Implications of Yukawa unification for the Higgs sector in supersymmetric grand-unified models

The SU(5) unification-scale relation ${\mathit{h}}_{\mathit{b}}$=${\mathit{h}}_{\mathrm{\tau }}$ between the b-quark and τ-lepton Yukawa couplings severely constrains tanβ and the t-quark mass (even more so if ${\mathit{h}}_{\mathit{t}}$=${\mathit{h}}_{\mathit{b}}$=${\mathit{h}}_{\mathrm{\tau }}$ holds) in supersymmetric models. We examine the implications of these constraints for the Higgs sector assuming universal soft breaking terms, and emphasize that both of these relations impose unique characteristics in terms of symmetries and of the spectrum. We further study the tanβ≊1 scenario, which is suggested by ${\mathit{h}}_{\mathit{b}}$=${\mathit{h}}_{\mathrm{\tau }}$, and, in particular, the loop-induced mass of the light Higgs boson. We compare the effective potential and renormalization group methods and stress the two-loop ambiguities in the calculation of the mass. These and a large enhancement to the loop correction due to t-scalar left-right mixing considerably weaken the upper bound on the mass of the light Higgs boson that has been reported. Nevertheless, we find that for this scenario the Higgs boson is probably lighter than 110 GeV, and typically lighter than 100 GeV. Thus, it is in the mass range that may be relevant for the CERN Large Electron-Positron Collider (LEP II). Our numerical results are presented in a self-contained manner. We also discuss the global symmetries of the Higgs potential, the issue of false (color-breaking) vacua, which may be important for tanβ≊1, two-loop calculations, and the effect of an additional Higgs singlet. We show that the approximate constraints that are often used to eliminate color-breaking vacua are not always relevant.