  Quark splitting in non-trivial $\theta$-vacuum with a given helicity is
investigated in pQCD with a modified quark propagator. We found that the quark
splitting functions were modified by the presence of a topologically
non-trivial QCD background field, though there is no explicit helicity flip
associated with the radiative processes. The interaction with the topological
non-trivial field leads to the degeneracy of the quark splitting functions for
left- and right-handed quarks. Such degeneracy can lead to imbalance of left-
and right-handed quarks in quark jet showers. We also discuss phenomenological
consequences of such imbalance if there exists non-trivial topological gluon
field configuration in heavy-ion collisions.
