Abstract
In this article, we take X(4140) as the diquark–antidiquark type $$cs\bar{c}\bar{s}$$ tetraquark state with $$J^{PC}=1^{++}$$ , and we study the mass and pole residue with the QCD sum rules in detail by constructing two types of interpolating currents. The numerical results $$M_{X_{L,+}}=3.95\pm 0.09\,\mathrm{GeV}$$ and $$M_{X_{H,+}}=5.00\pm 0.10\,\mathrm{GeV}$$ disfavor assigning the X(4140) to the $$J^{PC}=1^{++}$$ diquark–antidiquark type $$cs\bar{c}\bar{s}$$ tetraquark state. Moreover, we obtain the masses of the $$J^{PC}=1^{+-}$$ diquark–antidiquark type $$cs\bar{c}\bar{s}$$ tetraquark states as a byproduct. The present predictions can be confronted to the experimental data in the future.
Highlights
In 2009, the CDF collaboration observed X (4140) for the first time in the J/ψφ invariant mass distribution in the e√xclusive B+ → J/ψ φ K + decays in p pcollisions at s = 1.96 TeV with a statistical significance of more than 3.8σ [1]
In 2013, the CMS collaboration confirmed X (√4140) in the B± → J/ψφ K ± decays in pp collisions at s = 7 TeV collected with the CMS detector at the LHC with a statistical significance of more than 5σ [3], the D0 collaboration confirmed X (4140) in the B+ → J/ψφ K + decays with a statistical significance of 3.1σ based on the data sample corresponding to a√n integrated luminosity of 10.4 fb−1 of p pcollisions at s = 1.96 TeV [4]
The LHCb collaboration performed the first full amplitude analysis of the decays B+ → J/ψφ K + with J/ψ → μ+μ−, φ → K + K − with a data sample corresponding to an integrate√d luminosity of 3 fb−1 of pp collision data collected at s = 7 and 8 TeV with the LHCb detector, and observed that the data cannot be described by a model that contains only excited kaon states decaying into φ K + [24,25]
Summary
Before the work [34], we performed a systematic study of the mass spectrum of the axial-vector hidden-charm and hidden-bottom tetraquark states using the QCD sum rules, and obtained the ground state masses Mcqcq = 4.32 ± 0.18 GeV and Mcscs = 4.40 ± 0.16 GeV [39], and the mass breaking effect Mcscs − Mcqcq = 80 MeV, which is much smaller than the experimental value MX (4140) − MX (3872) = 275 MeV. We take X (4140) as the axial-vector cscstetraquark state, construct the diquark–antidiquark type axial-vector currents, calculate the contributions of the vacuum condensates up to dimension 10 in the operator product expansion in a consistent way, use the empirical energy-scale formula to determine the ideal energy scales of the QCD spectral densities, and study the ground state masses and pole residues in detail 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 axial-vector cscstetraquark states in Sect. 2; in Sect. 3, we present the numerical results and discussions; Sect. 4 is reserved for our conclusion
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