Higgs boson mass limits in perturbative unification theories

Motivated in part by recent demonstrations that electroweak unification into a simple group may occur at a low scale, we detail the requirements on the Higgs boson mass if the unification is to be perturbative. We do this for the standard model effective theory, minimal supersymmetry, and next-to-minimal supersymmetry with an additional singlet field. Within the standard model framework, we find that perturbative unification with ${\sin }^2{\theta }_W=1/4\mathrm{}$ occurs at $\Lambda =3.8\mathrm{}\hspace{0.5em}\mathrm{TeV}$ and requires $m_h\lesssim 460\hspace{0.5em}\mathrm{GeV},$ whereas perturbative unification with ${\sin }^2{\theta }_W=3/8\mathrm{}$ requires $m_h\lesssim 200\hspace{0.5em}\mathrm{GeV}.$ In supersymmetry, the presentation of the Higgs boson mass predictions can be significantly simplified, yet remain meaningful, by using a single supersymmetry breaking parameter ${\Delta }_S.$ We present Higgs boson mass limits in terms of ${\Delta }_S$ for the minimal supersymmetric model and the next-to-minimal supersymmetric model. We show that in next-to-minimal supersymmetry, the Higgs boson mass upper limit can be as large as $500\hspace{0.5em}\mathrm{GeV}$ even for moderate supersymmetry masses if the perturbative unification scale is low $\left(\Lambda \simeq 10\hspace{0.5em}\mathrm{TeV}\right).\mathrm{}$