Abstract
We apply the QCD sum rule method to systematically study the fully strange tetraquark states with the quantum number JPC = 2++. We construct both the diquark–antidiquark and mesonic–mesonic currents and calculate both their diagonal and off-diagonal correlation functions. Based on the obtained results, we further construct three mixing currents that are nearly non-correlated. We use one mixing current to extract the mass of the lowest-lying state to be GeV, which can be used to explain f2(2010) as a fully strange tetraquark state of JPC = 2++. This state was observed by BESIII in the ϕϕ channel, and we propose to confirm it in the η(′)η(′) channel.
1 Introduction
Many exotic hadrons were observed in particle experiments during the past 20 years [1], some of which are good candidates for the fully strange tetraquark states [2–22]. Especially the BESIII collaboration performed a partial wave analysis of the J/ψ → γϕϕ decay in 2016 [23]. They observed three tensor resonances, namely, f2(2010), f2(2300), and f2(2340) in the ϕϕ invariant mass spectrum, whose masses and widths were measured to be
These three resonances contain many strangeness components, so they are possible fully strange tetraquark states of JPC = 2++. With a large amount of the J/ψ sample, BESIII collaborations are still examining the physics happening in this energy region, and more rich-strangeness signals are expected in the coming future. Similar experiments can also be performed by Belle-II, COMPASS, GlueX, and PANDA, etc.
The fully strange tetraquark states are interesting from two aspects. Experimentally, their widths are possibly not very broad [possibly at the order of (100 MeV)], so they are capable of being observed. Theoretically, their internal structures are simpler than other tetraquark states due to the Pauli principle’s restriction on identical strangeness quarks and antiquarks, which limits their potential number and makes them easier to be observed. In the past 15 years, we have applied the QCD sum rule method to study the fully strange tetraquark states with quantum numbers JPC = 0−+/1±±/4+− [24–32]. More theoretical studies can be found in [33–42].
In this paper, we shall study the fully strange tetraquark states with the quantum number JPC = 2++. We shall systematically construct both the diquark–antidiquark and mesonic–mesonic currents. We shall apply the method of QCD sum rules to study these currents as a whole, and extract the mass of the lowest-lying state to be GeV. Our results suggest that the f2(2010) can be explained as the fully strange tetraquark state of JPC = 2++, while it is not easy to interpret the f2(2300) and f2(2340) as such states.
This paper is organized as follows. In Section 2, we systematically construct the fully strange tetraquark states with the quantum number JPC = 2++. We use these currents to perform QCD sum rule analyses in Section 3, where we calculate both their diagonal and off-diagonal correlation functions. Based on the obtained results, we use the diquark–antidiquark currents to perform numerical analyses in Section 4, and we use their mixing currents to perform numerical analyses in Section 5. The obtained results are summarized and discussed in Section 6.
2 Interpolating currents
The fully strange tetraquark currents with quantum numbers JPC = 0−+/1±±/4+− have been systematically constructed in [24–30, 43]. In this section, we follow the same approach to construct the fully strange tetraquark currents with the quantum number JPC = 2++. We observe three independent diquark–antidiquark currents.where a and b are color indices, C = iγ2γ0 is the charge-conjugation operator, and the symbol represents symmetrizing and subtracting trace terms in the set {μν}. Among these currents, and have the antisymmetric color structure , and has the symmetric color structure , so the internal structure of and is more stable than that of . Moreover, the first current only contains the S-wave diquark field and the S-wave antidiquark field , so it has a more stable internal structure that may lead to a better sum rule result. In addition, the diquark field of JP = 1± contains both S- and P-wave components, so the third current may also lead to a good sum rule result; the second current contains the P-wave diquark field , so its predicted mass is probably larger. In the present study, we only consider tetraquark currents without derivatives, and more JPC = 2++ currents can be constructed when using derivatives. However, their internal structures are not so stable, and their predicted masses are probably also larger.
In addition to the aforementioned diquark–antidiquark currents, we find six mesonic–mesonic currents.
We can verify the following relations through the Fierz rearrangement, so the number of independent mesonic–mesonic currents is also three.
Moreover, we can use the Fierz rearrangement to relate the diquark–antidiquark and mesonic–mesonic currents.
Therefore, the diquark–antidiquark and mesonic–mesonic constructions are equivalent to each other, when the local currents are investigated. We shall use this Fierz identity to study the decay behaviors at the end of this paper.
3 QCD sum rule analysis
The QCD sum rule method is a powerful and successful non-perturbative method [44, 45]. In this section, we apply it to study the currents and calculate their two-point correlation functionsat both the hadron and quark–gluon levels. Here, , and the symbol denotes symmetrizing and subtracting trace terms in the two sets {μν} and {μ′ν′}, respectively.
At the hadron level, we generally assume that the currents (i = 1⋯3) couple to the states Xn (n = 1⋯N) throughwhere fin is the decay constant and ϵμν is the symmetric and traceless polarization tensor. Then, we use the dispersion relation to express Πij(q2) aswhere is the physical threshold and is the phenomenological spectral density. We parameterize it for the states Xn and a continuum contribution aswhere Mn is the mass of Xn and ⋯ is contributed by the continuum. It should be noted that the widths of Xn are not taken into account in the present study, and the two-meson thresholds are also not taken into account, such as the ϕϕ threshold. The Fierz rearrangement given in Eq. 13 indicates that the tetraquark currents can easily couple to two mesons, which causes some difficulties in extracting the correct information about the resonance when the two-meson thresholds contribute significantly. The authors of [46] suggest that the four-quark diagrams with no singularity at (mi is the quark mass) are relevant to two free mesons but not relevant to the four-quark state. However, the validity of this criterion is still not clear.
At the quark–gluon level, we apply the method of the operator product expansion (OPE) to calculate Eq. 15 and extract the OPE spectral density . In this study, we take into account the Feynman diagrams shown in Figure 1 and perform the calculations up to the twelfth dimension, where we consider the perturbative term, the strange quark mass ms, the quark condensate , the double-gluon condensate , the quark–gluon mixed condensate , and their combinations. We do not consider some other condensates, such as and the diagrams with up/down quark loops, since their calculations are difficult. The vacuum saturation is assumed for higher-dimensional operators, i.e., and . We calculate all the diagrams proportional to and , where we find the D = 6 term and the D = 8 term to be important. We partly calculate the diagrams proportional to , and we find their contributions to be small.
FIGURE 1
Finally, we perform the Borel transformation at both the hadron and quark–gluon levels. After approximating the continuum using ρij(s) above the threshold value s0, we arrive at the sum rule equation
The explicit sum rule equations extracted from the currents are as follows:
For completeness, we have calculated both the diagonal and off-diagonal correlation functions. We shall investigate them using two steps, the single-channel analysis and the multi-channel analysis, in the following sections.
4 Single-channel analysis
In this section, we perform the single-channel analysis. To perform this, we simply neglect the off-diagonal correlation functions; i.e., we assume ρij(s)|i≠j = 0 so that only ρii(s) ≠ 0. Under this assumption, any two of the three currents cannot mainly couple to the same state X; otherwise,
This allows us to further assume that the three currents couple separately to the three states X1,2,3 throughwith fii ≠ 0 and fij = 0 for i, j = 1⋯3 and i ≠ j.
Now, we can parameterize the diagonal spectral density ρii(s) as one-pole dominance for the state Xi and a continuum contribution. This simplifies Eq. 19 to be
It can be used to calculate Mi through
We use the spectral density ρ11(s) given in Eq. 20 as an example to perform the single-channel numerical analysis. We take the following values for various sum rule parameters [1, 47–53]:
Equation 29 states that mass M1 depends on two free parameters, the threshold value s0, and the Borel mass MB. We consider three aspects to determine their working regions: a) the convergence of OPE, b) the sufficient amount of pole contribution, and c) the stability of the mass dependence on these two parameters.
First, we investigate the convergence of OPE, which is the cornerstone of a reliable QCD sum rule analysis. We require the D = 12/10/8 terms to be less than 5%/10%/20%, respectively.
Figure 2 shows that through the dashed curves, the lower bound of the Borel mass is determined to be GeV2.
FIGURE 2
Second, we investigate the one-pole-dominance assumption by requiring the pole contribution to be larger than 40%:
Figure 2 shows that through the solid curve, the upper bound of the Borel mass is determined to be GeV2 when setting s0 = 6.5 GeV2. Altogether, we determine the Borel window to be 1.53 GeV2 for s0 = 6.5 GeV2. We redo the same procedures and find that there are non-vanishing Borel windows when GeV2.
Third, we investigate the stability of the mass dependence on s0 and MB. As shown in Figure 3, we find a mass minimum around s0 ≈ 3 GeV2, and the mass dependence on s0 is moderate inside the region 5.5 GeV2 ≤ s0 ≤ 7.5 GeV2. As shown in Figure 4, the mass dependence on MB is rather weak inside the Borel window 1.53 GeV2.
FIGURE 3
FIGURE 4
Altogether, we determine our working regions to be 5.5 GeV2 ≤ s0 ≤ 7.5 GeV2 and 1.53 GeV2, where the mass of X1 is calculated to be
Its central value is obtained by setting s0 = 6.5 GeV2 and GeV2, and its uncertainty is due to the Borel mass MB, the threshold value s0, and various sum rule parameters listed in Eq. 30.
We follow the same procedures to study the other two currents, and , separately. The obtained results are shown in Table 1. We shall further study the three currents as a whole and perform the multi-channel analysis in the next section.
TABLE 1
| Current | [GeV2] | Working regions | Pole [%] | Mass [GeV] | |
|---|---|---|---|---|---|
| s0 [GeV2] | |||||
| 5.4 | 1.53–1.77 | 6.5 ± 1.0 | 40–53 | ||
| 12.4 | 2.19–2.65 | 13.5 ± 3.0 | 40–52 | ||
| 6.2 | 1.24–1.43 | 7.0 ± 1.0 | 40–53 | ||
| 5.6 | 1.71–1.81 | 6.0 ± 1.0 | 40–45 | ||
| 12.7 | 2.16–2.78 | 14.0 ± 3.0 | 40–55 | ||
| 12.2 | 2.19–2.69 | 13.5 ± 3.0 | 40–53 | ||
QCD sum rule results for the fully strange tetraquark states with the quantum number JPC =2++, extracted from the diquark–antidiquark currents and their mixing currents .
5 Multi-channel analyses
In this section, we perform the multi-channel analyses. To perform this, we do not neglect the off-diagonal correlation functions any more, i.e., ρij(s)|i≠j ≠ 0. When setting s0 = 6.0 GeV2 and GeV2, the 3 × 3 matrix becomes
Therefore, and are strongly correlated with each other, and the off-diagonal terms are indeed non-negligible.
In order to diagonalize the 3 × 3 matrix , we construct three mixing currents where is the transition matrix.
We use to denote the correlation functions extracted from the mixing currents . This 3 × 3 matrix becomeswhen settingas well as s0 = 6.0 GeV2 and GeV2. Therefore, the off-diagonal terms of are negligible, and the three mixing currents are nearly non-correlated around here. Moreover, the two correlation functions, and , are both negative around s ≈ 6.0 GeV2. This suggests that they are both non-physical around here, and the masses extracted from them should be significantly larger than GeV GeV.
Now, we can use the procedures applied in the previous section on the currents to study their mixing currents . The obtained results are shown in Table 1. Particularly, the mass extracted from the current isas shown in Figures 5, 6 with respect to the threshold value s0 and the Borel mass MB.
FIGURE 5
FIGURE 6
6 Summary and discussions
In this paper, we use the QCD sum rule method to study the fully strange tetraquark states with the quantum number JPC = 2++. We systematically construct their interpolating currents and find three independent diquark–antidiquark currents, denoted as . We calculate both their diagonal and off-diagonal correlation functions. Based on the obtained results, we construct three mixing currents that are nearly non-correlated and denoted as . We use both the diquark–antidiquark currents and the mixing currents to perform QCD sum rule analyses. The obtained results are shown in Table 1.
Particularly, we use the mixing current to evaluate the mass of the lowest-lying state to be GeV, while the masses extracted from the other two mixing currents, and , are significantly larger than 3.0 GeV. The fully strange tetraquark states of JPC = 2++ naturally decay into the ϕϕ channel, where the BESIII collaboration observed three tensor resonances, namely, f2(2010), f2(2300), and f2(2340) [23]. Accordingly, our results suggest that the f2(2010) can be explained as the fully strange tetraquark state of JPC = 2++, while it is not easy to interpret the f2(2300) and f2(2340) as such states.
In this paper, we also systematically construct the fully strange mesonic–mesonic currents of JPC = 2++, which can be related to the diquark–antidiquark currents through the Fierz rearrangement. In particular, we can apply Eqs. 37 and 39, and Eq. 14 to transform the mixing current to be
This Fierz identity suggests that the lowest-lying state dominantly decays into the S-wave ϕ(1020)ϕ(1020) channel through the mesonic–mesonic current , while it can also decay into the D-wave η(′)η(′) channel through . Accordingly, we propose to confirm the f2(2010) in the η(′)η(′) channel in the future Belle-II, BESIII, COMPASS, GlueX, and PANDA experiments. In addition, more possible decay patterns can be obtained by annihilating an pair into a gluon, which then transits into the final states with a pair of strange mesons, such as .
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding authors.
Author contributions
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Funding
This project was supported by the National Natural Science Foundation of China under grant no. 12075019, the Jiangsu Provincial Double-Innovation Program under grant no. JSSCRC2021488, and the Fundamental Research Funds for the Central Universities.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
exotic hadron, tetraquark state, QCD sum rules, Fierz rearrangement, interpolating current
Citation
Dong R-R, Su N, Chen H-X and Cui E-L (2023) QCD sum rule study on the fully strange tetraquark states of JPC = 2++. Front. Phys. 11:1184103. doi: 10.3389/fphy.2023.1184103
Received
11 March 2023
Accepted
22 May 2023
Published
07 June 2023
Volume
11 - 2023
Edited by
Alberto Martinez Torres, University of São Paulo, Brazil
Reviewed by
Feng-Kun Guo, Chinese Academy of Sciences (CAS), China
Alessandro Pilloni, National Institute of Nuclear Physics of Rome, Italy
Updates
Copyright
© 2023 Dong, Su, Chen and Cui.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Hua-Xing Chen, hxchen@seu.edu.cn; Er-Liang Cui, erliang.cui@nwafu.edu.cn
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.