aa r X i v : . [ h e p - t h ] D ec Preprint typeset in JHEP style - HYPER VERSION
Amplitude Relations in Non-linear Sigma Model
Gang Chen a , Yi-Jian Du b,c ∗ a Department of Physics, Nanjing University 22 Hankou Road, Nanjing 210093, China b Department of Physics and Center for Field Theory and Particle Physics, Fudan University,220 Handan Road, Shanghai 200433, P.R China c Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA [email protected]; [email protected]
Abstract:
In this paper, we investigate tree-level scattering amplitude relations in U ( N ) non-linear sigmamodel. We use Cayley parametrization. As was shown in the recent works [23, 24], both on-shell amplitudesand off-shell currents with odd points have to vanish under Cayley parametrization. We prove the off-shell U (1) identity and fundamental BCJ relation for even-point currents. By taking the on-shell limits of theoff-shell relations, we show that the color-ordered tree amplitudes with even points satisfy U (1)-decouplingidentity and fundamental BCJ relation, which have the same formations within Yang-Mills theory. Wefurther state that all the on-shell general KK, BCJ relations as well as the minimal-basis expansion arealso satisfied by color-ordered tree amplitudes. As a consequence of the relations among color-orderedamplitudes, the total 2 m -point tree amplitudes satisfy DDM form of color decomposition as well as KLTrelation. Keywords:
Amplitude relations, non-linear sigma model. ∗ Corresponding author ontents
1. Introduction 12. Preparation: Feynman rules and Berends-Giele recursion 4
3. Off-shell and on-shell U (1) identity from Berends-Giele recursion 5
4. Off-shell and on-shell fundamental BCJ relation from Berends-Giele recursion 11
5. General KK, BCJ relations, minimal-basis expansion and formulations of total ampli-tudes 18
6. Conclusion 21A. Convention of notation 21B. Eight-point diagrams 22C. Diagrams contribute to J ( { B } ) J ( { B } ) . . . J ( { B M } )
1. Introduction
One of the most significant progresses of scattering amplitudes in recent years is the discovery of newamplitude relations. The new relation (BCJ relation) was firstly proposed in Yang-Mills theory by Bern,Carrasco and Johansson[1]. Using BCJ relation in addition with KK relation which was earlier suggested– 1 –y Kleiss and Kuijf [2], one can simplify the calculations on color-ordered amplitudes at tree level. Inparticular, these relations provide a reduction of the basis of n -point tree-level amplitudes to a minimalbasis of ( n − , the proof of general BCJ relationwas given in [11]). The minimal-basis expansion has been proved [11] via so-called general BCJ relation.KK and BCJ relations in Yang-Mills theory can be regarded as results of color-kinematic duality [1]. In[1], it was pointed that one could express the amplitudes by Feynman-like diagrams with only cubic verticesand establish a duality between color factors and kinematic factors. Once the color factors satisfy somealgebraic property (antisymmetry and Jacobi identity), so do the corresponding kinematic factors. In fact,KK relation among color-ordered amplitudes can be considered as a result of antisymmetry of kinematicfactors, while, BCJ relation is a result of Jacobi identity. The kinematic factors in Yang-Mills theorycan be constructed from pure spinor string theory [12]. They can also be constructed by area-preservingdiffeomorphism algebra [13, 14] or a more general diffeomorphism algebra[15]. A further understanding ofthe kinematic algebra is the construction of color-dual decomposition and trace-like objects [16, 17, 18].It is interesting that KK and BCJ relations can be found not only in pure Yang-Mills theory butalso in other theories. For example, relations for amplitudes with gauge field coupled with matter wasinvestigated in [19]. In N = 4 super Yang-Mills theory, the super-amplitudes are also proven to satisfy KKand BCJ relations [20]. In [21], the KK and BCJ relations was proven to hold for color-scalar amplitudes.Though these amplitude relations are found in different theories, they have similar forms with the relationsin Yang-Mills theory. This is because the color-kinematic duality implies that different theories with colorfactors satisfying the same algebraic properties should have the similar form of amplitude relations. Whenthe algebraic properties are changed, the amplitude relations should also be changed. This can be furthersupported by the amplitude relations in three dimensional supper symmetric theory with 3-algebra [22].In this case, the algebraic properties of color factors are changed to the properties of 3-algebra, the formof amplitude relations are also changed to agree with the algebraic structure.Beyond the fundamental field theory, there are lots of interesting low energy effective theories whichare also widely used in the phenomenology of low energy physics. One of them is the well-known SU ( N )non-linear sigma model. This theory describes the low energy dynamics of the Goldstone Bosons underthe chiral symmetry breaking SU ( N ) L × SU ( N ) R → SU ( N ). In this paper, we focus on the relations oftree-level amplitudes in U ( N ) non-linear sigma model. For on-shell amplitudes, the result can apply to the SU ( N ) model directly. In recent works [23, 24], U (1)-decoupling identity was discussed via the decouplingof U (1) field from interaction, and color-order reversed relation was also pointed in [24]. These resultsencourage us to study the full amplitude relations in non-linear sigma model systematically. We expect Other approaches to fundamental BCJ relation can be found in [9], [10]. – 2 –hat there should be KK and BCJ relations, which share the same forms with the relations in Yang-Millstheory. This is because the color factors in these two cases satisfy the same algebraic properties. However,the kinematic factors which share the same algebraic properties cannot easy to construct because of theinfinity of the number of vertices in non-linear sigma model. The general amplitude relations are also notobvious along the decoupling argument in [23, 24]. In fact, the arguments on U (1)-decoupling identityin [23, 24] are valid for only on-shell amplitudes. When we consider the even-point off-shell currentsconstructed by Feynman rules, the U (1)-field under Cayley parametrization [23, 24] do not decouple frominteraction. This is quite different from in case of Yang-Mills theory where both on-shell amplitudes andoff-shell currents satisfy KK relation (the KK relation in off-shell case in Yang-Mills theory was proven inthe appendix of [15]). Furthermore, the highly nontrivial relations-BCJ relations seem hard to obtainedfrom this argument. One may hope to prove the relations by using the nontrivial extension of BCFWrecursion in non-linear sigma model [23, 24] and follow the similar proof within Yang-Mills case [8, 11],but it will be not easy to use the Even(odd)-shift form of the BCFW recursion [23, 24] to prove even if thesimple case- U (1) decoupling identity.In this work, we will use Berends-Giele recursion under Cayley parametrization to study the rela-tions. Since the odd-point amplitudes vanish [23, 24], we only need to study the relations for even-pointamplitudes. We conjecture and prove U (1) identity and fundamental BCJ relation for even-point off-shellcurrents. We will find that, the left hand side of the the U (1) identity and fundamental BCJ relation mustequal to terms proportional to ( p ) , where p is the momentum of the off-shell leg. When we turn ourattention to on-shell amplitudes, we should multiply the current by p and take the on-shell limit p → U (1)-decoupling identity and fundamental BCJ relations for on-shell amplitudes. We willleave the proof of general off-shell relations in future work.Though it will be hard to derive off-shell general BCJ relation from either Berends-Giele recursionor BCFW recursion, [25] provides another method to prove the general KK and BCJ relations. It waspointed out that all the on-shell general KK and general BCJ relations can be generated by the fundamentalBCJ relation as well as cyclic symmetry. In non-linear sigma model, at on-shell case, both fundamentalBCJ relation and cyclic symmetry are satisfied, thus we also have general KK and general BCJ relations.Since the general KK and BCJ relations are satisfied, consequent results such as minimal-basis expansion,Del Duca-Dixon-Maltoni(DDM) color decomposition [5] and the (2 n − n -point amplitudes can be derived.The structure of this paper is following. In section 2, we provide a short review of Feynman rules andBerends-Giele recursion in non-linear sigma model. In section 3, we will prove the off-shell U (1) identity. Although, in non-linear sigma model, one may use flavor factor instead of color factor, as was done in [24] for physicalreason, we will use color through this paper for convenience. We hope this will not make any confusion. Berends-Giele recursion was firstly given in Yang-Mills theory in [26]. The Berends-Giele recursion in non-linear sigmamodel was proposed in the recent work [23, 24]. In off-shell case, we use ’ U (1) identity’ instead of ’ U (1)-decoupling identity’ because in the off-shell case, the U (1) gaugefield in general cannot decouple. Only in the on-shell case, the U (1) gauge field decouples. – 3 –e first give some examples then the general proof. In section 4, we will prove the off-shell BCJ relation.We also give examples before general proof. After taking the on-shell limits of the off-shell KK and BCJrelations, we can obtain the U (1)-decoupling identity and fundamental BCJ relation for on-shell amplitudesimmediately. In section 5, we use the conclusions of the work [25] to state that all the on-shell general KKand BCJ relations can be generated by the on-shell fundamental BCJ relation as well as cyclic symmetry.Thus the on-shell general KK and general BCJ relations are naturally satisfied. We also point out that theminimal-basis expansion of color-ordered amplitudes, DDM color decomposition and the (2 m − m -point total amplitudes are also satisfied. In section 6, we summarize this work.Useful diagrams and convention of notations are included in appendix.
2. Preparation: Feynman rules and Berends-Giele recursion
In this section, we review the Feynman rules and Berends-Giele recursion in non-linear sigma model whichare useful through this paper. Most of the notations follow the recent works [23, 24].
Lagrangian
The Lagrangian of U ( N ) non-linear sigma model is given as L = F ∂ µ U ∂ µ U † ) , (2.1)where F is a constant. As in [23, 24], we use Cayley parametrization. Under Cayley parametrization U isdefined as U = 1 + 2 ∞ X n =1 (cid:18) F φ (cid:19) n . (2.2)Here φ = √ φ a t a and t a are generators of U ( N ) Lie algebra. Trace form of color decomposition
The total tree amplitudes can be given in terms of color-ordered amplitudes by trace form of colordecomposition M (1 a , . . . , n a n ) = X σ ∈ S n − Tr( T a T a σ . . . T a σn ) A (1 , σ ) . (2.3)Since the traces have cyclic symmetry, the color-ordered amplitudes also satisfy cyclic symmetry A (1 , , . . . , n ) = A ( n, , . . . , n − . (2.4) Feynman rules for color-ordered amplitudes
Vertices in color-ordered Feynman rules under Cayley parametrization (2.2) are V n +1 = 0 , V n +2 = (cid:18) − F (cid:19) n n X i =0 p i +1 ! = (cid:18) − F (cid:19) n n X i =0 p i +2 ! , (2.5)where momentum conservation has been considered.– 4 – .2 Berends-Giele recursion In the Feynman rule given by the previous subsection, one can construct tree-level currents with oneoff-shell line through Berends-Giele recursion J (2 , ..., n )= iP ,n n X m =4 X j 3. Off-shell and on-shell U (1) identity from Berends-Giele recursion In this section, we prove the U (1) identity satisfied by even-point currents. The identity is given as X σ ∈ OP ( { α } S { β ,...,β m } ) J ( { σ } ) = 12 F X divisions { β }→{ B } , { B } J ( { B } ) J ( { B } ) , (3.1)where, on the left hand side, we sum over all the possible permutations with keeping the relative ordersin { β } set and there is only one element α in { α } set. On the right hand side, we divide the ordered set { β , . . . , β m } into two nonempty subsets. In each subset, there are odd number of β ’s. For example, ifthere are six β ’s, there are three possible divisions { B } = { β } , { B } = { β , . . . , β } ; { B } = { β , β , β } , { B } = { β , β , β } and { B } = { β , . . . , β } , { B } = { β } .When we want to get the on-shell relations between amplitudes from the identity (3.1), we shouldmultiply both sides of (3.1) by p = ( p α + p β + · · · + p β m ) and take the limit p → 0. Since the righthand side are products of currents which are finite when p goes to zero, after multiplied by p , the righthand side has to vanish under p → 0. Then we arrive at on-shell U (1)-decoupling identity immediately X σ ∈ OP ( { α } S { β ,...,β m } ) A (1 , { σ } ) = 0 . (3.2) In this paper, an n -point current is mentioned as the current with n − – 5 –t is worth comparing the U (1) identities in non-linear sigma model and in Yang-Mills theory. In Yang-Mills theory, U (1)-decoupling identities in both on-shell and off-shell cases have the same form. Thus, inboth off-shell and on-shell cases, the identities can be understood as the decoupling of U (1)-gauge field.However, in non-linear sigma model, the U (1) field can only decouple in the on-shell case. In off-shell case,at least for the choice of Cayley parametrization, we get sum of products of two sub-currents. In otherwords, only when taking the on-shell limit, the U (1) field decouples.Before proving the identity (3.1), let us have a look at two examples. In four-point case, the U(1)-identity is J ( α , β , β ) + J ( β , α , β ) + J ( β , β , α ) = 12 F J ( β ) J ( β ) = 12 F . (3.3)This is easy to prove by substituting the four-point vertex into the left hand side directly J ( α , β , β ) + J ( β , α , β ) + J ( β , β , α )= − F ip i (cid:2) ( p + p β ) + ( p + p α ) + ( p + p β ) (cid:3) = − F ip i (cid:2) p + p α + p β + p β (cid:3) = 12 F . (3.4)where 1 is the off-shell line and we have used the on-shell conditions p α = 0, p β = 0, p β = 0. Now let us skip the proof of six-point U (1) identity and show how to use lower-point identity to proveeight-point U (1) identity. The eight-point U (1) identity is given as X σ ∈ OP ( { α } S { β ,...,β } ) J ( { σ } )= 12 F [ J ( β ) J ( β , . . . , β ) + J ( β , β , β ) J ( β , β , β ) + J ( β , . . . , β ) J ( β )]= 12 F [ J ( β , . . . , β ) + J ( β , β , β ) J ( β , β , β ) + J ( β , . . . , β )] . (3.5)To prove this relation, we first show the explicit expression of Fig. 1 and Fig. 2. • Fig. 1 can be expressed asFig. 1= − F ip i (cid:2) p + p α + p B + p B (cid:3) J ( { B } ) J ( { B } )– 6 – igure 1: Sum of diagrams with α connected with the off-shell leg directly via four-point vertex in U (1) identity. Figure 2: A diagram with lower-point substructure of U (1) identity. = 12 F J ( { B } ) J ( { B } ) + 12 F p (cid:2) p B J ( { B } ) (cid:3) J ( { B } ) + 12 F J ( { B } ) 1 p (cid:2) p B J ( { B } ) (cid:3) = 12 F J ( { B } ) J ( { B } )+ 1 p X divisions { B }→{ B } ... { B i +1 } (cid:18) − F (cid:19) i +1 V i +2 ( − P B ,B i +1 , P B , . . . P B i ) × J ( { B } ) J ( { B } ) . . . J ( { B i +1 } )+ 1 p X divisions { B }→{ B } ... { B i +1 } (cid:18) − F (cid:19) i +1 V i +2 ( − P B ,B i +1 , P B , . . . P B i ) × J ( { B } ) J ( { B } ) . . . J ( { B i +1 } ) , (3.6)where we have used the on-shell condition p α = 0. p B i denotes the sum of momenta of the on-shelllines in the set { B i } . • Fig. 2 can be expressed explicitly by using lower-point U (1) identitiesFig. 2 = X divisions { B i }→{ B i }{ B i } (cid:18) − F (cid:19) M p V ( p , p B , . . . , p B i − , p B i , p B i +1 , . . . , p B M − )– 7 – J ( { B } ) . . . J ( { B i − } ) J ( { B i } ) J ( { B i } ) J ( { B i +1 } ) . . . J ( { B M − } ) . (3.7)By Berends-Giele recursion, we can express the left hand side of eight-point U (1) identity (3.5) by sumof the diagrams in Fig. 7. We can always use (3.7) to reduce sum of the terms with sub-currents containingboth α and elements in { β } into products of currents with only β element. Thus the left hand side of (3.5)can be expressed in terms of J ( { B } ) . . . J ( { B M } ), where { B } . . . { B M } is an nontrivial division of { β } .Each subset of this division must containing odd number of β elements because the odd-point current mustvanish. We can classify the products of sub-currents into three categories according to different number ofsub-currents • six sub-currents: J ( β ) . . . J ( β ) • four sub-currents: J ( β ) J ( β ) J ( β ) J ( β , β , β ), J ( β ) J ( β ) J ( β , β , β ) J ( β ), J ( β ) J ( β , β , β ) J ( β ) J ( β ) and J ( β , β , β ) J ( β ) J ( β ) J ( β ) • two sub-currents: J ( β ) J ( β , . . . , β ), J ( β , β , β ) J ( β , β , β ) and J ( β , . . . , β ) J ( β ).Now let us discuss these contributions one by one. (i) Six sub-currents: J ( β ) J ( β ) J ( β ) J ( β ) J ( β ) J ( β ) = 1. There are three parts of contributions A , B and C in this case. A part is (A.1) in Fig. 7 and can be given as A = i ip (cid:18) − F (cid:19) (cid:2) ( p α + p β + p β + p β ) + ( p β + p β + p β + p β ) + ( p β + p α + p β + p β ) + ( p β + p β + p β + p β ) + ( p β + p β + p α + p β ) + ( p β + p β + p β + p β ) + ( p β + p β + p β + p α ) (cid:3) . (3.8) B part is sum of (B.5), (B.6), (B.7), (B.8) and (B.9) in Fig. 7. Using the property (3.7), this part canbe given as B = (cid:18) F (cid:19) i ip (cid:2) ( p α + p β + p β + p β + p β ) + ( p β + p β + p β ) + ( p β + p α + p β + p β + p β ) + ( p β + p β + p β ) + ( p β + p β + p α + p β + p β ) (cid:3) . (3.9) C part gets contributions from the diagrams (C.1) and (C.3). Particularly, we apply the property (3.6)to these two diagrams, then we find that the division { β , β , β , β , β } → { β } , { β } , { β } , { β } , { β } of(C.1) and the division { β , β , β , β , β } → { β } , { β } , { β } , { β } , { β } of (C.3) contribute to this case. C can be expressed as C = ip i (cid:18) F (cid:19) ( p β + p β + p β ) + ip i (cid:18) F (cid:19) ( p β + p β + p β ) . (3.10)– 8 –onsidering all three parts, we find that A + B + C = 1 p (cid:18) F (cid:19) p α = 0 , (3.11)where we have used the on-shell condition of α . (ii) Four sub-currents: There are four different products of sub-currents J ( β , β , β ) J ( β ) J ( β ) J ( β ), J ( β ) J ( β , β , β ) J ( β ) J ( β ), J ( β ) J ( β ) J ( β , β , β ) J ( β ) and J ( β ) J ( β ) J ( β ) J ( β , β , β ). Now let usconsider J ( β , β , β ) J ( β ) J ( β ) J ( β ) as an example. The contributions of this case can also be classifiedinto three parts A , B , C . A part is given by (B.1) in Fig. 7 and can be expressed explicitly A = i ip (cid:18) F (cid:19) (cid:2) ( p α + p β + p β ) + ( p β + p β + p β + p β + p β ) + ( p β + p β + p β + p α + p β ) + ( p β + p β + p β + p β + p β ) + ( p β + p β + p β + p β + p α ) (cid:3) . (3.12) B part get contributions from (C.4), (C.11) and (C.12) in Fig. 7. Particularly, we apply the prop-erty (3.7) to (C.4), (C.11) and (C.12). Then (C.11), (C.12) and the division { β , β , β , β , β } →{ β } , { β } , { β } , { β } , { β } of (C.4) contribute to B . Thus B can be given as B = − (cid:18) F (cid:19) i ip ( p α + p β + p β + p β + p β + p β ) − (cid:18) F (cid:19) i ip ( p β + p β + p β + p β ) − (cid:18) F (cid:19) i ip ( p β + p β + p β + p α + p β + p β ) . (3.13) C part gets contributions from (C.2) and (C.3). Particularly, when applying (3.6) to (C.2) and (C.3).The divisions { β , β , β } → { β } , { β } , { β } of (C.2) and { β , β , β , β , β } → { β , β , β } , { β } , { β } of(C.3) contribute to this case. Thus C part is given as C = − i ip (cid:18) F (cid:19) (cid:2) ( p β + p β ) + ( p β + p β + p β + p β ) (cid:3) . (3.14)Taking all three parts into account, we get A + B + C = 0 , (3.15)where we have used on-shell condition of α . Following a similar way, we find that the other products offour sub-currents also cancel out. (iii) Two sub-currents: There are three non-vanishing products of sub-currents J ( β ) J ( β , . . . , β ), J ( β , β , β ) J ( β , β , β ) and J ( β , . . . , β ) J ( β ). They can only get contributions from the three diagrams(C.1), (C.2) and (C.3). Particularly, we apply the property (3.6) to (C.1), (C.2) and (C.3). In this case,we need to keep the terms that of ( p ) in these three diagrams. Then we get12 F [ J ( β ) J ( β , . . . , β ) + J ( β , β , β ) J ( β , β , β ) + J ( β , . . . , β ) J ( β )] , (3.16)– 9 – α p B i +1 p B i s α B i +1 s α B i s B i +1 B j +1 s B iB j s B i +1 B j Type-A ( M + 1) 2( M − i ) 2 i M − i i n M − j ) ( i < j )0 (Otherwise) n i ( i < j )0 (Otherwise) n j − i ) − i < j )0 (Otherwise)Type-B − M − M − i ) + 1 − i + 1 − M + i − i n − M − j ) + 1 ( i < j )0 (Otherwise) n − i + 1 ( i < j )0 (Otherwise) n − j − i ) + 1 ( i < j )0 (Otherwise)Type-C 0 -1 -1 0 0 n − i < j )0 (Otherwise) n − i < j )0 (Otherwise) 0 Table 1: Coefficients of J ( { B } ) . . . J ( { B M } ) in U (1) identity. Here s α B u denotes 2 p α · P β p ∈{ B u } p β p ! , s B u B v denotes 2 P β p ∈{ B u } p β p ! · P β q ∈{ B v } p β q ! , u , v can be 2 i or 2 i + 1. For B i +1 , i runs from 0 to M − 1, while for B i , i runs from 1 to M . which is just the right hand side of the U (1) identity for eight-point currents.Therefore, after considering all the cases (i) (ii) and (iii), we get the U (1) identity (3.5) for eight-pointcurrents. Having shown the proof of the eight-point example, let us extend the proof to the general form of U (1)identity. In general, one can always express the left hand side of (3.1) by lower-point sub-currents viaBerends-Giele recursion (2.6). As in the eight-point examples, we can collect the diagrams with sameoff-shell momenta of sub-currents together. Then we can use the property (3.7) to reduce the diagramscontaining a substructure of U (1) identity (as shown in Fig. 2). After these reductions, the sub-currentscontaining both α and { β } elements are reduced to products of sub-currents with only elements in { β } set. Furthermore, we can apply (3.6) to a four-point structure in Fig. 1. After these reductions, we shouldread out the coefficients of J ( { B } ) . . . J ( { B M } ) for an arbitrary nontrivial division { β , . . . , β m } →{ B } . . . { B M } .For M > 1, as shown in the eight-point case, there are always three types of contributions Type-A,Type-B and Type-C in Fig. 8. The notations in these diagrams are defined by Fig. 5.For Type-A diagrams in Fig. 8, we can always use Feynman rules and momentum conservation to avoidthe appearance of the momentum of the off-shell leg 1 and express the coefficient of J ( { B } ) . . . J ( { B M } )by the on-shell momenta.For Type-B diagrams in Fig. 8, as have mentioned, we should substitute (3.7) into these diagrams toreduce them and keep the right divisions that can produce J ( { B } ) . . . J ( { B M } ). For example, we shouldkeep the division { B , B } → { B } , { B } in the first diagram and keep the division { B , B } → { B } , { B } in the second diagram, and so on. For convenience, we also express the vertices in Type-B diagrams bythe on-shell momenta via momentum conservation.For Type-C diagrams in Fig. 8, we should apply (3.6). For the first diagram of Type-C, we should keepthe division { B , . . . , B M } → { B } . . . { B M } while, for the second diagram we should keep the division { B , . . . , B M − } → { B } . . . { B M − } . – 10 –hen we can collect all the coefficients in the three types in Table 1. In Table 1, we have left a totalfactor ip i (cid:0) − F (cid:1) M apart. Thus, the total coefficient of J ( { B } ) J ( { B } ) . . . J ( { B M } ) is ip i (cid:0) − F (cid:1) M p α .Since p α = 0, the J ( { B } ) J ( { B } ) . . . J ( { B M } ) must vanish.For M = 1, there are only two sub-currents in the products. In this case, we only need to considerthe terms with ( p ) in the diagrams of the form in Fig. 1. We should sum over all the possible { B } and { B } and get p ip i (cid:18) − F (cid:19) X divisions { β }→{ B } , { B } J ( { B } ) J ( { B } ) , (3.17)which is just the right hand side of the off-shell U (1) identity (3.1). 4. Off-shell and on-shell fundamental BCJ relation from Berends-Giele recursion Having proven the U (1) identity, let us consider a more nontrivial relation-fundamental BCJ relation- innon-linear sigma model. Since the odd-point currents and amplitudes must vanish, we only need to considerthe relations for even-point currents and amplitudes. Being different from U (1) identity, fundamental BCJrelation has non-trivial coefficients accompanying with the currents or amplitudes. The general formula ofoff-shell fundamental BCJ relation is given as X σ ∈ OP ( { α } S { β ,...,β m − } ) X ξ σi <ξ α s α σ i J ( { σ } , β m )= − F X divisions { β }→{ B } , { B } X β i ∈{ B } s α β i J ( { B } ) J ( { B } ) , (4.1)where we use ξ i to denote the position of the leg i in permutation σ , we define ξ = 0, thus we alwayshave a s α in the coefficients for each currents on the left hand side. On the right hand side, we sum overall the possible divisions of the ordered set { β } into two sub-ordered sets { B } and { B } . Since J ( { B } )or J ( { B } ) must vanish when { B } or { B } have even number, the divisions that survive are those withboth odd number of elements in { B } and { B } . Since the right hand side is finite under p → 0, aftermultiplying p and taking the on-shell limit p → X σ ∈ OP ( { α } S { β ,...,β m − } ) X ξ σi <ξ α s α σ i A (1 , { σ } , β m ) = 0 . (4.2)The left hand side of fundamental BCJ relation can be understood as following. We move one externalleg α from the position next to the leg 1 to the position in front of the leg β m . For each position, we canwrite down a corresponding current(or amplitude) accompanied by a kinematic factor P ξ σi <ξ α s α σ i . Thenwe sum over all the currents with coefficients.Before giving the general proof of the relation (4.1), let us have a look at two examples.– 11 – .1 Four-point example The simplest example is the four-point fundamental BCJ relation s α J ( α , β , β ) + ( s α + s α β ) J ( β , α , β ) = − (cid:18) F (cid:19) s α β J ( β ) J ( β ) . (4.3)To see this, we write the currents on the left hand side of BCJ relation (4.3) explicitly via Feynman rules s α J ( α , β , β ) + ( s α + s α β ) J ( β , α , β )= − (cid:18) F (cid:19) i ip (cid:2) s α ( p α + p β ) + ( s α + s α β )( p β + p β ) (cid:3) J ( β ) J ( β )= − (cid:18) F (cid:19) s α β J ( β ) J ( β )= − (cid:18) F (cid:19) s α β , (4.4)where we have used momentum conservation and on-shell conditions of α , β and β . Thus we haveproved the fundamental BCJ relation (4.3) at four-point. The four-point example in above subsection just provides a starting point of an inductive proof. In thissubsection, we skip the proof of fundamental BCJ relation at six-point and assume that the relation (4.1)is satisfied at both four- and six- points. We will show how to prove the eight-point relation recursively.Fundamental BCJ relation for eight-point currents is given as X σ ∈ OP ( { α } S { β ,...,β } ) X ξ σi <ξ α s α σ i J ( { σ } , β )= − F X divisions { β ,...,β }→{ B } , { B } X β i ∈{ B } s α β i J ( { B } ) J ( { B } ) , (4.5)where, on the right hand side, we sum over three nonzero divisions { β , . . . , β } → { β }{ β , β , β , β , β } , { β , . . . , β } → { β , β , β }{ β , β , β } and { β , . . . , β } → { β , β , β , β , β }{ β } .To prove this relation, we first show the explicit expressions of Fig. 3 and Fig. 4: • We first consider the sum of the two diagrams in Fig. 3. If we divide the ordered set { β , . . . , β m } into two ordered subsets { B } and { B } , then Fig. 3 is given asFig. 3= 12 F p (cid:2) s α ( p α + p B ) + ( s α + s α B )( p B + p B ) (cid:3) J ( { B } ) J ( { B } )= 12 F p ( s α p B − s α B p ) J ( { B } ) J ( { B } )– 12 – igure 3: Sum of diagrams with α connected with the off-shell leg directly via four-point vertex in BCJ relation. Figure 4: A diagram with lower-point substructure of BCJ relation. = − F p s α B p J ( { B } ) J ( { B } )+ 12 F p s α X divisions { B }→{ B } ... { B i +1 } (cid:18) − F (cid:19) i V i +2 ( − P B ,B i +1 , P B , . . . P B i +1 ) × J ( { B } ) J ( { B } ) . . . J ( { B i +1 } ) . (4.6) • Now let us consider Fig. 4. The left hand side of Fig. 4 can be reexpressed by the right hand sideof Fig. 4 by considering momentum conservation and on-shell condition of α . Since the first andsecond terms of the right hand side of Fig. 4 have substructures of fundamental BCJ relation and U (1) identity respectively, we can further reduce them by lower-point relations. Then we haveFig. 4 = 12 F X divisions { B i }→{ B i }{ B i } (cid:18) − F (cid:19) M − p ( s α B i + s α B i +1 + · · · + s α B M − )– 13 – V ( p , p B , . . . , p B i − , p B i , p B i +1 , . . . , p B M − ) × J ( { B } ) . . . J ( { B i − } ) J ( { B i } ) J ( { B i } ) J ( { B i +1 } ) . . . J ( { B M − } ) . (4.7)A special case is i = 2 M − 1. In this case, α i cannot be moved to the position next to the last elementof { B M − } . This case can also be included in Fig. 4 by considering momentum conservation andon-shell condition of α . Thus the property (4.7) also holds.With the above two properties, one can prove the eight-point fundamental BCJ relation (4.5). We canwrite the left hand side of eight-point fundamental BCJ relation by lower-point currents via Berends-Gielerecursion (2.6). The left hand side of (4.5) is given as sum of the diagrams in Fig. 7. For the diagrams inFig. 7, we can apply (4.7) to (B.5)-(B.9), (C.4)-(C.12) and apply (4.6) to (C.1), (C.2), (C.3). It is easy tosee that the left hand side of eight-point fundamental BCJ relation can be expressed in terms of productsof currents of the form J ( { B } ) J ( { B } ) . . . J ( { B M } ) after considering the property (4.7) and J ( α ) = 1,where { B } . . . { B M } are non-vanishing divisions of the ordered set { β , . . . , β } . Then we can read offthe coefficients for each division and prove the relation.The divisions can be classified in following cases • six sub-currents: J ( β ) . . . J ( β ) • four sub-currents: J ( β ) J ( β ) J ( β ) J ( β , β , β ), J ( β ) J ( β ) J ( β , β , β ) J ( β ), J ( β ) J ( β , β , β ) J ( β ) J ( β ) and J ( β , β , β ) J ( β ) J ( β ) J ( β ) • two sub-currents: J ( β ) J ( β , . . . , β ), J ( β , β , β ) J ( β , β , β ) and J ( β , . . . , β ) J ( β ).We can calculate the coefficients for these divisions one by one: (i) Six sub-currents: J ( β ) J ( β ) J ( β ) J ( β ) J ( β ) J ( β ) = 1. This case get contributions from threeparts A , B and C . A part is (A.1) in Fig. 7 and can be given as A = i ip (cid:18) − F (cid:19) h s α ( p α + p β + p β + p β ) + ( s α + s α β )( p β + p β + p β + p β ) +( s α + s α β + s α β )( p β + p α + p β + p β ) +( s α + s α β + s α β + s α β )( p β + p β + p β + p β ) +( s α + s α β + s α β + s α β + s α β )( p β + p β + p α + p β ) +( s α + s α β + s α β + s α β + s α β + s α β )( p β + p β + p β + p β ) i . (4.8) B part is the sum of (B.5), (B.6), (B.7), (B.8) and (B.9) in Fig. 7. Using the property (4.7), we get B = (cid:18) F (cid:19) i ip h − ( s α β + s α β + s α β + s α β + s α β )( p α + p β + p β + p β + p β ) – 14 – ( s α β + s α β + s α β + s α β )( p β + p β + p β ) − ( s α β + s α β + s α β )( p β + p α + p β + p β + p β ) − ( s α β + s α β )( p β + p β + p β ) − s α β ( p β + p β + p α + p β + p β ) i . (4.9) C part is the division { β , β , β , β , β } → { β } , { β } , { β } , { β } , { β } of (C.1). Particularly, this part isgiven as C = − ip i (cid:18) F (cid:19) ( s α β + s α β + s α β + s α β + s α β + s α β )( p β + p β + p β ) . (4.10)Considering momentum conservation and on-shell condition p α = 0, we can see A + B + C = 0. (ii) Four sub-currents: There are four different products of sub-currents J ( β , β , β ) J ( β ) J ( β ) J ( β ), J ( β ) J ( β , β , β ) J ( β ) J ( β ), J ( β ) J ( β ) J ( β , β , β ) J ( β ) and J ( β ) J ( β ) J ( β ) J ( β , β , β ). Let us take J ( β , β , β ) J ( β ) J ( β ) J ( β ) as an example. J ( β , β , β ) J ( β ) J ( β ) J ( β ) gets contributions from threeparts A , B and C . A part is the contribution of (B.1) in Fig. 7 and given as A = i ip (cid:18) F (cid:19) h s α ( p α + p β + p β ) + ( s α + s α β + s α β + s α β )( p β + p β + p β + p β + p β ) +( s α + s α β + s α β + s α β + s α β )( p β + p β + p β + p α + p β ) +( s α + s α β + s α β + s α β + s α β + s α β )( p β + p β + p β + p β + p β ) i . (4.11) B is the sum of the (C.11), (C.12) in Fig. 7 and the division { β , β , β , β } → { β , β , β } , { β } of(C.4) in Fig. 7. Particularly, we have B = − (cid:18) F (cid:19) i ip h − ( s α β + s α β )( p β + p β + p β + p β ) − s α β ( p β + p β + p β + p α + p β + p β ) − ( s α β + s α β + s α β )( p α + p β + p β + p β + p β + p β ) i . (4.12) C part gets contribution of division { β , β , β } → { β } , { β } , { β } of (C.2). This part is given as C = − i ip (cid:18) F (cid:19) h − ( s α β + s α β + s α β + s α β + s α β + s α β )( p β + p β ) i . (4.13)After some calculations and considering momentum conservation and on-shell conditions of the on-shellexternal lines, we get A + B + C = 0. Following similar calculations, we find that coefficients for the otherproducts of four-currents also vanish. (iii) Two sub-currents – 15 –n this case, only the terms that of ( p ) in (C.1), (C.2) and (C.3) contribute and the sum of thesecontributions is given as 12 F h − ( s α β + s α β + s α β + s α β + s α β ) J ( β , . . . , β ) − ( s α β + s α β + s α β ) J ( β , β , β ) J ( β , β , β ) − s α β J ( β , . . . , β ) i . (4.14)After considering all the cases (i), (ii) and (iii), we find that only the productions of two sub-currents areleft and this part is just the right hand side of eight-point fundamental BCJ relation. Now let us consider the general proof of fundamental BCJ relation (4.1). As shown in the eight-pointexample, we can always express the left hand side of the relation (4.1) by Berends-Giele recursion (2.6)and collect the diagrams with same off-shell momenta of sub-currents(e.g., for eight point case the diagramsare given by Fig. 7). After applying (4.6) and (4.7), the left hand side of (4.1) can be written in terms of J ( { B } ) . . . J ( { B M } ), where { B } . . . { B M } are nontrivial divisions of the ordered set { β } . To prove therelation (4.1), we should read off the coefficient for each division. Then we should show that the coefficientsmust vanish for divisions with M > M = 1.For given M ( M > J ( { B } ) . . . J ( { B M } ) can be classified into threetypes (this is similar with the eight-point example) Type-A, Type-B and the first diagram of Type-C inFig. 8. The notations in these diagrams are defined by Fig. 6.For Type-A diagram in Fig. 8 we can use momentum conservation and on-shell condition of α torewrite the coefficient in each term into a form independent of momentum of the off-shell line 1. Forexample, if we consider the diagram with α between { B i } and { B i +1 } , the coefficient is rewritten as s α + s α B + · · · + s α B i = − ( s α B i +1 + · · · + s α B M ) . (4.15)The vertex is also written in the form independent of the momentum of off-shell leg.For Type-B diagrams in Fig. 8, we should write down the expression of each diagram by (4.7) andpick out the appropriate division such that we can get { B } . . . { B M } . For example, for the first diagramin Type-B in Fig. 8, we should keep the division { B , B } → { B }{ B } , for the second diagram we shouldkeep the division { B , B } → { B }{ B } and so on. We also write the coefficients and vertices as formsindependent of the momentum of the off-shell leg 1 via momentum conservation and on-shell condition of α . For Type-C diagrams in Fig. 8, we should write down the expression of each diagram by (4.6) andkeep the divisions such that we can get { B } . . . { B M } . Only the first diagram of Type-C contributes. Since the odd-point current must vanish, the number of elements in each subset must be odd so that the product isnonzero. – 16 – α B i +1 × k B j +1 s α B i +1 × k B j s α B i × k B j +1 s α B i × k B j Type-A n − i − j ) ( i > j )0 ( i ≤ j ) n − j ( i ≥ j ) − (2 i + 1) ( i < j ) n − (2 i − j − 1) ( i > j )0 ( i ≤ j ) n − j ( i > j ) − i ( i ≤ j )Type-B n i − j ) ( i > j )0 ( i ≤ j ) n j − i ≥ j )2 i ( i < j ) n i − j − i > j )0 ( i ≤ j ) n j − i > j )2 i − i ≤ j )Type-C 0 1 0 1 Table 2: Coefficients of J ( { B } ) . . . J ( { B M } ) in fundamental BCJ relation: Coefficients of the form s α B i × p B j with arbitrary i and j . s α B i +1 × s α B j +1 s α B i +1 × s α B j s α B i +1 × s B j +1 B l +1 s α B i +1 × s B jB l s α B i +1 × s B j +1 B l Type-A n − ( i − j ) ( i > j )0 ( i ≤ j ) n − j ( i ≥ j ) − ( i + 1) ( i < j ) n − i − l ) ( j < l < i )0 Otherwise n − (2 i + 1) ( i ≤ j < l ) − j ( j < l, j < i ) ( − i − j ) ( j < i < l ) − (2 l − j − 1) ( j < l ≤ i )0 Otherwise Type-B n i − j ( i > j )0 ( i ≤ j ) n j ( i ≥ j ) i ( i < j ) n i − l ) ( j < l < i )0 Otherwise n i ( i ≤ j < l )2 j − j < l, j < i ) ( i − j ) ( j < i < l )(2 l − j − 1) ( j < l ≤ i )0 Otherwise Type-C 0 0 0 1 0 Table 3: Coefficients of J ( { B } ) . . . J ( { B M } ) in fundamental BCJ relation: Coefficients of the form s α B i +1 × . . . . s α B i × s α B j +1 s α B i × s α B j s α B i × s B j +1 B l +1 s α B i × s B jB l s α B i × s B j +1 B l Type-A n − ( i − j − 1) ( i > j + 1)0 Otherwise n − j ( i > j ) − i ( i ≤ j ) n − (2 i − l − 1) ( j < l < i )0 Otherwise n − i ( i ≤ j < l ) − j ( j < l, j < i ) ( − (2 i − j − 1) ( j < i ≤ l ) − (2 l − j − 1) ( j < l < i )0 Otherwise Type-B n i − j ( i > j )0 ( i ≤ j ) n j ( i > j ) i ( i ≤ j ) n i − l − j < l < i )0 Otherwise n i − i ≤ j < l )2 j − j < l, j < i ) ( i − j − j < i ≤ l )2 l − j − j < l < i )0 Otherwise Type-C 0 0 0 1 0 Table 4: Coefficients of J ( { B } ) . . . J ( { B M } ) in fundamental BCJ relation: Coefficients of the form s α B i × . . . . We should keep the division { B , . . . , B M } → { B } . . . { B M } of the first diagram of Type-C. We alsouse momentum conservation to rewrite s α as − ( s α B + · · · + s α B M ) and write the vertices in (4.6) byfunctions of momentums of on-shell legs.After these steps, we can read off the coefficient of J ( { B } ) . . . J ( { B M } ) explicitly. They are shownin tables 2, 3, 4. The columns of tables 2, 3, 4, except for the second column of table 3 and the first columnof table 4, are canceled out. The sum of the second column of table 3 is given as ( i ≥ j ) − i < j ) , (4.16)while, the sum of the first column of table 4 is given as ( i > j )0 ( i ≤ j ) . (4.17)Since s α β i +1 × s α β j and s α β i × s α β j +1 can be related by i ⇔ j , we should interchange i and j in thefirst column of table 4. Then we can see these two nonzero contributions cancel with each other. Therefore,all the contributions of divisions with M > M = 1, the ordered set { β } is only divided into two ordered subsets. In this case, weonly need to consider the terms of ( p ) in diagrams shown in Fig. 4 (which is the first term of the secondline of (4.6)) with all the possible nontrivial divisions { β } → { B }{ B } . The sum of these terms preciselygives the right hand side of the fundamental BCJ relation (4.1).– 17 – . General KK, BCJ relations, minimal-basis expansion and formulations of total am-plitudes Having proven the U (1)-decoupling identity and fundamental BCJ relation in non-linear sigma model, letus now extend these relations to more general cases. In this section, we first state that the general KK andBCJ relations as well as minimal-basis expansion are all satisfied by color-ordered tree amplitudes. Thenwe will show that tree-level total amplitudes satisfy DDM form of color decomposition and KLT relation .All these discussions are parallel within Yang-Mills theory, thus we will only present the main points ofthe statements. Details can be found in the works [25], [11], [5] and [21]. General KK and BCJ relations KK relation and general BCJ relation can be considered as extensions of U (1)-decoupling identity andfundamental BCJ relation. In non-linear sigma model, KK relation for 2 m -point amplitudes is given as X σ ∈ OP ( { α ,...,α r } S { β ,...,β s } ) A (1 , { σ } , m ) = ( − r A (1 , { β } , m, { α } T ) , (5.1)where r + s = 2 m − 2. General BCJ relation is given as X σ ∈ OP ( { α ...α r } S { β ,...,β s } ) r X l =1 X ξ σi <ξ αl s α l σ i A (1 , { σ } , m ) = 0 . (5.2)From (5.1) and (5.2), we can see, if there is only one element in { α } , the relations turns back to the U (1)-decoupling identity (3.2) and the fundamental BCJ relation (4.2) with 2 m → β m .In principle, one can follow the similar steps in sections 3 and 4 to prove the general KK , BCJ relations(5.1), (5.2) for off-shell currents and then take on-shell limits to get the relations among color-ordered on-shell amplitudes. However, it is not easy to generalize the off-shell KK and BCJ relations in this way. Thisis because there are nontrivial products of sub-currents on the right hand side of the relations. When thereare more elements in { α } set, the forms of the right hand side may containing both divisions of { α } setand divisions of { β } set. Thus the formulations may become highly complicated.Fortunately, once we know the fundamental BCJ relation (4.2) in addition with cyclic symmetry (2.4),we have another way to prove the on-shell general KK and BCJ relations. This method was firstly proposedin [25] where general KK and BCJ relations in Yang-Mills theory are generated by so-called primaryrelations . The main point is that once the amplitudes satisfy a)cyclic symmetry as well as b)fundamentalBCJ relation , all the general KK and BCJ relations can be reexpressed as linear combinations of a set offundamental BCJ relations, and thus the general KK, BCJ relations must hold. Though the discussions in[25] was firstly found by monodromy relations in string theory, as stated in [25], all these arguments can We emphasize that the consequent relations that will be derived in this section are all for on-shell amplitudes. GeneralKK and BCJ relations for off-shell currents will be discussed in future work. – 18 –e extended to field theory. Since the fundamental BCJ relation(4.2) in non-linear sigma model has thesame form within Yang-Mills theory, all the steps in [25] are also valid in non-linear sigma model. Thusthe KK and BCJ relations must be satisfied by color-ordered tree amplitudes in non-linear sigma model.Details of this proof can be found in [25]. Minimal-basis expansion Since KK and general BCJ relations are both satisfied by even-point color ordered tree amplitudes.We are ready now for reduce the number of independent even-point color ordered tree amplitudes as inYang-Mills theory. Apparently, one can use KK relation in addition with cyclic symmetry to reduce thenumber of independent 2 m -point amplitudes to (2 m − m − m external legs. One can follow the same recursive procedure that given by section 4 of the paper[11] to prove the minimal-basis expansion, because we have the general BCJ relation (5.2) of the sameform within Yang-Mills theory. In Yang-Mills theory, amplitude relations imply various formations of total amplitudes. As we have dis-cussed, in non-linear sigma model, event-point color ordered tree amplitudes satisfy KK and BCJ relations,which have the same formations within Yang-Mills theory. Thus we expect that the total amplitudes canhave the same expressions within Yang-Mills theory. Particularly, the total amplitudes should satisfy DDMcolor decomposition as well as KLT relation . DDM form of color decomposition An immediate result of KK relation is that the total amplitudes satisfy Del Duca-Dixon-Maltoni(DDM)form of color decomposition which was firstly proven in Yang-Mills theory[5] M (1 , . . . , m ) = X σ ∈ S m − f a a σ a i . . . f a i m − a σ m − a m A (1 , σ, m ) . (5.3)The main points to prove DDM form of color decomposition are a) KK relations (5.1) and b) the followingrelations between trace factors and color factors in DDM form f a a σ a i . . . f a i m − a σ m − a m = Tr( T [ T σ , [ ..., [ T σ m − , T m ] ... ]]) . (5.4)We can express any color-ordered amplitude in (2.3) by KK relation, and collect the color coefficients ofeach amplitude in KK basis. Using the above relation between traces and the color factors in DDM form,we can prove the DDM form of color decomposition(5.3). Details of the proof can be found in [5]. KLT relation Another result is Kawai-Lewellen-Tye(KLT) relation[29]. In non-linear sigma model, total amplitudescan be expressed in terms of products of two color-ordered tree amplitudes A and e A , where A denote the KLT relation in Yang-Mills theory was suggested in [32] and the general proof can be found in[21]. – 19 –olor-ordered tree amplitudes in non-linear sigma model and e A denote the color-ordered tree amplitudesof scalar with cubic vertex f abc . As in Yang-Mills theory, the KLT relation has many formations[30, 31].For example the formulation manifests (2 m − symmetries is given as M (1 , , . . . , m ) = X γ,φ ∈ S m − A (2 m, γ, S [ γ | φ ] e A (1 , φ, m ) s ... (2 m − . (5.5)This relation can be proved by following the same steps within the subsection 6.3 of the paper [21]. Thisis because that the two critical points- the DDM color decomposition and the generalized BCJ relation forcolor scalar theory -are all satisfied.Another formulation which manifests (2 m − symmetries is given as M (1 , , . . . , m ) = ( − X γ,φ ∈ S m − A (1 , γ, m − , m ) S [ φ | γ ] e A (2 m − , m, φ, , (5.6)or equivalently M (1 , , . . . , m ) = ( − X γ,φ ∈ S m − A (1 , γ, m − , m ) S [ γ | φ ] p n − e A (1 , m − , φ, m ) . (5.7)This formulation seems not easy to prove along the same line in Yang-Mills theory (See section 6.1 of [21]),because the boundary behavior of the amplitudes of non-linear sigma model is not good enough. However,we also expect that the (2 m − M (1 , , , 4) = − A (1 , , , s e A (4 , , , . (5.8)To prove this relation, we express e A (4 , , , 3) explicitly by Feynman rules in color scalar theory. Thus theright hand side is expressed as − A (1 , , , s (cid:20) f e f e s + f e f e s (cid:21) . (5.9)Using antisymmetry of f abc as well as four-point BCJ relation s A (1 , , , 4) + ( s + s ) A (1324) = 0which have been proven in the previous sections, we reexpress the right hand side as f e f e A (1 , , , 4) + f e f e A (1 , , , . (5.10)This is just the DDM form of color decomposition of four-point total tree amplitude. Thus the four-pointKLT relation manifest (4 − . Conclusion In this work, we have discussed the tree-level amplitude relations in non-linear sigma model. We haveproven the off-shell version of U (1) identity and fundamental BCJ relation under Cayley parametrization.After taking on-shell limits, we got the U (1)-decoupling identity and the fundamental BCJ relation foron-shell amplitudes. We stated that the general KK and BCJ relations were also satisfied by even-pointtree amplitudes in non-linear sigma model. Two consequent results of KK and BCJ relations were given asthe minimal-basis expansion for color-ordered amplitudes and KLT relation for total amplitudes. Thoughthe procedure of proof in this work seems complicated, the relations are quite consistent with the coloralgebra. We hope these results can be useful in particle phenomenology. The algebraic interpretation ofthese relations and the dual decompositions of amplitudes deserve further work. A. Convention of notation Figure 5: Convention in section 3 In this paper, we use a diagram containing a curved arrow line to denote sum of diagrams for short.Since we encounter similar structures when considering U (1) identity and fundamental BCJ relation, weonly use the same diagrams expressions but let the curved arrow line have different meanings for conve-nience. The meaning of curved arrow line for section 3 and section 4 are given by Fig. 5 and Fig. 6respectively. – 21 – igure 6: Convention in section 4 B. Eight-point diagrams The left hand side of eight-point U (1) identity and eight-point fundamental BCJ relation can be expressedby Fig. 7 with the convention of notation defined by Fig. 5 and Fig. 6.– 22 – . Diagrams contribute to J ( { B } ) J ( { B } ) . . . J ( { B M } ) The diagrams contribute to J ( { B } ) J ( { B } ) . . . J ( { B M } ) in U (1) identity and fundamental BCJ relationare given by Fig. 8. Acknowledgements Y. J. Du is pleased to thank Prof. Yong-Shi Wu for helpful discussions. He would also like to thank theUniversity of Utah for hospitality. Y. J. Du is supported in part by the NSF of China Grant No.11105118,China Postdoctoral Science Foundation No.2013M530175 and the Fundamental Research Funds for theCentral Universities of Fudan University No.20520133169. The research of Gang Chen has been supportedin parts by the Jiangsu Ministry of Science and Technology under contract BK20131264 and by the SwedishResearch Links programme of the Swedish Research Council (Vetenskapsradets generella villkor) undercontract 348-2008-6049. Gang Chen also acknowledges 985 Grants from the Ministry of Education, andthe Priority Academic Program Development for Jiangsu Higher Education Institutions (PAPD). References [1] Z. Bern, J. J. M. Carrasco and H. Johansson, “New Relations for Gauge-Theory Amplitudes,” Phys. Rev. D , 085011 (2008) [arXiv:0805.3993 [hep-ph]].[2] R. Kleiss and H. Kuijf, “MULTI - GLUON CROSS-SECTIONS AND FIVE JET PRODUCTION ATHADRON COLLIDERS,” Nucl. Phys. B , 616 (1989).[3] N. E. J. Bjerrum-Bohr, P. H. Damgaard and P. Vanhove, “Minimal Basis for Gauge Theory Amplitudes,”Phys. Rev. Lett. , 161602 (2009) [arXiv:0907.1425 [hep-th]].[4] S. Stieberger, “Open & Closed vs. Pure Open String Disk Amplitudes,” arXiv:0907.2211 [hep-th].[5] V. Del Duca, L. J. Dixon and F. Maltoni, “New color decompositions for gauge amplitudes at tree and looplevel,” Nucl. Phys. B , 51 (2000) [arXiv:hep-ph/9910563].[6] R. Britto, F. Cachazo and B. Feng, “New Recursion Relations for Tree Amplitudes of Gluons,” Nucl. Phys. B , 499 (2005) [arXiv:hep-th/0412308].[7] R. Britto, F. Cachazo, B. Feng and E. Witten, “Direct Proof Of Tree-Level Recursion Relation In Yang-MillsTheory,” Phys. Rev. Lett. , 181602 (2005) [arXiv:hep-th/0501052].[8] B. Feng, R. Huang and Y. Jia, “Gauge Amplitude Identities by On-shell Recursion Relation in S-matrixProgram,” Phys. Lett. B , 350 (2011) [arXiv:1004.3417 [hep-th]].[9] H. Tye and Y. Zhang, “Remarks on the identities of gluon tree amplitudes,” Phys. Rev. D (2010) 087702[arXiv:1007.0597 [hep-th]].[10] F. Cachazo, “Fundamental BCJ Relation in N=4 SYM From The Connected Formulation,” arXiv:1206.5970[hep-th].[11] Y. X. Chen, Y. J. Du and B. Feng, “A Proof of the Explicit Minimal-basis Expansion of Tree Amplitudes inGauge Field Theory,” JHEP (2011) 112 [arXiv:1101.0009 [hep-th]]. – 23 – 12] C. R. Mafra, O. Schlotterer and S. Stieberger, “Explicit BCJ Numerators from Pure Spinors,” JHEP (2011) 092 [arXiv:1104.5224 [hep-th]].[13] R. Monteiro and D. O’Connell, “The Kinematic Algebra From the Self-Dual Sector,” JHEP (2011) 007[arXiv:1105.2565 [hep-th]].[14] N. E. J. Bjerrum-Bohr, P. H. Damgaard, R. Monteiro and D. O’Connell, “Algebras for Amplitudes,” JHEP (2012) 061 [arXiv:1203.0944 [hep-th]].[15] C. -H. Fu, Y. -J. Du and B. Feng, “An algebraic approach to BCJ numerators,” JHEP (2013) 050[arXiv:1212.6168 [hep-th]].[16] Z. Bern and T. Dennen, “A Color Dual Form for Gauge-Theory Amplitudes,” Phys. Rev. Lett. (2011)081601 [arXiv:1103.0312 [hep-th]].[17] Y. -J. Du, B. Feng and C. -H. Fu, “The Construction of Dual-trace Factor in Yang-Mills Theory,” JHEP (2013) 057 [arXiv:1304.2978 [hep-th]].[18] C. -H. Fu, Y. -J. Du and B. Feng, “Note on Construction of Dual-trace Factor in Yang-Mills Theory,”arXiv:1305.2996 [hep-th].[19] T. Sondergaard, “New Relations for Gauge-Theory Amplitudes with Matter,” Nucl. Phys. B (2009) 417[arXiv:0903.5453 [hep-th]].[20] Y. Jia, R. Huang and C. -Y. Liu, “ U (1)-decoupling, KK and BCJ relations in N = 4 SYM,” Phys. Rev. D (2010) 065001 [arXiv:1005.1821 [hep-th]].[21] Y. -J. Du, B. Feng and C. -H. Fu, “BCJ Relation of Color Scalar Theory and KLT Relation of GaugeTheory,” JHEP (2011) 129 [arXiv:1105.3503 [hep-th]].[22] T. Bargheer, S. He and T. McLoughlin, “New Relations for Three-Dimensional Supersymmetric ScatteringAmplitudes,” Phys. Rev. Lett. (2012) 231601 [arXiv:1203.0562 [hep-th]].[23] K. Kampf, J. Novotny and J. Trnka, “Recursion Relations for Tree-level Amplitudes in the SU(N) Non-linearSigma Model,” arXiv:1212.5224 [hep-th].[24] K. Kampf, J. Novotny and J. Trnka, “Tree-level Amplitudes in the Nonlinear Sigma Model,” JHEP (2013) 032 [arXiv:1304.3048 [hep-th]].[25] Q. Ma, Y. -J. Du and Y. -X. Chen, “On Primary Relations at Tree-level in String Theory and Field Theory,”JHEP (2012) 061 [arXiv:1109.0685 [hep-th]].[26] F. A. Berends and W. T. Giele, “Recursive Calculations for Processes with n Gluons,” Nucl. Phys. B (1988) 759.[27] F. A. Berends and W. T. Giele, “Multiple Soft Gluon Radiation in Parton Processes,” Nucl. Phys. B (1989) 595.[28] N. E. J. Bjerrum-Bohr, P. H. Damgaard, B. Feng and T. Sondergaard, “Gravity and Yang-Mills AmplitudeRelations,” Phys. Rev. D (2010) 107702 [arXiv:1005.4367 [hep-th]].[29] H. Kawai, D. C. Lewellen and S. H. H. Tye, “A Relation Between Tree Amplitudes of Closed and OpenStrings,” Nucl. Phys. B (1986) 1. – 24 – 30] N. E. J. Bjerrum-Bohr, P. H. Damgaard, B. Feng and T. Sondergaard, “Gravity and Yang-Mills AmplitudeRelations,” Phys. Rev. D (2010) 107702 [arXiv:1005.4367 [hep-th]].[31] N. E. J. Bjerrum-Bohr, P. H. Damgaard, B. Feng and T. Sondergaard, “Proof of Gravity and Yang-MillsAmplitude Relations,” JHEP (2010) 067 [arXiv:1007.3111 [hep-th]].[32] Z. Bern, A. De Freitas and H. L. Wong, “On the coupling of gravitons to matter,” Phys. Rev. Lett. (2000)3531 [hep-th/9912033]. – 25 – igure 7: Diagrams for eight-point U(1) identity(with curved arrow line defined by Fig. 3) or fundamental BCJrelation(with curved arrow line defined by Fig. 4) – 26 – igure 8: The three types of diagrams contribute to J B J B . . . J B M ..