Constraints and generalized gauge transformations on tree-level gluon and graviton amplitudes

Writing the fully color dressed and graviton amplitudes, respectively, as and , where is a set of Kleiss-Kuijf color ordered basis, , and are the similarly ordered numerators and color coefficients, we show that the propagator matrix M has ( n − 3)( n − 3)! independent eigenvectors with zero eigenvalue, for n -particle processes. The resulting equations are relations among the color ordered amplitudes. The freedom to shift and similarly for where f j are ( n − 3)( n − 3)! arbitrary functions, encodes generalized gauge transformations. They yield both BCJ amplitude and KLT relations, when such freedom is accounted for. Furthermore, f j can be promoted to the role of effective Lagrangian vertices in the field operator space.