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

TL;DR

This work uncovers the origin of constraints and generalized gauge freedom in tree-level gluon and graviton amplitudes by analyzing the propagator matrix M in the Kleiss-Kuijf basis. The authors show that M possesses $(n-3)(n-3)!$ zero-eigenvalue modes, whose associated null vectors generate channel-by-channel gauge transformations that yield BCJ relations and, via the KLT framework, gravity amplitudes as a product of gauge-theory numerators. At four and five points, the rank of M matches $(n-3)!$ (1 for 4pt, 2 for 5pt), with corresponding constraint spaces guiding the structure of color-ordered amplitudes and their gravity counterparts. The results offer a constructive view of gauge freedom as a tool for reorganizing amplitudes, and they suggest promoting the gauge parameters to effective Lagrangian vertices, connecting on-shell amplitude relations to semi-local space-time interactions. These insights pave the way for systematic higher-point constructions and a deeper understanding of color-kinematics duality.

Abstract

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.