Two-particle contributions and nonlocal effects in the QCD sum rules for the axialvector tetraquark candidate $Z_c(3900)$

In this article, we study the $Z_c(3900)$ with the QCD sum rules in details by including the two-particle scattering state contributions and nonlocal effects between the diquark and antidiquark constituents. The two-particle scattering state contributions cannot saturate the QCD sum rules at the hadron side, the contribution of the $Z_c(3900)$ plays an un-substitutable role, we can saturate the QCD sum rules with or without the two-particle scattering state contributions. If there exists a barrier or spatial separation between the diquark and antidiquark constituents, the Feynman diagrams can be divided into the factorizable and nonfactorizable diagrams. The factorizable diagrams consist of two colored clusters and lead to a stable tetraquark state. The nonfactorizable Feynman diagrams correspond to the tunnelling effects, which play a minor important role in the QCD sum rules, and are consistent with the small width of the $Z_c(3900)$. It is feasible to apply the QCD sum rules to study the tetraquark states, which begin to receive contributions at the order $\mathcal{O}(\alpha_s^0)$, not at the order $\mathcal{O}(\alpha_s^2)$.