No gauge cancellation at high energy in the five-vector $R_{\xi }$ gauge

Abstract

We propose a novel $R_{\xi }$ gauge in the five-vector (5V) framework within the Abelian Higgs model. In the Cartesian basis of the complex Higgs field, the 5V $R_{\xi }$ gauge ensures non-divergent tree-level amplitudes for each Feynman diagram in the high-energy limit. This framework pinpoints the origin of high-energy divergences in tree-level amplitudes for each diagram, providing a criterion for quantifying the degree of divergences from other gauges. The 5V description necessitates treating the Goldstone field as the fifth gauge-field component, offering deeper insight into the dynamics of massive gauge bosons, particularly its longitudinal mode. The impact of this framework is demonstrated by rigorously comparing tree-level amplitudes from the 5V $R_{\xi }$ gauge with those from the conventional 4V $R_{\xi }$ gauge and the Feynman diagram gauge, the latter of which exhibits no gauge cancellation, similar to the 5V $R_{\xi }$ gauge.

Comments: 25 pages, 7 figures, 2 tables. v1: Pre-publication version. v2: Published version with minor typo corrections. v3: Improved readability and clarified key arguments; corrected the gauge-fixing term to be consistent throughout the paper