Leptogenesis and Dirac neutrino masses in SO(10)

Within the scheme of Baryogenesis through Leptogenesis in the single flavor approximation, we emphasize the role of the Dirac neutrino mass $m_D$ in the SO(10) Grand Unification scheme, using the Higgs representations for the fermions masses ${\bf 10}$ and $\overline{\bf 126}$. Both representations are needed to get relations among the fermion masses, e.g. $m_b = m_\tau$, $m_\mu = 3 m_s$,... Taking these two representations allow to relax the condition $m_D = m_u$ obtained with the representation ${\bf 10}$ alone and to have general Dirac masses $m_D \not = m_u$. On the other hand, we assume that the antisymmetric representation ${\bf 120}$ is absent. Within this hypothesis, the unitary matrices $V^L$ and $V^R$ that diagonalize $m_D$ through $m_D = V^{L \dagger} m_D^{diag} V^R$ still satisfy the relation $V^R = V^{L*}$. Under these conditions, unlike the case of keeping only the ${\bf 10}$ representation, we obtain in parameter space several examples of the heavy neutrino spectrum consistent with the SO(10) unification scale, and a value for the baryon asymmetry $Y_B$ in agreement with data. For illustration, we detail a case that gives the heavy spectrum with masses $M_1 = 6.26 \times 10^{10}\ {\rm GeV}$, $M_2 = 2.23 \times 10^{12}\ {\rm GeV}$, $M_3 = 4.57 \times 10^{14}\ {\rm GeV}$ and, in the strong washout regime, the baryon asymmetry $Y_B = 8.50 \times 10^{-11}$ consistent with the data.