Axion-neutrino interplay in a gauged two-Higgs-doublet model

We propose a gauged two-Higgs-doublet model featuring an anomalous Peccei-Quinn symmetry, $U(1{)}_{PQ}$. Dangerous tree-level flavor-changing neutral currents, common in two-Higgs-doublet models, are forbidden by the extra gauge symmetry, $U(1{)}_{X}$. In our construction, the solutions to the important issues of neutrino masses, dark matter, and the strong $CP$ problems are interrelated. Neutrino masses are generated via a Dirac seesaw mechanism and are suppressed by the ratio of the $U(1{)}_{X}$ and the $U(1{)}_{PQ}$ breaking scales. Naturally small neutrino masses suggest that the breaking of $U(1{)}_{X}$ occurs at a relatively low scale, which may lead to observable signals in near-future experiments. Interestingly, spontaneous symmetry breaking does not lead to mixing between the $U(1{)}_{X}$ gauge boson, ${Z}^{\ensuremath{'}}$, and the standard $Z$. For the expected large values of the $U(1{)}_{PQ}$ scale, the associated axion becomes ``invisible,'' with Dine-Fischler-Srednicki-Zhitnitsky-like couplings, and may account for the observed abundance of cold dark matter. Moreover, a viable parameter space region, which falls within the expected sensitivities of forthcoming axion searches, is identified. We also observe that the flavor-violating process of kaon decaying into pion plus axion, ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}a$, is further suppressed by the $U(1{)}_{X}$ scale, providing a rather weak lower bound for the axion decay constant ${f}_{a}$.