  In this note, we examine the scattering of two identical fermions in theories
where fermionic fields minimally coupled to higher derivative gravity. In
particular, we consider the extension of general relativity with $R^2$
corrections or non-local terms. We expand the action of fermions around the
flat space background and obtain two fermion-one graviton vertex. Then, by
considering the scattering amplitude of two fermions, we calculate the
non-relativistic limit and that obtain the potential for two fermion-fermion
interaction which would be the usual Newtonian potential corrected with a
Yukawa-like term. At the end, we briefly discuss the astronomical effects of
such Yukawa-like potential by computing the gravitational pressure of a
spherical star and use it for a white dwarf to obtain quantum corrections of
Chandrasekhar radius.
