Abstract
We treat the coupled Dirac-Einstein system in the framework of effective field theories in order to quantize the gravitational field at long distances in a consistent manner. In the Dirac-Einstein system we consider the leading post-Newtonian and quantum corrections to the nonrelativistic scattering amplitude of charged spin-$\frac{1}{2}$ fermions. We extract the relevant vertex rules from the action appropriate to the one-loop level calculations and find the nonrelativistic scattering matrix potential for two massive charged spin-$\frac{1}{2}$ fermions. Our focus is kept only on the nonanalytic parts contributing to the potential which are known to generate the long-range interactions.
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