  Recent dark matter (DM) direct searches place very stringent constraints on
the possible DM candidates proposed in extensions of the Standard Model. There
are however models where these constraints are avoided. One of the simplest and
most striking examples comes from a straightforward Higgs portal pseudoscalar
DM model featured with a softly broken $U(1)$ symmetry. In this model the
tree-level DM-nucleon scattering cross section vanishes in the limit of zero
momentum-transfer. It has also been argued that the leading-order DM-nucleon
cross-section appears at the one-loop level. %, which is too small to be
constrained experimentally. In this work 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. We have concluded that, in agreement
with expectations, the amplitude for the scattering process is UV finite and
approaches zero in the limit of vanishing DM masses. Moreover, we made clear
that the finite DM velocity correction at tree-level is subdominant with
respect to the one-loop contribution. Based on the analytic formulae, our
numerical studies show that, for a typical choice of model parameters, the DM
nuclear recoiling cross section is well below ${\cal O}(10^{-50}~{\rm cm}^2)$,
which indicates that the DM direct detection signal in this model naturally
avoids the present strong experimental limits on the cross-section.
