Top-quark-mass prediction from supersymmetric grand unified theories

We consider a supersymmetric grand-unified-theory (GUT) framework motivated by SO(10) or ${\mathrm{E}}_6$ unification in which the parameter $tan\beta \left(\equiv \frac{v_2}{v_1}\right)$ of the minimal supersymmetric standard model is constrained by the condition that the Yukawa couplings $h_t$, $h_b$, and $h_{\tau }$ are all equal at the GUT scale. With ${\alpha }_s\left(M_Z\right)=0.106\mathrm{}\pm 0.006$, the estimate for the $b$-quark mass, which depends on $tan\beta$, lies in the "observed" range $m_b\left(m_b\right)=4.25\mathrm{}\pm 0.10$ GeV, provided that the top-quark mass is ${142}_{-49\mathrm{}}^{+26\mathrm{}}$ GeV.