A graphic approach to identities induced from multi-trace Einstein-Yang-Mills amplitudes

Symmetries of Einstein-Yang-Mills (EYM) amplitudes, together with the recursive expansions, induce nontrivial identities for pure Yang-Mills amplitudes. In the previous work [1], we have already proven that the identities induced from tree level single-trace EYM amplitudes can be precisely expanded in terms of BCJ relations. In this paper, we extend the discussions to those identities induced from all tree level multi-trace EYM amplitudes. Particularly, we establish a refined graphic rule for multi-trace EYM amplitudes and then show that the induced identities can be fully decomposed in terms of graph-based BCJ relations (hence traditional BCJ relations).


Introduction
It has been proven that any tree level multi-trace Einstein-Yang-Mills (EYM) amplitude A (m,s) with m gluon traces and s gravitons can be expanded in terms of the amplitudes A (m ,s ) with s + m < s + m [2] recursively. For A (1,s) with only one trace, these relations turn to the earlier proposed recursive expansions of single-trace EYM amplitudes [3][4][5]. When applying the recursive expansions repeatedly until there is no graviton and only one gluon trace in each amplitude, we finally express an arbitrary EYM amplitude as a combination of tree level color-ordered Yang-Mills (YM) ones. Such pure-YM expansions, in cases of amplitudes with only a few gravitons and/or gluon traces, were earlier suggested in [6][7][8][9]. In a recent work [10], the pure-YM expansion of single-trace EYM amplitudes into BCJ basis was studied.
A remarkable result of the expansions of multi-trace EYM amplitudes is that nontrivial identities for color-ordered YM amplitudes can be induced when the gauge invariance conditions or cyclic symmetries of traces are imposed [2]. The coefficients in such induced identities generally contain factors of the Lorentz contractions · , · k and k · k where µ are half polarizations of gravitons and k µ are external momenta. These identities guaranteed the localities in the Britto-Cachazo-Feng-Witten (BCFW) [11,12] proof of the recursive expansions [2,3] and played a crucial role in the proof [13] of the equivalence between distinct approaches [14][15][16] to nonlinear sigma model amplitudes. On another hand, color-ordered YM amplitudes also satisfy Kleiss-Kuijf (KK) [17] and Bern-Carrasco-Johansson (BCJ) relations [18] which does not involve any · or · k factor.
Although the identities induced from multi-trace EYM amplitudes have quite different forms from the BCJ relations, it seems that they can be related with each other: (i). First, the relationship between the identities induced from single-trace EYM amplitudes and BCJ relations have already been founded [1], while the multi-trace amplitudes can be obtained through replacing gravitons by gluon traces in an appropriate way [2]. (ii). Second, explicit examples in [2] support that the identities induced from multi-trace EYM amplitudes can be expanded in terms of BCJ relations. (iii). Third, the fact that color-kinematic duality [18], which is the structure behind BCJ relations, can be resulted from gauge invariance [19] implies that the identities induced from gauge invariance can also be related with BCJ relations. Nevertheless, the full relationships between identities induced from multi-trace EYM amplitudes and BCJ relations are still unclear yet.
In this paper, we complete our study on identities induced from an arbitrary multi-trace EYM amplitude. We show that all these induced identities can be expanded in terms of BCJ relations. The main idea is sketched as follows. To incorporate more gluon traces, we introduce the refined graphic rule which involves more line styles and more types of components than that of the identities induced from single-trace amplitudes [1]. Then we show that all graphs for the identities induced from multi-trace amplitudes can be obtained by connecting two mutually disjoint connected subgraphs (which are called the final upper and lower blocks) via a line corresponding to k ·k factor. All those graphs corresponding to a same configuration of the final upper and lower blocks together contribute a combination of the graph-based BCJ relations [1] 1 which can be further written in terms of the traditional BCJ relations.
The structure of this paper is following. In section 2, we review the expansions of multi-trace EYM amplitudes, BCJ relations and induced identities, which are helpful for the full paper. The refined graphic rule is given in section 3. Two instructive examples are displayed in section 4 and the general pattern is given in section 5. To avoid tedious discussions, we just describe the pattern by emphasizing the new features of identities induced from multi-trace amplitudes without a proof. We claim that the general proof of the pattern in section 5 can be obtained by a parallel discussion with [1]. Conclusions and further discussions are presented in section 6. The proof of a helpful relation is given in appendix A.

Expansions of multi-trace EYM amplitudes, induced identities and BCJ relations
In this section, we review the recursive/pure-YM expansions of multi-trace EYM amplitudes, the induced identities and the BCJ relations for YM amplitudes. With these notations, one can write down the following recursive expansion for tree level multi-trace EYM amplitudes [2]:

Expansions of multi-trace EYM amplitudes
if H a = t t t 0 is a gluon trace: i.e., we sum over all possible choices of the ordered pair of gluons {a i , b i } ⊂ t t t i for all traces t t t i ∈ H H H A . If the fiducial element H a is also a trace, namely t t t 0 , we should further fix an arbitrary gluon b 0 ∈ t t t 0 (we call it fiducial gluon), then sum over all choices of a 0 = b 0 in this trace. Since a gluon trace has cyclic symmetry, it can always be written in the form a i , X X We also sum over permutations β β β i (for a given {a i , b i }) and β β β 0 (for a given a 0 = b 0 ) which satisfy Here, X X X i and Y Y Y i are the two ordered sets of gluons separated by a i and b i ) for the given choice of pair Similarly, if the fiducial element H a is also a trace, say t t t 0 , it must be replaced by a permutation {a 0 , β β β 0 , b 0 } where β β β 0 satisfying eq. (2.5). In other words, each trace in eq. (2.6) is replaced by a permutation of KK basis with fixed two ends. Then the summation over σ σ σ means summing over all possible shuffle permutations in eq. (2.6) after this replacement. , H a } and given σ σ σ satisfying eq. (2.6), the coefficient C(1, σ σ σ, r) in eq. (2.1) is written as where [Eq:EFY3] (2.10) The notation σ −1 (x) denotes the position of x in the permutation σ σ σ 3 and the gluon 1 is always considered as the leftmost element in σ σ σ.
The recursive expansion eq. (2.1), where the fiducial element H a is chosen as a graviton or a gluon trace, corresponds to the type-I or -II relation in [2]. If there is only one trace, the relation eq. (2.1) turns to the expansion of single-trace EYM amplitudes [3]. As examples, we display the two types of expansions for the double-trace EYM amplitude A(1, 2 . . . r|2 2 2 h 1 ) with one graviton explicitly: where the factor (−1) |2 2 2,a 2 ,b 2 | and the sum over β β β 2 ∈ KK[2 2 2, a 2 , b 2 ] are hidden in the summation notations with a tilde.
When the recursive expansion eq. (2.1) is applied repeatedly, one finally expands the multi-trace EYM amplitude in terms of pure YM ones [2]. Instead of collecting the coefficients for amplitudes in each permutation (as shown in [2][3][4]), we introduce a distinct form where amplitudes corresponding to each graph is collected: Here, we summed over all possible graphs which will be defined in the next section.

Identities induced from multi-trace EYM amplitudes
Symmetries of multi-trace EYM amplitudes, together with the expansions eq. (2.11), induce identities for EYM amplitudes with fewer gravitons and/or traces. There are two symmetries under consideration in this paper: the gauge invariance and the cyclicity of any trace.
Muti-trace EYM amplitudes satisfy gauge invariance condition, which states that the amplitudes have to vanish once half polarization hx of a graviton h x is replaced by its momentum k hx . If this replacement is performed on the RHS of the recursive expansion eq. (2.1), we should consider two distinct situations: Therefore, the only nontrivial identity induced by the gauge invariance condition of a graviton is eq. (2.12) which is called type-I identity in [1].
Another identity (called type-II identity in [2]) is induced from the type-II expansion where the fiducial element H a is a gluon trace t t t 0 . In particular, we notice that the fiducial gluon b 0 ∈ t t t 0 can be chosen arbitrarily in the fiducial trace. This arbitrariness is essentially caused by the cyclicity of a trace [ where σ σ σ is defined by (2.14) Here we remove b 0 from the trace t t t 0 first, then shuffle these permutations as eq. (2.6).
When the recursive expansion eq. (2.1) for amplitudes with fewer gravitons and/or gluon traces are applied repeatedly, the two types of relations eq. (2.12) and eq. (2.13) induce nontrivial relations between pure YM amplitudes, which will be studied in detail in the coming sections.
In [1], the following graph-based BCJ relation for YM amplitudes was proposed Here, γ γ γ is an arbitrary permutation of elements in {2, . . . , r −1} and T is an arbitrary connected tree graph. When a node a is chosen as the leftmost element, the tree graph T establishes permutations ζ ζ ζ ∈ T | a as follows (i). For two adjacent nodes x and y, if x is nearer to a than y, we have ζ −1 (x) < ζ −1 (y), (ii). If there are subtree structures attached to a same node, we should shuffle the permutations established by these subtrees together. The factor f a is a relative sign depending on the node a. This factor is determined as follows: (i). Choose an arbitrary node c and require f c = 1, (ii). For arbitrary two adjacent nodes c 1 and c 2 , we have f c 1 = −f c 2 . As already proven in [1], the graph based BCJ relation eq. (2.16) can always be written as combination of the traditional ones eq. (2.15).

Refined graphic rule for multi-trace EYM amplitudes
In this section, we introduce the refined graphic rule for the expansion eq. (2.11).  Step-1 Define a reference order of elements in H H H ≡ (Tr Tr Tr s \ 1 1 1) ∪ H: where the position of each element (i.e. graviton or gluon trace) is called the weight.
Step-2 We consider elements (i.e. gravitons and gluon traces) in H H H ≡ (Tr Tr Tr s \ 1 1 1) ∪ H and the gluons in the trace 1 1 1 as nodes. A chain, which starts from the highest-weight element H ρ(l) , passes through some other elements j 1 , ..., j u (j 1 , . . . , j u ∈ H H H \ H ρ(l) ) an ends at w ∈ R = 1 1 1 \ {r} = {1, . . . , r − 1} can be constructed Here, H ρ , the elements j 1 , ..., j u and w are correspondingly called the starting, internal and ending elements of this chain. Redefine the ordered set R by removing the elements which have been used: and redefine the set R: where each gluon trace in {j 1 , . . . , j r , H ρ(l) } is treated as a single object.
Step-3 Repeat the above step by using the new defined R and R until the ordered set R is empty. We then get a tree graph.
Step-4 Refine all the graphs constructed by the above steps via replacing gravitons and gluon traces by  Fig. 4 (d)) whose arrow points to a i . We also connect adjacent gluons in the trace 1 1 1 = {1, 2, . . . , r} by type-4 lines whose arrows point to the gluon 1. (iv). If a trace plays as the ending element of a chain, it means any gluon belonging to this trace can be the ending node.
Step-5 For a given graph F, the coefficient C F in eq. (2.11) can be read off as the product of all factors corresponding to the type-1,-2 and -3 lines (see Fig. 3). The sign associated to such a graph gets two distinct contributions (i). (−1) N (F ) , where N (F) is the number of arrows pointing away from the root 1; (ii). Each trace t t t i contributes a (−1) |t t t i ,a i ,b i | for given a i and b i .
Step-6 Collect amplitudes A(1, σ σ σ F , r) for a given graph F. In any graph, the gluons 1 and r in the trace 1 1 1 = {1, 2, . . . , r} are always treated as the first and the last elements. The gluon 1 can be considered as the root of any graph. Permutations σ σ σ F are determined as follows: (i). If two adjacent nodes x and y are connected by a line (of any style), they must live on a path towards the root 1. If x is nearer to 1 than y on this path, we have (σ σ σ F ) −1 (x) < (σ σ σ F ) −1 (y). (ii). If there are several branches attached to a node, the relative order is defined by shuffling the branches together.
When summing over all possible graphs constructed by the above steps, (i.e., (i). summing over all graphs with given {a i , b i } pairs and given permutations β β β ∈ KK[t t t i , a i , b i ] for all traces, (ii). summing over all possible permutations β β β ∈ KK[t t t i , a i , b i ] for given {a i , b i } pairs in all traces, (iii). summing over all possible choices of the {a i , b i } pairs for internal traces and all possible choices of a i ∈ t t t i , a i = b i for starting traces), we finally arrive the expansion eq. (2.11).

Expressing the induced identities by refined graphs
Having shown the refined graphic rule, we are now ready to express the induced identities eq. (2.12) and eq. (2.13) as identities between pure YM amplitudes. We just need to replace the highest-weight element (i.e. the fiducial element in the first-step recursive expansion) in the following way: • If the highest-weight element H ρ(l) in the reference order eq. (3.1) is a graviton h a , the identity eq. (2.12), which is induced from the gauge invariance condition of h a , implies: The corresponding graphs can be obtained by the replacement Fig. 5 (a).
• If the highest-weight element H ρ(l) in the reference order eq. (3.1) is a gluon trace t t t 0 , the identity Figure 5. The type-I induced identity eq. (3.13) is obtained by replacing ha by k ha (shown by (a)) in eq. (2.11), where the graviton h a is the highest-weight element in the reference order. The type-II induced identity eq. (3.14) is obtained by performing the replacement (b) on eq. (2.11). The sum over all β β β ∈ KK[t t t 0 , a 0 , b 0 ] with the sign (−1) |t t t0,a0,b0| can be expressed by (c), where the full trace t t t 0 is written as This means that we consider {a 0 , KK[t t t 0 , a 0 , b 0 ]} (for a 0 ∈ t t t 0 ), which does not contain b 0 , as possible permutations of elements in the trace (see Fig. 5 (b)) and then apply the refined graphic rule, the total contribution must vanish. There is an interesting property: when we sum over β β β ∈ KK[t t t 0 , a 0 , b 0 ] with the sign (−1) |t t t 0 ,a 0 ,b 0 | , the trace after the replacement In the following, we always use the RHS of Fig. 5 (c) to stand for the highest-weight element in the type-II identity eq. (3.14).

Induced identities as combinations of BCJ relations: examples
In the previous section, we have introduced the refined graphic rule, by which, the induced identities eq. (3.13) and eq. (3.14) are expressed as summations over all possible refined graphs. Now we take the type-I and type-II identities, which are induced from the double-trace EYM amplitude A(1, · · · , r|2 2 2||h 1 ) (eq. (3.13)) with one graviton, as explicit examples. Through these examples, we will see identities of both types are combinations of graph-based BCJ relations (thus traditional BCJ relations).

Example-1: the type-I identity induced from
When the reference order is chosen as R = {2 2 2, h 1 } and h 1 is replaced by k h 1 , the expansion eq. (2.11) for the double-trace amplitude A(1, 2, · · · , r|2 2 2 h 1 ) induces a type-I identity eq. and (c) with the replacement h 1 → k h 1 . The total contribution for these graphs then reads where σ σ σ (a) , σ σ σ (b) and σ σ σ (c) are already given in eq. (3.5), eq. (3.6) and eq. (3.7) respectively. Then the full expression of the RHS of the identity (3.13) induced from A(1, · · · , r|2 2 2 h 1 ) is given by To investigate the relationship between the induced identity and BCJ relations, we define standard basis set for a gluon trace i i i by the set of permutations   where each graph stands for its full contribution and recall that a summation with tilde is defined by, e.g., The LHS is an internal trace structure defined by the refined graphic rule. On the RHS, the trace i i i is expressed by the standard basis with one end c i ∈ i i i fixed. The node a i (b i ) on both sides of eq. (4.5) must be connected to a same node outside the trace i i i via a line of the same style (i.e., type-2 or type-3 line). We leave the proof of eq. (4.5) in the appendix. Now we apply the relation eq. (4.5) to the first term of eq. (4.4) for a given l. Since the c i in eq. (4.5) can be chosen arbitrarily, we just choose the c i ∈ i i i (in this example i i i = 2 2 2) as the fiducial gluon c 2 in the starting trace of Fig. 6 (b) and (c). Then the summation T (a) in the first term of eq. (4.4) splits Figure 8. The I 1 defined by eq. (4.7) can be expanded in terms of graph-based BCJ relations with respect to graphs of the structure T 1 . into where T (a1) and T (a2) are the corresponding contributions of Fig. 7 (a1) and (a2). Substituting eq. (4.6) into eq. (4.4) and introducing the contribution of spurious graph Fig. 7 (a3) by 0 = T (a3) −T (a3) , we rewrite eq. (4.4) as the sum of I 1 and I 2 which are respectively defined by and Let us analyze I 1 and I 2 separately: • For I 1 , we find that the expression in the parenthesis in eq. (4.7) can be obtained by summing over all those graphs which are constructed by connecting an arbitrary gluon l ∈ {1, 2, . . . , r − 1} = 1 1 1 \ r with an arbitrary node a ∈ T 1 (T 1 is the tree graph shown by Fig. 8) via a type-3 line. The kinematic factor for such a graph is (k h 1 · k b 2 )(k a 2 · k l ). Given b 2 ∈ 2 2 2 (b 2 = c 2 ) in eq. (4.7), there is an overall sign (−1) |i i i,a i ,b i | which has been extracted out and absorbed into the summation notation with a tilde. The sign for any term inside the square brackets is collected as follows: (i). According to the refined graphic rule, any graph F with N (F) arrows pointing away from the root 1 is associated by a sign (−1) N (F ) . (ii). Each of (a1) and (a3) has an extra sign (−1). Based on this analysis of signs, we have the following observations: (1). Once a ∈ T 1 has been chosen, the sign is independent of the choice of l ∈ {1, 2, . . . , r − 1} = 1 1 1 \ r and all permutations established by the graph with any l satisfy {1, σ σ σ ∈ {2, . . . , r − 1} ¡T 1 | a , r}.
Hence we can collect the coefficients (k a 2 · k l ) together as (−k a · Y a (σ σ σ)) for each σ σ σ. (2). On another hand, two graphs with adjacent a ∈ T 1 are associated by opposite signs.
Thus I 1 turns to where (−1) F |x 0 denotes the sign for the graph with a = x 0 ∈ T 1 . The f a for any a ∈ T 1 is fixed by (i).
) if x 1 and x 2 are two adjacent nodes. Therefore, the expression in the square brackets is just the LHS of the graph-based BCJ relation B(1|T 1 , {2, · · · , r − 1}|r). Consequently, I 1 is a combination of BCJ relations (as proven in [1], all graph-based BCJ relations are combination of traditional BCJ relations eq. (2.15)).
• For I 2 , all the permutations established by the graphs Fig. 6   Coefficient for each permutation σ σ σ ∈ {h 1 } ¡ γ γ γ is collected as (k a 2 · k l )(k h 1 · Y h 1 (σ σ σ)) (with a minus sign). Hence I 2 turns to in which γ γ γ satisfies eq. (4.10). Apparently, the expression in the square brackets is just a special case (which is usually called fundamental BCJ relation) of the LHS of BCJ relation eq. (2.15). (If we consider the single node h 1 as a tree T 2 , the expression can also be understood as the LHS of a graph-based BCJ relation B(1|T 2 , {2, · · · , r − 1}|r)). Figure 10. The I 1 and I 2 in eq. (4.14) can be expanded in terms of graph-based BCJ relations with graphs of the structures T 3 and T 4 respectively.

Example-2: the type-II identity induced from
When the reference order for the expansion eq. (2.1) of the double-trace EYM amplitude A(1, 2, · · · , r|2 2 2 h 1 ) is chosen as R = {h 1 , 2 2 2} (i,e., the trace 2 2 2 is chosen as the highest-weight element), the arbitrariness of the choice of fiducial gluon in this trace induces a type-II identity eq. (3.14). All graphs for this identity are constructed by (i). first removing the fiducial gluon from the highest-weight trace 2 2 2 (this is equivalent to the replacement Fig. 5 (c)); (ii). then applying the refined graphic rule. Supposing that all gluons in the trace 2 2 2 are in the cyclic order d 1 , d 2 , . . . , d |2 2 2| and the gluon d |2 2 2| is chosen as the fiducial one, typical graphs for the type-II induced identity are then presented by Fig. 9 (a), (b), (c), (d) with the following contributions: Once all graphs are summed over, we arrive the RHS of the type-II induced identity eq. (3.14) for the amplitude A(1, 2, · · · , r|2 2 2 h 1 ): (4.14) where contributions of all graphs of the form Fig. 9 (a), (b) and (c), (d) were collected as I 1 and I 2 respectively. Now we analyze I 1 and I 2 separately.

Comments on examples in section 4
To close this section, we comment on some crucial features of the examples: (i). Expanding traces which are not the highest-weight element by standard basis In example-1, the trace 2 2 2 is not the highest-weight element in the reference order R. It plays as an internal trace in Fig. 6 (a) and a starting trace in Fig. 6 (b), (c). In the latter case, the trace 2 2 2 was already expressed by standard basis, while in the former case, it is expanded according to the relation eq. (4.5). The fixed node for the expansion of the trace 2 2 2 in Fig. 6 (a) was conveniently chosen as the fiducial node in Fig. 6 (b) and (c).

(ii). Skeletons and components
We define skeletons by removing all type-3 lines from the graphs where all traces, except the highest-weight element in R (if it is a trace), are already expressed by the standard basis. Since each graph defined by the refined graphic rule is a connected tree graph, its skeleton must be a disconnected one. Each maximally connected subgraph of a skeleton is called a component. A typical skeleton in example-1 is given by Fig. 11 (a) (for b 2 ∈ 2 2 2, b 2 = c 2 ) which consists of three components. Skeletons in example-2 have two distinct structures, Fig. 11 (b) and (c), each of which has two components. From the examples, we can see any skeleton must has at least two components which involve the highest-weight element (graviton or trace) and the trace 1 1 1 respectively.
(iii). The final upper and lower blocks Any graph in the examples can be reproduced by connecting a type-3 line between the final upper and lower blocks which are two mutually disjoint connected subgraphs containing the highest-weight element and the trace 1 1 1 respectively. In example-2, each of the skeletons Fig. 11 (b) and (c) already consists of only two disjoint connected subgraphs U and L which serve as the final upper and lower blocks. In example-1, there are three components in a skeleton Fig. 11  (a). A typical configuration of the final upper and lower blocks is constructed when we connect any b 2 (b 2 ∈ 2 2 2, b 2 = c 2 ) in Fig. 11 (a) to either (i) h 1 (see Fig. 12 (a)) or (ii) an element in {1, . . . , r − 1} via a type-3 line (see Fig. 12 (b)).
(iv). Physical and spurious graphs For a given configuration of the final upper and lower blocks U and L , we can connect two nodes x ∈ U and y ∈ L via a type-3 line. Then a fully connected graph is constructed. In example-2, the graphs Fig. 9 (a), (b) (and Fig. 9 (c), (d)) are reproduced by connecting the final upper and lower blocks in Fig. 11 (b) (and Fig. 11 (c)) via a type-3 line. Similarly, the graphs Fig. 7 (a1), Fig. 6 (c) (and Fig. 7 (a2), Fig. 6 (b)) in example-1 are constructed from Fig. 12 (a) (and (b)). All the graphs Fig. 6 (b), (c), Fig. 7 (a1), (a2) and Fig. 9 (a)-(d) are graphs directly defined by the refined graphic rule (in standard basis). These graphs are called physical graphs. The spurious graph Fig. 7 (a3), which is not defined by refined graphic rule, can be reproduced from either Fig. 12 (a) (connecting b 2 ∈ U with l ∈ L ) or Fig. 12 (b) (connecting h ∈ U with b 2 ∈ L ). The former case is associated with a minus to ensure the cancellation of the same skeleton constructed by distinct ways. Therefore, for any skeleton, the sum of all physical graphs can be given by (1) summing over all possible configurations of the final upper and lower blocks U , L , (2) for a given U and L , connecting two nodes x ∈ U and y ∈ L \ {r} via a type-3 line and summing over all possible choices of x, y.
(v). Induced identities as combinations of graph-based BCJ relations A crucial observation on the examples is that the sum over all graphs corresponding to a same configuration of the final upper and lower blocks (i.e. the I 1 and I 2 in either example) are combinations of graph-based BCJ relations (thus traditional BCJ relations).

Induced identities as combinations of BCJ relations: the general pattern
In this section, we extend our discussions to (both type-I and type-II) identities induced from arbitrary multi-trace EYM amplitudes. Inspired by the examples, we rewrite the sum over all physical graphs (graphs directly defined by the refined graphic rule) F in eq. (3.13) and eq. (3.14) by (1) summing over skeletons F where traces (except the highest-weight element if it is a trace) are expanded by standard basis, (2) summing over all physical graphs F for a given skeleton F . The RHS of the induced identities eq. (3.13) and eq. (3.14) are then uniformly arranged as  where P [F ] is the factor containing polarizations (i.e. product of factors of the form · and · k) for the skeleton F and K [F \F ] stands for the the product of all type-3 lines for the graph F. The overall signs (−1) |t t t i ,a i ,b i | which are introduced by all traces t t t i (except the fiducial trace t t t 0 if it is the highest-weight element in the reference order because this trace is already replaced according to Fig. 5 (c)) are absorbed by the summation notation F . The factor (−) F is the remaining sign associated to this graph. All amplitudes for the graph F are those with the permutations 1, σ σ σ F , r, where σ σ σ F denotes permutations established by F. Once the relationship between the expression in the square brackets eq. (5.1) (for any skeleton F ) and BCJ relations have been founded, we know the full relationship between the induced identities eqs. (3.13), (3.14) and BCJ relations. In the remaining part of this section, we classify components and show the proper construction rule for the final upper and lower blocks (as mentioned in the previous section) U and L which are two mutually disjoint connected graphs. We then show that the expression in the square brackets I[F ] for a given skeleton F can be written as where I[ F | U , L ] for a given configuration of U and L is a combination of the graph-based BCJ relations. The U ⊕ L denotes the union of the mutually disjoint U and L . For conciseness, the proof of the general pattern will not be presented. We claim that this pattern can be proven following a parallel discussion in the single-trace case [1] but with the enlarged classes of components.

Classification of components
As shown by our previous work [1], components for identities induced from single-trace amplitudes can be classified according to their internal structures. This classification are extended to multi-trace case when more internal structures are incorporated: • Type-I components Components containing a type-1 line and possible type-2 lines pointing towards the two ends of the type-1 line are called type-IA components (see Fig. 13 (a)). Components containing a gluon trace in the standard basis, with possible type-2 lines pointing to the nodes of this trace, are called type-IB components (see Fig. 13 (b)). The latter is a new feature for identities induced from multi-trace amplitudes, while the former already exists in the discussions on identities induced from single-trace amplitudes (see [1]). The kernel is defined by the type-1 line in a type-IA component and the type-4 line (i.e. the dashed arrow line) that is attached to the ending node of the trace in standard basis in a type-IB component. For a given reference order, any type-IA and -IB component is divided into two parts by the kernel: the part involving the highest-weight node (although a trace is considered as a single object in the reference order, the fixed node c is always considered as the highest-weight node of this trace and it carries the weight of the full trace in the reference order) of this component is called the top side, while the opposite side is called the bottom side.
• Type-II and Type-III components In any graph for the induced identities, there always exists a component that contains the highest-weight element (graviton or gluon trace) in the reference order. If the highest-weight element is a graviton (gluon trace) this component is called the type-IIA (type-IIB) component (see Fig. 14 (a), (b)). The component involving the trace 1 1 1 is called the type-III component (see Fig. 14 (c)).
Apparently, both type-II and type-III components appear once and only once in a skeleton. Type-I components may also be involved in a skeleton. All graphs corresponding to a given skeleton are reproduced by connecting type-3 lines between components in an appropriate way.

The construction of the final upper and lower blocks
Having classified the components, we are now ready for constructing all configurations of the final upper and lower blocks U , L in eq. (5.2): • Step-1 For any skeleton F , we define the reference order R C of all type-IA and -IB components by the relative order of the highest-weight nodes therein. We also define the upper block U and lower block L as the components containing the highest-weight element (graviton or trace) and the trace 1 1 1 respectively. At the beginning, the upper and the lower blocks are nothing but the type-II and the type-III components. • Step-2 Supposing the reference order of components is R C = {C 1 , C 2 , . . . , C N }, pick out the highestweight component C N as well as arbitrary components C a 1 , C a 2 , ..., C a i (the relative order of these components is not necessary the same relative order in R C ). Construct a chain of components towards either the upper block or the lower block as follows , the two ends x and y of the type-3 line between the components C N and C a i must belong to (C N ) b and (C a i ) t respectively. After this step, we redefine the reference order of components as well as the upper and lower blocks by: Step-3 Repeating step-2 with the new defined R C , U and L until the ordered set R becomes empty, we get a graph with only two mutually disjoint subgraphs: the final upper and lower blocks.

Induced identities as combinations of graph-based BCJ relations
For a given configuration of the final upper and lower blocks U and L , a fully connected graph is produced by connecting a type-3 line between arbitrary two nodes x ∈ U and y ∈ L \ {r}. Such a graph is either a physical or a spurious one. As shown by Fig. 15, a spurious graph has the structure in which the path starting from the highest-weight element and ending at the root passes through single sides of some type-IA or/and type-IB components (for convenience, such a component is called a spurious component). single-trace amplitudes, T r(U x , L ) vanishes and the sign eq. (5.5) turns to the one given by [1].
When all factors P [F ] are extracted from (both physical and spurious) graphs F with the same skeleton F , the remaining contributions of all F ⊃ F then becomes eq. (5.2). If all the (k · k) factors (i.e. the factor K [U ⊕L \F ] ) for a given configuration of the final upper and lower blocks are also extracted out, we further get I[ F | U , L ] (in eq. (5.2)), which is expressed by combination of amplitudes associated with a factor k x · k y (x ∈ U , y ∈ L \ {r}) and a sign eq. (5.5). Permutations in amplitudes are those established by the tree graphs. Therefore, I[ F | U , L ] can be written as where the sum over all permutations established by a tree graph F (for the final upper and lower blocks U , L and given x ∈ U , y ∈ L ) split into three summations: (i). the sum over all relative orders γ γ γ established by the final lower block L where 1 is considered as the leftmost node, (ii). the sum over all relative orders ζ ζ ζ established by the final upper block U (when x ∈ U is considered as the leftmost node), (iii). the sum over all those permutations σ σ σ ∈ ζ ζ ζ ¡γ γ γ such that σ −1 (y) < σ −1 (x) is satisfied.
To relate the I[ F | U , L ] with the graph-based BCJ relation eq. (2.16), we rearrange eq. (5.6) based on the following observations: (i). For a fixed x ∈ U and a given permutation σ σ σ, only those graphs with σ −1 (y) < σ −1 (x) contribute nonzero factors k x · k y (with the same sign (−) F ). Thus the total factor for permutation σ σ σ is collected as k x · Y x (σ σ σ) with the same sign. (ii). As shown by Fig. 16, two graphs with x = x 1 and x = x 2 , where x 1 , x 2 ∈ U are two adjacent nodes, must be associated with opposite signs. Then eq. (5.6) is finally written as where the summation over γ γ γ was extracted out because it is independent of the choice of x ∈ U . The factor (−) F |x 0 is the sign eq. (5.5) for an arbitrary x 0 ∈ U . The sign f x is defined by (i). 1 for x = x 0 , (ii). f x 1 = −f x 2 for adjacent x 1 and x 2 (x 1 , x 2 ∈ U ). Therefore, the expression in the square brackets in eq. (5.7) is nothing but the LHS of the graph-based BCJ relation: B(1|U, γ γ γ|r) where U is the final upper block.

Conclusions
In this paper, we have shown that the identities eq. (3.13) and eq. (3.14) induced from arbitrary tree level multi-trace EYM amplitude can be expressed as a combination of graph based BCJ relations (thus traditional BCJ relations). To extend the discussions on identities induced from single-trace amplitudes [1] to arbitrary multi-trace case, we enlarged the set of graphs by including more structures (i.e., more line styles and more types of components) in the refined graphic rule. The relation eq. (4.5) which expands internal traces by standard basis played a key role in this approach.
There are several directions that deserve further study: (i). How to understand the induced identities from string theory? A relationship between string theory and the Cachazo-He-Yuan [27][28][29][30] version of field theory was established in [31,32]. This work may provide an effective method for the study at string theory level. (ii). In [33], a unified web of expansions of amplitudes was established, which inspires that the induced identities exist in various theories. It is worth studying identities induced from other theories. (iii). Loop-level extension is another interesting direction.
A Proof of eq. (4.5) To prove eq. (4.5), we focus on a term with a given choice of {a i , b i } ⊂ i i i in eq. (4.5) and prove the following stronger relation: with a sign (−1) |i i i,a i ,b i | = (−1) j−1 (the overall sign coming from T | x has been neglected). Here x denotes the nearest to b i node in T and T | x are the permutations established by the tree T . The notation b i ≺ x means the position of b i in the permutation is less than that of x. The second line of eq. (A.2), where we shuffled T | x with the full trace and required the node b i is always on the left of the node x, is apparently equivalent to the first line.
On the RHS of eq. (A.1), we assume that the c i is chosen as d l (l > j). The case with l < j follows from a similar discussion. Then permutations in the first term on the RHS of eq. (A.1) is given by a i denotes the number of arrows, which point away from root, in the trace i i i.
According to the relative orders between d |i i i|−q and d |i i i|−q+1 , permutations σ σ σ (q) in eq. (A.6) splits into σ σ σ (q) A ≡ σ σ σ (q) | d |i i i|−q ≺d |i i i|−q+1 and σ σ σ Comments on the proof: In the above proof, the tree structure T | x is always treated separately from the trace in each step (i.e. it is shuffled with the full trace with a proper constraint, as shown in eqs.