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The amplitudes for $2\rightarrow 2$ scattering processes involvinglongitudinally polarized gauge bosons $( W_L^\pm, Z_L )$ and the Higgs bosonare analyzed up to two loops. Assuming $M_H >> M_W$, the trilinear Higgscoupling, $\lambda v$, is dominant for energies of $\sqrt{s}$ < 1.5 -- 2 $M_H$.For larger values of $\sqrt{s}$, the quartic coupling, $\lambda$, becomesdominant, allowing for a simpler calculation of higher-order corrections. Thehigh-energy amplitudes display a large logarithmic dependence on $\sqrt{s}$which can be resummed using renormalization group techniques. The resummationof leading-log terms is sufficient for Higgs masses of less than 350 GeV. For350 < $M_H$ < 450 GeV, a next-to-leading-log resummation is necessary. For evenlarger values of $M_H$, the perturbative approach fails completely sincetwo-loop terms become in magnitude larger than one-loop terms. Choosing the$\overline{{\rm MS}}$ renormalization scheme instead of the OMS scheme, thecoefficients of the perturbative series increase in magnitude, making thebreakdown of perturbation theory even more apparent. In conclusion, theStandard Model cross sections presented here have very large uncertainties if$M_H\gtrsim 450$ GeV and $\sqrt{s} \gtrsim 2 M_H$, reducing the sensitivity tocontributions from new physics significantly.Comment: 30 pages, Revtex using epsf, uuencoded, packed Postscript figures.  The analysis has been updated to include the 3-loop running coupling. Four  figures updated, one figure added to show the maximal energy allowed as a  function of the Higgs mass. Final version as to appear in Phys. Rev.

physical review. d. particles, fields, gravitation, and cosmology/physical review. d. particles and fields

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Large uncertainties in the cross section of elasticWL+WL−scattering