Mass spectrum and strong decays of strangeonium in a constituent quark model

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Qi Li, Long-Cheng Gui, Ming-Sheng Liu, Qi-Fang Lü and Xian-Hui Zhong. Mass spectrum and strong decays of strangeonium in a constituent quark model[J]. Chinese Physics C, 2021, 45(2): 023116. doi: 10.1088/1674-1137/abcf22
Qi Li, Long-Cheng Gui, Ming-Sheng Liu, Qi-Fang Lü and Xian-Hui Zhong. Mass spectrum and strong decays of strangeonium in a constituent quark model[J]. Chinese Physics C, 2021, 45(2): 023116.  doi: 10.1088/1674-1137/abcf22 shu
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Mass spectrum and strong decays of strangeonium in a constituent quark model

    Corresponding author: Long-Cheng Gui, guilongcheng@hunnu.edu.cn
    Corresponding author: Xian-Hui Zhong, zhongxh@hunnu.edu.cn
  • 1. Department of Physics, Hunan Normal University, Changsha 410081, China
  • 2. Synergetic Innovation Center for Quantum Effects and Applications (SICQEA), Changsha 410081, China
  • 3. Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Changsha 410081, China

Abstract: In this work we calculate the mass spectrum of strangeonium up to the $ 3D $ multiplet within a nonrelativistic linear potential quark model. Furthermore, using the obtained wave functions, we also evaluate the strong decays of the strangeonium states with the $ ^3P_0 $ model. Based on our successful explanations of the well established states $ \phi(1020) $, $ \phi(1680) $, $ h_1(1415) $, $ f'_2(1525) $, and $ \phi_3(1850) $, we further discuss the possible assignments of strangeonium-like states from experiments by combining our theoretical results with observations. It is found that some resonances, such as $ f_2(2010) $ and $ f_2(2150) $, listed by the Particle Data Group, and $ X(2062) $ and $ X(2500) $, newly observed by BESIII, may be interpreted as strangeonium states. The possibility of $ \phi(2170) $ as a candidate for $ \phi(3S) $ or $ \phi(2D) $ cannot be excluded. We expect our results to provide useful references for looking for the missing $ s\bar{s} $ states in future experiments.

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    I.   INTRODUCTION
    • The strangeonium ($ s\bar{s} $) states, one of the types of quarkonium state predicted in the quark model, lie between the light $ q\bar{q} $ states and heavy charmonium ($ c\bar{c} $) states. The $ s\bar{s} $ states provide a bridge for systematically exploring quantum chromodynamics (QCD) for light and heavy quarks. Furthermore, the study of $ s\bar{s} $ states is associated with the related topic of non-$ q\bar{q} $ states (glueballs, hybrids, and tetraquarks etc.) with the same quantum numbers as conventional $ q\bar{q} $ systems [1]. To confirm a non-$ q\bar{q} $ state from experiments, one needs good knowledge of the conventional $ q\bar{q} $ states. However, at present data for the $ s\bar{s} $ spectrum are rather scarce [1, 2]. There are only a few experimentally well established resonances, $ \phi(1020) $, $ \phi(1680) $, $ h_1(1415) $, $ f_2'(1525) $ and $ \phi_{3}(1850) $, which are widely accepted as $ s\bar{s} $ states. Besides some low-lying $ 1P $- and $ 1D $-wave states, many $ s\bar{s} $ states predicted in the quark model have yet to be established. For a long time, information on the $ s\bar{s} $ states was mainly extracted from the $ \gamma p $, $ K^-p $, $ \pi^-p $, and $ e^+e^- $ reactions. The lack of data may be due to the fact that these experiments do not efficiently produce $ s\bar{s} $ states.

      The BESIII experiment provides a powerful platform for the study of $ s\bar{s} $ states, with the world's largest $ J/\psi $ and $ \psi(2S) $ samples, which are well suited to study the $ s\bar{s} $ spectrum via their decays [2-4]. Recently, the BESIII Collaboration not only confirmed many $ s\bar{s} $ candidates observed in previous experiments, but also found some new $ s\bar{s} $ candidates by the decays of $ J/\psi $ and $ \psi(2S) $. For example, in 2019, evidence of a new $ 1^{+} $ resonance $ X(2060) $ with a mass of $ M = (2062.8\pm13.1\pm7.2) $ MeV [or $ 1^{-} $ resonance $ X(2000) $ with $ M = (2002.1\pm27.5\pm15.0) $ MeV] was observed in $ J/\psi\to \phi \eta \eta' $ at BESIII [5]. This resonance may be a candidate for the $ 2^1P_1 $ (or $ 3^3S_1 $ [6]) $ s\bar{s} $ state. In 2018, by an amplitude analysis of the process $ J/\psi\to \gamma K_SK_S $, several isoscalar $ 0^{++} $ and $ 2^{++} $ states around $ 1.3-2.5 $ GeV were extracted with a high significance by the BEIII Collaboration; the broad $ 0^{++} $ state, with a mass of $ M = (2411\pm17) $ MeV, and the broad $ 2^{++} $state, with a mass of $ M = (2233\pm 34 $$ ^{+9}_{-25}) $ MeV, might be candidates for the $ 3^3P_0 $ and $ 1^3F_2 $ $ s\bar{s} $ states, respectively [7]. In 2016, several isoscalar $ 0^{-+} $, $ 0^{++} $ and $ 2^{++} $ states around $ 2.0-2.4 $ GeV were observed in $ J/\psi\to \gamma \phi\phi $ at BESIII [8]. The $ f_2(2010) $ confirmed in this process might be a candidate for the $ 2^3P_2 $ $ s\bar{s} $ state, while the newly observed resonance $ X(2500) $ might be a candidate for a higher $ 0^{-+} $ $ s\bar{s} $ state [9]. The world's most precise resonance parameters for $ h_1(1415) $ were also determined by a recent measurement of $ J/\psi \to \eta' K\bar{K}\pi $ at BESIII [10]. Recently, the vector meson resonance $ \phi(2170) $ (often denoted $ Y (2175) $ in the literature) was also confirmed in the $ K^+(1460)K^- $, $ K^+_1(1400)K^- $, $ K^+_1(1270)K^- $, and $ \phi\eta' $ final states by the BESIII Collaboration [11, 12]. This state might be a candidate for the $ 3^3S_1 $ or $ 2^3D_1 $ $ s\bar{s} $ state [6, 13-16]. It should be mentioned that some forthcoming experiments from other collaborations, including COMPASS, BelleII, GlueX, and PANDA, will also provide more opportunities to study the $ s\bar{s} $ states.

      Theoretically, the $ s\bar{s} $ mass spectrum has been widely discussed within various quark models, including the relativized quark model [17, 18], the nonrelativistic covariant oscillator quark model [19], the QCD-motivated relativistic quark model [20], the nonrelativistic constituent quark model constrained in the study of the $ NN $ phenomenology and the baryon spectrum [21], the nonrelativistic constituent quark potential model [22], the extended Nambu-Jona-Lasinio quark model [23, 24], the Regge trajectory approach [15, 25], the modified relativized quark model [6], the framework of the Bethe-Salpeter equation [26-28], and so on. Furthermore, the strong decay properties of strangeonia have been studied within the pseudoscalar emission model [17], the flux-tube breaking model [29, 30], the $ ^3P_0 $ model [16], the corrected $ ^3P_0 $ model [31], the relativistic quark model framework [26], and so on. However, a systematic study of both the $ s\bar{s} $ mass spectrum and their decays by combining recent experimental progress is not found in the literature. An early review of the status of the $ s\bar{s} $ spectrum can be found in Ref. [32].

      Stimulated by recent notable progress in experiments, we carry out a systematic study of both the mass spectrum and strong decay properties of the $ s\bar{s} $ system. First, we calculate the mass spectrum up to the mass region of $ 3D $-wave states within a nonrelativistic constituent quark potential model by partially adopting the model parameters determined by the $ \Omega $ spectrum [33]. As done in the literature, e.g. Refs. [34-36], the spin-dependent potentials are dealt with non-perturbatively so that the effects of the spin-dependent interactions on the wave-functions can be included. More importantly, with the widely used $ ^3P_0 $ model [37-39] we further analyze the Okubo-Zweig-Iizuka (OZI)-allowed two-body strong decays of the $ s\bar{s} $ states by using wave functions obtained from the potential model, which are crucial to identify the nature of the resonances observed in experiments. We obtain successful explanations of both the mass and strong decay properties for the well established states $ \phi(1020) $, $ \phi(1680) $, $ h_1(1415) $, $ f'_2(1525) $, and $ \phi_3(1850) $. We find that (i) the $ f_2(2010) $ and $ f_2(2150) $ listed by the Particle Data Group (PDG) [1] might be candidates for the $ 2^3P_2 $ and $ 1^3F_2 $ $ s\bar{s} $ states, respectively; (ii) the $ 4^{++} $ resonance $ f_4(2210) $ first observed in the reaction $ K^-p\to K^+K^- \Lambda $ by the LASS Collaboration [40] might be an assignment of the $ 1^3F_4 $ $ s\bar{s} $ state; (iii) the $ f_0(2410) $ observed in $ J/\psi\to K_SK_S $ at BESIII [7] may favor the assignment of the $ 3^3P_0 $ $ s\bar{s} $ state; (iv) the newly observed resonances $ X(2500) $ [8] and $ X(2062) $ [5] from BESIII may be identified as the $ 4^1S_0 $ and $ 2^1P_1 $ $ s\bar{s} $ states, respectively; and (v) the possibility of $ \phi(2170) $ as a candidate for $ \phi(3S) $ or $ \phi(2D) $cannot be excluded, as the strong decay properties are very sensitive to its mass.

      This paper is organized as follows. In Sec. II, the mass spectrum is calculated within a nonrelativistic linear potential model. Then, by using the obtained spectrum the OZI-allowed two-body strong decays of the $ s\bar{s} $ states are estimated in Sec. III within the $ ^3P_0 $ model. In Sec. IV, we discuss the properties of the $ s\bar{s} $ states by combining our predictions with the experimental observations or other model predictions. Finally, a summary is given in Sec. V.

    II.   MASS SPECTRUM
    • To calculate the $ s\bar{s} $ mass spectrum, we adopt a nonrelativistic potential model [34-36, 41, 42]. In this model, the effective quark-antiquark potential is written as the sum of the spin-independent term $ H_0(r) $ and spin-dependent term $ H_{\rm sd}(r) $:

      $ V(r) = H_0(r)+H_{\rm sd}(r), $

      (1)

      where

      $ H_0(r) = -\frac{4}{3}\frac{\alpha_s}{r}+br+C_{0} $

      (2)

      includes the standard color Coulomb interaction and the linear confinement. The spin-dependent part $ H_{\rm sd}(r) $ can be expressed as [43]:

      $ H_{\rm sd}(r) = H_{SS}+H_{T}+H_{LS}, $

      (3)

      where

      $ H_{SS} = \frac{32\pi\alpha_s}{9m_qm_{\bar{q}}}\tilde{\delta}_\sigma(r){\bf{S}}_{q}\cdot {\bf{S}}_{\bar{q}} $

      (4)

      is the spin-spin contact hyperfine potential. Here, we take $ \tilde{\delta}_\sigma(r) = (\sigma/\sqrt{\pi})^3 {\rm e}^{-\sigma^2r^2} $ , as suggested in Ref. [41]. The tensor potential $ H_T $ is:

      $ H_{T} = \frac{4}{3}\frac{\alpha_s}{m_qm_{\bar{q}}}\frac{1}{r^3}\left(\frac{3{{S}}_{q}\cdot {{r}}{{S}}_{\bar{q}}\cdot {{r}}}{r^2}-{{S}}_{q}\cdot{{S}}_{\bar{q}}\right). $

      (5)

      For convenience in the calculations, the potential of the spin-orbit interaction $ H_{LS} $ is decomposed into a symmetric part $ H_{\rm sym} $ and antisymmetric part $ H_{\rm anti} $,

      $ H_{LS} = H_{\rm sym}+H_{\rm anti}, $

      (6)

      with

      $ H_{\rm sym} = \frac{{ S}_{+}\cdot {{L}}}{2}\left[\left(\frac{1}{2m_{\bar{q}}^{2}}+\frac{1}{2m_{q}^{2}}\right)\left(\frac{4}{3}\frac{\alpha_{s}}{r^{3}}-\frac{b}{r}\right) +\frac{8\alpha_{s}}{3m_{q}m_{\bar{q}}r^{3}}\right], $

      (7)

      $ H_{\rm anti} = \frac{{ S}_{-}\cdot {{L}}}{2}\left(\frac{1}{2m_{q}^{2}}-\frac{1}{2m_{\bar{q}}^{2}}\right)\left(\frac{4}{3}\frac{\alpha_{s}}{r^{3}}-\frac{b}{r}\right). $

      (8)

      In these equations, $ {{L}} $ is the relative orbital angular momentum of the $ q\bar{q} $ system; $ {{S}}_q $ and $ {{S}}_{\bar{q}} $ are the spins of the quark q and antiquark $ \bar{q} $, respectively, and $ {{S}}_{\pm}\equiv{{S}}_q\pm {{S}}_{\bar{q}} $; $ m_q $ and $ m_{\bar{q}} $ are the masses of the quark q and antiquark $ \bar{q} $, respectively; $ \alpha_s $ is the running coupling constant of QCD; r is the distance between the quark q and antiquark $ \bar{q} $; and the constant $ C_0 $ is the zero point energy. The six parameters in the above equations ($ \alpha_s $, b, $ \sigma $, $ m_q $, $ m_{\bar{q}} $ $ C_0 $) are determined by fitting the spectrum.

      Recently, the nonrelativistic potential model has been applied to study the $ \Omega $ baryon spectrum [33]. In order to be consistent with the $ \Omega $ spectrum, we set the parameters $ \alpha_s $, $ \sigma $ and $ m_s $ to the same values as those determinations in Ref. [33]. The studies in Refs. [17, 44] show that the parameters b and $ C_0 $ for a $ q\bar{q} $ system might be slightly different from the $ qqq $ system, so in the present work we reasonably adjust b and $ C_0 $ to better describe the masses of $ s\bar{s} $ states $ \phi(1020) $, $ \phi(1680) $ and $ \phi_{3}(1850) $. Our parameters are listed in Table 1, where they are compared to those of the $ \Omega $ baryon spectrum.

      This work Ref. [33]
      $ m_s $/GeV 0.600 Same
      $ \alpha_s $ 0.770 Same
      $ \sigma $/GeV 0.600 Same
      b/GeV$ ^2 $ 0.135 0.110
      $ C_0 $/GeV $ -0.519 $ $ -0.694 $

      Table 1.  Parameters of the nonrelativistic potential model.

      We solve the Schrödinger equation by using the three-point difference central method [45] from the centre ($ r = 0 $) to the outside ($ r\to \infty $), point by point. This method has been successfully applied to the $ b\bar{b} $, $ \bar{b}c $ and $ c\bar{c} $ systems [34-36]. To overcome the singular behavior of $ 1/r^3 $ in the spin-dependent potentials, we introduce a cutoff distance $ r_{c} $ in the calculation. In a small range $ r\in(0,r_{c}) $, we let $ 1/r^3 = 1/r_c^3 $. With this treatment, one can deal with spin-dependent potentials nonperturbatively, so that the effects of the spin-dependent potentials on the wave-functions can be included. Considering the fact that the mass of $ 1^3D_1 $ is sensitive to the cutoff distance $ r_c $, the mass of $ 1^3D_1 $ is used to determine the value of $ r_c $. It should be pointed out that the $ 1^3D_1 $ state has still not been established experimentally. Thus, we adopt a theoretical mass of $ 1^3D_1 $ predicted with a perturbation method, i.e., we let $ H = H_{0}+H' $, where $ H' $ is a part which contains the $ 1/r^{3} $ term. With the perturbation method one can obtain a fairly accurate mass, although one cannot include the effects of the spin-dependent interactions on the wave-functions. By solving the equation $ H_{0}|\psi_{n}^{(0)}\rangle = E_{0}|\psi_{n}^{(0)}\rangle $, we get the energy $ E_{0} $ and wave function $ |\psi_{n}^{(0)}\rangle $. Then, the mass of $ 1^3D_1 $, 1809 MeV, is worked out with $ M = 2m_s+E_{0}+ \langle\psi_{n}^{(0)}|H'|\psi_{n}^{(0)}\rangle $. Finally, with this predicted mass the cutoff distance $ r_c $ is determined to be $ 0.546 $ fm.

      Our predicted $ s\bar{s} $ mass spectrum is shown in Fig. 1. For comparison, our results together with some other model predictions and measurements are also listed in Table 2. From the table, one can see that our predictions with the nonrelativistic potential model are in reasonable agreement with the predictions of the relativized quark model [6, 17, 18], relativistic quark models [20], and nonrelativistic quark models [19, 21], although some model dependencies exist in the predictions for the higher excitations with $ n\geqslant 3 $. To understand why acceptable results can be provided by relativistic as well as by nonrelativistic approaches, there have been some studies of the connections between relativistic, semirelativistic, and nonrelativistic potential models of quarkonium using an interaction composed of an attractive Coulomb potential and a confining power-law term [46].

      Figure 1.  (color online) The strangeonium spectrum predicted with a nonrelativistic linear potential quark model. The shaded areas correspond to the experimental masses and their uncertainties, which are taken from the Particle Data Group [1] and BESIII Collaboration [5, 8].

      $n^{2S+1}L_{J}$ $J^{PC}$ Ours XWZZ [18] EFG [20] SIKY [19] GI [17] Pang [6] VFV [21] Observed state [1]
      $1 ^3S_{1}$ $1^{–}$ 1017 1009 1038 1020 1020 1030 1020 $\phi(1020)$
      $1 ^1S_{0}$ $0^{-+}$ 797 657 743 690 960 $\cdots$ 956 $\cdots$
      $2 ^3S_{1}$ $1^{–}$ 1699 1688 1698 1740 1690 1687 1726 $\phi(1680)$
      $2 ^1S_{0}$ $0^{-+}$ 1619 1578 1536 1440 1630 $\cdots$ 1795 $\cdots$
      $3 ^3S_{1}$ $1^{–}$ 2198 2204 2119 2250 $\cdots$ 2149 $\cdots$ $\cdots$
      $3 ^1S_{0}$ $0^{-+}$ 2144 2125 2085 1970 $\cdots$ $\cdots$ $\cdots$ $\cdots$
      $4 ^3S_{1}$ $1^{–}$ 2623 2627 2472 2540 $\cdots$ 2498 $\cdots$ $\cdots$
      $4 ^1S_{0}$ $0^{-+}$ 2580 2568 2439 2260 $\cdots$ $\cdots$ $\cdots$ $X(2500)$ [8]
      $1 ^3P_{2}$ $2^{++}$ 1513 1539 1529 1480 1530 $\cdots$ 1556 $f_2'(1525)$
      $1 ^3P_{1}$ $1^{++}$ 1492 1480 1464 1430 1480 $\cdots$ 1508 $f_1(1420)?$
      $1 ^3P_{0}$ $0^{++}$ 1373 1355 1420 1180 1360 $\cdots$ $\cdots$ $f_0(1370)$
      $1 ^1P_{1}$ $1^{+-}$ 1462 1473 1485 1460 1470 $\cdots$ 1511 $h_1(1415)$
      $2 ^3P_{2}$ $2^{++}$ 2030 2046 2030 2080 2040 $\cdots$ 1999 $f_2(2010)$
      $2 ^3P_{1}$ $1^{++}$ 2027 2027 2016 2020 2030 $\cdots$ $\cdots$ $\cdots$
      $2 ^3P_{0}$ $0^{++}$ 1971 1986 1969 1800 1990 $\cdots$ $\cdots$ $\cdots$
      $2 ^1P_{1}$ $1^{+-}$ 1991 2008 2024 2040 2010 $\cdots$ 1973 $X(2062)$ [5]
      $3 ^3P_{2}$ $2^{++}$ 2466 2480 2412 2540 $\cdots$ $\cdots$ $\cdots$ $\cdots$
      $3 ^3P_{1}$ $1^{++}$ 2470 2468 2403 2480 $\cdots$ $\cdots$ $\cdots$ $\cdots$
      $3 ^3P_{0}$ $0^{++}$ 2434 2444 2364 2280 $\cdots$ $\cdots$ $\cdots$ $f_0(2410)$ [7]
      $3 ^1P_{1}$ $1^{+-}$ 2435 2449 2398 2490 $\cdots$ $\cdots$ $\cdots$ $\cdots$
      $1 ^3D_{3}$ $3^{–}$ 1822 1897 1950 1830 1900 $\cdots$ 1875 $\phi_3(1850)$
      $1 ^3D_{2}$ $2^{–}$ 1840 1904 1908 1810 1910 $\cdots$ $\cdots$ $\cdots$
      $1 ^3D_{1}$ $1^{–}$ 1809 1883 1845 1750 1880 1869 $\cdots$ $\cdots$
      $1 ^1D_{2}$ $2^{-+}$ 1825 1893 1909 1830 1890 $\cdots$ 1853 $\cdots$
      $2 ^3D_{3}$ $3^{–}$ 2285 2337 2338 2360 $\cdots$ $\cdots$ $\cdots$ $\cdots$
      $2 ^3D_{2}$ $2^{–}$ 2297 2348 2323 2330 $\cdots$ $\cdots$ $\cdots$ $\cdots$
      $2 ^3D_{1}$ $1^{–}$ 2272 2342 2258 2260 $\cdots$ 2276 $\cdots$ $\cdots$
      $2 ^1D_{2}$ $2^{-+}$ 2282 2336 2321 2340 $\cdots$ $\cdots$ $\cdots$ $\cdots$
      $3 ^3D_{3}$ $3^{–}$ 2691 2725 2727 $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$
      $3 ^3D_{2}$ $2^{–}$ 2701 2734 2667 $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$
      $3 ^3D_{1}$ $1^{–}$ 2681 2732 2607 $\cdots$ $\cdots$ 2593 $\cdots$ $\cdots$
      $3 ^1D_{2}$ $2^{-+}$ 2685 2723 2662 $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$
      $1 ^3F_{4}$ $4^{++}$ 2078 2202 2286 2130 2200 $\cdots$ $\cdots$ $f_4(2210)$ [40]
      $1 ^3F_{3}$ $3^{++}$ 2128 2234 2215 2120 2230 $\cdots$ $\cdots$ $\cdots$
      $1 ^3F_{2}$ $2^{++}$ 2146 2243 2143 2090 2240 $\cdots$ $\cdots$ $f_2(2150)$
      $1 ^1F_{3}$ $3^{+-}$ 2111 2223 2209 2130 2220 $\cdots$ $\cdots$ $\cdots$
      $2 ^3F_{4}$ $4^{++}$ 2503 2596 2657 $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$
      $2 ^3F_{3}$ $3^{++}$ 2543 2623 2585 $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$
      $2 ^3F_{2}$ $2^{++}$ 2552 2636 2514 $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$
      $2 ^1F_{3}$ $3^{+-}$ 2528 2613 2577 $\cdots$ $\cdots$ $\cdots$ $\cdots$ $\cdots$

      Table 2.  Predicted masses of the $s\bar{s}$ states in this work compared with other predictions and observations.

    III.   STRONG DECAYS
    • In this work the Okubo-Zweig-Iizuka (OZI)-allowed two-body strong decays of the $ s\bar{s} $ states are calculated with the widely used $ ^{3}P_0 $ model [37-39]. In this model, one assumes that a quark-antiquark pair is produced from the vacuum with the quantum number $ 0^{++} $ and the initial meson decay takes place via the rearrangement of the four quarks as shown in Fig. 2. In the nonrelativistic limit, the transition operator is expressed as:

      Figure 2.  The meson two-body strong decay process $ A \to BC $ in the $ ^3P_0 $ model.

      $\begin{aligned}[b] \hat{T} =& -3 \gamma \sqrt{96 \pi} \sum_{m}^{} \langle 1 m 1 -m| 0 0 \rangle \int_{}^{} {\rm d}{{p}}_3 {\rm d}{{p}}_4 \delta^3 ({{p}}_3 + {{p}}_4) \\ & \times {\cal{Y}}_1^m \left(\frac{{{p}}_3 - {{p}}_4}{2}\right) \chi_{1-m}^{34} \phi_0^{34} \omega_0^{34} b_{3i}^\dagger ({{p}}_3) d_{4j}^\dagger ({{p}}_4) \ , \end{aligned}$

      (9)

      where $ \gamma $ is a dimensionless constant that denotes the strength of the quark-antiquark pair creation with momenta $ {{p}}_3 $ and $ {{p}}_4 $ from vacuum; $ b_{3i}^\dagger ({{p}}_3) $ and $ d_{4j}^\dagger({{p}}_4) $ are the creation operators for the quark and antiquark, respectively; the subscripts i and j are the SU(3)-color indices of the created quark and antiquark; $ \phi_0^{34} = (u\bar u + d\bar d +s \bar s)/\sqrt 3 $ and $ \omega_{0}^{34} = \dfrac{1}{\sqrt{3}} \delta_{ij} $ correspond to flavor and color singlets, respectively; $ \chi_{{1,-m}}^{34} $ is a spin triplet state; and $ {\cal{Y}}_{\ell m}({{k}})\equiv |{{k}}|^{\ell}Y_{\ell m}(\theta_{{{k}}},\phi_{{{k}}}) $ is the $ \ell $-th solid harmonic polynomial.

      For an OZI-allowed two-body strong decay process $ A\to B+C $, the helicity amplitude $ {\cal{M}}^{M_{J_A}M_{J_B} M_{J_C}}({{P}}) $ can be worked out by:

      $ \langle BC|T| A\rangle = \delta({{P}}_A-{{P}}_B-{{P}}_C){\cal{M}}^{M_{J_A}M_{J_B} M_{J_C}}({{P}}). $

      (10)

      In the center-of-mass (c.m.) frame of the initial meson A, the helicity amplitude can be written as:

      $ \begin{aligned}[b] {\cal{M}}^{M_{J_A}M_{J_B} M_{J_C}}({{P}}) =& \gamma \sqrt{96 \pi} \sum_{\begin{subarray}{l} M_{L_A},M_{S_A}, M_{L_B},M_{S_B}, M_{L_C},M_{S_C},m \end{subarray}} \langle L_A M_{L_A}; S_A M_{S_A} | J_A M_{J_A} \rangle \times \langle 1\;m;1\;-m|\;0\;0 \rangle\langle L_B M_{L_B}; S_B M_{S_B} | J_B M_{J_B} \rangle \\& \times\langle L_C M_{L_C}; S_C M_{S_C} | J_C M_{J_C} \rangle\times \langle \chi^{1 3}_{S_B M_{S_B}}\chi^{2 4}_{S_CM_{S_C}} | \chi^{1 2}_{S_A M_{S_A}} \chi^{3 4}_{1 -\!m} \rangle\\ & \times[\langle\phi^{13}_B \phi^{2 4}_C | \phi^{1 2}_A \phi^{3 4}_0\rangle I^{M_{L_A},m}_{M_{L_B},M_{L_C}}({{P}}) +(-1)^{S_A+S_B+S_C+1}\langle\phi^{2 4}_B \phi^{1 3}_C | \phi^{1 2}_A \phi^{3 4}_0\rangle I^{M_{L_A},m}_{M_{L_B},M_{L_C}}({-P)}], \end{aligned}$

      (11)

      with the integral in momentum space:

      $ \begin{aligned}[b] I^{M_{L_A},m}_{M_{L_B},M_{L_C}}({{P}}) =& \int {\rm d}^3{{p}}_3 \Psi^*_{n_B L_B M_{L_B}}\left(\frac{m_3 {{P}}}{m_1+m_3}-{{p}}_3 \right){\cal{Y}}_{1m}({{p}}_3)\\& \times\Psi^*_{n_C L_C M_{L_C}}\left(\frac{-m_3 {{P}}}{m_2+m_3}+{{p}}_3 \right)\Psi_{n_A L_A M_{L_A}}\left({{P}}-{{p}}_3 \right). \end{aligned}$

      (12)

      In the above equations, ($ J_{A} $, $ J_{B} $ and $ J_{C} $), ($ L_A $, $ L_B $ and $ L_C $) and ($ S_A $, $ S_B $ and $ S_C $) are the quantum numbers of the total angular momentum, orbital angular momentum and total spin for hadrons $ A,\,B,\,C $, respectively; in the c.m. frame of hadron A, the momenta $ {{P}}_B $ and $ {{P}}_C $ of mesons B and C satisfy $ {{P}}_B = -{{P}}_C\equiv {{P}} $; $ m_1 $ and $ m_2 $ are the constituent quark masses of the initial hadron A; $ m_3 $ is the mass of the anti-quark created from vacuum; $ \Psi_{n_A L_A M_{L_A}} $, $ \Psi_{n_B L_B M_{L_B}} $ and $ \Psi_{n_C L_C M_{L_C}} $ are the radial wave functions of hadrons A, B and C, respectively, in momentum space, while $ \phi^{12}_A $, $ \phi^{13}_B $ and $ \phi^{24}_C $ ($ \chi^{1 2}_{S_A M_{S_A}} $, $ \chi^{1 3}_{S_B M_{S_B}} $ and $ \chi^{2 4}_{S_C M_{S_C}} $) are the flavor (spin) wave functions of hadrons A, B and C, respectively; $ \langle \phi^{1 3}_B \phi^{2 4}_C | \phi^{1 2}_A \phi^{3 4}_{00}\rangle $ and $ \langle \chi^{1 3}_{S_B M_{S_B}} \chi^{2 4}_{S_C M_{S_C}}| \chi^{1 2}_{S_A M_{S_A}} \chi^{3 4}_{1 -\!m} \rangle $ are the flavor and spin matrix elements, respectively; and$ \langle L_A M_{L_A}; S_A M_{S_A} | J_A M_{J_A} \rangle $ and $ \langle L_B M_{L_B}; S_B M_{S_B} | J_B M_{J_B} \rangle $, $ \langle L_C M_{L_C}; S_C M_{S_C} | J_C M_{J_C} \rangle $ and $ \langle 1\;m;1\;-m|\;0\;0 \rangle $ are the corresponding Clebsch-Gordan coefficients.

      With the Jacob-Wick formula [47], the helicity amplitudes $ {\cal{M}}^{M_{J_A}M_{J_B} M_{J_C}}({{P}}) $ can be converted to the partial wave amplitudes $ {\cal{M}}^{JL} $ via

      $\begin{aligned}[b] {{\cal{M}}}^{J L}(A\rightarrow BC) =& \frac{\sqrt{4\pi (2 L+1)}}{2 J_A+1} \sum_{M_{J_B},M_{J_C}} \langle L 0 J M_{J_A}|J_A M_{J_A}\rangle\\& \times \langle J_B M_{J_B} J_C M_{J_C} | J M_{J_A} \rangle {\cal{M}}^{M_{J_A} M_{J_B} M_{J_C}}({{P}}), \end{aligned}$

      (13)

      where $ M_{J_A} = M_{J_B}+M_{J_C} $, $ {{J}}\equiv {{J}}_B+{{J}}_C $ and $ {{J}}_{A} \equiv {{J}}_{B}+ {{J}}_C+{{L}} $. More details of the $ ^3P_0 $ model can be found in our recent paper [48].

      To partly remedy the inadequacy of the nonrelativistic wave function as the momentum $ {{P}} $ increases, the partial width of the $ A\to B+C $ process is calculated with a semirelativistic phase space [29, 30]:

      $ \Gamma = 2\pi |{{P}}| \frac{M_B M_C}{M_A}\sum_{JL}\Big |{\cal{M}}^{J L}\Big|^2, $

      (14)

      where $ M_A $ is the mass of the initial hadron A, while $ M_B $ and $ M_C $ stand for the masses of final hadrons B and C, respectiIn our calculation, the masses of the final hadrons B and C appearing in the phase space are given "mock" values as suggested in Ref. [29]. These are worked out by:

      $ \begin{aligned}[b] \tilde{M}(fnL) = &\frac{1}{4}M(fnL;S = 0,J = L)\\ & +{\sum\limits_{m = -1,0,1}\frac{2(L+m)+1}{4(2L+1)} M(fnL;S = 1,J = L+m)}, \end{aligned} $

      (15)

      where $ \tilde{M} $, M, n, L, S, m stand for the mock mass, mass, principal quantum number, orbital quantum number, spin quantum number, and the third component of spin momentum, respectively. The masses and mock masses of final meson states are shown in Table 3. The masses of the initial hadrons, where known experimentally, are taken from the PDG [1]; otherwise their masses are taken from our potential model predictions as listed in Table 2.

      Meson $n^{2S+1}L_J$ $J^{P(C)}$ Mass/MeV $\tilde{M}$/MeV Flavor function
      $K$ $1 ^1S_{0}$ $0^{-}$ 494 793 $K^+ = u\bar{s},\,K^- = \bar{u}s,\,K^0 = d\bar{s},\,\bar{K^0} = \bar{d}s$
      $K^*$ $1 ^3S_{1}$ $1^{-}$ 896 793 $K^{*+} = u\bar{s},\,K^{*-} = \bar{u}s,\,K^{*0} = d\bar{s},\,\bar{K^{*0}} = \bar{d}s$
      $K(1460)$ $2 ^1S_{0}$ $0^{-}$ 1460 1580 $K^+ = u\bar{s},\,K^- = \bar{u}s,\,K^0 = d\bar{s},\,\bar{K^0} = \bar{d}s$
      $K^*(1410)$ $2 ^3S_{1}$ $1^{-}$ 1580 1580 $K^{*+} = u\bar{s},\,K^{*-} = \bar{u}s,\,K^{*0} = d\bar{s},\,\bar{K^{*0}} = \bar{d}s$
      $K^*_0(1430)$ $1 ^3P_{0}$ $0^{+}$ 1425 1381 $K^{*+}_0 = u\bar{s},\,K^{*-}_0 = \bar{u}s,\,K^{*0}_0 = d\bar{s},\,\bar{K^{*0}_0} = \bar{d}s$
      $K_1(1270)$ $\cos\theta_{1P}|1\;^{1}P_1\rangle +\sin\theta_{1P}|1\;^{3}P_1\rangle$ $1^{+}$ 1272 1381 $K^+_1 = u\bar{s},\,K^-_1 = \bar{u}s,\,K^0_1 = d\bar{s},\,\bar{K^0_1} = \bar{d}s$
      $K_1(1400)$ $ -\sin\theta_{1P}|1\;^{1}P_1\rangle +\cos\theta_{1P}|1\;^{3}P_1\rangle$ $1^{+}$ 1403 1381 $K^+_1 = u\bar{s},\,K^-_1 = \bar{u}s,\,K^0_1 = d\bar{s},\,\bar{K^0_1} = \bar{d}s$
      $\theta_{1P} = 45^\circ$ [1,49]
      $K^*_2(1430)$ $1 ^3P_{2}$ $2^{+}$ 1426 1381 $K^{*+}_2 = u\bar{s},\,K^{*-}_2 = \bar{u}s,\,K^{*0}_2 = d\bar{s},\,\bar{K^{*0}_2} = \bar{d}s$
      $K^*(1680)$ $1 ^3D_{1}$ $1^{-}$ 1718 1756 $K^{*+} = u\bar{s},\,K^{*-} = \bar{u}s,\,K^{*0} = d\bar{s},\,\bar{K^{*0}} = \bar{d}s$
      $K_2(1770)$ $\cos\theta_{1D}|1^{1}D_2\rangle +\sin\theta_{1D}|1^{3}D_2\rangle$ $2^{-}$ 1773 1756 $K^+_2 = u\bar{s},\,K^-_2 = \bar{u}s,\,K^0_2 = d\bar{s},\,\bar{K^0_2} = \bar{d}s$
      $K_2(1820)$ $ -\sin\theta_{1D}|1^{1}D_2\rangle +\cos\theta_{1D}|1^{3}D_2\rangle$ $2^{-}$ 1819 1756 $K^+_2 = u\bar{s},\,K^-_2 = \bar{u}s,\,K^0_2 = d\bar{s},\,\bar{K^0_2} = \bar{d}s$
      $\theta_{1D} = 45^\circ$ [49]
      $K^*_3(1780)$ $1 ^3D_{3}$ $3^{-}$ 1776 1756 $K^{*+}_3 = u\bar{s},\,K^{*-}_3 = \bar{u}s,\,K^{*0}_3 = d\bar{s},\,\bar{K^{*0}_3} = \bar{d}s$
      $\eta$ $1 ^1S_{0}$ $0^{-+}$ 548 793 $\cos\theta_1(\dfrac{u\bar{u}+d\bar{d} }{\sqrt{2} })-\sin\theta_1({s\bar{s} }),\,\theta_1 = 39.3^\circ$
      $\eta'$ $1 ^1S_{0}$ $0^{-+}$ 958 793 $\sin\theta_1(\dfrac{u\bar{u}+d\bar{d} }{\sqrt{2} })+\cos\theta_1({s\bar{s} }),\,\theta_1 = 39.3^\circ$
      $\phi(1020)$ $1 ^3S_{1}$ $1^{–}$ 1020 964 $s\bar{s}$
      $f_0(1373)$ $1 ^3P_{0}$ $0^{++}$ 1373 1488 $s\bar{s}$
      $f_1(1492)$ $1 ^3P_{1}$ $1^{++}$ 1492 1488 $s\bar{s}$
      $h_1(1415)$ $1 ^1P_{1}$ $1^{+-}$ 1416 1488 $s\bar{s}$
      $f_2'(1525)$ $1 ^3P_{2}$ $2^{++}$ 1525 1488 $s\bar{s}$

      Table 3.  Masses, mock masses (denoted with $\tilde{M}$) and flavor wave functions for the final meson states. The meson masses for the well established states are taken from the PDG [1]; otherwise they are taken from our theoretical estimations. The mock masses, which correspond to the calculated mass of the meson in the spin-independent $q\bar{q}$s potential, are calculated with Eq. (15). The mixing angles $\theta_{1P} = 45^\circ$ and $\theta_{1D} = 45^\circ$ for the $1P$- and $1D$-wave kaon meson states are determined according to the decay properties [1, 49].

      In the calculations, the wave functions of the initial and final states are taken from our potential model predictions. To obtain the wave functions for the final kaon meson states, we fit the masses of the well established states K, $ K^* $, $ K(1460) $, $ K_2(1430) $, $ K^*(1680) $, and $ K_3(1780) $. The potential model parameters for the kaon meson spectrum are determined to be $ \alpha_s = 0.885 $, $ b = 0.1383 $ GeV$ ^2 $, $ m_s = 0.6 $ GeV, $ m_u = 0.45 $ GeV, $ \sigma = 0.669 $ GeV, $ r_c = 0.59 $ fm and $ C_0 = -0.524 $ GeV. The pair creation strength $ \gamma = 0.360 $ is obtained by fitting the width of $ \phi(1680) $ from the PDG [1]. The OZI-allowed two-body strong decay properties, such as the partial decay width, total decay width, branching fraction, and partial wave amplitude, are calculated for the $ s\bar{s} $ states listed in Table 2. Our results are listed in Tables 4-19.

      State Mode Ours Ref. [16] Ref. [6]
      $ \Gamma_{\rm th} $ [$ \Gamma_{\exp} $]/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $) $ \Gamma_{\rm th} $/MeV $ Br $(%) $ \Gamma_{\rm th} $/MeV $ Br $(%)
      $ 1 ^3S_{1}[1020] $ $ KK $ 4.09 [3.5] 100 $ ^1P_1 $ = −0.079 2.5 59 $ \cdots $ $ \cdots $
      $ 2 ^3S_{1} $$ [1680] $ $ KK $ 8.06 5 $ ^1P_1 $ = 0.062 89 23.54 15.5 10.33
      $ KK^*(892) $ 132 79 $ ^3P_1 $ = −0.215 245 64.81 117 78
      $ \phi\eta $ 26.3 16 $ ^3P_1 $ = −0.537 44 11.64 16.7 11.13
      Total 167 [$ 150\pm50 $] 100 378 100 150 100
      $ 3 ^3S_{1} $[2198]a $ KK $ 1.70 0.6 $ ^1P_1 $ = 0.027 0 0 11.9 5.29
      $ [2050^b/2188^c] $ $ KK^*(892) $ 29.9 10 $ ^3P_1 $ = −0.086 20 5.29 60 26.67
      $ KK(1460) $ 24.5 8 $ ^1P_1 $ = 0.075 29 7.67 $ \cdots $ $ \cdots $
      $ KK^*(1410) $ 107 36 $ ^3P_1 $ = −0.188 93 24.60 48.4 21.51
      $ K^*(892)K^*(892) $ 26.9 9 $ ^1P_1 $ = 0.029 102 26.98 22.7 10.09
      $ ^5P_1 $ = −0.130
      $ KK^*_2(1430) $ 51.6 17 $ ^5D_1 $ = 0.113 9 2.38 39.8 17.69
      $ \phi\eta $ 9.67 3 $ ^3P_1 $ = −0.231 21 5.56 6.66 2.96
      $ \phi\eta' $ 0.67 0.2 $ ^3P_1 $ = −0.062 11 2.91 0.0862 0.04
      $ KK_1(1270) $ 23.0 8 $ ^3S_1 $ = 0.002 58 15.34 31.4 13.96
      $ ^3D_1 $ = −0.067
      $ KK_1'(1400) $ 1.35 0.5 $ ^3S_1 $ = 0.013 26 6.88 4.36 1.94
      $ ^3D_1 $ = 0.012
      $ K^*(892)K_1(1270) $ 9.45 3 $ ^3S_1 $ = −0.054 $ \cdots $ $ \cdots $ $ \cdots $ $ \cdots $
      $ ^3D_1 $ = −0.029
      $ ^5D_1 $ = −0.050
      $ \eta h_1(1415) $ 9.50 3 $ ^3S_1 $ = 0.076 8 2.12 $ \cdots $ $ \cdots $
      $ ^3D_1 $ = 0.202
      Total 295 100 378 100 225 100
      aMass (MeV) adopted in present work.
      bMass (MeV) adopted in Ref. [16].
      cMass (MeV) adopted in Ref. [6].

      Table 4.  Strong decay properties for the $ 1S $-, $ 2S $-, $ 3S $-wave vector $ s\bar s $ states. $ \Gamma_{\rm th} $ and $ Br $ stand for the partial widths and branching fractions of the strong decay processes, respectively. The experimental widths $ \Gamma_{\rm exp} $ are taken from the PDG [1]. To know the contributions of different partial waves to a decay process, the partial wave amplitudes of every decay mode (denoted with Amps.) are also given in the table. For comparison, some other predictions with the $ \; ^{3}P_{0} $ model [6, 16] are also listed.

      Mode State $ \Gamma_{\rm th} $ [$ \Gamma_{\exp} $]/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $) State $ \Gamma_{\rm th} $ [$ \Gamma_{\exp} $]/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $)
      $ KK $ $ 1 ^3P_{0} $ 281 83 $ ^1S_0 $ = −0.392 $ 1 ^3P_{2} $ 40.3 70 $ ^1D_2 $ = 0.142
      $ KK^*(892) $ (1373) $ \cdots $ $ \cdots $ (1525) 12.2 21 $ ^3D_2 $ = −0.076
      $ \eta\eta $ 56.2 17 $ ^1S_0 $ = 0.332 5.45 9 $ ^1D_2 $ = −0.096
      $ \eta\eta' $ $ \cdots $ $ \cdots $ 0.01 0.02 $ ^1D_2 $ = −0.011
      Total 338 [200-500] 100 58 [$ 86\pm5 $] 100
      $ KK^*(892) $ $ 1 ^3P_{1} $ 381 100 $ ^3S_1 $ = −0.453 $ 1 ^1P_{1} $ 141 100 $ ^3S_1 $ = −0.361
      (1492) $ ^3D_1 $ = 0.042 (1423) $ ^3D_1 $ = −0.019
      Total 381 100 141 [$ 90\pm 15 $] 100
      $ KK $ $ 2 ^3P_{0} $ 29.2 3.4 $ ^1S_0 $ = −0.113 $ 2 ^3P_{2} $ 5.03 3 $ ^1D_2 $ = 0.047
      $ KK^*(892) $ (1971) $ \cdots $ $ \cdots $ (2030) 1.59 1 $ ^3D_2 $ = 0.021
      $ KK(1460) $ 297 35 $ ^1S_0 $ = −0.498 7.65 5 $ ^1D_2 $ = 0.055
      $ K^*(892)K^*(892) $ 71.2 8.4 $ ^1S_0 $ = 0.054 24.5 17 $ ^1D_2 $ = −0.045
      $ ^5D_0 $ = −0.249 $ ^5S_2 $ = −0.060
      $ ^5D_2 $ = 0.119
      $ KK^*_2(1430) $ $ \cdots $ $ \cdots $ 35.3 24 $ ^5P_2 $ = −0.113
      $ ^5F_2 $ = −0.025
      $ \eta\eta $ 4.53 0.5 $ ^1S_0 $ = 0.080 0.13 0.1 $ ^1D_2 $ = −0.014
      $ \eta\eta' $ 4.09 0.5 $ ^1S_0 $ = 0.101 0.16 0.1 $ ^1D_2 $ = 0.020
      $ \eta'\eta' $ 2.17 0.3 $ ^1S_0 $ = −0.070 0.07 0.1 $ ^1D_2 $ = 0.011
      $ KK_1(1270) $ 430 51 $ ^3P_0 $ = −0.339 26.1 18 $ ^3P_2 $ = 0.006
      $ ^3F_2 $ = 0.079
      $ KK_1(1400) $ 10.7 1.3 $ ^3P_0 $ = 0.069 43.4 29 $ ^3P_2 $ = −0.122
      $ ^3F_2 $ = −0.005
      $ \eta f_1(1420) $ $ \cdots $ $ \cdots $ 3.61 2 $ ^3P_2 $ = −0.185
      $ ^3F_2 $ = 0.015
      Total 849 100 147 [$ 202^{+67}_{-62} $] 100
      $ KK^*(892) $ $ 2 ^3P_{1} $ 39.5 13 $ ^3S_1 $ = −0.097 $ 2 ^1P_{1} $ 31.9 18 $ ^3S_1 $ = −0.067
      (2027) $ ^3D_1 $ = −0.034 (1991) $ ^3D_1 $ = 0.064
      $ K^*(892)K^*(892) $ 50.7 16 $ ^5D_1 $ = −0.203 30.4 17 $ ^3S_1 $ = 0.020
      $ ^3D_1 $ = 0.162
      $ KK^*_0(1430) $ 0.97 0.3 $ ^1P_1 $ = 0.019 15.3 9 $ ^1P_1 $ = −0.084
      $ KK^*_2(1430) $ 110 35 $ ^5P_1 $ = −0.204 82.7 46 $ ^5P_1 $ = −0.196
      $ ^5F_1 $ = 0.022 $ ^5F_1 $ = −0.017
      $ \phi\eta $ $ \cdots $ $ \cdots $ 10.5 6 $ ^3S_1 $ = 0.121
      $ ^3D_1 $ = −0.228
      $ \phi\eta' $ $ \cdots $ $ \cdots $ 2.75 2 $ ^3S_1 $ = −0.247
      $ ^3D_1 $ = −0.017
      $ KK_1(1270) $ 93.0 29 $ ^3P_1 $ = 0.150 3.33 2 $ ^3P_1 $ = −0.029
      $ KK_1(1400) $ 8.36 3 $ ^3P_1 $ = 0.054 1.90 1 $ ^3P_1 $ = −0.028
      $ \eta f_0(1370) $ 0.76 0.2 $ ^1P_1 $ = 0.073 $ \cdots $ $ \cdots $
      $ \eta f_1(1420) $ 12.4 4 $ ^3P_1 $ = 0.350 $ \cdots $ $ \cdots $
      Total 315 100 179 100

      Table 5.  Strong decay properties for the $ 1P $ and $ 2P $-wave $ s\bar s $ states.

      State Mode Ours Ref. [16] Ref. [6]
      $ \Gamma_{\rm th} $ [$ \Gamma_{\exp} $]/MeV $ Br $(%) Amps./(GeV$ ^{-1/2} $) $ \Gamma_{\rm th} $/MeV $ B_{r} $(%) $ \Gamma_{\rm th} $/MeV $ Br $(%)
      $ 1 ^3D_{1} $(1809)a $ KK $ 30.5 4 $ ^1P_1 $ = −0.117 65 9.97 40.8 7.46
      ($ 1850^{\rm b}/1869^{\rm c} $) $ KK^*(892) $ 42.0 6 $ ^3P_1 $ = −0.113 75 11.50 57.8 10.57
      $ K^*(892)K^*(892) $ 1.02 0.1 $ ^1P_1 $ = −0.048 5 0.77 11.5 2.10
      $ ^5P_1 $ = 0.022
      $ ^5F_1 $ = 0.003
      $ \phi\eta $ 13.2 2 $ ^3P_1 $ = −0.322 29 4.45 13.6 2.49
      $ KK_1(1270) $ 620 88 $ ^3S_1 $ = −0.588 478 73.31 423 77.33
      $ ^3D_1 $ = −0.021
      Total 707 100 652 100 547 100
      $ 1 ^3D_{2} $(1840)$ ^{\rm a} $ $ KK^*(892) $ 88.5 69 $ ^3P_2 $ = −0.145 151 70.56 $ \cdots $ $ \cdots $
      $ ^3F_2 $ = 0.073
      ($1850^{\rm b}$) $ K^*(892)K^*(892) $ 6.90 5 $ ^5P_2 $ = 0.107 7 3.27 $ \cdots $ $ \cdots $
      $ ^5F_2 $ = 0.013
      $ \phi\eta $ 27.2 21 $ ^3P_2 $ = −0.440 53 24.77 $ \cdots $ $ \cdots $
      $ ^3F_2 $ = 0.101
      $ KK_1(1270) $ 5.79 5 $ ^3D_2 $ = 0.050 2 0.93 $ \cdots $ $ \cdots $
      Total 128 100 214 100 $ \cdots $ $ \cdots $
      $ 1 ^3D_{3} $(1850)$ ^{\rm a} $ $ KK $ 20.2 23 $ ^1F_3 $ = 0.095 45 43.27 $ \cdots $ $ \cdots $
      ($1854^{\rm b}$) $ KK^*(892) $ 28.2 32 $ ^3F_3 $ = −0.091 24 23.08 $ \cdots $ $ \cdots $
      $ K^*(892)K^*(892) $ 35.7 41 $ ^1F_3 $ = 0.005 32 30.77 $ \cdots $ $ \cdots $
      $ ^5P_3 $ = −0.233
      $ ^5F_3 $ = −0.012
      $ \phi\eta $ 2.26 3 $ ^3F_3 $ = −0.129 3 2.88 $ \cdots $ $ \cdots $
      $ KK_1(1270) $ 0.60 1 $ ^3D_3 $ = 0.015 0 0 $ \cdots $ $ \cdots $
      $ ^3G_3 $ = 0.002
      Total 87 [$ 87^{+28}_{-23} $] 100 104 100 $ \cdots $ $ \cdots $
      $ 1 ^1D_{2} $(1825)$ ^{\rm a} $ $ KK^*(892) $ 72.5 89 $ ^3P_2 $ = −0.120 111.7 90.08 $ \cdots $ $ \cdots $
      ($1842^{\rm b}$) $ ^3F_2 $ = −0.086
      $ K^*(892)K^*(892) $ 8.36 10 $ ^3P_2 $ = −0.129 12.1 9.76 $ \cdots $ $ \cdots $
      $ ^3F_2 $ = −0.007
      $ KK_1(1270) $ 0.52 1 $ ^3D_2 $ = −0.016 0.2 0.16 $ \cdots $ $ \cdots $
      Total 81.4 100 124 100 $ \cdots $ $ \cdots $
      aMass (MeV) adopted in present work.
      bMass (MeV) adopted in Ref. [16].
      cMass (MeV) adopted in Ref. [6].

      Table 6.  Strong decay properties for the $ 1D $-wave $ s\bar s $ states. For comparison, some other predictions with the $ \; ^{3}P_{0} $ model [6, 16] are also listed.

      Mode State $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps./(GeV$ ^{-1/2} $) State $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps./(GeV$ ^{-1/2} $)
      $ KK $ $ 1 ^3F_{2} $ 5.40 1.8 $ ^1D_2 $ = 0.048 $ 1 ^3F_{4} $ 7.8 11 $ ^1G_4 $ = −0.058
      $ KK^*(892) $ (2146) 12.2 4 $ ^3D_2 $ = 0.056 (2078) 16 23 $ ^3G_4 $ = 0.064
      $ KK(1460) $ 9.37 3 $ ^1D_2 $ = −0.049 0.13 0.2 $ ^1G_4 $ = 0.006
      $ KK^*(1410) $ 0.57 0.2 $ ^3D_2 $ = −0.016 $<0.01 $ $<0.01 $ $ ^3G_4\sim $0
      $ K^*(892)K^*(892) $ 19.4 6.3 $ ^1D_2 $ = 0.054 33 47 $ ^1G_4 $ = −0.017
      $ ^5D_2 $ = −0.041 $ ^5D_4 $ = 0.152
      $ ^5G_2 $ = −0.094 $ ^5G_4 $ = 0.034
      $ KK^*_2(1430) $ 28.4 9.2 $ ^5P_2 $ = −0.081 3.87 5.5 $ ^5F_4 $ = −0.035
      $ ^5F_2 $ = −0.034 $ ^5H_4 $ = −0.002
      $ \eta\eta $ 1.10 0.4 $ ^1D_2 $ = −0.039 1.07 1.5 $ ^1G_4 $ = 0.039
      $ \eta\eta' $ 3.69 1.2 $ ^1D_2 $ = −0.092 0.66 0.9 $ ^1G_4 $ = 0.040
      $ \eta'\eta' $ 0.93 0.3 $ ^1D_2 $ = −0.033 0.01 0.01 $ ^1G_4 $ = 0.003
      $ \phi\phi $ 0.69 0.2 $ ^1D_2 $ = −0.031 0.36 0.5 $ ^1G_4\sim $0
      $ ^5D_2 $ = 0.024 $ ^5D_4 $ = −0.038
      $ ^5G_2 $ = 0.011 $ ^5G_4\sim $0
      $ KK_1(1270) $ 190 62 $ ^3P_2 $ = −0.192 2.48 3.5 $ ^3F_4 $ = 0.021
      $ ^3F_2 $ = −0.048 $ ^3H_4 $ = 0.011
      $ KK_1(1400) $ 5.97 1.9 $ ^3P_2 $ = 0.039 4.72 6.8 $ ^3F_4 $ = −0.037
      $ ^3F_2 $ = −0.001 $ ^3H_4 = -0.001 $
      $ \eta f_1(1420) $ 26.5 8.6 $ ^3P_2 $ = −0.385 0.09 0.1 $ ^3F_4 $ = −0.026
      $ ^3F_2 $ = −0.031 $ ^3H_4\sim $0
      $ \eta f_2'(1525) $ 3.89 1.3 $ ^5P_2 $ = 0.184 $<0.01 $ $<0.01 $ $ ^5F_4\sim $0
      $ ^5F_2 $ = 0.017 $ ^5H_4\sim $0
      Total 308 100 69.8 100

      $ KK^*(892) $ $ 1 ^3F_{3} $ 32.6 13 $ ^3D_3 $ = 0.066 $ 1 ^1F_{3} $ 32 18 $ ^3D_3 $ = 0.058
      (2128) $ ^3G_3 $ = −0.063 (2111) $ ^3G_3 $ = 0.070
      $ KK^*(1410) $ 0.53 0.2 $ ^3D_3 $ = −0.016 0.17 0.1 $ ^3D_3 $ = −0.010
      $ ^3G_3 $ = 0.001 $ ^3G_3 $ = −0.001
      $ K^*(892)K^*(892) $ 18.6 7.6 $ ^5D_3 $ = −0.091 21 12 $ ^3D_3 $ = 0.110
      $ ^5G_3 $ = −0.069 $ ^3G_3 $ = 0.057
      $ KK^*_0(1430) $ 1.24 0.5 $ ^1F_3 $ = 0.019 1.2 0.7 $ ^1F_3 $ = −0.019
      $ KK^*_2(1430) $ 137 56 $ ^5P_3 $ = −0.196 107 60 $ ^5P_3 $ = −0.171
      $ ^5F_3 $ = −0.003 $ ^5F_3 $ = −0.042
      $ ^5H_3 $ = 0.004 $ ^5H_3 $ = −0.004
      $ \phi\eta $ $ \cdots $ $ \cdots $ 7.13 4 $ ^3D_3 $ = −0.169
      $ ^3G_3 $ = −0.113
      $ \phi\eta' $ $ \cdots $ $ \cdots $ 1.16 0.7 $ ^3D_3 $ = −0.091
      $ ^3G_3 $ = −0.012
      $ \phi\phi $ 0.81 0.3 $ ^5D_3 $ = 0.046 $ \cdots $ $ \cdots $
      $ ^5G_3 $ = 0.006
      $ KK_1(1270) $ 36 15 $ ^3F_3 $ = 0.087 6.6 3.7 $ ^3F_3 $ = −0.038
      $ KK_1(1400) $ 0.43 0.2 $ ^3F_3 $ = 0.011 1.5 0.8 $ ^3F_3 $ = −0.020
      $ \eta f_0(1370) $ 0.56 0.2 $ ^1F_3 $ = 0.054 $ \cdots $ $ \cdots $
      $ \eta f_1(1420) $ 0.79 0.3 $ ^3F_3 $ = 0.068 $ \cdots $ $ \cdots $
      $ \eta f_2'(1525) $ 16 6.5 $ ^5P_3 $ = 0.401 $ \cdots $ $ \cdots $
      $ ^5F_3\sim $0
      $ ^5H_3\sim $0
      Total 245 100 178 100

      Table 7.  Strong decay properties for the 1F-wave states.

      Mode State $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $) State $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $)
      $ KK $ $ 2 ^3D_{1} $ 7.8 2.4 $ ^1P_1 $ = −0.057 $ 2 ^3D_{3} $ 7.7 5.7 $ ^1F_3 $ = 0.057
      $ KK^*(892) $ (2272) 7.7 2.4 $ ^3P_1 $ = −0.043 (2285) 3.6 2.7 $ ^3F_3 $ = −0.030
      $ KK(1460) $ 38 12 $ ^1P_1 $ = −0.089 25 19 $ ^1F_3 $ = 0.072
      $ KK^*(1410) $ 32 9.8 $ ^3P_1 $ = −0.092 18 13 $ ^3F_3 $ = −0.067
      $ K^*(892)K^*(892) $ 28 8.6 $ ^1P_1 $ = −0.023 17 12 $ ^1F_3 $ = −0.014
      $ ^5P_1 $ = 0.010 $ ^5P_3 $ = −0.095
      $ ^5F_1 $ = −0.128 $ ^5F_3 $ = 0.032
      $ KK^*_2(1430) $ 25 7.7 $ ^5D_1 $ = 0.075 15 11 $ ^5D_3 $ = 0.042
      $ ^5G_3 $ = 0.039
      $ \phi\eta $ 1.6 0.5 $ ^3P_1 $ = −0.092 0.01 0.01 $ ^3F_3 $ = −0.006
      $ \phi\eta' $ $<0.01 $ $<0.01 $ $ ^3P_1 $ = −0.005 0.2 0.2 $ ^3F_3 $ = 0.035
      $ KK_1(1270) $ 88 27 $ ^3S_1 $ = −0.120 24 17 $ ^3D_3 $ = 0.007
      $ ^3D_1 $ = 0.041 $ ^3G_3 $ = −0.065
      $ KK_1(1400) $ 8.9 2.8 $ ^3S_1 $ = 0.020 11 8.3 $ ^3D_3 $ = 0.048
      $ ^3D_1 $ = −0.039 $ ^3G_3 $ = 0.007
      $ K^*(892)K_1(1270) $ 67 21 $ ^3S_1 $ = 0.072 7.9 5.8 $ ^3D_3 $ = −0.017
      $ ^3D_1 $ = 0.093 $ ^3G_3 $ = −0.011
      $ ^5D_1 $ = −0.105 $ ^5D_3 $ = −0.046
      $ ^5G_3 $ = −0.015
      $ KK^*(1680) $ 0.03 0.01 $ ^3P_1 $ = −0.004 0.03 0.02 $ ^3F_3 $ = 0.003
      $ KK_2(1770) $ 5.6 1.7 $ ^5P_1 $ = −0.094 0.02 0.02 $ ^5P_3 $ = 0.004
      $ ^5F_1\sim $0 $ ^5F_3 $ = 0.001
      $ ^5H_3\sim $0
      $ KK^*_3(1780) $ $<0.01 $ $<0.01 $ $ ^7F_1\sim $0 2.2 1.6 $ ^7P_3 $ = −0.045
      $ ^7F_3 $ = −0.002
      $ ^7H_3\sim $0
      $ \eta h_1(1415) $ 13 4.0 $ ^3S_1 $ = 0.144 4.7 3.5 $ ^3D_3 $ = 0.125
      $ ^3D_1 $ = −0.189 $ ^3G_3 $ = 0.068
      Total 322 100 136 100

      Table 8.  Strong decay properties for the $ 2 ^3D_{1} $ and $ 2 ^3D_{3} $ states.

      Mode State $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $) State $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $)
      $ KK^*(892) $ $ 2 ^3D_{2} $ 15 6.5 $ ^3P_2 $ = −0.058 $ 2 ^1D_{2} $ 11 5.2 $ ^3P_2 $ = −0.048
      (2297) $ ^3F_2 $ = 0.016 (2282) $ ^3F_2 $ = −0.017
      $ KK^*(1410) $ 71 30 $ ^3P_2 $ = −0.118 57 27 $ ^3P_2 $ = −0.100
      $ ^3F_2 $ = 0.061 $ ^3F_2 $ = −0.069
      $ K^*(892)K^*(892) $ 11 4.6 $ ^5P_2 $ = 0.044 12 5.6 $ ^3P_2 $ = 0.060
      $ ^5F_2 $ = −0.067 $ ^3F_2 $ = 0.060
      $ KK^*_0(1430) $ $<0.01 $ $<0.01 $ $ ^1D_2\sim $0 9.2 4.4 $ ^1D_2 $ = 0.045
      $ KK^*_2(1430) $ 50 21 $ ^5S_2 $ = −0.088 56 27 $ ^5S_2 $ = −0.070
      $ ^5D_2 $ = 0.031 $ ^5D_2 $ = 0.069
      $ ^5G_2 $ = −0.046 $ ^5G_2 $ = 0.052
      $ \phi\eta $ 4.4 1.9 $ ^3P_2 $ = −0.152 $ \cdots $ $ \cdots $
      $ ^3F_2 $ = −0.019
      $ \phi\eta' $ 1.2 0.5 $ ^3P_2 $ = −0.067 $ \cdots $ $ \cdots $
      $ ^3F_2 $ = −0.041
      $ \phi\phi $ $ \cdots $ $ \cdots $ 1.6 0.7 $ ^3P_2 $ = −0.003
      $ ^3F_2 $ = −0.051
      $ KK_1(1270) $ 3.7 1.6 $ ^3D_2 $ = −0.026 0.01 $<0.01 $ $ ^3D_2 $ = −0.001
      $ KK_1(1400) $ 7.9 3.4 $ ^3D_2 $ = −0.041 0.2 0.1 $ ^3D_2 $ = 0.007
      $ K^*(892)K_1(1270) $ 23 9.8 $ ^1D_2 $ = 0.016 30 14 $ ^1D_2 $ = −0.018
      $ ^3D_2 $ = −0.032 $ ^3D_2 $ = −0.004
      $ ^5S_2 $ = 0.012 $ ^5S_2 $ = 0.028
      $ ^5D_2 $ = −0.076 $ ^5D_2 $ = 0.095
      $ ^5G_2 $ = 0.019 $ ^5G_2 $ = 0.018
      $ KK^*(1680) $ 7.2 3.1 $ ^3P_2 $ = 0.052 11 5.1 $ ^3P_2 $ = −0.066
      $ ^3F_2 $ = 0.003 $ ^3F_2 $ = −0.003
      $ KK_2(1770) $ 9.2 4.0 $ ^5P_2 $ = 0.076 0.1 0.1 $ ^5P_2 $ = −0.011
      $ ^5F_2 $ = 0.005 $ ^5F_2\sim $0
      $ KK^*_3(1780) $ 29 13 $ ^7P_2 $ = −0.140 8.9 4.3 $ ^7P_2 $ = −0.095
      $ ^7F_2 $ = −0.001 $ ^7F_2 $ = −0.001
      $ ^7H_2\sim $0 $ ^7H_2\sim $0
      $ \eta f_0(1370) $ $ \cdots $ $ \cdots $ 4.0 1.9 $ ^1D_2 $ = 0.127
      $ \eta f_1(1420) $ $ \cdots $ $ \cdots $ 0.7 0.34 $ ^3D_2 $ = 0.055
      $ \eta h_1(1415) $ 0.7 0.3 $ ^3D_2 $ = 0.056 $ \cdots $ $ \cdots $
      $ \eta f_2'(1525) $ $ \cdots $ $ \cdots $ 7.6 3.6 $ ^5S_2 $ = 0.034
      $ ^5D_2 $ = −0.192
      $ ^5G_2 $ = −0.048
      Total 232 100 208 100

      Table 9.  Strong decay properties for the $ 2 ^3D_{2} $ and $ 2 ^1D_{2} $ states.

      Mode Case I (2079 MeV) Case II (2135 MeV) Case III (2175 MeV)
      $ 3 ^3S_{1} $ $ 2 ^3D_{1} $ $ 3 ^3S_{1} $ $ 2 ^3D_{1} $ $ 3 ^3S_{1} $ $ 2 ^3D_{1} $
      $ KK $ 2.90 (1.8%) 9.9 (6.3%) 2.4 (1%) 9.6 (4.5%) 1.9 (0.7%) 1.9 (0.7%)
      $ KK^*(892) $ 27 (17%) 5.2 (3.3 %) 29 (12%) 6.3 (3%) 30 (11 %) 30 (11 %)
      $ KK(1460) $ 35 (22%) 59 (38%) 35 (15%) 66 (31%) 29 (10 %) 29 (10%)
      $ KK^*(1410) $ 2.1 (1.4%) 0.58 (0.37%) 62 (26%) 17 (7.9%) 96 (35%) 96 (35%)
      $ K^*(892)K^*(892) $ 2.9 (1.8%) 14 (9.2%) 11 (4.9%) 20 (9.2%) 21 (7.5%) 21 (7.5%)
      $ KK^*_2(1430) $ 28 (18%) 11 (7.3%) 42 (18%) 18 (8.5 %) 49 (18 %) 49 (18%)
      $ \phi\eta $ 5.6 (3.6%) 0.34 (0.22%) 7.5 (3.2%) 0.64 (0.3%) 8.9 (3.2%) 8.9 (3.2%)
      $ \phi\eta' $ 0.01 (0.01%) 0.3 (0.19%) 0.07 (0.03%) 0.23 (0.1%) 0.36 (0.13%) 0.36 (0.13%)
      $ KK_1(1270) $ 35 (22%) 44 (28%) 31 (13%) 65 (30%) 27 (9.6%) 27 (9.6%)
      $ KK_1(1400) $ 12 (7.6%) 4.1 (2.6 %) 4.9 (2.1%) 6.1 2.9%) 2.2 (0.8 %) 2.2 (0.8%)
      $ K^*(892)K_1(1270) $ $ \cdots $ $ \cdots $ $ \cdots $ $ \cdots $ 3.2 (1.1%) 3.2 (1.1%)
      $ \eta h_1(1415) $ 7 (4.5%) 8.3 (5.3%) 8 (3.4%) 5 (2.4%) 8.9 (3.2%) 8.9 (3.2%)
      Total width 157 (100%) 158 (100%) 233 (100%) 213 (100%) 276 (100%) 276 (100%)

      Table 10.  Partial and total strong decay widths (MeV) for the $ \phi(2170) $ as candidate for $ 3 ^3S_{1} $ and $ 2 ^3D_{1} $, respectively. Case I, Case II, Case III stand for the results from taking the mass of $ \phi(2170) $ as 2079, 2135 and 2175 MeV respectively. The values in parentheses are branching fractions.

      Mode State $ \Gamma_{\rm th} $/MeV $ Br$(%) Amps. /(GeV$ ^{-1/2} $) State $ \Gamma_{\rm th} $/MeV $ Br$(%) Amps. /(GeV$ ^{-1/2} $)
      $ KK $ $ 3 ^3P_{0} $ 9.2 2.7 $ ^1S_0 $ = −0.062 $ 3 ^3P_{2} $ 3.0 2.1 $ ^1D_2 $ = 0.035
      $ KK^*(892) $ (2434) $ \cdots $ $ \cdots $ $ \cdots $ (2466) 0.1 0.1 $ ^3D_2 $ = 0.005
      $ KK(1460) $ 30 8.6 $ ^1S_0 $ = −0.072 3.7 2.6 $ ^1D_2 $ = 0.025
      $ KK^*(1410) $ $ \cdots $ $ \cdots $ $ \cdots $ 3.2 2.2 $ ^3D_2 $ = 0.025
      $ K^*(892)K^*(892) $ 39 12 $ ^1S_0 $ = −0.025 15 10 $ ^1D_2 $ = −0.021
      $ ^5D_0 $ = −0.145 $ ^5S_2 $ = −0.068
      $ ^5D_2 $ = 0.056
      $ K^*(892)K(1460) $ $ \cdots $ $ \cdots $ $ \cdots $ 17 12 $ ^3D_2 $ = −0.075
      $ K^*(892)K^*(1410) $ $ \cdots $ $ \cdots $ $ \cdots $ $ \cdots $ $ \cdots $ $ \cdots $
      $ KK^*_2(1430) $ $ \cdots $ $ \cdots $ $ \cdots $ 7.8 5.4 $ ^5P_2 $ = −0.037
      $ ^5F_2 $ = 0.009
      $ K^*(892)K^*_0(1430) $ 4.5 1.3 $ ^3P_0 $ = −0.041 0.7 0.5 $ ^3P_2 $ = 0.015
      $ K^*(892)K^*_2(1430) $ 31 9.0 $ ^3P_0 $ = 0.035 19 13 $ ^3P_2 $ = −0.001
      $ ^7F_0 $ = −0.102 $ ^3F_2 $ = −0.017
      $ ^5P_2 $ = 0.007
      $ ^5F_2 $ = 0.017
      $ ^7P_2 $ = 0.053
      $ ^7F_2 $ = 0.054
      $ \eta\eta $ 1.7 0.5 $ ^1S_0 $ = 0.047 0.1 0.04 $ ^1D_2 $ = −0.009
      $ \eta\eta' $ 2.9 0.8 $ ^1S_0 $ = 0.078 0.01 0.01 $ ^1D_2 $ = 0.004
      $ \eta'\eta' $ 0.3 0.1 $ ^1S_0 $ = 0.016 0.05 0.03 $ ^1D_2 $ = 0.006
      $ \phi\phi $ 3.8 1.1 $ ^1S_0 $ = −0.018 1.4 1.0 $ ^1D_2 $ = 0.013
      $ ^5D_0 $ = 0.071 $ ^5S_2 $ = 0.021
      $ ^5D_2 $ = −0.035
      $ KK_1(1270) $ 87 25 $ ^3P_0 $ = −0.120 4.4 3.0 $ ^3P_2 $ = −0.001
      $ ^3F_2 $ = 0.027
      Continued on next page

      Table 11.  Strong decay properties for the $ 3 ^3P_{0} $ and $ 3 ^3P_{2} $ $ s\bar s $ states.

      Mode State $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $) State $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $)
      $ KK^*(892) $ $ 3 ^3P_{1} $ 12 4.1 $ ^3S_1 $ = −0.051 $ 3 ^1P_{1} $ 12 4.6 $ ^3S_1 $ = −0.038
      $ ^3D_1 $ = −0.016 $ ^3D_1 $ = 0.037
      $ KK^*(1410) $ (2470) 31 10 $ ^3S_1 $ = −0.074 (2435) 28 10 $ ^3S_1 $ = −0.050
      $ ^3D_1 $ = −0.025 $ ^3D_1 $ = 0.056
      $ K^*(892)K^*(892) $ 20 6.7 $ ^5D_1 $ = −0.105 18 6.6 $ ^3S_1 $ = −0.040
      $ ^3D_1 $ = 0.092
      $ K^*(892)K(1460) $ 12 4.0 $ ^3S_1 $ = 0.013 18 6.6 $ ^3S_1 $ = 0.032
      $ ^3D_1 $ = 0.061 $ ^3D_1 $ = 0.077
      $ KK^*_0(1430) $ 0.2 0.1 $ ^1P_1 $ = 0.006 6.4 2.4 $ ^1P_1 $ = −0.035
      $ KK^*_2(1430) $ 28 9.6 $ ^5P_1 $ = −0.072 39 15 $ ^5P_1 $ = −0.080
      $ ^5F_1 $ = −0.013 $ ^5F_1 $ = 0.033
      $ K^*(892)K^*_0(1430) $ 0.9 0.3 $ ^3P_1 $ = −0.017 0.03 0.01 $ ^3P_1 $ = −0.003
      $ K^*(892)K^*_2(1430) $ 31 10 $ ^3P_1 $ = −0.007 25 9.2 $ ^3P_1 $ = −0.002
      $ ^5P_1 $ = 0.012 $ ^5P_1 $ = 0.075
      $ ^5F_1 $ = −0.017 $ ^5F_1 $ = 0.059
      $ ^7F_1 $ = −0.098
      $ \phi\eta $ $ \cdots $ $ \cdots $ $ \cdots $ 4.7 1.7 $ ^3S_1 $ = 0.084
      $ ^3D_1 $ = −0.129
      $ \phi\eta' $ $ \cdots $ $ \cdots $ $ \cdots $ 1.5 0.6 $ ^3S_1 $ = 0.020
      $ ^3D_1 $ = −0.079
      Continued on next page

      Table 12.  Strong decay properties for the $ 3 ^3P_{1} $ and $ 3 ^1P_{1} $ $ s\bar s $ states.

      Mode State $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $) State $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $)
      $ KK $ $ 4 ^1S_{0} $ $ \cdots $ $ \cdots $ $ \cdots $ $ 4 ^3S_{1} $ 0.42 0.18 $ ^1P_1 $ = 0.013
      $ KK^*(892) $ (2580) 42 10 $ ^3P_0 $ = −0.098 (2623)$ ^a $ 12 5.5 $ ^3P_1 $ = −0.053
      $ KK(1460) $ $ \cdots $ $ \cdots $ $ \cdots $ 0.49 0.22 $ ^1P_1 $ = 0.009
      $ KK^*(1410) $ 46 11 $ ^3P_0 $ = −0.090 19 8.3 $ ^3P_1 $ = −0.057
      $ K^*(892)K^*(892) $ 30 7.3 $ ^3P_0 $ = −0.126 29 13 $ ^1P_1 $ = 0.027
      $ ^5P_1 $ = −0.119
      $ K^*(892)K(1460) $ 6.4 1.6 $ ^3P_0 $ = −0.039 11 4.9 $ ^3P_1 $ = 0.050
      $ K^*(892)K^*(1410) $ 35 8.5 $ ^3P_0 $ = 0.111 2.4 1.0 $ ^1P_1 $ = −0.006
      $ ^5P_1 $ = −0.026
      $ KK^*_0(1430) $ 4.2 1.0 $ ^1S_0 $ = 0.027 $ \cdots $ $ \cdots $ $ \cdots $
      $ KK^*_2(1430) $ 44 11 $ ^5D_0 $ = 0.088 14 6.0 $ ^5D_1 $ = 0.048
      $ K^*(892)K^*_0(1430) $ $ \cdots $ $ \cdots $ $ \cdots $ 5.9 2.6 $ ^3S_1 $ = 0.038
      $ K^*(892)K^*_2(1430) $ 0.1 0.03 $ ^5D_0 $ = −0.005 7.4 3.2 $ ^3D_1 $ = −0.007
      $ ^5D_1 $ = 0.009
      $ ^7D_1 $ = 0.041
      $ \phi\eta $ $ \cdots $ $ \cdots $ $ \cdots $ 5.4 2.4 $ ^3P_1 $ = −0.161
      $ \phi\eta' $ $ \cdots $ $ \cdots $ $ \cdots $ 3.2 1.4 $ ^3P_1 $ = −0.111
      $ \phi\phi $ 5.4 1.3 $ ^3P_0 $ = 0.082 $ \cdots $ $ \cdots $ $ \cdots $
      $ KK_1(1270) $ $ \cdots $ $ \cdots $ $ \cdots $ 2.7 1.2 $ ^3S_1 $ = 0.002
      $ ^3D_1 $ = −0.021
      $ KK_1(1400) $ $ \cdots $ $ \cdots $ $ \cdots $ 0.1 0.04 $ ^3S_1 $ = −0.001
      $ ^3D_1 $ = 0.004
      $ K^*(892)K_1(1270) $ 38 9.3 $ ^1S_0 $ = −0.016 31 14 $ ^3S_1 $ = −0.006
      $ ^5D_0 $ = 0.087 $ ^3D_1 $ = −0.039
      $ ^5D_1 $ = −0.067
      $ K^*(892)K_1(1400) $ 61 15 $ ^1S_0 $ = −0.122 14 6.0 $ ^3S_1 $ = −0.056
      $ ^5D_0 $ = −0.004 $ ^3D_1 $ = 0.003
      $ ^5D_1 $ = 0.005
      $ K_1(1270)K_1(1270) $ 13 3.2 $ ^3P_0 $ = 0.070 16 7.0 $ ^1P_1 $ = −0.001
      $ ^5P_1 $ = −0.015
      $ ^5F_1 $ = 0.062
      $ KK^*(1680) $ 0.46 0.1 $ ^3P_0 $ = −0.009 0.68 0.3 $ ^3P_1 $ = 0.011
      $ KK_2(1770) $ $ \cdots $ $ \cdots $ $ \cdots $ 9.6 4.2 $ ^5P_1 $ = −0.001
      $ ^5F_1 $ = −0.043
      $ KK_2(1820) $ $ \cdots $ $ \cdots $ $ \cdots $ 0.14 0.1 $ ^5P_1 $ = −0.001
      $ ^5F_1 $ = 0.005
      $ KK^*_3(1780) $ 43 11 $ ^7F_0 $ = 0.095 11 5.1 $ ^7F_1 $ = 0.047
      $ K^*(892)K^*(1680) $ $ \cdots $ $ \cdots $ $ \cdots $ 1.3 0.6 $ ^1P_1 $ = 0.022
      $ ^3P_1 $ = 0.029
      $ ^5P_1 $ = 0.012
      $ \eta f_0(1370) $ 3.3 0.8 $ ^1S_0 $ = 0.103 $ \cdots $ $ \cdots $ $ \cdots $
      Continued on next page

      Table 13.  Strong decay properties for the $ 4S $-wave $ s\bar s $ states.

      Mode State $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $) State $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $)
      $ KK $ $ 2 ^3F_{2} $ 2.3 0.46 $ ^1D_2 $ = 0.031 $ 2 ^3F_{4} $ 5.3 3.6 $ ^1G_4 $ = −0.047
      $ KK^*(892) $ (2552) 3.9 0.79 $ ^3D_2 $ = 0.030 (2503) 6.4 4.4 $ ^3G_4 $ = 0.038
      $ KK(1460) $ 1.3 0.27 $ ^1D_2 $ = 0.015 13 8.7 $ ^1G_4 $ = −0.046
      $ KK^*(1410) $ 6.0 1.2 $ ^3D_2 $ = 0.033 19 13 $ ^3G_4 $ = 0.061
      $ K^*(892)K^*(892) $ 1.9 0.38 $ ^1D_2 $ = 0.024 9.9 6.8 $ ^1G_4 $ = −0.002
      $ ^5D_2 $ = −0.018 $ ^5D_4 $ = 0.073
      $ ^5G_2 $ = 0.009 $ ^5G_4 $ = 0.004
      $ K^*(892)K(1460) $ 12 2.5 $ ^3D_2 $ = −0.056 2.2 1.5 $ ^3G_4 $ = −0.025
      $ K^*(892)K^*(1410) $ 5.9 1.2 $ ^1D_2 $ = 0.035 3.2 2.2 $ ^1G_4 $ = −0.001
      $ ^5D_2 $ = −0.026 $ ^5D_4 $ = 0.046
      $ ^5G_2 $ = −0.022 $ ^5G_4 $ = 0.002
      $ KK^*_2(1430) $ 5.4 1.1 $ ^5P_2 $ = −0.028 4.7 3.3 $ ^5F_4 $ = −0.001
      $ ^5F_2 $ = 0.013 $ ^5H_4 $ = 0.029
      $ K^*(892)K^*_0(1430) $ 6.7 1.4 $ ^3F_2 $ = −0.042 1.2 0.79 $ ^3F_4 $ = 0.019
      $ K^*(892)K^*_2(1430) $ 18 3.7 $ ^3P_2\sim $0 16 11 $ ^3F_4 $ = −0.011
      $ ^3F_2\sim $0 $ ^3H_4 $ = −0.004
      $ ^5P_2\sim $0 $ ^5F_4 $ = 0.001
      $ ^5F_2 $ = 0.034 $ ^5H_4 $ = 0.004
      $ ^7P_2\sim $0 $ ^7P_4 $ = 0.017
      $ ^7F_2 $ = −0.045 $ ^7F_4 $ = 0.065
      $ ^7H_2 $ = −0.042 $ ^7H_4 $ = 0.009
      $ \eta\eta $ 0.41 0.08 $ ^1D_2 $ = −0.023 0.46 0.32 $ ^1G_4 $ = 0.025
      $ \eta\eta' $ 1.3 0.26 $ ^1D_2 $ = −0.050 0.31 0.21 $ ^1G_4 $ = 0.025
      $ \eta'\eta' $ 0.52 0.1 $ ^1D_2 $ = −0.021 0.01 0.01 $ ^1G_4 $ = 0.003
      Continued on next page

      Table 14.  Strong decay properties for the $ 2 ^3F_{2} $ and $ 2 ^3F_{4} $ states.

      State Mode $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $) Mode $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $)
      $ 2 ^3F_{3} $ (2543) $ KK^*(892) $ 11 4 $ ^3D_3 $ = 0.035 $ K^*(892)K_1(1400) $ 13 4.9 $ ^1F_3 $ = −0.009
      $ ^3G_3 $ = −0.035 $ ^3F_3 $ = 0.054
      $ KK^*(1410) $ 28 10 $ ^3D_3 $ = 0.039 $ ^5P_3 $ = −0.007
      $ ^3G_3 $ = −0.060 $ ^5F_3 $ = −0.017
      $ K^*(892)K^*(892) $ 3.4 1.3 $ ^5D_3 $ = −0.042 $ ^5H_3 $ = −0.003
      $ ^5G_3 $ = −0.007 $ KK^*(1680) $ 2.8 1.1 $ ^3D_3 $ = −0.023
      $ K^*(892)K(1460) $ 20 7.4 $ ^3D_3 $ = −0.064 $ ^3G_3 $ = −0.005
      $ ^3G_3 $ = 0.033 $ KK_2(1770) $ 17 6.3 $ ^5D_3 $ = −0.049
      $ K^*(892)K^*(1410) $ 7.2 2.7 $ ^5D_3 $ = −0.054 $ ^5G_3 $ = −0.035
      $ ^5G_3 $ = −0.014 $ KK_2(1820) $ 1.6 0.62 $ ^5D_3 $ = −0.019
      $ KK^*_0(1430) $ 1.3 0.47 $ ^1F_3 $ = 0.015 $ ^5G_3 $ = −0.005
      $ KK^*_2(1430) $ 34 13 $ ^5P_3 $ = −0.068 $ KK^*_3(1780) $ 41 15 $ ^7S_3 $ = −0.078
      $ ^5F_3 $ = 0.007 $ ^7D_3 $ = 0.053
      $ ^5H_3 $ = −0.038 $ ^7G_3 $ = −0.001
      $ K^*(892)K^*_0(1430) $ 3.3 1.2 $ ^3F_3 $ = −0.030 $ ^7I_3 $ = −0.006
      $ K^*(892)K^*_2(1430) $ 24 9 $ ^3F_3 $ = −0.018 $ \eta f_0(1370) $ 0.24 0.09 $ ^1D_3 $ = 0.028
      $ ^5P_3 $ = −0.003 $ \eta f_1(1420) $ 1.3 0.48 $ ^3F_3 $ = −0.066
      $ ^5F_3 $ = 0.023 $ \eta f_2'(1525) $ 6.5 2.4 $ ^5P_3 $ = 0.146
      $ ^5H_3 $ = −0.007 $ ^5F_3 $ = −0.020
      $ ^7P_3 $ = 0.004 $ ^5H_3 $ = 0.055
      $ ^7F_3 $ = −0.070 $ \eta'f_0(1370) $ 0.01 $<0.01 $ $ ^1F_3 $ = −0.005
      $ ^7H_3 $ = −0.027 $ \eta' f_1(1420) $ 0.86 0.32 $ ^3F_3 $ = −0.058
      $ \phi\phi $ 1.03 0.39 $ ^5D_3 $ = 0.029 $ \eta' f_2'(1525) $ 1.1 0.43 $ ^5P_3 $ = −0.085
      $ ^5G_3 $ = −0.022 $ ^5F_3 $ = −0.004
      $ KK_1(1270) $ 4.2 1.6 $ ^3F_3 $ = 0.026 $ ^5H_3 $ = 0.001
      $ KK_1(1400) $ 1.7 0.63 $ ^3F_3 $ = −0.017 $ \phi h_1(1415) $ 3.3 1.2 $ ^1F_3 $ = −0.015
      $ K^*(892)K_1(1270) $ 40 15 $ ^1F_3 $ = 0.007 $ ^3F_3 $ = 0.052
      $ ^3F_3 $ = −0.013 $ ^5P_3 $ = 0.080
      $ ^5P_3 $ = 0.077 $ ^5F_3 $ = 0.008
      $ ^5F_3 $ = −0.025 $ ^5H_3 $ = −0.005
      $ ^5H_3 $ = 0.040 Total 266 100
      $ 2 ^1F_{3} $ $ KK^*(892) $ 10 4.5 $ ^3D_3 $ = 0.031 $ K^*(892)K_1(1270) $ 38 16 $ ^1F_3 $ = −0.012
      (2528) $ ^3G_3 $ = 0.038 $ ^3F_3 $ = −0.002
      $ KK^*(1410) $ 31 13 $ ^3D_3 $ = 0.036 $ ^5P_3 $ = −0.063
      $ ^3G_3 $ = 0.067 $ ^5F_3 $ = 0.046
      $ K^*(892)K^*(892) $ 5 2.1 $ ^3D_3 $ = 0.052 $ ^5H_3 $ = 0.044
      $ ^3G_3 $ = 0.004 $ K^*(892)K_1(1400) $ 15 6.7 $ ^1F_3 $ = −0.061
      $ K^*(892)K(1460) $ 16 6.8 $ ^3D_3 $ = 0.056 $ ^3F_3 $ = 0.019
      $ ^3G_3 $ = 0.034 $ ^5P_3 $ = 0.005
      $ K^*(892)K^*(1410) $ 6.5 2.8 $ ^3D_3 $ = 0.056 $ ^5F_3 $ = −0.007
      $ ^3G_3 $ = 0.008 $ ^5H_3 $ = −0.003
      Continued on next page

      Table 15.  Strong decay properties for the $ 2 ^3F_{3} $ and $ 2 ^1F_{3} $ states.

      State Mode $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $) Mode $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $)
      $ 3 ^3D_{3} $ (2691) $ KK $ 6.1 3.8 $ ^1F_3 $ = 0.050 $ KK^*(1680) $ 0.04 0.02 $ ^3F_3 $ = −0.003
      (2691) $ KK^*(892) $ 3.8 2.4 $ ^3F_3 $ = −0.029 $ KK_2(1770) $ 6.4 4.0 $ ^5P_3\sim $ = 0
      $ KK(1460) $ 13 8.1 $ ^1F_3 $ = 0.044 $ ^5F_3 $ = −0.003
      $ KK^*(1410) $ 6.8 4.3 $ ^3F_3 $ = −0.033 $ ^5H_3 $ = −0.034
      $ K^*(892)K^*(892) $ 7.2 4.5 $ ^1F_3 $ = 0.004 $ KK_2(1820) $ 8.4 5.3 $ ^5P_3 $ = −0.038
      $ ^5P_3 $ = −0.059 $ ^5F_3 $ = 0.013
      $ ^5F_3 $ = 0.010 $ ^5H_3 $ = 0.003
      $ K^*(892)K(1460) $ 0.63 0.4 $ ^3F_3 $ = 0.011 $ KK^*_3(1780) $ 6.8 4.3 $ ^7P_3 $ = −0.023
      $ K^*(892)K^*(1410) $ 22 13 $ ^1F_3 $ = −0.024 $ ^7F_3 $ = 0.010
      $ ^5P_3 $ = −0.047 $ ^7H_3 $ = 0.024
      $ ^5F_3 $ = 0.052 $ K^*(892)K^*(1680) $ 0.72 0.45 $ ^1F_3 $ = 0.009
      $ KK^*_2(1430) $ 2.8 1.7 $ ^5D_3 $ = 0.008 $ ^3F_3 $ = 0.008
      $ ^5G_3 $ = 0.020 $ ^5P_3 $ = 0.011
      $ K^*(892)K^*_0(1430) $ 1.5 0.9 $ ^3D_3 $ = 0.018 $ ^5F_3\sim $0
      $ K^*(892)K^*_2(1430) $ 19 12 $ ^3D_3 $ = −0.007 $ K^*(892)K_2(1770) $ 0.84 0.52 $ ^3F_3\sim 0 $
      $ ^3G_3 $ = 0.009 $ ^5P_3 $ = −0.007
      $ ^5D_3\sim $0 $ ^5F_3 $ = −0.001
      $ ^5G_3 $ = −0.008 $ ^5H_3\sim $0
      $ ^7S_3 $ = −0.032 $ ^7P_3 $ = −0.023
      $ ^7D_3 $ = 0.050 $ ^7F_3 $ = −0.004
      $ ^7G_3 $ = −0.022 $ ^7H_3\sim $0
      $ \phi\eta $ 0.1 0.05 $ ^3F_3 $ = −0.019 $ K^*(892)K^*_3(1780) $ 21 13 $ ^5P_3 $ = −0.003
      $ \phi\eta' $ 0.03 0.02 $ ^3F_3 $ = 0.010 $ ^5F_3 $ = −0.001
      $ KK_1(1270) $ 13 7.9 $ ^3D_3 $ = 0.009 $ ^5H_3\sim $0
      $ ^3G_3 $ = −0.043 $ ^7P_3 $ = 0.006
      $ KK_1(1400) $ 1.9 1.2 $ ^3D_3 $ = 0.017 $ ^7F_3 $ = 0.001
      $ ^3G_3 $ = 0.004 $ ^7H_3\sim $0
      $ K^*(892)K_1(1270) $ 3.1 1.9 $ ^3D_3 $ = −0.010 $ ^9P_3 $ = 0.127
      $ ^3G_3 $ = −0.001 $ ^9F_3 $ = 0.007
      $ ^5D_3 $ = −0.022 $ ^9H_3\sim $0
      $ ^5G_3 $ = 0.001 $ \eta h_1(1415) $ 0.82 0.52 $ ^3D_3 $ = 0.045
      $ K^*(892)K_1(1400) $ 5.6 3.5 $ ^3D_3 $ = −0.031 $ ^3G_3 $ = 0.024
      $ ^3G_3 $ = −0.001 $ \eta' h_1(1415) $ 0.37 0.23 $ ^3D_3 $ = 0.029
      $ ^5D_3 $ = 0.015 $ ^3G_3 $ = −0.016
      $ ^5G_3 $ = −0.002 $ \phi f_0(1370) $ 0.26 0.16 $ ^3D_3 $ = 0.022
      $ K_1(1270)K_1(1270) $ 4.2 2.6 $ ^1F_3 $ = 0.005 $ \phi f_1(1420) $ 0.35 0.22 $ ^3D_3\sim $0
      $ ^5P_3 $ = −0.004 $ ^3G_3 $ = 0.016
      $ ^5F_3 $ = 0.016 $ ^5D_3 $ = 0.005
      $ ^5H_3 $ = 0.022 $ ^5G_3 $ = 0.020
      $ K_1(1270)K_1(1400) $ 0.14 0.08 $ ^1F_3 $ = −0.003 $ \phi f_2'(1525) $ 3.6 2.2 $ ^3D_3 $ = −0.003
      $ ^3F_3 $ = 0.003 $ ^3G_3 $ = −0.008
      $ ^5P_3 $ = −0.008 $ ^5D_3 $ = 0.006
      $ ^5F_3 $ = −0.001 $ ^5G_3 $ = 0.008
      $ ^5H_3\sim $0 $ ^7S_3 $ = −0.079
      $ ^7D_3 $ = 0.047
      $ ^7G_3 $ = 0.022
      Total 160 100

      Table 16.  Strong decay properties for the $ 3 ^3D_{3} $ state.

      State Mode $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $) Mode $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $)
      $ 3 ^3D_{1} $(2681) $ KK $ 2.8 0.82 $ ^1P_1 $ = −0.034 $ K_1(1270)K_1(1400) $ 3.0 0.87 $ ^1P_1 $ = 0.046
      $ KK^*(892) $ 3.3 0.94 $ ^3P_1 $ = −0.027 $ ^3P_1 $ = 0.024
      $ KK(1460) $ 4.1 1.2 $ ^1P_1 $ = −0.025 $ ^5P_1 $ = 0.011
      $ KK^*(1410) $ 4.6 1.3 $ ^3P_1 $ = −0.027 $ ^5F_1\sim $0
      $ K^*(892)K^*(892) $ 16 4.7 $ ^1P_1 $ = −0.021 $ KK^*(1680) $ 0.08 0.02 $ ^3P_1 $ = −0.004
      $ ^5P_1 $ = 0.009 $ KK_2(1770) $ 59 17 $ ^5P_1 $ = −0.102
      $ ^5F_1 $ = −0.088 $ ^5F_1 $ = 0.018
      $ K^*(892)K(1460) $ 5.1 1.5 $ ^3P_1 $ = 0.032 $ KK_2(1820) $ 2.9 0.85 $ ^5P_1 $ = 0.003
      $ K^*(892)K^*(1410) $ 64 18 $ ^1P_1 $ = −0.002 $ ^5F_1 $ = −0.024
      $ ^5P_1 $ = 0.001 $ KK^*_3(1780) $ 4.2 1.2 $ ^7F_1 $ = 0.028
      $ ^5F_1 $ = −0.128 $ K^*(892)K^*(1680) $ 15 4.5 $ ^1P_1 $ = −0.065
      $ KK^*_2(1430) $ 7.4 2.1 $ ^5D_1 $ = 0.035 $ ^3P_1 $ = 0.044
      $ K^*(892)K^*_0(1430) $ 3.5 1.01 $ ^3D_1 $ = −0.028 $ ^5P_1 $ = −0.005
      $ K^*(892)K^*_2(1430) $ 33 9.5 $ ^3S_1\sim $0 $ ^5F_1 $ = −0.008
      $ ^3D_1 $ = −0.001 $ K^*(892)K_2(1770) $ 6.7 1.9 $ ^3P_1 $ = −0.009
      $ ^5D_1 $ = 0.021 $ ^5P_1 $ = 0.079
      $ ^7D_1 $ = −0.017 $ ^5F_1 $ = 0.002
      $ ^7G_1 $ = 0.081 $ ^7F_1 $ = −0.003
      $ \phi\eta $ 0.79 0.23 $ ^3P_1 $ = −0.061 $ K^*(892)K^*_3(1780) $ 0.3 0.09 $ ^5P_1 $ = 0.018
      $ \phi\eta' $ 0.22 0.06 $ ^3P_1 $ = −0.029 $ ^5F_1\sim $0
      $ KK_1(1270) $ 29 8.4 $ ^3S_1 $ = −0.063 $ ^7F_1 $ = 0.001
      $ ^3D_1 $ = 0.020 $ ^9F_1 $ = −0.001
      $ KK_1(1400) $ 2.0 0.57 $ ^3S_1 $ = 0.012 $ ^9H_1\sim $0
      $ ^3D_1 $ = −0.013 $ \eta h_1(1415) $ 7.2 2.1 $ ^3S_1 $ = 0.128
      $ K^*(892)K_1(1270) $ 51 15 $ ^3S_1 $ = −0.038 $ ^3D_1 $ = −0.079
      $ ^3D_1 $ = 0.059 $ \eta' h_1(1415) $ 0.15 0.04 $ ^3S_1 $ = −0.012
      $ ^5D_1 $ = −0.068 $ ^3D_1 $ = −0.017
      $ K^*(892)K_1(1400) $ 1.0 0.3 $ ^3S_1 $ = 0.002 $ \phi f_0(1370) $ 0.73 0.21 $ ^3D_1 $ = −0.036
      $ ^3D_1 $ = 0.012 $ \phi f_1(1420) $ 1.3 0.37 $ ^3S_1 $ = 0.043
      $ ^5D_1 $ = −0.009 $ ^3D_1 $ = −0.020
      $ K_1(1270)K_1(1270) $ 15 4.4 $ ^1P_1 $ = 0.001 $ ^5D_1 $ = −0.018
      $ ^5P_1 $ = −0.024 $ \phi f_2'(1525) $ 2.3 0.67 $ ^3S_1 $ = −0.041
      $ ^5F_1 $ = −0.049 $ ^3D_1 $ = 0.013
      $ ^5D_1 $ = 0.017
      $ ^7D_1 $ = −0.023
      $ ^7G_1 $ = −0.058
      Total 346 100

      Table 17.  Strong decay properties for the $ 3 ^3D_{1} $ state.

      State Mode $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $) Mode $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $)
      $ 3 ^3D_{2} $ (2701) $ KK^*(892) $ 5.2 1.9 $ ^3P_2 $ = −0.031 $ KK^*(1680) $ 3.1 1.2 $ ^3P_2 $ = 0.021
      $ ^3F_2 $ = 0.013 $ ^3F_2 $ = 0.009
      $ KK^*(1410) $ 14 5.3 $ ^3P_2 $ = −0.040 $ KK_2(1770) $ 7.6 2.9 $ ^5P_2 $ = 0.034
      $ ^3F_2 $ = 0.027 $ ^5F_2 $ = 0.014
      $ K^*(892)K^*(892) $ 4.6 1.7 $ ^5P_2 $ = 0.031 $ KK_2(1820) $ 1.4 0.52 $ ^5P_2 $ = 0.002
      $ ^5F_2 $ = −0.037 $ ^5F_2 $ = −0.017
      $ K^*(892)K(1460) $ 12 4.4 $ ^3P_2 $ = 0.047 $ KK^*_3(1780) $ 30 11 $ ^7P_2 $ = −0.066
      $ ^3F_2 $ = 0.011 $ ^7F_2 $ = 0.007
      $ K^*(892)K^*(1410) $ 33 12 $ ^5P_2 $ = 0.021 $ ^7H_2 $ = −0.032
      $ ^5F_2 $ = −0.087 $ K^*(892)K^*(1680) $ 4.6 1.7 $ ^3P_2 $ = 0.037
      $ KK^*_0(1430) $ 0.22 0.08 $ ^1D_2 $ = 0.006 $ ^3F_2 $ = −0.012
      $ KK^*_2(1430) $ 17 6.3 $ ^5S_2 $ = −0.047 $ ^5P_2 $ = −0.008
      $ ^5D_2 $ = 0.013 $ ^5F_2 $ = −0.004
      $ ^5G_2 $ = −0.020 $ K^*(892)K_2(1770) $ 17 6.3 $ ^3P_2 $ = −0.001
      $ K^*(892)K^*_0(1430) $ 2.4 0.91 $ ^3D_2 $ = −0.023 $ ^3F_2 $ = 0.002
      $ K^*(892)K^*_2(1430) $ 22 8.3 $ ^3D_2 $ = −0.014 $ ^5P_2 $ = −0.021
      $ ^5S_2 $ = −0.007 $ ^5F_2 $ = −0.007
      $ ^5D_2 $ = 0.020 $ ^7P_2 $ = −0.097
      $ ^5G_2 $ = 0.010 $ ^7F_2 $ = −0.005
      $ ^7D_2 $ = −0.045 $ ^7H_2 $ = 0.001
      $ ^7G_2 $ = 0.046 $ K^*(892)K^*_3(1780) $ 5.0 1.9 $ ^5P_2 $ = −0.014
      $ \phi\eta $ 1.7 0.65 $ ^3P_2 $ = −0.090 $ ^5F_2 $ = −0.002
      $ ^3F_2 $ = −0.013 $ ^7P_2 $ = 0.052
      $ \phi\eta' $ 1.3 0.49 $ ^3P_2 $ = −0.065 $ ^7F_2 $ = 0.004
      $ ^3F_2 $ = −0.026 $ ^7H_2\sim $0
      $ \phi\phi $ $ \cdots $ $ \cdots $ $ ^9F_2 $ = −0.012
      $ ^9H_2 $ = −0.001
      $ KK_1(1270) $ 0.06 0.02 $ ^3D_2 $ = 0.003 $ \eta f_0(1370) $ $ \cdots $ $ \cdots $
      $ KK_1(1400) $ 3.2 1.2 $ ^3D_2 $ = −0.023 $ \eta f_1(1420) $ $ \cdots $ $ \cdots $
      $ K^*(892)K_1(1270) $ 28 11 $ ^1D_2 $ = 0.009 $ \eta h_1(1415) $ 0.17 0.06 $ ^3D_2 $ = 0.023
      $ ^3D_2 $ = −0.017 $ \eta f_2'(1525) $ $ \cdots $ $ \cdots $
      $ ^5S_2 $ = 0.058
      $ ^5D_2 $ = −0.039
      $ ^5G_2 $ = −0.003 $ \eta'f_0(1370) $ $ \cdots $ $ \cdots $
      $ K^*(892)K_1(1400) $ 6.2 2.3 $ ^1D_2 $ = −0.008 $ \eta' f_1(1420) $ $ \cdots $ $ \cdots $
      $ ^3D_2 $ = 0.032 $ \eta' h_1(1415) $ 0.17 0.06 $ ^3D_2 $ = 0.022
      $ ^5S_2 $ = −0.006 $ \eta' f_2'(1525) $ $ \cdots $ $ \cdots $
      $ ^5D_2 $ = −0.013
      $ ^5G_2 $ = 0.002
      $ K^*_0(1430)K_1(1270) $ 0.14 0.05 $ ^3P_2 $ = −0.013 $ \phi f_0(1370) $ 0.53 0.20 $ ^3D_2 $ = −0.031
      $ ^3F_2\sim $0 $ \phi f_1(1420) $ 1.0 0.39 $ ^3D_2 $ = −0.008
      $ K_1(1270)K_1(1270) $ 34 13 $ ^5P_2 $ = −0.002 $ ^5S_2 $ = −0.023
      $ ^5F_2 $ = 0.079 $ ^5D_2 $ = −0.028
      $ K_1(1270)K_1(1400) $ 7.9 3.0 $ ^3P_2 $ = 0.059 $ ^5G_2 $ = −0.025
      $ ^3F_2 $ = −0.004 $ \phi h_1(1415) $ $ \cdots $ $ \cdots $
      $ ^5P_2 $ = −0.010
      $ ^5F_2\sim $0
      $ K_1(1270)K^*_2(1430) $ 2.7 1.0 $ ^3P_2 $ = −0.006
      $ ^3F_2\sim $0
      $ ^5P_2 $ = 0.002 $ \phi f_2'(1525) $ 1.8 0.68 $ ^3D_2 $ = −0.008
      $ ^5F_2\sim $0 $ ^5S_2 $ = −0.030
      $ ^7P_2 $ = −0.059 $ ^5D_2 $ = 0.005
      $ ^7F_2\sim $0 $ ^5G_2 $ = −0.010
      $ ^7H_2\sim $0 $ ^7D_2 $ = −0.035
      $ ^7G_2 $ = −0.046
      Total 268 100

      Table 18.  Strong decay properties for the $ 3 ^3D_{2} $ state.

      State Mode $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $) Mode $ \Gamma_{\rm th} $/MeV $ Br $(%) Amps. /(GeV$ ^{-1/2} $)
      $ 3 ^1D_{2} $(2685) $ KK^*(892) $ 4.0 2 $ ^3P_2 $ = −0.026 $ KK^*(1680) $ 7.1 3.6 $ ^3P_2 $ = −0.032
      $ ^3F_2 $ = −0.014 $ ^3F_2 $ = 0.013
      $ KK^*(1410) $ 12 6 $ ^3P_2 $ = −0.035 $ KK_2(1770) $ 0.24 0.12 $ ^5P_2 $ = −0.007
      $ ^3F_2 $ = −0.028 $ ^5F_2\sim $0
      $ K^*(892)K^*(892) $ 5.8 2.9 $ ^3P_2 $ = −0.042 $ KK_2(1820) $ 0.25 0.13 $ ^5P_2 $ = −0.007
      $ ^3F_2 $ = 0.034 $ ^5F_2 $ = −0.002
      $ K^*(892)K(1460) $ 9.0 4.5 $ ^3P_2 $ = −0.038 $ KK^*_3(1780) $ 32 16 $ ^7P_2 $ = −0.063
      $ ^3F_2 $ = 0.019 $ ^7F_2 $ = 0.022
      $ K^*(892)K^*(1410) $ 25 13 $ ^3P_2 $ = −0.021 $ ^7H_2 $ = 0.037
      $ ^3F_2 $ = 0.077 $ K^*(892)K^*(1680) $ 1.5 0.76 $ ^3P_2 $ = −0.023
      $ KK^*_0(1430) $ 3.9 2 $ ^1D_2 $ = 0.026 $ ^3F_2 $ = −0.006
      $ KK^*_2(1430) $ 17 8.4 $ ^5S_2 $ = −0.039 $ ^5P_2 $ = 0.002
      $ ^5D_2 $ = 0.028 $ ^5F_2 $ = 0.004
      $ ^5G_2 $ = 0.022 $ K^*(892)K_2(1770) $ 7.1 3.6 $ ^3P_2 $ = −0.006
      $ K^*(892)K^*_0(1430) $ 0.27 0.13 $ ^3D_2 $ = 0.008 $ ^3F_2 \sim $0
      $ K^*(892)K^*_2(1430) $ 15 7.7 $ ^3D_2 $ = −0.014 $ ^5P_2 $ = −0.003
      $ ^5S_2 $ = −0.011 $ ^5F_2 \sim $0
      $ ^5D_2 $ = 0.039 $ ^7P_2 $ = 0.077
      $ ^5G_2 $ = −0.039 $ ^7F_2 $ = 0.005
      $ ^7H_2\sim $0
      $ K^*(892)K^*_3(1780) $ 5.9 3 $ ^5P_2 $ = −0.006
      $ \phi\eta $ $ \cdots $ $ \cdots $ $ \cdots $ $ ^5F_2 $ = −0.001
      $ ^7P_2 $ = 0.074
      $ \phi\eta' $ $ \cdots $ $ \cdots $ $ \cdots $ $ ^7F_2 $ = 0.003
      $ ^7H_2\sim $0
      $ \phi\phi $ 1.5 0.78 $ ^3P_2 $ = 0.023
      $ ^3F_2 $ = −0.036
      $ KK_1(1270) $ 0.13 0.06 $ ^3D_2 $ = −0.004 $ \eta f_0(1370) $ 1.4 0.72 $ ^1D_2 $ = 0.066
      $ KK_1(1400) $ $<0.01 $ $<0.01 $ $ ^3D_2 $ = −0.001 $ \eta f_1(1420) $ 0.2 0.1 $ ^3D_2 $ = 0.025
      $ K^*(892)K_1(1270) $ 26 13 $ ^1D_2 $ = −0.008 $ \eta h_1(1415) $ $ \cdots $ $ \cdots $ $ \cdots $
      $ ^3D_2 $ = −0.002 $ \eta f_2'(1525) $ 4.2 2.1 $ ^5S_2 $ = 0.075
      $ ^5S_2 $ = −0.045 $ ^5D_2 $ = −0.092
      $ ^5D_2 $ = 0.051 $ ^5G_2 $ = 0.016
      $ ^5G_2 $ = −0.009 $ \eta'f_0(1370) $ 0.61 0.31 $ ^1D_2 $ = 0.041
      $ K^*(892)K_1(1400) $ 6.3 3.2 $ ^1D_2 $ = −0.032 $ \eta' f_1(1420) $ 0.11 0.06 $ ^3D_2 $ = 0.018
      $ ^3D_2 $ = 0.016 $ \eta' h_1(1415) $ $ \cdots $ $ \cdots $ $ \cdots $
      $ ^5S_2 $ = 0.004 $ \eta' f_2'(1525) $ 0.50 0.25 $ ^5S_2 $ = −0.034
      $ ^5D_2 $ = −0.006 $ ^5D_2 $ = −0.005
      $ ^5G_2 $ = 0.003 $ ^5G_2 $ = 0.026
      $ K^*_0(1430)K_1(1270) $ $ \cdots $ $ \cdots $ $ \cdots $ $ \phi f_0(1370) $ $ \cdots $ $ \cdots $ $ \cdots $
      $ \phi f_1(1420) $ $ \cdots $ $ \cdots $ $ \cdots $
      $ K_1(1270)K_1(1270) $ 8.7 4.4 $ ^3P_2 $ = 0.020
      $ ^3F_2 $ = 0.036
      $ K_1(1270)K_1(1400) $ 1.0 0.5 $ ^3P_2 $ = 0.026
      $ ^3F_2 $ = 0.001 $ \phi h_1(1415) $ 1.4 0.69 $ ^1D_2 $ = 0.004
      $ ^5P_2 $ = −0.006 $ ^3D_2 $ = 0.016
      $ ^5F_2 $ = −0.001 $ ^5S_2 $ = −0.024
      $ K_1(1270)K^*_2(1430) $ $ \cdots $ $ \cdots $ $ \cdots $ $ ^5D_2 $ = −0.014
      $ ^5G_2 $ = 0.041
      Total 198 100

      Table 19.  Strong decay properties for the $ 3 ^1D_{2} $ state.

    IV.   DISCUSSION
    • In this work we focus only on states which can be approximately considered pure $ s\bar{s} $ states. Considering the fact that for the low-lying pseudoscalar isoscalar states with $ J^{PC} = 0^{-+} $ there may exist a strong flavor mixing between $ n\bar{n} = (u\bar{u}+d\bar{d})/\sqrt{2} $ and $ s\bar{s} $ [17, 21, 26, 50-52], we omit discussions about these states in the present work.

    • A.   Well-established vector $ s\bar{s} $ states

      1.   $ \phi(1020) $
    • The $ \phi(1020) $ resonance, as the lowest S-wave vector $ s\bar{s} $ state $ 1^3S_1 $, was first observed in a bubble chamber experiment at Brookhaven in 1962 [53]. In a previous study with the standard relativistic phase space, the width of $ \phi \to KK $ was predicted to be $ \Gamma[\phi(1020) \to KK]\simeq 2.5 $ MeV [16], which is clearly smaller than the measured value $ 3.5 $ MeV [1]. To include some relativistic corrections to the phase space, we adopted the "mock meson" method in our calculations. The partial decay width of $ \phi \to KK $ is predicted to be $ \Gamma[\phi(1020) \to KK]\simeq 4.1 $ MeV, so our result is in good agreement with the measured value of $ 3.5 $ MeV [1].

    • 2.   $ \phi(1680) $
    • As the $ 2^3S_1 $ $ s\bar{s} $ state, the $ \phi(1680) $ was first discovered in $ e^+e^- \to K_SK^\pm\pi^\mp $ [54]. Both the mass and width can be well understood within the quark model. Our predictions of the strong decay properties are shown in Table 4. It shows that the predicted decay width $ \Gamma_{\rm total} = 167 $ MeV is in agreement with the measured value $ 150\pm 50 $ MeV [1]. Furthermore, our calculation shows that the decay of $ \phi(1680) $ is governed by the $ KK^*(892) $ mode, and its branching fraction can reach up to 81%, which is close to the other predictions [6, 16]. In addition, our predicted partial width ratio:

      $ \frac{\Gamma (KK)}{\Gamma (KK^*)}\approx0.06 $

      (16)

      is comparable with the DM1 measured value $ 0.07\pm0.01 $ [54]. Our prediction of

      $ R_{\eta\phi/KK^*} = \frac{\Gamma (\eta\phi)}{\Gamma (KK^*)}\approx0.20 $

      (17)

      is consistent with the predictions in Refs. [6, 16, 31]. However, it is about two times smaller than the measured value of 0.37 from the $ BaBar $ Collaboration [55]. To clarify the inconsistency in the ratio $ R_{\eta\phi/KK^*} $, more accurate measurements are expected to be carried out in future experiments.

    • B.   $ 1P $-wave $ s\bar{s} $ states

      1.   $ f_2'(1525) $
    • The $ f_2'(1525) $ resonance listed by the PDG is widely accepted as the $ 1^3P_2 $ $ s\bar{s} $ state. Both the mass and decay properties can be reasonably understood in the quark model [16, 17, 56]. Considering the $ f_2'(1525) $ as the $ 1^3P_2 $ $ s\bar{s} $ state, we calculate its OZI-allowed two-body strong decays by using the wave function obtained from our potential model. Our results are listed in Table 5. It is found that our predicted width,

      $ \Gamma_{\rm total} \simeq 58 \; \mathrm{MeV} $

      (18)

      is slightly smaller than the average value, $ \Gamma_{\rm exp} = (86\pm 5 )$ MeV, from the PDG [1]. The decay of $ f_2'(1525) $ is governed by the $ KK $ mode. Its branching fraction can reach up to

      $ Br[f_2'(1525)\to KK]\simeq 70 \%, $

      (E19)

      which is close to the value of 76% predicted in Refs. [16, 56] and the measured value of 88% from the PDG [1]. Furthermore, the branching fraction for the $ \eta\eta $ channel is predicted to be

      $ Br[f_2'(1525)\to \eta\eta]\simeq 9 \%. $

      (20)

      We also find that the branching fraction ratio,

      $ R_{\eta\eta/KK} = \frac{\Gamma (\eta\eta)}{\Gamma (KK)}\approx 13 \% $

      (21)

      is close to the average value of 11.5% from the PDG [1].

      It should be pointed out the $ KK^*(892) $ is another important decay mode of $ f_2'(1525) $, and the branching fraction may reach up to

      $ Br[f_2'(1525)\to KK^*(892)]\simeq 21 \%. $

      (22)

      A fairly large decay rate into the $ KK^*(892) $ final state is also predicted in Refs. [16, 57]. This important decay mode is hoped to be measured in future experiments. A small $ n\bar{n} = (u\bar{u}+d\bar{d})/\sqrt{2} $ component may exist in the $ f_2'(1525) $ resonance for a tiny decay rate into the $ \pi\pi $ decay channel [57].

    • 2.   $ h_1(1415) $
    • The $ h_1(1415) $ resonance is a convincing candidate for the $ 1^1P_1 $ $ s\bar{s} $ state in the quark model [21]. The recent BESIII measurements have greatly improved the accuracy of the observed mass and width of $ h_1(1415) $ by using the $ \chi_{cJ} $ [58] and $ J/\psi $ [10] decays. The most precise mass and width of $ h_1(1415) $ are measured to be $ M = (1423.2\pm 9.4) $ MeV and $ \Gamma = (90.3\pm 27.3) $ MeV, respectively [10]. Our predicted mass of $ M = 1462 $ MeV, together with other quark model predictions for the $ 1^1P_1 $ $ s\bar{s} $ state (see Table 2), is consistent with the observation. Due to strong suppression by the phase space factor, $ KK^*(892) $ is the only dominant decay mode. Considering the $ h_1(1415) $ as the $ 1^1P_1 $ $ s\bar{s} $ state with the physical mass $ M = 1416 $ MeV, the total width is predicted to be:

      $ \Gamma_{\rm total}\simeq\Gamma[h_1(1415)\to KK^*(892)]\simeq 141 \; \mathrm{MeV},$

      (23)

      which is comparable with the newest data, $ (90.3\pm 27.3) $ MeV from the BESIII Collaboration [10], and the average value of $ (90\pm 15) $ MeV from the PDG [1].

      It should be mentioned that in some works the $ h_1(1415) $ was suggested to be a dynamically generated resonance [59, 60], or that the triangle singularity might be relevant here [61].

    • 3.   The $ 1^3P_1 $ $ s\bar{s} $ state
    • The situation for the $ 1^3P_1 $ $ s\bar{s} $ state is ambiguous. In theory, its mass is estimated to be $ \sim 1.4-1.5 $ GeV [17, 19-21]. In this mass range there are two candidates, $ f_1(1420) $ and $ f_1(1510) $, although the $ f_1(1510) $ has yet to be firmly established. There are some long-standing puzzles about the nature of these $ 1^{++} $ isovector mesons [1, 32, 61, 62].

      Considering the $ f_1(1420) $ as the $ 1^3P_1 $ $ s\bar{s} $ state, our predicted mass $ M = 1492 $ MeV is about $ 70 $ MeV larger than the measured value of $ 1426 $ MeV [1]. In Ref. [62], it is mentioned that the physical resonance could be shifted to the $ f_1(1420) $ mass due to the presence of the $ KK^*(892) $ threshold, through a mechanism similar to that suggested in Ref. [63]. On the other hand, we analyze the decay properties of the $ f_1(1420) $ as the assignment of the $ 1^3P_1 $ state. Due to strong suppression by the phase space factor, $ KK^*(892) $ is the only dominant decay mode. The decay width is predicted to be:

      $ \Gamma_{\rm total}\simeq \Gamma[f_1(1420)\to KK^*(892)]\simeq 296 \; \mathrm{MeV},$

      (24)

      which is too large to be comparable with the measured width $ \Gamma_{\rm exp.}\simeq 55 $ MeV [1]. Our conclusion is consistent with the study in Ref. [16]. It should be mentioned that the resonance envelope may be distorted by the nearby $ KK^*(892) $ threshold, which may lead to a strong suppression of the resonance width [16]. To know the $ KK^*(892) $ threshold on the width, more theoretical studies are needed. In Ref. [64], the $ f_1(1420) $ was suggested to be a mixed $ 1^3P_1 $ state containing sizeable $ n\bar{n} $ components. Moreover, some unconventional explanations, such as a $ KK^* $ molecule [65], hybrid state [66], or the manifestation of the $ KK^* $ and $ \pi a_0(980) $ decay modes of the $ f_1(1285) $ [67], have also been proposed in the literature. More discussions on $ f_1(1420) $ can be found in Refs. [1, 32, 62].

      The $ f_1(1510) $ competes with the $ f_1(1420) $ to be the $ 1^3P_1 $ $ s\bar{s} $ state [1]. Considering the $ f_1(1510) $ as the $ 1^3P_1 $ $ s\bar{s} $ state, our predicted mass $ M = 1492 $ MeV is very close to the measured average value of $ (1518\pm 5) $ MeV [1]. In this case the decay width is predicted to be

      $ \Gamma_{\rm total}\simeq \Gamma[f_1(1510)\to KK^*(892)]\simeq 383\;\mathrm{MeV}, $

      (25)

      which is also too large to be comparable with the measured width $ \Gamma_{\rm exp.}\simeq (73\pm 25) $ MeV [1]. The $ f_1(1510) $ may not be well established [62]. If the $ f_1(1510) $ is well established, it may not be identified as the $ 1^3P_1 $ $ s\bar{s} $ state because of its narrow decay width.

      The world's largest $ J/\psi $ samples at BESIII may offer an opportunity for establishing the $ 1^3P_1 $ $ s\bar{s} $ state by observing the $ J/\psi $ radiative decays $ J/\psi\to \gamma X, X\to KK^*(892) $.

    • 4.   The $ 1^3P_0 $ $ s\bar{s} $ state
    • The $ 1^3P_0 $ $ s\bar{s} $ state is still not established. The identification of the isoscalar $ 0^{++} $ mesons observed from experiments is a long-standing puzzle [1]. In the quark potential model, the $ 1^3P_0 $ state should be the lightest of the three $ 1^3P_J $ states for a strong negative tensor interaction, which has been confirmed in the observations of the $ \chi_{cJ}(1P) $ and $ \chi_{bJ}(1P) $ ($ J = 0,1,2 $) states [1]. The mass splitting between $ \chi_{c2}(1P) $ and $ \chi_{c0}(1P) $ can reach up to $ \sim 150 $ MeV [1]. Considering this fact, we may conclude that the mass of the $ 1^3P_0 $ $ s\bar{s} $ state should be obviously lighter than the mass of the $ f_2'(1525) $ state ($ 1^3P_2 $ $ s\bar{s} $ state), which indicates that the $ f_0(1500) $ and $ f_0(1710) $ resonances listed by the PDG [1] cannot be identified as the $ 1^3P_0 $ $ s\bar{s} $ state.

      Our predicted mass for the $ 1^3P_0 $ $ s\bar{s} $ state is $ \sim 1370 $ MeV, which is consistent with the predictions in Refs. [17, 18, 20, 21]. Concerning the mass, the $ f_0(1370) $ is likely a candidate for the $ 1^3P_0 $ $ s\bar{s} $ state. However, the $ f_0(1370) $ decays mostly into pions ($ 2\pi $ and $ 4\pi $), which suggests it favors an $ n\bar{n} $ structure [1]. It should be mentioned that the experimental situation for $ f_0(1370) $ is rather fluid [1]. The $ f_0(1370) $ resonance may actually correspond to two different states with dominant $ n\bar{n} $ and $ s\bar{s} $ contents, respectively [21, 68]. Thus, some resonances with a mass around $ 1370 $ MeV observed in the $ KK $ channel might be good candidates for the $ 1^3P_0 $ $ s\bar{s} $ state.

      With our predicted mass $ M = 1373 $ MeV and wave function from the potential model, we study the strong decay properties of the $ 1^3P_0 $ $ s\bar{s} $ state. It is found that this state has a broad width,

      $ \Gamma_{\rm total}\simeq 338 \; \mathrm{MeV}, $

      (26)

      and it dominantly decays into the $ KK $ and $ \eta\eta $ final states with branching fractions ~83% and ~17%, respectively. The partial width ratio between $ \eta\eta $ and $ KK $ is predicted to be:

      $ R_{\eta\eta/KK} = \frac{\Gamma(\eta\eta)}{\Gamma(KK)}\simeq 0.20, $

      (27)

      which can be tested in future experiments. Recently, a scalar resonance $ f_0(1370) $ has been established in the $ KK $ final state from the $ J/\psi\to \gamma K^+K^-,K_S^0K_S^0 $ processes by using the data from CLEO [69] and BESIII [7], respectively. The measured mass and width of $ f_0(1370) $ are $ M = (1350\pm 9^{+12}_{-2}) $ MeV and $ \Gamma = (231\pm 21^{+28}_{-48}) $ MeV, respectively [7]. Both our predicted mass and width for the $ 1^3P_0 $ $ s\bar{s} $ state are consistent with the observations from CLEO and BESIII. One point which should be emphasized is that no obvious evidence for $ f_0(1370) $ is observed in the $ \pi\pi $ spectra of the $ J/\psi\to \gamma \pi^+\pi^-,\pi^0\pi^0 $ processes [69], which indicates that the scalar resonance $ f_0(1370) $ may be dominated by the $ s\bar{s} $