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 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 recursively expanded in terms of the amplitudes A (m ,s ) with s + m < s + m [2]. In the special case m = 1, these relations give rise to the earlier proposed expansions of single-trace EYM amplitudes [3][4][5]. When the recursive expansions are applied repeatedly until there is no graviton and only one gluon trace, an arbitrary tree-level EYM amplitude is finally expressed as a combination of tree JHEP05(2020)008 level color-ordered Yang-Mills (YM) ones. Such pure-YM expansions precisely coincide with the earlier studies on amplitudes with only a few gravitons and/or gluon traces [6][7][8][9]. The pure YM expansions of EYM amplitudes, together with the gauge invariance conditions of gravitons or the cyclic symmetries of gluon traces, induce nontrivial identities for color-ordered YM amplitudes [2]. These identities guaranteed the localities in the Britto-Cachazo-Feng-Witten (BCFW) [10,11] proof of the recursive expansions [2,3] and played a crucial role in the proof [12] of the equivalence between distinct approaches [13][14][15] to nonlinear sigma model amplitudes.
A prominent feature of the identities induced from multi-trace EYM amplitudes is that the coefficients therein generally contain factors of all the three types of Lorentz contractions · , ·k and k·k where µ are half polarizations of gravitons and k µ are external momenta. This is quite different from the known Kleiss-Kuijf (KK) [16] and Bern-Carrasco-Johansson (BCJ) relations [17] whose coefficients at most involve k ·k factors. Nevertheless, several clues imply that the identities induced from multi-trace EYM amplitudes can be related with BCJ relations: (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 (as pointed in [2]). (ii). Second, as demonstrated in [2], examples with a few gluon traces and gravitons provided evidence of the connection between the induced identities and BCJ relations. (iii). Third, the fact that color-kinematic duality [17], which is the underlaid structure of the BCJ relations for YM amplitudes, can be resulted from gauge invariance [18] also implies that the identities induced from gauge invariance can be related with BCJ relations.
In the current paper, we extend our discussions in [1] to multi-trace cases and show that all these identities induced from multi-trace EYM amplitudes can be expanded in terms of BCJ relations. The main idea is sketched as follows: (i) Refined graphic rule. We first introduce a refined graphic rule 1 for the coefficients in pure-YM expansions of multi-trace EYM amplitudes, where the three types of factors · , · k and k · k are presented by distinct types of lines (as already introduced in the study of single-trace case [1]) and a new type of line is invented to record the relative order of gluons in a gluon trace. The induced identities are then expressed through a summation over connected tree graphs which are built of the four types of lines, while color-ordered YM amplitudes corresponding to each graph can be collected in a proper way.
(ii) Skeletons and components. To relate the induced identities with BCJ relations, we split gluon traces in an appropriate way and then remove all the k · k lines from the graphs. After that, a physical graph (i.e. a graph defined by the refined graphic rule) turns to a disconnected one which is called skeleton and consists of disjoint components. The summation over all physical graphs is thus given by summing over JHEP05(2020)008 all skeletons and summing over all the physical graphs corresponding to a given skeleton.
(iii) The final upper and lower blocks. For a given skeleton, the summation over all possible physical graphs can further be arranged by the following two steps: (a). Connect components via k · k lines properly such that the skeleton becomes a graph with only two disjoint maximally connected subgraphs, which are called the final upper and lower blocks, (b). Connect the final upper and lower blocks into a physical graph via a k · k line. Spurious graphs, which are not defined by the refined graphic rule, can also be introduced for a given configuration of the final upper and lower blocks. When associated with proper signs, all spurious graphs cancel out. Then the summation over all physical graphs for a given final upper and lower blocks can be reexpressed by a summation over all physical and spurious graphs.
(iv) Expressing an induced identity by BCJ relations. We finally find that all contributions of the physical and spurious graphs, corresponding to a given configuration of the final upper and lower blocks, together can be written as a combination of the graphbased BCJ relations [1] which have been proven to be combinations of the traditional BCJ relations [17]. 2 The structure of this paper is the following. In section 2, we introduce a refined graphic rule for the expansion of multi-trace EYM amplitudes and then express the induced identities by this rule. We further show two examples in section 3 which support the fact that the contributions of all those graphs corresponding to a given skeleton can be written as a combination of graph-based BCJ relations. In section 4, the general pattern of skeletons and components are studied. We then provide the general construction of the final upper and lower blocks for a given skeleton. The pattern of spurious graphs is also discussed. We finally show how to express the contributions of all (physical and spurious) graphs for a given final upper and lower blocks in terms of the graph-based BCJ relations. This work is summarized in section 5. A review of the background knowledge, the proof of the splitting trace relation and the pattern of the signs for graphs are included in the appendix.
Convention of notations The notations in this paper are gathered as follows.
• Permutations and sets: permutations are denoted by boldface Greek letters: σ σ σ, α α α, β β β, γ γ γ, ζ ζ ζ, etc. The i-th element in σ σ σ is denoted by σ(i). The position of an element a in σ σ σ is expressed by σ −1 (a). The inverse permutation of elements in an ordered set X X X is denoted by X X X T . Shuffle permutations of two ordered sets X X X and Y Y Y are written denote the difference of the sets A A A and B B B.
• Gravitons and gluon traces: gluon traces are denoted by boldface numbers 1 1 1, 2 2 2, . . . , or boldface lowercase Latin letters t t t, i i i . . . If a trace t t t i can be written as t t t i = JHEP05(2020)008  • Graphs: graphs are denoted by F , G or T . The notation G stands for the skeleton of a graph G. Reference order and root set are respectively expressed by R and R.
Components of a skeleton are given by A , B, C . . . , while a chain of components is denoted by CH. The reference order of components is given by R C . The final upper and lower blocks are respectively presented by U and L whose disjoint union is U ⊕ L .

Refined graphic rule for multi-trace EYM amplitudes and the induced identities
In this section, we present a refined graphic rule, by which one expresses a tree level multi-trace EYM amplitude A(1, 2, . . . , r|2 2 2| . . . |m m m H) with m gluon traces 1 1 1 ≡ {1, 2, . . . , r}, 2 2 2, . . . , m m m and s gravitons H ≡ {h 1 , h 2 , . . . , h s } in terms of (m + s)-point tree level color-ordered YM amplitudes: where we have summed over all possible connected tree graphs F . Each graph F defines a coefficient C F and proper permutations 1, σ σ σ F , r (of all elements in 1 1 1 ∪ 2 2 2 ∪ . . . ∪ m m m ∪ H) according to the refined graphic rule. The expansion (2.1) is obtained by applying the recursive expansion (A.1) (see [2]) iteratively and it is essentially equivalent to the graphic expansion given in [2]. Two examples which are helpful for understanding the refined graphic rule are given in this section. We then provide two identities that are respectively induced by the gauge invariance condition of a graviton and the cyclic symmetry of a gluon trace.

Refined graphic rule
To illustrate the refined graphic rule for the expansion (2.1), it is helpful to consider the gluon trace 1  Step-1. Define a reference order of elements in H H H as the following ordered set:

2)
JHEP05(2020)008  where each H i stands for an element (i.e. a graviton or a gluon trace) of H H H and the position of H i in R is called its weight. Apparently, H ρ(l=m+s−1) is the highest-weight element in the reference order (2.2), while H ρ(1) is the lowest-weight one. We also define the root set R by collecting elements of the trace 1 1 1: where the last element r ∈ 1 1 1 is always excluded. 3 Step-2. Pick out the highest-weight element H ρ(l) as well as other elements j 1 , . . . , j u (not necessary in the same relative order in R) from the ordered set R, then construct a chain towards an element w in the root set R: In the above chain, the H ρ(l) , j 1 , . . . , j u and w are respectively mentioned as the starting element (graviton or trace), the internal elements (gravitons and/or traces) and the ending element (also mentioned as the root of the chain). The special case that a chain with no internal element H ρ(l) → w is allowed. The contribution of the chain (2.4) and the graphic expression are evaluated in the following way: • Polarizations and momenta. Each half polarization µ of a graviton is expressed by a solid line connected to the graviton. The momentum k µ of any node (graviton or gluon) is presented by a solid arrow line pointing to the node. Here, if an arrow points away from the direction of root, an extra minus should be dressed.
JHEP05(2020)008 • Starting and internal gravitons. If the starting element H ρ(l) is a graviton h i , it contributes a half polarization µ h i presented by the structure figure 1 (a). If an internal element is a graviton h i ∈ {j 1 , . . . , j u }, it contributes a strength tensor µ h i and is expressed by the structure figure 1 (b).
• Starting and internal traces. To express a gluon trace t t t i , we select an ordered pair of gluons where a i and b i play as the first and the last elements of the trace respectively. Then we arrange other gluons of t t t i in a relative order Supposing that the trace can be written as Supposing a i is nearer to root than b i , we draw a dashed arrow line between any two adjacent gluons in the permutation (2.5). Each arrow points towards the direction of the node a i as shown by figure 2, thus it also points towards the root. If the trace t t t i plays as the starting element H ρ(l) of the chain (2.4), it is presented by the structure figure 2 (a) and contributes a −k µ a i to the coefficient. If t t t i ∈ {j 1 , . . . , j u } is an internal trace, it should be expressed by the structure figure 2 (b) and contributes a −k µ b i k ν a i . As shown by figure 2 (c), the special trace 1 1 1 = {1, 2, . . . , r} is presented via connecting adjacent gluons by dashed arrow lines whose arrows point towards the direction of the first gluon 1.
• Lorentz contractions and line styles. Contracting the Lorentz indices accompanying with adjacent elements, we get the contribution of the chain (2.4) in which

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There are three types of lines figure 3 (a), (b) and (c), which are resulted by the Lorentz contraction (2.6) and correspond to · , · k and k · k. Recalling that dashed arrow lines figure 3 (d) between gluons in a same trace have been introduced, we have four types of lines in all.
Redefine the ordered set R by removing the elements which have been used: and redefine the root set R by: Here each element in R is either a graviton or a gluon (this is different from the ordered set R where a trace is considered as a single element). If j i or H ρ(l) in eq. (2.10) is a graviton, it always stands for a single-element set {j i } or {H ρ(l) }.
Step-3. Repeat the above step by using the new defined R and R until the ordered set R becomes empty. Then a fully connected tree graph F which is rooted at the gluon 1 ∈ 1 1 1 is produced.
Step-4. For a given graph F , the coefficient C F in eq. (2.1) can be read off as the product of all factors corresponding to the type-1,-2 and -3 lines (see figure 3). The sign associated with such a graph gets two distinct contributions (i). (−1) N (F ) , where N (F ) is the number of arrows pointing away from the gluon 1; Step-5. Collect amplitudes A(1, σ σ σ F , r) for a given graph F . In any graph F , the gluons 1 (i.e. the root) and r in the trace 1 1 1 = {1, 2, . . . , r} are always treated as the first and the last elements. Permutations σ σ σ F are determined as follows: (i). Two adjacent nodes x and y which are connected by a line (of any style) must live on a path towards the root 1. If x is nearer to 1 than y on this path, we have (σ F ) −1 (x) < (σ F ) −1 (y) where (σ F ) −1 (a) denotes the position 5 of a. (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 with fixed b i 's 6 ), we is reasonable to define j = (σ F ) −1 (a). 6 Notice that the refined graphic rule is given by applying the recursive expansion (A.1) iteratively. The bi of the starting trace in each step of recursive expansion can be chosen freely (see section A). This implies that the bi's for the same trace, which plays as the starting element of different chains, can be chosen differently. In this paper, we fix the bi ∈ t t ti as the same one in all the graphs where the trace t t ti plays as a starting element.
JHEP05(2020)008 finally arrive the expansion (2.1). In the coming subsection, we show a concrete example to explain this rule.

Induced identities by refined graphs
As pointed in [2], one can induce a nontrivial identity (see eq. (A.9) or eq. (A.10)) of EYM amplitudes by imposing each of the following conditions on the recursive expansion (A.1): (i). the gauge invariance condition of a graviton, (ii). the cyclic symmetry of a gluon trace. When the EYM amplitudes on the l.h.s. of eq. (A.9) and eq. (A.10) are further expanded repeatedly according to eq. (A.1), we finally arrive identities for pure YM amplitudes.
Such an identity can also be obtained by imposing the condition (i) or (ii) on eq. (2.1) straightforwardly. As shown in the appendix, we only need to study identities which are induced by conditions of the highest-weight element. Those identities induced by conditions of elements other than the highest-weight one are essentially treated as cases with a smaller H H H set. If the highest-weight element H ρ(l) in the reference order (2.2) is a graviton h a , the gauge invariance condition of h a states that the expansion (2.1) vanishes when µ ha is replaced by k µ ha : The corresponding graphs can be obtained by the replacement figure 5 (a). 7 If the highest-weight element H ρ(l) in the reference order (2.2) is a gluon trace t t t 0 , the following identity is induced by the replacement (2.20) JHEP05(2020)008 for any a 0 ∈ t 0 This identity is understood as follows: when we consider {a 0 , KK[t t t 0 , a 0 , b 0 ]} (for a 0 ∈ t t t 0 ) (which does not contain the fixed gluon b 0 ∈ t t t 0 ) as the highest-weight element H ρ(l) (see figure 5 (b)) and then apply the refined graphic rule, the total contribution of the r.h.s. of eq. (2.1) must vanish. As stated in [2], this identity is essentially a result of the cyclic symmetry of the trace t t t 0 . It is worth pointing out an interesting property [2]: when we sum In the coming sections, we always use the r.h.s. of figure 5 (c) to stand for the highest-weight element in the identity (2.21).
In the next section, we show that the identities (2.19) and (2.21) induced from the double-trace EYM amplitude A(1, · · · , r|2 2 2 h 1 ) can be expanded in terms of graph-based BCJ relations.
To investigate the relationship between the induced identity and BCJ relations, we define the standard basis set for a gluon trace i i i by the set of permutations we do not sum over the end node c i ). Graphically, any two adjacent nodes for a given permutation in the standard basis set , c i } are connected by a dashed arrow line pointing towards the node b i . A starting trace defined by the refined graphic rule is already expressed by the standard basis because an end node of this trace is already fixed. Moreover, all internal traces can be expanded by the standard basis, according to the following nontrivial property: where each graph stands for its full contribution (including coefficients and amplitudes). The l.h.s. of the above equation is an internal trace structure defined by the refined graphic rule. On the r.h.s. , 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. (3.5) must be connected to a same node outside the trace i i i via the same type of line (i.e., type-2 or type-3 line). The summation notations with JHEP05(2020)008 Figure 7. When the identity (3.5) is applied, the contribution of figure 6 (a) splits into (a1) and (a2) which are expressed by standard basis. The graph (a1) comes from the second term of eq. (3.5), thus it must be associated with an extra minus. The graph (a3) is a spurious graph which cannot be directly obtained by the refined graphic rule and the identity (3.5). It can be considered as the a i = b i supplement to both terms on the r.h.s. of eq. (3.5). a tilde in eq. (3.5) was already defined by eq. (2.15). If a dashed arrow points away from the root, an extra minus should be dressed. We leave the proof of eq. (3.5) in appendix B. Now we apply the relation (3.5) to the first term of eq. (3.4) for a given l. Since the c i in eq. (3.5) can be chosen arbitrarily, we just choose the c i ∈ i i i (in this example i i i = 2 2 2) as the fixed gluon c 2 in the trace 2 2 2 of figure 6 (b) and (c) for convenience. Then the summation where T (a1) and T (a2) are corresponding to figure 7 (a1) and (a2). Substituting eq. (3.6) into eq. (3.4) and introducing the contribution of spurious graph figure 7 (a3) by 0 = T (a3) − T (a3) , we rewrite eq. (3.4) as the sum of I 1 and I 2 which are respectively defined by and In the following, we analyze I 1 and I 2 in turn and prove that both of them can be expanded in terms of graph-based BCJ relations.
(i) It is easy to see the graphs figure 7 (a1), (a3) and figure 6 (c) corresponding to the three terms of eq. (3.7) can be reproduced by drawing a type-3 line between a gluon l ∈ {1, 2, . . . , r − 1} = 1 1 1 \ {r} and a node a ∈ T 1 where T 1 is the tree structure figure 8. Particularly, a is given by a 2 ∈ 2 2 2 (a 2 = b 2 ) for figure 7 (a1), b 2 for figure 7 (a3) and h 1 for figure 6 (c). The kinematic factors of figure 7 (a1), (a3) and figure 6 JHEP05(2020)008 (c) can then be uniformly given by (k h 1 · k b 2 )(k a · k l ). Let us count the sign: given , there is an overall sign (−1) |2 2 2,b 2 ,c 2 | which has already been absorbed into the summation notation with a tilde The sign for each term inside the square brackets is collected as follows: (1). According to the refined graphic rule, any graph F with N (F ) arrows (for both solid and dashed arrow lines) pointing away from the root 1 is associated with a sign (−1) N (F ) . (2). Each of (a1) and (a3) has an extra sign (−1). The above observations further lead to the following pattern: • Once a ∈ T 1 has been chosen, the sign is independent of the choice of l ∈ {1, 2, . . . , r − 1} = 1 1 1 \ {r} because neither N (F ) nor the extra sign in eq. (3.7) relies on l. On another hand, two graphs with adjacent a ∈ T 1 are associated with opposite signs.
• All permutations established by the graph with any given a ∈ T 1 have the form (3.9) Here T 1 | a is introduced as the relative orders between nodes of the tree T 1 when the node a is considered as the leftmost one. According to the refined graphic rule, when we connect a type-3 line between a ∈ T 1 and l ∈ 1 1 1 \ {r}, l must be nearer to the root 1 than a. In other words, for a given permutation σ σ σ in eq. (3.9), l ∈ {1, . . . , r − 1} can be any node satisfying σ −1 (l) < σ −1 (a). Then the total coefficient (which comes from the type-3 line between a and l) for the k µ l (the momentum of the root 1 is always included in this summation).
Altogether, I 1 in eq. (3.7) can be reexpressed by where (−) F (x 0 ) denotes the sign for the graph F with a = x 0 (x 0 ∈ T 1 ). The f a for any a ∈ T 1 is fixed as (i). f x 0 = 1, (ii). f x 1 = −f x 2 if x 1 and x 2 are two adjacent JHEP05(2020)008 nodes in T 1 . Therefore, the expression in the square brackets is just the l.h.s. of the graph-based BCJ relation (A.13) which has been proven to be a combination of traditional BCJ relations (A.12) (see [1]).
(ii) For I 2 , all the permutations established by the graphs figure 6 (b), figure 7 (a2) and Hence I 2 turns to in which γ γ γ satisfy eq. (3.11). Apparently, the expression in the square brackets is just the l.h.s. of a special case of BCJ relation (A.12), which can also be understood as the graph-based BCJ relation (A.13) when the tree graph T is the single node h 1 .
3.2 Example-2: the identity (2.21) induced from A(1, 2, · · · , r|2 2 2 h 1 ) When the reference order for the expansion (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 the highest-weight element), the cyclic symmetry of the trace 2 2 2 induces the identity ( in which, (3.14) Once all graphs are summed over, we arrive the r.h.s. of the induced identity eq. where contributions of all graphs of the form figure 9 (a), (b) and (c), (d) were collected as I 1 and I 2 respectively. Now we prove that both I 1 and I 2 in eq. (3.15) can be expanded in terms of BCJ relations. For the I 1 part in eq. (3.15), the summation over j = 1, . . . , |2 2 2| − 1 is nothing but just the summation over all nodes a 2 ∈ T 3 where T 3 is the tree graph figure 10 (a). All permutations established by the graphs figure 9 (a), (b) with a given a 2 ∈ T 3 and a given l ∈ {1, . . . , r − 1} have the form 1, σ σ σ ∈ (T 3 | a 2 ) ¡γ γ γ, r . Here, γ γ γ ∈ {2, . . . , l, {h 1 } ¡{l+1,...,r − 1}} and T 3 | a 2 denotes the relative orders of nodes in T 3 when a 2 is considered as the leftmost one. Based on a similar discussion with the example-1 in section 3.1, we find the following patterns: (1). The coefficients (k a 2 · k l ) (l ∈ {1, . . . , r − 1} for figure 9 (a) and l = h 1 for figure 9 (b)) corresponding to a same σ σ σ ∈ (T 3 | a 2 ) ¡ γ γ γ with different choices of l are collected as (−k a 2 · Y a 2 (σ σ σ)); (2). Any two graphs, where a 2 ∈ T 3 are chosen as adjacent nodes, have opposite signs. Then the I 1 part JHEP05(2020)008  .17) Here (−) F (x 0 ) is the sign for a graph with a 2 = x 0 ∈ T 3 . Since (−) F (x 0 ) has been extracted as an overall sign, we have f x 0 = 1. If x 1 , x 2 ∈ T 3 are two adjacent nodes, we have Obviously, the expression in the square brackets is nothing but (up to a total minus) the l.h.s. of a graph-based BCJ relation (A.13). As a result, I 1 is a combination of traditional BCJ relations.

Common features of the examples
Now let us extract some common features from the examples, which will be extended to general cases in the next section.
(i) Expressing traces by standard basis. In example-1, the trace 2 2 2 played as an internal trace in figure 6 (a) and a starting trace in either figure 6 (b) or (c). In the latter cases, the trace 2 2 2 was already expressed by the standard basis, i.e., one end of the trace, JHEP05(2020)008 the gluon c 2 , was fixed. In the former case, both ends of the trace 2 2 2 were not fixed (in other words both are summed over). In order to expand the trace 2 2 2 in figure 6 (a) by the standard basis, we have made used of the splitting trace relation (3.5), in which the fixed node was conveniently chosen as the same element (i.e. c 2 ) with that in figure 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 graph. Each maximally connected subgraph of a skeleton is called a component. (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 \ {r} (recalling that the gluon r is always excluded) via a type-3 line. Then a fully connected graph is constructed. In example-2, the graphs figure

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All the graphs figure 6 (b), (c), figure 7 (a1), (a2) and figure 9 (a)-(d) are graphs in standard basis which are directly defined by the refined graphic rule. These graphs are called physical graphs. The spurious graph figure 7 (a3), which is not defined by the refined graphic rule, can be reproduced from either figure 12 (a) (connecting b 2 ∈ U with l ∈ L \ {r}) or figure 12 (b) (connecting h 1 ∈ U with b 2 ∈ L \ {r}). In the former case, a minus sign is introduced so that the spurious graph constructed by distinct ways cancel with one another. 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 (in other words summing over all possible physical and spurious graphs corresponding to U and L ).
(v) Induced identities as combinations of graph-based BCJ relations. A crucial observation is that the sum over all the graphs corresponding to a given configuration of the final upper and lower blocks U and L (i.e. either I 1 or I 2 in each example) is a combination of graph-based BCJ relations (A.13).
In the next section, we extend these observations to general cases and show that both identities (2.19) and (2.21) can be expanded in terms of BCJ relations.

General study
To investigate the general induced identities (2.19) and (2.21) in a unified way, we write them as where the graphs G are obtained by imposing the replacement figure 5 (a) or (b), which corresponds to eq. (2.19) or eq. (2.21), on the graphs F in eq. (2.1). When we introduce skeletons G by deleting all type-3 lines from the graphs G and expressing all traces by standard basis according to eq. (3.5), eq. (4.1) is rearranged as In the above equation, the summation notation G means that all possible skeletons G are summed over and the signs (−1) |t t t i ,a i ,c i | and/or (−1) |t t t i ,b i ,c i | accompanying to the traces in each G are absorbed. The factor P [G ] denotes the kinematic factor corresponding to the skeleton G . Since a skeleton does not involve any type-3 line, P [G ] only consists of · and · k factors. In the expression inside the square brackets, all possible physical graphs G (i.e. graphs generated by refined graphic rule with all internal traces expressed by the standard basis) containing the skeleton G and all permutations σ σ σ G for each graph G are summed over. The factor K [G\G ] in eq. (4.2) stands for the product of all k · k factors that JHEP05(2020)008 Figure 13. The graph (a) is a chain that does not involve the highest-weight element H ρ(l) in the reference order R. The graph (b) is a chain involving the highest-weight element. Each node (except for the ending node w) outside the boxed structures in the chains (a) and (b) is a graviton and we define F µν (+) ≡ k µ ν , F µν (−) ≡ − ν k µ . The structures shown by the graph (c) are not allowed. Each boxed structure in the chain (a) and in an internal sector of the chain (b) can be (1). two gravitons connected together by a type-1 line (as shown by the graph (d)) or (2). a gluon trace in standard basis (as shown by (e) and (f) which correspond to the two terms in eq. (3.5)). The node c in (e), (f) denotes the fixed node of a trace t t t. The boxed structure in the starting sector of (b) can be either a graviton (as shown by (g)) or a gluon trace t t t → {d 1 , d 2 , . . . , d |t t t|−1 } (as shown by (h)). are presented by the type-3 lines in G. Those signs caused by arrows pointing away from the root and the extra signs caused by the second term of eq. (3.5) are all collected as (−) G .
As observed in section 3.3, all physical graphs G involving a given skeleton G can be generated by connecting the components via type-3 lines in a proper way: (i). first generate all possible configurations of the final upper and lower blocks U ⊕ L ; (ii). then connect a type-3 line between two nodes x ∈ U and y ∈ L \ {r} appropriately. One should take care of the step (ii) because spurious graphs may also be produced. Nevertheless, the spurious graphs in fact all cancel out in the examples. Hence we suppose that the expression in the square brackets in eq. (4.2) can be generally written as  Here, K [U ⊕L \G ] is the product of all k · k factors corresponding to the given configuration of the final upper and lower blocks U , L , while k x · k y is the factor corresponding to the type-3 line between U and L . The first summation in eq. (4.3) is taken over all possible configurations of the final upper and lower blocks U , L for the skeleton G . In the square brackets, all choices of nodes x ∈ U and y ∈ L \{r} as well as the permutations σ σ σ G defined by the (physical or spurious) graph G (determined by U , L , x and y) are summed over. The sign for the (physical or spurious) graph G is denoted by (−) G . In this section, we study eq. (4.3) schematically. We first classify components of skeletons, then show how to construct the final upper and lower blocks from a given skeleton G . After that, we show all spurious graphs cancel out. Thus the summation over all possible choices of x ∈ U and y ∈ L \{r} is equivalent to summing over all possible physical graphs for given U and L . At last, we demonstrate that the expression inside the square brackets in eq. (4.3) is a combination of BCJ relations.

Skeletons and components
When all type-3 lines (i.e. k·k factors) are removed, a graph G becomes a skeleton G . To analyze possible structures of components which are maximally connected subgraphs of G , we JHEP05(2020)008 should look into the inner structure of a chain via expanding eq. (2.6) by eq. (2.7), eq. (2.8) and eq. (3.5). According to whether the starting node is the highest-weight element in the reference order R (defined in eq. (2.2)) or not, we carry out the discussion as follows: (ii) If the starting node of a chain is the highest-weight element H ρ(l) , its corresponding graph must have the general pattern figure 13 (b), where the highest-weight element (graviton or trace) is already replaced according to figure 5 (a) (for a graviton) or figure 5 (b) and (c) (for a gluon trace). An important feature is the chain figure 13 (b) has at least two sectors, which follows from the fact that structures in figure 13 (c) are forbidden.
Having the above discussions, sectors of chains can be easily classified as figure 14. In a full graph G, nodes of any sector may play as the ending nodes of other chains. When all type-3 lines are removed, each sector in a skeleton thus can be attached by the type-3 sectors figure 14 (c) (or equivalently ending sectors) of other chains. Consequently, components in a skeleton can be classified by the following way.  figure 15 (b), the component is called a type-IB component. We define the kernel of a type-I component by (i). the type-1 line of a type-IA component (see figure 15 (a)), (ii). the type-4 line that is attached to the unfixed end node of the trace (in standard basis) inside a type-IB component (see figure 15 (b)). 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 part is called the bottom side.
• Type-II component. A component consisting of a type-II sector figure 14 (a) and possible type-III sectors whose arrows point towards the type-II sector If the structure in the box of the type-II sector is figure 13 (g) (i.e. the highest-weight element is a graviton), this component is called a type-IIA component (see figure 16 (a)). If the structure in the box is a gluon trace figure 13 (h), the component is called a type-IIB component (see figure 16 (b)).
• Type-III component. A component consisting of the trace 1 1 1 and possible type-III sectors with arrows pointing towards the trace 1 1 1 (see figure 16 (c)).
Since the chain figure 13 (b) that is lead by the highest weight node has at least two sectors, a skeleton must at least contain two components: the type-II and the type-III components. In general, Type-I components may also be involved in a skeleton. All those graphs corresponding to a given skeleton are reproduced by connecting type-3 lines between components of a skeleton G in an appropriate way.
Multi-trace from single-trace. It is worth pointing out that the refined graphic rule given in section 2 can be obtained from the rule for identities induced from single-trace amplitudes (which was presented in [1]) by an appropriate replacement. Particularly, we consider a single-trace amplitude A (1, 2, . . . , r H)   For identities induced from a single-trace amplitude, the highest-weight element (graviton) is further given by k µ (see figure 5 (a)). In the corresponding multi-trace case, the highestweight element can either be a graviton (if it is not replaced by a trace) or be replaced by a gluon trace. The latter is described via replacing the node (the highest-weight graviton for identity induced from single-trace amplitude) by the r.h.s. of figure 5 (c). Now we look into the inner structure of a chain by further expanding internal gravitons on the l.h.s. of eq. (4.4) according to . On the r.h.s. , internal gravitons are also expanded by F µν (+) −F µν (−) , while internal gluon traces are expanded according to the relation eq. (3.5). Although this expansion of gluon trace does not affect the tensor F µν for a given a i and b i (see eq. (2.8)), it splits the graphs corresponding to the trace into the standard basis. Hence, the replacement h i → i i i (for an internal graviton h i ) is achieved graphically through replacing the first (second) graph in figure 1 (b) by the first (second) graph on the r.h.s. of eq. (3.5) (for a given {a i , b i }). It follows that the general chain structures figure 13 (a) and (b) are obtained from those chain structures for the single-trace case [1] (where only type-IA, type-IIA and type-III sectors are allowed) by incorporating more types of sectors: type-IB and type-IIB sectors which reflect structures of gluon traces. Consequently, the full classification of components in multi-trace cases can be given by enlarging the families of the type-I and the type-II components that were defined in [1]: Type-I components → Type-IA or Type-IB components, Type-II components → Type-IIA or Type-IIB components, (4.6) where the type-IIA and type-IIB components are respectively the highest-weight components of the identities (2.19) and (2.21). All the above discussions allow us to borrow some crucial conclusions from the single-trace case [1]: (i) When keeping track of chains in the single-trace case [1], one can build all possible physical graphs corresponding to a skeleton by connecting type-3 lines between components properly (see appendix D in [1]). This construction can be immediately generalized to multi-trace cases by the enlargement (4.6).
(ii) As proved in [1] (see sections 6.1, 6.2 and appendix D of [1]), all the physical graphs in the single-trace case, which are corresponding to a given skeleton and are constructed by the above step, can be reproduced by (1). constructing the final upper and lower blocks, (2). connecting the final upper and lower blocks via a type-3 line appropriately. In multi-trace cases, we just follow the same construction rule but enlarging the type-I and type-II classes of components according to (4.6).
(iii) As pointed in section 6.3 of [1], in the single-trace case, the sum over all physical graphs which are produced by (ii) can be further written as the sum of all physical JHEP05(2020)008 and spurious graphs (those graphs which are not directly constructed from the refined graphic rule). The latter all cancel out after summation. Again, the spurious graphs for single-trace induced identities can be straightforwardly extended to multi-trace cases by the help of (4.6) and they all cancel out (we have seen this cancellation by the examples in section 3).
In the coming two subsections, we display the construction rule of the final upper and lower blocks as well as the construction of physical and spurious graphs without a proof. In fact, all the proofs follow from discussions parallel with those in the single-trace case [1].

The final upper and lower blocks
Now we provide the general rule for constructing all possible configurations of the final upper and lower blocks corresponding to a given skeleton G : • Step-1. For any skeleton G , we define the reference order R C of all type-I components (including type-IA and type-IB components) by the relative order of the highest-weight nodes therein. In other words, the weight (i.e. the position in R C ) of a component inherits from its highest-weight node. For example, suppose there are three type-I components (IA and/or IB) C 1 , C 2 , and C 3 with the corresponding highest-weight nodes (graviton or a gluon) a 1 , a 2 and a 3 . If the weights W a i have the relation W a 2 < W a 1 < W a 3 , the reference order of these components is then given by the ordered set R C = {C 2 , C 1 , C 3 }. We further define the upper block U and lower block L as the components respectively containing the highest-weight element (graviton or trace) and the trace 1 1 1. 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 highest-weight 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 Here the subscripts t and b respectively denote the top and bottom sides of a type-I component, which are separated by a comma. 8 The double arrow line '↔' between two components stands for the type-3 line (i.e. k · k), which connects any two nodes belonging to the corresponding regions. For example, if the chain of components has the form [ , 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 JHEP05(2020)008 Figure 17.
A typical spurious graph where the path starting from the highest-weight element and ending at the root 1 passes through some single sides of type-I (IA or IB) components C a1 , C a2 , . . . , C am (which are called spurious components). and lower blocks by: (4.8) • Step-3. Repeating step-2 with the new defined R C , U and L iteratively until the ordered set R becomes empty, we get a graph with only two mutually disjoint subgraphs: the final upper and lower blocks U and L .

Physical and spurious graphs
For a given configuration of the final upper and lower blocks U and L which are constructed previously, a fully connected graph G in eq. (4.3) is produced by connecting arbitrary two nodes x ∈ U and y ∈ L \ {r} via a type-3 line. As pointed in section 3, such a graph can be either a physical graph or a spurious one. We have already stated that physical and spurious graphs can be obtained from those in single-trace case [1] by the enlargement (4.6). As a result, a spurious graph has the structure figure 17, where the chain starting form the highest-weight element (graviton for the identity (2.19) and gluon for the identity (2.21)) and ending at the root 1 passes through single sides of some type-IA and/or type-IB components C a 1 , C a 2 , . . . , C am which respectively belong to the chains of components CH 1 , . . . , CH m .

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In order to display more details of spurious graphs, we define the weight W i of a chain CH a i by the weight of the starting component (equivalently the highest-weight component) of CH a i . As pointed in [1], if the lowest-weight chain among CH 1 , . . . , CH m in figure 17 is CH l , we must have W 1 > · · · > W l−1 > W l and W l < W l+1 < · · · < W m . Following a discussion which is parallel with that in [1], we conclude that all the chains CH 1 , . . . , CH l−1 (and structures attached to them) belong to the final lower block L , while CH l+1 , . . . , CH m (and structures attached to them) belong to the final upper block U . Only the lowestweight chain CH l (among CH l+1 , . . . , CH m ) can live in either L or U . Correspondingly, the type-3 line (colored by red in figure 17) on either the l.h.s. or the r.h.s. of C a l is considered as the one between the final upper and lower blocks. Thus a given spurious graph is corresponding to two distinct configurations of U and L . In other words, all spurious graphs must appear in pairs! This fact allows us to associate a pair of spurious graphs with opposite signs so that all spurious graphs cancel out.
By the help of the above discussion, we now determine the sign (−) G in eq. (4.3) for a (physical or spurious) graph G:

Expanding induced identities in terms of BCJ relations
We are now ready to show the expression in the square brackets in eq. (4.3), i.e.
for a given configuration of the final upper and lower blocks U , L is the l.h.s. of the graphbased BCJ relation (A.13) (hence a combination of traditional BCJ relations (A.12)). Here a graph G in the above expression is a physical or a spurious graph which is constructed by connecting two nodes x ∈ U , y ∈ L \ {r} via a type-3 line. Our discussion is carried out by the follow steps: (i) According to the refined graphic rule, permutations σ σ σ G in eq. (4.10) are independent of the line styles in G. Therefore, all lines in G can be replaced by dashed lines (with no arrow) which only characterize the relative positions of nodes in G, as shown by figure 18. Apparently, for a given configuration of the final upper and lower blocks U , L and a given choice of x ∈ L , the corresponding permutations σ σ σ G satisfy σ σ σ G ∈ ζ ζ ζ ¡ γ γ γ y≺x for ζ ζ ζ ∈ U x and γ γ γ ∈ L 1 \ {1, r} , (4.11) where U x and L 1 denote the permutations established by U and L when x ∈ U and the root 1 ∈ L are the leftmost elements respectively. Since 1 and r are fixed as the first and the last elements in eq. (4.10), they should be excluded from σ σ σ G . The y ≺ x means y situates before x in the permutation, i.e. (σ G ) −1 (y) < (σ G ) −1 (x).
Noting that the choice of y is independent of the relative orders ζ ζ ζ and γ γ γ, we rewrite the summations in eq. (4.10) as follows x∈U y∈L \{r} σ σ σ G → x∈U ζ ζ ζ∈U |x γ γ γ∈L | 1 \{1,r} y∈L \{r} σ σ σ∈[ζ ζ ζ¡γ γ γ]|y≺x . (4.12) (ii) For a given permutation σ σ σ ∈ ζ ζ ζ ¡ γ γ γ, one can collect together the coefficients k x · k y in eq. (4.11) with different choices of y ∈ L \ {r}. Specifically, only those y satisfying y ≺ x in σ σ σ have nonzero contributions and the sign (4.9) is totally independent of y. Hence, all the k x · k y factors for a given σ σ σ are collected as k x · Y x (σ σ σ) where Y µ x (σ σ σ) ≡ y≺x k µ y (the gluon 1 is always included as the leftmost y in this summation). Meanwhile, the last two summations in eq. (4.12) turns into y∈L \{r} σ σ σ∈[ζ ζ ζ¡γ γ γ]|y≺x → σ σ σ∈ζ ζ ζ¡γ γ γ . (4.13) (iii) As illustrated by appendix C, the sign (−) G in eq. (4.10) has the following pattern. Two graphs with x = x 1 and x = x 2 where x 1 , x 2 ∈ U are adjacent to each other must be associated with opposite signs. Hence one can extract the sign eq. (4.9) for a graph with x = x 0 ∈ U as an overall factor and then the sign for an arbitrary choice of node x is given by where f x is defined by (i). f x 0 = 1, (ii). f x 1 = −f x 2 for two adjacent nodes x 1 and When all the above are taken into account, eq. (4.10) is finally expressed by where the summation over γ γ γ was extracted out because it is independent of the choice of x ∈ U . The expression in the square brackets in eq. (4.15) is nothing but the l.h.s. of the graph-based BCJ relation (A.13).

Conclusions
In this paper, we provided the refined graphic rule for expanding tree level multi-trace EYM amplitudes in terms of color-ordered YM amplitudes. When the gauge invariance condition of a graviton and the cyclic symmetry of a gluon trace were imposed, this expansion induced two identities (2.19) and (2.21) respectively. By extending the analysis for the singletrace case [1] to an arbitrary multi-trace induced identity, we demonstrated that eq. (2.19) and eq. (2.21) can finally be expressed as a combination of graph-based BCJ relations (thus traditional BCJ relations). There are several related topics that deserve further study: (i). First, how to understand the induced identities from the view of string theory? String theory studies of the expansions of EYM amplitudes have been established in [6,9,33,34], while BCJ relations have also been proven in string theory [31,35]. Hence it is reasonable to expect a stringtheory approach to both induced identities and graph-based BCJ relations. (ii). Second, it is worth investigating the induced identities in various theories systematically. In [36], JHEP05(2020)008 a unified web of expansions of amplitudes was founded with the help of the unifying relation [37], which inspires that the induced identities may exist in many other theories. (iii). Third, the YM expansion of EYM amplitudes, which have been used in this paper, is in KK basis [16]. As pointed in [38], this expansion can be extended to BCJ basis [17]. We expect that the refined graphic rule can also be generalized to expansions in BCJ basis [17]. (iv). Last but not least, a kinematic algebra for constructing BCJ numerators in the MHV sector was proposed [39]. It seems that distinct sectors of numerators are corresponding to graphs with different numbers of type-IA kernels. Thus, one may provide a general rule for constructing all sectors of BCJ numerators, with the help of refined graphic rule.

A.2 Identities induced from multi-trace EYM amplitudes
Symmetries of multi-trace EYM amplitudes, together with the recursive expansion (A.1), induce nontrivial identities for EYM amplitudes with fewer gravitons and/or gluon traces [2]. There are two symmetries under consideration in this paper: the gauge invariance condition for a graviton and the cyclic symmetry of a gluon trace. Muti-trace EYM amplitude satisfies gauge invariance condition, which states that the amplitude has to vanish once half polarization µ hx of a graviton h x is replaced by the momentum k µ hx . If this replacement is performed on the r.h.s. of the recursive expansion (A.1), we should consider two distinct situations: • If the fiducial element H a is a graviton h x (i.e. eq. (A. Therefore, the only nontrivial identity induced by the gauge invariance condition of a graviton is eq. (A.9) which is called type-I identity in [2].
Another identity (called type-II identity in [2]) is induced from the expansion (A.1) where the fiducial element H a is a gluon trace t t t 0 . In particular, we notice that the end element b 0 ∈ t t t 0 can be chosen arbitrarily in the fiducial trace t t t 0 . This arbitrariness is essentially caused by the cyclic symmetry of the trace t t t 0 [2] and indicates the following identity

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Here we removed b 0 from the trace t t t 0 first, then shuffled these permutations according to eq. (A.6). The j 1 , j 2 , . . . , j u is a permutation of elements in H H H A . When the recursive expansion (A.1) for amplitudes with fewer gravitons and/or gluon traces are applied repeatedly, the two types of relations (A.9) and (A.10) respectively turn into the induced identities (2.19) and (2.21) for pure YM amplitudes.
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 by the following steps.
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. (B.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 r.h.s. of eq. (B.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 r.h.s. of eq. (B.1) is given by a i denotes the number of arrows pointing 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. (B.7) 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.  C The sign in eq. (4.10) In eq. (4.10), the graph G is constructed by connecting x ∈ U and y ∈ L via a type-3 line. As mentioned before, the sign for such a graph is dependent of the choice of x but independent of the choice of y. Now we show that two graphs with choosing adjacent x ∈ U must have opposite signs. To see this, we study all possible structures presented by figure 19 (a)-(e), where x = x 1 is adjacent to x = x 2 (x 1 , x 2 ∈ U ), as follows.
(i) As shown by figure 19 (a), nodes x 1 and x 2 connected by a type-3 line must belong to a same (top or bottom) side of a type-IA or type-IB component. In this case, each one of S(U x ), T r(U x , L ) and N (L ) is the same for choosing two adjacent x's. But the numbers N (U x ) in eq. (4.9) for x = x 1 and x = x 2 differ by one because the type-2 line between x 1 and x 2 has opposite directions for these two cases.
(ii) If x 1 and x 2 are connected by a type-4 line, they must belong to a same type-IB component. If the type-4 line is not the kernel of the type-IB component, the structure is given by figure 19 (b). In this case, x 1 and x 2 must belong to a same side of the component. The numbers N (U x ) for choosing x = x 1 and x = x 2 should differ by one because the type-4 line between x 1 and x 2 has opposite directions. Each of S(U x ), T r(U x , L ) and N (L ) in eq. (4.9) is the same for both choices of x.
(iii) If the two adjacent nodes x 1 and x 2 connected by a type-4 line belong to opposite sides of a type-IB component, the type-4 line must be the kernel of the type-IB component (as shown by figure 19 (c)). In this case, each of N (U x ), T r(U x , L ) and S(U x ) for x = x 1 and x = x 2 differ by one but N (L ) is the same.
(iv) If x 1 and x 2 are connected by a type-1 line which must be the kernel of a type-IA component (as shown by figure 19 (d)), only the number S(U x ) for choosing x = x 1 JHEP05(2020)008 and x = x 2 differ by one. Any of T r(U x , L ), N (U x ) and N (L ) is the same for the two choices of x.
(v) If x 1 and x 2 are connected by a type-3 line as shown by figure 19 (e), only S(U x ) for choosing x = x 1 and x = x 2 differ by one. Any other number is the same for the two choices.
Therefore, in all the above cases, S(U x )+T r(U x , L )+N (U x )+N (L )+1 for choosing x = x 1 and x = x 2 , where x 1 , x 2 ∈ U are two adjacent nodes, must differ by an odd number. In other words, two graphs with adjacent x's have opposite signs.
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