  We study a dark matter model with one singlet complex scalar and two Higgs
doublets. The scalar potential respects a softly broken global symmetry, which
makes the imaginary part of the singlet become a pseudo-Nambu-Goldstone boson
acting as a dark matter candidate. The pseudo-Nambu-Goldstone nature of the
boson leads to the vanishing of its tree-level scattering amplitude off
nucleons at zero momentum transfer. Therefore, although the interaction
strength could be sufficiently large to yield a viable relic abundance via
thermal mechanism, direct detection is incapable of probing this candidate. We
further investigate the constraints from Higgs measurements, relic abundance
observation, and indirect detection.
