Massive neutrinos and invisible axion minimally connected

We survey a few minimal scalar extensions of the standard electroweak model that provide a simple setup for massive neutrinos in connection with an invisible axion. The presence of a chiral $U(1)$ \`a la Peccei-Quinn drives the pattern of Majorana neutrino masses while providing a dynamical solution to the strong CP problem and an axion as a dark matter candidate. We paradigmatically apply such a renormalizable framework to type-II seesaw and to two viable models for neutrino oscillations where the neutrino masses arise at one and two loops, respectively. We comment on the naturalness of the effective setups as well as on their implications for vacuum stability and electroweak baryogenesis.