  We study the loop corrections to the scalar propagator and the fermionic
self-energy for the mass dimension one fermionic dark matter with the Yukawa
interaction. We find, in the former case, there is a non-vanishing
Lorentz-violating term while the later is Lorentz-invariant. Our study of the
fermionic loop correction shows that unitarity demands the fermionic mass must
be at least half of the bosonic mass and that the Lorentz-violating term makes
a non-trivial correction to the bosonic propagator. We discuss what these
results mean in the context of the Standard Model and the possibility of
bypassing the unitarity constraint. In the simplest scenario, within the
framework of standard quantum field theory, by identifying the scalar boson to
be the Higgs boson with a mass of 125 GeV, the mass of the fermion must be at
least 62.5 GeV.
