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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 $|λ^0_j>$ with zero eigenvalue, for $n$-particle processes. The resulting equations $<λ^0_j|A> = 0$ are relations among the color ordered amplitudes. The freedom to shift $|N> \to |N> +\sum_j f_j|λ^0_j>$ 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.