  The WIMPs are considered one of the favorable dark matter (DM) candidates,
but as the upper bounds on the interactions between DM and standard model (SM)
particles obtained by the upgraded facilities of DM direct detections get lower
and lower. Researchers turn their attentions to search for less massive DM
candidates, i.e. light dark matter of MeV scale. The recently measured
anomalous transition in $^8$Be suggests that there exists a vectorial boson
which may mediate the interaction between DM and SM particles. Based on this
scenario, we combine the relevant cosmological data to constrain the mass range
of DM, and have found that there exists a model parameter space where the
requirements are satisfied, a range of $10.4 \lesssim m_{\phi} \lesssim $ 16.7
MeV for scalar DM, and $13.6 \lesssim m_{V} \lesssim $ 16.7 MeV for vectorial
DM is demanded. Then a possibility of directly detecting such light DM
particles via the DM-electron scattering is briefly studied in this framework.
