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

TL;DR

This paper explores the structure of tree-level gluon and graviton amplitudes, revealing new gauge freedom and relations like BCJ and KLT through eigenvector analysis of the propagator matrix.

Contribution

It demonstrates the existence of zero eigenvalue eigenvectors of the propagator matrix and connects gauge transformations to amplitude relations like BCJ and KLT.

Findings

Identifies $(n-3)(n-3)!$ eigenvectors with zero eigenvalue.

Derives relations among color-ordered amplitudes from eigenvector equations.

Shows gauge freedom corresponds to shifts in numerator functions, leading to known amplitude relations.

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 $|\lambda ^0_j>$ 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.