Mass spectrum of the vector hidden charmed and bottomed tetraquark states

The Babar, Belle, CLEO, D0, CDF and FOCUS collaborations have discovered (or confirmed) a large number of charmonium-like states X(3940), X(3872), Y (4260), Y (4008), Y (3940), Y (4325), Y (4360), Y (4660), etc, and revitalized the interest in the spectroscopy of the charmonium states [1, 2, 3, 4]. Many possible assignments for those states have been suggested, such as multiquark states (whether the molecular type or the diquarkantidiquark type), hybrid states, charmonium states modified by nearby thresholds, threshold cusps, etc [1, 2, 3, 4]. The Z+(4430) observed in the ψ′π+ decay mode is the most interesting subject [5], it can’t be a pure cc̄ state due to the positive charge. We can distinguish the multiquark states from the hybrids or charmonia with the criterion of non-zero charge. The two resonance-like structures (thereafter we will denote them as Z1(4050) and Z2(4250) respectively) in the πχc1 invariant mass distribution near 4.1GeV are also particularly interesting [6]. Their quark contents must be some special combinations of the cc̄ud̄, just like the Z+(4430), they cannot be the conventional mesons. In Refs.[7, 8], we assume that the hidden charmed mesons Z1(4050) and Z2(4250) are vector (and scalar) tetraquark states, and study their masses using the QCD sum rules. The numerical results indicate that the masses of the vector hidden charmed tetraquark states are about MZ = (5.12± 0.15)GeV or MZ = (5.16± 0.16)GeV, while the masses of the scalar hidden charmed tetraquark states are bout MZ = (4.19±0.17)GeV. The scalar hidden charmed tetraquark states may have smaller masses than the corresponding vector states. In Ref.[9], we study the mass spectrum of the scalar hidden charmed and bottomed tetraquark states using the QCD sum rules. In this article, we extend our previous work to study the mass spectrum of the vector hidden charmed and bottomed tetraquark states. In the QCD sum rules, the operator product expansion is used to expand the timeordered currents into a series of quark and gluon condensates which parameterize the long distance properties of the QCD vacuum. Based on the quark-hadron duality, we can obtain copious information about the hadronic parameters at the phenomenological side [10, 11]. The mass is a fundamental parameter in describing a hadron, whether or not there exist those hidden tetraquark configurations is of great importance itself, because it provides a