Anomaly-free flavor models for Nambu-Goldstone bosons and the 3.5 keV X-ray line signal

We pursue a possibility that a pseudo-Nambu-Goldstone boson is lurking around or below the intermediate scale. To this end we consider an anomaly-free global flavor symmetry, and construct models where the pseudo-Nambu-Goldstone boson is coupled preferentially to leptons. The experimental and astrophysical bounds derived from couplings to photons and nucleons are significantly relaxed. If sufficiently light, the pseudo-Nambu-Goldstone boson contributes to dark matter, and interestingly, it generally decays into photons through couplings arising from threshold corrections. We show that the recent hint for the X-ray line at about $3.5$ keV can be explained by the decay of such pseudo-Nambu-Goldstone boson of mass about $7$ keV with the decay constant of order $10^{10}$ GeV, if the electron is charged under the flavor symmetry.