Two-loop unitarity constraints on the Higgs boson coupling

Abstract

We use the results of Maher {\em et al.\/} (preceding paper) to construct the matrix of $j=0$ partial-wave two-body and $2\to 3$ 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 limit $\sqrt{s}\gg M_H\gg M_W$. We show explicitly that the energy dependence of the $2\to 2$ amplitudes can be completely absorbed into a running quartic Higgs 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 ${\lambda }_s\text{alt}2.3$, 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 ${\lambda }_s\text{agt}2.3$. The bound ${\lambda }_s\text{alt}2.3$ for a weakly interacting theory translates to a physical Higgs mass $M_H\text{alt}380$ 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.

Comments: 29 pages, 6 figures available by request to LD, RevTEX 3.0, MAD/TH/92-7