Equivalence theorem redux

The equivalence theorem states that amplitudes involving longitudinal vector bosons are equal to those with the corresponding unphysical scalars in the limit $\frac{M_W^2}{s}\to 0$. There are two ways to approach this limit, depending on whether $\frac{M_W}{M_H}\to 0$ or $\frac{M_H^2}{s}\to 0$. We show that the theorem has a different physical interpretation in each limit, but its validity in both depends only on the wave-function renormalization of the unphysical Goldstone bosons. We derive a condition that the renormalization parameters must satisfy in order for the theorem to hold. We show that this condition is satisfied in the first limit, appropriate to the heavy-Higgs-boson regime, if momentum subtraction at a scale $m\ll M_H$ is used. With this prescription, the theorem is true to lowest nonzero order in $g$ and to all orders in the Higgs-boson coupling.