# X

Authors: Bing Chen; Ke-Wei Wei; Ailin Zhang
Journal: Advances in High Energy Physics
 (2013)

---

## Abstract

The mass spectrum and strong decays of theX(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⟶ρf0980π and J/ψ⟶ρb11235π in the future. The latter channel is very important for testing the hybrid scenario.


---

## Body

## 1. Introduction

An isoscalar resonant structure ofX(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
(1)M=1877.3±6.3-7.4+3.4MeV,Γ=57±12-4+19MeV.
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)→a0±(980)π∓)·ℬ(a0±(980)→ηπ±)=[1.5±0.26(stat)-0.36+0.72(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 ofX(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
(2)M=1842±8MeV,Γ=225±14MeV.Theη2(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) A prior production of glueball in theJ/ψ→γXG; (b) a prior production of hybrid in the J/ψ→ωXH.
(a)
(b)Furthermore, the branching ratioℛ1=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 processJ/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 of0-+ 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 ofKK-π, 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 theη2(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 forX(1870) and η2(1870). Some valuable suggestions for the experiments in future are also needed.The paper is organized as follows. In Section2, 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 P30 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
(3)H=p→q2+mq2+p→q-2+mq-2+Vqq-(r).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]
(4)Vqq-(r)=-43αsr+σr+C.The strong coupling constantαs, 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 hc(1P) meson with respect to the center gravity of the χc(1P) mesons is much small: Mcog(χc)-M(hc)=-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:
(5)[p→q2+mq2+p→q-2+mq-2+Vqq-(r)]ψ(r)=Eψ(r).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(6)ψnl(r,β)=β3/22(2n-1)!Γ(n+l+(1/2))(βr)le-β2r2/2Ln-1l+1/2(β2r2)
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(7)ψnl(p,β)=(-1)nβ3/22(2n-1)!Γ(n+l+(1/2))(pβ)le-p2/2β2Ln-1l+1/2(p2β2).The wave functions ofψnl(r,β) and ψnl(p,β) meet the normalization conditions:
(8)∫0∞‍ψnl2(r,β)r2dr=1,∫0∞‍ψnl2(p,β)p2dp=1.In the variational approach, the correspondingM-nl are given by minimizing the expectation value of H(9)ddβEnl(β)=0,
where
(10)Enl(β)≡〈H〉nl=〈ψnl|H|ψnl〉.When all the parameters of the potential model are known, the values of the harmonic oscillator parameterβ- 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 ofVs→·s→(r) for the 1S mesons. The following Gaussian-smeared contact hyperfine interaction [36] is taken for convenience:
(11)Vqq-s→·s→(r)=32παs9mq2(κπ)3e-κ2r2S→q·S→q-.In this work, we choose the model parameters as follows:mu=md=0.220 GeV, ms=0.428 GeV, αs=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 αst′ 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 ofM-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 parameterβ- (unit: GeV−1) of the states 2S, 3S, 4S, 1P, 2P, 3P, 1D, 2D, 3D, 1F, 2F, 1G, and 1H.
States
M
-
n
l
(
n
n
-
)
β
-
Expt. [2]
M
-
n
l
(
s
s
-
)
β
-
Expt. [2]
1
S
—
0.44/0.34
—
—
0.42/0.39
—
2
S
1.399
0.310
1.389
1.631
0.330
1.629
3
S
1.859
0.295
2.069
0.310
4
S
2.240
0.290
2.436
0.300
1
P
1.252
0.310
1.257
1.460
0.340
1.478
2
P
1.711
0.294
1.926
0.315
3
P
2.110
0.290
2.308
0.300
1
D
1.661
0.280
1.672
1.883
0.300
2
D
2.067
0.276
2.272
0.292
3
D
2.417
0.275
2.609
0.288
1
F
1.924
0.277
2.128
0.295
2
F
2.287
0.275
2.478
0.290
1
G
2.161
0.275
2.350
0.292
1
H
2.377
0.273
2.554
0.287Obviously, the spin-averaged masses of the2S, 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 1996-9+10 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 β-s 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 Vconf(r) is determinant for the properties of higher excited states. In [37], the masses for higher excited states with σ=0.143 GeV2 and αs=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 ofη′(31S0), f1′(23P1), η2′(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 f1′(23P1), η2′(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 for31S0(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].
States
η
′
(
3
1
S
0
)
f
1
′
(
2
3
P
1
)
η
2
′
(
1
1
D
2
)
η
2
(
2
1
D
2
)
[20]
—
2030
1890
2130†
[25]
2085
2016
1909
1960
[26]
2099
1988
1851
—
[27]
—
—
1853
1863Regge 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
(12)M2=1.38(4)n+1.12(4)J-1.25(4).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].Nowh1(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 f1′(21P1). The mass of X(1870) is too large for the η2′(11D2) state in the RTs. η2(1645) has been assigned as the 11D2nn- meson [2]. Since M2(X(1870))-M2(η2(1640))=0.91-0.03+0.04 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))=2.16-0.05+0.06 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].
Four possible states forX(1870)
η
′
(
3
1
S
0
)
f
1
′
(
2
3
P
1
)
η
2
′
(
1
1
D
2
)
η
2
(
2
1
D
2
)
η
′
(
1475
)
f
1
(
1420
)
h
1
(
1380
)
η
2
(
1645
)
μ
2
=
1
.
34
-
0.03
+
0.04
μ
2
=
1
.
38
-
0.03
+
0.04
β
2
=
1
.
6
-
0.06
+
0.06
μ
2
=
0
.
91
-
0.03
+
0.04As mentioned in Section1, 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-+nn-g states by these models are collected in Table 4.Table 4
The masses predicted forηH(0+0-+), fH(0+1++) and ηH(0+2-+) hybrid states in [11–15].
States
η
H
(
0
+
0
-
+
)
f
H
(
0
+
1
+
+
)
η
H
(
0
+
2
-
+
)
Bag [11, 12]
1.3
Heavier
1.9
Flux tube [13, 14]
1.7~1.9
1.7~1.9
1.7~1.9
Constituent gluon [15]
1.8~2.2
1.3~1.8
1.8~2.2In this section, the mass ofX(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 f1′(23P1), η2′(11D2), or η2(21D2) state with a reasonable uncertainty. In the RTs, X(1870) favors the η′(31S0) and f1′(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 ofqq- 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 qqq-q-  [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 Figure2). 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 the𝒮 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 theSU(3) flavor symmetry. We will show that a0(980)π, σ0η, and f0η are the main decay channels for the isoscalar nn- and the nn-g 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 the𝒮 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 byP=(-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
(13)(3⊗3)3-⊗(3-⊗3-)3=8⊕1.Because theSU(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],
(14)(f0σ0)=(cosϑsinϑ-sinϑcosϑ)(|8〉I=0|1〉I=0).
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
(15)(f0σ0)=(|12([su][s-u-]+[sd][s-d-])〉|[ud][u-d-]〉).It seems that the deviation from the ideal mixing angle of theσ(500) and f0(980) is small [48]. In the following calculations, we will treat them in the ideal mixing scheme.Under theSU(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, gA88, gA81, gA18, gB88, and gB11, corresponding to five different channels
(16)|8〉qq-⟶|8〉𝒮⊗|8〉qq-:gA88,|8〉qq-⟶|8〉𝒮⊗|1〉qq-:gA81,|8〉qq-⟶|1〉𝒮⊗|8〉qq-:gA18,|1〉qq-⟶|8〉𝒮⊗|8〉qq-:gB88,|1〉qq-⟶|1〉𝒮⊗|1〉qq-:gB11.In order to determine the relations between these coupling constants, we will assume the process that theqq- or qq-g 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
(17)gA81:gA18:gB88:gB11:gA88=2:-25(5+1):-25(5+1):-25(5+1):1≈1.41:-2.05:-2.89:-2.05:1.00.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, θp, as follows:
(18)(η(548)η′(958))=(cosθp-sinθpsinθpcosθp)(|8〉I=0|1〉I=0).The mixing angleθp has been measured by various means. However, there is still uncertainty for θp. An excellent fit to the tensor meson decay widths was performed under the SU(3) symmetry, and θp≃-17° was obtained [23]. In our calculation, θp is taken as -17°. The excited mixtures of nn- and ss- are denoted as
(19)(ξξ′)=(cosθsinθ-sinθcosθ)(|1〉I=0|8〉I=0).In this scheme, the ideal mixing occurs with the choice ofθ=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 coefficientsζ2 of ξ and ξ′ in the ideal mixing.
Decaychannels
a
0
π
σ
η
κ
K
f
0
η
σ
η
′
ζ
2
[
n
n
-
(
g
)
]
2.17
3.56
0.47
1.07
0.44
ζ
2
[
s
s
-
(
g
)
]
0.00
0.00
0.72
1.08
0.00Figure 3
The coefficientsζ2 of the isoscalar meson ξ versus the mixing angle θ.Figure 4
The coefficientsζ2 of the isoscalar meson ξ′ versus the mixing angle θ.Here the mixing ofη(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 Figures3 and 4, the primary decay channels of a ss- or ss-g 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 nn-g 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 nn-g predominant state.Of course, theSU(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 nn-g candidate. In the following subsection, we will explore the two-body strong decays of X(1870) within the P30 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 ofη2(1870) and X(1870)

In [24], the P30 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 strengthγ 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 P30 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/mqβ, where mq 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=|k→3-k→4| 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 theP30 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 ofX(1870) will be investigated in the P30 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 operator𝒯^ of the P30 model is depicted as
(20)𝒯^=-3γ∑m‍〈1,m;1,-m∣0,0〉×∬‍d3k→3d3k→4δ3(k→3+k→4)×𝒴1m(k→3-k→42)ω0(3,4)φ0(3,4)χ1,-m(3,4)d3i†(k→3)d4j†(k→4),
where the ω0(3,4) and φ0(3,4) are the color and flavor wave functions of the q3q-4 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 P30 state. Then χ1,-m(3,4) represents the pair production in a spin triplet state. The solid harmonic polynomial 𝒴1m(k→)≡|k→|𝒴1m(θk,ϕk) reflects the momentum-space distribution of the q3q-4.The helicity amplitudeℳMJA,MJB,MJC(p) of A→B+C is given by
(21)〈BC|𝒯^|A〉=δ3(P→A-P→B-P→C)ℳMJA,MJB,MJC(p),
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 ℳMJA,MJB,MJC(p) can be constructed in the L-S basis easily [62, 63]. The mock state for A meson is
(22)|A(nA2SA+1LAJAMJA(P→A))〉≡2EA∑MLAMSA〈LAMLASA∣JAMJA〉ωA12ϕA12χSAMSA120000000000000×∫dP→AψnALAMLA(k→1,k→2)|q1(k→1)q2(k→2)〉.To obtain the analytical amplitudes, the SHO wave functions are usually employed forψnALAMLA(k→1,k→2). For comparison with experiments, one obtains the partial decay width ℳJL(p) via the Jacob-Wick formula [68]:
(23)ℳLS(p)=2L+12JA+1∑MJB,MJC‍〈L0JMJA∣JAMJA〉×〈JB,MJBJC,MJC∣JMJA〉ℳMJA,MJB,MJC(p).Finally, the decay widthΓ(A→BC) is derived analytically in terms of the partial wave amplitudes
(24)Γ(A⟶BC)=2πEBECMAp∑LS‍|ℳLS(p)|2.More technical details of theP30 model can be found in [63]. The inherent uncertainties of the P30 decay model itself have been discussed in [28, 69, 70].The dimensionless parameterγ will be fixed by the 8 well-measured partial decay widths which are listed in Table 6. The ℳLS amplitudes of these decay channels are presented explicitly in Appendix A.Table 6
Values ofγ in different channels and comparison with the results given in [28]. Here, ρ(770), a2(1320), f1′(1420), f2(1270), f2′(1525) and ρ3(1690) have been studied.
Decay channels
p (GeV)
γ (103)
γ [28]
ρ
→
π
π
0.362
17.8
9.18
a
2
→
η
π
0.535
11.5
—
f
2
→
π
π
0.622
7.8
7.13
ρ
3
→
π
π
0.833
4.2
—
f
1
′
→
K
*
K
-
0.158
4.9
—
f
2
→
K
K
-
0.401
2.9
6.11
a
2
→
K
K
-
0.434
2.3
3.91
f
2
′
→
K
K
-
0.579
2.0
5.66As 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 Table6 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 f2′→KK- are roughly equal.In the following calculations, we assume that the values ofγ 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 ofγ(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 factorℱ (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 Figure6. The last column for the flavor multiplicity factor ℱ. Here, |η〉=(|nn-〉-|ss-〉)/2 and |η′〉=(|nn-〉+|ss-〉)/2 have been taken for simplicity.
Decay channels
ℐ
flavor
(
d
1
)
ℐ
flavor
(
d
1
)
ℱ
ρ
→
π
π
+
1
/
2
-
1
/
2
1
f
2
→
π
π
-
1
/
2
-
1
/
2
3/2
f
2
→
K
K
0
-
1
/
2
2
f
1
′
→
K
*
K
+1
0
4
f
2
′
→
K
K
+1
0
2
a
2
→
K
K
0
−1
1
a
2
→
η
π
+
1
/
2
+
1
/
2
1
η
2
→
ω
ω
-
1
/
2
-
1
/
2
1/2
η
2
→
a
i
π
-
1
/
2
-
1
/
2
3
η
2
→
f
i
η
+
1
/
2
+
1
/
2
1Two topological diagrams for aqq- meson decay in the P30 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.
(a)
(b)The partial decay widths ofX(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 ofX(1870) and compared with results from [16, 24].
Decaychannels
η
2
(
2
1
D
2
)
η
H
(
0
+
0
-
+
)
f
H
(
0
+
1
+
+
)
η
H
(
0
+
2
-
+
)
Our
[24]
Our
[16]
Our
[16]
Our
[16]
K
*
K
0.5
17.7
19.3
12.6
10
5
4.9
24.1
18.0
3.2
2.0
1.0
ρ
ρ
12.9
52.2
56.8
×
×
×
×
×
×
×
×
×
ω
ω
4.2
16.9
18.4
×
×
×
×
×
×
×
×
×
K
*
K
*
0.2
2.1
2.3
×
×
×
×
×
×
×
×
×
a
0
(
1450
)
π
16.0
2.4
2.6
56.3
70
175
0.5
×
6
0.5
0.0
0.6
a
1
(
1260
)
π
0.0
15.2
16.6
×
×
×
57.3
14
232
×
0.3
×
f
1
(
1280
)
η
0.0
0.0
0.0
×
×
×
2.5
—
—
×
0.0
×
a
2
(
1320
)
π
54.2
102.5
111.6
8.8
1
16
35.1
5.0
179.4
26.7
25.1
67
f
2
(
1275
)
η
15.1
17.5
19.0
0.0
×
×
1.0
—
—
4.6
0.0
0.0
∑
‍
Γ
i  Expt (MeV)
103.3
26.5
246.7
77.7
81
196
101.3
43.1
435.4
35.0
27.4
68.6
225
±
14 [2]
57
±
12
-
4
+
19 [1]
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 the3P0 model from [24] in the left column and the right column by the flux-tube model. In [16], the masses are taken as 1.8 GeV for the 0-+, 1++, and 2-+ hybrid states.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 P30 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 supportX(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 ofX(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 ηH(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 Table8, X(1870) is impossible to be the ηH(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 ηH(0+0-+) state. We can exclude the possibility of X(1870) as the ηH(0+0-+) hybrid state preliminarily.The assignment forX(1870) as the fH(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 fH(0+1++) hybrid is much small. So the possibility of X(1870) as a fH(0+1++) hybrid cannot be excluded. We suggest to detect the decay channel of a1(1260)π because this channel is forbidden for the ηH(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 fH(0+1++) and ηH(0+2-+) for X(1870).Finally, ifη2(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.


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

Despite many theoretical efforts, the scalar nonet ofqq- 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 qqq-q-  [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 Figure2). 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 the𝒮 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 theSU(3) flavor symmetry. We will show that a0(980)π, σ0η, and f0η are the main decay channels for the isoscalar nn- and the nn-g 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 the𝒮 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 byP=(-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
(13)(3⊗3)3-⊗(3-⊗3-)3=8⊕1.Because theSU(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],
(14)(f0σ0)=(cosϑsinϑ-sinϑcosϑ)(|8〉I=0|1〉I=0).
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
(15)(f0σ0)=(|12([su][s-u-]+[sd][s-d-])〉|[ud][u-d-]〉).It seems that the deviation from the ideal mixing angle of theσ(500) and f0(980) is small [48]. In the following calculations, we will treat them in the ideal mixing scheme.Under theSU(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, gA88, gA81, gA18, gB88, and gB11, corresponding to five different channels
(16)|8〉qq-⟶|8〉𝒮⊗|8〉qq-:gA88,|8〉qq-⟶|8〉𝒮⊗|1〉qq-:gA81,|8〉qq-⟶|1〉𝒮⊗|8〉qq-:gA18,|1〉qq-⟶|8〉𝒮⊗|8〉qq-:gB88,|1〉qq-⟶|1〉𝒮⊗|1〉qq-:gB11.In order to determine the relations between these coupling constants, we will assume the process that theqq- or qq-g 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
(17)gA81:gA18:gB88:gB11:gA88=2:-25(5+1):-25(5+1):-25(5+1):1≈1.41:-2.05:-2.89:-2.05:1.00.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, θp, as follows:
(18)(η(548)η′(958))=(cosθp-sinθpsinθpcosθp)(|8〉I=0|1〉I=0).The mixing angleθp has been measured by various means. However, there is still uncertainty for θp. An excellent fit to the tensor meson decay widths was performed under the SU(3) symmetry, and θp≃-17° was obtained [23]. In our calculation, θp is taken as -17°. The excited mixtures of nn- and ss- are denoted as
(19)(ξξ′)=(cosθsinθ-sinθcosθ)(|1〉I=0|8〉I=0).In this scheme, the ideal mixing occurs with the choice ofθ=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 coefficientsζ2 of ξ and ξ′ in the ideal mixing.
Decaychannels
a
0
π
σ
η
κ
K
f
0
η
σ
η
′
ζ
2
[
n
n
-
(
g
)
]
2.17
3.56
0.47
1.07
0.44
ζ
2
[
s
s
-
(
g
)
]
0.00
0.00
0.72
1.08
0.00Figure 3
The coefficientsζ2 of the isoscalar meson ξ versus the mixing angle θ.Figure 4
The coefficientsζ2 of the isoscalar meson ξ′ versus the mixing angle θ.Here the mixing ofη(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 Figures3 and 4, the primary decay channels of a ss- or ss-g 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 nn-g 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 nn-g predominant state.Of course, theSU(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 nn-g candidate. In the following subsection, we will explore the two-body strong decays of X(1870) within the P30 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 ofη2(1870) and X(1870)

In [24], the P30 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 strengthγ 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 P30 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/mqβ, where mq 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=|k→3-k→4| 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 theP30 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 ofX(1870) will be investigated in the P30 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 operator𝒯^ of the P30 model is depicted as
(20)𝒯^=-3γ∑m‍〈1,m;1,-m∣0,0〉×∬‍d3k→3d3k→4δ3(k→3+k→4)×𝒴1m(k→3-k→42)ω0(3,4)φ0(3,4)χ1,-m(3,4)d3i†(k→3)d4j†(k→4),
where the ω0(3,4) and φ0(3,4) are the color and flavor wave functions of the q3q-4 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 P30 state. Then χ1,-m(3,4) represents the pair production in a spin triplet state. The solid harmonic polynomial 𝒴1m(k→)≡|k→|𝒴1m(θk,ϕk) reflects the momentum-space distribution of the q3q-4.The helicity amplitudeℳMJA,MJB,MJC(p) of A→B+C is given by
(21)〈BC|𝒯^|A〉=δ3(P→A-P→B-P→C)ℳMJA,MJB,MJC(p),
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 ℳMJA,MJB,MJC(p) can be constructed in the L-S basis easily [62, 63]. The mock state for A meson is
(22)|A(nA2SA+1LAJAMJA(P→A))〉≡2EA∑MLAMSA〈LAMLASA∣JAMJA〉ωA12ϕA12χSAMSA120000000000000×∫dP→AψnALAMLA(k→1,k→2)|q1(k→1)q2(k→2)〉.To obtain the analytical amplitudes, the SHO wave functions are usually employed forψnALAMLA(k→1,k→2). For comparison with experiments, one obtains the partial decay width ℳJL(p) via the Jacob-Wick formula [68]:
(23)ℳLS(p)=2L+12JA+1∑MJB,MJC‍〈L0JMJA∣JAMJA〉×〈JB,MJBJC,MJC∣JMJA〉ℳMJA,MJB,MJC(p).Finally, the decay widthΓ(A→BC) is derived analytically in terms of the partial wave amplitudes
(24)Γ(A⟶BC)=2πEBECMAp∑LS‍|ℳLS(p)|2.More technical details of theP30 model can be found in [63]. The inherent uncertainties of the P30 decay model itself have been discussed in [28, 69, 70].The dimensionless parameterγ will be fixed by the 8 well-measured partial decay widths which are listed in Table 6. The ℳLS amplitudes of these decay channels are presented explicitly in Appendix A.Table 6
Values ofγ in different channels and comparison with the results given in [28]. Here, ρ(770), a2(1320), f1′(1420), f2(1270), f2′(1525) and ρ3(1690) have been studied.
Decay channels
p (GeV)
γ (103)
γ [28]
ρ
→
π
π
0.362
17.8
9.18
a
2
→
η
π
0.535
11.5
—
f
2
→
π
π
0.622
7.8
7.13
ρ
3
→
π
π
0.833
4.2
—
f
1
′
→
K
*
K
-
0.158
4.9
—
f
2
→
K
K
-
0.401
2.9
6.11
a
2
→
K
K
-
0.434
2.3
3.91
f
2
′
→
K
K
-
0.579
2.0
5.66As 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 Table6 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 f2′→KK- are roughly equal.In the following calculations, we assume that the values ofγ 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 ofγ(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 factorℱ (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 Figure6. The last column for the flavor multiplicity factor ℱ. Here, |η〉=(|nn-〉-|ss-〉)/2 and |η′〉=(|nn-〉+|ss-〉)/2 have been taken for simplicity.
Decay channels
ℐ
flavor
(
d
1
)
ℐ
flavor
(
d
1
)
ℱ
ρ
→
π
π
+
1
/
2
-
1
/
2
1
f
2
→
π
π
-
1
/
2
-
1
/
2
3/2
f
2
→
K
K
0
-
1
/
2
2
f
1
′
→
K
*
K
+1
0
4
f
2
′
→
K
K
+1
0
2
a
2
→
K
K
0
−1
1
a
2
→
η
π
+
1
/
2
+
1
/
2
1
η
2
→
ω
ω
-
1
/
2
-
1
/
2
1/2
η
2
→
a
i
π
-
1
/
2
-
1
/
2
3
η
2
→
f
i
η
+
1
/
2
+
1
/
2
1Two topological diagrams for aqq- meson decay in the P30 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.
(a)
(b)The partial decay widths ofX(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 ofX(1870) and compared with results from [16, 24].
Decaychannels
η
2
(
2
1
D
2
)
η
H
(
0
+
0
-
+
)
f
H
(
0
+
1
+
+
)
η
H
(
0
+
2
-
+
)
Our
[24]
Our
[16]
Our
[16]
Our
[16]
K
*
K
0.5
17.7
19.3
12.6
10
5
4.9
24.1
18.0
3.2
2.0
1.0
ρ
ρ
12.9
52.2
56.8
×
×
×
×
×
×
×
×
×
ω
ω
4.2
16.9
18.4
×
×
×
×
×
×
×
×
×
K
*
K
*
0.2
2.1
2.3
×
×
×
×
×
×
×
×
×
a
0
(
1450
)
π
16.0
2.4
2.6
56.3
70
175
0.5
×
6
0.5
0.0
0.6
a
1
(
1260
)
π
0.0
15.2
16.6
×
×
×
57.3
14
232
×
0.3
×
f
1
(
1280
)
η
0.0
0.0
0.0
×
×
×
2.5
—
—
×
0.0
×
a
2
(
1320
)
π
54.2
102.5
111.6
8.8
1
16
35.1
5.0
179.4
26.7
25.1
67
f
2
(
1275
)
η
15.1
17.5
19.0
0.0
×
×
1.0
—
—
4.6
0.0
0.0
∑
‍
Γ
i  Expt (MeV)
103.3
26.5
246.7
77.7
81
196
101.3
43.1
435.4
35.0
27.4
68.6
225
±
14 [2]
57
±
12
-
4
+
19 [1]
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 the3P0 model from [24] in the left column and the right column by the flux-tube model. In [16], the masses are taken as 1.8 GeV for the 0-+, 1++, and 2-+ hybrid states.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 P30 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 supportX(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 ofX(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 ηH(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 Table8, X(1870) is impossible to be the ηH(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 ηH(0+0-+) state. We can exclude the possibility of X(1870) as the ηH(0+0-+) hybrid state preliminarily.The assignment forX(1870) as the fH(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 fH(0+1++) hybrid is much small. So the possibility of X(1870) as a fH(0+1++) hybrid cannot be excluded. We suggest to detect the decay channel of a1(1260)π because this channel is forbidden for the ηH(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 fH(0+1++) and ηH(0+2-+) for X(1870).Finally, ifη2(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 ofX(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, bothX(1870) and η2(1870) could be the η2′(11D2), f1′(23P1), and η2(21D2) states. In RTs, the possible assignments are the η(31S0), f1′(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 ann- quarkonium or a nn-g 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 nn-g 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 ofX(1870) is computed in the P30 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 1P30qq- meson (see Figure 7). Then the uncertainties of the coupling vertex for 1P30(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 theX(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 ηH(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.



---

*Source: 217858-2013-11-24.xml*