  In the 5-component representation of weak bosons, the first four components
make a Lorentz four vector, representing the transverse and longitudinal
polarizations excluding the scalar component of the weak bosons, whereas its
fifth component corresponds to the Goldstone boson. We obtain the $5\times 5$
component propagators of off-shell weak bosons, proposed previously and named
after the Goldstone boson equivalence theorem, by starting from the
unitary-gauge representation of the tree-level scattering amplitudes, and by
applying the BRST (Becchi--Rouet--Stora--Tyutin) identities to the two
sub-amplitudes connected by each off-shell weak-boson line. By replacing all
weak boson vertices with those among the off-shell 5-component wavefunctions,
we arrive at the expression of the electroweak scattering amplitudes, where the
magnitude of each Feynman amplitude has the correct on-shell limits for all
internal propagators, and hence with no artificial gauge cancellation among
diagrams. Although our derivation is limited to the tree-level only, it allows
us to study the properties of each Feynman amplitude separately, and then learn
how they interfere in the full amplitudes. We implement the 5-component weak
boson propagators and their vertices in the numerical helicity amplitude
calculation code HELAS (Helicity Amplitude Subroutines), so that an automatic
amplitude generation program such as MadGraph can generate the scattering
amplitudes without gauge cancellation. We present results for several
high-energy scattering processes where subtle gauge-theory cancellation among
diagrams takes place in all the other known approaches.
