Modification of the equivalence theorem due to loop corrections

We show that, because of mixing between longitudinal vector bosons and pseudo-Goldstone scalars and because of differences in external leg amputations, there are loop corrections to the equivalence theorem, which relates S-matrix elements of $W^{\pm }$ and Z to those of their respective pseudo-Goldstone partners. We derive general compact expressions in the form of multiplicative factors to account for these effects. An explicit one-loop calculation is performed to demonstrate the consistency of our formulation. In the limit when the physical Higgs and/or the top quark become very massive, we obtain a result that there is no $m_H^2$, ln$m_H^2$, $m_t^2$, or ln$m_t^2$ dependence in these factors if on-shell or finite-momentum subtractions are used for renormalization. However, we have not been able to, nor do we think it likely to, arrive at this conclusion for higher-order loops.