Further investigation on the precise formulation of the equivalence theorem

Based on a systematic analysis of the renormalization schemes in the general $R_{\xi }$ gauge, the precise formulation of the equivalence theorem for longitudinal weak boson scatterings is given both in the ${\mathrm{SU}\mathrm{}\left(2\right)\mathrm{}}_L$ Higgs theory and in the realistic SU(2) × 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 $\xi$ 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 $\xi$ 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 $\xi$ dependence of $C_{\mathrm{mod}}$ are significant even in the large-$m_H$ limit. Therefore care should be taken when applying the equivalence theorem.