Neutrino Mass in Non-Supersymmetric $SO(10)$ GUT

We study a prediction on neutrino observables in a non-supersymmetric renormalizable $SO(10)$ GUT model that contains a ${\bf 10}$ complex scalar field and a ${\bf 126}$ scalar field whose Yukawa couplings with ${\bf 16}$ matter fields provide the quark and charged lepton Yukawa couplings, neutrino Dirac Yukawa coupling and Majorana mass for the singlet neutrinos. The $SO(10)$ breaking is achieved in two steps by a ${\cal O}(10^{15})$ GeV VEV of a ${\bf 54}$ real scalar field and a ${\cal O}(10^{14})$ GeV VEV of the ${\bf 126}$ field. First, we analyze the gauge coupling unification conditions and determine the VEV of the ${\bf 126}$ field. Next, we constrain the Yukawa couplings of the ${\bf 10}$ and ${\bf 126}$ fields at the scale of the ${\bf 126}$ field's VEV from experimental data on quark and charged lepton masses and quark flavor mixings. Then we express the active neutrino mass with the above Yukawa couplings and the ${\bf 126}$ field's VEV based on the Type-1 seesaw mechanism, and fit neutrino oscillation data, thereby deriving a prediction on poorly or not measured neutrino observables. What distinguishes our work from previous studies is that we do not assign Peccei-Quinn charges on visible sector fields so that the ${\bf 10}$ scalar field and its complex conjugate both have Yukawa couplings with ${\bf 16}$ matter fields. From the fitting of neutrino oscillation data, we find that not only the normal neutrino mass hierarchy, but also the inverted hierarchy can be realized. We also reveal that in the normal hierarchy case, the Dirac CP phase of the neutrino mixing matrix $\delta_{CP}$ is likely in the ranges of $-2.4<\delta_{\rm CP}<-1.2$ and $1.2<\delta_{\rm CP}<2.4$, and not in the region with $\delta_{\rm CP}\sim\pi$, and that in the normal hierarchy case, $\theta_{23}$ is likely in the upper octant and in the range of $0.50\lesssim\sin^2\theta_{23}\lesssim0.55$.