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 .