Reanalysis of the $Y(3940)$, $Y(4140)$, $Z_c(4020)$, $Z_c(4025)$ and $Z_b(10650)$ as molecular states with QCD sum rules

In this article, we calculate the contributions of the vacuum condensates up to dimension-10 in the operator product expansion, and study the $J^{PC}=0^{++}$, $1^{+-}$, $2^{++}$ $D^{\ast }{\bar{D}}^{\ast }$, $D_s^{\ast }{\bar{D}}_s^{\ast }$, $B^{\ast }{\bar{B}}^{\ast }$, $B_s^{\ast }{\bar{B}}_s^{\ast }$ molecular states with the QCD sum rules. In calculations, we use the formula $\mu =\sqrt{M_{X\mathrm{/}Y\mathrm{/}Z}^2-(2{\mathbb{M}}_Q{)}^2}$ to determine the energy scales of the QCD spectral densities. The numerical results favor assigning the $Z_c(4020)$ and $Z_c(4025)$ as the $J^{PC}=0^{++}$, $1^{+-}$ or $2^{++}$ $D^{\ast }{\bar{D}}^{\ast }$ molecular states, the $Y(4140)$ as the $J^{PC}=0^{++}$ $D_s^{\ast }{D}_s^{\ast }$ molecular state, the $Z_b(10650)$ as the $J^{PC}=1^{+-}$ $B^{\ast }{\bar{B}}^{\ast }$ molecular state, and disfavor assigning the $Y(3940)$ as the ($J^{PC}=0^{++}$) molecular state. The present predictions can be confronted to the experimental data in the futures.