  We investigate the long-distance, low-energy, leading quantum corrections to
gravitational potential for scalarized neutron star (NS) binary systems, by
treating general relativity as an effective field theory. We neglect the
extended scales of two star components and treat them as heavy point particles,
which gravitationally interact with each other via the exchanges of both
gravitons and scalar particles, because of the settled scalar configurations
inside the stars. Accordingly, the gravitational potential includes both
Newtonian potential and scalar-modified Newtonian-like part. We, in the
non-relativistic limit, calculate the non-analytic corrections to the modified
gravitational potential directly from the sum of all exchanges of both
gravitons and scalar particles to one-loop order. The appropriate vertex rules
are extracted from the effective Lagrangian. Our calculations demonstrate that
either the graviton exchanges or the exchanges of scalar particles contribute
to both classical relativistic corrections and quantum corrections to the
gravitational potential of the scalarized NS binaries.
