  We study the fermionic dark matter (DM) particle interacting with Standard
Model quarks via a light pseudoscalar mediator. We consider separately the
scenarios for which the DM-pseudoscalar coupling is $CP$ conserving or $CP$
violating. We show that taking a contact interaction is not suitable, even when
the mediator has a mass of the same order of magnitude as the typical momentum
transfer at the direct-detection experiments, such that the allowed DAMA region
is excluded or considerably modified by the correct relic density requirement.
The DAMA result seems to indicate that the $CP$-violating interaction is
dominant at direct searches. We find that, if the proton-to-neutron effective
coupling ratio is $-60\sim -40$, the exclusion limits set by SuperCDMS,
XENON100, and LUX are highly suppressed, and the DAMA signal can thus be easily
reconciled with these null measurements. For this model, the allowed region
determined by the DAMA signal and correct relic density can successfully
satisfy the conditions required by the thermal equilibrium, big bang
nucleosynthesis, and DM self-interactions. The results of future measurements
on flavor physics will provide important constraints on the related models.
Precise measurements performed by COUPP, PICASSO, SIMPLE and KIMS should be
able to test this model in the near future.
