I am going to compare two simple models of dark matter (DM): a vector and a scalar DM model. Both models require the presence of two physical Higgs bosons h_1 and h_2 which come from mixed components of the standard Higgs doublet H and a complex singlet S. In the vector model, the extra U(1) symmetry is spontaneously broken by the vacuum of the complex field S. This leads to a massive gauge boson X_mu that is a DM candidate stabilized by the dark charge conjugation symmetry. On the other hand, in the scalar model the gauge group remains the standard one. The DM A is the imaginary component of S and the stabilizing symmetry is also the dark charge conjugation. In this case, in order to avoid spontaneous breaking, the U(1) symmetry is broken explicitly, but softly, in the scalar potential. In the scalar model the tree-level DM-nucleon scattering cross section vanishes in the limit of zero momentum-transfer. We have calculated the exact cross section in the zero momentum-transfer at the leading-order i.e., at the one-loop level of perturbative expansion. The possibility to disentangle the two models has been investigated.