Analysis of the tetraquark and hexaquark molecular states with the QCD sum rules

In this article, we construct the color-singlet-color-singlet type currents and the color-singlet-color-singlet-color-singlet type currents to study the scalar $D^{\ast }{\overline{D}}^{\ast }$, $D^{\ast }{D}^{\ast }$ tetraquark molecular states and the vector $D^{\ast }{D}^{\ast }{\overline{D}}^{\ast }$, $D^{\ast }{D}^{\ast }{D}^{\ast }$ hexaquark molecular states with the QCD sum rules in details. In calculations, we choose the pertinent energy scales of the QCD spectral densities with the energy scale formula $\mu =\sqrt{M_T^2-{\left(2M_c\right)}^2}$ and $\sqrt{M_H^2-{\left(3M_c\right)}^2}$ for the tetraquark and hexaquark molecular states respectively in a consistent way. We obtain stable QCD sum rules for the scalar $D^{\ast }{\overline{D}}^{\ast }$, $D^{\ast }{D}^{\ast }$ tetraquark molecular states and the vector $D^{\ast }{D}^{\ast }{\overline{D}}^{\ast }$ hexaquark molecular state, but cannot obtain stable QCD sum rules for the vector $D^{\ast }{D}^{\ast }{D}^{\ast }$ hexaquark molecular state. The connected (nonfactorizable) Feynman diagrams at the tree level (or the lowest order) and their induced diagrams via substituting the quark lines make positive contributions for the scalar $D^{\ast }{D}^{\ast }$ tetraquark molecular state, but make negative or destructive contributions for the vector $D^{\ast }{D}^{\ast }{D}^{\ast }$ hexaquark molecular state. It is of no use or meaningless to distinguish the factorizable and nonfactorizable properties of the Feynman diagrams in the color space in the operator product expansion so as to interpret them in terms of the hadronic observables, we can only obtain information about the short-distance and long-distance contributions. (paper)