$SO(10)$ theory on the plateau: the importance of being renormalizable

We revisit a minimal renormalisable $SO(10)$ grand unified theory, with the Higgs representation $45_{\rm H}$, $126_{\rm H}$ and complex $10_{\rm H}$, responsible for the unification, intermediate and the weak scale symmetry breaking, respectively. We perform the study of unification constraints and find that it allows for the Left-Right symmetric scale to be accessible even at the LHC, and the Quark-Lepton unification scale as low as its phenomenological limit around $10^5\,$GeV. Moreover, one can have neutron - anti neutron oscillations at the level of the present day sensibility in both of the above cases, while in the former case one can have simultaneously neutrinoless double beta decay induced by new light scalar states, reachable today - with both electrons emerging as left-handed, as in the neutrino exchange through its possible Majorana mass. We also discuss a recently raised issue of the fine-tuning of the light Higgs mass and its potential conflict with low intermediate mass scales.