Towards completing the standard model: Vacuum stability, electroweak symmetry breaking, and dark matter

We study the standard model (SM) in its full perturbative validity range between ${\mathrm{\ensuremath{\Lambda}}}_{\mathrm{QCD}}$ and the $U{(1)}_{Y}$ Landau pole, assuming that a yet unknown gravitational theory in the UV does not introduce additional particle thresholds, as suggested by the tiny cosmological constant and the absence of new stabilizing physics at the electroweak scale. We find that, due to dimensional transmutation, the SM Higgs potential has a global minimum at ${10}^{26}\text{ }\text{ }\mathrm{GeV}$, invalidating the SM as a phenomenologically acceptable model in this energy range. We show that extending the classically scale invariant SM with one complex singlet scalar $S$ allows us to (i) stabilize the SM Higgs potential, (ii) induce a scale in the singlet sector via dimensional transmutation that generates the negative SM Higgs mass term via the Higgs portal, (iii) provide a stable $CP$-odd singlet as the thermal relic dark matter due to $CP$-conservation of the scalar potential, and (iv) provide a degree of freedom that can act as an inflaton in the form of the $CP$-even singlet. The logarithmic behavior of dimensional transmutation allows one to accommodate the large hierarchy between the electroweak scale and the Landau pole, while understanding the latter requires a new nonperturbative view on the SM.