Heavy neutrino threshold effects in low energy phenomenology

Right handed neutrinos with mass of ${\cal O} \; (10^{12}-10^{13})$ GeV are required to implement the see-saw mechanism and generate neutrino masses capable of playing a role in structure formation. Moreover models of fermion masses often relate the Yukawa couplings involving these neutrinos to the up-quark Yukawa couplings. Here we study the effects of such couplings on the radiative corrections to quark masses. We find that $b-\tau$ equality at $M_{GUT}$ may still give the correct $m_b/m_{\tau}$-- ratio at low energies, but only if there is large $\mu-\tau$ mixing in the charged leptonic sector. We propose specific mass matrix ``textures'' dictated by a $U(1)$ family symmetry whose structure preserves $m_b=m_{\tau}$ at $M_{GUT}$. In these schemes, due to the large $\nu_{\mu} - \nu_{\tau}$ mixing, it is possible to give a simultaneous solution to the solar neutrino deficit and the atmospheric neutrino problem.