  If dark matter (DM) is a fermion and its interactions with the standard model
particles are mediated by pseudoscalar particles, the tree-level amplitude for
the DM-nucleon elastic scattering is suppressed by the momentum transfer in the
non-relativistic limit. At the loop level, on the other hand, the
spin-independent contribution to the cross section appears without such
suppression. Thus, the loop corrections are essential to discuss the
sensitivities of the direct detection experiments for the model prediction. The
one-loop corrections were investigated in the previous works. However, the
two-loop diagrams give the leading order contribution to the DM-gluon effective
operator ($\bar{\chi} \chi G^{a}_{\mu \nu} G^{a \mu \nu} $) and have not been
correctly evaluated yet. Moreover, some interaction terms which affect the
scattering cross section were overlooked. In this paper, we show the cross
section obtained by the improved analysis and discuss the region where the
cross section becomes large.
