Two-loop renormalization constants and high-energy 2 → 2 scattering amplitudes in the Higgs sector of the standard model

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-boson 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 ${\Pi }_i^0\left(M_i^2\right)$, the renormalization constants $Z_i$, and the bare quartic Higgs boson 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.