Seesaw fermion masses in an SO(10) grand unified theory

In this work we study an SO(10) GUT model with minimum Higgs representations belonging only to the $\mathbf{210}$ and $\mathbf{16}$ dimensional representations of SO(10). We add a singlet fermion $S$ in addition to the usual $\mathbf{16}$ dimensional representation containing quarks and leptons. There are no Higgs bi-doublets and so charged fermion masses come from one-loop corrections. Consequently all the fermion masses, Dirac and Majorana, are of the seesaw type. We minimize the Higgs potential and show how the left-right symmetry is broken in our model where it is assumed that a $D$-parity odd Higgs field gets a vacuum expectation value at the grand unification scale. From the renormalization group equations we infer that in our model unification happens at ${10}^{15}\text{ }\text{ }\mathrm{GeV}$ and left-right symmetry can be extended up to some values just above ${10}^{11}\text{ }\text{ }\mathrm{GeV}$. The Yukawa sector of our model is completely different from most of the standard grand unified theories and we explicitly show how the Yukawa sector will look in the different phases and briefly comment on the running of the top-quark mass. We end with a brief analysis of lepton number asymmetry generated from the interactions in our model.