Abstract
Some hadrons have the exotic quantum numbers that the traditional \(\bar{q} q\) mesons and qqq baryons can not reach, such as \(J^{PC} = 0^{--}/0^{+-}/1^{-+}/2^{+-}/3^{-+}/4^{+-}\), etc. We investigate for the first time the exotic quantum number \(J^{PC}=4^{+-}\), and study the fully-strange tetraquark states with such an exotic quantum number. We systematically construct all the diquark–antidiquark interpolating currents, and apply the method of QCD sum rules to calculate both the diagonal and off-diagonal correlation functions. The obtained results are used to construct three mixing currents that are nearly non-correlated, and we use one of them to extract the mass of the lowest-lying state to be \(2.85^{+0.19}_{-0.22}\) GeV. We apply the Fierz rearrangement to transform this mixing current to be the combination of three meson–meson currents, and the obtained Fierz identity suggests that this state dominantly decays into the P-wave \(\phi (1020) f_2^\prime (1525)\) channel. This fully-strange tetraquark state of \(J^{PC}=4^{+-}\) is a purely exotic hadron to be potentially observed in future particle experiments.
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1 Introduction
In the past twenty years many candidates of exotic hadrons were observed in particle experiments, which can not be well explained in the traditional quark model [1]. Most of them still have the “traditional” quantum numbers that the traditional \(\bar{q} q\) mesons and qqq baryons can also reach, making them not so easy to be clearly identified as exotic hadrons. However, there are some “exotic” quantum numbers that the traditional hadrons can not reach, such as the spin-parity quantum numbers \(J^{PC} = 0^{--}/0^{+-}/1^{-+}/2^{+-}/3^{-+}/4^{+-}/\cdots \). The hadrons with such exotic quantum numbers are of particular interests, since they can not be explained as traditional hadrons any more. Their possible interpretations are compact multiquark states [2,3,4,5], hadronic molecules [6,7,8], glueballs [9,10,11,12,13,14,15,16], and hybrid states [17,18,19,20,21,22,23,24,25,26], etc.
Among these exotic quantum numbers, the states of \(J^{PC} = 1^{-+}\) have been extensively studied in the literature [2, 3, 5,6,7, 17,18,19,20,21,22,23,24,25,26], since they are predicted to be the lightest hybrid states [27]. Up to now there have been four structures observed in experiments with \(J^{PC} = 1^{-+}\), including three isovector states \(\pi _1(1400)\) [28], \(\pi _1(1600)\) [29], and \(\pi _1(2015)\) [30] as well as one isoscalar state \(\eta _1(1855)\) [31]. Besides, the states of \(J^{PC} = 0^{--}/0^{+-}/2^{+-}/3^{-+}\) have also been studied to some extent [4, 8,9,10,11,12,13,14,15,16]. These theoretical and experimental studies have significantly improved our understanding on the non-perturbative behaviors of the strong interaction in the low energy region. However, there has not been any investigation on the exotic quantum number \(J^{PC} = 4^{+-}\) yet.
In this paper we shall investigate for the first time the exotic quantum number \(J^{PC}=4^{+-}\), and study the fully-strange tetraquark states with such an exotic quantum number. We shall work within the diquark–antidiquark picture, and systematically construct all the diquark–antidiquark currents of \(J^{PC} = 4^{+-}\), as depicted in Fig. 1a. 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 \(2.85^{+0.19}_{-0.22}\) GeV.
Besides, we shall also systematically construct all the meson–meson currents of \(J^{PC}=4^{+-}\), as depicted in Fig. 1b. We shall relate these currents and the diquark–antidiquark currents through the Fierz rearrangement. The obtained Fierz identity suggests that the lowest-lying state dominantly decays into the P-wave \(\phi (1020) f_2^\prime (1525)\) channel. Accordingly, we propose to search for it in the \(X \rightarrow \phi (1020) f_2^\prime (1525) \rightarrow \phi K \bar{K}\) decay process. With a large amount of \(J/\psi \) sample, the BESIII collaboration are intensively studying the physics happening around here. Such experiments can also be performed by Belle-II, COMPASS, GlueX, and PANDA, etc. Accordingly, this fully-strange tetraquark state of \(J^{PC}=4^{+-}\) is a purely exotic hadron to be potentially observed in future particle experiments.
This paper is organized as follows. In Sect. 2 we systematically construct the local fully-strange tetraquark currents with the exotic quantum number \(J^{PC} = 4^{+-}\). We use them to perform QCD sum rule analyses in Sect. 3, where we calculate both their diagonal and off-diagonal two-point correlation functions. Based on the obtained results, we use the three single currents to perform numerical analyses in Sect. 4, while their mixing currents are investigated in Sect. 5. Section 6 is a summary.
2 Fully-strange tetraquark currents
As the first step, we construct the local fully-strange tetraquark currents with the exotic quantum number \(J^{PC} = 4^{+-}\). This quantum number can not be reached by simply using one quark and one antiquark, and moreover, we need two quarks and two antiquarks together with at least two derivatives to reach such a quantum number.
As depicted in Fig. 1, there are two possible configurations, the diquark–antidiquark configuration and the meson–meson configuration. When investigating the former configuration, the two covalent derivative operators \(D_\alpha (\equiv \partial _\alpha + i g_s A_\alpha )\) and \(D_\beta \) can be either inside the diquark/antidiquark field or between them:
where \(\big [ A {\overset{\leftrightarrow }{D}}_\alpha B \big ] \equiv A [D_\alpha B] - [D_\alpha A] B\), \(a \cdots d\) are color indices, and \(\varGamma _{1\cdots 8}\) are Dirac matrices. The internal orbital angular momenta contained in these currents are
After carefully examining all the possible combinations, we find that only the \(\eta \) currents can reach \(J^{PC} = 4^{+-}\), as depicted in Fig. 2, while the \(\eta ^{\prime }/\eta ^{\prime \prime }/\eta ^{\prime \prime \prime }\) currents can not. Altogether, we can construct three independent diquark–antidiquark currents of \(J^{PC} = 4^{+-}\):
The symbol \(\mathcal {S}\) denotes symmetrization and subtracting the trace terms in the set \(\{\alpha _1 \cdots \alpha _J\}\). Among these currents, \(\eta ^{1}_{\cdots }\) and \(\eta ^{3}_{\cdots }\) have the antisymmetric color structure \([ss]_{{\bar{3}}_C} [\bar{s} \bar{s}]_{{3}_C}\), and \(\eta ^{2}_{\cdots }\) has the symmetric color structure \([ss]_{{6}_C}[\bar{s} \bar{s}]_{ {\bar{6}}_C}\).
After similarly investigating the meson–meson configuration, we can also construct three independent meson–meson currents of \(J^{PC} = 4^{+-}\):
As depicted in Fig. 2, the internal orbital angular momenta contained in these currents are
After applying the Fierz rearrangement, we obtain
This Fierz identity will be used to study the decay behaviors later.
3 QCD sum rule analysis
We apply the QCD sum rule method [32, 33] to study the fully-strange tetraquark currents \(\eta ^{1,2,3}_{\alpha _1\alpha _2\alpha _3\alpha _4}\) with the exotic quantum number \(J^{PC} = 4^{+-}\). This non-perturbative method has been successfully applied to study various conventional and exotic hadrons in the past 50 years [34].
We generally assume that the current \(\eta ^{i}_{\alpha _1\alpha _2\alpha _3\alpha _4}\) (\(i=1\cdots 3\)) couples to the fully-strange tetraquark states \(X_n\) (\(n=1 \cdots N\)) through
where \(f_{in}\) is the \(3 \times N\) matrix for the decay constants, and \(\epsilon _{\alpha _1\alpha _2\alpha _3\alpha _4}\) is the traceless and symmetric polarization tensor satisfying
with \(\tilde{g}_{\mu \nu } = g_{\mu \nu } - q_\mu q_\nu / q^2\). The symbol \(\mathcal {S}^\prime \) denotes symmetrization and subtracting the trace terms in the sets \(\{\alpha _1\alpha _2\alpha _3\alpha _4\}\) and \(\{\beta _1\beta _2\beta _3\beta _4\}\).
Based on Eq. (7), we can investigate both the diagonal and off-diagonal correlation functions:
At the hadron level we express \(\varPi _{ij}(q^2)\) using the dispersion relation as
with \(s_< = 16 m_s^2\) the physical threshold. We parameterize the spectral density \(\rho ^\mathrm{phen}_{ij}(s)\) for the states \(X_n\) together with a continuum contribution as
with \(M_n\) the mass of \(X_n\).
At the quark–gluon level we calculate \(\varPi _{ij}(q^2)\) using the method of operator product expansion (OPE), and extract the OPE spectral density \(\rho _{ij}(s) \equiv \rho ^\mathrm{OPE}_{ij}(s)\) [35]. In the calculations we take into account the Feynman diagrams depicted in Fig. 3. We consider the perturbative term, the strange quark mass \(m_s\), the quark condensate \(\langle \bar{s} s \rangle \), the quark–gluon mixed condensate \(\langle g_s \bar{s} \sigma G s \rangle \), the gluon condensate \(\langle g_s^2 GG \rangle \), and their combinations. We calculate all the diagrams proportional to \(g_s^{N=0}\) and \(g_s^{N=1}\), where we find the \(D=6\) term \(\langle \bar{s} s \rangle ^2\) and the \(D=8\) term \(\langle \bar{s} s \rangle \langle g_s \bar{s} \sigma G s \rangle \) to be important. We partly calculate the diagrams proportional to \(g_s^{N\ge 2}\), whose contributions are found to be small. Especially, we have not taken into account the radiative corrections in our QCD sum rule calculations.
Then we perform the Borel transformation at both the hadron and quark–gluon levels. After approximating the continuum using \(\rho _{ij}(s)\) above the threshold value \(s_0\), we obtain the sum rule equation
We shall investigate it through two steps, the single-channel analysis and the multi-channel analysis, as follows.
4 Single-channel analysis
To perform the single-channel analysis, we neglect the off-diagonal correlation functions by setting \(\rho _{ij}(s)|_{i \ne j} = 0\) so that only \(\rho _{ii}(s) \ne 0\). This assumption means that the three currents \(\eta ^{1,2,3}_{\alpha _1\alpha _2\alpha _3\alpha _4}\) are “non-correlated”, and any two of them can not mainly couple to the same state X, otherwise,
Accordingly, we assume that there are three states \(X_{1,2,3}\) corresponding to the three currents \(\eta ^{1,2,3}_{\alpha _1\alpha _2\alpha _3\alpha _4}\) through
After parameterizing the spectral density \(\rho _{ii}(s)\) as one pole dominance for the state \(X_i\) together with a continuum contribution, Eq. (12) is simplified to be
which can be used to calculate \(M_i\) through
We use the spectral density \(\rho _{11}(s)\) extracted from the current \(\eta ^1_{\alpha _1\alpha _2\alpha _3\alpha _4}\) as an example to perform the numerical analysis. We take the following values for various QCD parameters [1, 36,37,38,39,40,41,42]:
As shown in Eq. (16), the mass \(M_1\) of the state \(X_1\) depends on two free parameters, the Borel mass \(M_B\) and the threshold value \(s_0\). We investigate three aspects to find their proper working regions: (a) the convergence of OPE, (b) the sufficient amount of the pole contribution, and (c) the mass dependence on these two parameters.
Firstly, we investigate the convergence of OPE, which is the cornerstone for a reliable QCD sum rule analysis. We require the \(D=12\) terms (CVG\(_{12}\)) to be less than 5%, the \(D=10\) terms (CVG\(_{10}\)) to be less than 10%, and the \(D=8\) terms (CVG\(_{8}\)) to be less than 20%:
As depicted in Fig. 4 using the dashed curves, the lower bound of the Borel mass is determined to be \(M_B^2 > 2.40\) GeV\(^2\).
Secondly, we investigate the one-pole-dominance assumption by requiring the pole contribution (PC) to be larger than 40%:
As depicted in Fig. 4 using the solid curve, the upper bound of the Borel mass is determined to be \(M_B^2 < 2.65\) GeV\(^2\) when setting \(s_0 = 16.0\) GeV\(^2\). Altogether the Borel window is determined to be 2.40 GeV\(^2< M_B^2 < 2.65\) GeV\(^2\) for \(s_0 = 16.0\) GeV\(^2\). Redoing the same procedures, we find that there are non-vanishing Borel windows for \(s_0 > s_0^\mathrm{min} = 14.6\) GeV\(^2\). Accordingly, we choose \(s_0\) to be slightly larger, and determine its working region to be 13.0 GeV\(^2< s_0 < 19.0\) GeV\(^2\).
Thirdly, we show the mass \(M_1\) in Fig. 5, and investigate its dependence on \(M_B\) and \(s_0\). It is stable against \(M_B\) inside the Borel window 2.40 GeV\(^2< M_B^2 < 2.65\) GeV\(^2\), and its dependence on \(s_0\) is moderate insider the working region 13.0 GeV\(^2< s_0 < 19.0\) GeV\(^2\), where the mass is calculated to be
Its uncertainty is due to \(s_0\) and \(M_B\) as well as various QCD parameters listed in Eq. (17).
We repeat the same procedures to study the other two currents \(\eta ^2_{\alpha _1\alpha _2\alpha _3\alpha _4}\) and \(\eta ^3_{\alpha _1\alpha _2\alpha _3\alpha _4}\). The obtained results are summarized in Table 1.
5 Multi-channel analysis
To perform the multi-channel analysis, we take into account the off-diagonal correlation functions, which are actually non-zero, i.e., \(\rho _{ij}(s)|_{i \ne j} \ne 0\). It is interesting to see how large they are, so we choose \(s_0 = 11.0\) GeV\(^2\) and \(M_B^2 = 1.85\) GeV\(^2\) to obtain
Hence, \(\eta ^{1}_{\cdots }\) and \(\eta ^{3}_{\cdots }\) are strongly correlated with each other, making the off-diagonal terms of \(\rho _{ij}(s)\) non-negligible, as depicted in Fig. 6 using the solid curve.
To diagonalize the \(3\times 3\) matrix \(\rho _{ij}(s)\), we construct three mixing currents \(J^{1,2,3}_{\alpha _1\alpha _2\alpha _3\alpha _4}\):
with \(\mathbb {T}_{3\times 3}\) the transition matrix.
We apply the method of operator product expansion to extract the spectral densities \(\rho ^\prime _{ij}(s)\) from the mixing currents \(J^{1,2,3}_{\alpha _1\alpha _2\alpha _3\alpha _4}\). After choosing
we obtain
at \(s_0 = 11.0\) GeV\(^2\) and \(M_B^2 = 1.85\) GeV\(^2\). Hence, the off-diagonal terms of \(\rho ^\prime _{ij}(s)\) are negligible around here, suggesting that the three mixing currents \(J^{1,2,3}_{\alpha _1\alpha _2\alpha _3\alpha _4}\) are nearly non-correlated around here, as depicted in Fig. 6 using the dashed curve. Moreover, Eq. (26) indicates that the QCD sum rule result from \(J^3_{\alpha _1\alpha _2\alpha _3\alpha _4}\) is non-physical around here due to its negative correlation function. Besides, Eq. (25) indicates that \(J^{2}_{\cdots }\) is almost the same as \(\eta ^{2}_{\cdots }\), while \(J^{1}_{\cdots }\) and \(J^{3}_{\cdots }\) are mainly from the recombination of \(\eta ^{1}_{\cdots }\) and \(\eta ^{3}_{\cdots }\).
We use the procedures previously applied on the diquark–antidiquark currents \(\eta ^{1,2,3}_{\alpha _1\alpha _2\alpha _3\alpha _4}\) to study their mixing currents \(J^{1,2,3}_{\alpha _1\alpha _2\alpha _3\alpha _4}\). The obtained results are also summarized in Table 1. Especially, the mass extracted from the current \(J^1_{\alpha _1\alpha _2\alpha _3\alpha _4}\) is significantly reduced to be
For completeness, we show it in Fig. 7 as a function of the threshold value \(s_0\) and the Borel mass \(M_B\).
6 Conclusion
In this paper we apply the method of QCD sum rules to study the fully-strange tetraquark states with the exotic quantum number \(J^{PC}=4^{+-}\). We work within the diquark–antidiquark picture and systematically construct their interpolating currents. We calculate both the diagonal and off-diagonal correlation functions. The obtained results are used to construct three mixing currents that are nearly non-correlated. We use the mixing current \(J^1_{\alpha _1\alpha _2\alpha _3\alpha _4}\) to evaluate the mass of the lowest-lying state to be \(2.85^{+0.19}_{-0.22}\) GeV.
In this paper we also systematically construct the fully-strange meson–meson currents of \(J^{PC}=4^{+-}\), and relate them to the diquark–antidiquark currents through the Fierz rearrangement. Especially, we can apply Eqs. (24) and (6) to transform the mixing current \(J^1_{\alpha _1\alpha _2\alpha _3\alpha _4}\) to be
This Fierz identity suggests that the lowest-lying state dominantly decays into the P-wave \(\phi (1020) f_2^\prime (1525)\) channel through the meson–meson current \(\xi ^1_{\alpha _1\alpha _2\alpha _3\alpha _4}\), given that the operator \(\bar{s}_b \gamma _{\alpha _2} s_b\) of \(I^GJ^{PC}=0^-1^{--}\) well couples to the vector meson \(\phi (1020)\) and the operator \(\mathcal {S}[\bar{s}_a \gamma _{\alpha _1} {\overset{\leftrightarrow }{D}}_{\alpha _3} s_a]\) of \(I^GJ^{PC}=0^+2^{++}\) well couples to the \(f_2^\prime (1525)\) meson. Accordingly, we propose to search for it in the \(X \rightarrow \phi (1020) f_2^\prime (1525) \rightarrow \phi K \bar{K}\) decay process in the future Belle-II, BESIII, COMPASS, GlueX, and PANDA experiments.
This is the first study on the exotic quantum number \(J^{PC}=4^{+-}\), and the above lowest-lying fully-strange tetraquark state of \(J^{PC}=4^{+-}\) is a purely exotic hadron to be potentially observed in future experiments. Its theoretical and experimental studies will continuously improve our understanding on the non-perturbative behaviors of the strong interaction in the low energy region.
Data Availability Statement
This manuscript has data included as electronic supplementary material. The online version of this article contains supplementary material, which is available to authorized users.
References
P.A. Zyla et al. [Particle Data Group], Review of Particle Physics. PTEP 2020, 083C01 (2020). https://doi.org/10.1093/ptep/ptaa104
H.X. Chen, A. Hosaka, S.L. Zhu, \(I^G J^{PC}=1^- 1^{-+}\) tetraquark states. Phys. Rev. D 78, 054017 (2008). https://doi.org/10.1103/PhysRevD.78.054017
H.X. Chen, A. Hosaka, S.L. Zhu, \(I^G J^{PC}=0^+ 1^{-+}\) tetraquark state. Phys. Rev. D 78, 117502 (2008). https://doi.org/10.1103/PhysRevD.78.117502
W. Zhu, Y.R. Liu, T. Yao, Is \(J^{PC}=3^{-+}\) molecule possible? Chin. Phys. C 39, 023101 (2015). https://doi.org/10.1088/1674-1137/39/2/023101
Z.R. Huang, W. Chen, T.G. Steele, Z.F. Zhang, H.Y. Jin, Investigation of the light four-quark states with exotic \(J^{PC}=0^{-}\). Phys. Rev. D 95, 076017 (2017). https://doi.org/10.1103/PhysRevD.95.076017
X. Zhang, J.J. Xie, Prediction of possible exotic states in the \(\eta \bar{K}K^*\) system. Chin. Phys. C 44, 054104 (2020). https://doi.org/10.1088/1674-1137/44/5/054104
X.K. Dong, Y.H. Lin, B.S. Zou, Interpretation of the \(\eta _1(1855)\) as a \(K\bar{K}_1(1400)+ c.c.\) molecule. Sci. China Phys. Mech. Astron. 65, 261011 (2022). https://doi.org/10.1007/s11433-022-1887-5
T. Ji, X.K. Dong, F.K. Guo, B.S. Zou, Prediction of a narrow exotic hadronic state with quantum numbers \(J^{PC} = 0^{-}\). Phys. Rev. Lett. 129(10), 102002 (2022). https://doi.org/10.1103/PhysRevLett.129.102002
C.J. Morningstar, M.J. Peardon, The Glueball spectrum from an anisotropic lattice study. Phys. Rev. D 60, 034509 (1999). https://doi.org/10.1103/PhysRevD.60.034509
Y. Chen et al., Glueball spectrum and matrix elements on anisotropic lattices. Phys. Rev. D 73, 014516 (2006). https://doi.org/10.1103/PhysRevD.73.014516
V. Mathieu, N. Kochelev, V. Vento, The physics of glueballs. Int. J. Mod. Phys. E 18, 1 (2009). https://doi.org/10.1142/S0218301309012124
H.B. Meyer, Glueball regge trajectories. arXiv:hep-lat/0508002
E. Gregory, A. Irving, B. Lucini, C. McNeile, A. Rago, C. Richards, E. Rinaldi, Towards the glueball spectrum from unquenched lattice QCD. JHEP 1210, 170 (2012). https://doi.org/10.1007/JHEP10(2012)170
A. Athenodorou, M. Teper, The glueball spectrum of SU(3) gauge theory in \(3+1\) dimensions. JHEP 11, 172 (2020). https://doi.org/10.1007/JHEP11(2020)172
C.F. Qiao, L. Tang, Finding the \(0^{-}\) glueball. Phys. Rev. Lett. 113, 221601 (2014). https://doi.org/10.1103/PhysRevLett.113.221601
A. Pimikov, H.J. Lee, N. Kochelev, P. Zhang, V. Khandramai, Exotic glueball \(0^{\pm -}\) states in QCD sum rules. Phys. Rev. D 96, 114024 (2017). https://doi.org/10.1103/PhysRevD.96.114024
J.M. Frere, S. Titard, A new look at exotic decays: \(\tilde{\rho }(1^{-+}, I = 1) \rightarrow \eta ^\prime \pi \) versus \(\rho \pi \). Phys. Lett. B 214, 463–466 (1988). https://doi.org/10.1016/0370-2693(88)91395-0
P.R. Page, E.S. Swanson, A.P. Szczepaniak, Hybrid meson decay phenomenology. Phys. Rev. D 59, 034016 (1999). https://doi.org/10.1103/PhysRevD.59.034016
C.W. Bernard et al., [MILC Collaboration], Exotic mesons in quenched lattice QCD. Phys. Rev. D 56, 7039–7051 (1997). https://doi.org/10.1103/PhysRevD.56.7039
J.J. Dudek, R.G. Edwards, M.J. Peardon, D.G. Richards, C.E. Thomas, Highly excited and exotic meson spectrum from dynamical lattice QCD. Phys. Rev. Lett. 103, 262001 (2009). https://doi.org/10.1103/PhysRevLett.103.262001
J.J. Dudek et al., [Hadron Spectrum Collaboration], Toward the excited isoscalar meson spectrum from lattice QCD. Phys. Rev. D 88, 094505 (2013). https://doi.org/10.1103/PhysRevD.88.094505
K.G. Chetyrkin, S. Narison, Light hybrid mesons in QCD. Phys. Lett. B 485, 145 (2000). https://doi.org/10.1016/S0370-2693(00)00621-3
H.X. Chen, Z.X. Cai, P.Z. Huang, S.L. Zhu, Decay properties of the \(1^{-+}\) hybrid state. Phys. Rev. D 83, 014006 (2011). https://doi.org/10.1103/PhysRevD.83.014006
Z.R. Huang, H.Y. Jin, T.G. Steele, Z.F. Zhang, Revisiting the b\(_1\pi \) and \(\rho \pi \) decay modes of the 1\(^{-+}\) light hybrid state with light-cone QCD sum rules. Phys. Rev. D 94, 054037 (2016). https://doi.org/10.1103/PhysRevD.94.054037
L. Qiu, Q. Zhao, Towards the establishment of the light \(J^{P(C)}=1^{-(+)}\) hybrid nonet. Chin. Phys. C 46, 051001 (2022). https://doi.org/10.1088/1674-1137/ac567e
X.Y. Wang, F.C. Zeng, X. Liu, Production of the \(\eta _1(1855)\) through kaon induced reactions under the assumptions that it is a molecular or a hybrid state. Phys. Rev. D 106(3), 036005 (2022). https://doi.org/10.1103/PhysRevD.106.036005
C.A. Meyer, E.S. Swanson, Hybrid mesons. Prog. Part. Nucl. Phys. 82, 21–58 (2015). https://doi.org/10.1016/j.ppnp.2015.03.001
D. Alde et al., [LAPP Collaboration], Evidence for a \(1^{-+}\) Exotic Meson. Phys. Lett. B 205, 397 (1988). https://doi.org/10.1016/0370-2693(88)91686-3
G.S. Adams et al., [E852 Collaboration], Observation of a new \(J^{PC} = 1^{-+}\) exotic state in the reaction \(\pi ^-p\rightarrow \pi ^+ \pi ^- \pi ^- p\) at \(18\) GeV/c. Phys. Rev. Lett. 81, 5760–5763 (1998). https://doi.org/10.1103/PhysRevLett.81.5760
J. Kuhn et al., [E852 Collaboration], Exotic meson production in the \(f_1(1285) \pi ^-\) system observed in the reaction \(\pi ^-p \rightarrow \eta \pi ^+ \pi ^- \pi ^- p\) at \(18\) GeV/c. Phys. Lett. B 595, 109–117 (2004). https://doi.org/10.1016/j.physletb.2004.05.032
M. Ablikim et al., [BESIII Collaboration], Observation of an isoscalar resonance with exotic \(J^{PC}=1^{-+}\) quantum numbers in \(J/\psi \rightarrow \gamma \eta \eta ^{\prime }\). arXiv:2202.00621 [hep-ex]
M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, QCD and resonance physics. Theoretical foundations. Nucl. Phys. B 147, 385 (1979). https://doi.org/10.1016/0550-3213(79)90022-1
L.J. Reinders, H. Rubinstein, S. Yazaki, Hadron properties from QCD sum rules. Phys. Rep. 127, 1 (1985). https://doi.org/10.1016/0370-1573(85)90065-1
M. Nielsen, F.S. Navarra, S.H. Lee, New charmonium states in QCD sum rules: a concise review. Phys. Rep. 497, 41–83 (2010). https://doi.org/10.1016/j.physrep.2010.07.005
All the spectral densities calculated in the present study are given in the supplementary file “OPE.nb”
K.C. Yang, W.Y.P. Hwang, E.M. Henley, L.S. Kisslinger, QCD sum rules and neutron proton mass difference. Phys. Rev. D 47, 3001–3012 (1993). https://doi.org/10.1103/PhysRevD.47.3001
S. Narison, QCD as a theory of hadrons (from partons to confinement). Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 17, 1 (2002)
V. Gimenez, V. Lubicz, F. Mescia, V. Porretti, J. Reyes, Operator product expansion and quark condensate from lattice QCD in coordinate space. Eur. Phys. J. C 41, 535–544 (2005). https://doi.org/10.1140/epjc/s2005-02250-9
M. Jamin, Flavor-symmetry breaking of the quark condensate and chiral corrections to the Gell–Mann–Oakes–Renner relation. Phys. Lett. B 538, 71–76 (2002). https://doi.org/10.1016/S0370-2693(02)01951-2
B.L. Ioffe, K.N. Zyablyuk, Gluon condensate in charmonium sum rules with three-loop corrections. Eur. Phys. J. C 27, 229–241 (2003). https://doi.org/10.1140/epjc/s2002-01099-8
A.A. Ovchinnikov, A.A. Pivovarov, QCD sum rule calculation of the quark gluon condensate. Sov. J. Nucl. Phys. 48, 721 (1988)
J.R. Ellis, E. Gardi, M. Karliner, M.A. Samuel, Renormalization scheme dependence of Pade summation in QCD. Phys. Rev. D 54, 6986–6996 (1996). https://doi.org/10.1103/PhysRevD.54.6986
Acknowledgements
We thank Wei Chen, Er-Liang Cui, and Hui-Min Yang for useful discussions. This project is supported by the National Natural Science Foundation of China under Grant no. 12075019, and the Fundamental Research Funds for the Central Universities.
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Dong, RR., Su, N. & Chen, HX. Highly excited and exotic fully-strange tetraquark states. Eur. Phys. J. C 82, 983 (2022). https://doi.org/10.1140/epjc/s10052-022-10955-0
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DOI: https://doi.org/10.1140/epjc/s10052-022-10955-0