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

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.

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) $:

where

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

where

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:

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} $,

with

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.

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.

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:

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

with the integral in momentum space:

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

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]:

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:

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.