  In this paper we consider a model of Poincar\'e gauge theory (PGT) in which a
translational gauge field and a Lorentz gauge field are actually identified
with the Einstein's gravitational field and a pair of ``Yang-Mills'' field and
its partner, respectively.In this model we re-derive some special solutions and
take up one of them. The solution represents a ``Yang-Mills'' field without its
partner field and the Reissner-Nordstr\"om type spacetime, which are generated
by a PGT-gauge charge and its mass.It is main purpose of this paper to
investigate the interaction of massless Dirac fields with those fields. As a
result, we find an interesting fact that the left-handed massless Dirac fields
behave in the different manner from the right-handed ones. This can be
explained as to be caused by the direct interaction of Dirac fields with the
``Yang-Mills'' field. Accordingly, the phenomenon can not happen in the
behavior of the neutrino waves in ordinary Reissner-Nordstr\"om geometry. The
difference between left- and right-handed effects is calculated quantitatively,
considering the scattering problems of the massless Dirac fields by our
Reissner-Nordstr\"om type black-hole.
