  We calculate the complete matrix of two-body scattering amplitudes for the
scattering of longitudinally polarized gauge bosons $W_L^\pm$, $Z_L$ and Higgs
bosons to two loops in the high-energy, heavy-Higgs limit $\sqrt{s}\gg M_H\gg
M_W$. Use of the Goldstone boson equivalence theorem reduces the problem to one
involving only the scalar fields $w^\pm$, $z$ (the Goldstone bosons of the
original theory) and the Higgs boson. Renormalization of the scattering
amplitudes requires the calculation of the self-energy functions $\Pi
_i^0(M_i^2)$, the renormalization constants $Z_i$, and the bare quartic Higgs
coupling $\lambda _0$ to two loops. The results will be useful in other
calculations. To facilitate the calculations, we introduce a powerful new
technique for evaluating integrals over Feynman parameters in dimensional
regularization which is based on a Barnes' type representation of the binomial
expansion. We also collect some useful integrals which extend the tables given
by Devoto and Duke.
