  We study phenomenological implications of an axion that arises as a pseudo
Nambu-Goldstone boson due to the spontaneous breaking of anomaly-free global
flavor symmetry. One interesting possibility for such anomaly-free axion to
explain dark matter (DM) is when it has a mass of order keV and an intermediate
scale decay constant, since it can be explored through direct search
experiments, X-ray observations, various stellar cooling processes, and the
misalignment mechanism naturally explains the DM abundance. As a concrete
renormalizable model of such axion, we consider an extended Higgs sector with
global flavor symmetry, which consists of three Higgs doublet fields and three
singlet Higgs fields with $U(1)_{\rm B-L}$ charges. We identify viable
parameter regions that satisfy theoretical bounds on the Higgs potential and
various experimental limits on this model, and evaluate the mass spectra of the
axion and extra Higgs bosons. We find that even an anomaly-free axion can
generally couple to photons through mixing with CP-odd Higgs, and that its
strength depends on the vacuum expectation values of the Higgs doublets as well
as the axion mass. As a result, the ratios of the vacuum expectation values of
the Higgs doublets are tightly constrained to satisfy the X-ray constraints. We
show the favored parameter region where axion DM explains the XENON1T excess.
We also demonstrate that the axion-electron coupling is correlated with the
extra Higgs boson masses and mixing angles for CP-even Higgs bosons. Thus, if
the axion is detected in future observations, the extra Higgs boson masses and
the coupling of the standard model-like Higgs boson with the weak gauge bosons
are restricted. This is a good example of the synergy between searches for the
axion DM and the BSM around the electroweak scale.
