𝑋(1870) and 𝜂₂(1870): Which Can Be Assigned as a Hybrid State?

Abstract

The mass spectrum and strong decays of the 𝑋(1870) and 𝜂₂(1870) are analyzed. Our results indicate that 𝑋(1870) and 𝜂₂(1870) are the two different resonances. The narrower 𝑋(1870) seems likely a good hybrid candidate. We support the 𝜂₂(1870) as the 𝜂₂(2¹𝐷₂) quarkonium. We suggest to search the isospin partner of 𝑋(1870) in the channels of 𝐽/𝜓⟶𝜌𝑓₀(980)𝜋 and 𝐽/𝜓⟶𝜌𝑏₁(1235)𝜋 in the future. The latter channel is very important for testing the hybrid scenario.

1. Introduction

An isoscalar resonant structure of 𝑋(1870) was observed by the BESIII Collaboration with a statistical significance of 7.2𝜎 in the processes 𝐽/𝜓→𝜔𝑋(1870)→𝜔𝜂𝜋⁺𝜋⁻ recently [1]. Its mass and width were given as 𝑀=1877.3±6.3^+3.4_−7.4MeV,Γ=57±12⁺¹⁹₋₄MeV. (1) Here the first errors are statistical and the second ones are systematic. The product branching fraction of ℬ(𝐽/𝜓→𝜔𝑋(1870))⋅ℬ(𝑋(1870)→𝑎^±₀(980)𝜋^∓)⋅ℬ(𝑎^±₀(980)→𝜂𝜋^±)=[1.5±0.26(stat)^+0.72_−0.36(syst)]×10⁻⁴ was also presented [1]. But the quantum numbers of 𝑋(1870) are still unknown, then the partial wave analysis is required in future.

The mass of 𝑋(1870) is consistent with the 𝜂₂(1870), but the width is much narrower than the 𝜂₂(1870). In the tables of the Particle Data Group (PDG) [2], the available mass and width of 𝜂₂(1870) are 𝑀=1842±8MeV,Γ=225±14MeV. (2)

The 𝜂₂(1870) has been observed in 𝛾𝛾 reactions [3, 4], 𝑝𝑝 annihilation [5–8], and radiative 𝐽/𝜓 decays [9]. It should be stressed that radiative 𝐽/𝜓 decay channels (Figure 1(a)) and 𝑝𝑝 annihilation processes are the ideal glueball hunting grounds. But the glueball production is suppressed in 𝛾𝛾 reaction. By contrast, the hadronic 𝐽/𝜓 decay are considered “hybrid rich” (Figure 1(b)).

(a)

(b)

Figure 1 

(a) A prior production of glueball in the 𝐽/𝜓→𝛾𝑋𝐺; (b) a prior production of hybrid in the 𝐽/𝜓→𝜔𝑋𝐻.

Furthermore, the branching ratio ℛ₁=Γ(𝜂₂(1870)→𝑎₂(1320)𝜋)/Γ(𝜂₂(1870)→𝑎₀(980)𝜋)=32.6±12.6 reported by the WA102 Collaboration indicates that the decay channel of 𝑎₀(980)𝜋 is tiny for 𝜂₂(1870) [7]. This has been confirmed by an extensive reanalysis of the Crystal Barrel data [10]. Differently, the analysis of BESIII Collaboration indicates that the 𝑋(1870) primarily decays via the 𝑎₀(980)𝜋 channel [1]. Then the present measurements of the decay widths, productions, and decay properties suggest that 𝜂₂(1870) and 𝑋(1870) are two different isoscalar mesons.

If the production process 𝐽/𝜓→𝜔𝑋(1870) is mainly hadronic, the quantum numbers of 𝑋(1870) should be 0⁺0⁻⁺, 0⁺1⁺⁺, or 0⁺2⁻⁺. One notices that the predicted masses for the light 0⁺0⁻⁺, 0⁺1⁺⁺ and 0⁺2⁻⁺ hybrids overlap 1.8 GeV in the Bag model [11, 12], the flux tube model [13, 14], and the constituent gluon model [15]. In addition, the decay width of isoscalar 2⁻⁺ hybrid is expected to be narrow [16]. Therefore, 𝑋(1870) becomes a possible 2⁻⁺ hybrid candidate.

In addition, the predicted masses of 0⁻⁺ and 2⁻⁺ glueball are much higher than 1.8 GeV by lattice gauge theory [17–19]. Therefore, 𝑋(1870) is not likely to be a glueball state. Moreover, the molecule and four-quark 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 𝜂₂(1870) is still ambiguous.(i)Since no lines of evidence have been found in the decay mode of 𝐾𝐾𝜋, the 𝜂₂(1870) disfavors the 1¹𝐷₂𝑠𝑠 quarkonium assignment. The mass of 𝜂₂(1870) seems much smaller for the 2¹𝐷₂𝑛𝑛 (𝑛𝑛≡(𝑢𝑢+𝑑𝑑)/√2) state in the Godfrey-Isgur (GI) quark model [20]. Therefore the 𝜂₂(1870) has been assigned as the 2⁻⁺ hybrid state [10, 21–23]. (ii)However, Li and Wang pointed out that the mass, production, total decay width, and decay pattern of the 𝜂₂(1870) do not appear to contradict with the picture of it as being the conventional 2¹𝐷₂𝑛𝑛 state [24].

Therefore, systematical study of the mass spectrum and strong decay properties is urgently required for 𝑋(1870) and 𝜂₂(1870). Some valuable suggestions for the experiments in future are also needed.

The paper is organized as follows. In Section 2, the masses of 𝑋(1870) and 𝜂₂(1870) will be explored in the GI relativized quark model and the Regge trajectories (RTs) framework. In Section 3, the decay processes that an isoscalar meson decays into light scalar (below 1 GeV) and pseudoscalar mesons will be discussed. The two-body strong decays of 𝑋(1870) and 𝜂₂(1870) will be calculated within the ³𝑃₀ model and the flux-tube model. Finally, our discussions and conclusions will be presented in Section 4.

2. Mass Spectrum

In the Godfrey-Isgur relativized potential model [20], the Hamiltonian 𝐻 consists of a central potential and the kinetic term in a “relativized” form 𝐻=𝑝²_𝑞+𝑚²_𝑞+𝑝²_𝑞+𝑚²_𝑞+𝑉_𝑞𝑞(𝑟). (3)

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 [29] 𝑉_𝑞𝑞(𝑟)4=−3𝛼_𝑠𝑟+𝜎𝑟+𝐶. (4)

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 [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 [30]. 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: 𝑀_cog(𝜒_𝑐)−𝑀(ℎ_𝑐)=−0.02±0.19±0.13 MeV [31]. However, for the light isovector mesons 𝑎₀(1450), 𝑎₁(1260), 𝑎₂(1320), and 𝑏₁(1235), the hyperfine shift is 76.7±44.4 MeV. Here the masses of 𝑎₀, 𝑎₁, 𝑎₂, and 𝑏₁ are taken from PDG [2]. For the complexities of nonperturbative interactions, 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: 𝑝²_𝑞+𝑚²_𝑞+𝑝²_𝑞+𝑚²_𝑞+𝑉_𝑞𝑞(𝑟)𝜓(𝑟)=𝐸𝜓(𝑟). (5)

Here we employ a variational approach described in [32] to solve (5). This variational approach has been applied well in solving the Salpeter equation for 𝑐𝑠 [33], 𝑐𝑐 and 𝑏𝑏 [34] mass spectrum.

In the calculations, the basic simple harmonic oscillator (SHO) functions are taken as the trial wave functions. It is given by 𝜓_𝑛𝑙𝑟,𝛽=𝛽^3/22(2𝑛−1)!Γ(𝑛+𝑙+(1/2))𝛽𝑟^𝑙𝑒^{−𝛽2𝑟2/2}𝐿^𝑙+1/2_𝑛−1𝛽²𝑟² (6) 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 𝜓_𝑛𝑙=()𝑝,𝛽−1^𝑛𝛽^3/22(2𝑛−1)!Γ((𝑝𝑛+𝑙+1/2))𝛽^𝑙𝑒^{−𝑝2/2𝛽2}𝐿^𝑙+1/2_𝑛−1𝑝²𝛽². (7)

The wave functions of 𝜓_𝑛𝑙(𝑟,𝛽) and 𝜓_𝑛𝑙(𝑝,𝛽) meet the normalization conditions: ^∞₀𝜓²_𝑛𝑙𝑟,𝛽𝑟²𝑑𝑟=1,^∞₀𝜓²_𝑛𝑙𝑝,𝛽𝑝²𝑑𝑝=1. (8)

In the variational approach, the corresponding 𝑀_𝑛𝑙 are given by minimizing the expectation value of 𝐻𝑑𝐸𝑑𝛽^𝑛𝑙𝛽=0, (9) where 𝐸_𝑛𝑙𝛽≡⟨𝐻⟩_𝑛𝑙=𝜓_𝑛𝑙|𝐻|𝜓_𝑛𝑙. (10)

When all the parameters of the potential model are known, the values of the harmonic oscillator parameter 𝛽 can be fixed directly. With the values of 𝛽, all the spin-averaged masses 𝑀_𝑛𝑙 will be obtained easily. 𝑀_𝑛𝑙 obtained in this way trend to be better for the higher-excited states [35].

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_{⃗⃗𝑠𝑠⋅}(𝑟) for the 1𝑆 mesons. The following Gaussian-smeared contact hyperfine interaction [36] is taken for convenience: 𝑉^⃗⃗𝑠_𝑞^𝑠⋅_𝑞(𝑟)=32𝜋𝛼_𝑠9𝑚²_𝑞𝜅√𝜋³𝑒^{−𝜅2𝑟2}𝑆_𝑞⋅𝑆_𝑞. (11)

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 GeV², 𝜅=0.37 GeV, and 𝐶=−0.37 GeV. We take the smaller value of 𝜎 here rather than the value in [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 [30]. The Gaussian smearing parameter 𝜅 seems a little smaller than that in [20]. However, the 𝜅 is usually fitted by the hyperfine splitting of low-excited 𝑛𝑆 states in the works of literature with a certain arbitrariness.

The values of 𝑀_𝑛𝐿 and 𝛽 for the states 2𝑆, 3𝑆, 4𝑆, 1𝑃, 2𝑃, 3𝑃, 1𝐷, 2𝐷, 3𝐷, 1𝐹, 2𝐹, 1𝐺, and 1𝐻 are listed in Table 1. The experimental masses for the relative mesons are taken from PDG [2].

Table 1 

The spin-averaged mass (unit: GeV) and the harmonic oscillator parameter 𝛽 (unit: GeV⁻¹) of the states 2𝑆, 3𝑆, 4𝑆, 1𝑃, 2𝑃, 3𝑃, 1𝐷, 2𝐷, 3𝐷, 1𝐹, 2𝐹, 1𝐺, and 1𝐻.

Obviously, the spin-averaged masses of the 2𝑆, 1𝑃, 1𝐷𝑛𝑛 and 1𝑃, and 2𝑆𝑠𝑠 mesons are consistent with the experimental data. Indeed, the predicted masses of higher excited states here are also reasonable, for example, 𝑎₄(2040) and 𝑓₄(2050) are very possible the 𝐹-wave 𝑛𝑛 isovector and isoscalar mesons with the masses of 1996⁺¹⁰₋₉ MeV and 2018±11 MeV, 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 [37]. The trend that a higher excited state corresponds to a smaller 𝛽 coincides with [38–40]. For considering the spin-spin contact hyperfine interaction, there are two 𝛽s for the 1𝑆 mesons. The larger one corresponds to the 1¹𝑆₀ state and the smaller one the 1³𝑆₁ state.

As shown in [37, 41], the confinement potential 𝑉_conf(𝑟) is determinant for the properties of higher excited states. In [37], the masses for higher excited states with 𝜎=0.143 GeV² and 𝛼_𝑠=0 are closer to experimental data than the results given in [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 100 MeV.

The masses of 𝜂^′(3¹𝑆₀), 𝑓^′₁(2³𝑃₁), 𝜂^′₂(1¹𝐷₂), and 𝜂₂(2¹𝐷₂) are usually within 1.8~2.1 GeV in various quark potential models [20, 25–27] (see in Table 2). The predicted spin-averaged masses of 3𝑆(𝑠𝑠), 2𝑃(𝑠𝑠), 1𝐷(𝑠𝑠), and 2𝐷(𝑛𝑛) are also within this mass regions (bold ones in Table 1). Due to the uncertainty of the potential models, absolute deviation from experimental data are usually about 100~150 MeV for the higher excited states. Compared with these predicted masses, 𝑋(1870) disfavors the 𝜂^′(3¹𝑆₀) assignment for its low mass. But the possibilities of 𝑓^′₁(2³𝑃₁), 𝜂^′₂(1¹𝐷₂), and 𝜂₂(2¹𝐷₂) still exist. Here we do not consider the possibility of 𝑋(1870) as the 𝜂(3¹𝑆₀) state because 𝜂(1760) looks more like a good 𝜂(3¹𝑆₀) candidate [42–44].

Table 2 

The masses predicted for 3¹𝑆₀(𝜂^′), 2³𝑃₁(𝜂^′), 1¹𝐷₂(𝜂^′), and 2¹𝐷₂(𝜂) in [20, 25–27]. The value denoted by “^†” was predicted for the mass of isovector 2¹𝐷₂ state in [20].

Regge trajectories (RTs) are another useful tool for studying the mass spectrum of the light flavor mesons. In [45], the authors fitted the RTs for all light-quark meson states listed in the PDG tables. A global description was constructed as 𝑀²=1.38(4)𝑛+1.12(4)𝐽−1.25(4). (12)

Here, 𝑛 and 𝐽 mean the the radial and angular-momentum quantum number. Recently, the authors of [45] repeated their fits with the subset mesons of the paper [46]. They found a little smaller averaged slopes of 𝜇²=1.28(5) GeV² and 𝛽²=1.09(6) GeV² to be compared with 𝜇²=1.38(4) GeV² and 𝛽²=1.12(4) GeV² in (12). Here the 𝜇² and 𝛽² are the weighed averaged slopes for radial and angular-momentum RTs [45, 47].

Now ℎ₁(1380), 𝑓₁(1420), and 𝜂^′(1475) have been established as the 1¹𝑃₁, 1³𝑃₁, and 2¹𝑆₀𝑠𝑠 states in PDG [2]. With the differences between the mass squared of 𝑋(1870) and these states (Table 3), 𝑋(1870) could be assigned for the 𝜂^′(3¹𝑆₀) and 𝑓^′₁(2¹𝑃₁). The mass of 𝑋(1870) is too large for the 𝜂^′₂(1¹𝐷₂) state in the RTs. 𝜂₂(1645) has been assigned as the 1¹𝐷₂𝑛𝑛 meson [2]. Since 𝑀²(𝑋(1870))−𝑀²(𝜂₂(1640))=0.91^+0.04_−0.03 GeV² which is much smaller than 1.38(4) GeV², 𝑋(1870) looks unlike the 2¹𝐷₂𝑛𝑛 state for its low mass. However, the difference of 𝑀²(𝑋(1870))−𝑀²(ℎ₁(1170))=2.16^+0.06_−0.05 GeV² matches the slopes 2.37(11) GeV² well. Then the RTs can not exclude the possibility of 𝑋(1870) as the 2¹𝐷₂𝑛𝑛 state.

Table 3 

𝑋(1870) calculated in RTs for different states. The masses of 𝜂^′(1475), 𝑓₁(1420), ℎ₁(1380), and 𝜂₂(1645) are taken from PDG [2].

As mentioned in Section 1, 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 4.

Table 4 

The masses predicted for 𝜂_𝐻(0⁺0⁻⁺), 𝑓_𝐻(0⁺1⁺⁺) and 𝜂_𝐻(0⁺2⁻⁺) hybrid states in [11–15].

In this section, the mass of 𝑋(1870) has been studied in the GI quark potential model and the RTs framework. In the GI quark potential model, 𝑋(1870) can be interpreted as the 𝑓^′₁(2³𝑃₁), 𝜂^′₂(1¹𝐷₂), or 𝜂₂(2¹𝐷₂) state with a reasonable uncertainty. In the RTs, 𝑋(1870) favors the 𝜂^′(3¹𝑆₀) and 𝑓^′₁(2³𝑃₁) assignments. But the 𝜂₂(2¹𝐷₂) assignment can not be excluded thoroughly. 𝑋(1870) is also a good hybrid state candidate. Since the masses of 𝑋(1870) and 𝜂₂(1870) are nearly equal, the possible assignments of 𝑋(1870) also suit 𝜂₂(1870). The investigations of the strong decay properties will be more helpful to distinguish the 𝜂₂(1870) and 𝑋(1870).

3. The Strong Decay

3.1. The Final Mesons Include the Scalar Mesons below 1 GeV

Despite many theoretical efforts, the scalar nonet of 𝑞𝑞 mesons has never established well. The lowest-lying scalar mesons including 𝜎(500) (or 𝑓₀(600)), 𝜅(800), 𝑎₀(980), and 𝑓₀(980) are difficult to be described as 𝑞𝑞 states; for example, 𝑎₀(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, For examples, two clusters of two quarks and two antiquarks [48], particular quasi-molecular states [49–51], and uncorrelated four-quark states 𝑞𝑞𝑞𝑞  [52–54] 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 most works of the literature (as illustrated in Figure 2). In the following, we will denote this nonet as “𝒮” multiplet for convenience.

Figure 2 

The “𝒮” nonet below 1 GeV shown in 𝑌-𝐼3 plane.

Due to the unclear nature of the 𝒮 mesons, it seems much difficult to study the decay processes when the final mesons include a 𝒮 member. As an approximation, 𝑎₀(980), 𝜎(500), and 𝑓₀(980) were treated as 1³𝑃₀𝑞𝑞 mesons in [44, 55]. In [24, 43], this kind of decay channel was ignored. However, this kind of decay mode may be predominant for some mesons. For example, the observations indicate that 𝑓₁(1285), 𝜂(1405), and 𝑋(1870) primarily decay via the 𝑎₀(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 𝑎₀(980)𝜋, 𝜎₀𝜂, and 𝑓₀𝜂 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 [56, 57], the 𝒮 mesons consist of a “good” diquark and a “good” antidiquark. When 𝑢, 𝑑 quarks form a “good” diquark, it means that the two light quarks, 𝑢 and 𝑑, could be treated as a quasiparticle in color 3, flavor 3, 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)^𝐿12−34 [58] where the 𝐿₁₂₋₃₄ refers to the relative angular momentum between two clusters. Thus the 𝒮 mesons are the lightest tetraquark states in the diquark-antidiquark model with 𝐿₁₂₋₃₄=0. The 𝒮 nonet in the full set of flavor representations is (3⊗3)₃⊗3⊗3₃=8⊕1. (13)

Because the SU(3) flavor symmetry is not exact, the two physical isoscalar mesons, 𝜎₀ and 𝑓₀, are usually the mixing states of the |8⟩_𝐼=0 and |1⟩_𝐼=0 states [48], 𝑓₀𝜎₀=cos𝜗sin𝜗|8⟩−sin𝜗cos𝜗|1⟩^𝐼=0_𝐼=0. (14) When the mixing angle 𝜗 equals the so-called ideal mixing angle, that is, 𝜗=54.74^∘, the composition of the 𝜎(500) and 𝑓₀(980) is 𝑓₀𝜎₀=1√2[𝑠𝑢][𝑠𝑢]+[𝑠𝑑]𝑠𝑑[𝑢𝑑]𝑢𝑑. (15)

It seems that the deviation from the ideal mixing angle of the 𝜎(500) and 𝑓₀(980) is small [48]. 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 has a different coupling constant. Particularly, when a quarkonium decays into 𝒮 and 𝑞𝑞 mesons, there are five independent coupling constants, that is, 𝑔_𝐴88, 𝑔_𝐴81, 𝑔_𝐴18, 𝑔_𝐵88, and 𝑔_𝐵11, corresponding to five different channels |8⟩_𝑞𝑞⟶|8⟩_𝒮⊗|8⟩_𝑞𝑞∶𝑔_𝐴88,|8⟩_𝑞𝑞⟶|8⟩_𝒮⊗|1⟩_𝑞𝑞∶𝑔_𝐴81,|8⟩_𝑞𝑞⟶|1⟩_𝒮⊗|8⟩_𝑞𝑞∶𝑔_𝐴18,|1⟩_𝑞𝑞⟶|8⟩_𝒮⊗|8⟩_𝑞𝑞∶𝑔_𝐵88,|1⟩_𝑞𝑞⟶|1⟩_𝒮⊗|1⟩_𝑞𝑞∶𝑔_𝐵11. (16)

In order to determine the relations between these coupling constants, we will assume the process that the 𝑞𝑞 or 𝑞𝑞𝑔 meson decays into a 𝒮 and another 𝑞𝑞 mesons obey the OZI (Okubo-Zweig-Iizuka) rule; that is, the two quarks in the mother meson go into two daughter mesons, respectively. Therefore, there are four forbidden processes: 𝑋((1/√2)(𝑢𝑢−𝑑𝑑))↛𝑎₀+𝑠𝑠, 𝑋((1/√2)(𝑢𝑢+𝑑𝑑))↛𝜎₀+𝑠𝑠, 𝑋(𝑠𝑠)↛𝑓₀+(1/√2)(𝑢𝑢+𝑑𝑑) and 𝑋(𝑠𝑠)↛𝜎₀+𝑠𝑠. With the help of the SU(3) Clebsch-Gordan coefficients [59], the ratios between the five coupling constants are extracted as 𝑔_𝐴81∶𝑔_𝐴18∶𝑔_𝐵88∶𝑔_𝐵11∶𝑔_𝐴88=√2∶−25√5+1∶−2√5√∶−5+125√5+1∶1≈1.41∶−2.05∶−2.89∶−2.05∶1.00. (17)

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: 𝜂()𝜂548_′()=958cos𝜃_𝑝−sin𝜃_𝑝sin𝜃_𝑝cos𝜃_𝑝|8⟩_𝐼=0|1⟩_𝐼=0. (18)

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^∘ was obtained [23]. In our calculation, 𝜃_𝑝 is taken as −17^∘. The excited mixtures of 𝑛𝑛 and 𝑠𝑠 are denoted as 𝜉𝜉^′=cos𝜃sin𝜃|1⟩−sin𝜃cos𝜃|8⟩^𝐼=0_𝐼=0. (19)

In this scheme, the ideal mixing occurs with the choice of 𝜃=35.3^∘. When 𝜉 and 𝜉^′ decay into a 𝒮 and pseudoscalar mesons, the relations of decay amplitudes are governed by the coefficients 𝜁² which are model-independent in the limitation of SU(3)_𝑓 symmetry. With the coupling constants in hand, the coefficients 𝜁² of 𝜉 and 𝜉^′ versus the mixing angle 𝜃 are shown in the Figures 3 and 4. When 𝜉 and 𝜉^′ occur in the ideal mixing, the values of 𝜁² are presented in Table 5. In the factorization framework, the decay difference of a hybrid and excited 𝑞𝑞 mesons comes from the spatial contraction [60]. Then the coefficients 𝜁² for hybrid states are the same as these of 𝑞𝑞 quarkonia.

Table 5 

The coefficients 𝜁² of 𝜉 and 𝜉^′ in the ideal mixing.

Figure 3 

The coefficients 𝜁2 of the isoscalar meson 𝜉 versus the mixing angle 𝜃.

Figure 4 

The coefficients 𝜁2 of the isoscalar meson 𝜉′ versus the mixing angle 𝜃.

Here the mixing of 𝜂(548) and 𝜂^′(958) has been considered. It is sure that the 𝜁² are zero for the processes 𝜉^′→𝑎₀𝜋, 𝜉^′→𝜎𝜂, and 𝜉^′→𝜎𝜂^′, since they are OZI-forbidden. 𝜁² of 𝜉^′→𝑓₀𝜂^′ has not been considered in Table 5 since 𝑋(1870) lies below the threshold of 𝑓₀𝜂^′.

As illustrated in Figures 3 and 4, the primary decay channels of a 𝑠𝑠 or 𝑠𝑠𝑔 predominant excitation are 𝑓₀𝜂 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 decays via the 𝑎₀(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 Figure 1(b), it should also be 𝑛𝑛 or 𝑛𝑛𝑔 predominant state.

Of course, the SU(3)_𝑓 symmetry breaking will affect the ratios of these channels listed in Table 5, because the three-momentum of the these products are different. However, the coefficients 𝜁² have presented the valuable information for these specific decay channels. When 𝜂₂(1870) occupies the 2¹𝐷₂𝑛𝑛 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 ³𝑃₀ model and the flux-tube model. Of course, the analysis of 𝑋(1870) also suits 𝜂₂(1870) for their nearly equal masses.

3.2. The Strong Decays of 𝜂₂(1870) and 𝑋(1870)

In [24], the ³𝑃₀ model [61–63] and the flux-tube model [64] were employed to study the two-body strong decays of 𝜂₂(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 [28, 65]. 𝛾 may also depend on the reduced mass of quark-antiquark pair of the decaying meson [66]. Firstly, the relations of the ³𝑃₀ model to “microscopic” QCD decay mechanisms have been studied in [65]. There, the authors found that the constant 𝛾 corresponds approximately to the dimensionless combination, 𝜎/𝑚_𝑞𝛽, 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 [28]. It was found that 𝛾 is dependent on the relative momentum of the created 𝑞𝑞 pair, and the form of 𝛾(𝑘)=𝐴+𝐵exp(−𝐶𝑘²) with 𝑘=|𝑘₃−𝑘₄| was suggested. Thirdly, Segovia et al. proposed that 𝛾 is a function of the reduced mass of quark-antiquark pair of the decaying meson [66]. 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 ³𝑃₀ 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 extract 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 ³𝑃₀ model where the strength 𝛾 will be extracted by fitting the experimental data. The SHO wave function scale parameter, 𝛽s, will be borrowed from Table 1 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 nonrelativistic limit, the transition operator 𝒯 of the ³𝑃₀ model is depicted as 𝒯=−3𝛾_𝑚⟨⟩×1,𝑚;1,−𝑚∣0,0𝑑³𝑘₃𝑑³𝑘₄𝛿³𝑘₃+𝑘₄×𝒴^𝑚₁𝑘₃−𝑘₄2𝜔₀^(3,4)𝜑₀^(3,4)𝜒^(3,4)_1,−𝑚𝑑^†_3𝑖𝑘₃𝑑^†_4𝑗𝑘₄, (20) where the 𝜔₀^(3,4) and 𝜑₀^(3,4) are the color and flavor wave functions of the 𝑞₃𝑞₄ pair created from vacuum. Thus, 𝜔₀^(3,4)=(𝑅𝑅+𝐺𝐺+𝐵𝐵)/√3, 𝜑₀^(3,4)=(𝑢𝑢+𝑑𝑑+𝑠𝑠)/√3 are color and flavor singlets. The pair is also assumed to carry the quantum numbers of 0⁺⁺, suggesting that they are in a ³𝑃₀ state. Then 𝜒^(3,4)_1,−𝑚 represents the pair production in a spin triplet state. The solid harmonic polynomial 𝒴^𝑚₁(𝑘)≡|𝑘|𝒴^𝑚₁(𝜃_𝑘,𝜙_𝑘) reflects the momentum-space distribution of the 𝑞₃𝑞₄.

The helicity amplitude ℳ^{𝑀𝐽𝐴,𝑀𝐽𝐵,𝑀𝐽𝐶}(𝑝) of 𝐴→𝐵+𝐶 is given by 𝐵𝐶𝒯𝐴=𝛿³𝑃_𝐴−𝑃_𝐵−𝑃_𝐶ℳ^{𝑀𝐽𝐴,𝑀𝐽𝐵,𝑀𝐽𝐶}𝑝, (21) where 𝑝 represents the momentum of the outgoing meson in the rest frame of the meson 𝐴. When the mock state [67] is adopted to describe the spatial wave function of a meson, the helicity amplitude ℳ^{𝑀𝐽𝐴,𝑀𝐽𝐵,𝑀𝐽𝐶}(𝑝) can be constructed in the 𝐿−𝑆 basis easily [62, 63]. The mock state for 𝐴 meson is 𝐴𝑛_𝐴^2𝑆𝐴+1𝐿^{𝐽𝐴𝑀𝐽𝐴}_𝐴𝑃_𝐴≡2𝐸_{𝐴𝑀𝐿𝐴𝑀𝑆𝐴}𝐿_𝐴𝑀_𝐿𝐴𝑆_𝐴∣𝐽_𝐴𝑀_𝐽𝐴𝜔_𝐴¹²𝜙_𝐴¹²𝜒_𝑆¹²_{𝐴𝑀𝑆𝐴}×𝑑𝑃_𝐴𝜓^{𝐿𝐴𝑀𝐿𝐴}_𝑛𝐴𝑘₁,𝑘₂𝑞₁𝑘₁𝑞₂𝑘₂. (22)

To obtain the analytical amplitudes, the SHO wave functions are usually employed for 𝜓^{𝐿𝐴𝑀𝐿𝐴}_𝑛𝐴(𝑘₁,𝑘₂). For comparison with experiments, one obtains the partial decay width ℳ^𝐽𝐿(𝑝) via the Jacob-Wick formula [68]: ℳ_𝐿𝑆𝑝=√2𝐿+12𝐽_𝐴+1_{𝑀𝐽𝐵,𝑀𝐽𝐶}𝐿0𝐽𝑀_𝐽𝐴∣𝐽_𝐴𝑀_𝐽𝐴×𝐽_𝐵,𝑀_𝐽𝐵𝐽_𝐶,𝑀_𝐽𝐶∣𝐽𝑀_𝐽𝐴ℳ^{𝑀𝐽𝐴,𝑀𝐽𝐵,𝑀𝐽𝐶}𝑝. (23)

Finally, the decay width Γ(𝐴→𝐵𝐶) is derived analytically in terms of the partial wave amplitudes Γ()𝐸𝐴⟶𝐵𝐶=2𝜋^𝐵𝐸_𝐶𝑀_𝐴𝑝_𝐿𝑆ℳ_𝐿𝑆𝑝². (24)

More technical details of the ³𝑃₀ model can be found in [63]. The inherent uncertainties of the ³𝑃₀ decay model itself have been discussed in [28, 69, 70].

The dimensionless parameter 𝛾 will be fixed by the 8 well-measured partial decay widths which are listed in Table 6. The ℳ_𝐿𝑆 amplitudes of these decay channels are presented explicitly in Appendix A.

Table 6 

Values of 𝛾 in different channels and comparison with the results given in [28]. Here, 𝜌(770), 𝑎₂(1320), 𝑓^′₁(1420), 𝑓₂(1270), 𝑓^′₂(1525) and 𝜌₃(1690) have been studied.

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, and the other includes 𝑠𝑠→𝑛𝑠+𝑠𝑛 and 𝑛𝑛→𝑛𝑠+𝑠𝑛. The values of 𝛾 here are a little different from these given in [28] where an 𝐴𝐿1 potential (for details of 𝐴𝐿1 potential, see [35]) 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 6 that 𝛾 decrease with 𝑝 increase. In addition, our calculation indicates that 𝛾 depends on flavors of both the decaying meson and the produced quark pairs. For example, values of 𝛾 fixed by 𝑎₂→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+𝐵exp(−𝐶𝑝²), for the creation vertex. The function of the creation vertex here is different from the one used in [28]. With the four decay channels listed in the fifth column of Table 6, we fix the function as 𝛾(𝑝)=1.8+4exp(−10𝑝²). For the processes of 𝑛𝑛→𝑛𝑛+𝑛𝑛 (the first column of Table 6), we fix the creation vertex function as 𝛾(𝑝)=3.0+25exp(−4𝑝²). The dependence of 𝛾 on the momentum 𝑝 is plotted in Figure 5. Obviously the functions can describe the dependence of 𝛾 and 𝑝 well. The functions of creation vertex given here need further test.

Figure 5 

The functions of 𝛾(𝑝)=𝐴+𝐵exp(−𝐶𝑝2) in different decay processes. The symbols of red “●” and black “■” denote 𝛾 values determined by the experimental data.

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 7). This ℱ factor also incorporates the statistical factor 1/2 if the final state mesons 𝐵 and 𝐶 are identical (as illustrated in Figure 6). More details of ℱ can be found in the Appendix A of [71].

Table 7 

The second and third columns for the flavor weight factors corresponding to two topological diagrams shown in Figure 6. The last column for the flavor multiplicity factor ℱ. Here, |𝜂⟩=(|𝑛𝑛⟩−|𝑠𝑠⟩)/√2 and |𝜂^′⟩=(|𝑛𝑛⟩+|𝑠𝑠⟩)/√2 have been taken for simplicity.

(a)

(b)

Figure 6 

Two topological diagrams for a 𝑞𝑞 meson decay in the 3𝑃0 decay model. We refer to the left one as 𝑑1 where the produced quark goes into meson 𝐶 and 𝑑2 where it goes into 𝐵.

The partial decay widths of 𝑋(1870) are shown in Table 8 except the channels of 𝒮+𝑃 mesons. 𝑎₂(1320)𝜋 and 𝑓₂(1275)𝜂 are large channels for the 𝜂₂(2¹𝐷₂)𝑛𝑛 state in our work and [24], which are consistent with the experimental observations of the 𝜂₂(1870). The partial widths of 𝐾^∗𝐾, 𝜌𝜌, and 𝜔𝜔 are narrower in our work than the expectations from [24]. 𝜂₂(1870) has been observed by the BES Collaboration in the radiative decay channel of 𝐽/𝜓→𝛾𝜂𝜋𝜋 [24]. However, no apparent 𝜂₂(1870) signals were detected in the channels of 𝐽/𝜓→𝛾𝜌𝜌 [72] and 𝐽/𝜓→𝛾𝜔𝜔 [73, 74]. Therefore, improved experimental measurements of the radiative 𝐽/𝜓 decay channels are needed for the 𝜂₂(1870) in future.

Table 8 

The partial widths of 𝑋(1870) and compared with results from [16, 24].

Next, we will evaluate the partial widths of “𝒮+𝑃” channels which have not been listed in Table 8. The scheme is proposed as follows. As illustrated in Figure 7, we assume that 𝑋(1870) decays into 𝑎₀(980)𝜋 via a virtual intermediate 1³𝑃₀𝑞𝑞 meson. We notice that the 𝜂(1295) also dominantly decays into the 𝜂𝜋𝜋 [1]. Its three-body decay can occur via three intermediate processes: 𝜂(1295)→𝜂𝜎/𝑎₀(980)𝜋/𝜂(𝜋𝜋)_𝑆-wave→𝜂𝜋𝜋 [2]. With the ratio Γ(𝑎₀(980)𝜋)/Γ(𝜂𝜋𝜋)=0.65±0.10 and Γ(𝜂(1295))=55±5 MeV, the partial width of 𝜂(1295) decaying into 𝑎₀(980)𝜋 is estimated no more than 45 MeV. By the ³𝑃₀ model, the ratio of (Γ(𝑋(1870)→𝑎₀(1³𝑃₀)𝜋))/(Γ(𝜂(1295)→𝑎₀(1³𝑃₀)𝜋)) can be reached easily. If the uncertainty of the coupling vertex of 𝜀(1³𝑃₀(𝑞𝑞)→𝑎₀(980)) (see in Figure 7) is assumed to be canceled in the ratio of (Γ(𝑋(1870)→𝑎₀(1³𝑃₀)𝜋)⋅𝜀(1³𝑃₀(𝑞𝑞)→𝑎₀(980)))/(Γ(𝜂(1295)→𝑎₀(1³𝑃₀)𝜋)⋅𝜀(1³𝑃₀(𝑞𝑞)→𝑎₀(980))), the value of (Γ(𝑋(1870)→𝑎₀(980)𝜋))/(Γ(𝜂(1295)→𝑎₀(980)𝜋)) can be extracted roughly. Although the assumption above seems a little rough, we just need to evaluate magnitudes of these decay channels.

Figure 7 

The diagram for the “𝒮+𝑃” channels through a virtual intermediate 13𝑃0𝑞𝑞 meson.

𝜂(1295) is proposed to be the first radial excited state of 𝜂(550). Then the total decay widths of Γ(𝑋(1870)→𝒮+𝑃) are evaluated no more than 12.6 MeV and Γ(𝑋(1870)→𝑎₀(980)𝜋)≤3.8 MeV. The BESIII Collaboration claimed that 𝑋(1870) primarily decay via 𝑎₀(980)𝜋 [1]. The small partial width of Γ(𝑋(1870)→𝑎₀(980)𝜋) also indicates that the 𝑋(1870) can not be interpreted as the 2¹𝐷₂𝑞𝑞 state.

In addition, our results do not support 𝑋(1870) as the 𝜂₂(2¹𝐷₂)𝑛𝑛 state since its observed decay width is much smaller than the theoretical estimate. The 𝑎₂(1320)𝜋 is the largest decay channel in our numerical results and in [24] for the 𝜂₂(2¹𝐷₂)𝑛𝑛 state (Table 8). If the partial width of 𝑎₀(980)𝜋 channel is as large as 𝑎₂(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 8 for the comparison. Details of the flux model are collected in Appendix B.

Two groups of the partial widths predicted in [16] are quoted in the Table 8. The left column was given by the flux tube decay model of Isgur, Kokoski, and Paton (IKP) with the “standard parameters” [75]. The right column was by the developed flux tube decay model of Swanson-Szczepaniak (SS). In [16], the masses are taken as 1.8  GeV for the 0⁻⁺, 1⁺⁺, and 2⁻⁺ hybrid states.

For a hybrid meson, 𝑋(1870) seems most possible to be the 𝜂_𝐻(0⁺2⁻⁺) state because the total widths which exclude the channels of 𝒮+𝑃 are much narrow in our work and in [16]. It is consistent with the narrow width of 𝑋(1870).

As shown in Table 8, 𝑋(1870) is impossible to be the 𝜂_𝐻(0⁺0⁻⁺) hybrid state. The predicted width in both our work and in [16] is broader. In addition, 𝜂𝜋 is a visible channel for both 𝑎₀(1450). A week signal was found in the region of 1200∼1400 MeV in the analysis of 𝜂𝜋^± (Figure 2(b) of [1]), which contradicts the large 𝑎₀(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 𝑎₁(1260)𝜋 is rather broad in our results and in the IKP model. If the partial width of 𝑎₀(980)𝜋 channel is as large as 𝑎₁(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 cannot be excluded. We suggest to detect the decay channel of 𝑎₁(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 Table 8). Then the channel of 𝑎₁(1260)𝜋 can discriminate the states 𝑓_𝐻(0⁺1⁺⁺) and 𝜂_𝐻(0⁺2⁻⁺) for 𝑋(1870).

Finally, if 𝜂₂(1870) is the 𝜂₂(2¹𝐷₂) state, its decay width is predicted about 100 MeV which is much smaller than the experiments. However, the difference can be explained by the remedy of mixing effect. If 𝑋(1870) and 𝜂₂(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 𝜂₂(1870). The broad decay width of 𝜂₂(1870) could be explained naturally. On the other hand, 𝜂₂(1870) has been observed in the channel of 𝑎₀(980)𝜋. However, this channel seems much small if 𝜂₂(1870) is a pure 2¹𝐷₂𝑛𝑛 meson. The mixing effect will also enlarge this partial width. Here, we do not plan to discuss the mixing of 𝑋(1870) and 𝜂₂(1870) further for the complex mechanism.

4. 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 𝜂₂(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 𝜂₂(1870) are analyzed.

Firstly, the mass spectrum is studied in the GI potential model and the RTs framework. In the GI potential model, both 𝑋(1870) and 𝜂₂(1870) could be the 𝜂^′₂(1¹𝐷₂), 𝑓^′₁(2³𝑃₁), and 𝜂₂(2¹𝐷₂) states. In RTs, the possible assignments are the 𝜂(3¹𝑆₀), 𝑓^′₁(2³𝑃₁) and 𝜂₂(2¹𝐷₂) states. For the mass spectrum, they are also good hybrid candidates since the masses overlap the predictions given by different models (see Table 4).

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 that the processes obey the OZI rule. We find that the channels of 𝑎₀𝜋, 𝜎𝜂, and 𝑓₀𝜂 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.

Thirdly, the two-body strong decay of 𝑋(1870) is computed in the ³𝑃₀ model. As the 𝜂₂(2¹𝐷₂) quarkonium, the predicted width of 𝑋(1870) looks much larger than the observations. The broad resonance, 𝜂₂(1870), can be a natural candidate for the 2¹𝐷₂𝑛𝑛 meson. There, we fix the creation strength, 𝛾, in two kinds of processes: (1) 𝑛𝑛→𝑛𝑛+𝑛𝑛; (2) 𝑛𝑛→𝑛𝑠+𝑠𝑛 and 𝑠𝑠→𝑛𝑠+𝑠𝑛. The functions of creation vertex are determined as 𝛾(𝑝)=3.0+25exp(−4𝑝²) and 𝛾(𝑝)=1.8+4exp(−10𝑝²), respectively. Meanwhile, the SHO wave function scale, 𝛽s, is obtained by the GI potential model.

We have evaluated the magnitude of the partial widths of “𝒮+𝑃” channels by the ratio, (Γ(𝑋(1870)→𝑎₀(980)𝜋))/(Γ(𝜂(1295)→𝑎₀(980)𝜋)), under a rather crude assumption that 𝜂(1295)/𝑋(1870)→𝑎₀(980)𝜋 through a virtual intermediate 1³𝑃₀𝑞𝑞 meson (see Figure 7). Then the uncertainties of the coupling vertex for 1³𝑃₀(𝑞𝑞)→𝑎₀(980) are assumed to be canceled in the ratio. The total widths of “𝒮+𝑃” are evaluated no more than 12.6 MeV and Γ(𝑋(1870)→𝑎₀(980)𝜋)≤3.8 MeV. Since 𝑋(1870) primarily decays via 𝑎₀(980)𝜋, it also indicated that the 𝑋(1870) cannot be interpreted as the 2¹𝐷₂𝑛𝑛 state.

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 [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 cannot exclude the possibility of 0⁺1⁺⁺. A precise measurement of 𝑎₁(1260)𝜋 is suggested to pin down this uncertainty.

Finally, some important arguments and useful suggestions are given as follows. (1) If 𝜂₂(1870) is the 𝜂₂(2¹𝐷₂) state, the broad 𝜋₂(1880) should be isovector partner of 𝜂₂(1870). 𝜋₂(1880) has been interpreted as the conventional 2¹𝐷₂𝑞𝑞 meson in [76]. Indeed, the decay channel of 𝜔𝜌 is large enough for 𝜋₂(1880) [77]. This observation disfavors the 𝜋₂(1880) as a 2⁻⁺ hybrid candidate for the selection rule that a hybrid meson decaying into two 𝑆-wave mesons is strongly suppressed [78]. (2) If 𝑋(1870) is a hybrid meson, we suggest to search its isospin partner in the decay channels of 𝐽/𝜓→𝜌𝑓₀(980)𝜋 and 𝐽/𝜓→𝜌𝑏₁(1235)𝜋, which are accessible at BESIII, Belle, and BABAR Collaborations. The decay channel of 𝑏₁(1235)𝜋 is forbidden for the 𝜋₂(2¹𝐷₂) quarkonium due to the “spin selection rule” [60, 79]. We also suggest to search the 𝜂₂(1870) in the decay channels of 𝐽/𝜓→𝛾𝜌𝜌 and 𝐽/𝜓→𝛾𝜔𝜔 since these channels are forbidden for the hybrid production.

Appendices

A. The Expressions of Amplitudes

We have omitted an exponential factor in the following decay amplitudes ℳ_𝐿𝑆 for compactness: exp−2𝜆𝜇−𝜈²𝑝4𝜇², (A.1) where we defined 1𝜇=21𝛽²_𝐴+1𝛽²_𝐵+1𝛽²_𝐶,𝑚𝜈=¹𝑚+𝑚₁𝛽²_𝐵+𝑚₂𝑚+𝑚₂𝛽²_𝐶,𝑚𝜆=²¹𝑚+𝑚₁²𝛽²_𝐵+𝑚²₂𝑚+𝑚₂²𝛽²_𝐶𝑚,𝜂=¹𝑚+𝑚₁. (A.2) For 1³𝑆₁→1¹𝑆₀+1¹𝑆₀, ℳ₁₀=2𝜇−𝜈8√3𝜋^5/4𝜇^5/2𝛽_𝐴𝛽_𝐵𝛽_𝐶^3/2𝑝. (A.3) For 1³𝑃₂→1¹𝑆₀+1¹𝑆₀, ℳ₂₀=2𝜇𝛽_𝐵^3/2−𝑝²𝜈²+2𝜇1−𝑝²𝜈𝛽_𝐶^3/28√15𝜋^5/4𝜇^7/2𝛽_𝐴^5/2𝛽_𝐵^3/2𝛽³_𝐶. (A.4) For 1³𝑃₂→1³𝑆₁+1¹𝑆₀, ℳ₂₁=−3/2ℳ₂₀.

For 1³𝑃₁→1³𝑆₁+1¹𝑆₀, ℳ₀₁=4𝜇𝛽_𝐵^3/2+𝑝²𝜈²+2𝜇1−𝑝²𝜈𝛽_𝐶^3/224𝜋^5/4𝜇^7/2𝛽_𝐴^5/2𝛽_𝐵^3/2𝛽³_𝐶,ℳ₂₁=2𝜇−𝜈𝜈24√2𝜋^5/4𝜇^7/2𝛽_𝐴^5/2𝛽_𝐵^3/2𝛽³_𝐶𝑝². (A.5) For 1³𝐷₃→1¹𝑆₀+1¹𝑆₀, ℳ₃₀=−2𝜇−𝜈16√35𝜋^5/4𝜇^9/2𝛽_𝐴^9/2𝛽_𝐵^3/2𝛽_𝐶^3/2𝜈²𝑝³. (A.6) For 2¹𝑆₀→1³𝑃₀+1¹𝑆₀, ℳ₁₁=196𝜋^5/4𝜇^11/2𝛽_𝐴^7/2𝛽_𝐵^5/2𝛽_𝐶^3/2×−24𝑝²𝜂𝜇³−𝑝⁴𝜈⁴+2𝑝²𝜇𝜈²−10+𝑝²1+𝜂𝜈−4𝜇²15−5𝑝²1+𝜂𝜈+𝑝⁴𝜂𝜈²+6𝜇²×4𝑝²𝜂𝜇²+𝑝²𝜈²−2𝜇−3+𝑝²1+𝜂𝜈𝛽²_𝐴. (A.7) For 2¹𝐷₂→1³𝑆₁+1¹𝑆₀, ℳ₁₁=−𝑝⁴𝜈⁴+2𝑝²𝜈²𝜇𝜈𝑝²−14+28𝜇²𝜈𝑝²−5+14𝜇²𝑝²𝜈²−2𝜈𝑝²−5𝜇𝛽²_𝐴×160√21𝜋^5/4𝜇^13/2𝛽_𝐴^11/2𝛽_𝐵^3/2𝛽_𝐶^3/2−1ℳ𝜈𝑝,₃₁=−28𝜇²+𝜈³𝑝²−2𝜇𝜈𝜈𝑝²𝜇−9+142𝜇−𝜈²𝛽²_𝐴×160√14𝜋^5/4𝜇^13/2𝛽_𝐴^11/2𝛽_𝐵^3/2𝛽_𝐶^3/2−1×𝜈²𝑝³. (A.8) For 2¹𝐷₂→1³𝑆₁+1³𝑆₁, ℳ^′₁₁=√2ℳ₁₁ and ℳ^′₃₁=√2ℳ₃₁.

For 2¹𝐷₂→1³𝑃₀+1¹𝑆₀ℳ₂₀1=−192√35𝜋^5/4𝜇^15/2𝛽_𝐴^11/2𝛽_𝐵^5/2𝛽_𝐶^3/2×𝑝⁴𝜈⁵−2𝑝²𝜇𝜈³−18+𝑝²1+𝜂𝜈+4𝜇²𝜈63−11𝑝²1+𝜂𝜈+𝑝⁴𝜂𝜈²+56𝜇³−2+𝜂−2+𝑝²𝜈−14𝜇²𝑝²𝜈³−2𝜇𝜈−7+𝑝²1+𝜂𝜈+4𝜇²−2+𝜂−2+𝑝²𝜈𝛽²_𝐴𝜈𝑝². (A.9) For 2¹𝐷₂→1³𝑃₁+1¹𝑆₀, ℳ₂₁𝑝=−²𝜈²−14𝜇²𝛽²_𝐴+14𝜇16√35𝜋^5/4𝜇^11/2𝛽_𝐴^11/2𝛽_𝐵^5/2𝛽_𝐶^3/2×𝜂−1𝜈𝑝². (A.10) For 2¹𝐷₂→1³𝑃₂+1¹𝑆₀, ℳ₀₂=1480√14𝜋^5/4𝜇^15/2𝛽_𝐴^11/2𝛽_𝐵^5/2𝛽_𝐶^3/2×𝑝⁶𝜇⁶−2𝑝⁴𝜇𝜈⁴𝑝²1+𝜂𝜈−21+4𝑝²𝜇²𝜈²105−14𝑝²1+𝜂𝜈+𝑝⁴𝜂𝜈²+56𝜇³15−5𝑝²1+𝜂𝜈+𝑝⁴𝜂𝜈²−14𝜇²𝑝⁴𝜈⁴−2𝑝²𝜇𝜈²−10+𝑝²1+𝜂𝜈+4𝜇²15−5𝑝²1+𝜂𝜈+𝑝⁴𝜂𝜈²𝛽²_𝐴,ℳ₂₂1=−672√5𝜋^5/4𝜇^15/2𝛽_𝐴^15/2𝛽_𝐵^5/2𝛽_𝐶^3/2×𝑝⁴𝜈⁵−2𝑝²𝜇𝜈³−18+𝑝²1+𝜂𝜈+2𝜇²𝜈126−25𝑝²1+𝜂𝜈+2𝑝⁴𝜂𝜈²+28𝜇³−7+𝜂−7+2𝑝²𝜈−14𝜇²𝑝²𝜈³−2𝜇𝜈−7+𝑝²1+𝜂𝜈+2𝜇²−7+𝜂−7+2𝑝²𝜈𝛽²_𝐴𝜈𝑝²,ℳ₄₂1=−1120𝜋^5/4𝜇^15/2𝛽_𝐴^11/2𝛽_𝐵^5/2𝛽_𝐶^3/2×𝜈𝑝²𝜈³−36𝜇²+2𝜇𝜈11−𝑝²𝜈+2𝜂𝜇28𝜇²−𝑝²𝜈³+2𝜇𝜈𝑝²𝜈−9−14𝜇²2𝜇−𝜈2𝜂𝜇−𝜈𝛽²_𝐴𝜈²𝑝⁴. (A.11) For 2¹𝐷₂→2³𝑃₁+1¹𝑆₀ℳ₂₁=1160√14𝜋^5/4𝜇^15/2𝛽_𝐴^11/2𝛽_𝐵^9/2𝛽_𝐶^3/2×−112𝜂𝜇³+252𝜇²𝜈+56𝑝²𝜂²𝜇³𝜈−80𝑝²𝜂𝜇²𝜈²+36𝑝²𝜇𝜈³+4𝑝⁴𝜂²𝜇²𝜈³−4𝑝⁴𝜂𝜇𝜈⁴+𝑝⁴𝜈⁵−10𝜇²𝜈14𝜇+𝑝²𝜈²𝛽²_𝐵−14𝜇²𝛽²_𝐴14𝜇𝜈+4𝑝²𝜂²𝜇²𝜈+𝑝²𝜈³−4𝜂𝜇2𝜇+𝑝²𝜈²−10𝜇²𝜈𝛽²_𝐵×𝜂−1𝑝². (A.12)

𝑚₁ and 𝑚₂ 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 the same as those in Section 2. The above amplitudes, ℳ_𝐿𝑆, can be reduced further in the approximation of 𝑚₁=𝑚₂=𝑚 and 𝛽_𝐴=𝛽_𝐵=𝛽_𝐶=𝛽. The reduced ℳ_𝐿𝑆 are consistent with those given by [71] except for an unimportant factor, −2^9/2, since this factor can be absorbed into the coefficient 𝛾.

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 ³𝑃₀ decay model but with an essential difference. The flux tube model extends 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 are not taken into account, the partial decay width Γ_𝐿𝑆(𝐻→𝐵𝐶) is given by the flux model as [79] Γ_𝐿𝑆()=𝑝𝐻⟶𝐵𝐶2𝐽_𝐴𝜋𝑀+1^𝐵𝑀_𝐶𝑀_𝐴ℳ_𝐿𝑆(𝐻⟶𝐵𝐶)², (B.1) where 𝑀_𝐴, 𝑀_𝐵, 𝑀_𝐶 are the “mock-meson” masses of 𝐴, 𝐵, 𝐶 [64]. When a hybrid meson decays into 𝑃-wave and pseudoscalar mesons, the partial wave amplitude ℳ_𝐿(𝐻→𝐵𝐶) (with 𝑆=𝑆_𝐵) is given as the following form: ℳ_𝐿()=𝜙𝐻⟶𝐵𝐶_𝐵𝜙_𝐶∣𝜙_𝐴𝜙₀𝑎𝑐̃9√312𝐴⁰₀₀𝑓𝑏𝜋𝜅√𝑏1+𝑓𝑏/(2𝛽²)²×2𝜋33Γ2𝛽+𝛿𝛽^𝐴3/2+𝛿ℳ_𝐿(𝐻⟶𝐵𝐶). (B.2)

The flavor matrix element ⟨𝜙_𝐵𝜙_𝐶∣𝜙_𝐴𝜙₀⟩ has been discussed before. ℳ_𝐿(𝐻→𝐵𝐶) are listed in Table 9 for the states of 𝜂_𝐻(0⁺0⁻⁺), 𝑓_𝐻(0⁺1⁺⁺), and 𝜂_𝐻(0⁺2⁻⁺).

Table 9 

Partial wave amplitudes ℳ_𝐿(𝐻→𝐵𝐶) for an initial hybrid 𝐻 decaying into a 𝑃-wave and pseudoscalar mesons.

Here the ℳ_𝑆, ℳ_𝐷, ℳ_𝑃𝑖, and ℳ_𝐹 are defined as ℳ_𝑆=−(3ℎ₀−𝑔₁+4ℎ₂), ℳ_𝐷=(𝑔₁+5ℎ₂), ℳ_𝑃1=−𝑖(2𝑔₀+3ℎ₁−𝑔₂), ℳ_𝑃2=−𝑖(𝑔₀+𝑔₂), ℳ_𝑃4=−𝑖(10𝑔₀+9ℎ₁+𝑔₂), and ℳ_𝐹=−3𝑖(𝑔₂+ℎ₃). The analytical expressions of 𝑔_𝑖 and ℎ_𝑖 are given as 𝑔_𝑛=2^3+𝛿𝑀^𝑛𝑚(𝑀+𝑚)^𝑛+12𝛽²_𝐴+𝛽^{2−(𝑛+𝛿+3)/2}Γ𝑛+𝛿+32₁×𝐹₁𝑛+𝛿+32𝑀,𝑛+1,−𝑀+𝑚²×𝑝²2𝛽²_𝐴+𝛽²𝑝^𝑛+1,ℎ_𝑛=2^3+𝛿𝛽²𝑀𝑀+𝑚^𝑛2𝛽²_𝐴+𝛽^{2−(𝑛+𝛿+4)/2}Γ𝑛+𝛿+42₁×𝐹₁𝑛+𝛿+42𝑀,𝑛+1,−𝑀+𝑚²𝑝²2𝛽²_𝐴+𝛽²𝑝^𝑛, (B.3) where ₁𝐹₁[⋅] are the confluent hypergeometric functions. Here we do not 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; for example, 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 [79] and take the combination (𝑎𝑐/̃9√3)(1/2)𝐴⁰₀₀𝑓𝑏/𝜋 as 0.64 which was fixed by the conventional mesons [64], 𝑀=𝑚=𝑚_𝑢,𝑑=330 MeV, 𝑀_𝐵^𝐼=0=𝑀_𝐵^𝐼=1=1250 MeV, 𝑀_𝐶^𝐼=0=720 MeV, 𝑀_𝐶^𝐼=1=850 MeV, 𝛽_𝐴=0.27 GeV, 𝛿=0.62, 𝑏=0.18 GeV^2, and 𝜅=0.9. Final states containing 𝜋 have 𝛽=0.36 GeV; otherwise 𝛽=0.40 GeV.

Acknowledgments

Bing Chen thanks Jun-Long Tian and D. V. Bugg for very helpful discussions. This work is supported by the Key Program of the He’nan Educational Committee of China (nos. 13A140014 and 12B140001), the National Natural Science Foundation of China under Grant no. 11305003, 11075102, 11005003, and U1204115, the Innovation Program of Shanghai Municipal Education Commission under Grant no. 13ZZ066, and the Program of He’nan Technology Department (no. 11147201).