Constraints and Generalized Gauge Transformations on Tree-Level Gluon and Graviton Amplitudes

Writing the fully color dressed and graviton amplitudes, respectively, as ${\bf A}=<C|A> =<C|M|N>$ and ${\bf A}_{gr}= <\tilde N|M|N>$, where $|A>$ is a set of Kleiss-Kuijf color-ordered basis, $|N>,$|\tilde N> $and$|C>$are the similarly ordered numerators and color coefficients, we show that the propagator matrix$M$has$(n-3)(n-3)!$independent eigenvectors$|\lambda ^0_j>$with zero eigenvalue, for$n$-particle processes. The resulting equations$<\lambda ^0_j|A> = 0$are relations among the color ordered amplitudes. The freedom to shift$|N> \to |N> +\sum_j f_j|\lambda ^0_j>$and similarly for$|\tilde N>$, where$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.