  We study the nonrelativistic limit of the quantum theory of a Chern-Simons
field minimally coupled to a scalar field with quartic self-interaction. The
renormalization of the relativistic model, in the Coulomb gauge, is discussed.
We employ a procedure to calculate scattering amplitudes for low momenta that
generates their $|p|/m$ expansion and separates the contributions coming from
high and low energy intermediary states. The two body scattering amplitude is
calculated up to order $p^2/m^2$. It is shown that the existence of a critical
value of the self-interaction parameter for which the 2-particle scattering
amplitude reduces to the Aharonov-Bohm one is a strictly nonrelativistic
feature. The subdominant terms correspond to relativistic corrections to the
Aharonov-Bohm scattering. A nonrelativistic reduction scheme and an effective
nonrelativistic Lagrangian to account for the relativistic corrections are
proposed.
