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Predictive Neutrino Spectrum in Minimal $SO(10)$ Grand Unification

We show that minimal SO(10) Grand Unification models where the fermions have Yukawa couplings to only one (complex) {\bf 10} and one {\bf 126} of Higgs scalars lead to a very predictive neutrino spectrum. This comes about since the standard model doublet contained in the {\bf 126} of Higgs (needed for the see--saw mechanism) receives an induced vacuum expectation value at tree--level, which, in addition to correcting the bad asymptotic mass relations $m_d=m_e$ and $m_s=m_μ$, also relates the Majorana neutrino mass matrix to observables in the charged fermion sectors. We find that (i) the $ν_e-ν_μ$ mixing angle relevant for the solar neutrinos can be considerably smaller than the Cabibbo angle and lies in the range ${\rm sin}θ_{e μ}= 0-0.3$, (ii) $ν_e-ν_τ$ mixing is sin$θ_{e τ} \simeq 3|V_{td}| \simeq 0.05$, (iii) the $ν_μ-ν_τ$ mixing angle is large, ${\rm sin}θ_{μτ} \simeq 3|V_{cb}|=0.12-0.16$, and (iv) $m_{ν_τ}/m_{ν_μ} \ge 10^3$, implying that $ν_μ-ν_τ$ oscillations should be accessible to forthcoming experiments.