QCD sum rule studies of \(s s {\bar{s}} {\bar{s}}\) tetraquark states with \(J^{PC} = 1^{+}\)
Abstract
We apply the method of QCD sum rules to study the structure X newly observed by the BESIII Collaboration in the \(\phi \eta ^\prime \) mass spectrum in 2.0–2.1 GeV region in the \(J/\psi \rightarrow \phi \eta \eta ^\prime \) decay. We construct all the \(s s {\bar{s}} {\bar{s}}\) tetraquark currents with \(J^{PC} = 1^{+}\), and use them to perform QCD sum rule analyses. One current leads to reliable QCD sum rule results and the mass is extracted to be \(2.00^{+0.10}_{0.09}\) GeV, suggesting that the structure X can be interpreted as an \(s s {\bar{s}} {\bar{s}}\) tetraquark state with \(J^{PC} = 1^{+}\). The Y(2175) can be interpreted as its \(s s {\bar{s}} {\bar{s}}\) partner having \(J^{PC} = 1^{}\), and we propose to search for the other two partners, the \(s s {\bar{s}} {\bar{s}}\) tetraquark states with \(J^{PC} = 1^{++}\) and \(1^{+}\), in the \(\eta ^\prime f_0(980)\), \(\eta ^\prime K {\bar{K}}\), and \(\eta ^\prime K {\bar{K}}^*\) mass spectra.
1 Introduction
 1.After assuming X to have the spinparity quantum numbers \(J^P = 1^\), its mass and decay width are determined to be$$\begin{aligned} M_{1^}= & {} 2002.1 \pm 27.5 \pm 15.0~\mathrm{MeV},\nonumber \\ \varGamma _{1^}= & {} 129 \pm 17 \pm 7~\mathrm{MeV}. \end{aligned}$$(1)
 2.After assuming X to have the spinparity quantum numbers \(J^P = 1^+\), its mass and decay width are determined to be$$\begin{aligned} M_{1^+}= & {} 2062.8 \pm 13.1 \pm 4.2~\mathrm{MeV},\nonumber \\ \varGamma _{1^+}= & {} 177 \pm 36 \pm 20~\mathrm{MeV}. \end{aligned}$$(2)
Because the structure X was observed in the \(\phi \eta ^\prime \) mass spectrum but not reported in the \(\phi \eta \) mass spectrum [1], it may contain large \({\bar{s}} s {\bar{s}} s\) component. This makes it a good candidate of exotic hadrons in the light sector [3, 4, 5, 6, 7, 8, 9]. Another similar candidate is the Y(2175), which was first observed by the BaBar Collaboration in the \(\phi f_0(980)\) invariant mass spectrum [10, 11, 12, 13], and later confirmed in the BESII [14], BESIII [15, 16], and Belle [17] experiments. The Y(2175) may also contain large \({\bar{s}} s {\bar{s}} s\) component, but its measured mass and width are significantly different from those of X [1].
In our previous studies [18, 19] we have applied the method of QCD sum rules to systematically study the \(s s {\bar{s}} {\bar{s}}\) tetraquark states with \(J^{PC} = 1^{{}{}}\). There we found two independent \(s s {\bar{s}} {\bar{s}}\) tetraquark currents with \(J^{PC} = 1^{{}{}}\), and the masses are evaluated to be \(2.34 \pm 0.17\) GeV and \(2.41 \pm 0.25\) GeV, not far from each other [19]. These two values are both significantly larger than the first mass value listed in Eq. (1), suggesting that the structure X is difficult to be interpreted as an \(s s {\bar{s}} {\bar{s}}\) tetraquark state of \(J^{PC} = 1^{{}{}}\). Instead, the Y(2175) can be well interpreted as an \(s s {\bar{s}} {\bar{s}}\) tetraquark state of \(J^{PC} = 1^{{}{}}\) [18, 19]. Moreover, the above two mass values are extracted from two diagonalized currents, which do not strongly correlate to each other and may couple to two different physical states: one is the Y(2175), and the other is around 2.4 GeV. There have been some evidences for the latter structure in the previous experiments [11, 14, 15, 17], and we refer to Ref. [19] for detailed discussions.
In the present study we follow the same approach to study the \(s s {\bar{s}} {\bar{s}}\) tetraquark states with \(J^{PC} = 1^{+}\), and examine whether the structure X can be explained. Again, we shall find that there are two independent \(s s {\bar{s}} {\bar{s}}\) tetraquark currents with \(J^{PC} = 1^{+}\), which we shall use to perform QCD sum rule analyses. The internal structures of exotic hadrons are always complicated. For each internal structure we can construct the relevant interpolating current, and there are usually many interpolating currents when studying multiquark states. In this case, the only two independent currents make it possible to study their mixing. Note that we have done this in Ref. [19] when studying the \(s s {\bar{s}} {\bar{s}}\) tetraquark states with \(J^{PC} = 1^{{}{}}\). By doing this we can carefully examine the relations between physical states and the relevant interpolating currents, and further understand the internal structures of exotic hadrons.
This paper is organized as follows. In Sect. 2, we systematically construct the \(s s {\bar{s}} {\bar{s}}\) tetraquark currents with \(J^{PC} = 1^{+}\), using both diquark/antidiquark fields and quarkantiquark pairs. These currents are then used to perform QCD sum rule analyses in Sect. 3, and numerical analyses in Sect. 4. Their mixing are investigated in Sect. 5. Section 6 is a summary.
2 Interpolating currents
In the following we shall use \(\eta _{1\mu }\) and \(\eta _{2\mu }\) to perform QCD sum rule analyses.
3 QCD sum rule analyses
4 Numerical analyses
 First we study the convergence of the operator product expansion. After taking \(s_0\) to be \(\infty \) and the integral subscript \(16 m_s^2\) to be zero, we obtain the numerical series of the OPE as a function of \(M_B\):From this equation, we clearly see that the OPE convergence is quite good: the dimension 12 terms \(\big (\sim M_B^{2}\big )\) are significantly smaller than the dimension 10 terms \(\big (\sim M_B^{0}\big )\), which are again significantly smaller than the dimension 8 terms \(\left( \sim M_B^{2}\right) \). Numerically, we show the ratio$$\begin{aligned}&\varPi _{\eta _2\eta _2}\left( M_B^2, \infty \right) \nonumber \\&\quad = +\,\, 2.0 \times 10^{6} M_B^{10}  2.2 \nonumber \\&\qquad \times 10^{8} M_B^8 + \,\,3.0 \times 10^{6} M_B^6\nonumber \\&\qquad +\,\, 6.1 \times 10^{6} M_B^4  6.5 \times 10^{6} M_B^2\nonumber \\&\qquad + \,\,6.2 \times 10^{7} M_B^{0} + 7.5 \times 10^{8} M_B^{2}. \end{aligned}$$(20)in Fig. 2 as a function of the Borel mass \(M_B\). We find it to be smaller than 5% in the regions \(1.6\hbox { GeV}^2< M_B^2< 2.0\hbox { GeV}^2\) and \(5.5\hbox { GeV}^2< s_0< 6.5\hbox { GeV}^2\).$$\begin{aligned} \text{ CVG } \equiv {\varPi ^{\mathrm{Dim}=10+12}_{\eta _2\eta _2}\left( M_B^2, s_0 \right) \over \varPi _{\eta _2\eta _2}\left( M_B^2, s_0 \right) }, \end{aligned}$$(21)
 Then we study the pole contribution, defined asWe show it as a function of the Borel mass \(M_B\) in Fig. 3. We find it to be 30% < PC < 58% in the regions \(1.6\hbox { GeV}^2< M_B^2< 2.0\hbox { GeV}^2\) and \(5.5\hbox { GeV}^2< s_0< 6.5\hbox { GeV}^2\). This amount of pole contribution is acceptable when one applies the method of QCD sum rules to study multiquark states.$$\begin{aligned} \text{ PC } \equiv {\varPi _{\eta _2\eta _2}\left( M_B^2, s_0 \right) \over \varPi _{\eta _2\eta _2}(M_B^2, \infty )}. \end{aligned}$$(22)

Finally we study the mass dependence on the two free parameters, the Borel mass \(M_B\) and the threshold value \(s_0\). To clearly see this, we show \(M_{\eta _2}\), the mass extracted from the current \(\eta _{2\mu }\), in Fig. 4 as a function of \(M_B\) and \(s_0\).
In the left panel we show \(M_{\eta _2}\) as a function of the Borel mass \(M_B\), and find it quite stable in the Borel window \(1.6\hbox { GeV}^2< M_B^2< 2.0\hbox { GeV}^2\). Comparing this figure with Figs. 2 and 3, we find that one can obtain a still larger pole contribution by choosing a smaller Borel mass (as shown in Fig. 3), but at the same time the convergence of OPE would become worse (as shown in Fig. 2) and the mass dependence on the Borel mass would become stronger (as shown in the left panel of Fig. 4). Considering all these behaviours, we find it suitable to fix the Borel window to be \(1.6\hbox { GeV}^2< M_B^2< 2.0\hbox { GeV}^2\).
In the right panel we show \(M_{\eta _2}\) as a function of the threshold value \(s_0\). We find that the mass curves moderately depend on the threshold value \(s_0\). Especially, we evaluate the mass to be 1.94 GeV\(<M_{\eta _2}<2.06\) GeV in the region \(5.5\hbox { GeV}^2< s_0< 6.5\hbox { GeV}^2\). This uncertainty is about 6%, quite typical in QCD sum rule studies.
5 Mixing of currents
In the previous section we have used the two single \(s s {\bar{s}} {\bar{s}}\) tetraquark currents with \(J^{PC} = 1^{+}\), \(\eta _{1\mu }\) and \(\eta _{2\mu }\), to perform QCD sum rule analyses. In this section we further study their mixing. We shall follow the procedures used in Ref. [19], where the mixing of two \(s s {\bar{s}} {\bar{s}}\) tetraquark currents with \(J^{PC} = 1^{{}{}}\) is carefully investigated.
We use \(J_{1\mu }\) and \(J_{2\mu }\) to perform QCD sum rule analyses, and the results obtained are almost the same as those extracted from \(\eta _{1\mu }\) and \(\eta _{2\mu }\): (a) \(J_{1\mu }\) does not lead to reliable QCD sum rule results because \(\varPi _{J_1J_1}(M_B^2, s_0)\) is negative in the region \(s_0< 10\hbox { GeV}^2\), and (b) the mass extracted from \(J_{2\mu }\) is about 2.00 GeV, the same as the one extracted from \(\eta _{2\mu }\).
6 Summary and discussions
In this work we systematically construct all the \(s s {\bar{s}} {\bar{s}}\) tetraquark currents with the quantum numbers \(J^{PC} = 1^{+}\). We find there are two independent ones (\(\eta _{1\mu }\) and \(\eta _{2\mu }\)), which are then used to perform QCD sum rule analyses. The sum rules extracted from \(\eta _{1\mu }\) and \(\eta _{2\mu }\) are much different from each other: a) \(\eta _{1\mu }\) does not lead to reliable results because \(\varPi _{\eta _1\eta _1}(M_B^2, s_0)\) is negative, and so nonphysical, in the region \(s_0< 10\hbox { GeV}^2\), and b) \(\eta _{2\mu }\) leads to reliable results and the mass is extracted to be \(2.00^{+0.10}_{0.09}\) GeV, consistent with the second mass value listed in Eq. (2), \(2062.8 \pm 13.1 \pm 4.2\) MeV. The mixing between \(\eta _{1\mu }\) and \(\eta _{2\mu }\) has been taken into account, and the results are the same. Hence, our results suggest that the structure X observed at BESIII [1] has the spinparity quantum numbers \(J^P = 1^{+}\), and it can be interpreted as an \(s s {\bar{s}} {\bar{s}}\) tetraquark state.
Masses extracted from the vector and axialvector tetraquark currents. Possible experimental candidates are listed for comparisons. We use q to denote an up or down quark, and s to denote a strange quark. The mass value \(2.00^{+0.10}_{0.09}\) GeV denoted by \(^\dagger \) is obtained in the present study
Contents  \(J^{PC} = 1^{+}\)  \(J^{PC} = 1^{{}{}}\)  \(J^{PC} = 1^{++}\)  \(J^{PC} = 1^{+}\)  

(Isospin)  Theo. (GeV)  Exp.  Theo. (GeV)  Exp.  Theo. (GeV)  Exp.  Theo. (GeV)  Exp. 
\(q q {\bar{q}} {\bar{q}}\)  1.47–1.66 [22]  –  1.60–1.73 [22]  \(\rho (1570)\)  1.51–1.63 [22]  \(\begin{array}{c}a_1(1640) \\ a_1(1420) \end{array}\)  \(\sim 1.6\) [20]  \(\pi _1(1600)\) 
\((I = 1)\)  \(\rho (1700)\)  
\(q s {\bar{q}} {\bar{s}}\)  1.91–2.13 [22]  \(\rho (1900)\)  \(\sim 2.0\) [20]  \(\pi _1(2015)\)  
\((I = 1)\)  \(\rho (2150)\)  
\(s s {\bar{s}} {\bar{s}}\)  \(2.00 ^{+0.10}_{0.09}~^\dagger \)  X(2063)  \(2.34 \pm 0.17\) [19]  Y(2175)  –  –  –  – 
(I \(=\) 0)  \(2.41 \pm 0.25\) [19]  Y(2470) 
Besides these isoscalar \(s s {\bar{s}} {\bar{s}}\) tetraquark states, in Ref. [22] we have systematically constructed all the isovector tetraquark currents of \(I^GJ^{PC} = 1^+1^{+}/1^+1^{{}{}}/1^1^{++}/1^1^{+}\), and found a onetoone correspondence among them, i.e., for every tetraquark current of \(I^GJ^{PC} = 1^+1^{+}\) one can construct a corresponding one of \(I^GJ^{PC} = 1^+1^{{}{}}\), etc. These tetraquark currents have been used to perform QCD sum rule analyses in Refs. [20, 22], and the results are summarized in Table 1, where q denotes an up or down quark, and s denotes a strange quark. Note that the sum rule results do not have the above onetoone correspondence, for examples: a) there are four \(q q {\bar{q}} {\bar{q}}\) currents and four \(q s {\bar{q}} {\bar{s}}\) currents with \(I^GJ^{PC} = 1^+1^{+}\), and the masses extracted from these currents are all around 1.471.66 GeV; b) there are also four \(q q {\bar{q}} {\bar{q}}\) currents and four \(q s {\bar{q}} {\bar{s}}\) currents with \(I^GJ^{PC} = 1^+1^{{}{}}\), but the masses extracted from the former four are around 1.601.73 GeV and the masses extracted from the latter four are around 1.912.13 GeV. This behaviour may relate to their internal structures, such as internal orbital excitations.
Similarly, there is a onetoone correspondence among the \(s s {\bar{s}} {\bar{s}}\) tetraquark currents with \(J^{PC} = 1^{+}/1^{{}{}}/1^{++}/1^{+}\). Those with \(J^{PC} = 1^{+}\) and \(1^{{}{}}\) have been used to perform QCD sum rule analyses in the present study as well as in Refs. [18, 19]. The results are also summarized in Table 1. From this table, we propose to search for the \(s s {\bar{s}} {\bar{s}}\) tetraquark states with \(J^{PC} = 1^{++}\) and \(1^{+}\) in future experiments. We are now studying them following the same approach used in the present study. Their masses may also be around 2.02.4 GeV, and the possible decay channels to observe them are \(\eta ^\prime f_0(980)\), \(\eta ^\prime K {\bar{K}}\), and \(\eta ^\prime K {\bar{K}}^*\), etc.
When studying light tetraquark states, it is usually difficult to determine the experimental signal as a genuine fourquark state other than a conventional \({\bar{q}} q\) meson, because the signal always has a quite large decay width. For example, besides the \(s s {\bar{s}} {\bar{s}}\) tetraquark state of \(J^{PC} = 1^{+}\), there are many other possible interpretations to explain the structure X, such as the second radial excitation of \(h_1(1380)\) having \(I(J^P) = 0(1^+)\) [2]. However, with the large amount of data collected at BESIII, this problem may be partly solved, and it is promising to continuously study light exotic hadrons. Together with those studies on charmoniumlike XYZ states, our understudying on the nature of exotic hadrons can be significantly improved.
Notes
Acknowledgements
We thank Professor ShiLin Zhu for useful discussions. This project is supported by the National Natural Science Foundation of China under Grants Nos. 11575017, 11722540, and 11761141009, the Fundamental Research Funds for the Central Universities, and the Chinese National Youth Thousand Talents Program.
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