Predictive fermion mass matrixAnsätzein nonsupersymmetric SO(10) grand unification

We investigate the status of predictive fermion mass Ansa$iuml—tze in nonsupersymmetric SO(10) grand unification which make use of the grand unification scale conditions ${\mathit{m}}_{\mathit{e}}$=${\mathit{m}}_{\mathit{d}}$/3, ${\mathit{m}}_{\mathrm{\mu }}$=3${\mathit{m}}_{\mathit{s}}$, and ‖${\mathit{V}}_{\mathit{c}\mathit{b}}$‖= √${\mathit{m}}_{\mathit{c}}$/${\mathit{m}}_{\mathit{t}}$ in nonsupersymmetric SO(10) grand unification. The gauge symmetry below an intermediate symmetry-breaking scale ${\mathit{M}}_{\mathit{I}}$ is assumed to be that of the standard model with either one Higgs doublet or two Higgs doublets. We find in both cases that a maximum of 5 standard model parameters may be predicted within 1σ experimental ranges. We find that the standard model scenario predicts the low energy ‖${\mathit{V}}_{\mathit{c}\mathit{b}}$‖ to be in a range which includes its experimental midvalue 0.044 and which for a large top mass can extend to lower values than the range resulting in the supersymmetric case. In the two Higgs standard model case, we identify the regions of parameter space for which unification of the bottom quark and τ lepton Yukawa couplings is possible at grand unification scale. In fact, we find that unification of the top, bottom, and τ Yukawa couplings is possible with the running b-quark mass within the 1σ preferred range ${\mathit{m}}_{\mathit{b}}$=4.25±0.1 GeV provided ${\mathrm{\alpha }}_{3\mathit{c}}$(${\mathit{M}}_{\mathit{Z}}$) is near the low end of its allowed range. In this case, one may make 6 predictions which include ‖${\mathit{V}}_{\mathit{c}\mathit{b}}$‖ within its 90% confidence limits. However unless the running mass ${\mathit{m}}_{\mathit{b}}$>4.4 GeV, third generation Yukawa coupling unification requires the top mass to be greater than 180 GeV. We compare these nonsupersymmetric cases to the case of the minimal supersymmetric standard model embedded in the SO(10) grand unified group. We also give an example of a possible mechanism, based on induced vacuum expectation values and a broken U(1${)}^3$ symmetry for generating the observed hierachy of masses and a mass matrix texture.