X(1870) and $\eta_2(1870)$: Which can be assigned as a hybrid state?

The mass spectrum and strong decays of the X(1870) and $\eta_2(1870)$ are analyzed. Our results indicate that X(1870) and $\eta_2(1870)$ are the two different resonances. The narrower X(1870) seems likely a good hybrid candidate. We support the $\eta_2(1870)$ as the $\eta_2(2^1D_2)$ quarkonium. We suggest to search the isospin partner of X(1870) in the channels of $J/\psi\rightarrow\rho f_0(980)\pi$ and $J/\psi\rightarrow\rho b_1(1235)\pi$ in the future. The latter channel is very important for testing the hybrid scenario.

In addition, the predicted masses of 0 −+ and 2 −+ glueball are much higher than 1.8 GeV by lattice gauge theory [17][18][19]. Therefore, X(1870) is not likely to be a glueball state. Moreover, the molecule and fourquark states are not expected in this region [16]. Then the unclear structure X(1870) looks more like a good hybrid candidate. But the actual situation is much complicated because the nature of η 2 (1870) is still ambiguous: (i) Since no evidences have been found in the decay mode of KKπ, the η 2 (1870) disfavors the 1 1 D 2 ss quarkonium assignment. The mass of η 2 (1870) seems much smaller for the 2 1 D 2 nn (nn ≡ (uū+dd)/ √ 2) state in the Godfrey-Isgur (GI) quark model [20]. Therefore the η 2 (1870) has been assigned as the 2 −+ hybrid state [10,[21][22][23].
(ii) However, Li and Wang pointed out that the mass, production, total decay width, and decay pattern of the η 2 (1870) do not appear to contradict with the picture of it as being the conventional 2 1 D 2 nn state [24].
Therefore, systematical study of the mass spectrum and strong decay properties is urgently required for X(1870) and η 2 (1870). Some valuable suggestions for the experiments in future are also needed.
The paper is organized as follows. In Sec.II, the masses of X(1870) and η 2 (1870) will be explored in the GI relativized quark model and the Regge trajectories (RTs) framework. In Sec.III, the decay processes that a isoscalar meson decays into light scalar (below 1 GeV) and pseudoscalar mesons will discussed. The two-body strong decays X(1870) and η 2 (1870) will be calculated within the 3 P 0 model and the flux-tube model. Finally, our discussions and conclusions will be presented in Sec.IV.

II. MASS SPECTRUM
In the Godfrey-Isgur relativized potential model [20], the Hamiltonian consists of the central potential and a kinetic term in a "relativized" form The funnel-shaped potentials which include a color coulomb term at short distances and a linear scalar confining term at large distances are usually incorporated as the zeroth-order potential. The typical funnel-shaped potential was proposed by the Cornell group (Cornell potential) with the form [25] V qq (r) = − 4 3 The strong coupling constant α s , the string tension σ and the constant C are the model parameters which can be fixed by the well established experimental states. The remaining spin-dependent terms for mass shifts are usually treated as the leading-order perturbations which include the spin-spin contact hyperfine interaction, spin-orbit and tensor interactions and a longer-ranged inverted spin-orbit term. They arise from one gluon exchange (OGE) forces and the assumed Lorentz scalar confinement. The expressions for these terms may be found in Ref. [20].
It should be pointed out that the nonperturbative contribution may dominate for the hyperfine splitting of light mesons, which is not like the heavy quarkonium [26]. For example, the hyperfine shift of the h c (1P) meson with respect to the center gravity of the χ c (1P) mesons is much small: [27]. However, for the light isovector mesons a 0 (1450), a 1 (1260), a 2 (1320), and b 1 (1235), the hyperfine shift is 76.7 ± 44.4 MeV. Here the masses of a 0 , a 1 , a 2 , and b 1 are taken from PDG [2]. For the complexities of nonperturbative interactions, then we are not going to calculate the hyperfine splitting. Now, the spin-averaged mass,M nl , of nL multiplet can be obtained by solving the spinless Salpeter equation Here we employ a variational approach described in Ref. [28] to solve the Eq.(3). This variational approach has been applied well in solving the Salpeter equation for cs [29], cc and bb [30] mass spectrum.
In the calculations, the basic simple harmonic oscillator (SHO) functions are taken as the trial wave functions. It is given by in the position space. Here the SHO function scale β is the variational parameter. By the Fourier transform, the SHO radial wave function in the momentum is The wave functions of ψ nl (r, β) and ψ nl (p, β) meet the normalization conditions: In the variational approach, the correspondingM nl are given by minimizing the expectation value of H where When all the parameters of the potential model are known, the values of the harmonic oscillator parameterβ can be fixed directly. With the values ofβ, all the spin-averaged massM nl will be obtained easily.M nl obtained in this way trend to be better for the higher-excited states [31].
It is unreasonable to treat the spin-spin contact hyperfine interaction as a perturbation for the ground states, because the mass splitting between pseudoscalar mesons and vector mesons are much large. Then we consider the contributions of V s· s (r) for the 1S mesons. The following Gaussian-smeared contact hyperfine interaction [32] is taken for convenience, In this work, we choose the model parameters as follows: m u = m d = 0.220 GeV, m s = 0.428 GeV, α s = 0.6, σ = 0.143 GeV 2 , κ = 0.37 GeV, and C = −0.37 GeV. We take the smaller value of σ here rather than the value in Ref. [20]. The smaller σ was obtained by the relation between the slope of the Regge trajectory for the Salpeter equation α ′ and the slope α ′ st in the string picture [26]. The Gaussian smearing parameter κ seems a little smaller than that in Ref. [20]. However, the κ is usually fitted by the hyperfine splitting of low-excited nS states in the literatures with a certain arbitrariness.
The values ofM nL andβ for the states 2S , 3S , 4S , 1P, 2P, 3P, 1D, 2D, 3D, 1F, 2F, 1G and 1H are listed in Table I. The experimental masses for the relative mesons are taken from PDG [2].  Obviously, the spin-averaged masses of the 2S , 1P, 1D nn and 1P, 2S ss mesons are consistent with the experimental data. Indeed, the predicted masses of higher excited states here are also reasonable, e.g., a 4 (2040) and f 4 (2050) are very possible the F−wave nn isovector and isoscalar mesons with the masses of 1996 +10 −9 MeV and 2018 ± 11MeV, respectively [2]. The predicted spin-averaged mass of 1F is not incompatible with experiments. Our results are also overall in good agreement with the expectations from Ref. [33]. The trend that a higher excited state corresponds to a smallerβ coincides with Ref. [34][35][36]. For considering the spin-spin contact hyperfine interaction, there are twoβs for the 1S mesons. The larger one corresponds to the 1 1 S 0 state, the smaller one the 1 3 S 1 state.
As shown in Ref. [33,37], the confinement potential V con f (r) is determinant for the properties of higher excited states. In Ref. [33], the masses for higher excited states with σ = 0.143GeV 2 and α s = 0 are closer to experimental data than the results given in Ref. [20]. Then we ignored the Coulomb interaction for 1D, 2D, 1F, 1G and 1H states. In this way,M nl for these states increase about 100MeV.
Four possible states for X(1870) As mentioned in the Introduction, X(1870) is also a good hybrid candidate since its mass overlaps the predictions given by different models. The predicted masses for 0 + 0 −+ , 0 + 1 ++ and 0 + 2 −+ nng states by these models are collected in Table  IV.
Since the masses of X(1870) and η 2 (1870) are nearly equal, the possible assignments of X(1870) also suit η 2 (1870). The investigations of the strong decay properties will be more helpful to distinguish the η 2 (1870) and X(1870).

A. The final mesons include the scalar mesons below 1 GeV
Despite many theoretical efforts, the scalar nonet of qq mesons has never well-established. The lowest-lying scalar mesons including σ(500) (or f 0 (600)), κ(800), a 0 (980) and f 0 (980) are difficult to be described as qq states, e.g., a 0 (980) is associated with nonstrange quarks in the qq scheme. If this is true, its high mass and decay properties are difficult to be understood simultaneously. So interpretations as exotic states were triggered, i.e., as two clusters of two quarks and two antiquarks [47], particular quasimolecular states [48], and uncorrelated four quark states qqqq [49][50][51] have been proposed.
Though the structures of these scalar mesons below 1 GeV are still in dispute, the viewpoint that these scalar mesons can constitute a complete nonet states has been reached in the most literatures (as illustrated in Fig.2). In the following, we will denote this nonet as " S " multiplet for convenience. Due to the unclear nature of the S mesons, it seems much difficult to study the decay processes when the final mesons includes a S member. As an approximation, a 0 (980), σ(500) and f 0 (980) were treated as 1 3 P 0 qq mesons in Refs. [43,52]. In Refs. [24,42], this kind of decay channel was ignored. However, this kind of decay mode maybe predominant for some mesons. For example, the observations indicate that f 1 (1285), η(1405) and X(1870) primarily decay via the a 0 (980)π channel [1].
In what follows, we will extract some useful information about this kind of decay mode by the SU(3) flavor symmetry. We will show that a 0 (980)π, σ 0 η and f 0 η are the main decay channels for the isoscalar nn and the nng mesons when they decay primarily through " S + P" mesons, where the sign "P" denotes a light pseudoscalar meson. This will explain why X(1870) has been first observed in the ηπ + π − channel.
We noticed that the S nonet could be interpreted like the qq nonet in the diquark-antidiquark scenario. In Wilczek and Jaffe's terminology [53,54], the S mesons consist of a "good" diquark and a "good" antidiquark. When u, d quarks forms a "good" diquark, it means that the two light quarks, u and d, could be treated as a quasiparticle in color3, flavor3 and the spin singlet. The "good" u, d diquark is usually denoted as [ud].
In the diquark-antidiquark limit, the parity of a tetraquark is determined by P = (−1) L 12−34 [55] where the L 12−34 refer to the relative angular momentum between two clusters. Thus the S mesons are the lightest tetraquark states in the diquarkantidiquark model with L 12−34 = 0. The S nonet in the full set of flavor representations is Because the SU(3) flavor symmetry is not exact, the two physical isoscalar mesons, σ 0 and f 0 , are usually the mixing states of the |8 I=0 and |1 I=0 states [47], When the mixing angle ϑ equals the so-called ideal mixing angle, i.e., ϑ = 54.74 • , the composition of the σ(500) and f 0 (980) are It seems that the deviation from the ideal mixing angle of the σ(500) and f 0 (980) is small [47]. In the following calculations, we will treat them in the ideal mixing scheme.
Under the SU (3) flavor assumption, all the members of the octet have the same basic coupling constant in one type of reaction, while the singlet member have a different coupling constant. Particularly, when a quarkonium decays into S and qq mesons, there are five independent coupling constants, i.e., g A88 , g A81 , g A18 , g B88 and g B11 , corresponding to five different channels In order to determine the relations between these coupling constants, we shall assume the process that the qq or qqg meson decays into a S and another qq mesons obeys the OZI (Okubo-Zweig-Iizuka) rule, i.e., the two quarks in the mother meson go into two daughter mesons, respectively. Therefore, there are four forbidden processes: With the help of the SU(3) Clebsch−Gordon coefficients [56], the ratios between the five coupling constants are extracted as It is well known that the physical states, η(548) and η ′ (958) are the mixture of the SU(3) flavor octet and singlet. They can be written in terms of a mixing angle, θ p , as follows The mixing angle θ p has been measured by various means. However, there is still uncertainty for θ p . An excellent fit to the tensor meson decay widths was performed under the SU(3) symmetry, and θ p ≃ −17 o was obtained [23]. In our calculation, θ p is taken as −17 o . The excited mixtures of nn and ss are denoted as In this scheme, the ideal mixing occurs with the choice of θ = 35.3 o . When ξ and ξ ′ decay into a S and pseudoscalar mesons, the relations of decay amplitudes are governed by the coefficients ζ 2 which are model-independent in the limitation of SU(3) f symmetry. With the coupling constants in hand, the coefficients ζ 2 of ξ and ξ ′ versus the mixing angle θ are shown in the Fig.3 and Fig.4. When ξ and ξ ′ occurs in the ideal mixing, the values of ζ 2 are presented in Table V. In the factorization framework, the decay difference of a hybrid and excited qq mesons comes from the spatial contraction [57]. Then the coefficients ζ 2 for hybrid states are same as these of qq quarkoniums.  Here the mixing of η(548) and η ′ (958) has been considered. It is sure that the ζ 2 are zero for the processes, ξ ′ → a 0 π, ξ ′ → ση and ξ ′ → ση ′ , since they are OZI-forbidden. ζ 2 of ξ ′ → f 0 η ′ hasn't been considered in Table V since X(1870) lies below the threshold of f 0 η ′ .
As illustrated in the Fig.3 and Fig.4, the primary decay channels of a ss or ssg predominant excitation are f 0 η and κK. If the deviation of θ from the ideal mixing angle is not large, X(1870) should be a nn or nng predominant state since X(1870) primarily decay via the a 0 (980)π channel. At present, only the ground 0 −+ and the 0 ++ isoscalar mesons deviate from the ideal mixing distinctly. In addition, if the X(1870) is produced via a diagram of Fig.1 [B], its should also be nn or nng predominant state.
Of course, the SU(3) f symmetry breaking will effect the ratios of these channels listed in Table V, because the threemomentum of the these products are different. However, the coefficients ζ 2 have presented the valuable information for these specific decay channels. When η 2 (1870) occupies the 2 1 D 2 nn state, X(1870) becomes a good nng candidate. In the following subsection, we will explore the two-body strong decays of X(1870) within the 3 P 0 model and the flux-tube model. Of course, the analysis of X(1870) also suit η 2 (1870) for their nearly equal masses. In Ref. [24], the 3 P 0 model [58][59][60] and the flux-tube model [61] were employed to study the two-body strong decays of η 2 (1870). There, the pair production (creation) strength γ and the simple harmonic oscillator (SHO) wave function scale parameter, βs, were taken as constants.
However, a series of studies indicate that the strength γ may depend on both the flavor and the relative momentum of the produced quarks [62,63]. γ may also depend on the reduced mass of quark-antiquark pair of the decaying meson [64]. Firstly, the relations of the 3 P 0 model to "microscopic" QCD decay mechanisms have been studied in Ref. [62]. There, the authors found that the constant γ corresponds approximately to the dimensionless combination, σ/m q β, where m q is the mass of produced quark, β means the meson wave function scale and σ is the string tension. Secondly, the momentum dependent manner of γ has been studied in Ref. [63]. It was found that γ is dependent on the relative momentum of the created qq pair, and the form of γ(k) = A + B exp(−Ck 2 ) with k = | k 3 − k 4 | was suggested. Thirdly, J.Segovia, et al., proposed that γ is a function of the reduced mass of quark-antiquark pair of the decaying meson [64]. Based on the first and third points above, γ will depend on the flavors of both the decaying meson and produced pairs. In our calculations, we will treat the γ as a free parameter and fix it by the well-measured partial decay widths.
In addition, the amplitudes given by the 3 P 0 model and the flux-tube model often contain the nodal-type Gaussian form factors which can lead to a dynamic suppression for some channels. Then the values of β are important to exact the decay width for the higher excited mesons in these two strong decay models.
In the following, the two-body strong decay of X(1870) will be investigated in the 3 P 0 model where the strength γ will be extracted by fitting the experimental data. The SHO wave function scale parameter, βs, will be borrowed from the Table  I which are extracted by the GI relativized potential model. We will also check the possibility of X(1870) as a possible hybrid state by the flux-tube model.
In the non relativistic limit, the transition operatorT of the 3 P 0 model is depicted aŝ where p represents the momentum of the outgoing meson in the rest frame of the meson A. When the mock state [65] is adopted to describe the spatial wave function of a meson, the helicity amplitude M M J A ,M J B ,M J C (p) can be constructed in the L − S basis easily [59,60]. The mock state for A meson is Finally, the decay width Γ(A → BC) is derived analytically in terms of the partial wave amplitudes More technical details of the 3 P 0 model can be found in Ref. [60]. The inherent uncertainties of the 3 P 0 decay model itself have been discussed in the Refs. [63,67,68].
The dimensionless parameter γ will be fixed by the 8 wellmeasured partial decay widths which are listed in TableVI  As mentioned before, γ may depend on the flavors of both the decaying meson and produced pairs. Then we divide the 8 decay channels into two groups: one is nn → nn + nn, the other includes ss → ns+ sn and nn → ns+ sn. The values of γ here are a little different from these given in Ref. [63] where an AL1 potential (for details of AL1 potential, see Ref. [31]) was selected to determine the meson wave functions. Of course, the meson wave function given by different potentials will influence the values of γ.
It is clear in Table VI that γ decrease with p increase. In addition, our calculation indicates that γ depend on flavors of both the decaying meson and the produced quark pairs. For example, values of γ fixed by a 2 → KK and f ′ 2 → KK are roughly equal.
In the following calculations, we assume that the values of γ corresponding to the processes of ss → ns + sn and nn → ns + sn are determined by one function. Similarly, we take the function, γ(p) = A + B exp(−C p 2 ), for the creation vertex. The function of the creation vertex here is different with the one used in the Ref [63]. With the four decay channels listed in fifth column of Table VI, we fix the function as γ(p) = 1.8 + 4 exp(−10p 2 ). For the processes of nn → nn + nn (the first column of Table VI), we fix the creation vertex function as γ(p) = 3.0 + 25 exp(−4p 2 ). The dependence of γ on the momentum p are plotted in the Fig. 5. Obviously the functions can describe the dependence of γ and p well. The functions of creation vertex given here need further test.
Since we neglected the mass splitting within the isospin multiplet, the partial width into the specific charge channel should be multiplied by the flavor multiplicity factor F (Table   VII). This F factor also incorporates the statistical factor 1/2 if the final state mesons B and C are identical (as illustrated in Fig.6). More details of F can be found in the Appendix A of Ref. [69]. Decay  Fig.6. The last column for the the flavor multiplicity factor F . Here, |η = (|nn − |ss )/ √ 2 and |η ′ = (|nn + |ss )/ √ 2 have been taken for simplicity.
Decay   [16,24]. The symbol "×" indicates that the decay modes are forbidden and "−" denotes that the decay channels can be ignored. Here, we collected the results given by the 3 P 0 model from Ref. [24] in the left column, the right column by the flux-tube model. In Ref. [16], the masses are taken as 1.8 GeV for the 0 −+ , 1 ++ and 2 −+ for the hybrid states.
In addition, our results do not support X(1870) as the η 2 (2 1 D 2 ) nn state since its observed decay width is much smaller than the theoretical estimate. The a 2 (1320)π is the largest decay channel in our numerical results and in Ref. [24] for the η 2 (2 1 D 2 ) nn state (Table VIII). If the partial width of a 0 (980)π channel is as large as a 2 (1320)π, the predicted width of X(1870) will be much larger than the observed value.
We adopt the flux tube model to check the possibility of X(1870) as a hybrid meson. The partial widths are also listed in Table VIII for the comparison. Details of the flux model are collected in the Appendix B.
Two groups of the partial widths predicted in the Ref. [16] are quoted in the Table VIII. The left column was given by the flux tube decay model of Isgur, Kokoski, and Paton (IKP) with the "standard parameters" [73]. The right column was by the developed flux tube decay model of Swanson-Szczepaniak (SS). In Ref. [16], the masses are taken as 1.8 GeV for the 0 −+ , 1 ++ and 2 −+ for the hybrid states.
For a hybrid meson, X(1870) seems most possible to be the η H (0 + 2 −+ ) state because the total widths exclude the channels of S + P are much narrow in our work and in Ref. [16]. It is consistent with the narrow width of X(1870).
As shown in Table VIII, X(1870) is impossible to be the η H (0 + 0 −+ ) hybrid state. The predicted width in both our work and in Ref. [16] are broader. In addition, ηπ is a visible channel for both a 0 (1450). A week signal was found in the region of 1200∼1400MeV in the analysis of ηπ ± (Fig.2(b) of Ref. [1]), which contradicts the large a 0 (1450)π channel of the η H (0 + 0 −+ ) state. We can exclude the possibility of X(1870) as the η H (0 + 0 −+ ) hybrid state preliminarily.
The assignment for X(1870) as the f H (0 + 1 ++ ) hybrid seems impossible since the theoretical width of a 1 (1260)π is rather broad in our results and in the IKP model. If the partial width of a 0 (980)π channel is as large as a 1 (1260)π, the total widths of X(1870) will be much broader than the experimental value. But the width given by the SS flux tube decay model for the f H (0 + 1 ++ ) hybrid is much small. So the possibility of X(1870) as a f H (0 + 1 ++ ) hybrid can not be excluded. We suggest to detect the decay channel of a 1 (1260)π because this channel is forbidden for the η H (0 + 2 −+ ) state in the IKP flux tube decay model and very small in the SS flux tube decay model (see TableVIII). Then the channel of a 1 (1260)π can discriminate the state f H (0 + 1 ++ ) and η H (0 + 2 −+ ) for X(1870).
Finally, if η 2 (1870) is the η 2 (2 1 D 2 ) state, its decay width is predicted about 100MeV which is much smaller than the experiments. However, the difference can be explained by the remedy of mixing effect. If X(1870) and η 2 (1870) have the same quantum numbers, 0 + 2 −+ , they should mix with each other with a visible mixing angle. Then the interference enhancement will enlarge the width of η 2 (1870). The broad decay width of η 2 (1870) could be explained naturally. On the other hand, η 2 (1870) has been observed in the channel of a 0 (980)π. However, this channel seems much small if η 2 (1870) is a pure 2 1 D 2 nn meson. The mixing effect will also enlarge this partial width. Here, we don't plan to discuss the mixing of X(1870) and η 2 (1870) further for the complex mechanism.

IV. DISCUSSIONS AND CONCLUSIONS
A isoscalar resonant structure of X(1870) was observed by BESIII in the channels J/ψ → ωX(1870) → ωηπ + π − recently. Although the mass of X(1870) is consistent with the η 2 (1870), the production, decay width and decay properties are much different. In this paper, the mass spectrum and strong decays of the X(1870) and η 2 (1870) are analyzed.
Firstly, the mass spectrum are studied in the GI potential model and the RTs framework. In the GI potential model, both X(1870) and η 2 (1870) could be the η ′ 2 (1 1 D 2 ), f ′ 1 (2 3 P 1 ) and η 2 (2 1 D 2 ) states. In RTs, the possible assignments are the η(3 1 S 0 ), f ′ 1 (2 3 P 1 ) and η 2 (2 1 D 2 ) states. For the mass spectrum, they are also good hybrid candidates since the masses overlap the predictions given by different models (see TableIV).
Secondly, the processes of a nn quarkonium or a nng hybrid meson decaying into the "S+ P" mesons are studied under the SU(3) f symmetry and the diquark-antidiquark description of the S mesons. We assumed the processes obey the OZI rule.
We find that the channels of a 0 π, ση and f 0 η are the dominant when a nn quarkonium or a nng hybrid meson decays primarily through this kind of processes. This result can explain why X(1870) has been first observed in the ηππ channel.
We also study the X(1870) as a hybrid state in the flux tube model. Our results agree well with most of predictions given by Ref. [16]. X(1870) looks most like the η H (0 + 2 −+ ) state for the narrow predicted width, which is consistent with the experiments. But we can't exclude the possibility of 0 + 1 ++ . A precise measurement of a 1 (1260)π is suggested to pin down this uncertainty.
(A15) m 1 and m 2 are the masses of quarks in the decaying meson A. m is the mass of the created quark from the vacuum. For calculating the decay widths, the masses of quarks are taken as: m u = m d = 0.220 GeV, m s = 0.428 GeV, which are as same as these in the Section II. The above amplitudes, M LS , can be reduced further in the approximation of m 1 = m 2 = m and β A = β B = β C = β. The reduced M LS are consistent with these given by Ref. [69] except for an unimportant factor, −2 9/2 , since this factor can be absorbed into the coefficient γ.

Appendix B: Hybrid decay in the flux tube model
The flux tube model was motivated by the strong coupling expansion of the lattice QCD. In this model, decay occurs when the flux-tube breaks at any point along its length, with a qq pair production in a relative J PC = 0 ++ state. It is similar to the 3 P 0 decay model but with an essential difference. The flux tube model extend the nonrelativistic constituent quark model to include gluonic degrees of freedom in a very simple and intuitive way, where the gluonic field is regarded as tubes of color flux. Then it can be extended to the hybrid research. When the hybrid mesons are assumed to be narrow, and the threshold effects aren't taken into account, the partial decay width Γ LS (H → BC) is given by the flux model as [77] whereM A ,M B ,M C are the "mock-meson" masses of A, B, C [61]. When a hybrid meson decay into P-wave and pseudoscalar mesons, the partial wave amplitude M L (H → BC) (with S = S B ) is given as the following form The flavor matrix element φ B φ C |φ A φ 0 have been discussed before.M L (H → BC) are listed in Table IX for the states of η H (0 + 0 −+ ), f H (0 + 1 ++ ) and η H (0 + 2 −+ ). Here the M S , M D , M P i and M F are defined as M S = −(3h 0 −g 1 + 4h 2 ), M D = (g 1 + 5h 2 ), M P 1 = −i(2g 0 + 3h 1 −g 2 ), M P 2 = −i(g 0 +g 2 ), M P 4 = −i(10g 0 + 9h 1 +g 2 ) and M F = −3i(g 2 +h 3 ). The analytical expressions ofg i andh i are given as where 1 F 1 [· · · ] are the confluent hypergeometric functions.
Here we don't take account of the decay channels of H → 2S + 1S because they are forbidden by the conservation laws, or the "spin selection rule", or the phase space, e.g., the decay channel of π(1300) + π is forbidden for the f H (0 + 1 ++ ) state by the "spin selection rule". In this work, we choose to follow the Refs. [77] and take the combination (ac/9