The Goldstone boson equivalence theorem tells us that the amplitude for emission/absorption of a longitudinally polarized gauge boson is equal to the amplitude for emission/absorption of the corresponding Goldstone boson at high energy.
I'm wondering what's the physical meaning of this theorem. Is there any relation between equivalence theorem and Higgs mechanism?
Probably after 3 years you've answered this question, but it might be useful for others. I think you got it. At high energies, you can choose to view the spectrum as a heavy vector gauge boson with 3 degrees of freedom (3 independent polarizations) or view the spectrum as a massless vector gauge boson (2 degrees of freedom) with an extra massless Goldstone boson (1 degree of freedom) for a total of 3 degrees of freedom. The Higgs mechanism is exactly this process. You have a global $SU(2)_L \times U(1)_Y$ symmetry that in the standard model which gets broken to $U(1)_{EM}$. $SU(2)$ has 3 generators, and $U(1)$ has 1 generator. So at the end of the day, it would seem as if you lose 3 degrees of freedom, but not really, as you also produce 3 massless Goldstone bosons. These in turn give mass terms to W+ W- and Z bosons. At high energies, you can think of W+ W- and Z as massive, or as massless with these 3 goldstone bosons.
Practically, the advantage of all this is that although in a cross section calculation, thinking of massless Goldstone bosons adds more diagrams to a given process, they are much easier to calculate.
This theorem states that, at energies large compared with the $\text{W}$-boson mass, an amplitude involving external longitudinal vector bosons is equivalent to the same amplitude with the external longitudinal vector bosons replaced by the corresponding Goldstone bosons of the $R_\xi $ gauge. This theorem has been applied at both the tree and one-loop levels in calculating longitudinal vector-boson scattering
