VISH$\nu$: a unified solution to five SM shortcomings with a protected electroweak scale

We propose a Standard Model extension, coined VISH$\nu$ (Variant-axIon Seesaw Higgs $\nu$-trino), that is an $N_{\text{DW}} = 1$ variation of its predecessor, the $\nu$DFSZ model. In accounting for the origin of neutrino masses, dark matter and the baryon asymmetry of the universe, VISH$\nu$ inherits the explanatory power of $\nu$DFSZ while, of course, resolving the strong $CP$ problem. In both models, the electroweak scale is naturally protected from a high seesaw scale that is identified with the Peccei-Quinn (PQ) spontaneous symmetry breaking scale. Through a flavour variant coupling structure, VISH$\nu$ evades a domain wall problem, extending the cosmological reach of the $\nu$DFSZ to include a viable period of inflation. The primary focus of this paper is on the inflationary dynamics of VISH$\nu$ and their naturalness in the sense of radiative stability. We find that non-minimal gravitational couplings, generically developed by the VISH$\nu$ scalar fields, naturally support a viable inflaton field which typically has both a PQ scalar and Higgs component. An axion mass window [$40\mu\text{eV}, \sim 2\text{meV}$] accessible to forthcoming searches, results for the case where PQ symmetry is restored during the (p)reheating phase.