  We propose a new mechanism to explain neutrino masses with lepton number
conservation, in which the Dirac neutrino masses are generated at the two-loop
level involving a dark matter candidate. In this model, branching ratios of
lepton flavor violating decays of the Higgs boson can be much larger than those
of lepton flavor violating decays of charged leptons. If lepton flavor
violating decays of the Higgs boson are observed at future collider experiments
without detecting lepton flavor violating decays of charged leptons, most of
the models previously proposed for tiny neutrino masses are excluded while our
model can still survive. We show that the model can be viable under constraints
from current data for neutrino experiments, searches for lepton flavor
violating decays of charged leptons and dark matter experiments.
