Quantum nature of the minimal potentially realistic $\mathrm{SO}(10)$ Higgs model

We study several aspects of the quantum structure of the minimal potentially realistic renormalizable $\mathrm{SO}(10)$ Higgs model in which the $\mathbf{45}\oplus \mathbf{126}$ scalars spontaneously break the symmetry down to the Standard Model group $\mathrm{SU}(3)_{c}\times \mathrm{SU}(2)_{L}\times \mathrm{U}(1)_{Y}$. With a complete information about the one-loop corrections to the masses of all scalars in the theory and the one-loop beta functions governing the running of all dimensionless scalar self-couplings, the domains of the parameter space where the model can be treated perturbatively are established, along with improved bounds from the requirements of the SM vacuum stability and gauge coupling unification. We demonstrate that the model is fully consistent and potentially realistic only in very narrow regions of the parameter space corresponding to the breaking chains with well pronounced $\mathrm{SU}(4)_{C}\times \mathrm{SU}(2)_L\times \mathrm{U}(1)_R$ and $\mathrm{SU}(3)_{c}\times \mathrm{SU}(2)_L\times \mathrm{SU}(2)_R\times \mathrm{U}(1)_{B-L}$ intermediate symmetries, with a clear preference for the former case. Barring accidental fine-tunings in the scalar sector, this makes it possible to provide a very sharp prediction for the position of the unification scale and the value of the associated gauge coupling, with clear implications for the phenomenology of grand unified models based on this structure.