Forests, groves and Higgs bosonsGauge invariance classes in spontaneously broken gauge theories

We determine the gauge invariance classes of tree level Feynman diagrams in spontaneously broken gauge theories, providing a proof for the formalism of gauge and flavor flips. We find new gauge invariance classes in theories with a nonlinearly realized scalar sector. In unitarity gauge, the same gauge invariance classes correspond to a decomposition of the scattering amplitude into pieces that satisfy the relevant Ward identities individually. In theories with a linearly realized scalar sector in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_\xi$\end{document} gauge, no additional non-trivial gauge invariance classes exist compared to the unbroken case.