Amplitude Relations in Non-linear Sigma Model

In this paper, we investigate tree-level scattering amplitude relations in $U(N)$ non-linear sigma model. We use Cayley parametrization. As was shown in the recent works [23,24] both on-shell amplitudes and off-shell currents with odd points have to vanish under Cayley parametrization. We prove the off-shell $U(1)$ identity and fundamental BCJ relation for even-point currents. By taking the on-shell limits of the off-shell relations, we show that the color-ordered tree amplitudes with even points satisfy $U(1)$-decoupling identity and fundamental BCJ relation, which have the same formations within Yang-Mills theory. We further state that all the on-shell general KK, BCJ relations as well as the minimal-basis expansion are also satisfied by color-ordered tree amplitudes. As a consequence of the relations among color-ordered amplitudes, the total $2m$-point tree amplitudes satisfy DDM form of color decomposition as well as KLT relation.


Introduction
One of the most significant progresses of scattering amplitudes in recent years is the discovery of new amplitude relations. The new relation (BCJ relation) was firstly proposed in Yang-Mills theory by Bern, Carrasco and Johansson [1]. Using BCJ relation in addition with KK relation which was earlier suggested by Kleiss and Kuijf [2], one can simplify the calculations on color-ordered amplitudes at tree level. In particular, these relations provide a reduction of the basis of n-point tree-level amplitudes to a minimal basis of (n − 3)! independent ones [2]. Tree-level amplitude relations in Yang-Mills theory have been studied in both string theory and field theory. In string theory, both KK and BCJ relations can be considered as so-called monodromy relations [3,4]. In field theory, KK relation was firstly proved via new color decompositions [5], while, both KK and BCJ relations have been proved by BCFW recursion [6,7](the proof of KK relation and fundamental BCJ relation can be found in [8] 1 , the proof of general BCJ relation was given in [11]). The minimal-basis expansion has been proved [11] via so-called general BCJ relation.
KK and BCJ relations in Yang-Mills theory can be regarded as results of color-kinematic duality [1]. In [1], it was pointed that one could express the amplitudes by Feynman-like diagrams with only cubic vertices and establish a duality between color factors and kinematic factors. Once the color factors satisfy some algebraic property (antisymmetry and Jacobi identity), so do the corresponding kinematic factors. In fact, KK relation among color-ordered amplitudes can be considered as a result of antisymmetry of kinematic factors, while, BCJ relation is a result of Jacobi identity. The kinematic factors in Yang-Mills theory can be constructed from pure spinor string theory [12]. They can also be constructed by area-preserving diffeomorphism algebra [13,14] or a more general diffeomorphism algebra [15]. A further understanding of the kinematic algebra is the construction of color-dual decomposition and trace-like objects [16,17,18].
It is interesting that KK and BCJ relations can be found not only in pure Yang-Mills theory but also in other theories. For example, relations for amplitudes with gauge field coupled with matter was investigated in [19]. In N = 4 super Yang-Mills theory, the super-amplitudes are also proven to satisfy KK and BCJ relations [20]. In [21], the KK and BCJ relations was proven to hold for color-scalar amplitudes. Though these amplitude relations are found in different theories, they have similar forms with the relations in Yang-Mills theory. This is because the color-kinematic duality implies that different theories with color factors satisfying the same algebraic properties should have the similar form of amplitude relations. When the algebraic properties are changed, the amplitude relations should also be changed. This can be further supported by the amplitude relations in three dimensional supper symmetric theory with 3-algebra [22]. In this case, the algebraic properties of color factors are changed to the properties of 3-algebra, the form of amplitude relations are also changed to agree with the algebraic structure.
Beyond the fundamental field theory, there are lots of interesting low energy effective theories which are also widely used in the phenomenology of low energy physics. One of them is the well-known SU (N ) non-linear sigma model. This theory describes the low energy dynamics of the Goldstone Bosons under the chiral symmetry breaking SU (N ) L × SU (N ) R → SU (N ). In this paper, we focus on the relations of tree-level amplitudes in U (N ) non-linear sigma model. For on-shell amplitudes, the result can apply to the SU (N ) model directly. In recent works [23,24], U (1)-decoupling identity was discussed via the decoupling of U (1) field from interaction, and color-order reversed relation was also pointed in [24]. These results encourage us to study the full amplitude relations in non-linear sigma model systematically. We expect that there should be KK and BCJ relations, which share the same forms with the relations in Yang-Mills theory. This is because the color factors 2 in these two cases satisfy the same algebraic properties. However, the kinematic factors which share the same algebraic properties cannot easy to construct because of the infinity of the number of vertices in non-linear sigma model. The general amplitude relations are also not obvious along the decoupling argument in [23,24]. In fact, the arguments on U (1)-decoupling identity in [23,24] are valid for only on-shell amplitudes. When we consider the even-point off-shell currents constructed by Feynman rules, the U (1)-field under Cayley parametrization [23,24] do not decouple from interaction. This is quite different from in case of Yang-Mills theory where both on-shell amplitudes and off-shell currents satisfy KK relation (the KK relation in off-shell case in Yang-Mills theory was proven in the appendix of [15]). Furthermore, the highly nontrivial relations-BCJ relations seem hard to obtained from this argument. One may hope to prove the relations by using the nontrivial extension of BCFW recursion in non-linear sigma model [23,24] and follow the similar proof within Yang-Mills case [8,11], but it will be not easy to use the Even(odd)-shift form of the BCFW recursion [23,24] to prove even if the simple case-U (1) decoupling identity.
In this work, we will use Berends-Giele recursion 3 under Cayley parametrization to study the relations. Since the odd-point amplitudes vanish [23,24], we only need to study the relations for even-point amplitudes. We conjecture and prove U (1) identity 4 and fundamental BCJ relation for even-point off-shell currents. We will find that, the left hand side of the the U (1) identity and fundamental BCJ relation must equal to terms proportional to (p 2 1 ) 0 , where p 1 is the momentum of the off-shell leg. When we turn our attention to on-shell amplitudes, we should multiply the current by p 2 1 and take the on-shell limit p 2 1 → 0. Then we get the U (1)-decoupling identity and fundamental BCJ relations for on-shell amplitudes. We will leave the proof of general off-shell relations in future work.
We first give some examples then the general proof. In section 4, we will prove the off-shell BCJ relation. We also give examples before general proof. After taking the on-shell limits of the off-shell KK and BCJ relations, we can obtain the U (1)-decoupling identity and fundamental BCJ relation for on-shell amplitudes immediately. In section 5, we use the conclusions of the work [25] to state that all the on-shell general KK and BCJ relations can be generated by the on-shell fundamental BCJ relation as well as cyclic symmetry. Thus the on-shell general KK and general BCJ relations are naturally satisfied. We also point out that the minimal-basis expansion of color-ordered amplitudes, DDM color decomposition and the (2m − 2)! formula of KLT relation for 2m-point total amplitudes are also satisfied. In section 6, we summarize this work. Useful diagrams and convention of notations are included in appendix.

Preparation: Feynman rules and Berends-Giele recursion
In this section, we review the Feynman rules and Berends-Giele recursion in non-linear sigma model which are useful through this paper. Most of the notations follow the recent works [23,24].

Lagrangian
The Lagrangian of U (N ) non-linear sigma model is given as where F is a constant. As in [23,24], we use Cayley parametrization. Under Cayley parametrization U is defined as Here φ = √ 2φ a t a and t a are generators of U (N ) Lie algebra.

Trace form of color decomposition
The total tree amplitudes can be given in terms of color-ordered amplitudes by trace form of color decomposition Since the traces have cyclic symmetry, the color-ordered amplitudes also satisfy cyclic symmetry where momentum conservation has been considered.
There is at least one odd-point vertex for current with odd-point lines(including the off-shell line) and the odd-point vertices are zero. As a result, we have J(2, . . . , 2m + 1) = 0, (2.7) for (2m + 1)-point amplitudes. The currents with even-points in general are nonzero and are built up by only odd numbers of even-point sub-currents. Since odd-point currents have to vanish, in all following sections of this paper, we just need to discuss on the relations among even-point currents.

Off-shell and on-shell U(1) identity from Berends-Giele recursion
In this section, we prove the U (1) identity satisfied by even-point currents. The identity is given as where, on the left hand side, we sum over all the possible permutations with keeping the relative orders in {β} set and there is only one element α 1 in {α} set. On the right hand side, we divide the ordered set {β 1 , . . . , β 2m } into two nonempty subsets. In each subset, there are odd number of β's. For example, if there are six β's, there are three possible divisions When we want to get the on-shell relations between amplitudes from the identity (3.1), we should multiply both sides of (3.1) by p 2 1 = (p α 1 + p β 1 + · · · + p β 2m ) 2 and take the limit p 2 1 → 0. Since the right hand side are products of currents which are finite when p 2 1 goes to zero, after multiplied by p 2 1 , the right hand side has to vanish under p 2 1 → 0. Then we arrive at on-shell U (1)-decoupling identity immediately It is worth comparing the U (1) identities in non-linear sigma model and in Yang-Mills theory. In Yang-Mills theory, U (1)-decoupling identities in both on-shell and off-shell cases have the same form. Thus, in both off-shell and on-shell cases, the identities can be understood as the decoupling of U (1)-gauge field. However, in non-linear sigma model, the U (1) field can only decouple in the on-shell case. In off-shell case, at least for the choice of Cayley parametrization, we get sum of products of two sub-currents. In other words, only when taking the on-shell limit, the U (1) field decouples.
Before proving the identity (3.1), let us have a look at two examples.

Four-point example
In four-point case, the U(1)-identity is This is easy to prove by substituting the four-point vertex into the left hand side directly where 1 is the off-shell line and we have used the on-shell conditions p 2 α 1 = 0, p 2 β 1 = 0, p 2 β 2 = 0.

Eight-point example
Now let us skip the proof of six-point U (1) identity and show how to use lower-point identity to prove eight-point U (1) identity. The eight-point U (1) identity is given as (3.5) To prove this relation, we first show the explicit expression of Fig. 1 and Fig. 2.
• Fig. 1 can be expressed as where we have used the on-shell condition p 2 α 1 = 0. p B i denotes the sum of momenta of the on-shell lines in the set {B i }.
• Fig. 2 can be expressed explicitly by using lower-point U (1) identities By Berends-Giele recursion, we can express the left hand side of eight-point U (1) identity (3.5) by sum of the diagrams in Fig. 7. We can always use (3.7) to reduce sum of the terms with sub-currents containing both α 1 and elements in {β} into products of currents with only β element. Thus the left hand side of (3.5) can be expressed in terms of J( Each subset of this division must containing odd number of β elements because the odd-point current must vanish. We can classify the products of sub-currents into three categories according to different number of sub-currents • two sub-currents: J(β 1 )J(β 2 , . . . , β 6 ), J(β 1 , β 2 , β 3 )J(β 4 , β 5 , β 6 ) and J(β 1 , . . . , β 5 )J(β 6 ). Now let us discuss these contributions one by one.
(i) Six sub-currents: There are three parts of contributions A, B and C in this case.
A part is (A.1) in Fig. 7 and can be given as B part is sum of (B.5), (B.6), (B.7), (B.8) and (B.9) in Fig. 7. Using the property (3.7), this part can be given as C part gets contributions from the diagrams (C.1) and (C.3). Particularly, we apply the property (3.6) to these two diagrams, then we find that the division 3) contribute to this case. C can be expressed as (3.10) Considering all three parts, we find that where we have used the on-shell condition of α 1 .
(ii) Four sub-currents: There are four different products of sub-currents J(β 1 , as an example. The contributions of this case can also be classified into three parts A, B, C. A part is given by (B.1) in Fig. 7 and can be expressed explicitly (3.12) B part get contributions from (C.4), (C.11) and (C.12) in Fig. 7. Particularly, we apply the property (3.7) to (C.4), (C.11) and (C.12). Then (C.11), (C.12) and the division Thus B can be given as 3) contribute to this case. Thus C part is given as (3.14) Taking all three parts into account, we get where we have used on-shell condition of α 1 . Following a similar way, we find that the other products of four sub-currents also cancel out.
Therefore, after considering all the cases (i) (ii) and (iii), we get the U (1) identity (3.5) for eight-point currents.

General proof
Having shown the proof of the eight-point example, let us extend the proof to the general form of U (1) identity. In general, one can always express the left hand side of (3.1) by lower-point sub-currents via Berends-Giele recursion (2.6). As in the eight-point examples, we can collect the diagrams with same off-shell momenta of sub-currents together. Then we can use the property (3.7) to reduce the diagrams containing a substructure of U (1) identity (as shown in Fig. 2). After these reductions, the sub-currents containing both α 1 and {β} elements are reduced to products of sub-currents with only elements in {β} set. Furthermore, we can apply (3.6) to a four-point structure in Then we can collect all the coefficients in the three types in Table 1. In Table 1, we have left a total factor i p 2 . . . J({B 2M }) must vanish. For M = 1, there are only two sub-currents in the products. In this case, we only need to consider the terms with (p 2 1 ) 0 in the diagrams of the form in Fig. 1. We should sum over all the possible {B 1 } and {B 2 } and get which is just the right hand side of the off-shell U (1) identity (3.1).

Off-shell and on-shell fundamental BCJ relation from Berends-Giele recursion
Having proven the U (1) identity, let us consider a more nontrivial relation-fundamental BCJ relation-in non-linear sigma model. Since the odd-point currents and amplitudes must vanish, we only need to consider the relations for even-point currents and amplitudes. Being different from U (1) identity, fundamental BCJ relation has non-trivial coefficients accompanying with the currents or amplitudes. The general formula of off-shell fundamental BCJ relation is given as where we use ξ i to denote the position of the leg i in permutation σ, we define ξ 1 = 0, thus we always have a s α 1 1 in the coefficients for each currents on the left hand side. On the right hand side, we sum over all the possible divisions of the ordered set {β} into two sub-ordered sets {B 1 } and {B 2 }. Since J({B 1 }) or J({B 2 }) must vanish when {B 1 } or {B 2 } have even number, the divisions that survive are those with both odd number of elements in {B 1 } and {B 2 }. Since the right hand side is finite under p 2 1 → 0, after multiplying p 2 1 and taking the on-shell limit p 2 1 → 0 we get the on-shell relation for amplitudes The left hand side of fundamental BCJ relation can be understood as following. We move one external leg α 1 from the position next to the leg 1 to the position in front of the leg β 2m . For each position, we can write down a corresponding current(or amplitude) accompanied by a kinematic factor ξσ i <ξα 1 s α 1 σ i . Then we sum over all the currents with coefficients.
Before giving the general proof of the relation (4.1), let us have a look at two examples.

Four-point example
The simplest example is the four-point fundamental BCJ relation To see this, we write the currents on the left hand side of BCJ relation (4.3) explicitly via Feynman rules where we have used momentum conservation and on-shell conditions of α 1 , β 1 and β 2 . Thus we have proved the fundamental BCJ relation (4.3) at four-point.
To prove this relation, we first show the explicit expressions of Fig. 3 and Fig. 4: • We first consider the sum of the two diagrams in Fig. 3. If we divide the ordered set {β 1 , . . . , β 2m } into two ordered subsets {B 1 } and {B 2 }, then Fig. 3 is given as • Now let us consider Fig. 4. The left hand side of Fig. 4 can be reexpressed by the right hand side of Fig. 4 by considering momentum conservation and on-shell condition of α 1 . Since the first and second terms of the right hand side of Fig. 4 have substructures of fundamental BCJ relation and U (1) identity respectively, we can further reduce them by lower-point relations. Then we have (4.7) A special case is i = 2M − 1. In this case, α i cannot be moved to the position next to the last element of {B 2M −1 }. This case can also be included in Fig. 4 by considering momentum conservation and on-shell condition of α 1 . Thus the property (4.7) also holds.
With the above two properties, one can prove the eight-point fundamental BCJ relation (4.5). We can write the left hand side of eight-point fundamental BCJ relation by lower-point currents via Berends-Giele recursion (2.6). The left hand side of (4.5) is given as sum of the diagrams in Fig. 7. For the diagrams in We can calculate the coefficients for these divisions one by one: (i) Six sub-currents: J(β 1 )J(β 2 )J(β 3 )J(β 4 )J(β 5 )J(β 6 ) = 1. This case get contributions from three parts A, B and C.
A part is (A.1) in Fig. 7 and can be given as  6 } of (C.1). Particularly, this part is given as A part is the contribution of (B.1) in Fig. 7
(iii) Two sub-currents In this case, only the terms that of (p 2 1 ) 0 in (C.1), (C.2) and (C.3) contribute and the sum of these contributions is given as (4.14) After considering all the cases (i), (ii) and (iii), we find that only the productions of two sub-currents are left and this part is just the right hand side of eight-point fundamental BCJ relation.

General proof
Now let us consider the general proof of fundamental BCJ relation (4.1). As shown in the eight-point example, we can always express the left hand side of the relation (4.1) by Berends-Giele recursion (2.6) and collect the diagrams with same off-shell momenta of sub-currents(e.g., for eight point case the diagrams are given by Fig. 7). After applying (4.6) and (4.7), the left hand side of (4.1) can be written in terms of For Type-A diagram in Fig. 8 we can use momentum conservation and on-shell condition of α 1 to rewrite the coefficient in each term into a form independent of momentum of the off-shell line 1. For example, if we consider the diagram with α 1 between {B i } and {B i+1 }, the coefficient is rewritten as The vertex is also written in the form independent of the momentum of off-shell leg. For Type-B diagrams in Fig. 8, we should write down the expression of each diagram by (4.7) and pick out the appropriate division such that we can get {B 1 } . . . {B 2M }. For example, for the first diagram in Type-B in Fig. 8, we should keep the division {B 1 , B 2 } → {B 1 }{B 2 } , for the second diagram we should keep the division {B 2 , B 3 } → {B 2 }{B 3 } and so on. We also write the coefficients and vertices as forms independent of the momentum of the off-shell leg 1 via momentum conservation and on-shell condition of For Type-C diagrams in Fig. 8, we should write down the expression of each diagram by (4.6) and keep the divisions such that we can get {B 1 } . . . {B 2M }. Only the first diagram of Type-C contributes. with arbitrary i and j.    table 3 and the first column  of table 4, are canceled out. The sum of the second column of table 3 is given as , (4.16) while, the sum of the first column of table 4 is given as Since s α 1 β 2i+1 × s α 1 β 2j and s α 1 β 2i × s α 1 β 2j+1 can be related by i ⇔ j, we should interchange i and j in the first column of table 4. Then we can see these two nonzero contributions cancel with each other. Therefore, all the contributions of divisions with M > 1 at last must vanish. For division with M = 1, the ordered set {β} is only divided into two ordered subsets. In this case, we only need to consider the terms of (p 2 1 ) 0 in diagrams shown in Fig. 4 (which is the first term of the second line of (4.6)) with all the possible nontrivial divisions {β} → {B 1 }{B 2 }. The sum of these terms precisely gives the right hand side of the fundamental BCJ relation (4.1).

General KK, BCJ relations, minimal-basis expansion and formulations of total amplitudes
Having proven the U (1)-decoupling identity and fundamental BCJ relation in non-linear sigma model, let us now extend these relations to more general cases. In this section, we first state that the general KK and BCJ relations as well as minimal-basis expansion are all satisfied by color-ordered tree amplitudes. Then we will show that tree-level total amplitudes satisfy DDM form of color decomposition and KLT relation 7 . All these discussions are parallel within Yang-Mills theory, thus we will only present the main points of the statements. Details can be found in the works [25], [11], [5] and [21]. In principle, one can follow the similar steps in sections 3 and 4 to prove the general KK , BCJ relations (5.1), (5.2) for off-shell currents and then take on-shell limits to get the relations among color-ordered onshell amplitudes. However, it is not easy to generalize the off-shell KK and BCJ relations in this way. This is because there are nontrivial products of sub-currents on the right hand side of the relations. When there are more elements in {α} set, the forms of the right hand side may containing both divisions of {α} set and divisions of {β} set. Thus the formulations may become highly complicated.
Fortunately, once we know the fundamental BCJ relation (4.2) in addition with cyclic symmetry (2.4), we have another way to prove the on-shell general KK and BCJ relations. This method was firstly proposed in [25] where general KK and BCJ relations in Yang-Mills theory are generated by so-called primary relations. The main point is that once the amplitudes satisfy a)cyclic symmetry as well as b)fundamental BCJ relation, all the general KK and BCJ relations can be reexpressed as linear combinations of a set of fundamental BCJ relations, and thus the general KK, BCJ relations must hold. Though the discussions in [25] was firstly found by monodromy relations in string theory, as stated in [25], all these arguments can 7 We emphasize that the consequent relations that will be derived in this section are all for on-shell amplitudes. General KK and BCJ relations for off-shell currents will be discussed in future work. be extended to field theory. Since the fundamental BCJ relation(4.2) in non-linear sigma model has the same form within Yang-Mills theory, all the steps in [25] are also valid in non-linear sigma model. Thus the KK and BCJ relations must be satisfied by color-ordered tree amplitudes in non-linear sigma model. Details of this proof can be found in [25].

Minimal-basis expansion
Since KK and general BCJ relations are both satisfied by even-point color ordered tree amplitudes. We are ready now for reduce the number of independent even-point color ordered tree amplitudes as in Yang-Mills theory. Apparently, one can use KK relation in addition with cyclic symmetry to reduce the number of independent 2m-point amplitudes to (2m − 2)!. As in Yang-Mills theory, BCJ relations provide further constraints. One can use general BCJ relations to express the amplitudes in KK basis by only (2m − 3)! independent amplitudes. The explicit formation of minimal-basis expansion is Eq. (4.22) in [1] with 2m external legs. One can follow the same recursive procedure that given by section 4 of the paper [11] to prove the minimal-basis expansion, because we have the general BCJ relation (5.2) of the same form within Yang-Mills theory.

Formulations of total amplitudes
In Yang-Mills theory, amplitude relations imply various formations of total amplitudes. As we have discussed, in non-linear sigma model, event-point color ordered tree amplitudes satisfy KK and BCJ relations, which have the same formations within Yang-Mills theory. Thus we expect that the total amplitudes can have the same expressions within Yang-Mills theory. Particularly, the total amplitudes should satisfy DDM color decomposition as well as KLT relation 8 .

DDM form of color decomposition
An immediate result of KK relation is that the total amplitudes satisfy Del Duca-Dixon-Maltoni(DDM) form of color decomposition which was firstly proven in Yang-Mills theory [5] M (1, . . . , 2m) = σ∈S 2m−2 f a 1 aσ 2 a i 1 . . . f a i 2m−3 aσ 2m−1 a 2m A(1, σ, 2m). We can express any color-ordered amplitude in (2.3) by KK relation, and collect the color coefficients of each amplitude in KK basis. Using the above relation between traces and the color factors in DDM form, we can prove the DDM form of color decomposition(5.3). Details of the proof can be found in [5].

KLT relation
Another result is Kawai-Lewellen-Tye(KLT) relation [29]. In non-linear sigma model, total amplitudes can be expressed in terms of products of two color-ordered tree amplitudes A and A, where A denote the color-ordered tree amplitudes in non-linear sigma model and A denote the color-ordered tree amplitudes of scalar with cubic vertex f abc . As in Yang-Mills theory, the KLT relation has many formations [30,31].
For example the formulation manifests (2m − 2)! symmetries is given as This relation can be proved by following the same steps within the subsection 6.3 of the paper [21]. This is because that the two critical points-the DDM color decomposition and the generalized BCJ relation for color scalar theory-are all satisfied. Another formulation which manifests (2m − 3)! symmetries is given as This formulation seems not easy to prove along the same line in Yang-Mills theory (See section 6.1 of [21]), because the boundary behavior of the amplitudes of non-linear sigma model is not good enough. However, we also expect that the (2m − 3)! formulation have the same form within Yang-Mills theory. In this paper, we just take the four-point KLT relation as an example This is just the DDM form of color decomposition of four-point total tree amplitude. Thus the four-point KLT relation manifest (4 − 3)! = 1 symmetry is proved. We leave the general proof of this formula for future discussion. Though KLT relation was suggested in gravity and then in Yang-Mills theory, it is not surprising that the double-copy formula can also exist in a scalar theory such as non-linear sigma model. An example for KLT relation of scalar amplitudes can be found in bosonic string theory where the closed string tachyon amplitudes at tree level can be expressed by double copy of open string tachyon amplitudes [29]. Actually, the non-linear sigma model also have the similar double-copy structure when considering the color part and the kinematic part as the two copies.

Conclusion
In this work, we have discussed the tree-level amplitude relations in non-linear sigma model. We have proven the off-shell version of U (1) identity and fundamental BCJ relation under Cayley parametrization. After taking on-shell limits, we got the U (1)-decoupling identity and the fundamental BCJ relation for on-shell amplitudes. We stated that the general KK and BCJ relations were also satisfied by even-point tree amplitudes in non-linear sigma model. Two consequent results of KK and BCJ relations were given as the minimal-basis expansion for color-ordered amplitudes and KLT relation for total amplitudes. Though the procedure of proof in this work seems complicated, the relations are quite consistent with the color algebra. We hope these results can be useful in particle phenomenology. The algebraic interpretation of these relations and the dual decompositions of amplitudes deserve further work. In this paper, we use a diagram containing a curved arrow line to denote sum of diagrams for short. Since we encounter similar structures when considering U (1) identity and fundamental BCJ relation, we only use the same diagrams expressions but let the curved arrow line have different meanings for convenience. The meaning of curved arrow line for section 3 and section 4 are given by Fig. 5 and Fig. 6 respectively.