Gauge invariance induced relations and the equivalence between distinct approaches to NLSM amplitudes

In this paper, we derive generalized Bern-Carrasco-Johansson relations for color-ordered Yang-Mills amplitudes by imposing gauge invariance conditions and dimensional reduction appropriately on the new discovered graphic expansion of Einstein-Yang-Mills amplitudes. These relations are also satisfied by color-ordered amplitudes in other theories such as color-scalar theory, bi-scalar theory and nonlinear sigma model (NLSM). As an application of the gauge invariance induced relations, we further prove that the three types of BCJ numerators in NLSM , which are derived from Feynman rules, Abelian Z-theory and Cachazo-He- Yuan formula respectively, produce the same total amplitudes. In other words, the three distinct approaches to NLSM amplitudes are equivalent to each other.


Introduction
Color-kinematic duality (BCJ duality), which was suggested by Bern Carrasco and Johansson [1,2], provides a deep insight into the study of scattering amplitudes. According to BCJ duality, full color-dressed Yang-Mills amplitudes are expressed by summing over trivalent (Feynman-like) diagrams, each of which is associated with a color factor and a kinematic factor (BCJ numerator) sharing the same algebraic properties (i.e., antisymmetry and Jacobi identity). Once the color factors are replaced by BCJ numerators of another copy Yang-Mills amplitude, we obtain a gravity amplitude.
A significant consequence of BCJ duality is that tree-level color-ordered Yang-Mills amplitudes satisfy BCJ relations where the coefficients for amplitudes are functions of Mandelstam variables. Together with the earlier proposed Kleiss-Kuijf [3] (KK) relations, BCJ relations reduce the number of independent colorordered Yang-Mills amplitudes to (n − 3)! (see the field theory proofs [4,5] and string theory approaches [6,7]). Though BCJ relations are first discovered in Yang-Mills theory, they actually hold for amplitudes in many other theories including: bi-scalar theory, NLSM [8], which can be uniformly described in the framework of CHY formulation [9][10][11][12]. It was pointed out that fundamental BCJ relation can be regarded as the most elementary one since the minimal basis [4] and a set of more general BCJ relations [5,6] are generated by them [13]. Nevertheless, in some situations, one may encounter BCJ relations which have much more complicated forms than knowns ones. Such relations can be neither directly understood as a result of fundamental relations nor straightforwardly proved by Britto-Cachazo-Feng-Witten [14,15] recursion or CHY formula. Therefore, a new approach to nontrivial BCJ relations is required.
Apart from the BCJ relations for amplitudes, the construction of BCJ numerators in various theories is also an important direction. In NLSM, there are three distinct constructions of BCJ numerators, all of which are polynomial functions of Mandelstam variables. (i) A construction based on off-shell extended BCJ relation (see [8]) was suggested by Fu and one of the current authors [16] (DF). In DF approach, the set of half-ladder numerators with the first and the last lines fixed (which serves as a basis of BCJ numerators) are expressed by proper combinations of momentum kernels [17][18][19][20][21][22]. Since the off-shell extended BCJ relation [8] was proved by the use of Berends-Giele recursion (Feyman rules), the DF type BCJ numerators can be essentially regarded as a result of Feyman rules. (ii) A much more compact construction of BCJ numerators in NLSM, which was based on Abelian Z theory, was provided by Carrasco, Mafra and Schlotterer (CMS) [23]. A half ladder numerator of CMS type is elegantly expressed by only one momentum kernel. (iii) In a more recent work [24], a graphic approach to polynomial BCJ numerators (DT type numerator) in NLSM, which was based on CHY formula was proposed by Teng and one current author. All the three distinct constructions given above must produce the same scattering amplitudes in NLSM, but this equivalence is still not proven explicitly.
In this paper, we derive highly nontrivial generalized BCJ relations (gauge invariance induced relations) by imposing gauge invariance conditions and CHY-inspired dimensional reduction on the recent discovered graphic expansion of color-ordered Einstein-Yang-Mills (EYM) amplitudes [24]. Expansion of EYM amplitudes was first proposed in [25] and further studied in [24,[26][27][28][29][30][31][32]. In the series work [24,29,31,32], general recursive expansion for all tree-level EYM amplitudes and the graphic expansion of EYM amplitudes in terms of pure Yang-Mills ones were established. When gauge invariance condition for the so-called fiducial graviton is imposed, the recursive expansion of EYM amplitudes induces relations between those amplitudes with fewer gravitons. Equivalently, when the graphic expansion [24] is considered, such gauge invariance induced relation implies a relation between color ordered Yang-Mills amplitudes whose coefficients are functions of both momenta and polarizations. To induce amplitude relations where all coefficients are functions of Mandelstam variables, one should convert all polarizations in the coefficients into momenta. In the current paper, we propose gauge invariance induced relations based on the following two crucial observations: (i) One can impose the gauge invariance conditions for several gravitons simultaneously. (ii) The gauge invariance conditions are independent of dimensions. With these two critical observations in hand and inspired by the dimensional reduction in CHY formula [12], we define (d + d)-dimensional polarizations and momenta whose nonzero components are expressed by only d-dimensional momenta. Imposing the gauge invariance in (d + d) dimensions on the graphic expansion [24] of single-trace EYM amplitudes, we naturally induce nontrivial amplitude relations where all coefficients are polynomials of Mandelstam variables (in d dimensions). In the framework of CHY formula, such relations become nontrivial relations between Parke-Taylor factors. As a consequence, the gauge invariance induced relations hold for not only color-ordered Yang-Mills amplitudes but also color-ordered amplitudes in other theories such as bi-scalar theory and NLSM.
An interesting application of our gauge invariance induced relation is the proof of equivalence between different approaches to NLSM amplitudes. Full color-dressed NLSM amplitudes can be spanned in terms of bi-scalar amplitudes via dual Del Duca-Dixon-Maltoni (DDM) [33] decomposition (The dual DDM decomposition for Yang-Mills amplitudes are given in [11,22,[34][35][36][37][38][39][40][41], for NLSM amplitudes are provided in [8,16,23,24]), in which the coefficients are half-ladder BCJ numerators with fixing the first and the last lines. Although the three distinct approaches: Feyman rules, Abelian Z theory and CHY formula provide different types of half-ladder BCJ numerators, they must produce the same NLSM amplitudes through the dual DDM decomposition. This equivalence condition then requires nontrivial relations between color-ordered bi-scalar amplitudes. By using the gauge invariance induced relations and defining partial momentum kernel, we prove that the three distinct constructions of BCJ numerators produce the same NLSM amplitudes precisely. In other words, the equivalence between the three different approaches to NLSM amplitudes is explicitly proven. The relation between main results of this paper is provided as gauge invariance + dimensional reduction The structure of this paper is given as follows. In section 2, we provide a review of the background knowledge including CHY formula, the recursive expansion and the graphic expansion of EYM amplitudes. In section 3, we induce generalized BCJ relations by combining gauge invariance conditions and dimensional reduction. Partial momentum kernel, which is important for the discussions in this paper, is introduced in section 3. A review of the three distinct constructions of BCJ numerators in NLSM is provided in section 4. In section 5, we prove the equivalence between CMS type and DT type numerators by inducing identities expressed by partial momentum kernel. The proof of equivalence between DF type and CMS type numerators is given in section 6. We summarized this paper in section 7. Complicated graphs and proofs are included by appendices.

A review of CHY formula and the expansion of EYM amplitudes
In this section, we review the CHY formula [9][10][11][12]42] for various theories and the recursive/graphic expansion of EYM amplitudes which will be used in the coming sections.

CHY formula
CHY formula expresses a tree level on-shell amplitude with n massless particles by integration over n scattering variables z i where dΩ CHY is Möbius invariant measure which contains the condition that scattering variables satisfy the following scattering equations Here k i denotes the momenta of the particle i. The integrand I L I R in (2.1) relies on theories. An important feature is that the CHY formula is independent of dimensions.
The matrix Ψ H is the one obtained by removing those rows and columns with respect to gluons in Ψ.

The CHY integrand for NLSM amplitudes
The CHY integrands for color-ordered NLSM amplitudes are obtained by dimensional reduction strategy [12]. In particular, I NLSM L has the same expression with I YM L , while I NLSM R is obtained by extending I YM R to (d + d + d)-dimensions and defining momenta and polarizations as follows: (2.10) The matrix Ψ (d+d+d) is thus written as where the A, B, C are defined via replacing the polarizations and momenta in (2.9) by the (d + d + d)dimensional ones E and K correspondingly. With the explicit components given in (2.10), we immediately arrive C = 0, A = A and B = B. As a consequence, the reduced Pfaffian Pf Ψ (d+d+d) is factorized into: By a further replacement a → k a , we reduce Pf Ψ (d+d+d) to the final expression of the NLSM integrand To sum up, NLSM amplitudes are obtained by performing the following replacements on Yang-Mills amplitudes a ∈ {1, n} and b ∈ {2 . . . n − 1} , or vice versa (2.14)

Expansions of EYM amplitudes
Tree level color-ordered EYM amplitude can be expressed recursively by ones with fewer gravitons and/or fewer traces. One can repeat this expansion until all amplitudes become pure Yang-Mills ones, then the expansion coefficients are constructed by graphic rules. Now we review the expansions of single-trace EYM amplitudes. The expansions of multi-trace amplitudes can be found in [32].
Assuming the permutation of elements of given h h h is where F µν a is the linearized field strength of particle a F µν a ≡ k µ a ν a − µ a k ν a (2.18) and Y i 1 denotes the sum of all momenta of gluons in the original gluon set which appear on the left hand side of i 1 . An explicit example is given by the expansion of the single-trace EYM amplitude A(1, 2, . . . , r h 1 , h 2 , h 3 ) with r gluons and three gravitons. By choosing h 3 as the fiducial graviton and summing over the five terms in (2.16), we finally express the single-trace EYM amplitude with three gravitons by those amplitudes with two, one and no graviton: Here, we summed over all possible permutations obtained by merging together the original gluon set {2, . . . , r − 1} and the set of gluons ('half gravitons') which come from the graviton set H. The relative order of gluons should be preserved, while the 'perms' under the summation notation means that all possible relative orders of elements in H should be considered. Given order σ σ σ, the full coefficient C(1, σ σ σ, r) can be determined by the following graphic rule 2 .
Graphic rule for the expansion of EYM amplitudes: (1) Define a reference order ρ ρ ρ of gravitons, then all gravitons are arranged into an ordered set (2) Pick the last graviton h ρ(s) in the ordered set R, an arbitrary gluon l ∈ {1, 2, . . . , r − 1} (noting that the gluon r is not considered here) as well as gravitons ). Now consider each particle in the set {l, h i 1 , h i 2 , . . . , h i j , h ρ(s) } as a node, we define a chain starting from the node h ρ(s) and ending at the node l. The graviton h ρ(s) here is mentioned as a the starting point of this chain, while the gluon l is mentioned as a root. All other gravitons on this chain are mentioned as internal nodes of this chain. The factor associated to this chain is , h ρ(s ) } starting from h ρ(s ) and ending at l . This chain is associated with a factor (4) Repeating the above steps until the ordered set R becomes empty, we get a graph ('forest') with gluons as roots of trees 4 . For a given graph F, the product of the factors accompanied to all chains produces a term C [F ] (σ σ σ) in the coefficient C(1, σ σ σ, r) in (2.20). Thus the final expression of C(1, σ σ σ, r) is given by summing over all possible graphs defined above The expansions of Pfaffians in the CHY formula of single-trace EYM amplitudes It is worth closing this section by translating the expansions (2.15), (2.20) of EYM amplitudes into the language of CHY formulation (see [31]). In CHY formulation, the recursive expansion (2.15) reflects

Gauge invariance induced relations
In this section, we induce nontrivial generalized BCJ relations for color-ordered Yang-Mills amplitudes (also bi-scalar amplitudes and color-ordered NLSM amplitudes) by combining gauge invariance conditions with CHY inspired dimensional reductions. The coefficients of amplitudes in the gauge invariance induced relations are polynomials of Mandelstam variables.

Inducing generalized BCJ relations by gauge invariance and dimensional reduction
In the pure Yang-Mills expansion (2.20) of EYM amplitude A(1, 2, . . . , r H), each term C [F ] (1, σ σ σ, r) (see (2.25)) of the expansion coefficient C(1, σ σ σ, r) is expressed as a product of Lorentz invariants · k, · and k · k and constructed by the grapic rule in section 2. For a given graph in the expansion of C(1, σ σ σ, r), the graviton h can be either an internal node or a starting point of a chain. In the former case, the gauge invariance condition is naturally encoded by F µν h | h →k h = 0, thus this contribution has to vanish. The only nontrivial contributions are those graphs in which the graviton h plays as the starting point of a chain. The gauge invariance condition is then reduced to where G σ σ σ H [h] denotes the set of graphs for permutation σ σ σ, where h plays as starting point of a chain. As shown by examples in [29,32] ( similar discussions on the gauge invariance relations can be found in [25,26,30,[43][44][45]), (3.2) is generated by known BCJ relations, thus it is not new relation beyond known BCJ relations. Nevertheless, a systematical study on the connection between (3.2) and the standard KK and BCJ relations still deserves future work.
Coefficients in the relation (3.2) still contain polarizations. To induce a relation where coefficients are only functions of Mandelstam variables s ij = k i · k j , we should 'turn' all polarizations in the expansion of coefficients to momenta. One reasonable approach to realize this point is combining gauge invariance conditions with dimensional reduction inspired by CHY formulation. Our discussion is based on the following crucial observations: (1) Gauge invariance conditions for more than one graviton can be imposed simultaneously. This can be understood from two different aspects. (i) Since the pure Yang-Mills expansion (2.20) is obtained by applying the recursive expansion (2.15) repeatedly, we can take gauge invariance condition for (2.15) instead. If we replace ha by k ha for more than one graviton h a ∈ A ⊆ H (A consists of at least two gravitons) on the RHS of (2.15), there is at most one graviton plays as the fiducial one. The polarizations of the rest of the gravitons belonging to A are contained by either F µν or an EYM amplitude with fewer gravitons. When replacing ha by k ha for all h a ∈ A on the RHS of (2.15), every term has to vanish due to the antisymmetry of F µν or/and the gauge invariance condition for EYM amplitudes with fewer gravitons (as an inductive assumption). (ii) In the language of CHY formula (2.1), polarizations are packaged into (reduced) Pffafians. When the replacement h → k h for a given graviton h ∈ H is imposed, the Ψ H matrix becomes degenerate because two rows/columns coincide with each other (Noting the diagonal entry C haha for C matrix vanishes due to scattering equation (2.2)) as shown by the left matrix in the following · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · If we take gauge invariance conditions for more than one graviton, e.g. h a and h b , the matrix Ψ is also degenerate for the same reason (see the right matrix in (3.3)), thus the Pfaffian has to vanish.
(2) The gauge invariance conditions are independent of dimensions. This is because the statements (i) and (ii) in (1) hold for arbitrary dimension space.
Having (1) and (2) are satisfied. According to our observations (1) and (2) Once the coefficients C(1, σ σ σ, r) in the above equation are expressed by graphs (see eq. (2.25)) and the gauge invariance conditions are imposed, a chain in which any h a ∈ A ⊆ H plays as an internal node vanishes due to the antisymmetry of the Thus only those graphs where all h a ∈ A play as starting points of chains survive. The relation (3.5) then turns to Here, G σ σ σ H [A] denotes the set of graphs corresponding to the permutation σ σ σ, where all elements in the nonempty subset A play as starting points of chains (Note that other elements in H may also be starting points of chains).
The equation (3.6) does not rely on details of (d + d)-dimensional polarizations E and momenta K, only the conditions (3.4) are required. Thus, we can assign details of polarizations and momenta in (d + d) dimensions appropriately s.t. (3.4) is satisfied. A reasonable definition inspired by the dimensional reduction strategy (see (2.10)) in the CHY formula is which apparently satisfies (3.4). With this assignment, the coefficients in the gauge invariance condition (3.6) become polynomial functions of Mandelstam variables. When the coefficients C(1, σ σ σ, r) in (d + d) dimensions are expressed by the graphic rules and E ha in C(1, σ σ σ, r) are replaced by K ha (h a ∈ A ⊆ H), chains in the graphs are classified into two types: A chain of this type has to vanish if its length is odd, because we cannot avoid a factor of the form E i · K j which is zero in the definition (3.7). Thus the length of nonvanishing type-1 chains must be even. When plugging the components (3.7) into an even-length chain of type-1, we get a chain expressed by d-dimensional Mandelstam variables associated with a factor (−1) j+1 2 , where j is odd. Since the length L of this chain is j +1, the prefactor can be given by (−1) (3.10) A chain of this type vanishes if its length is even, for an even-length type-2 chain must contain a vanishing factor of the form E i · K j . Thus the length of nonvanishing type-2 chains are odd. Inserting the choice of (d + d)-dimensional polarizations and momenta (3.7) into an odd-length chain of this type, we arrive associated with a factor (−1) j 2 , where j is even. The prefactor for this chain is further expressed by the length L of the chain as (−1) Collecting all nonzero chains together, we induce the following relation for PT factors in d dimensions from the (d + d)-dimensional gauge invariance condition (3.5): (3.13) To translate the gauge invariance induced relation (3.12) for Parke-Taylor factors into amplitude relation, we consider the expression where I R can be I BS R , I YM In the gauge invariance induced relation (3.15), the nonempty subset A cannot contain only one element because the total length of all chains is an even number 2. If A contains for example h 1 , i.e., there is an odd-length chain started by h 1 , we must have another odd-length chain started by h 2 so that the total length of all chains is even. Thus the nonempty subset A of H can only be chosen as {h 1 , h 2 } while h 1 and h 2 are starting points of two length-1 chains in this example. The graph (b) which contains a length-2 chain does not appear in our gauge invariance induced relation. The relation (3.15) s ai . This relation is in agreement with a fundamental BCJ relation.

H
If A contains only one element h 1 . Then h 1 must leads to a length-1 chain while h 3 must leads to a length-2 chain s h 3 h 2 s h 2 a with an internal node h 2 . Among the graphs in figure 2, only (a5) (for the relative order {h 1 , h 2 , h 3 }), (c3), (c4) (for the relative order {h 2 , h 1 , h 3 }) and (d2), (d4), (d6) (for the relative order This relation is consistent with a fundamental BCJ relation.
Again, the vanish of RHS can be considered as a result of fundamental BCJ relation.
can start either a length-3 chain or a length-1 chain. In the former case, both h 1 and h 2 must be internal nodes of the length-3 chain ((a6) and (c6) in figure 2), while in the latter case h 2 must start a length-2 chain with h 1 as the internal node ((a2), (b3), (e5) and (e6) in figure 2). All together, the relation (3.15) turns to which is not as trivial as previous examples. One can check this identity by expanding all amplitudes in terms of BCJ basis amplitudes.

Three distinct constructions of BCJ numerators in NLSM
Now let us review the DF, CMS and DT types of BCJ numerators which correspond to the Feynman diagram approach, Abelian Z theory and CHY formula.

The DF type numerators
The DF type BCJ numerator was derived by applying off-shell extended BCJ relation [8,16], which is based on Berends-Giele recursion (thus Feynman diagrams). The explicit expression of DF type BCJ numerator is given by a proper combination of momentum kernel: 5 where we summed over permutations ρ ρ ρ in Γ which is defined as the collection of permutations satisfying the following conditions. For any a ∈ {2, . . . , n − 1}, we assume b (c) is the nearest element on the LHS (RHS) of a in the permutation ρ ρ ρ, which satisfies 6 . The permutations ρ ρ ρ in the DF type numerator The CMS type numerators The CMS type BCJ numerator, which comes from Abelian Z theory [23], expresses each numerator in dual DDM decomposition by only one momentum kernel: n CMS 1|σ σ σ|n = (−1)  It is worthy emphasizing that both DF and CMS types BCJ numerators manifest the relabeling symmetry of n − 2 elements, i.e., n 1|σ(2),...,σ(n−1)|n can be obtained from n 1|2,...,n−1|n by the replacement 2, 3, . . . , n − 1 → σ(2), σ(3), . . . , σ(n − 1). 5 We adjust the total sign by (−1) to agree with the CMS type numerators. 6 Here 1 and n are correspondingly considered as the first and the last elements in both permutations σ σ σ and ρ ρ ρ. There is always a particle n (maybe not the nearest ) on the RHS and LHS of a in the permutation ρ ρ ρ s.t. σ −1 (n) = n > σ −1 (a) in the sense of cyclicity, see [16].

The DT type numerators
Being different from the previous two constructions, the DT type numerator which is based on the graphic expansion of amplitudes and the dimensional reduction in CHY formula is not a symmetric form. This type of BCJ numerators are expanded by graphic rule instead of momentum kernels. The construction of n DT 1|σ σ σ|n is given by . This chain is associated with a factor Remove this chain from the ordered set R → R = R \ {i 1 , i 2 , . . . , i j , ρ(s)} ≡ {ρ (1), . . . , ρ (s )}.
• Repeat the above steps until the ordered set R becomes empty. Each new even-length chain is attached to nodes which have been used and associated with a factor. Collecting the factors corresponding to all chains in a graph and summing over all possible graphs (noting that the total phase factor is (−1) is adjusted by (−1) to agree with that in CMS type. This adjustment does not affect our discussions.

The equivalence between DT and CMS constructions of NLSM amplitudes
The DT and the CMS types of numerators produce the same amplitude if and only if the second equality in (4.2) holds. Substituting (4.10) and (4.6) into (4.2), we arrive the following relation for bi-scalar amplitudes A(1, σ σ σ, n) (ii) The number of external particles is not limited to be even. Amplitudes with odd number external particles are also under consideration.
Having the above generalizations, we will prove the following two relations

Expressing partial momentum kernel by graphs
The partial momentum kernel S H [α α α ∈ {2, . . . , r − 1} ¡ σ σ σ H |2, . . . , r − 1, σ σ σ H ] can be conveniently expanded by the graphic rule in section (2.2), when replacing the factors ha · F h i 1 · · · · · F h i j · k b for each chain by  We now consider the partial momentum kernel where H contains three elements and σ σ σ H in this example is chosen as σ σ σ H = {h 1 , h 3 , h 2 }. From the definition (3.24), (5.8) is given by the product of three factors This partial momentum kernel can be obtained as follows: • Define a reference order of elements in H, e.g., R = {h 1 , h 2 , h 3 }.
• Pick the last element h 3 in the ordered set R = {h 1 , h 2 , h 3 } and pick a term from the factor corresponding to h 3 in (5.9). Such a term has the form s h 3 j , where j can be any element in , then a chain s h 3 h 1 s h 1 k started from h 3 towards k have been constructed. We take the j = h 1 case for instance and continue our discussion.
• Remove the starting node h 3 and the internal node h 1 of the chain which have been already constructed, from the ordered set R = {h 1 , h 2 , h 3 } and redefine R as R → R = {h 2 }. Construct a chain started from the element h 2 in R towards l ∈ {h 1 , h 3 } ∪ {1, 2, . . . , r − 1}. Then we have a factor s h 2 l . For example, we choose l = h 1 .
• Remove h 2 from R , then the set R becomes empty. Putting the chains obtained together, we arrive a term s h 3 h 1 s h 1 k s h 2 h 1 corresponding to the graph (b4) of figure 2.
• The full partial momentum kernel in this example is obtained by summing over all possible graphs constructed by the above steps (displayed by the graphs (b1) ∼ (b6) in figure 2).
Again, we emphasize that the reference order R can be chosen arbitrarily. If we change the reference order, only the chains are changed, the structure of graphs and the final expression of partial momentum kernel are not changed. Now we extend our discussions to the graphic expansion of any partial momentum kernel with the form: (5.10) • Define a reference order R = {h ρ(1) , h ρ(2) , . . . , h ρ(s) } for elements in the set H (assume there are s elements in the set H). Pick h ρ(s) and an arbitrary term s h ρ(s)h i j (σ −1 (h i j ) < σ −1 (h ρ(s) )) from the factor corresponding to h ρ(s) . Then pick an arbitrary term ) from the factor corresponding to h i j . Next, pick a term of the form ) from the factor corresponding to h i j−1 , and so on. This procedure is terminated at a factor s h i 1 l where l belongs to the set {1, 2, . . . , r−1}. Putting all factors together, we get a chain s h ρ(s)h i j s h i j h i j−1 . . . s h i 1 l . Redefine R by removing the internal nodes and the starting point of the chain which was already constructed: • We construct a chain from h ρ (s ) towards an element l ∈ {1, 2, . . . , r} .., h i 1 in the partial momentum kernel (5.10). The we get another chain • Repeat the above steps until the R set becomes empty. Then putting all chains together, we get a graph. The sum of all possible graphs gives the partial momentum kernel (5.10).

Proof of the relations (5.3) and (5.4)
We have already shown that the equivalence condition (5.1) is a special case of the relation (5.3) with even s. In addition, we also have the relation (5.4) with odd s. Now let us prove both relations (5.3) and (5.4) by expanding the partial momentum kernels into graphs.  α α 2, . . . , r − 1, h 2 , h 1 A(1, α α α, r). (5.11) Expanding the partial momentum kernels into graphs (see figure 1), we rewrite the above expression as Example-2: H = {h 1 , h 2 , h 3 , h 4 } Inspired by the previous example with s = 2, one can expand all partial momentum kernels on the LHS of (5.3) in terms of graphs for a given reference order R. For the case H = {h 1 , h 2 , h 3 , h 4 }, the total length of all chains of each expansion graph should equal to 4. On the other hand, the total length L total of all chains is given by where L odd and L even denote the total lengths of all odd-and even-length chains, respectively. If a graph contains odd number of odd-length chains, the total length must be odd according to the above equation. This conflicts with the fact L total = 4. Therefore, the number of odd-length chains must be even.  If j = 0, the set {h i 1 h i 2 , . . . , h i j } ⊆ H is nonempty. Such a term has to vanish due to the gauge invariance induced relation (3.15) for the nonempty subset A with even number of elements. The first term in (5.18) (the case j = 0) is given by summing over all graphs consisting of only even length chains, which is the RHS of (5.3).

The proof of (5.4)
The first nontrivial example of (5.4) for odd s is given by H = {h 1 , h 2 , h 3 }. Let us study this case before the general proof of (5.4).
Example: H = {h 1 , h 2 , h 3 } We expand the partial momentum kernels on the LHS of (5.4) in terms of graphs for a fixed reference order R = {h ρ(1) , h ρ(2) , h ρ(3) }. For a given graph, the total length of all chains must be 3. As a consequence, the number of odd length chains in each graph must be odd (in this example it can be 1 or 3). Thus the LHS of (5.4) for s = 3 is decomposed into The combination of amplitudes in the LHS of (5.4) leads to D [F ] (1, α α α, r) A(1, α α α, n), (5.22) in which, all terms must vanish due to the gauge invariance induced relation (3.15) for A with odd number of elements. Thus the relation (5.4) is proven.

The equivalence between DF and CMS constructions of NLSM amplitudes
The equivalence between DF and CMS constructions of NLSM amplitudes, i.e., the first equality of (4.2) can be explicitly expressed by the following amplitude relation where Γ is defined in section 4. In this section, we will prove the relation (6.1). The identity (5.6) (as a consequence (5.4)) with odd s is crucial for the proof. To show the pattern, let us first discuss the fourand six-point examples as a warmup.

Warm-up examples
Now we take the cases with n = 4 and n = 6 as examples.

Four-point example
The simplest example is the four-point case, which have already been discussed in [23] and [24]. The LHS of the relation (6.1) for n = 4 is explicitly written as which is the RHS of (6.1) for four-point case.

Six-point example
The relation (6.1) for six-point amplitudes is much more nontrivial. By substituting the six-point numerators of DF type (4.5) into the LHS of (6.1), we get To prove this expression equals to the RHS of (6.1) with n = 6, we perform our discussions by the following steps.

Conclusions
In this paper, we derived highly-nontrivial generalized BCJ relation (3.15) by imposing gauge invariance and dimensional reduction on the graphic expansion of EYM amplitudes. Two additional relations (5.3) and (5.4) expressed by partial momentum kernels are consequent results of the gauge invariance induced relation (3.15). As an application, we proved the equivalence between amplitudes constructed by three different types of BCJ numerators. Thus the three approaches (Feynman rules, Abelian Z theory and CHY formula) to NLSM amplitudes are equivalent to each other. This way we prove the CHY formula of NLSM directly instead of relying on incomplete evidence, like the enhanced soft behavior [46].
There are several further directions. (i) First, generalized BCJ relations induced from the gauge invariance of multi-trace amplitudes deserves further consideration. (ii) Second, it seems that the CHYinspired dimensional reduction is not the unique way to reduce the Lorentz invariants to pure Mandelstam variables. Along the line of unifying relation [47], one can also turn the polarizations to momenta. In addition, other formulations of gauge invariance identities were depicted in [26,30,[43][44][45]. Thus it will be interesting to give a more comprehensive understanding of the gauge invariance induced relations by considering [47] and [26,30,[43][44][45] 8 . (iii) As we have seen, the gauge invariance induced relations bridge the DF type BCJ numerators of NLSM amplitudes and the compact CMS type ones . Maybe they will help us to find compact polynomial BCJ numerators of YM amplitudes which are independent of any reference ordering from that of DF type. We know the sum of BCJ numerators of all possible reference orderings satisfy this requirement, but how about more compact ones? (iv) Last but not least, the gauge invariance induced relations should also exists in string theory. How about their applications in string amplitudes? B Proof of (6.23) To prove the relation (6.23), we consider the LHS for a given j: ρ ρ ρ∈S n−2 T (ρ(2)| . . . |ρ(2j)|ρ(2j + 1), . . . , ρ(n − 1)).