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in 2008 , bern , carrasco , and johansson discovered a novel set of linear relations satisfied by tree - level color - ordered amplitudes in gauge theories@xcite . they arrived at these relations by writing the tree - level @xmath0-gluon amplitude as a sum over @xmath5 diagrams assembled from cubic vertices  =   _ i c_i   n_i d_i [ cubicdecompintro ] where the color factor @xmath6 associated with the diagram is composed of group theory structure constants @xmath7 , the denominator @xmath8 consists of the product of the inverse propagators associated with the diagram , and the kinematic numerator @xmath9 depends on the momenta and polarizations of the gluons . all contributions from diagrams with quartic vertices are redistributed among the cubic diagrams . by virtue of the jacobi identity @xmath10 satisfied by the structure constants , the color factors @xmath6 obey a set of jacobi relations of the form c_i + c_j + c_k  =   0 . because of these linear dependences , the kinematic numerators @xmath9 are not uniquely defined , but can undergo _ generalized gauge transformations _ @xmath11 which leave unchanged @xcite . the authors of ref . @xcite conjectured that there exists a generalized gauge in which the kinematic numerators satisfy the same algebraic relations as the color factors ; in particular , they can be made to satisfy kinematic jacobi relations n_i + n_j + n_k  =   0 . [ kinematicjacobi ] from this assumption of _ color - kinematic duality _ , they demonstrated the existence of new relations ( subsequently known as bcj relations ) satisfied by the color - ordered amplitudes @xmath12 . these relations can be derived from the _ fundamental bcj relation _ ( and permutations thereof ) @xcite _ b=3^n ( _ c=1^b-1 k_2 k_c ) a(1 , 3 , , b-1 , 2 , b , , n )  =   0 [ fundbcjintro ] where @xmath13 are the ( outgoing ) momenta of the gluons . besides color - kinematic duality , these relations rely on the properties of the propagator matrix @xcite , constructed from the inverse denominators @xmath14 ( see sec .  [ sec : prop ] for a precise definition ) . specifically , as a consequence of momentum conservation , this @xmath15 matrix has rank @xmath16 , and consequently possesses a set of @xmath17 eigenvectors with eigenvalue zero . the bcj relations ( [ fundbcjintro ] ) were subsequently proven using string - theory techniques @xcite and bcfw on - shell recursion @xcite , providing evidence for the conjecture of tree - level color - kinematic duality . bern et al . conjectured that color - kinematic duality also applies to the integrands of loop - level amplitudes @xcite ; while not proven , this conjecture has been tested for @xmath18 supersymmetric yang - mills theory through four loops @xcite , and for pure yang - mills theory through two loops @xcite . despite the fact that the bcj relations for @xmath0-gluon amplitudes have been definitively established , interest in tree - level kinematic numerators continues , not least because the numerators that are naturally generated by feynman rules generally do not obey the relations ( [ kinematicjacobi ] ) except in the case of four - point amplitudes @xcite . many approaches have been developed to obtain kinematic numerators that obey color - kinematic duality directly from a lagrangian approach @xcite .    in this paper , we introduce a new set of symmetries obeyed by tree - level gauge - theory amplitudes , associated with each external gluon in the amplitude . these symmetries act on the color factors @xmath6 of the amplitude , shifting them by momentum - dependent quantities . since color factors do not carry any momentum dependence , this is a purely formal operation ; we prove , however , that the tree - level @xmath0-gluon amplitude is invariant under these shifts by writing it in an alternative form known as the _ radiation vertex expansion _ @xcite . we then show that the bcj relations ( [ fundbcjintro ] ) follow as an immediate consequence of the color - factor symmetry of the @xmath0-gluon amplitude . although bcj relations have been previously established , our results reveal a more direct connection to the symmetries of the lagrangian formulation of gauge theory and its feynman rules ( gauge and poincar invariance ) and provide a basis for generalizations . let us describe this symmetry in a bit more detail , reserving a full description for section [ sec : cfs ] . given a tree - level @xmath0-gluon color factor @xmath6 , the choice of one of the external gluon legs @xmath19 divides the diagram in two at its point of attachment . let @xmath20 denote the subset of the remaining legs on one side of this point ; it does not matter which side we choose . the shift of the color factor @xmath6 associated with gluon @xmath19 must satisfy _ a c_i   _ c k_a k_c . [ colorfactorshiftintro ] choosing to sum over the complement of @xmath20 gives the same result ( up to sign ) due to momentum conservation . the constants of proportionality in are then constrained by requiring that the shifted color factors respect all the jacobi relations satisfied by @xmath6 for any values of the momenta . consider the case where @xmath19 is one of the legs involved in the jacobi identity ( see fig . [ fig : one ] ) . imagine that each of the three graphs in fig . [ fig : one ] is embedded in a larger tree diagram , the same for each . denote the color factors associated with each diagram by @xmath21 , where @xmath22 ; these obey the jacobi relation @xmath23 . by , the shifts of these color factors are _ a = _ ( r ) k_a k_(r ) where @xmath24 is the momentum flowing out of each leg . requiring @xmath25 implies that @xmath26 is independent of @xmath27 , as a result of momentum conservation and masslessness of the gluon . a more detailed description of the color - factor shifts is given in sec . [ sec : cfs ] . the symmetry we have introduced has roots in the radiation symmetry @xcite that underlies the general radiation zero theorem @xcite . in theories with local gauge couplings and spins @xmath28 , all single - photon tree amplitudes vanish if the ratios @xmath29 are all equal , where @xmath13 is the photon momentum , and @xmath30 labels external particles with momentum @xmath31 and charge @xmath32 . these spin - independent zeros have spin - dependent counterparts where @xmath32 are replaced by numerators @xmath33 , closely related to the kinematic numerators @xmath9 in . the underlying radiation symmetry refers to invariance under @xmath34 as well as @xmath35 for arbitrary @xmath36 and @xmath37 . the extension to nonabelian `` charges '' has also been considered and the details behind a nonabelian radiation vertex expansion discussed . the general color - factor symmetry introduced here , however , incorporates crucial nonabelian constraints ( jacobi relations ) on @xmath36 , which lead to a complete set of bcj relations , and have not heretofore been developed . nevertheless , since the color - factor symmetry relies on the presence of massless gauge bosons , we may regard it as a generalized radiation symmetry . we also introduce in this paper the _ cubic vertex expansion _ of an @xmath0-point amplitude @xmath38 with respect to one of the gluons @xmath19 . consider the set of cubic diagrams @xmath39 that contribute to the @xmath40-point amplitude of all the particles in @xmath38 except for gluon @xmath19 . we show that , for any @xmath19 , the amplitude @xmath38 can be written as a triple sum over the legs @xmath27 of the vertices @xmath41 of the cubic diagrams @xmath39 :  =   _ i _ v 1 _ s=1 ^ 3 d_(a , i , v , s ) . here @xmath42 is the product of inverse propagators that branch off from leg @xmath27 of vertex @xmath41 of diagram @xmath39 , @xmath43 is the color factor of the @xmath0-point diagram obtained by attaching gluon @xmath19 to leg @xmath27 of vertex @xmath41 of diagram @xmath39 ( as in fig .  [ fig : one ] ) , and @xmath44 is the @xmath0-point kinematic numerator associated with that color factor . the shift of @xmath43 associated with gluon @xmath19 is @xmath45 , where as explained above @xmath46 is independent of @xmath27 . since the alternative radiation vertex expansion shows that the amplitude @xmath38 is invariant under the color - factor shift , we may conclude from the cubic vertex expansion of @xmath38 that _ i _ v _ ( a , i , v ) _ s=1 ^ 3 d_(a , i , v , s ) n_(a , i , v , r )   =   0 . [ sumoverdeltaintro ] note that this constraint on the kinematic numerators , less stringent than the kinematic jacobi relations ( which state that @xmath47 for each vertex ) , is nonetheless sufficient to imply the bcj relations ( [ fundbcjintro ] ) . moreover , unlike the kinematic jacobi relations , the condition ( [ sumoverdeltaintro ] ) is invariant under generalized gauge transformations . a constraint of precisely the form ( [ sumoverdeltaintro ] ) was derived in refs . @xcite for the five - gluon amplitude using the monodromy properties of string theory amplitudes . we show in this paper that more general gauge - theory amplitudes , with both gluons and massless or massive particles in an arbitrary representation of the gauge group and with arbitrary spin @xmath28 , are also invariant under the color - factor symmetry . consequently , the kinematic numerators of these amplitudes obey a constraint analogous to . we further show that color - factor symmetry implies bcj relations for the color - ordered amplitudes of a class of @xmath0-point amplitudes involving @xmath48 gluons and a pair of particles in an arbitrary representation of the gauge group and arbitrary spin , as previously conjectured in refs . @xcite . bcj relations for the primitive amplitudes of a more general class of amplitudes containing gluons and an arbitrary number of pairs of differently flavored fundamentals ( based on a proper decomposition developed by melia @xcite and johansson and ochirov @xcite ) were conjectured by johansson and ochirov @xcite , and subsequently proven using bcfw on - shell recursion by de la cruz , kniss , and weinzierl @xcite . in a sequel to this paper @xcite , we prove that these bcj relations also follow as a direct consequence of the color - factor symmetry . the amplitudes of the theory of massless bi - adjoint scalars with cubic interactions@xcite also exhibit invariance under color - factor symmetry , as we show using the cubic vertex expansion . in this case , the color - factor shifts are associated with each massless adjoint scalar in the amplitude . as a consequence , we demonstrate the reduced rank of the propagator matrix for the @xmath0-gluon gauge - theory amplitude by deriving the set of its null eigenvectors .    finally , we generalize the cubic vertex expansion and color - factor symmetry to loop - level amplitudes containing at least one external gluon . we exhibit an independent set of shifts that act on the color factors of one - loop @xmath0-gluon amplitudes and which depend on the loop momentum as well as external momenta . these one - loop amplitudes are invariant under color - factor shifts in theories that admit a color - kinematic - dual representation of numerators . the color - factor symmetry also implies certain relations among the integrands of one - loop color - ordered amplitudes that were previously uncovered in refs . @xcite . the contents of this paper are as follows . in sec . [ sec : cfs ] we define the color - factor shift for the @xmath0-gluon amplitude and derive the bcj relations as a consequence of the invariance of the amplitude under this shift . we also introduce the cubic vertex expansion , and use it to derive a gauge - invariant constraint on the kinematic numerators of the @xmath0-gluon amplitude . we introduce an analogous set of shifts of the kinematic numerators , and show that they correspond to a generalized gauge transformation . in sec . [ sec : fourgluon ] , we prove the invariance of the four - gluon amplitude under the color - factor symmetry , and in sec .  [ sec : rvegluon ] , we extend this to the @xmath0-gluon amplitude by using the radiation vertex expansion . in sec . [ sec : fund ] , we define the color - factor shift for more general amplitudes , and derive the bcj relations for the class of amplitudes containing @xmath48 gluons and a pair of fundamentals . in sec .  [ sec : fourfund ] , we prove the invariance of the four - point amplitude with two gluons and a pair of massive fundamentals of arbitrary spin @xmath28 under the color - factor symmetry , and in sec .  [ sec : rvefund ] , we extend this to a general @xmath0-point amplitude containing gluons and fundamentals . in sec . [ sec : prop ] , we prove the invariance of the amplitudes of the bi - adjoint scalar theory under the color factor symmetry , and derive the null eigenvectors of the propagator matrix . in sec . [ sec : loop ] , we generalize the cubic vertex expansion and color - factor symmetry to loop - level amplitudes , and derive a constraint on the integrands of one - loop color - ordered amplitudes . section [ sec : concl ] contains a discussion and conclusions . in appendix [ sec : app ] , we write the shifts for all the color factors of the five - gluon amplitude , and derive the explicit constraint on the kinematic numerators that follow from the color - factor symmetry . [ sec : cfs ]    we begin this section by introducing the color - factor symmetry in the simplest context , the tree - level four - gluon amplitude _ 4  =   c_s n_s s + c_t n_t t + c_u n_u u [ fourpointamplitude ] where c_s  =   f__1 _ 2 f__3 _ 4 , c_t  =   f__1 _ 4 f__2 _ 3 , c_u  =   f__1 _ 3 f__4 _ 1 and @xmath49 , @xmath50 , and @xmath51 are mandelstam variables . we define the four - point color - factor shift to act as c_s   c_s +  s , c_t   c_t +  t , c_u c_u +  u [ fourpointcfs ] where @xmath36 is arbitrary . preserves the jacobi relation @xmath52 by virtue of momentum conservation @xmath53 . the statement that is invariant under implies the kinematic jacobi relation n_s+ n_t+n_u  =  0 . [ fourpointkinematicjacobi ] it is well - known @xcite that is satisfied in the case of the four - gluon amplitude , and we will show this explicitly in sec . [ sec : fourgluon ] . this serves as proof of the invariance of the four - gluon amplitude under the color - factor shift . recall that the kinematic numerators @xmath9 are not uniquely defined by because a generalized gauge transformation n_s n_s +  s , n_t n_t +  t , n_u n_u +  u [ fourpointggt ] ( with @xmath37 arbitrary ) leaves unchanged by virtue of the jacobi identity @xmath52 . in the case of the four - gluon amplitude , however , the sum @xmath54 is well - defined : it is invariant under the generalized gauge transformation ( [ fourpointggt ] ) due to momentum conservation . the four - gluon amplitude can be written in terms of color - ordered amplitudes as _ 4  =   c_s a(1,2,3,4 ) - c_u a(1,3,2,4 ) . [ fourgluonbcj ] invariance of under the shift ( [ fourpointcfs ] ) immediately implies   _ 4  =   s a(1,2,3,4 ) - u a(1,3,2,4 )  =   0 which is the four - gluon bcj relation @xcite . next we turn to tree - level @xmath0-gluon amplitudes with @xmath55 , which may be written as a sum over diagrams composed of cubic vertices ( referred to as cubic diagrams ) @xcite  =   _ i c_i   n_i d_i . [ cubicdecomp ] associated with each cubic diagram @xmath56 is a color factor @xmath6 obtained by sewing together structure constants @xmath57 . among these color factors @xmath6 we may identify the subset of half - ladder color factors @xmath58 defined by _ ,_n-3 f__(1 ) _ ( 2 ) _ 1 f__1 _ ( 3 ) _ 2 f__n-3 _ ( n-1 ) _ ( n ) , s_n . [ halfladder ] the color factors @xmath6 are not independent but obey a set of jacobi relations . using the procedure outlined in ref . @xcite , the jacobi identity @xmath59 may be repeatedly applied to reduce each @xmath6 to a linear combination of half - ladder color factors c_i  =   _ s_n-2 m_i , 1n _ 1 n , _ 1 n   _ 1 ( 2 ) ( n-1 ) n [ dependent ] where @xmath60 denotes a permutation of @xmath61 . the @xmath62 half - ladders @xmath63 form an independent set . alternatively , @xmath64 may be computed by rewriting @xmath6 using @xmath65 t^\tc ) $ ] , reducing the resulting expression to a linear combination of single traces , and then identifying the coefficient of @xmath66 ( see ref . @xcite ) . we now define a set of momentum - dependent shifts , associated with each external gluon @xmath19 in the amplitude , that act on the color factors @xmath6 . the action of the shift @xmath67 associated with gluon @xmath19 is constrained by two requirements : ( i ) that it preserve all the jacobi relations satisfied by @xmath6 , and ( ii ) that it satisfy _ a c_i   _ c k_a k_c [ colorfactorshift ] where @xmath20 denotes the subset of the external particles on one side of the point at which @xmath19 is attached to @xmath6 . in particular , if @xmath6 is a color factor in which gluon @xmath19 is attached to an external leg @xmath68 , the shift is proportional to @xmath69 , which is an inverse propagator in the associated feynman diagram . more generally , is related ( see ) to the propagators in the feynman diagram associated with @xmath6 . consider the subset of @xmath0-point color factors obtained from a given @xmath40-point cubic diagram @xmath39 by attaching gluon @xmath19 to it in all possible ways . one of these color factors has gluon @xmath19 attached to external leg @xmath70 of the @xmath40-point diagram ] ; define its shift to be @xmath71 . one may easily verify ( using the argument in the introduction ) that the conditions ( i ) and ( ii ) above uniquely fix the coefficients of the shifts of all the other color factors in this subset . the coefficients @xmath72 for different @xmath40-point diagrams are then constrained by jacobi relations among their color factors . \sigma(b+1 ) \cdots n } $ ] and @xmath73 , width=453 ] we now demonstrate that there is an @xmath16-parameter family of color - factor shifts associated with each gluon @xmath19 in the @xmath0-gluon amplitude . first choose @xmath74 , and consider the subset of half - ladder color factors @xmath75 , where @xmath76 denotes a permutation of @xmath77 . we define the color - factor shift associated with gluon @xmath19 to act on these half ladders as _ a   _ 1 a ( 2 ) ( n-1 ) n  =   _ a , k_a k_1 [ colorshiftdefinition ] where @xmath78 are a set of @xmath16 arbitrary , independent constants ( or functions ) for each @xmath19 . let @xmath79 \sigma(b+1 ) \cdots n}$ ] denote the color factor shown in fig . [ fig : two ] ; its shift is proportional to @xmath80 . this together with and the jacobi relation c_1 ( 2 ) ( b-1 ) [ a(b ) ] ( b+1 ) ( n-1 ) n  =   _ 1 ( 2 ) ( b-1 ) a ( b ) ( n-1 ) n - _ 1 ( 2 ) ( b ) a ( b+1 ) ( n-1 ) n implies that @xmath81 acts on the independent half - ladder color factors @xmath82 as , replace @xmath83 with @xmath84 , and the sum over @xmath30 should begin with 3 . in the case @xmath85 , replace @xmath86 with @xmath87 . ] _ a   _ 1 ( 2 ) ( b-1 ) a ( b ) ( n-1 ) n  =   _ a , ( k_a k_1 + _ c=2^b-1 k_a k_(c ) ) , a , b \{2 , , n-1 } , b a [ halfladdershift ] consistent with . the action on the remaining color factors is given by _ a c_i  =   _ s_n-2 m_i , 1n  _ a _ 1 n [ dependentshift ] which is also consistent with , as may be shown using the procedure described in ref . @xcite . the color - factor shifts associated with the gluons @xmath88 are not all independent . in particular , the @xmath16-parameter family of shifts associated with gluon @xmath89 are linear combinations of shifts associated with @xmath90 as a result of momentum conservation . , but we know the result must be true for all @xmath0 because , as we will see in sec .  [ sec : prop ] , the color - factor shifts correspond to null eigenvectors of the propagator matrix . since the @xmath91 propagator matrix has rank @xmath16 @xcite there are at most @xmath17 independent null eigenvectors . ] we may also define @xmath16-parameter families of shifts associated with gluons 1 and @xmath0 . these are also not independent of the others . thus the dimension of the ( abelian ) group of color - factor shifts is @xmath17 .      by using , the tree - level @xmath0-gluon amplitude ( [ cubicdecomp ] ) may be rewritten in the del duca - dixon - maltoni half - ladder decomposition @xcite  =   _ s_n-2 _ 1 n a(1 , ( 2 ) , , ( n-1 ) , n ) [ ddm ] where the coefficients a(1 , ( 2 ) , , ( n-1 ) , n )  =   _ i m_i , 1n   n_i d_i [ colorordered ] are color - ordered amplitudes belonging to the kleiss - kuijf basis @xcite . in the previous subsection , we defined the action of the color - factor symmetry associated with a given gluon @xmath19 . the variation of under the shift associated with @xmath92 gives  =   _ s_n-3 _ 2 , _ b=3^n ( k_1 k_2 + _ c=3^b-1 k_2 k_(c ) ) a(1 , ( 3 ) , , ( b-1 ) , 2 , ( b ) , , ( n-1 ) , n ) . in secs . [ sec : fourgluon ] and [ sec : rvegluon ] , we prove that @xmath38 is invariant under this shift . since @xmath93 are arbitrary and independent , this implies that _ b=3^n ( k_1 k_2 + _ c=3^b-1 k_2 k_(c ) ) a(1 , ( 3 ) , , ( b-1 ) , 2 , ( b ) , , ( n-1 ) , n )  =   0 [ fundbcj ] which is the fundamental bcj relation ( [ fundbcjintro ] ) . all other permutations of this relation can be obtained using the invariance of the amplitude under the color - factor shifts associated with gluons 3 through @xmath89 . it is known @xcite that the bcj relations reduce the number of independent color - ordered amplitudes from @xmath62 to @xmath94 . not surprisingly , the difference between these , @xmath17 , is the dimension of the group of color - factor shifts that leave the amplitude invariant .      in order to examine the implications of the color - factor symmetry for the kinematic numerators @xmath9 appearing in the cubic decomposition ( [ cubicdecomp ] ) , we introduce in this section the _ cubic vertex expansion _ of the amplitude with respect to one of the gluons . this expansion is similar to , but distinct from , the radiation vertex expansion @xcite that will be used in secs . [ sec : rvegluon ] and [ sec : rvefund ] to prove the invariance of @xmath0-point gauge - theory amplitudes under color - factor shifts . the cubic decomposition ( [ cubicdecomp ] ) is a sum over the @xmath5 cubic diagrams of an @xmath0-gluon amplitude , but for any @xmath95 it can be viewed as a sum over the @xmath96 cubic diagrams of an @xmath40-point function with external legs @xmath97 , to each of which gluon @xmath19 is attached in @xmath98 different ways . let us label these @xmath40-point cubic diagrams by @xmath39 and their denominators by @xmath99 . each @xmath40-point cubic diagram @xmath39 has @xmath100 vertices , the set of which we denote by @xmath101 . for each vertex @xmath102 , we can break @xmath99 into three factors @xmath103 , where @xmath42 is the product of propagators that branch off from leg @xmath27 of the vertex . if leg @xmath27 is an external leg of the diagram , then @xmath104 . we can attach gluon @xmath19 either to one of the @xmath89 external legs or to one of the @xmath105 internal lines of @xmath39 , yielding altogether @xmath98 of the terms in the sum ( [ cubicdecomp ] ) . let @xmath106 be the momentum running through one of the internal lines of @xmath39 . attaching gluon @xmath19 to this line will replace the factor @xmath107 in @xmath99 with @xmath108 . we split the inverse denominator into two terms using the identity  =   1 k^2 ( 2 k_a k )  +   1 ( -2 k_a k ) ( k+k_a)^2 [ splitprop ] and we associate each of the terms on the right hand side of the equation with one of the two vertices to which the internal line is connected . thus , with this doubling of internal line terms , we now have a total of @xmath109 terms for each @xmath39 ; namely , one term for each of the legs of each of the @xmath100 vertices of @xmath39 . we label this term by @xmath110 , and write the cubic vertex expansion of the @xmath0-gluon amplitude with respect to gluon @xmath19 as  =   _ i _ v v_(a , i ) 1 _ s=1 ^ 3 d_(a , i , v , s ) [ cve ] where @xmath43 are the color factors @xmath21 in fig . [ fig : one ] associated with each vertex @xmath111 , @xmath44 are the associated kinematic numerators , and @xmath112 denotes the momentum flowing out of leg @xmath27 . the @xmath43 and @xmath44 are equal to the @xmath6 and @xmath9 in up to signs ( such that @xmath113 ) . an explicit example of the cubic vertex expansion for the five - gluon amplitude is given in appendix a.    the @xmath114 freedom in the definition of @xmath43 is used to make the relative signs in the jacobi relation positive : c_(a , i , v , r )  =   0 . [ vertexjacobi ] the denominators in each triple also sum to zero k_a k_(a , i , v , r )  =   0 [ vertexmomentum ] by momentum conservation @xmath115 and the masslessness of the gluon @xmath116 . _ a priori _ , however , there is no reason for the kinematic numerators @xmath117 associated with each vertex to sum to zero . we will see in the next subsection , however , that the color - factor symmetry of the amplitude leads to a constraint on the sum of kinematic numerators . having introduced the cubic vertex expansion of the @xmath0-gluon amplitude with respect to gluon @xmath19 , we now consider the effect of a color - factor shift @xmath81 on the amplitude . the shift associated with gluon @xmath19 acts on the color factors appearing in the cubic vertex expansion ( [ cve ] ) as _ a   c_(a , i , v , r )  =   _ ( a , i , v )  k_a k_(a , i , v , r ) where @xmath118 is a linear combination of @xmath78 uniquely determined by the jacobi relations . these shifts respect by virtue of . the variation of under this shift gives _ a    =   12 _ i _ v v_(a , i ) _ ( a , i , v ) _ s=1 ^ 3 d_(a , i , v , s ) n_(a , i , v , r ) . [ variation ] we prove that @xmath119 in secs . [ sec : fourgluon ] and [ sec : rvegluon ] ; hence the color - factor symmetry implies the following constraint on the kinematic numerators _ i _ v v_(a , i ) _ ( a , i , v ) _ s=1 ^ 3 d_(a , i , v , s ) n_(a , i , v , r )   =   0 . [ sumoverdelta ] now _ if _ the constants @xmath118 were all independent , then we could conclude from this argument that @xmath120 , i.e. that the kinematic numerators necessarily obey jacobi relations . this would be in conflict with the well - known fact that the kinematic numerators obtained from feynman rules in general do _ not _ satisfy color - kinematic duality , where there is only one term in the sum ( [ sumoverdelta ] ) , may we conclude that @xmath121 . ] ; indeed this is not a valid inference from because the @xmath118 are not independent . the set of @xmath118 for all the vertices of a given diagram @xmath39 are equal ( up to signs ) because any two adjacent vertices share a common color factor . in fact , with an appropriate choice of signs for @xmath43 , the @xmath118 may be made independent of @xmath41 . the @xmath122 for different diagrams @xmath39 are further constrained by the jacobi relations among the color factors of @xmath39 . while does _ not _ imply that the numerators satisfy the kinematic jacobi relations , it _ does _ impose a set of ( gauge invariant ) conditions that the color - kinematic violations @xmath123 must satisfy . we wish to emphasize that , while the kinematic jacobi relations @xmath124 are not invariant under generalized gauge transformations ( hence the actual claim of color - kinematic duality is that _ there exists _ a generalized gauge in which they hold true ) , the conditions ( [ sumoverdelta ] ) are invariant under generalized gauge transformations . the argument for this is simple . a generalized gauge transformation is a transformation @xmath125 that leaves the amplitude ( [ cubicdecomp ] ) unchanged . hence by starting with @xmath126 and following the steps above ( since the condition @xmath127 is also gauge invariant ) , we obtain the same result ( [ sumoverdelta ] ) except with @xmath44 replaced with @xmath128 .    to obtain a more explicit form of , we would need to identify all the linear dependences among the @xmath118 required by the color jacobi relations . previously , we observed that the number of independent color - factor shifts was @xmath17 , parametrized by constants @xmath78 , where @xmath129 and @xmath76 denotes a permutation of @xmath77 . if we were to write @xmath118 in terms of these independent constants , would yield @xmath17 independent constraints on the @xmath130 . in appendix a , we carry out this procedure for the five - gluon amplitude . while the bcj relations ( [ fundbcj ] ) were originally derived as a consequence of the assumption of color - kinematic duality , it was known @xcite from early on that they are equivalent to a set of weaker conditions on the numerators . the conditions for five - gluon numerators were derived in refs . @xcite as a consequence of the monodromy properties of string - theory amplitudes . these conditions are equivalent to , as we show in appendix a. in this section , we have demonstrated that both and the bcj relations ( [ fundbcj ] ) are a consequence of the invariance of the amplitude under the color - factor symmetry . we have considered the effect on the amplitude of a shift of the color factors . one may ask what effect an analogous shift of the kinematic numerators would have . considered in refs .  @xcite . ] we show in this section that such a shift is simply a generalized gauge transformation . let us define the kinematic shift associated with leg @xmath19 , where @xmath74 , on the half - ladder numerator @xmath131 to be _ a   _ 1 ( 2 ) ( b-1 ) a ( b ) ( n-1 ) n  =   _ a , ( k_1 k_a + _ c=2^b-1 k_a k_(c ) ) , a , b \{2 , , n-1 } , b a [ kinematichalfladdershift ] where @xmath76 denotes a permutation of @xmath77 , and @xmath132 are a set of arbitrary constants ( or functions ) . the action on all other numerators @xmath9 is then defined by _ a n_i  =   _ s_n-2 m_i , 1n  _ a _ 1 n . [ kinematicdependentshift ] note that we have _ not _ assumed that the @xmath9 obey the jacobi relations ( [ kinematicjacobi ] ) . however , the numerator shifts defined by will satisfy _ a ( n_i + n_j + n_k )  =   0 so if the @xmath9 do satisfy kinematic jacobi relations , the shifted numerators will continue to do so , and if they do not , then the neither will the shifted numerators .    now consider the cubic vertex expansion of the @xmath0-gluon amplitude  =   _ i _ v v_(a , i ) 1 _ s=1 ^ 3 d_(a , i , v , s ) . the kinematic shift with respect to gluon @xmath19 acts on the numerators appearing in this expression as _ a   n_(a , i , v , r )  =   _ ( a , i , v ) k_a k_(a , i , v , r ) and therefore on the amplitude itself as _ a    =   1 2 _ i _ v v_(a , i ) _ ( a , i , v ) _ s=1 ^ 3 d_(a , i , v , s ) c_(a , i , v , r ) . this vanishes courtesy of ; hence the @xmath0-gluon amplitude is invariant under the shift of kinematic numerators . this is precisely the definition of a generalized gauge transformation @xcite . [ sec : fourgluon ]    in this section we prove that the tree - level four - gluon amplitude is invariant under the color - factor symmetry . in doing so , we develop some results that will be necessary for our more general proof of the invariance of the @xmath0-gluon amplitude in the next section .    the four - gluon amplitude can be constructed from a three - gluon vertex by attaching a fourth gluon to a propagator emanating from each of the legs of the vertex or to the vertex itself . this yields _ 4  =   [ fourgluonamplitude ] where c_(1 )  =   f__1 _ 4 f__2 _ 3 , c_(2 )  =   f__2 _ 4 f__3 _ 1 , c_(3 )  =   f__3 _ 4 f__1 _ 2 . [ fourgluoncolorfactors ] the kinematic numerator @xmath133 receives contributions from the diagram in which gluon 4 is attached to leg @xmath27 as well as from the four - gluon vertex . the color - factor symmetry ( associated with gluon 4 ) acts on the color factors ( [ fourgluoncolorfactors ] ) and the four - gluon amplitude ( [ fourgluonamplitude ] ) as _ 4  =   _ 4 k_4 k_r _ 4 _ 4  =   12 _ 4 . thus , by showing that @xmath134 , we will establish that @xmath135 . this we now proceed to do . the three - gluon vertex is t^\tc ) $ ] with @xmath136 , so that @xmath137 = f_{\ta\tb\tc } t^\tc$ ] . this differs from the standard textbook convention by a factor of @xmath138 . we use @xmath139 in this paper . ] - i g 2 f__1 _ 2 _ 3 v^_1 _ 2 _ 3 ( k_1 , k_2 , k_3 ) [ fv ] where v^_1 _ 2 _ 3 ( k_1 , k_2 , k_3 )  =   ^_1_2 ( k_2 - k_1)^_3 + ^_2_3 ( k_3 - k_2)^_1 + ^_3_1 ( k_1 - k_3)^_2 [ threegluonvertex ] and @xmath13 are outgoing momenta . in feynman gauge , the gluon propagator is @xmath140 . attaching gluon 4 to leg 1 yields the expression @xmath141    { v_{\nu}^{~\mu_2 \mu_3 } ( k_1 + k_4 , k_2 , k_3 ) } \ , . { \nonumber}\end{aligned}\ ] ] the contribution of this diagram to the four - gluon amplitude ( [ fourgluonamplitude ] ) is obtained by contracting with @xmath142 and dividing by @xmath56 . the first and second terms in the square brackets vanish using @xmath143 . the third term vanishes due to ( k_1 + k_4)^v_^  _ 2 _ 3 ( k_1 + k_4 , k_2 , k_3 ) _ 2_2 _ 3_3  =  0 [ wardone ] using @xmath144 . the contribution of the remaining terms of to the kinematic numerator @xmath145 can be written n_(1)|_leg 1_1 v^_2 _ 3 ( k_1 + k_4 , k_2 , k_3 ) _ 2_2 _ 3_3 [ attachleg ] where ( s_r^ ) ^_r _   =   i ( ^_r ^ _   - ^_r ^ _   ) [ s ] are the spin - one angular momentum matrices acting on gluon @xmath27 . these satisfy the lorentz algebra commutation relations  =   -i . attaching gluon 4 directly to the three - gluon vertex , we obtain the four - gluon vertex @xmath146 \ , . { \nonumber}\end{aligned}\ ] ] using , this can be recast as v__1 _ 2 _ 3 _ 4^_1 _ 2 _ 3 _ 4  =   - i g^2 2 ( c_(1 ) k_1_4 + c_(2 ) k_2_4 + c_(3 ) k_3_4 ) v^_1 _ 2 _ 3 ( k_1 , k_2 , k_3 ) . [ fourgluonvertexderiv ] the contribution of this vertex to the four - gluon amplitude is obtained by contracting with @xmath142 and dividing by @xmath56 . this contribution is then parceled out among the three terms in ; the contribution to @xmath145 is n_(1)|_vertex  =   - g^2 k_4 k_1 _ 4_4 k_1_4 v^_1 _ 2 _ 3 ( k_1 , k_2 , k_3 ) _ 1_1 _ 2_2 _ 3_3 . [ attachvertex ] the reader may be concerned about the use of for the following reason . the three - gluon vertex ( [ threegluonvertex ] ) can be rewritten using momentum conservation @xmath147 . for example , we can eliminate @xmath148 from , writing it as v^_1 _ 2 _ 3 ( k_1 , k_2 , k_3 )  =   ^_1_2 ( k_2 - k_1)^_3 + ^_2_3 ( -k_1 - 2 k_2)^_1 + ^_3_1 ( 2k_1 + k_2)^_2 . [ altthreegluonvertex ] the partial derivatives @xmath149 obtained from differ from those obtained from , so that gives @xmath150 \ , . \label{altfourgluonvertex}\end{aligned}\ ] ] nonetheless is equal to courtesy of the jacobi relation @xmath151 . simply corresponds to a different way of parceling the four - gluon vertex among the color factors @xmath21 , and the expressions @xmath133 will differ by a generalized gauge transformation ( [ fourpointggt ] ) . the amplitude ( [ fourgluonamplitude ] ) of course remains unchanged . using the fact that the three - gluon vertex ( [ threegluonvertex ] ) is linear in momenta , we rewrite using v^_2 _ 3 ( k_1 + k_4 , k_2 , k_3 )  =   ( 1 + k_4 k_1 ) v^_2 _ 3 ( k_1 , k_2 , k_3 ) [ taylor ] and then combine to obtain @xmath152    v^{\nu \mu_2 \mu_3 } ( k_1 , k_2 , k_3 ) \ve_{2\mu_2 } \ve_{3\mu_3 }   { \nonumber}\\ & \quad\quad + g^2 \ve_{1\mu_1 }   \left [ -i \ve_{4\alpha } k_{4\beta } ( s_1^{\alpha\beta } ) ^{\mu_1}_{~~ \nu }    \right ]    \left (   k_{4\gamma } { \partial \over \partial   k_{1\gamma } } \right )   v^{\nu \mu_2 \mu_3 } ( k_1 , k_2 , k_3 ) \ve_{2\mu_2 } \ve_{3\mu_3 } \end{aligned}\ ] ] where the orbital angular momentum operators are defined as l_r^  =   i ( k_r^ - k_r^ ) . [ l ] these satisfy the lorentz algebra commutation relations  =   -i . similar expressions are obtained for @xmath153 and @xmath154 . finally , defining the total angular momentum operator ( j_r^ ) ^_r _   =   l_r^ ^_r _   + ( s_r^ ) ^_r _   [ j ] we can write all the kinematic numerators as @xmath155    v(k_1 , k_2 , k_3 ) \label{nr}\end{aligned}\ ] ] where we have suppressed the polarization vectors @xmath156 , @xmath157 , and @xmath158 . the subscripts on @xmath159 and @xmath160 indicate on which polarization indices these operators act . note that under a gauge transformation @xmath161 of gluon 4 , the last two terms in vanish by virtue of the antisymmetry of @xmath162 and @xmath163 , and the first term gives @xmath164 in accord with . we observe that the operators @xmath165 and @xmath166 are the same as those appearing in the burnett - kroll form @xcite of the subleading terms of the low soft - photon theorem @xcite as applied to gluons @xcite . ( see refs . @xcite for recent derivations of the soft - gluon theorem from gauge invariance . ) the first two terms in correspond precisely to the leading and subleading terms in the @xmath167 expansion of the four - gluon amplitude ; the third term is higher order in the soft momentum . we emphasize , however , that , and the vertex expansion that we will derive in the next section , are exact , and not dependent on taking a soft limit . we will now show that the sum of the four - gluon kinematic numerators ( [ nr ] ) vanishes . this result , which has been known at least since 1980 @xcite inspired the conjecture of color - kinematic duality @xcite . we demonstrate it in a way that will facilitate the proof of color - factor symmetry of the @xmath0-gluon amplitude . using , we see that @xmath168 the first sum on the right - hand side of vanishes by momentum conservation @xmath169 together with @xmath170 . the second sum on the right hand side of , which may be written more explicitly as ( j_1^ ) ^_1 _   v^_2 _ 3 ( k_1 , k_2 , k_3 ) + ( j_2^ ) ^_2 _   v^_1 _ 3 ( k_1 , k_2 , k_3 ) + ( j_3^ ) ^_3 _   v^_1 _ 2 ( k_1 , k_2 , k_3 ) [ secondsum ] is the first - order lorentz transformation of the three - gluon vertex . this vanishes , as may be verified by explicit computation , because @xmath171 is a lorentz tensor . alternatively , we can define the spin - one angular momentum operator to act on polarization indices @xcite s_r^  =   i ( _ r^ - _ r^ ) [ alts ] in which case ( j_r^ ) v(k_1 , k_2 , k_3 )  =   i ( k_r^ - k_r^ + _ r^ - _ r^ ) v(k_1 , k_2 , k_3 ) [ lorentzscalar ] where v(k_1 , k_2 , k_3 )  =   _ 1 _ 2  _ 3 ( k_2 - k_1 ) + _ 2 _ 3  _ 1 ( k_3 - k_2 ) + _ 3 _ 1  _ 2 ( k_1 - k_3 ) . vanishes because @xmath172 is a lorentz - invariant function of @xmath173 and @xmath174 . the third sum in is proportional to @xmath175 this expression is antisymmetric in @xmath37 and @xmath60 . since it multiplies @xmath176 in , the whole expression vanishes . the cancellations that we have just exhibited were originally used in ref . @xcite to prove the radiation zero theorem . we have shown that the sum of kinematic numerators vanishes ( the kinematic jacobi relation ) , and thus have demonstrated the invariance of the four - gluon amplitude @xmath177 under the color - factor shift symmetry . we would like to emphasize that in proving the vanishing of we did not use that @xmath173 were on - shell , nor did we use @xmath178 for @xmath22 . thus we will be able to use these results in sec . [ sec : rvegluon ] for an off - shell three - gluon vertex @xmath171 . [ sec : rvegluon ]    we now turn to the proof that the tree - level @xmath0-gluon amplitude is invariant under a color - factor shift associated with any of the gluons . to do so , we employ a decomposition of the amplitude known as the radiation vertex expansion @xcite . this is a recursive approach which constructs an @xmath0-point amplitude by attaching a massless vector boson to all possible @xmath40-point diagrams . consider the set of all tree - level @xmath40-gluon diagrams with external legs @xmath97 constructed using feynman rules . label these diagrams by @xmath179 . please note that this set differs from the set of @xmath40-point diagrams appearing in the cubic vertex expansion ( cf . [ sec : cfs ] ) because it includes not only cubic diagrams but also diagrams with four - gluon vertices . ( in the cubic decomposition ( [ cubicdecomp ] ) , all diagrams containing quartic vertices are redistributed among the cubic diagrams . ) we will construct all possible @xmath0-gluon diagrams by attaching gluon @xmath19 to each @xmath40-gluon diagram @xmath179 in all possible ways . this includes : ( 1 ) attaching gluon @xmath19 to an external leg , ( 2 ) attaching gluon @xmath19 to an internal line , or ( 3 ) attaching gluon @xmath19 to one of the three - gluon vertices of @xmath179 to make a four - gluon vertex . we describe these in turn . first we single out one of the external legs , @xmath68 , of @xmath179 , denoting the resulting expression as _ b_b b^_b__b ( k_b , ) where @xmath180 and @xmath181 are the momentum and polarization vector of gluon @xmath68 , and @xmath182 denotes momenta belonging to gluons @xmath183 . attaching gluon @xmath19 to external leg @xmath68 and using we obtain @xmath184    b_{\tb \nu }   ( k_a + k_b , \cdots   ) \ , . { \nonumber}\end{aligned}\ ] ] contracting this with @xmath185 eliminates the first two terms in the square brackets . the third term is proportional to ( k_a + k_b)^ b _ ( k_a + k_b , ) . [ wardtwo ] this term does not vanish by itself as it did in the case of the four - gluon amplitude , but when we obtain the total @xmath0-gluon amplitude by including all @xmath40-point diagrams @xmath179 , the sum of such terms vanishes due to gauge invariance ( ward identity ) . therefore we are left with the last two terms in square brackets , which yield the two terms in the following expression - 2 g f__b _ a ( k_a + k_b)^2 _ b_b b _ ( k_a + k_b , ) . [ attachexternalgluon ] we set this expression aside for now . next we single out one of the internal lines of @xmath179 , which divides the diagram into two subdiagrams @xmath186 and @xmath187 , and splits the external legs @xmath188 into two complementary sets @xmath189 and @xmath190 . the contribution of the diagram can thus be written as b_^ ( -k , ) ( -i _ ) _ k^2 c_^ ( k , ) where @xmath191 is the momentum running through the line , and the @xmath182 in @xmath186 and @xmath187 denote momenta belonging to @xmath189 and @xmath190 respectively . attaching gluon @xmath19 to the line connecting the two subgraphs yields b _ ( -k , ) f__a v^_a ( k ,- k - k_a , k_a ) k^2 ( k+k_a)^2 c_(k+k_a , ) . writing the three - gluon vertex ( [ threegluonvertex ] ) as v^_a ( k , -k - k_a , k_a )  =   - ^ k_a^_a + ^_a k^ + ^_a ( k + k_a)^ - 2 ^ k^_a - 2 ( ^_a k_a^- ^_a k_a^ ) we see that the first term vanishes when contracted with @xmath192 . the second and third terms give terms proportional to k^b _ ( -k , ) , ( k+ k_a)^c _ ( k + k_a , ) . [ wardthree ] again these terms do not vanish by themselves , but when we include all @xmath40-gluon diagrams @xmath179 , @xmath193 will be replaced by the sum over all diagrams containing external legs @xmath189 plus one additional off - shell line , and similarly for @xmath194 , and these expressions will vanish by gauge invariance ( ward identity ) . we are thus left with - 2 i g b _ ( -k , ) f__a k^2 ( k+k_a)^2 c _ ( k+k_a , ) . now we use the identity ( [ splitprop ] ) to rewrite this as @xmath195 c_{\tc\nu } ( k+k_a ,   \cdots   )   \right\ } \label{attachinternalgluon } \\ & + \left\ {   - \sqrt2 g { f_{\tb \tc \ta_a }   \over 2 k_a \cdot k } b_{\tb \mu } ( \cdots ,   -k )    \big [   \eta^{\mu \nu } \ve_a \cdot k +   \left ( \ve_a^\mu k_a^\nu - \ve_a^\nu k_a^{\mu }   \right ) \big ] \right \}{-i \over ( k+k_a)^2 }    c_{\tc\nu } ( k+k_a ,   \cdots   )   \ , . { \nonumber}\end{aligned}\ ] ] we associate each of the terms in this equation with one of the two vertices to which the line is attached .      in the previous subsections , we showed that attaching gluon @xmath19 to an @xmath40-gluon diagram yields one term ( [ attachexternalgluon ] ) for each external leg and two terms ( [ attachinternalgluon ] ) for each internal line , or in other words , one term for each leg of each vertex of @xmath179 . we can therefore reorganize these terms into a sum over the legs of the vertices of each of the @xmath40-gluon diagrams @xmath179 . this is the radiation vertex expansion @xcite . first we choose one of the three - gluon vertices @xmath41 of @xmath179 ( if it has any ) . such a vertex divides the external legs into three non - overlapping subsets @xmath196 , @xmath197 such that @xmath198 . the contribution of diagram @xmath179 to the @xmath40-gluon amplitude can be expressed -i g 2 f__1_2_3 v^_1 _ 2 _ 3 ( k_1 , k_2 , k_3 ) _ r=1 ^ 3 a^__r _ r ( - k_r , ) [ threegluonfactors ] where @xmath199 is the momentum flowing out of each leg of the vertex , and @xmath182 in @xmath200 denotes momenta belonging to @xmath201 . if any of the legs is external , then @xmath202 is just @xmath203 . next we choose one of the four - gluon vertices @xmath41 of @xmath179 ( if it has any ) . such a vertex divides the external legs into four non - overlapping subsets @xmath208 , @xmath209 such that @xmath210 . the contribution of the diagram @xmath179 can be expressed as v__1 _ 2 _ 3 _ 4^_1 _ 2 _ 3 _ 4 _ r=1 ^ 4 a^__r _ r ( - k_r , ) . we now attach gluon @xmath19 to each of the legs of this four - gluon vertex . from we obtain @xmath211 \eta_{\nu\lambda } v_{\tb   \tc_2 \tc_3 \tc_4}^{\lambda \mu_2 \mu_3 \mu_4 }   { \nonumber}\\ + & { f _ { \tb \tc_{2 } \ta_a } \over 2 k_a \cdot k_2 } \big [   \eta^{\mu_2 \nu } \ve_a \cdot k_2 +   \left ( \ve_a^{\mu_2 } k_a^\nu - \ve_a^\nu k_a^{\mu_2 }   \right ) \big ] \eta_{\nu\lambda } v_{\tc_1 \tb \tc_3 \tc_4}^{\mu_1 \lambda \mu_3 \mu_4 }   { \nonumber}\\ + & { f _ { \tb \tc_{3 } \ta_a } \over 2 k_a \cdot k_3 } \big [   \eta^{\mu_3 \nu } \ve_a \cdot k_3 +   \left ( \ve_a^{\mu_3 } k_a^\nu - \ve_a^\nu k_a^{\mu_3 }   \right ) \big ] \eta_{\nu\lambda } v_{\tc_1 \tc_2 \tb \tc_4}^{\mu_1 \mu_2 \lambda \mu_4 }   { \nonumber}\\ + & { f _ { \tb \tc_{4 } \ta_a } \over 2 k_a \cdot k_4 } \big [   \eta^{\mu_4 \nu } \ve_a \cdot k_4 +   \left ( \ve_a^{\mu_4 } k_a^\nu - \ve_a^\nu k_a^{\mu_4 }   \right ) \big ] \eta_{\nu\lambda } v_{\tc_1 \tc_2 \tc_3 \tb}^{\mu_1 \mu_2 \mu_3 \lambda }   \bigg ) \ , . \label{attachlegfour}\end{aligned}\ ] ] one can not attach gluon @xmath19 to the four - gluon vertex itself since there are no five - gluon vertices . thus the contribution to the radiation vertex expansion from the four - gluon vertex is @xmath212 v_{\tb   \tc_2 \tc_3 \tc_4}^{\nu \mu_2 \mu_3 \mu_4 }   { \nonumber}\\ + & { f _ { \tb \tc_{2 } \ta_a } \over 2 k_a \cdot k_2 } \big [ \delta^{\mu_2 } _ { ~\nu } \ve_a \cdot k_2 -i \ve_{a\alpha } k_{a\beta } ( s_2^{\alpha\beta } ) ^{\mu_2 } _ { ~~\nu }   \big ] v_{\tc_1\tb   \tc_3 \tc_4}^{\mu_1 \nu \mu_3 \mu_4 }   { \nonumber}\\ + & { f _ { \tb \tc_{3 } \ta_a } \over 2 k_a \cdot k_3 } \big [ \delta^{\mu_3 } _ { ~\nu } \ve_a \cdot k_3 -i \ve_{a\alpha } k_{a\beta } ( s_3^{\alpha\beta } ) ^{\mu_3 } _ { ~~\nu }   \big ] v_{\tc_1   \tc_2 \tb \tc_4}^{\mu_1 \mu_2 \nu \mu_4 }   { \nonumber}\\ + & { f _ { \tb \tc_{4 } \ta_a } \over 2 k_a \cdot k_4 } \big [ \delta^{\mu_4 } _ { ~\nu } \ve_a \cdot k_4 -i \ve_{a\alpha } k_{a\beta } ( s_4^{\alpha\beta } ) ^{\mu_4 } _ { ~~\nu }   \big ] v_{\tc_1   \tc_2 \tc_3 \tb}^{\mu_1 \mu_2 \mu_3 \nu }   \bigg ) \ , . \label{fourgluoncontrib}\end{aligned}\ ] ]    to summarize this section , we have expressed an @xmath0-gluon amplitude as a sum over the vertices of all of the @xmath40-gluon diagrams @xmath179 , comprising a term ( [ threegluoncontrib ] ) for each three - gluon vertex of @xmath179 and a term ( [ fourgluoncontrib ] ) for each four - gluon vertex of @xmath179 . computing the variation of the radiation vertex expansion of the amplitude under a color - factor shift is somewhat more delicate than calculating the variation of the cubic vertex expansion of the amplitude ( as we did in sec . [ sec : cfs ] ) because each factor @xmath202 in can contain more than one color factor @xmath6 due to the possible presence of four - gluon vertices . first let us consider the contribution ( [ threegluoncontrib ] ) of a three - gluon vertex to the radiation vertex expansion , and for the moment let us assume that the subdiagrams corresponding to @xmath202 contain only three - gluon vertices . designate by @xmath213 with @xmath22 the color factor associated with each line of . thus , for example , @xmath214 is the product of @xmath215 and the structure constants from all the three - gluon vertices in @xmath216 . these color factors manifestly satisfy @xmath217 . the variation of @xmath213 under the color - factor shift associated with gluon @xmath19 is _ a   c _  =   _ ( a,,v )   k_a k_r which preserves @xmath217 . the variation of under the color - factor shift is therefore proportional to v(k_1 , k_2 , k_3 ) . [ threesums ] in sec . [ sec : fourgluon ] , we demonstrated that each of the three terms in vanishes . if the diagrams corresponding to @xmath202 contain four - gluon vertices , we can use to expand these expressions and then use the argument above to show that each separate contribution will vanish under the color - factor shift . therefore the contribution of the three - gluon vertices ( [ threegluoncontrib ] ) to the radiation vertex expansion is invariant under the color - factor shift associated with gluon @xmath19 . second let us consider the contribution ( [ fourgluoncontrib ] ) of a four - gluon vertex to the radiation vertex expansion . again we begin by assuming that the diagrams corresponding to @xmath202 contain only three - gluon vertices . we now expand the four - gluon vertices @xmath218 in into several terms , one of which is @xmath219 \left ( \eta^{\nu \mu_3 } \eta^{\mu_2 \mu_4 } -\eta^{\nu \mu_4 } \eta^{\mu_2 \mu_3 } \right ) { \nonumber}\\ + & { f _ { \tb \tc_{2 } \ta_a } f_{\tc_{1 } \tb \td } f_{\td \tc_3 \tc_4 } \over 2 k_a \cdot k_2 } \big [ \delta^{\mu_2}_{~ \nu } \ve_a \cdot k_2 -i \ve_{a\alpha } k_{a\beta } ( s_2^{\alpha\beta } ) ^{\mu_2}_{~~ \nu }   \big ] \left ( \eta^{\mu_1 \mu_3 } \eta^{\nu \mu_4 } -\eta^{\mu_1 \mu_4 } \eta^{\nu \mu_3 } \right ) { \nonumber}\\ + & { f _ { \tb \tc_{3 } \ta_a } f_{b \tc_{4 } \td } f_{\td \tc_1 \tc_2 } \over 2 k_a \cdot k_3 } \big [ \delta^{\mu_3}_{~ \nu } \ve_a \cdot k_3 -i \ve_{a\alpha } k_{a\beta } ( s_3^{\alpha\beta } ) ^{\mu_3}_{~~ \nu }   \big ] \left ( \eta^{\mu_1 \nu } \eta^{\mu_2 \mu_4 } -\eta^{\mu_1 \mu_4 } \eta^{\mu_2 \nu } \right ) { \nonumber}\\ + & { f _ { \tb \tc_{4 } \ta_a } f_{\tc_{3 } \tb \td } f_{\td \tc_1 \tc_2 } \over 2 k_a \cdot k_4 } \big [ \delta^{\mu_4}_{~ \nu } \ve_a \cdot k_4 -i \ve_{a\alpha } k_{a\beta } ( s_4^{\alpha\beta } ) ^{\mu_4}_{~~ \nu }   \big ] \left ( \eta^{\mu_1 \mu_3 } \eta^{\mu_2 \nu } -\eta^{\mu_1 \nu } \eta^{\mu_2 \mu_3 } \right ) \bigg ) \ , . \label{expandedfourgluoncontrib}\end{aligned}\ ] ] designate by @xmath213 with @xmath220 the color factor associated with each line of , including the structure constants from all the three - gluon vertices in @xmath221 . these color factors satisfy @xmath222 by virtue of f _ _ 1 _ a f_b _ 2 f__3 _ 4 + f _ _ 2 _ a f__1 f__3 _ 4 + f _ _ 3 _ a f_b _ 4 f__1 _ 2 + f _ _ 4 _ a f__3 f__1 _ 2  =  0 . the variation of @xmath213 under the color - factor shift associated with gluon @xmath19 is _ a   c _ =   _ ( a,,v )   k_a k_r . the variation of under the color - factor shift therefore contains two sums . the first ( _ a k_r ) ( ^_1 _ 3 ^_2 _ 4 -^_1 _ 4 ^_2 _ 3 ) [ variationfourglounleading ] vanishes by momentum conservation , @xmath223 , together with @xmath224 . the second @xmath225 is the first - order lorentz transformation of the tensor @xmath226 , which vanishes . the variation under the color - factor shift of the other two terms from the expansion of @xmath218 similarly vanishes . furthermore , the same argument applies when the diagrams corresponding to @xmath202 contain four - gluon vertices , by expanding these expressions using . therefore the contribution of the four - gluon vertices ( [ fourgluoncontrib ] ) to the radiation vertex expansion is invariant under the color - factor shift associated with gluon @xmath19 .    in fine , we have shown that each contribution to the radiation vertex expansion is invariant under the color - factor shift associated with gluon @xmath19 , and therefore the entire @xmath0-gluon amplitude is invariant under this shift . [ sec : fund ]    in secs .  [ sec : fourgluon ] and [ sec : rvegluon ] , we proved the color - factor symmetry of @xmath0-gluon amplitudes , from which follow the bcj relations for color - ordered amplitudes . color - factor symmetry is a property of a much larger class of tree - level gauge - theory amplitudes , namely those containing at least one gluon together with massless or massive particles in arbitrary representations of the gauge group with arbitrary spin @xmath28 , with the usual gauge - theory couplings . we will establish the invariance of this larger class of gauge - theory amplitudes under a color - factor shift in secs . [ sec : fourfund ] and [ sec : rvefund ] .    consider a tree - level @xmath0-point gauge - theory amplitude @xmath38 with gluons as well as particles @xmath227 and @xmath228 , either massless or massive , with spin zero , one - half , or one , in an arbitrary representation of the gauge group . for convenience , we refer to @xmath227 ( and @xmath228 ) as fundamentals ( and antifundamentals ) , but they can be in any representation . this amplitude has the cubic decomposition  =   _ i c_i   n_i d_i [ cubicdecompfund ] where we decorate the color factors , kinematic numerators , and denominators with primes to distinguish them from the analogous quantities for @xmath0-gluon amplitudes . the denominator @xmath229 now consists of the product of inverse propagators for both massless and massive particles . the color factor @xmath230 associated with each cubic diagram is obtained by sewing together @xmath231 vertices @xmath232 and @xmath233 vertices @xmath234 where @xmath235 denote the generators in the appropriate representation . contributions from feynman diagrams with quartic vertices ( either @xmath236 or @xmath237 in the case of a scalar or vector @xmath227 ) are parceled out among the cubic diagrams . the number of cubic diagrams in the sum ( [ cubicdecompfund ] ) will generally be fewer than for the @xmath0-gluon amplitude , as some will be excluded for violating fermion number or flavor symmetry . just as in the case of the @xmath0-gluon amplitude , the amplitude @xmath38 can be written in a cubic vertex expansion with respect to gluon @xmath19 :  =   _ i _ v 1 _ s=1 ^ 3 d_(a , i , v , s ) . the only difference between the derivation of this expression and that for the @xmath0-gluon amplitude given in sec . [ sec : cfs ] is that we must use the modified identity  =   1 ( 2 k_a k )  +   1 ( -2 k_a k ) [ ( k+k_a)^2 - m^2 ] [ massivesplitprop ] when gluon @xmath19 is attached to a propagator of a massive field .    as usual , the color - factor shift associated with gluon @xmath19 is defined by two requirements : ( i ) that it satisfy all the algebraic symmetries ( jacobi relations ) obeyed by the color factors @xmath230 , and ( ii ) that it satisfy _ a c_i   _ c k_a k_c where @xmath20 denotes the subset of the external particles on one side of the point at which @xmath19 is attached to @xmath230 . these together imply that _ a   c_(a , i , v , r )  =   _ ( a , i , v )  k_a k_(a , i , v , r ) . therefore the invariance of the amplitude under the color - factor shift implies the constraint _ i _ v _ ( a , i , v ) _ s=1 ^ 3 d_(a , i , v , s ) n_(a , i , v , r )   =   0 on the sums of kinematic numerators appearing in the cubic decomposition .    as in the case of @xmath0-gluon amplitudes , color - factor symmetry can be used to derive bcj relations among the color - ordered amplitudes associated with @xmath38 . bcj relations for @xmath0-point amplitudes with gluons and a single pair of massive fundamentals were conjectured in ref . @xcite and more generally for amplitudes containing an arbitrary number of pairs of fundamentals in ref . @xcite , based on the assumption of color - kinematic duality . a proof of these bcj relations using bcfw on - shell recursion was given in ref . @xcite . in order to derive these relations , however , it is necessary to write the amplitude in a _ proper _ decomposition @xcite , in terms of an independent set of color factors and gauge - invariant primitive amplitudes . for general amplitudes , this is a subtle problem , which was recently solved for the case of multiple pairs of distinct - flavor fundamentals by melia @xcite and johansson and ochirov @xcite . in a sequel to this paper @xcite , we review their solution , and then derive the bcj relations using the color - factor symmetry . there is one class of amplitudes , however , for which the story is practically identical to the @xmath0-gluon case , namely , @xmath0-point amplitudes with @xmath48 gluons and a single pair of fundamentals . for that case , an independent set of @xmath62 color factors is given by the half - ladders _1 n   ( t^_(2)t^_(3 ) ^_(n-1 ) ) ^_1 _   _ n . all other color factors @xmath238 can be reduced to half ladders c_i  =   _ s_n-2 m_i , 1n _1 n by repeatedly applying @xmath239^{\textsf{i}}_{~\textsf{j } }    \ , , $ ] similar to the case of @xmath0-gluon amplitudes @xcite . the coefficients @xmath64 are precisely the same as in the @xmath0-gluon case . the @xmath0-point amplitude can then be written in a proper decomposition @xcite ( _ 1 , g_2 , g_3 , , g_n-1 , _ n ) & = & _ s_n-2 _1 n  a ( 1 , ( 2 ) , , ( n-1 ) , n ) [ ddmfund ] where the primitive amplitudes are given by a ( 1 , ( 2 ) , , ( n-1 ) , n ) & = & _ i m_i , 1n   n_i d_i . we define an @xmath16-parameter family of shifts associated with each gluon @xmath74 via @xmath240 where @xmath60 is a permutation of @xmath61 , @xmath241 is a permutation of @xmath77 , and @xmath78 is a set of @xmath16 arbitrary constants for each @xmath19 . as in the case of the @xmath0-gluon amplitude , the dimension of the ( abelian ) group of color - factor shifts is @xmath17 . we show in secs . [ sec : fourfund ] and [ sec : rvefund ] that the amplitude @xmath242 is invariant under the color - factors shifts ( [ halfladdershiftfund ] ) . as a consequence , the color - ordered amplitudes defined in obey bcj relations that have the same form ( when expressed in terms of invariants @xmath69 , where @xmath13 is the momentum of a gluon ) as those for the @xmath0-gluon amplitude , namely _ b=3^n ( k_1 k_2 + _ c=3^b-1 k_2 k_(c ) ) a ( 1 , ( 3 ) , , ( b-1 ) , 2 , ( b ) , , ( n-1 ) , n )  =   0 together with all permutations of this equation with @xmath243 replaced by @xmath19 , as conjectured in refs .  @xcite . [ sec : fourfund ]    in this section , we prove that the tree - level four - point amplitude with two gluons and two massive particles in the fundamental representation of the gauge group with spin zero , one - half , or one is invariant under the color - factor symmetry . we will use these results for the proof of the invariance of the more general @xmath0-point amplitude in sec . [ sec : rvefund ] .    the four - point amplitude @xmath244 can be constructed from the @xmath245 vertex by attaching a gluon to a propagator emanating from each of the external legs of the vertex , or ( in the case of a spin - zero or spin - one fundamental ) to the vertex itself . this yields _ 4 ( _ 1 , _ 2 , g_3 , g_4 )  =   [ fourfundamplitude ] where the color factors c_(1 )  =   - ( t^_4 t^_3 ) ^_1 _   _ 2 , c_(2 )  =   ( t^_3 t^_4 ) ^_1 _   _ 2 , c_(3 )  =   f__4 _ 3 ( t^ ) ^_1 _   _ 2 [ fourfundcolorfactors ] obey @xmath246 using @xmath137 = f_{\ta\tb\tc } t^\tc $ ] . the color - factor symmetry ( associated with gluon 4 ) acts on the color factors ( [ fourfundcolorfactors ] ) and the four - point amplitude ( [ fourfundamplitude ] ) as _ 4  =   _ 4 k_4 k_r _ 4 _ 4  =   12 _ 4 . we will establish that @xmath135 by showing that @xmath247 , a result that has long been known @xcite . we will do this separately for spin zero , spin one - half , and spin one fundamentals . we begin with the case of a spin - one - half fundamental , which is simpler due to the absence of a @xmath248 vertex . the @xmath245 vertex . ] and dirac propagator are ( t^_3 ) ^_1 _   _ 2 ^_3 , . attaching gluon 4 to ( fermion ) leg 1 yields the expression @xmath249 \right ] \v^{\mu_3 }   \ , . \label{fermionlegone}\end{aligned}\ ] ] the contribution of this diagram to is obtained by sandwiching between @xmath250 and @xmath251 , contracting with @xmath252 , and dividing by @xmath56 . the first term in the square brackets vanishes using @xmath253 , leaving n_(1 )  =   g^2    @xmath254 satisfy the lorentz algebra commutation relations  =   -i . similarly , attaching gluon 4 to ( fermion ) leg 2 yields @xmath255 \right]\,.\end{aligned}\ ] ] using @xmath256 , we obtain n_(2 )  =   g^2    n_(3 )  =   g^2 _ 3_3 |u ( k_1 ) ^ u(-k_2 ) . the sum of the kinematic numerators is thus @xmath257 the first sum on the right - hand side of this equation vanishes by momentum conservation @xmath169 together with @xmath170 . the second sum vanishes because ^ ^_3 - ^_3 ^ + ( s_3^ ) ^_3 _   ^ is the first - order lorentz transformation of @xmath258 ( acting on both spinor indices as well as the vector index ) and hence vanishes . thus the sum of kinematic numerators for the amplitude @xmath244 is zero for spin - one - half fundamentals . next we turn to the case of spin - zero fundamentals . the @xmath245 vertex is ( t^_3 ) ^_1 _   _ 2 v^_3 ( k_1 , k_2 , k_3 ) , v^_3 ( k_1 , k_2 , k_3 )  =   ( k_1 - k_2)^_3 [ scalarscalargluonvertex ] where @xmath13 are outgoing momenta . the scalar propagator is @xmath259 . thus attaching gluon 4 to ( scalar ) leg 1 yields the expression @xmath260    { v^{\mu_3 } ( k_1 + k_4 , k_2 , k_3 ) } \,.\end{aligned}\ ] ] the contribution of this diagram to is obtained by contracting with @xmath252 and dividing by @xmath56 . the first term in the square brackets vanishes using @xmath170 , leaving n_(1)|_leg  =   g^2 _ 4 k_1 _ 3_3 v^_3 ( k_1 + k_4 , k_2 , k_3 ) . similarly , attaching gluon 4 to ( scalar ) leg 2 yields n_(2)|_leg  =   g^2 _ 4 k_2 _ 3_3 v^_3 ( k_1 , k_2+k_4 , k_3 ) . attaching gluon 4 to ( gluon ) leg 3 yields n_(3)|_leg  =   g^2 _ 3_3 v^ ( k_1 , k_2 , k_3 + k_4 ) . attaching gluon 4 directly to the @xmath245 vertex , we obtain the @xmath248 vertex ^_3 _ 4 ( \ { t^_3 , t^_4 } ) ^_1 _   _ 2 [ ssgg ] which can be written as - i g^2 2 ( c_(1 ) k_1_4 + c_(2 ) k_2_4 + c_(3 ) k_3_4 ) v^_3 ( k_1 , k_2 , k_3 ) . [ scalarscalargluongluonvertex ] the rest of the story proceeds exactly as in sec . [ sec : fourgluon ] , allowing us to write the kinematic numerators as @xmath261    v(k_1 , k_2 , k_3)\end{aligned}\ ] ] where we have suppressed the polarization vector @xmath158 . because two of the legs are scalars , we have @xmath262 , whereas @xmath263 is given by .    now consider the sum of kinematic numerators @xmath264 where the scalar - scalar - gluon vertex is @xmath265)^{\mu_3}$ ] with @xmath266 arbitrary due to momentum conservation . the first sum on the right - hand side of this equation vanishes as usual by momentum conservation . the second sum , which may be written more explicitly as l_1^ v^_3 ( k_1 , k_2 , k_3 ) + l_2^ v^_3 ( k_1 , k_2 , k_3 ) + ( j_3^ ) ^_3 _   v^ ( k_1 , k_2 , k_3 ) [ secondsumscalar ] is the first - order lorentz transformation of @xmath245 vertex . this vanishes , as may be seen by explicit computation , because @xmath267 is a lorentz tensor . the third term on the right - hand side of is _ 4 k_4 k_4 ( s_3^)^_3 _   k_3 v^(k_1 , k_2 , k_3 ) = ( k_4 ^ 2 _ 4^_3 - _ 4 k_4 k_4^_3 ) [ thirdtermscalar ] which automatically vanishes due to @xmath268 and @xmath269 . again , we emphasize that in proving the vanishing of we did not use that @xmath173 were on - shell , nor did we use @xmath178 , for @xmath22 . thus these results remain valid for an off - shell vertex @xmath267 . finally we consider massive spin - one fundamentals with @xmath270 vertex of the form ( t^_3 ) ^_1 _   _ 2 v^_1 _ 2 _ 3 ( k_1 , k_2 , k_3 ) [ spinonevertex ] where @xmath271 is given by . we emphasize that although we refer to the vector particles as fundamentals , they could be in any representation , including the adjoint , in which case @xmath272 and is equal to , except that now the vector boson is massive . the propagator for a massive spin - one particle is , p_(k ) = _ - k_k_m^2 . [ spinonepropagator ] in the case in which the vector boson gets its mass from a spontaneously - broken symmetry , is the propagator in unitary gauge ; this is most convenient for tree - level calculations as we need not compute contributions involving goldstone bosons .    attaching gluon 4 to leg 1 yields the expression v^_1 _ 4 ( k_1 , k_4 , -k_1-k_4 ) p _ ( k_1+k_4 ) v^_2 _ 3 ( k_1 + k_4 , k_2 , k_3 ) . the contribution of this diagram to is obtained by contracting with @xmath142 and dividing by @xmath56 , giving n_(1)|_leg  =   g^2 _ 1_1 v^_2 _ 3 ( k_1 + k_4 , k_2 , k_3 ) _ 2_2 _ 3_3 where we have used @xmath273 and @xmath274 but we did not use , which is not valid in this case because @xmath275 . analogous expressions are obtained for @xmath276 and @xmath277 . attaching gluon 4 directly to the @xmath245 vertex , we obtain the @xmath248 vertex - i g^2 2 ( c_(1 ) k_1_4 + c_(2 ) k_2_4 + c_(3 ) k_3_4 ) v^_1_2_3 ( k_1 , k_2 , k_3 ) . again , the rest of the story proceeds exactly as in sec . [ sec : fourgluon ] , allowing us to write the kinematic numerators as @xmath261    v(k_1 , k_2 , k_3)\end{aligned}\ ] ] where we have suppressed the polarization vectors @xmath174 for @xmath22 . note that the kinematic numerators @xmath278 have exactly the same form as the kinematic numerators for the four - gluon amplitude , even though the masses for particles 1 through 3 can be nonzero . the proof of the vanishing of the sum of numerators proceeds exactly as in sec . [ sec : fourgluon ] .    in this section , we have explicitly shown that the sum of kinematic numerators for four - point amplitudes @xmath279 vanishes ( where @xmath227 can have spin zero , one - half , or one ) and thus have demonstrated the invariance of the four - point amplitude under the color - factor symmetry associated with gluon 4 . as the results of the last subsection have shown , this result remains valid even when particles 1 through 3 are massive ; only the gluon associated with the color - factor symmetry need be massless . the results we have derived will be used in the next section to prove a more general result . [ sec : rvefund ]    in this section , we use the radiation vertex expansion to demonstrate the invariance under the color - factor symmetry of tree - level gauge - theory amplitudes containing at least one gluon together with massless or massive particles in arbitrary representations of the gauge group and with arbitrary spin @xmath28 . for concreteness , we focus on the @xmath0-point amplitude @xmath280 with @xmath48 gluons and a pair of fundamentals @xmath227 , but it will be clear that the proof applies to more general amplitudes . the proof is very similar to that given in sec . [ sec : rvegluon ] for @xmath0-gluon amplitudes , and so we only highlight the differences . the radiation vertex expansion constructs the @xmath0-point amplitude @xmath280 by attaching gluon @xmath281 to all possible @xmath40-point diagrams @xmath179 , with two fundamentals and @xmath100 gluons , in all possible ways , and reorganizing this as a sum over all the vertices of the @xmath40-point diagram . we have already shown in sec . [ sec : rvegluon ] that the contributions of the three- and four - gluon vertices are invariant under the color - factor symmetry , so we only need to demonstrate the same for vertices involving two fundamentals . we do this separately for fundamentals with spin zero , one - half , and one . to derive the contribution of the @xmath245 vertices to the radiation vertex expansion for spin - one - half fundamentals @xmath227 , we examine the effect of attaching a gluon to a fermion leg , either external or internal . first we single out ( fermion ) leg 1 , denoting the contribution of an @xmath40-point diagram @xmath179 to the amplitude as @xmath282 where @xmath182 denotes momenta belonging to @xmath283 . attaching gluon @xmath19 to external fermion leg @xmath70 , using @xmath253 , and contracting with @xmath284 , we obtain - 2 g ( t^_a)^_1 _   2 k_a k_1 |u ( k_1 ) c^ ( k_1+k_a , ) . [ attachexternalpsibar ] next , we single out ( fermion ) leg @xmath0 , denoting the contribution of the diagram @xmath179 to the amplitude as @xmath285 , where @xmath182 denotes momenta belonging to @xmath286 . attach gluon @xmath19 to external fermion leg @xmath0 , use @xmath287 , and contract with @xmath284 to obtain + 2 g b _ ( k_n + k_a , ) ( t^_a)^ _  _ n 2 k_a k_n u(-k_n ) . [ attachexternalpsi ] now we single out one of the internal fermion lines of @xmath179 , which divides the diagram into two subdiagrams @xmath288 and @xmath289 , and splits the external legs @xmath290 into two complementary sets @xmath189 and @xmath190 . the contribution of the diagram @xmath179 can thus be written as b _ ( -k , ) i ^ _   - m c^ ( k , ) where @xmath191 is the momentum running through the line , and the @xmath182 in @xmath186 and @xmath187 denote momenta belonging to @xmath189 and @xmath190 respectively . attaching gluon @xmath19 to the line connecting the two subgraphs and contracting with @xmath192 , we have - i g2 b _ ( -k , ) 1 - m ( t^_a)^ _   c^ ( k+k_a , ) . now we use the identity@xcite @xmath291\right )   \over    k_a \cdot k }   ~-~   { \left ( \ve_a \cdot k   + { 1\over 4 } \left [ \epsaslash , \kaslash \right ] \right ) \over   k_a\cdot k }   { 1 \over \kslash + \kaslash - m } \label{fermionpropidentity}\end{aligned}\ ] ] to rewrite this as @xmath292 c^{\tk } ( k + k_a ,   \cdots   )   \right\ } { \nonumber}\\ & +   \left\ {   \sqrt2 g { ( t^{\ta_a})^\tj_{~\tk } \over 2 k_a\cdot k }   ~b_{\tj } (   -k , \cdots )   \left [   \ve_a \cdot k   -i \ve_{a\alpha } k_{a\beta }   \sigma^{\alpha\beta }    \right ] \right\ } { i \over \kslash + \kaslash - m } c^{\tk } ( k+k_a ,   \cdots   )   \ , . \label{attachinternalfermion}\end{aligned}\ ] ] each term can be associated with one of the two vertices to which the line is attached . we now choose one of the @xmath245 vertices @xmath41 of @xmath179 . such a vertex divides the external legs into three non - overlapping subsets @xmath196 , @xmath197 such that @xmath293 . the contribution of the diagram @xmath179 to the @xmath40-point amplitude can be expressed as @xmath294 where @xmath199 is the momentum flowing out of each leg of the vertex , and the @xmath182 in @xmath186 , @xmath187 , and @xmath295 denote momenta belonging to @xmath296 , @xmath297 , and @xmath298 respectively . if either fermion leg is external , then @xmath299 or @xmath300 . if the gluon leg is external , then @xmath301 is @xmath302 . we now attach gluon @xmath19 to each of the legs of this @xmath303 vertex , either to an external leg or to an internal line . using the expressions above as well as those in sec . [ sec : rvegluon ] , we obtain @xmath304 \v^{\mu_3 }   + { ( t^{\tc_3 } t^{\ta_a } ) ^{\tj_1}_{~~ \tj_2 } \over 2 k_a \cdot k_2 } \v^{\mu_3 }   \big [ \ve_a \cdot k_2 + i \ve_{a\alpha } k_{a\beta }   \sigma^{\alpha\beta } \big ] { \nonumber}\\ & + { f _ { \ta_a \tc_{3 } \tb } ( t^{\tb } ) ^{\tj_1}_{~~ \tj_2 } \over 2 k_a \cdot k_3 } \big [   \eta^{\mu_3 \nu } \ve_a \cdot k_3 -i \ve_{a\alpha } k_{a\beta }   ( s_3^{\alpha\beta } ) ^{\mu_3\nu } \big ] \v_{\nu }   \bigg )   c^{\tj_2 } ( -k_2 ,   \cdots )     \label{fermioncontrib}\end{aligned}\ ] ] which is the contribution of the @xmath303 vertex to the radiation vertex expansion . we now wish to show that is invariant under the color - factor symmetry . as in sec . [ sec : rvegluon ] , we first assume that the subdiagrams corresponding to @xmath295 , @xmath186 , and @xmath187 contain no four - gluon vertices . designate by @xmath213 with @xmath22 the color factors associated with each of the three terms in , including factors of @xmath57 and @xmath305 in the subdiagrams . these color factors manifestly satisfy @xmath217 , and the variation of @xmath213 under the color - factor shift associated with gluon @xmath19 is _ a   c _  =   _ ( a,,v )   k_a k_r [ psicolorfactorshift ] which preserves @xmath217 . the variation of under is then proportional to ( _ a k_r ) ^_3 - i _ a k_a . [ variationspinhalf ] the first term vanishes by momentum conservation , and the second by the transformation properties of @xmath306 , as we saw in sec . [ sec : fourfund ] . if the subdiagrams @xmath295 , @xmath186 , and @xmath187 do contain four - gluon vertices , we can expand into individual pieces , each of which is invariant under the color - factor symmetry . together with the result from sec . [ sec : rvegluon ] that the contributions to the radiation vertex expansion from the three- and four - gluon vertices are also invariant , we have thus shown that the amplitude @xmath280 with spin - one - half fundamentals is invariant under the color - factor symmetry . in fact , this proof applies to an amplitude with an arbitrary number of pairs of fundamentals , and will be used in the sequel @xcite to prove the bcj relations @xcite for that class of amplitudes .      to derive the contribution of the @xmath245 and @xmath248 vertices to the radiation vertex expansion for spin - zero fundamentals @xmath227 , we examine the effect of attaching a gluon to a scalar leg , either external or internal . first we single out ( scalar ) leg 1 , denoting the contribution of an @xmath40-point diagram @xmath179 to the amplitude as @xmath307 where @xmath182 denotes momenta belonging to @xmath283 . attaching gluon @xmath19 to external scalar leg @xmath70 we obtain @xmath308    c^{\tj } ( k_1+k_a , \cdots ) \,.\end{aligned}\ ] ] contracting this with @xmath284 eliminates the first term in the square brackets , leaving - 2 g ( t^_a)^_1 _   2 k_a k_1 _ a k_1 c^ ( k_1+k_a , ) . similarly , attaching gluon @xmath19 to external scalar leg @xmath0 , we obtain + 2 g _ a k_n b _ ( k_n + k_a , ) ( t^_a)^ _  _ n 2 k_a k_n . next we single out one of the internal scalar lines of @xmath179 , which divides the diagram into two subdiagrams @xmath288 and @xmath289 , and splits the external legs into two complementary sets @xmath189 and @xmath190 . the contribution of the diagram @xmath179 can thus be written as b _ ( -k , ) i ^_k^2-m^2 c^ ( k , ) where @xmath191 is the momentum running through the line , and the @xmath182 in @xmath186 and @xmath187 denote momenta belonging to @xmath189 and @xmath190 respectively . attaching gluon @xmath19 to the line connecting the two subgraphs yields - i g2 b _ ( -k , ) ( t^_a)^ _   v^_a ( k ,- k - k_a , k_a ) c^ ( k+k_a , ) . contracting with @xmath192 and using @xmath143 , we have - 2 i g b _ ( -k , ) ( t^_a)^ _   _ a k c^ ( k+k_a , ) . now we use the identity ( [ massivesplitprop ] ) to rewrite this as @xmath309 we associate each of the terms in this equation with one of the two vertices to which the line is attached . next we choose one of the scalar - scalar - gluon vertices @xmath41 of @xmath179 ( if it has any ) . such a vertex divides the external legs into three non - overlapping subsets @xmath196 , @xmath197 such that @xmath293 . the contribution of the diagram @xmath179 can be expressed as @xmath310 where @xmath199 is the momentum flowing out of each leg of the vertex , and the @xmath182 in @xmath186 , @xmath187 , and @xmath295 denote momenta belonging to @xmath296 , @xmath297 , and @xmath298 respectively . if either scalar leg is external , then @xmath311 or @xmath312 . if the gluon leg is external , then @xmath301 is @xmath302 . we now attach gluon @xmath19 to each of the legs of this scalar - scalar - gluon vertex , either to an external leg or to an internal line . this yields @xmath313 v^{\mu_3 } ( k_1 + k_a , k_2 , k_3 ) + {   ( t^{\tc_3 } t^{\ta_a } ) ^{\tj_1}_{~~ \tj_2 }   \over 2 k_a \cdot k_2 } \big [   \ve_a \cdot k_2 \big ] v^{\mu_3 } ( k_1 , k_2 + k_a , k_3 ) { \nonumber}\\ & + {   f _ { \ta_a \tc_{3 } \tb } ( t^{\tb } ) ^{\tj_1}_{~~ \tj_2 }   \over 2 k_a \cdot k_3 } \big [   \eta^{\mu_3 \nu } \ve_a \cdot k_3 +   \left ( \ve_a^{\mu_3 } k_a^\nu - \ve_a^\nu k_a^{\mu_3 }   \right ) \big ] v_{\nu } ( k_1 , k_2 , k_3 + k_a ) \bigg )   c^{\tj_2 } ( -k_2 ,   \cdots )    \ , . \label{attachscalarleg}\end{aligned}\ ] ] we can also attach gluon @xmath19 directly to the scalar - scalar - gluon vertex itself . using , this yields @xmath314 we now use in , and combine as we did in sec . [ sec : fourfund ] . leaving the @xmath315 index implicit , we obtain the contribution of the scalar - scalar - gluon vertex to the radiation vertex expansion @xmath316 v(k_1 , k_2 , k_3 ) \label{threescalarcontrib } \\ & + { ( t^{\tc_3 } t^{\ta_a } ) ^{\tj_1}_{~~ \tj_2 } \over 2 k_a \cdot k_2 } \left [   \ve_a \cdot k_2   -i \ve_{a\alpha } k_{a\beta }   l_2^{\alpha\beta }   \right ] v(k_1 , k_2 , k_3 ) { \nonumber}\\ & + { f _ { \ta_a \tc_{3 } \tb } ( t^{\tb } ) ^{\tj_1}_{~~ \tj_2 } \over 2 k_a \cdot k_3 } \left [   \ve_a \cdot k_3   -i \ve_{a\alpha } k_{a\beta }   j_3^{\alpha\beta }   -i \ve_{a\alpha } k_{a\beta }   k_{a\gamma } s_3^{\alpha\beta }   { \partial \over \partial   k_{3\gamma } } \right ] v(k_1 , k_2 , k_3 ) \bigg )   c^{\tj_2 } ( -k_2 ,   \cdots )    { \nonumber}\end{aligned}\ ] ] where @xmath317 and @xmath318 act on the @xmath315 index of @xmath319 .    at the end of the last subsection , we discussed the color - factor symmetry acting on the @xmath245 vertex contribution to the radiation vertex expansion . under , the variation of is proportional to v(k_1 , k_2 , k_3 ) . [ threesumsfund ] in sec . [ sec : fourfund ] , we demonstrated that each of the three terms in vanishes . therefore the contributions of the @xmath245 vertices ( [ threescalarcontrib ] ) to the radiation vertex expansion are invariant under the color - factor shift associated with gluon @xmath19 . finally we choose one of the scalar - scalar - gluon - gluon vertices @xmath41 of @xmath179 ( if it has any ) . such a vertex divides the external legs into four non - overlapping subsets @xmath208 , @xmath209 such that @xmath210 . using , the contribution of diagram @xmath179 can be expressed as @xmath320 we now attach gluon @xmath19 to each of the legs of this scalar - scalar - gluon - gluon vertex , either to an external leg or to an internal line . this yields @xmath321 \eta^{\mu_3 \mu_4 }   + { ( \ { t^{\tc_3 } , t^ { \tc_4 } \ } t^{\ta_a } ) ^{\tj_1}_{~~ \tj_2 } \over 2 k_a \cdot k_2 } \big [ \ve_a \cdot k_2 \big ] \eta^{\mu_3 \mu_4 }   { \nonumber}\\ & + { f _ { \ta_a   \tc_{3 } \tb } ( \ { t^{\tb } , t^ { \tc_4 } \ } ) ^{\tj_1}_{~~ \tj_2 }   \over 2 k_a \cdot k_3 } \big [ \delta^{\mu_3 } _ { ~\nu } \ve_a \cdot k_3 -i \ve_{a\alpha } k_{a\beta } ( s_3^{\alpha\beta } ) ^{\mu_3 } _ { ~~\nu }   \big ] \eta^{\nu \mu_4 }   { \nonumber}\\ & + { f _ { \ta_a   \tc_{4 } \tb } ( \ { t^{\tc_3 } , t^ { \tb } \ } ) ^{\tj_1}_{~~ \tj_2 }   \over 2 k_a \cdot k_4 } \big [ \delta^{\mu_4 } _ { ~\nu } \ve_a \cdot k_4 -i \ve_{a\alpha } k_{a\beta } ( s_4^{\alpha\beta } ) ^{\mu_4 } _ { ~~\nu }   \big ] \eta^{\mu_3 \nu }   \bigg ) c^{\tj_2 } ( -k_2 ,   \cdots )   \ , . \label{fourscalarcontrib}\end{aligned}\ ] ] we now need to consider the variation under the color - factor symmetry of this contribution to the radiation vertex expansion . designate by @xmath213 with @xmath220 the color factor associated with each of the terms in , including the factors of @xmath57 and @xmath305 in the subdiagrams . these color factors satisfy @xmath222 by virtue of @xmath322 the variation of @xmath213 under the color - factor shift associated with gluon @xmath19 is _ a   c _ =   _ ( a,,v )   k_a k_r . [ scalarshift ] the variation of under is therefore proportional to ( _ a k_r ) ^_3 _ 4 -i _ a k_a . [ variationquarticspinzero ] , the first term vanishes by momentum conservation , @xmath223 , and @xmath224 . the second term is the first - order lorentz transformation of the tensor @xmath323 , which vanishes . thus each vertex involving scalars that contributes to the radiation vertex expansion is invariant under the color - factor shift associated with gluon @xmath19 . together with the result from sec . [ sec : rvegluon ] that the contributions from the three- and four - gluon vertices to the radiation vertex expansion are also invariant , we have shown that the full amplitude @xmath280 with spin - zero fundamentals is invariant under the color - factor shift . in fact , any amplitude built with @xmath245 and @xmath324 vertices for scalar @xmath227 will have the color - factor symmetry .      to derive the contribution of the @xmath245 and @xmath248 vertices to the radiation vertex expansion for spin - one fundamentals @xmath227 , we examine the effect of attaching a gluon to a massive vector leg , either external or internal . first we single out ( vector ) leg 1 , denoting the contribution of an @xmath40-point diagram @xmath179 to the amplitude as @xmath325 where @xmath326 denotes momenta belonging to @xmath283 . attaching gluon @xmath19 to external vector leg @xmath70 we obtain - g 2 ( t^_a)^_1 _   ( k_a + k_1)^2 - m_1 ^ 2 v^_1 _ a ( k_1 , -k_1 - k_a , k_a ) p _ ( k_1+k_a ) c^ ( k_1+k_a , ) . contracting with @xmath327 , we obtain - 2 g ( t^_a)^_1 _   2 k_a k_1 _ 1_1 c^_(k_1 + k_a , ) [ attachexternalmassivevector ] where we used @xmath328 and @xmath274 . we did _ not _ use the vanishing of . similar expressions result from attaching gluon @xmath19 to the other legs . by comparing , we observe that the expression is the same for a massless or a massive vector particle . next we single out one of the internal lines of @xmath179 , which divides the diagram into two subdiagrams @xmath186 and @xmath187 , and splits the external legs @xmath188 into two complementary sets @xmath189 and @xmath190 . the contribution of the diagram can thus be written as b_^ ( -k , ) ( -i ^ _ ) p_(k ) k^2 - m^2 c^ ( k , ) where @xmath191 is the momentum running through the line , and the @xmath182 in @xmath186 and @xmath187 denote momenta belonging to @xmath189 and @xmath190 respectively . attaching gluon @xmath19 to the line connecting the two subgraphs and contracting with @xmath284 , we obtain - i g2 b_^ ( -k , ) ( t^_a)^ _   p _ ( k ) _ a_a v^_a ( k ,- k - k_a , k_a ) p _ ( k+k_a ) c^ ( k+k_a , ) . now we use the identity @xcite @xmath329   [ ( k+k_a)^2-m^2 ] } \label{vectorpropidentity } \\ & = { 1 \over k_a \cdot k } \left\ {   { -   p_{\mu\lambda } ( k )   \left [   \delta^\lambda_{~\nu } \ve_a \cdot k +   \ve_a^\lambda k_{a\nu } - k_a^{\lambda } \ve_{a\nu } \right ] \over    k^2-m^2 }   + { \left [   \delta_\mu^{~\kappa } \ve_a \cdot k +   \ve_{a\mu } k_a^\kappa - k_{a\mu } \ve_a^\kappa \right ]   p_{\kappa\nu } ( k+k_a )   \over ( k+k_a)^2 -m^2 } \right\ } { \nonumber}\end{aligned}\ ] ] to rewrite this as @xmath330 c^{\tk}_{\nu } ( k+k_a ,   \cdots   )   \right\ } \\ & + \left\ {     \sqrt2 g b_{\tj \mu } ( \cdots ,   -k )    { ( t^{\ta_a})^\tj_{~\tk } \over   2 k_a \cdot k }   \big [   \eta^{\mu \kappa } \ve_a \cdot k +   \left ( \ve_a^\mu k_a^\kappa - \ve_a^\kappa k_a^{\mu }   \right ) \big ] \right \}{-i p_{\kappa\nu } ( k+k_a ) \over ( k+k_a)^2 -m^2 }    c^{\tk\nu } ( k+k_a ,   \cdots   )   \ , . { \nonumber}\end{aligned}\ ] ] we did _ not _ use the vanishing of . we associate each of the terms in curly brackets with one of the two vertices to which the line is attached .    for the rest of the discussion , we can be brief . the expressions for the contributions to the radiation vertex expansion from @xmath245 and @xmath324 vertices are similar to those for spin - zero fundamentals , except that we must include @xmath159 and @xmath160 terms for @xmath331 and 2 . the proof that these vertex contributions are invariant under the color - factor symmetry relies on some of the results from sec . [ sec : rvegluon ] . thus we have shown that the full amplitude @xmath280 with massive spin - one fundamentals is invariant under the color - factor shift . in fact , any amplitude built with @xmath245 and @xmath324 vertices with a massive vector particle @xmath227 will have the color - factor symmetry . we can go even further and state that any amplitude built from @xmath231 and @xmath332 vertices ( with @xmath227 having arbitrary spin @xmath28 ) where not only @xmath227 but also some of the gluons are massive ( through spontaneous symmetry breaking ) will be invariant under the color - factor shift . the only particle in the amplitude that _ must _ be massless is the gluon associated with the color - factor symmetry . [ sec : prop ]    the symmetry that we have introduced in this paper is possessed not only by gauge - theory amplitudes but also by the amplitudes of the much simpler theory @xcite of massless scalars @xmath333 transforming in the adjoint of the color group @xmath334 . these bi - adjoint scalars have only cubic interactions of the form f _ f_   ^ ^ ^ where @xmath335 and @xmath336 are the structure constants of @xmath337 and @xmath338 . the tree - level @xmath0-point amplitude is given by the sum over cubic diagrams _ n  =   _ i c_i c_i d_i . [ biadjointamp ] using , the bi - adjoint scalar tree amplitude ( [ biadjointamp ] ) can be written as _ n  =   _ s_n-2 _ s_n-2 _ 1 n  m ( 1 n| 1 n )  _ 1 n [ biadjointprop ] where m ( 1 n| 1 n )  =   _ i m_i,1n m_i , 1n d_i [ propagator ] are double - partial amplitudes of the bi - adjoint scalar theory @xcite . the @xmath339 are also the entries of @xmath15 propagator matrix defined in ref . vaman and yao argued that the propagator matrix has rank @xmath16 by virtue of momentum conservation , using explicit low @xmath0 examples . cachazo , he , and yuan confirmed this for general @xmath0 by expressing the double - partial amplitudes as a sum over the @xmath16 solutions of the scattering equations @xcite .    the cubic vertex expansion of the @xmath0-point bi - adjoint scalar amplitude with respect to external scalar @xmath19 ( see sec . [ sec : cfs ] ) is given by _ n  =   _ i _ v v_(a , i ) 1 _ s=1 ^ 3 d_(a , i , v , s ) [ biadjointvertexexpansion ] where c_(a , i , v , r )  =   0 , _ ( a , i , v , r )  =   0 . [ biadjointjacobi ] the color - factor shift with respect to massless scalar @xmath19 acts on the color factors appearing in as _ a   c_(a , i , v , r )  =   _ ( a , i , v )  k_a k_(a , i , v , r ) . the variation of under this shift _ a   _ n   =   12 _ i _ v v_(a , i ) _ ( a , i , v ) _ s=1 ^ 3 d_(a , i , v , s ) _ ( a , i , v , r ) vanishes by virtue of , thus establishing that the amplitudes of the bi - adjoint scalar theory possess the color - factor symmetry . now we consider the variation of the amplitude ( [ biadjointprop ] ) under the shift ( [ halfladdershift ] ) associated with @xmath92 , @xmath340 using the invariance of @xmath341 , together with independence of @xmath93 and @xmath342 , we obtain _ b=3^n ( k_1 k_2 + _ c=3^b-1 k_2 k_(c ) ) m(1 , ( 3 ) , , ( b-1 ) , 2 , ( b ) , , ( n-1 ) , n | 1 n )  =   0 [ propnullvectors ] i.e. , we have derived a set of @xmath16 null eigenvectors of the propagator matrix . other sets of null eigenvectors are obtained from the color - factor shifts associated the other massless scalars in the amplitude . since the @xmath15 propagator matrix is known @xcite to have rank @xmath16 , at most @xmath17 of these null eigenvectors can be independent . thus , the color - factor symmetry associated with @xmath100 of the massless scalars suffices to guarantee the reduced rank of the propagator matrix . this nicely explains a result found by one of the present authors in ref . @xcite , that the propagator matrix persists in having rank @xmath16 even when up to three of the external particles in the amplitude are massive , but if more than three particles are massive , the rank of the propagator matrix is greater . we now see that for @xmath343 massive and @xmath344 massless particles , the number of null eigenvectors generated by the color - factor symmetry will be @xmath345 , and therefore the rank of the propagator matrix will be @xmath346 for @xmath347 through @xmath348 . returning now to yang - mills theory , if we assume that the numerators of the @xmath0-gluon amplitude obey color - kinematic duality , i.e. they obey the same jacobi relations as @xmath6 , then they can similarly be expressed in terms of @xmath62 half - ladder numerators @xmath349 via n_i  =   _ s_n-2 m_i , 1n  * n*_1 n . [ nmn ] using , the color - ordered amplitudes in the kleiss - kuijf basis can be written in terms of the propagator matrix as a(1 , ( 2 ) , , ( n-1 ) , n )  =   _ s_n-2 m ( 1 n| 1 n )  * n*_1 n . [ amn ] thus the null eigenvectors of the propagator matrix ( [ propnullvectors ] ) imply that the color - ordered @xmath0-gluon amplitudes obey the bcj relations ( [ fundbcj ] ) .    as we saw in sec . [ sec : cfs ] , however , it is not necessary to require color - kinematic duality in order to prove the bcj relations . the bcj relations follow from the weaker constraint ( [ sumoverdelta ] ) , and both and the bcj relations are a consequence of the color - factor symmetry of the amplitude , which is established through the radiation vertex expansion . [ sec : loop ]    given the connection established in this paper between the color - factor symmetry of tree - level gauge - theory amplitudes and color - kinematic duality / bcj relations , it is naturally of great interest to see whether these ideas can be extended to loop level . in this section , we generalize the cubic vertex expansion introduced in sec . [ sec : cfs ] to loop - level amplitudes . we then define a set of shifts of one - loop color factors that depend on the loop momentum as well as the momenta of external particles , and ask whether the cubic vertex expansion of the one - loop amplitude is invariant under these shifts . it is obvious from the variation of the cubic vertex expansion ( [ variation ] ) that the tree - level amplitude will be invariant under a color - factor shift if the numerators satisfy color - kinematic duality . the reader may have noticed , however , that up until now we have taken great pains not to invoke color - kinematic duality to prove color - factor symmetry . we have chosen rather to show that it follows directly from a lagrangian approach . at loop level , we no longer have that luxury , at least at the current stage of development . instead we will explicitly invoke loop - level color - kinematic duality ( for the theories in which it has been shown to hold @xcite ) in order to demonstrate the color - factor symmetry of the one - loop amplitude . we then show that the invariance of the amplitude under color - factor shifts implies a set of relations among the integrands of its color - ordered amplitudes . 23}$ ] , @xmath3503}$ ] , @xmath351}$ ] , @xmath352 , @xmath353 , and @xmath354.,width=604 ]    to construct the cubic vertex expansion of an @xmath355-loop @xmath0-gluon amplitude with respect to gluon @xmath19 , we begin with the set of @xmath355-loop @xmath40-point cubic diagrams @xmath39 with external legs @xmath356 . for example , for the one - loop four - gluon amplitude , two of the three - point diagrams are shown in fig . [ fig : three ] ; the rest are obtained from relabelings of the external legs . then we attach gluon @xmath19 in all possible ways , either to the external legs or to the internal lines of @xmath39 . for example , attaching gluon 4 to the triangle diagram in fig . [ fig : three ] in all possible ways , we obtain the diagrams in fig . [ fig : four ] . the diagrams in fig . [ fig : four ] correspond to the following terms in the cubic decomposition of the one - loop four - point amplitude @xcite @xmath35723 } \,\ , n_{[14]23 } \over d_{[14]23 } }   ~+~ { c_{1[24]3 } \,\ , n_{1[24]3 } \over d_{1[24]3 } }   ~+~ { c_{12[34 ] } \,\ , n_{12[34 ] } \over d_{12[34 ] } } { \nonumber}\\ &   ~+~ { c_{1234 } \,\ , n_{1234 } \over d_{1234 } }   ~+~ { c_{1423 } \,\ , n_{1423 } \over d_{1423 } }   ~+~ { c_{1243 } \,\ , n_{1243 } \over d_{1243 } } \bigg ] \label{looptrianglecontrib}\end{aligned}\ ] ] where the denominators in are the products of inverse propagators associated with the diagrams . ( there could be different sets of denominators depending on the mass of the particle circulating in the loop . ) explicit definitions of the color factors are given below in . the terms obtained by attaching the gluon to an internal line  in this case , the last three terms of  are split into two by applying the identity ( [ splitprop ] ) or , in the case of massive internal lines , . the terms are then reorganized into a sum over the vertices of @xmath39 . for example , the terms in are reorganized into @xmath35823 } \,\ , n_{[14]23 } \over 2 k_4 \cdot k_1 } ~+~ { c_{1234 } \,\ , n_{1234 } \over 2 k_4 \cdot \ell } ~-~ { c_{1423 } \,\ , n_{1423 } \over 2 k_4 \cdot ( \ell+k_1 ) } \right ] { \nonumber}\\ & + { 1 \over \ell^2 ( \ell+k_1)^2 ( \ell - k_3)^2 } \left [      { c_{1[24]3 } \,\ , n_{1[24]3 } \over 2 k_4 \cdot k_2 } ~+~ { c_{1423 } \,\ , n_{1423 } \over 2 k_4 \cdot ( \ell+k_1 ) } ~-~ { c_{1243 } \,\ , n_{1243 } \over 2 k_4 \cdot ( \ell+k_1+k_2 ) } \right ] \label{loopfourpoint } \\ & + { 1 \over \ell^2 ( \ell+k_1)^2 ( \ell+k_1 + k_2)^2 } \left [      { c_{12[34 ] } \,\ , n_{12[34 ] } \over 2 k_4 \cdot k_3 } ~+~ { c_{1243 } \,\ , n_{1243 } \over 2 k_4 \cdot ( \ell+k_1+k_2 ) } ~-~ { c_{1234 } \,\ , n_{1234 } \over 2 k_4 \cdot ( \ell+k_1+k_2+k_3 ) } \right ] \bigg\ } { \nonumber}\end{aligned}\ ] ] where @xmath359 in the last term of the last line of differs from the label in fig . [ fig : four ] by a shift . if the particle circulating in the loop has mass @xmath360 , the expressions @xmath361 in ( those outside the square brackets ) are all replaced by @xmath362 . we hasten to remind the reader that @xmath363 is only one part of the one - loop four - gluon amplitude ; similar expressions are obtained by attaching gluon 4 to the other one - loop three - point diagrams @xmath39 . observe that if we apply a shift to the numerators @xmath36423 } & ~=~ \beta k_4 \cdot k_1 , & \delta_4 n_{1[24]3 } & ~=~ \beta k_4 \cdot k_2 , & \delta_4 n_{12[34 ] } & ~=~ \beta k_4 \cdot k_3 , { \nonumber}\\ \delta_4 n_{1234 }    & ~=~ \beta k_4 \cdot \ell , & \qquad   \delta_4 n_{1423 }     & ~=~ \beta k_4 \cdot ( \ell + k_1 ) , & \qquad \delta_4 n_{1243 }     & ~=~ \beta k_4 \cdot ( \ell + k_1+k_2 ) , \label{loopkinematicshift}\end{aligned}\ ] ] the expression ( [ loopfourpoint ] ) remains unchanged as a result of the jacobi identities 0  =   c_[14]23 + c_1234 - c_1423  =   c_1[24]3 + c_1423 - c_1243  =   c_12[34 ] + c_1243 - c_1234 [ loopcolorjacobi ] which means that corresponds to a generalized gauge transformation . \cdots}$ ] and @xmath365 , width=264 ]    the @xmath0-point generalization of , obtained by attaching gluon @xmath0 to the @xmath40-gon diagram , is given by @xmath366 \cdots } n_{\cdots [ bn ] \cdots } \over 2 k_n \cdot k_b } ~+~ {   c_{\cdots b-1,n , b \cdots } n_{\cdots b-1,n , b \cdots } \over 2 k_n \cdot \left ( \ell + \sum_{a=1}^{b-1}k_a \right )   } ~-~ {   c_{\cdots b , n , b+1 \cdots } n_{\cdots b , n , b+1 \cdots } \over 2 k_n \cdot \left ( \ell + \sum_{a=1}^{b } k_a \right )   } \right ] \bigg\ } \label{loopnpoint}\end{aligned}\ ] ] where @xmath3673 \cdots n }   & ~\equiv~   \sum_{\tb_1,\ldots,\tb_n }   f_{\ta_1 \ta_2 \tb_2 } f_{\tb_1 \tb_2 \tb_3 } f_{\tb_3 \ta_3 \tb_3 } \cdots   f_{\tb_n \ta_n \tb_1 } \ , . \label{loopcolorfactors}\end{aligned}\ ] ] the `` ring '' diagrams @xmath368 are cyclically symmetric and reflection symmetric c_12 n = ( -1)^n c_n21 . [ reflection ] the color factors appearing in are shown in fig . [ fig : five ] and satisfy the jacobi relations 0  =   c _  +   c_b-1,n , b  -   c_b , n , b+1 . [ loopnpointjacobi ] again , if a particle of mass @xmath360 is circulating in the loop , we replace terms of the form @xmath369 with @xmath370 . similar expressions are obtained by attaching gluon @xmath0 to the other one - loop @xmath40-point diagrams @xmath39 , including @xmath40-gon diagrams with other permutations of the external legs @xmath371 ,    recently , partial fraction identities similar to have been employed to recast one - loop amplitudes into a new form whose denominators contain factors linear in the loop momentum @xcite , somewhat analogous to . these new expressions are those that naturally emerge from a scattering - equation approach to loop - level amplitudes . next we consider the behavior of one - loop amplitudes under momentum - dependent shifts of its color factors . first we must define a set of shifts consistent with the requirements elucidated earlier in the paper . for the one - loop four - gluon amplitude , the color - factor shift associated with gluon 4 must satisfy _ 4 c_[14]23  =   k_4 k_1 , _ 4 c_1[24]3  =   k_4 k_2 , _ 4 c_12[34 ]  =   k_4 k_3 [ loopfourcommshift ] since these diagrams have gluon 4 attached to an external leg . requiring the color - factor shift to respect the jacobi relations ( [ loopcolorjacobi ] ) implies that _ 4 c_1423  =   _ 4 c_1234 + k_4 k_1 _ 4 c_1243  =   _ 4 c_1234 + k_4 ( k_1+k_2 ) . [ loopfourjacobishift ] unlike the tree - level case , however , these requirements alone are not sufficient to fix the values of all the shifts ; one of them , @xmath372 , remains arbitrary . in analogy with , we define the remaining shift to be _ 4 c_1234  =   k_4 [ loopfourshift ] where @xmath359 is the loop momentum . the effect of this shift on is @xmath37323 } + n_{1234 } -n_{1423 } \over \ell^2 ( \ell - k_2 -k_3)^2 ( \ell - k_3)^2 } ~+~ { n_{1[24]3 } + n_{1423 } -n_{1243 } \over \ell^2 ( \ell+k_1)^2 ( \ell - k_3)^2 } { \nonumber}\\ & ~+~ { n_{12[34 ] } + n_{1243 } -n_{1234 } \over \ell^2 ( \ell+k_1)^2 ( \ell+k_1 + k_2)^2 } \bigg ] \ , . \label{loopsumoverdelta}\end{aligned}\ ] ] similar expressions are obtained for the contributions to the cubic vertex expansion from the other three - point diagrams .    a goal consistent with the development in this paper would be to prove by some alternative means ( such as the radiation vertex expansion ) that @xmath374 vanishes under the one - loop color - factor shift . that would imply the vanishing of the r.  h.  s.  of plus the expressions obtained from the other three - point diagrams , imposing a gauge - invariant constraint on the one - loop kinematic numerators . but at this point in the development of the subject , we have no such proof , and therefore we will turn the argument around , and use the knowledge that there exist kinematic numerators for the one - loop four - gluon amplitude that obey color - kinematic duality 0  =   n_[14]23 + n_1234 - n_1423  =   n_1[24]3 + n_1423 - n_1243  =   n_12[34 ] + n_1243 - n_1234 [ loopkinematicjacobi ] for pure yang - mills theory ( with only gluons circulating in the loop ) @xcite as well as for theories with other particles circulating in the loop @xcite . in these cases , the kinematic jacobi identities ( [ loopkinematicjacobi ] ) imply that vanishes , as do the other contributions to the cubic vertex expansion . thus , the one - loop four - gluon amplitude in these theories is invariant under the color - factor shift specified by eqs . ( [ loopfourcommshift ] ) , ( [ loopfourjacobishift ] ) , and ( [ loopfourshift ] ) . the cubic vertex expansion of the one - loop four - point amplitude of the bi - adjoint scalar theory may be obtained by replacing the kinematic numerators @xmath9 with a second copy of the color factors @xmath375 . since the latter obey the one - loop color jacobi identities ( [ loopcolorjacobi ] ) , the bi - adjoint scalar one - loop four - point amplitude is also invariant under the color - factor shift . it is known that an independent basis of one - loop color factors are those associated with ring diagrams modulo cyclic permutations and reflections @xcite . therefore the one - loop @xmath0-gluon amplitude can be written _ n^(1 )  =   _ s_n-1/-5pt _ 2 c_1   i(1 , ( 2 ) , , ( n ) ) [ ringbasis ] where @xmath60 is a permutation of @xmath376 , @xmath377 denotes the reflection symmetry @xmath378 , and @xmath379 are the integrands of the one - loop color - ordered amplitudes . may be regarded as the result of a generalized gauge transformation in which the kinematic numerators associated with the non - ring color factors are set to zero @xcite . specializing to the one - loop four - gluon amplitude , gives _ 4^(1 )  =   . for theories whose numerators respect one - loop color - kinematic duality , the invariance of the one - loop amplitude under the color - factor shift implies the following condition on the integrands 0  =   . [ integrandfourpointconstraint ] relations of this form were first uncovered in refs . @xcite from the perspective of on - shell recursion relations , and revisited recently using monodromy relations in string theory @xcite . conversely , if the integrands of the one - loop amplitude of a theory can be shown to satisfy , then it follows that the one - loop amplitude is invariant under the color - factor shift , avoiding the need to invoke color - kinematic duality .    it is straightforward to generalize these considerations to one - loop @xmath0-gluon amplitudes . we may define a color - factor shift associated with any external gluon @xmath19 , but for simplicity of presentation we will focus on gluon @xmath0 . let @xmath241 denote a permutation of @xmath380 . the color factor @xmath381 \cdots}$ ] shown in fig . [ fig : five ] undergoes a shift _ k_n k_(b ) because gluon @xmath0 is attached to gluon leg @xmath382 . requiring the shifts to respect the jacobi relation ( [ loopnpointjacobi ] ) implies _ n c _ ( b ) n ( b+1 )  =   _ n c _ ( b-1 ) n ( b )  +   _ n c _ we must additionally define the shifts _ n c _ 1 ( 2 ) ( n-1 ) n = _ n , k_n [ additionalshift ] for a set of half ) . for example for the four - point case @xmath383 but @xmath384 , which is not of the form ( [ additionalshift ] ) . ] of the permutations @xmath241 , where @xmath385 are a set of @xmath386 independent arbitrary constants . together these conditions imply that the shifts of the ring color factors are given by _ n c_1 ( 2 ) ( b-1 ) n ( b ) ( n )  =   _ n , ( k_n + k_n k_1 + _ c=2^b-1 k_n k_(c ) ) , b \{2 , , n-1 } [ loopnpointshift ] and the shifts of all other one - loop color factors are fixed by requiring that they respect the jacobi relations . thus there is an @xmath386-dimensional family of one - loop color - factor shifts associated with gluon @xmath0 .    applying the shift ( [ loopnpointshift ] ) to , we obtain _ n , ( n-1)gon^(1 )   _ b=1^n-1 ( n _  +   n_b-1,n , b  -   n_b , n , b+1 ) _ c=1^b ( + _ a=1^c-1 k_a)^2 _ d = b+1^n-1 ( - _ a = d^n-1 k_a)^2 . [ loopsumoverdeltanpoint ] for @xmath18 supersymmetric yang - mills theory , numerators for the one - loop five - gluon @xcite and higher - point @xcite amplitudes have been constructed which satisfy color - kinematic duality 0  =   n _ +   n_b-1,n , b  -   n_b , n , b+1 and which therefore imply that vanishes . the shifts of the terms in the cubic vertex expansion obtained from other @xmath40-point diagrams @xmath39 similarly vanish . thus , we have established that these amplitudes possess one - loop color - factor symmetry . the one - loop @xmath0-point amplitudes of the bi - adjoint scalar theory also possess this symmetry because the second copy of the color factors @xmath375 obey jacobi identities ( [ loopnpointjacobi ] ) .    as we did above for the four - gluon amplitude , we can use this invariance to derive constraints on the integrands of color - ordered amplitudes . can be rewritten as _ n^(1 )  =   _ s_n-2/-5pt _ 2 _ b=2^n c_1 ( 2 ) ( b-1)n ( b ) ( n-1 )   i(1 , ( 2 ) , , ( b-1 ) , n , ( b ) , , ( n-1 ) ) invariance of this expression under together the independence of the parameters @xmath385 yields 0= _ b=2^n k_n ( + k_1 + _ c=2^b-1 k_(c ) )   i(1 , ( 2 ) , , ( b-1 ) , n , ( b ) , , ( n-1 ) ) [ integrandnpointconstraint ] the relations uncovered in refs .  @xcite . conversely , if we were to establish that the integrands of the color - ordered amplitudes of a given theory satisfy , we would have proven that the one - loop amplitude is invariant under the color - factor shift , independently of the assumption of color - kinematic duality . further study of this alternate path is merited . [ sec : concl ]    in this paper , we have introduced a new set of symmetries of gauge - theory amplitudes , which act as momentum - dependent shifts on the color factors appearing in the cubic decomposition of the amplitude . these symmetries are intimately linked to the presence of massless gauge bosons in the amplitude ( or massless adjoint scalars in the case of the bi - adjoint scalar theory ) and can be considered generalizations of the radiation symmetry of ref . we demonstrated that a wide class of tree - level gauge - theory amplitudes are invariant under these shifts , using a representation of the amplitude known as the radiation vertex expansion @xcite . we also introduced a related but distinct cubic vertex expansion of the amplitude , and used this to derive a set of gauge - independent constraints on the kinematic numerators appearing in the cubic decomposition of the amplitude . all known bcj relations for tree - level gauge - theory amplitudes @xcite follow as a direct consequence of the color - factor symmetry ( this paper and ref . @xcite ) . finally , we generalized the cubic vertex expansion and color - factor symmetry to loop level . we showed that one - loop amplitudes that satisfy color - kinematic duality are invariant under the one - loop color - factor symmetry , and derived a set of relations among the integrands of one - loop color - ordered amplitudes . let us take a look at the connection between the color - factor symmetry and more fundamental symmetries of the lagrangian , gauge and poincar invariance @xcite . the color - factor symmetry follows as a result of the vanishing of certain expressions , namely eqs . ( [ threesums ] ) , ( [ variationspinhalf ] ) , and ( [ threesumsfund ] ) , associated with the cubic vertices of a gauge - theory amplitude , and eqs . ( [ variationfourglounleading ] ) , ( [ variationfourglounsubleading ] ) , and ( [ variationquarticspinzero ] ) , associated with the quartic vertices . it is illustrative to examine the various contributions in a soft expansion in the gluon momentum @xmath13 , even though the color - factor symmetry is exact in @xmath13 . the leading term in the soft expansion corresponds to the @xmath387 term in each of eqs . ( [ threesums ] ) , ( [ variationfourglounleading ] ) , ( [ variationspinhalf ] ) , ( [ threesumsfund ] ) , and ( [ variationquarticspinzero ] ) . these are all proportional to @xmath388 where @xmath389 are the momenta flowing out of each leg of the vertex . this vanishes by @xmath224 together with momentum conservation @xmath390  a result of symmetry under spacetime translations . the subleading term in the soft expansion corresponds to the @xmath391 term in each of eqs . ( [ threesums ] ) , ( [ variationspinhalf ] ) , ( [ threesumsfund ] ) , and ( [ variationquarticspinzero ] ) , and to . these expressions are all given by a sum of angular momentum generators @xmath162 , which act as a first - order lorentz transformation on the relevant vertex factors . they vanish by lorentz invariance . thus the first two terms in the soft expansion vanish as a result of poincar invariance . it is a little more difficult to pin down the underlying symmetry responsible for the vanishing of the sub - subleading terms in , together with an analogous expression for spin - one fundamentals . the @xmath392 term in is proportional to @xmath393 and the @xmath392 term in is proportional to ( s_3^)^_3 _   k_3 v^(k_1 , k_2 , k_3 ) = ( ^_3 ^ -^_3 ^ ) . neither expression vanishes by itself but both do when contracted with @xmath394 for gluon @xmath19 . these identities , which go beyond the first - order poincar cancellation and are connected to yang - mills gauge symmetry , are key ingredients contributing to the color - factor symmetry .    returning to our discussion in the introduction of the connection of the color - factor symmetry with the photon radiation symmetry and radiation zeros ( for a collection of early references , see refs . @xcite ) , we have uncovered some additional analogs . for example , we can write a factorized form for the four - gluon amplitude ( [ fourpointamplitude ] ) _ 4 = s ( c_s s - c_t t ) ( n_s s - n_u u ) which vanishes at @xmath395 const . this is a non - abelian version of the original radiation zero studied almost forty years ago in @xmath396 and @xmath397 reactions , with a zero at @xmath398 constant . this original radiation factorization and its zero led to the prediction of a measurable experimental dip , which has now been confirmed @xcite . the analogous zero in the four - gluon amplitude is washed out , however , by the color averaging that must be performed in observable quark - gluon processes .    in the generalization to tree - level @xmath0-point amplitudes , the abelian radiation symmetry and existence of zeros for @xmath399 constant ( for photon momentum @xmath13 ) rest on having gauge - theory couplings , as noted earlier . the non - abelian color - factor symmetry uncovered in this paper can also lead to zeros in @xmath0-point amplitudes , but with an important complication . the invariance under @xmath400 for the attachment of a gluon with momentum @xmath13 can not be used to systematically cancel out the complete @xmath0-gluon amplitude using an overall common value for @xmath401 . because of the jacobi relations , that overall common value must vanish . there are in principle zeros , however , for separate islands of @xmath401 values . although they are again washed out by color - averaging , the generalized factorization coming from the color - factor symmetry remains useful for theoretical analysis of tree amplitudes . in a different direction , note that the bcj form of the gluon amplitudes has been utilized in the planar zeros studied recently in refs . @xcite . the analogs described above are a bridge to a final overall remark . it has been helpful to think of gluon emission or absorption as effecting a ( first - order ) transformation in both color and kinematic space simultaneously on the graph to which it is attached . in particular , the attachments lead to transformations of the various legs and vertices of the `` parent '' diagram in either momentum or space - time representations . all the parent wave functions end up transformed , and identities derived from for the different spins , , yield exactly the two terms expected from the propagator with its bilinearity in the wave functions . the cancellations highlighted throughout this paper arise precisely because we consider those theories whose amplitudes transform covariantly under color and kinematic transformations . adding all possible massless gluon attachments to a complete set of parent graphs leads to a sum of corresponding color shifts that vanishes because of invariance . such a picture should help in finding directions in the diagrammatic analyses of a variety of extensions of the gauge theories considered in this paper , supersymmetric and otherwise . this material is based upon work supported by the national science foundation under grants nos . phy14 - 16123 and pfi : bic 1318206 . rwb is also supported by funds made available through a cwru institute professorship chair . sgn gratefully acknowledges sabbatical support from the simons foundation ( grant no . 342554 to stephen naculich ) . he would also like to thank the michigan center for theoretical physics and the physics department of the university of michigan for generous hospitality and for providing a welcoming and stimulating sabbatical environment . [ sec : app ]    in this appendix , we use the five - gluon amplitude to provide an explicit example of the cubic vertex expansion ( [ cve ] ) introduced in sec .  [ sec : cfs ] , and the relations among kinematic numerators ( [ sumoverdelta ] ) resulting from the color - factor symmetry . the cubic decomposition ( [ cubicdecomp ] ) of the five - gluon amplitude is given by @xmath402 where @xmath58 are half - ladder color factors defined in . let us recast this amplitude in a cubic vertex expansion with respect to gluon 2 . we have already arranged the terms in so that each line corresponds to one of the four - gluon diagrams @xmath39 obtained by omitting gluon 2 . we rewrite the denominator of the third term of the first line as  =   1s_45 ( -s_12 - s_23 )  +   1s_13 ( -s_24 - s_25 ) and similarly the denominators of the third terms of the other two lines to obtain @xmath403 which is precisely of the form of the cubic vertex expansion ( [ cve ] ) .    the six independent five - gluon half - ladder color factors ( in the notation of ref . @xcite ) are @xmath404 according to , the color - factor shifts associated with gluon 2 act as @xmath405 where @xmath406 and @xmath407 are arbitrary constants . the nine remaining five - gluon color factors , and the action of the color - factor shifts thereon , are determined by the jacobi relations to be @xmath408 applying this shift to , we obtain @xmath409 which is precisely of the form of . the color - factor shift with respect to gluon 3 instead yields @xmath410 since @xmath411 , @xmath412 , @xmath413 , and @xmath414 are independent arbitrary constants , each expression in parentheses vanishes , yielding four independent constraint equations on the kinematic numerators of the five - gluon amplitude . no additional independent constraints are obtained from the color - factor symmetries associated with the other three gluons . these `` generalized jacobi relations '' for five - gluon amplitudes were previously derived in refs . @xcite by using the properties of string theory amplitudes . d0 collaboration , v.  m.  abazov et  al . , _ `` first study of the radiation - amplitude zero in @xmath419 production and limits on anomalous @xmath420 couplings at @xmath421 = 1.96- tev '' _ , , ` arxiv:0803.0030 ` .
we introduce a new set of symmetries obeyed by tree - level gauge - theory amplitudes involving at least one gluon 
 . the symmetry acts as a momentum - dependent shift on the color factors of the amplitude . using the radiation vertex expansion , we prove the invariance under this color - factor shift of the @xmath0-gluon amplitude , as well as amplitudes involving massless or massive particles in an arbitrary representation of the gauge group with spin zero , one - half , or one . 
 the bern - carrasco - johansson relations are a direct consequence of this symmetry . 
 we also introduce the cubic vertex expansion of an amplitude , and use it to derive a gauge - invariant constraint on the kinematic numerators of the amplitude . 
 we show that the amplitudes of the bi - adjoint scalar theory are invariant under the color - factor symmetry , and use this to derive the null eigenvectors of the propagator matrix . 
 we generalize the color - factor shift to loop level , and prove the invariance under this shift of one - loop @xmath0-gluon amplitudes in any theory that admits a color - kinematic - dual representation of numerators . 
 we show that the one - loop color - factor symmetry implies known relations among the integrands of one - loop color - ordered amplitudes . 
 bow - ph-162 + mctp-16 - 17 +    * bcj relations from a new symmetry of gauge - theory amplitudes *    1.5 cm    robert w. brown@xmath1 and stephen g. naculich@xmath2    0.5 cm @xmath1_department of physics + case western reserve university + cleveland , oh 44106 usa _    0.5 cm @xmath3_department of physics + bowdoin college + brunswick , me 04011 usa _    0.5 cm @xmath4_michigan center for theoretical physics ( mctp ) + department of physics + university of michigan + ann arbor , mi 48109 usa 
 _    1.5 cm