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

Abstract

Writing the fully color dressed and graviton amplitudes, respectively, as \( \textbf{A} = \left\langle {C} \mathrel{\left | {\vphantom {C A}} \right. } {A} \right\rangle = \left\langle {C} \mathrel{\left | {\vphantom {C {M\left| N \right.}}} \right. } {{M\left| N \right.}} \right\rangle \) and \( {\textbf{A}_{gr}} = \left\langle {{\tilde{N}}} \mathrel{\left | {\vphantom {{\tilde{N}} {M\left| N \right.}}} \right. } {{M\left| N \right.}} \right\rangle \) , where \( \left| A \right\rangle \) is a set of Kleiss-Kuijf color ordered basis, \( \left| N \right\rangle \), \( \left| {\tilde{N}} \right\rangle \) and \( \left| C \right\rangle \) are the similarly ordered numerators and color coefficients, we show that the propagator matrix M has (n − 3)(n − 3)! independent eigenvectors \( \left| {\lambda_j^0} \right\rangle \) with zero eigenvalue, for n-particle processes. The resulting equations \( \left| {\lambda_j^0} \right\rangle = 0 \) are relations among the color ordered amplitudes. The freedom to shift \( \left| N \right\rangle \to \left| N \right\rangle + \sum\nolimits_j {{f_j}\left| {\lambda_j^0} \right\rangle } \) and similarly for \( \left| {\tilde{N}} \right\rangle \) 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.