Analysis of the $Z_c(4020)$, $Z_c(4025)$, $Y(4360)$ and $Y(4660)$ as vector tetraquark states with QCD sum rules

In this article, we distinguish the charge conjugations of the interpolating currents, calculate the contributions of the vacuum condensates up to dimension-10 in the operator product expansion, and study the masses and pole residues of the $J^{PC}=1^{-\pm}$ hidden charmed tetraquark states with the QCD sum rules. We suggest a formula $\mu=\sqrt{M^2_{X/Y/Z}-(2{\mathbb{M}}_c)^2}$ with the effective mass ${\mathbb{M}}_c=1.8\,\rm{GeV}$ to estimate the energy scales of the QCD spectral densities of the hidden charmed tetraquark states, which works very well. The numerical results disfavor assigning the $Z_c(4020)$, $Z_c(4025)$, $Y(4360)$ as the diquark-antidiquark (with the Dirac spinor structure $C-C\gamma_\mu$) type vector tetraquark states, and favor assigning the $Z_c(4020)$, $Z_c(4025)$ as the diquark-antidiquark type $1^{+-}$ tetraquark states. While the masses of the tetraquark states with symbolic quark structures $c\bar{c}s\bar{s}$ and $c\bar{c}(u\bar{u}+d\bar{d})/\sqrt{2}$ favor assigning the $Y(4660)$ as the $1^{--}$ diquark-antidiquark type tetraquark state, more experimental data are still needed to distinguish its quark constituents. There are no candidates for the positive charge conjugation vector tetraquark states, the predictions can be confronted with the experimental data in the future at the BESIII, LHCb and Belle-II.