turned into photons. Let us consider a given partition ρ = {(i 1 , j 1 ), (i 2 , j 2 ), ⋯, (i m , j m )} with i 1 < i 2 < ⋯ < i m and i k < j k , ∀k for external photons. Under the action of corresponding ∏ (i k ,j k )∈ρ T̃ i k j k in the operator T̃ X 2m , all non-vanishing terms on the RHS of (15) must contain all (̃ i k ⋅̃ j k ) with (i k , j k ) ∈ ρ. It indicates that each pair in ρ should appear in one subset of the ordered splitting at nearby positions, according to the definition of L l and F 0 . More explicitly, if Or l includes i k , it must take the form {⋯, j k , i k , ⋯} or {⋯, i k , j k , ⋯}. If Or l does not contain j k , or contains j k but i k and j k are not nearby, the corresponding term on the RHS of (15) is annihilated by ∏ (i k ,j k )∈ρ T̃ i k j k . We use the notation Or p l to denote subsets under these proper ordered splittings. With the ordered splittings are determined, now we consider the effect of acting the operator on corresponding coefficients. For Or p 0 , since F 0 includes̃ µ 1 and̃ µ n at the first and last positions, the action of ∏ (i k ,j k )∈ρ T̃ i k j k turns the vectors (̃ 1 ⋅ f a ) µ and (f b ⋅̃ n ) µ into −k µ a and k µ b , respectively. For other pairs in Or p 0 , the tensors where p 0 is the number of photon-pairs in Or p 0 . Thus for the subset Or p 0 we get (−) p 0 −1 G 0 , where G 0 can be obtained from F 0 by the replacement If Or p 0 = ∅, we have (̃ 1 ⋅̃ n ) → 1. For other Or p l , if i k in one pair (i k , j k ) is at the last position of the subset, i.e., appears as {⋯, j k , i k }, then the vector (̃ i k ⋅ f j k ) µ will be turned into −k µ j k . For other cases, the tensors (f i k ⋅ f j k ) µν are turned into −k µ i k k ν j k . Thus, one can obtain (−) p l H l , where H l can be obtained from L l via the replacement Collecting these results together, we find the contribution of an individual splitting can be expressed as thus the full expansion is given by where the summation ∑ {Or p l } ρ is over all ordered splittings correspond to a special partition ρ of photons, and ∑ ρ is over all partitions due to the definition of the operator T̃ X 2m . When all external particles are photons, it can be verified straightforwardly the general formula (26) is reduced to the special one (20).
At the end of this subsection, we discuss the expansion of EM amplitudes that photons carry flavors. For this case, the operator T̃ X 2m is replaced by T̃ X 2m , and a contraction (̃ i k ⋅̃ j k ) is permitted only when two photons carry the same flavor. The constraints from δ I i k I j k lead to the conclusion only partitions satisfy δ I i k I j k = 1 for all i k , j k ∈ ρ provide nonvanishing contributions. Thus, the expansion of these amplitudes in the ordered splitting formula can be obtained from the formula (26) by restricting the summation ∑ ρ on proper partitions.

B. Expansion of EM amplitudes into KK basis
The formula (26) provides the expansion of tree EM amplitudes into pure YM amplitudes in the framework of ordered splitting. To obtain the expansion into KK basis, one can extract coefficients of KK basis with desired color-ordering from the obtained expansion in the ordered splitting formula. An alternative way is reconstructing corresponding ordered splittings from the desired color-ordering in KK basis directly, without requiring the given expansion in the ordered splitting formula, as will be discussed in this subsection. The procedure for the expansion of EYM amplitudes have been provided in [13,14,16]. The manipulation is similar but a little different for EM amplitudes. Now we propose the algorithm.
Assuming the color-ordering in KK basis is (1, 2, 3, ⋯, n − 1, n) 3 , and the reference ordering is chosen to be n ≺ i 1 ≺ i 2 ≺ ⋯ ≺ i n−1 . Subsequently, for a given partition ρ = {(i 1 , j 1 ), ⋯, (i m , j m )}, one can determine the corresponding ordered splittings as follows, • First step: List all possible ordered subsets Or p 0 = {1, γ 1 , γ 2 , ⋯, γ r , n}, respecting the color-ordering in KK basis, i.e., γ 1 < γ 2 < ⋯γ r . In addition, if a photon is included in Or p 0 , its partner in ρ should also be included in Or p 0 , and positions of two photons in Or p 0 are nearby.
• Second step: For each Or p 0 , remove its elements in {1, 2, ⋯, n}. Then, for remaining elements in {1, 2, ⋯, n} Or p 0 , we select the lowest element h 1 in the reference ordering and construct all possible ordered subsets Or p and regarding the color-ordering in KK basis, i.e., γ ′ 1 < γ ′ 2 < ⋯ < γ ′ r ′ < h 1 . The subset Or p 1 also satisfy the condition that each photon contained in Or p 1 occurs nearby its partner in ρ.
• Repeat the second step until the complete ordered splitting is achieved.
All proper ordered splittings for all partitions can be found via the manipulation mentioned above. After generating ordered splittings, the coefficient for the particular YM amplitude A YM (1, 2, ⋯, n) can be obtained by summing factors G 0 ∏ t l=1 H l over all correct ordered splittings and all partitions. Some simple examples will be presented in the next section to illustrate the algorithm more explicitly.
Until now, the expansion of EM amplitudes into KK basis can be formally represented as where σ stands for permutations among (n−2) elements. The coefficients C̃ (σ, m, ρ) depend on polarization vectors̃ , permutation σ, number of the photon-pairs m, as well as allowed partitions ρ. Of course, it also depend on external momenta although the dependence is implicit in the formula (27). Since the dependence on possible partitions, the formula (27) is correct for EM amplitudes whether or not photons carry flavor.

C. Expansions of φ 4 , sYMS and DBI amplitudes
In this subsection, we will identify that the coefficients C̃ (σ, m, ρ) in the expression (27) are also the coefficients in the expansion of φ 4 amplitudes into BS amplitudes, the expansion of sYMS amplitudes into BS amplitudes, as well as the expansion of DBI amplitudes into special extended DBI amplitudes.
Let us come to φ 4 theory whose amplitudes can be generated by acting the operator T [i ′ 1 ⋯i ′ n ] on EM amplitudes A ,̃ EM ({p} n ; ∅) that all external particles are photons without flavor, due to relations in Table I. If one set the LHS of (27) to be A ,̃ EM ({p} n ; ∅), and act the operator T [i ′ 1 ⋯i ′ n ] on two sides of (27) simultaneously, the LHS gives the φ 4 amplitude A φ 4 (i ′ 1 , ⋯, i ′ n ). For the RHS, since the operator T [i ′ 1 ⋯i ′ n ] is defined via polarization vectors i , it only affect on KK basis A YM (1, σ 2 , σ 3 , ⋯, σ n−1 , n) and transmute them into BS amplitudes A BS (1, σ 2 , σ 3 , ⋯, σ n−1 , n; i ′ 1 , ⋯, i ′ n ), as shown in Table I. Then, we get the expansion of φ 4 amplitudes into BS ones as Coefficients C(σ, m, ρ) are products of Lorentz contractions of external momenta, which can be understood as sewing 4-point vertexes of φ 4 theory into 3-point vertexes of BS theory by eliminating propagators.
In the above discussion, if we act the operator T [i ′ 1 ⋯i ′ n ] on EMf amplitudes A ,̃ EMf ({p} 2m ; {h} n−2m ) that photons carry flavors, the generated amplitudes are amplitudes of special YM theory which describe the low energy effective action of coincident D-brans. Thus we also have Notice that the allowed partitions ρ for (29) and (30) are different since constraints from δ I i k I j k for the second one.
Similarly, by acting the operator L ⋅ T [ab] on two sides of (27), one get the expansion of DBI amplitudes into the seDBI ones as If formulae (29), (30) and (31) are correct expansions, basis used in them must be complete and independent. Now we explain that there are KK-like relations among color-ordered BS and seDBI amplitudes, thus this condition is satisfied. As pointed out in [17], the KK relation can be derived by using differential operators. Indeed, one can regard the KK relation as the inference of the algebraical property of the differential operator T̃ [α]. To see this, we rewrite the operator T̃ [α] for length-n set [α] as The operator T α 1 αn ⋅ ∏ k i=2 T α i−1 α i αn generates the color-ordering (α 1 , α 2 , ⋯, α k , α n ) which is equivalent to (α n , α 1 , α 2 , ⋯, α k ) due to the cyclic symmetry, and the operator (−) n−k−1 ∏ n−1 j=k+1 T αnα j α j−1 can be interpreted as inserting {α n−1 , α n−2 , ⋯, α k+1 } between α n and α k in (α n , α 1 , α 2 , ⋯, α k ) [18][19][20]. Setting α n = 1, α k = n, {a} = {α 1 , ⋯, α k−1 }, {b} = {α k+1 , ⋯, α n−1 }, and applying this operator on the n-point gravitational amplitude A ,̃ G ({h} n ), one get the KK relation (13) immediately. Since the algebraical relation (32) is general, it is not surprising that similar relations exist among color-ordered amplitudes of other theories beyond YM. Replacing A ,̃ G ({h} n ) in the above derivation by , the KK-like relations for BS amplitudes and seDBI amplitudes can be obtained as and Thus, BS amplitudes and seDBI amplitudes with the color-ordering (1, {a} ¡{b} T , n) share the completeness and independence of KK basis, therefore can be chosen as proper basis for rational coefficients. Consequently, one can conclude that formulae (29), (30) and (31) are correct expansions for φ 4 , sYMS and DBI amplitudes, respectively.

IV. EXAMPLES
Next, we provide some examples to illustrate the expansions obtained in the previous section. Since the expansion of φ 4 amplitudes into BS amplitudes, the expansion of sYMS amplitudes into BS amplitudes, and the expansion of DBI amplitudes into seDBI amplitudes share the same coefficients with the expansion of EM amplitudes into KK basis, we only consider EM amplitudes in this section.
The formula of expansion depends on the choice of KK basis, i.e., the choice of two special legs which are fixed at two ends in the color-ordering, and the choice of reference ordering. For instance, if we chose the basis A YM (1, σ 3 , σ 4 , 2), and the reference ordering 2 ≺ 3 ≺ 4 ≺ 1, similar manipulation yields As a verification of self-consistency, we need to prove two expressions (35) and (36) are equivalent. We first use the observation that Z 3 = Z 4 = k 1 , together with the momentum conservation law and on shell condition k 2 i = 0, to turn (35) into and (36) into (1, 3, 4, 2) .
Using the ordered reserved identity together with the cyclic symmetry of color-ordering, we have therefore where we employ k 2 ⋅ (k 3 + k 4 ) = −(k 2 ⋅ k 1 ). Then, we apply the cyclic symmetry and ordered reserved identity again to get To continue, we use the well known fundamental BCJ relation to arrive Then we use the fundamental BCJ relation Putting it back to (41) we get Since we finally get thus although seems different, two expressions are indeed equivalent to each other. Now we turn to the case photons carry flavors. Suppose there are two flavors labeled by 1 and 2 of external photons, 1 is carried by 1 and 3, another one 2 is carried by 2 and 4. Then, the corresponding partition is {(1 1 , 3 1 ), (2 2 , 4 2 )}, thus only the splitting {{1 1 , 3 2 , 2 2 , 4 1 }} is allowed, which yields the expansion Until now expansions in this subsection are given in the ordered splitting formula. To get expansions into KK basis, one can identify coefficients of KK basis via the procedure proposed in subsection III B. Since 1 and 4 are fixed at two ends in the color-ordering of KK basis, there are two color-orderings (1,2,3,4) and (1,3,2,4) need to be considered.
where C(2, For these splittings, we have and the expansion in the framework of ordered splitting expresses One can construct ordered splittings for other color-orderings in a similar way. After de-picting G 0 and H l for each splitting, we get coefficients of KK basis as follows Thus, applying the general formula (26), we get the expansion in the ordered splitting formula Then we turn to the case photons carry flavors. Assuming there are two flavors labeled by 1 and 2, 1 is carried by photons 1 and 5, while 2 is carried by 3 and 4. One can find that the proper ordered splittings for the current case has only two candidates Finally, we consider the coefficients of KK basis. We choose the color-ordering (1, 2, 4, 3, 5) as the example to illustrate the algorithm proposed in subsection III B. The recursive construction of ordered splittings can be summarized as follows, At the first step, we drop the candidate Or p 0 = {1, 4, 2, 3, 5} which violates the colorordering (1, 2, 4, 3, 5). Before ending this subsection, we list the coefficients of KK basis for each partition:

V. SUMMARY
We demonstrate how to obtain the expansion of tree EM amplitudes into KK basis of tree YM amplitudes efficiently by applying proper differential operators in this paper. The coefficients for KK basis in the expansion are shared by the expansion of tree φ 4 amplitudes into tree BS amplitudes, the expansion of tree sYMS amplitudes into tree BS amplitudes, as well the expansion of tree DBI amplitudes into tree special extended DBI amplitudes, as have been explained in detail. These expansions exhibit the connections among amplitudes of different theories which are invisible from the angle of Feynman rules, and serve as the dual representations of unifying relations described by differential operators.
The method used in [17] and this paper can also be applied to other theories linked by differential operators. One of our future direction is to derive expansions of other theories via this method and constructe a complete web for expansions.
Interestingly, for expansions of EM amplitudes obtained in the current paper, the manifest gauge invariance is missing for all gravitons 4 . The lose of manifest gauge invariance is a general feature for expansions of amplitudes into KK basis. As discussed in [17], for the expansion of single-trace EYM amplitudes, the manifest gauge invariance for all gravitons can be ensured when expanding into BCJ basis rather than KK basis, with the cost that coefficients contain poles. For EM amplitudes, how to reproduce the manifest gauge invariance for gravitons is a significative problem.
The expansions of amplitudes not only provide the theoretical understanding of connections between different theories, but also benefits the practical calculations. For example, since the evaluation of YM amplitudes is much easier than that of EM amplitudes, one can calculate YM amplitudes at the first step, and get EM amplitudes through the expansions. The obtained EM amplitudes may be used to study the quantum corrections of the behavior of photons in the gravitational field, such as gravitational light bending and Hawking radiation.