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, $Z^{\prime }$, 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, $K^{+}\to {\pi }^{+}a$, is further suppressed by the $U(1{)}_X$ scale, providing a rather weak lower bound for the axion decay constant $f_a$.

• Received 17 June 2021
• Accepted 29 August 2021

DOI:https://doi.org/10.1103/PhysRevD.104.075014

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Published by the American Physical Society