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Elastic collisions with relativistic electrons from the blazar's jet canaccelerate dark matter (DM) particles in the DM spike surrounding thesupermassive black hole at its center. This can allow one to set stringentlimits on the DM-electron scattering cross section ($\bar{\sigma}_{e\chi}$) forDM masses less than 100 MeV. We consider DM particles boosted by energeticelectrons in the jets of the blazars TXS 0506+056 and BL Lacertae. Both vectorand scalar mediators for the scattering of electron and electrophilic fermionicDM are studied. We highlight that the ensuing energy dependency of the S-matrixfor the corresponding Lorentz structure of the vertex significantly modifiesthe constraints. We find that the revised exclusion limits are orders ofmagnitude stronger than the equivalent results for the simple constant crosssection assumption. Our limits are also assessed for the less cuspy spike.

Blazar boosted Dark Matter -- direct detection constraints on $\sigma_{e\chi}$ : Role of energy dependent cross sections