  Quantum chromodynamics is a non-Abelian gauge theory of strong interactions,
in which the parity symmetry can be violated by the non-trivial $\theta$-vacuum
tunneling effects. The $\theta$-vacuum induces the local parity odd domains.
Those reactions that occur in these domains can be affected by the tunneling
effects and quantities become parity odd. In this paper we consider the
fragmentation process where parity odd fragmentation functions are introduced.
We present the fragmentation functions by decomposing the quark-quark
correlator. Among the total 16 fragmentation functions, 8 of them are parity
conserved, and the others are parity violated. They have a one-to-one
correspondence. Positivity bounds of these one dimensional fragmentation
functions are shown. To be explicit, we also introduce a operator definition of
the parity odd correlator. According to the definition, we give a proof that
the parity odd fragmentation functions are local quantities and vanish when sum
over all the hadrons h.
