Abstract

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.

Highlights

  • The BESIII collaboration studied the process e+e− → (D∗ D ∗)±π ∓ at a center-of-mass energy of 4.26 GeV using a 827 pb−1 data sample obtained with the BESIII detector at the Beijing Electron Positron Collider, and observed a structure Z ± c (4025)near the (D∗ D ∗)±threshold in the π ∓ recoil mass spectrum [1]

  • In Ref. [16], we extend our previous works on the axialvector tetraquark states [44], distinguish the charge conjugations of the interpolating currents, calculate the contributions of the vacuum condensates up to dimension-10 and discard the perturbative corrections in the operator product expansion, study the Cγ5 − Cγμ type axial-vector hidden charmed tetraquark states with the QCD sum rules

  • In Refs. [45,46], we study the Cγμ − C and Cγμγ5 − Cγ5 type tetraquark states with the QCD sum rules by carrying out the operator product expansion to the vacuum condensates up to dimension-10 and setting the energy scale to be μ = 1 GeV

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Summary

Introduction

The BESIII collaboration studied the process e+e− → (D∗ D ∗)±π ∓ at a center-of-mass energy of 4.26 GeV using a 827 pb−1 data sample obtained with the BESIII detector at the Beijing Electron Positron Collider, and observed a structure. We study the diquark–antidiquark type vector tetraquark states in detail with the QCD sum rules, and we explore possible assignments of the Zc(4020), Zc(4025), Y (4360), and Y (4660) in the tetraquark scenario. [45,46], we study the Cγμ − C and Cγμγ5 − Cγ5 type tetraquark states with the QCD sum rules by carrying out the operator product expansion to the vacuum condensates up to dimension-10 and setting the energy scale to be μ = 1 GeV. We extend our previous works [16] to study the vector tetraquark states, distinguish the charge conjugations of the interpolating currents, calculate the contributions of the vacuum condensates up to dimension-10 and discard the perturbative corrections, study the masses and pole residues of the C − Cγμ type vector hidden charmed tetraquark states with the QCD sum rules. The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of the vector tetraquark states in Sect. 2; in Sect. 3, we present the numerical results and discussions; Sect. 4 is reserved for our conclusion

QCD sum rules for the vector tetraquark states
Numerical results and discussions
Conclusion
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