No gauge cancellation at high energy in the five-vector <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>R</mml:mi><mml:mi>ξ</mml:mi></mml:msub></mml:math> gauge

We propose a novel <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>R</a:mi><a:mi>ξ</a:mi></a:msub></a:math> gauge in the five-vector (5V) framework within the Abelian Higgs model. In the Cartesian basis of the complex Higgs field, the 5V <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>R</c:mi><c:mi>ξ</c:mi></c:msub></c:math> gauge ensures nondivergent 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>R</e:mi><e:mi>ξ</e:mi></e:msub></e:math> gauge with those from the conventional 4V <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>R</g:mi><g:mi>ξ</g:mi></g:msub></g:math> gauge and the Feynman diagram gauge, the latter of which exhibits no gauge cancellation, similar to the 5V <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>R</i:mi><i:mi>ξ</i:mi></i:msub></i:math> gauge.