Parity-odd Parton Distribution Functions from $\theta$-Vacuum

Quantum chromodynamics is a fundamental non-abelian gauge theory of strong interactions. The physical quantum chromodynamics vacuum state, $|\theta\rangle$, is a linear superposition of the $n$-vacua states with different topological numbers. Because of the configuration of the gauge fields, the tunneling events can induce the local parity-odd domains. Those interactions that occur in these domains can be affected by these effects. Considering the hadron (nucleon) system, we introduce the parity-odd parton distribution functions in order to describe the parity-odd structures inside the hadron in this paper. We obtain 8 parity-odd parton distribution functions at leading twist for spin-1/2 hadrons and present properties of these parton distribution functions. By introducing the parity-odd quark-quark correlator, we find the parity-odd effects vanish from the macroscopic point of view. Since the parity-odd effects are confined in small domains, we consider the high energy semi-inclusive deeply inelastic scattering process to investigate these effects by calculating the single spin asymmetries.