  Some extensions of the Standard Model provide Dark Matter candidate particles
which can have a dominant coupling with the lepton sector of the ordinary
matter. Thus, such Dark Matter candidate particles ($\chi^{0}$) can be directly
detected only through their interaction with electrons in the detectors of a
suitable experiment, while they are lost by experiments based on the rejection
of the electromagnetic component of the experimental counting rate. These
candidates can also offer a possible source of the 511 keV photons observed
from the galactic bulge. In this paper this scenario is investigated. Some
theoretical arguments are developed and related phenomenological aspects are
discussed. Allowed intervals and regions for the characteristic
phenomenological parameters of the considered model and of the possible
mediator of the interaction are also derived considering the DAMA/NaI data.
