  We consider the leading post-Newtonian and quantum corrections to the
non-relativistic scattering amplitude of charged scalars in the combined theory
of general relativity and scalar QED. The combined theory is treated as an
effective field theory. This allows for a consistent quantization of the
gravitational field. The appropriate vertex rules are extracted from the
action, and the non-analytic contributions to the 1-loop scattering matrix are
calculated in the non-relativistic limit. The non-analytical parts of the
scattering amplitude, which are known to give the long range, low energy,
leading quantum corrections, are used to construct the leading post-Newtonian
and quantum corrections to the two-particle non-relativistic scattering matrix
potential for two charged scalars. The result is discussed in relation to
experimental verifications.
