Two-loop unitarity constraints on the Higgs boson coupling

We use the results of Maher et al. (preceding paper) to construct the matrix of $j=0\mathrm{}$ partial-wave two-body and 2 → 3 scattering amplitudes for the scattering of longitudinally polarized gauge bosons $W_L^{\pm }$, $Z_L$ and Higgs bosons $H$ correct to two loops in the high-energy, heavy-Higgs-boson limit $\sqrt{s}\gg M_H\gg M_W$. We show explicitly that the energy dependence of the 2 → 2 amplitudes can be completely absorbed into a running quartic Higgs boson coupling ${\lambda }_s={\lambda }_s(s, M_H^2)$ and factors which involve small anomalous dimensions and remain near unity. After diagonalizing the matrix of partial-wave amplitudes we use an Argand-diagram analysis to show that the elastic scattering amplitudes are approximately unitary and weakly interacting for ${\lambda }_s\lesssim 2.3$, but that three-loop corrections are necessary to restore unitarity for larger values of ${\lambda }_s$. That is, the interactions in the Higgs sector of the standard model are effectively strong with respect to the perturbative expansion for ${\lambda }_s\gtrsim 2.3$. The bound ${\lambda }_s\lesssim 2.3$ for a weakly interacting theory translates to a physical Higgs boson mass $M_H\lesssim 380$ GeV if the bound is to hold for energies up to a few TeV, or $M_H\le 155$ GeV in perturbatively unified theories with mass scales of order ${10}^{16}$ GeV.