Baryogenesis via leptogenesis from quark-lepton symmetry\par and a compact heavy $N_R$ spectrum

By demanding a compact spectrum for the right-handed neutrinos and an approximate quark-lepton symmetry inspired from SO(10) gauge unification (assuming a Dirac neutrino mass matrix close to the up quark mass matrix), we construct a {\it fine tuning} scenario for baryogenesis via leptogenesis. We find two solutions with a normal hierarchy, with the lightest neutrino mass $m_1$ different from zero, providing an absolute scale for the spectrum. In the approximations of the model, there are three independent CP phases : $\delta_L$ (that we take of the order of the quark Kobayashi-Maskawa phase) and the two light neutrino Majorana phases $\alpha$ and $\beta$. A main conclusion is that, although this general scheme is rather flexible, in some regions of parameter space we find that the necessary baryogenesis with its sign is given in terms of the $\delta_L$ phase alone. The light Majorana phases can also be computed and turn out to be close of $\pi/2$ or small. Moreover, SO(10) breaks down to the Pati-Salam group $SU(4) \times SU(2) \times SU(2)$ at the expected natural intermediate scale of about $10^{10}-10^{11}\ GeV$. A prediction is done for the effective mass in $(\beta \beta)_{0\nu}$ decay, the $\nu_e$ mass and the sum of all light neutrino masses.