Perturbative unitarity and high-energy $W_L^{\pm }$, $Z_L$, $H$ scattering. One-loop corrections and the Higgs-boson coupling

We reexamine the unitarity constraints on the high-energy scattering of longitudinally polarized $W\text{'}\mathrm{s}$ and $Z\text{'}\mathrm{s}$ and Higgs bosons in the standard model including one-loop corrections. Using an Argand diagram analysis, we find that the $j=0\mathrm{}$ scattering amplitudes are approximately unitary and weakly interacting at order ${\lambda }^2$ for Higgs-boson couplings $\lambda (s,M_H^2)\lesssim 2$, but that corrections of order ${\lambda }^3$ or higher must be included to restore perturbative unitarity for larger values of $\lambda$. We show also that two-loop $\left[\mathrm{O}\right({\lambda }^3\left)\right]$ corrections cannot extend the range of validity of perturbation theory beyond $\lambda \approx 2.2$. An analysis of inelastic 2 → 4 scattering in the $W_L^{\pm }$, $Z_L$, $H$ system gives an independent but weaker limit $\lambda (s,M_H^2)\lesssim 5$. The limit $\lambda (s,M_H^2)<\mathrm{}2\mathrm{}$ translates to a physical-Higgsboson mass $M_H\lesssim 400$ GeV if the bound is to hold up to energies of a few TeV, or $M_H\lesssim 160$ GeV in perturbatively unified theories with a mass scale of order ${10}^{15}$ GeV. For masses much larger than these bounds, low-order perturbation theory fails and the Higgs sector of the standard model becomes effectively strongly interacting.