Further investigation on the precise formulation of the equivalence theorem

Abstract

Based on a systematic analysis of the renormalization schemes in the general ${R}_{\ensuremath{\xi}}$ gauge, the precise formulation of the equivalence theorem for longitudinal weak boson scatterings is given both in the ${\mathrm{SU}(2)}_{L}$ Higgs theory and in the realistic SU(2) \ifmmode\times\else\texttimes\fi{} U(1) electroweak theory to all orders in the perturbation for an arbitrary Higgs boson mass ${m}_{H}$. It is shown that there is generally a renormalization-scheme- and $\ensuremath{\xi}$ dependent modification factor ${C}_{\mathrm{mod}}$ and a simple formula for ${C}_{\mathrm{mod}}$ is obtained. Furthermore, a convenient particular renormalization scheme is proposed in which ${C}_{\mathrm{mod}}$ is exactly unity. Results of ${C}_{\mathrm{mod}}$ in other currently used schemes are also discussed especially on their $\ensuremath{\xi}$ and ${m}_{H}$ dependence through explicit one-loop calculations. It is shown that in some currently used schemes the deviation of ${C}_{\mathrm{mod}}$ from unity and the $\ensuremath{\xi}$ dependence of ${C}_{\mathrm{mod}}$ are significant even in the large-${m}_{H}$ limit. Therefore care should be taken when applying the equivalence theorem.