#  $X(1870)$ and $\eta_{2}(1870)$: Which can be assigned as a hybrid state?

Bing Chen111Corresponding author<EMAIL_ADDRESS>School of Physics and
Electrical Engineering, Anyang Normal University, Anyang 455000, China  Ke-Wei
Wei  School of Physics and Electrical Engineering, Anyang Normal University,
Anyang 455000, China  Ailin Zhang  Department of Physics, Shanghai University,
Shanghai 200444, China

###### Abstract

The mass spectrum and strong decays of the $X(1870)$ and $\eta_{2}(1870)$ are
analyzed. Our results indicate that $X(1870)$ and $\eta_{2}(1870)$ are the two
different resonances. The narrower $X(1870)$ seems likely a good hybrid
candidate. We support the $\eta_{2}(1870)$ as the $\eta_{2}(2^{1}D_{2})$
quarkonium. We suggest to search the isospin partner of $X(1870)$ in the
channels of $J/\psi\rightarrow\rho f_{0}(980)\pi$ and $J/\psi\rightarrow\rho
b_{1}(1235)\pi$ in the future. The latter channel is very important for
testing the hybrid scenario.

###### pacs:

12.38.Lg, 13.25.Jx

††preprint: AHEP(Hadron)/AYNU[2013]

##  I INTRODUCTION

A isoscalar resonant structure of $X(1870)$ was observed by the BESIII
Collaboration with a statistical significance of 7.2$\sigma$ in the processes
$J/\psi\rightarrow\omega X(1870)\rightarrow\omega\eta\pi^{+}\pi^{-}$ recently
Bes6 . Its mass and width were given as

$M=1877.3\pm 6.3^{+3.4}_{-7.4}$MeV, $\Gamma=57\pm 12^{+19}_{-4}$MeV.

Here the first errors are statistical and the second ones are systematic. The
product branching fraction of $\mathcal{B}(J/\psi\rightarrow\omega
X(1870))\cdot\mathcal{B}(X(1870)\rightarrow
a^{\pm}_{0}(980)\pi^{\mp})\cdot\mathcal{B}(a^{\pm}_{0}(980)\rightarrow\eta\pi^{\pm})=[1.5\pm
0.26(stat)^{+0.72}_{-0.36}(syst)]\times 10^{-4}$ was also presented Bes6 . 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 $\eta_{2}(1870)$, but the width
is much narrower than the $\eta_{2}(1870)$. In the tables of the Particle Data
Group (PDG) PDG , the available mass and width of $\eta_{2}(1870)$ are

_M_ = 1842 $\pm$ 8MeV, $\Gamma$ = 225 $\pm$ 14MeV.

The $\eta_{2}(1870)$ has been observed in $\gamma\gamma$ reactions Crystall ;
DESY , $p\bar{p}$ annihilation pp1 ; pp2 ; WA1021 ; WA1022  and radiative
$J/\psi$ decays Bes7 . It should be stressed that radiative $J/\psi$ decay
channels (Fig.1[A]) and $p\bar{p}$ annihilation prosesses are the ideal
glueball hunting grounds. But the glueball production is suppressed in
$\gamma\gamma$ reaction. By contrast, the hadronic $J/\psi$ decay are
considered ``hybrid rich'' (Fig.1[B]).

Figure 1: [A]. A prior production of glueball in the $J/\psi\rightarrow\gamma
X_{G}$; [B]. A prior production of hybrid in the $J/\psi\rightarrow\omega
X_{H}$.

Furthermore, the branching ratio
$\mathcal{R}_{1}=\frac{\Gamma(\eta_{2}(1870)\rightarrow
a_{2}(1320)\pi)}{\Gamma(\eta_{2}(1870)\rightarrow a_{0}(980)\pi)}=32.6\pm
12.6$ reported by the WA102 Collaboration indicates that the decay channel of
$a_{0}(980)\pi$ is tiny for $\eta_{2}(1870)$ WA1021 . This has been confirmed
by an extensive re-analysis of the Crystal Barrel data Bugg1 . Differently,
the analysis of BESIII Collaboration indicates that the $X(1870)$ primarily
decay via the $a_{0}(980)\pi$ channel Bes6 . Then the present measurements of
the decay widths, productions, and decay properties suggest that
$\eta_{2}(1870)$ and $X(1870)$ are two different isoscalar mesons.

If the production process $J/\psi\rightarrow\omega X(1870)$ 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 1.8GeV in
the Bag model Bag1 ; Bag2 , the flux-tube model tube1 ; tube2  and the
constituent gluon model gluon . In addition, the decay width of isoscalar
$2^{-+}$ hybrid is expected to be narrow Swanson . 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 latt1 ; latt2 ; latt3 . Therefore,
$X(1870)$ is not likely to be a glueball state. Moreover, the molecule and
fourquark states are not expected in this region Swanson . Then the unclear
structure $X(1870)$ looks more like a good hybrid candidate. But the actual
situation is much complicated because the nature of $\eta_{2}(1870)$ is still
ambiguous:

  1. (i)

Since no evidences have been found in the decay mode of $K\bar{K}\pi$, the
$\eta_{2}(1870)$ disfavors the $1^{1}D_{2}$ $s\bar{s}$ quarkonium assignment.
The mass of $\eta_{2}(1870)$ seems much smaller for the $2^{1}D_{2}$
$n\bar{n}$ ($n\bar{n}\equiv(u\bar{u}+d\bar{d})/\sqrt{2}$) state in the
Godfrey-Isgur (GI) quark model Isgur . Therefore the $\eta_{2}(1870)$ has been
assigned as the $2^{-+}$ hybrid state Bugg1 ; Bugg2 ; Klmept1 ; Amsler1 .

  2. (ii)

However, Li and Wang pointed out that the mass, production, total decay width,
and decay pattern of the $\eta_{2}(1870)$ do not appear to contradict with the
picture of it as being the conventional $2^{1}D_{2}$ $n\bar{n}$ state Li3 .

Therefore, systematical study of the mass spectrum and strong decay properties
is urgently required for $X(1870)$ and $\eta_{2}(1870)$. Some valuable
suggestions for the experiments in future are also needed.

The paper is organized as follows. In Sec.II, the masses of $X(1870)$ and
$\eta_{2}(1870)$ will be explored in the GI relativized quark model and the
Regge trajectories (RTs) framework. In Sec.III, the decay processes that a
isoscalar meson decays into light scalar (below 1 GeV) and pseudoscalar mesons
will discussed. The two-body strong decays $X(1870)$ and $\eta_{2}(1870)$ will
be calculated within the ${}^{3}P_{0}$ model and the flux-tube model. Finally,
our discussions and conclusions will be presented in Sec.IV.

##  II Mass spectrum

In the Godfrey-Isgur relativized potential model Isgur , the Hamiltonian
consists of the central potential and a kinetic term in a ``relativized'' form

$H=\sqrt{\vec{p}_{q}^{2}+m^{2}_{q}}+\sqrt{\vec{p}_{\bar{q}}^{2}+m^{2}_{\bar{q}}}+V_{q\bar{q}}(r).$
(1)

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
Eichten

$V_{q\bar{q}}(r)=-\frac{4}{3}\frac{\alpha_{s}}{r}+\sigma r+C.$ (2)

The strong coupling constant $\alpha_{s}$, the string tension $\sigma$ and the
constant _C_ are the model parameters which can be fixed by the well
established experimental states. The remaining spin-dependent terms for mass
shifts are usually treated as the leading-order perturbations which include
the spin-spin contact hyperfine interaction, spin-orbit and tensor
interactions and a longer-ranged inverted spin-orbit term. They arise from one
gluon exchange (OGE) forces and the assumed Lorentz scalar confinement. The
expressions for these terms may be found in Ref. Isgur .

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 Badalian . For example, the hyperfine shift of the $h_{c}(1P)$
meson with respect to the center gravity of the $\chi_{c}(1P)$ mesons is much
small: $M_{cog}(\chi_{c})-M(h_{c})=-0.02\pm 0.19\pm 0.13$MeV CLEO . However,
for the light isovector mesons $a_{0}(1450)$, $a_{1}(1260)$, $a_{2}(1320)$,
and $b_{1}(1235)$, the hyperfine shift is $76.7\pm 44.4$ MeV. Here the masses
of $a_{0}$, $a_{1}$, $a_{2}$, and $b_{1}$ are taken from PDG PDG . For the
complexities of nonperturbative interactions, then we are not going to
calculate the hyperfine splitting.

Now, the spin-averaged mass, $\bar{M}_{nl}$, of $nL$ multiplet can be obtained
by solving the spinless Salpeter equation

$[\sqrt{\vec{p}_{q}^{2}+m^{2}_{q}}+\sqrt{\vec{p}_{\bar{q}}^{2}+m^{2}_{\bar{q}}}+V_{q\bar{q}}(r)]\psi(r)=E\psi(r).$
(3)

Here we employ a variational approach described in Ref. var  to solve the
Eq.(3). This variational approach has been applied well in solving the
Salpeter equation for $c\bar{s}$ cs , $c\bar{c}$ and $b\bar{b}$ cc  mass
spectrum.

In the calculations, the basic simple harmonic oscillator (SHO) functions are
taken as the trial wave functions. It is given by

$\psi_{nl}(r,\beta)=\beta^{3/2}\sqrt{\frac{2(2n-1)!}{\Gamma(n+l+\frac{1}{2})}}(\beta
r)^{l}e^{-\frac{\beta^{2}r^{2}}{2}}L^{l+1/2}_{n-1}(\beta^{2}r^{2})$

in the position space. Here the SHO function scale $\beta$ is the variational
parameter.

By the Fourier transform, the SHO radial wave function in the momentum is

$\displaystyle\psi_{nl}(p,\beta)=\frac{(-1)^{n}}{\beta^{3/2}}\sqrt{\frac{2(2n-1)!}{\Gamma(n+l+\frac{1}{2})}}(\frac{p}{\beta})^{l}e^{-\frac{p^{2}}{2\beta^{2}}}L^{l+1/2}_{n-1}(\frac{p^{2}}{\beta^{2}}).$

The wave functions of $\psi_{nl}(r,\beta)$ and $\psi_{nl}(p,\beta)$ meet the
normalization conditions:

$\int^{\infty}_{0}\psi^{2}_{nl}(r,\beta)r^{2}dr=1;~{}~{}~{}~{}\int^{\infty}_{0}\psi^{2}_{nl}(p,\beta)p^{2}dp=1.$

In the variational approach, the corresponding $\bar{M}_{nl}$ are given by
minimizing the expectation value of $H$

$\frac{d}{d\beta}E_{nl}(\beta)=0.$ (4)

where

$E_{nl}(\beta)\equiv\langle
H\rangle_{nl}=\langle\psi_{nl}|H|\psi_{nl}\rangle.$ (5)

When all the parameters of the potential model are known, the values of the
harmonic oscillator parameter $\bar{\beta}$ can be fixed directly. With the
values of $\bar{\beta}$, all the spin-averaged mass $\bar{M}_{nl}$ will be
obtained easily. $\bar{M}_{nl}$ obtained in this way trend to be better for
the higher-excited states Roberts .

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 $V_{\vec{s}\cdot\vec{s}}(r)$ for the $1S$ mesons. The
following Gaussian-smeared contact hyperfine interaction ss  is taken for
convenience,

$V_{q\bar{q}}^{\vec{s}\cdot\vec{s}}(r)=\frac{32\pi\alpha_{s}}{9m^{2}_{q}}(\frac{\kappa}{\sqrt{\pi}})^{3}e^{-\kappa^{2}r^{2}}\vec{S}_{q}\cdot\vec{S}_{\bar{q}}.$
(6)

In this work, we choose the model parameters as follows: $m_{u}$ = $m_{d}$ =
0.220 GeV, $m_{s}=$ 0.428 GeV, $\alpha_{s}=$ 0.6, $\sigma=$ 0.143 GeV2,
$\kappa=$ $0.37$ GeV, and $C=$ $-0.37$ GeV. We take the smaller value of
$\sigma$ here rather than the value in Ref. Isgur . The smaller $\sigma$ was
obtained by the relation between the slope of the Regge trajectory for the
Salpeter equation $\alpha^{\prime}$ and the slope $\alpha^{\prime}_{st}$ in
the string picture Badalian . The Gaussian smearing parameter $\kappa$ seems a
little smaller than that in Ref. Isgur . However, the $\kappa$ is usually
fitted by the hyperfine splitting of low-excited $nS$ states in the
literatures with a certain arbitrariness.

The values of $\bar{M}_{nL}$ and $\bar{\beta}$ 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
PDG .

States | $\bar{M}_{nl}(n\bar{n})$ | $\bar{\beta}$ | Expt. PDG  | $\bar{M}_{nl}(s\bar{s})$ | $\bar{\beta}$ | Expt. PDG
---|---|---|---|---|---|---
1S | - | 0.44 0.34 | - | - | 0.42 0.39 | -
2S | 1.399 | 0.310 | 1.389 | 1.631 | 0.330 | 1.629
3S | 1.859 | 0.295 |  | $\underline{2.069}$ | 0.310 |
4S | 2.240 | 0.290 |  | 2.436 | 0.300 |
1P | 1.252 | 0.310 | 1.257 | 1.460 | 0.340 | 1.478
2P | 1.711 | 0.294 |  | $\underline{1.926}$ | 0.315 |
3P | 2.110 | 0.290 |  | 2.308 | 0.300 |
1D | 1.661 | 0.280 | 1.672 | $\underline{1.883}$ | 0.300 |
2D | $\underline{2.067}$ | 0.276 |  | 2.272 | 0.292 |
3D | 2.417 | 0.275 |  | 2.609 | 0.288 |
1F | 1.924 | 0.277 |  | 2.128 | 0.295 |
2F | 2.287 | 0.275 |  | 2.478 | 0.290 |
1G | 2.161 | 0.275 |  | 2.350 | 0.292 |
1H | 2.377 | 0.273 |  | 2.554 | 0.287 |
Table 1: The spin-averaged mass (unit: GeV) and the harmonic oscillator
parameter $\bar{\beta}$ (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 $2S$, $1P$, $1D$ $n\bar{n}$ and
$1P$, $2S$ $s\bar{s}$ mesons are consistent with the experimental data.
Indeed, the predicted masses of higher excited states here are also
reasonable, $e.g.$, $a_{4}(2040)$ and $f_{4}(2050)$ are very possible the
$F-$wave $n\bar{n}$ isovector and isoscalar mesons with the masses of
$1996^{+10}_{-9}$MeV and $2018\pm{11}$MeV, respectively PDG . The predicted
spin-averaged mass of $1F$ is not incompatible with experiments. Our results
are also overall in good agreement with the expectations from Ref. long1 . The
trend that a higher excited state corresponds to a smaller $\bar{\beta}$
coincides with Ref. Godfrey ; Close ; Li . For considering the spin-spin
contact hyperfine interaction, there are two $\bar{\beta}$s for the $1S$
mesons. The larger one corresponds to the $1^{1}S_{0}$ state, the smaller one
the $1^{3}S_{1}$ state.

As shown in Ref. long1 ; long2 , the confinement potential $V_{conf}(r)$ is
determinant for the properties of higher excited states. In Ref. long1 , the
masses for higher excited states with $\sigma=0.143$GeV2 and $\alpha_{s}=0$
are closer to experimental data than the results given in Ref. Isgur . Then we
ignored the Coulomb interaction for $1D$, $2D$, $1F$, $1G$ and $1H$ states. In
this way, $\bar{M}_{nl}$ for these states increase about 100MeV.

The masses of $\eta^{\prime}(3^{1}S_{0})$, $f^{\prime}_{1}(2^{3}P_{1})$,
$\eta^{\prime}_{2}(1^{1}D_{2})$ and $\eta_{2}(2^{1}D_{2})$ are usually within
$1.8\sim 2.1$GeV in various quark potential models Isgur ; Ebert ; Sorace ;
Vijande  (see in Table 2). The predicted spin-averaged masses of
$3S(s\bar{s})$, $2P(s\bar{s})$, $1D(s\bar{s})$ and $2D(n\bar{n})$ are also
within this mass regions (see in Table 1). Due to the uncertainty of the
potential models, absolute deviation from experimental data are usually about
100$\sim$150 MeV for the higher excited states. Comparing with these predicted
masses, $X(1870)$ disfavors the $\eta^{\prime}(3^{1}S_{0})$ assignment for its
low mass. But the possibilities of $f^{\prime}_{1}(2^{3}P_{1})$,
$\eta^{\prime}_{2}(1^{1}D_{2})$ and $\eta_{2}(2^{1}D_{2})$ still exist. Here
we don't consider the possibility of $X(1870)$ as the $\eta(3^{1}S_{0})$ state
because $\eta(1760)$ looks more like a good $\eta(3^{1}S_{0})$ candidate Li5 ;
Liu ; Yu .

States |  $\eta^{\prime}(3^{1}S_{0})$ |  $f^{\prime}_{1}(2^{3}P_{1})$ |  $\eta^{\prime}_{2}(1^{1}D_{2})$ |  $\eta_{2}(2^{1}D_{2})$
---|---|---|---|---
Ref. Isgur  |  $-$ |  2030 |  1890 |  2130†
Ref. Ebert  |  2085 |  2016 |  1909 |  1960
Ref. Sorace  |  2099 |  1988 |  1851 |  $-$
Ref. Vijande  |  $-$ |  $-$ |  1853 |  1863
Table 2: The masses predicted for $3^{1}S_{0}$($\eta^{\prime}$),
$2^{3}P_{1}$($\eta^{\prime}$), $1^{1}D_{2}$($\eta^{\prime}$) and
$2^{1}D_{2}$($\eta$) in Refs. Isgur ; Ebert ; Sorace ; Vijande .

Regge trajectories (RTs) is another useful tool for studying the mass spectrum
of the light flavor mesons. In Ref. Regge1 , the authors fitted the RTs for
all light-quark meson states listed in the PDG tables. A global description
was constructed as

$M^{2}=1.38(4)n+1.12(4)J-1.25(4).$ (7)

Here, _n_ and _J_ mean the the radial and angular-momentum quantum number.
Recently, the authors of Ref. Regge1  repeated their fits with the subset
mesons of the paper Regge2 . They found a little smaller averaged slopes of
$\mu^{2}=1.28(5)$GeV2 and $\beta^{2}=1.09(6)$GeV2, to be compared with
$\mu^{2}=1.38(4)$GeV2 and $\beta^{2}=1.12(4)$GeV2 in the Eq.(7). Here the
$\mu^{2}$ and $\beta^{2}$ are the weighted averaged slope for radial and
angular-momentum RTs Regge1 ; Regge3 .

Now $h_{1}(1380)$, $f_{1}(1420)$ and $\eta^{\prime}(1475)$ have been
established as the $1^{1}P_{1}$, $1^{3}P_{1}$ and $2^{1}S_{0}$ $s\bar{s}$
states in PDG PDG . With the differences between the mass squared of $X(1870)$
and these states (Table 3), $X(1870)$ could be assigned for the
$\eta^{\prime}(3^{1}S_{0})$ and $f^{\prime}_{1}(2^{1}P_{1})$. The mass of
$X(1870)$ is too large for the $\eta^{\prime}_{2}(1^{1}D_{2})$ state in the
RTs. $\eta_{2}(1645)$ has been assigned as the $1^{1}D_{2}$ $n\bar{n}$ meson
PDG . Since $M^{2}(X(1870))-M^{2}(\eta_{2}(1640))=0.91^{+0.04}_{-0.03}$GeV2
which is much smaller than $1.38(4)$GeV2, $X(1870)$ looks unlike the
$2^{1}D_{2}$ $n\bar{n}$ state for its low mass. However, the difference of
$M^{2}(X(1870))-M^{2}(h_{1}(1170))=2.16^{+0.06}_{-0.05}$GeV2 matches the
slopes $2.37(11)$GeV2 well. Then the RTs can't exclude the possibility of
$X(1870)$ as the $2^{1}D_{2}$ $n\bar{n}$ state.

Four possible states for _X_(1870)
---
$\eta^{\prime}(3^{1}S_{0})$ |  $f_{1}^{\prime}(2^{3}P_{1})$ |  $\eta^{\prime}_{2}(1^{1}D_{2})$ |  $\eta_{2}(2^{1}D_{2})$
$\eta^{\prime}(1475)$ |  $f_{1}(1420)$ |  $h_{1}(1380)$ |  $\eta_{2}(1645)$
$\mu^{2}=$1.34${}^{+0.04}_{-0.03}$ |  $\mu^{2}=$1.38${}^{+0.04}_{-0.03}$ |  $\beta^{2}=$1.60${}^{+0.06}_{-0.06}$ |  $\mu^{2}=$0.91${}^{+0.04}_{-0.03}$
Table 3: $X(1870)$ calculated in RTs for different states are shown. The
masses of $\eta^{\prime}(1475)$, $f_{1}(1420)$, $h_{1}(1380)$ and
$\eta_{2}(1645)$ are taken from PDG PDG .

As mentioned in the Introduction, $X(1870)$ is also a good hybrid candidate
since its mass overlaps the predictions given by different models. The
predicted masses for $0^{+}0^{-+}$, $0^{+}1^{++}$ and $0^{+}2^{-+}$
$n\bar{n}g$ states by these models are collected in Table 4.

States |  $\eta_{H}(0^{+}0^{-+})$ |  $f_{H}(0^{+}1^{++})$ |  $\eta_{H}(0^{+}2^{-+})$
---|---|---|---
Bag Bag1 ; Bag2  |  1.3 |  heavier |  1.9
Flux tube tube1 ; tube2  |  1.7$\sim$1.9 |  1.7$\sim$1.9 |  1.7$\sim$1.9
Constituent gluon gluon  |  1.8$\sim$2.2 |  1.3$\sim$1.8 |  1.8$\sim$2.2
Table 4: The masses predicted for $\eta_{H}(0^{+}0^{-+})$ $f_{H}(0^{+}1^{++})$
and $\eta_{H}(0^{+}2^{-+})$ hybrid states in Refs. Bag1 ; Bag2 ; tube1 ; tube2
; gluon .

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 $f^{\prime}_{1}(2^{3}P_{1})$,
$\eta^{\prime}_{2}(1^{1}D_{2})$ or $\eta_{2}(2^{1}D_{2})$ state with a
reasonable uncertainty. In the RTs, $X(1870)$ favors the
$\eta^{\prime}(3^{1}S_{0})$ and $f^{\prime}_{1}(2^{3}P_{1})$ assignments. But
the $\eta_{2}(2^{1}D_{2})$ assignment can't be excluded thoroughly. $X(1870)$
is also a good hybrid state candidate. Since the masses of $X(1870)$ and
$\eta_{2}(1870)$ are nearly equal, the possible assignments of $X(1870)$ also
suit $\eta_{2}(1870)$. The investigations of the strong decay properties will
be more helpful to distinguish the $\eta_{2}(1870)$ and $X(1870)$.

##  III The strong decay

###  III.1 The final mesons include the scalar mesons below 1 GeV

Despite many theoretical efforts, the scalar nonet of $q\bar{q}$ mesons has
never well-established. The lowest-lying scalar mesons including $\sigma(500)$
(or $f_{0}(600)$), $\kappa(800)$, $a_{0}(980)$ and $f_{0}(980)$ are difficult
to be described as $q\bar{q}$ states, _e.g_., $a_{0}(980)$ is associated with
nonstrange quarks in the $q\bar{q}$ scheme. If this is true, its high mass and
decay properties are difficult to be understood simultaneously. So
interpretations as exotic states were triggered, _i.e_., as two clusters of
two quarks and two antiquarks Maiani , particular quasimolecular states
molecule , and uncorrelated four quark states $qq\bar{q}\bar{q}$ tetraquark1 ;
tetraquark2 ; tetraquark3  have been proposed.

Though the structures of these scalar mesons below 1 GeV are still in dispute,
the viewpoint that these scalar mesons can constitute a complete nonet states
has been reached in the most literatures (as illustrated in Fig.2). In the
following, we will denote this nonet as `` $\mathcal{S}$ '' multiplet for
convenience.

Figure 2: The `` $\mathcal{S}$ '' nonet below 1 GeV shown in $Y-I_{3}$ plane.

Due to the unclear nature of the $\mathcal{S}$ mesons, it seems much difficult
to study the decay processes when the final mesons includes a $\mathcal{S}$
member. As an approximation, $a_{0}(980)$, $\sigma(500)$ and $f_{0}(980)$ were
treated as $1^{3}P_{0}$ $q\bar{q}$ mesons in Refs. Yu ; Liu1 . In Refs. Li3 ;
Liu , this kind of decay channel was ignored. However, this kind of decay mode
maybe predominant for some mesons. For example, the observations indicate that
$f_{1}(1285)$, $\eta(1405)$ and $X(1870)$ primarily decay via the
$a_{0}(980)\pi$ channel Bes6 .

In what follows, we will extract some useful information about this kind of
decay mode by the SU(3) flavor symmetry. We will show that $a_{0}(980)\pi$,
$\sigma_{0}\eta$ and $f_{0}\eta$ are the main decay channels for the isoscalar
$n\bar{n}$ and the $n\bar{n}g$ mesons when they decay primarily through ``
$\mathcal{S}$ $+$ P'' mesons, where the sign ``P'' denotes a light
pseudoscalar meson. This will explain why $X(1870)$ has been first observed in
the $\eta\pi^{+}\pi^{-}$ channel.

We noticed that the $\mathcal{S}$ nonet could be interpreted like the
$q\bar{q}$ nonet in the diquark-antidiquark scenario. In Wilczek and Jaffe's
terminology Jaffe ; Wilczek , the $\mathcal{S}$ mesons consist of a ``good''
diquark and a ``good'' antidiquark. When $u$, $d$ quarks forms a ``good''
diquark, it means that the two light quarks, _u_ and _d_ , could be treated as
a quasiparticle in color $\bar{3}$, flavor $\bar{3}$ 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
$P=(-1)^{L_{12-34}}$ Santopinto  where the $L_{12-34}$ refer to the relative
angular momentum between two clusters. Thus the $\mathcal{S}$ mesons are the
lightest tetraquark states in the diquark-antidiquark model with
$L_{12-34}=0$. The $\mathcal{S}$ nonet in the full set of flavor
representations is

$\displaystyle(3\otimes 3)_{\bar{3}}\otimes(\bar{3}\otimes\bar{3})_{3}=8\oplus
1$

Because the SU(3) flavor symmetry is not exact, the two physical isoscalar
mesons, $\sigma_{0}$ and $f_{0}$, are usually the mixing states of the
$|8\rangle_{I=0}$ and $|1\rangle_{I=0}$ states Maiani ,

$\displaystyle\begin{aligned} \left(\begin{array}[]{c}f_{0}\\\ \sigma_{0}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\cos\vartheta&\sin\vartheta\\\
-\sin\vartheta&\cos\vartheta\\\
\end{array}\right)\left(\begin{array}[]{c}|8\rangle_{I=0}\\\
|1\rangle_{I=0}\\\ \end{array}\right)\end{aligned}$ (8)

When the mixing angle $\vartheta$ equals the so-called ideal mixing angle,
$i.e.$, $\vartheta$ = 54.74∘, the composition of the $\sigma(500)$ and
$f_{0}(980)$ are

$\displaystyle\begin{aligned} \left(\begin{array}[]{c}f_{0}\\\ \sigma_{0}\\\
\end{array}\right)=\left(\begin{array}[]{c}|\frac{1}{\sqrt{2}}([su][\bar{s}\bar{u}]+[sd][\bar{s}\bar{d}])\rangle\\\
|[ud][\bar{u}\bar{d}]\rangle\\\ \end{array}\right).\end{aligned}$

It seems that the deviation from the ideal mixing angle of the $\sigma(500)$
and $f_{0}(980)$ is small Maiani . In the following calculations, we will
treat them in the ideal mixing scheme.

Under the SU(3) flavor assumption, all the members of the octet have the same
basic coupling constant in one type of reaction, while the singlet member have
a different coupling constant. Particularly, when a quarkonium decays into
$\mathcal{S}$ and $q\bar{q}$ mesons, there are five independent coupling
constants, $i.e.$, $g_{A88}$, $g_{A81}$, $g_{A18}$, $g_{B88}$ and $g_{B11}$,
corresponding to five different channels

$\begin{cases}\mid 8\rangle_{q\bar{q}}\rightarrow\mid
8\rangle_{\mathcal{S}}\otimes\mid 8\rangle_{q\bar{q}}:\hskip
28.45274ptg_{A88}\\\ \mid 8\rangle_{q\bar{q}}\rightarrow\mid
8\rangle_{\mathcal{S}}\otimes\mid 1\rangle_{q\bar{q}}:\hskip
28.45274ptg_{A81}\\\ \mid 8\rangle_{q\bar{q}}\rightarrow\mid
1\rangle_{\mathcal{S}}\otimes\mid 8\rangle_{q\bar{q}}:\hskip
28.45274ptg_{A18}\\\ \mid 1\rangle_{q\bar{q}}\rightarrow\mid
8\rangle_{\mathcal{S}}\otimes\mid 8\rangle_{q\bar{q}}:\hskip
28.45274ptg_{B88}\\\ \mid 1\rangle_{q\bar{q}}\rightarrow\mid
1\rangle_{\mathcal{S}}\otimes\mid 1\rangle_{q\bar{q}}:\hskip
28.45274ptg_{B11}\\\ \end{cases}$

In order to determine the relations between these coupling constants, we shall
assume the process that the $q\bar{q}$ or $q\bar{q}g$ meson decays into a
$\mathcal{S}$ and another $q\bar{q}$ mesons obeys the OZI (Okubo-Zweig-Iizuka)
rule, $i.e.$, the two quarks in the mother meson go into two daughter mesons,
respectively. Therefore, there are four forbidden processes:
$X(\frac{1}{\sqrt{2}}(u\bar{u}-d\bar{d}))\nrightarrow a_{0}+s\bar{s}$,
$X(\frac{1}{\sqrt{2}}(u\bar{u}+d\bar{d}))\nrightarrow\sigma_{0}+s\bar{s}$,
$X(s\bar{s})\nrightarrow f_{0}+\frac{1}{\sqrt{2}}(u\bar{u}+d\bar{d})$ and
$X(s\bar{s})\nrightarrow\sigma_{0}+s\bar{s}$. With the help of the SU(3)
Clebsch$-$Gordon coefficients Coefficients , the ratios between the five
coupling constants are extracted as

$\displaystyle g_{A81}:g_{A18}:g_{B88}:g_{B11}:g_{A88}$
$\displaystyle=\sqrt{2}:-\sqrt{\frac{2}{5}}(\sqrt{5}+1):-\frac{2}{\sqrt{5}}(\sqrt{5}+1):-\sqrt{\frac{2}{5}}(\sqrt{5}+1):1$
(9) $\displaystyle\approx 1.41:-2.05:-2.89:-2.05:1.00$

Figure 3: The coefficients $\zeta^{2}$ of the isoscalar meson $\xi$ versus the
mixing angle $\theta$. Figure 4: The coefficients $\zeta^{2}$ of the isoscalar
meson $\xi^{\prime}$ versus the mixing angle $\theta$.

It is well known that the physical states, $\eta(548)$ and
$\eta^{\prime}(958)$ are the mixture of the SU(3) flavor octet and singlet.
They can be written in terms of a mixing angle, $\theta_{p}$, as follows

$\displaystyle\begin{aligned} \left(\begin{array}[]{c}\eta(548)\\\
\eta^{\prime}(958)\\\
\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta_{p}&-\sin\theta_{p}\\\
\sin\theta_{p}&\cos\theta_{p}\\\
\end{array}\right)\left(\begin{array}[]{c}|8\rangle_{I=0}\\\
|1\rangle_{I=0}\\\ \end{array}\right)\end{aligned}$ (10)

The mixing angle $\theta_{p}$ has been measured by various means. However,
there is still uncertainty for $\theta_{p}$. An excellent fit to the tensor
meson decay widths was performed under the SU(3) symmetry, and
$\theta_{p}\simeq-17^{o}$ was obtained Amsler1 . In our calculation,
$\theta_{p}$ is taken as $-17^{o}$. The excited mixtures of $n\bar{n}$ and
$s\bar{s}$ are denoted as

$\displaystyle\begin{aligned} \left(\begin{array}[]{c}\xi\\\ \xi^{\prime}\\\
\end{array}\right)=\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\\\
\end{array}\right)\left(\begin{array}[]{c}|1\rangle_{I=0}\\\
|8\rangle_{I=0}\\\ \end{array}\right)\end{aligned}$ (11)

In this scheme, the ideal mixing occurs with the choice of $\theta=35.3^{o}$.
When $\xi$ and $\xi^{\prime}$ decay into a $\mathcal{S}$ and pseudoscalar
mesons, the relations of decay amplitudes are governed by the coefficients
$\zeta^{2}$ which are model-independent in the limitation of SU(3)f symmetry.
With the coupling constants in hand, the coefficients $\zeta^{2}$ of $\xi$ and
$\xi^{\prime}$ versus the mixing angle $\theta$ are shown in the Fig.3 and
Fig.4. When $\xi$ and $\xi^{\prime}$ occurs in the ideal mixing, the values of
$\zeta^{2}$ are presented in Table 5. In the factorization framework, the
decay difference of a hybrid and excited $q\bar{q}$ mesons comes from the
spatial contraction Burns . Then the coefficients $\zeta^{2}$ for hybrid
states are same as these of $q\bar{q}$ quarkoniums.

Decay |  $a_{0}\pi$ |  $\sigma\eta$ |  $\kappa K$ |  $f_{0}\eta$ |  $\sigma\eta^{\prime}$
---|---|---|---|---|---
channels
$\zeta^{2}[n\bar{n}(g)]$ |  2.17 |  3.56 |  0.47 |  1.07 |  0.44
$\zeta^{2}[s\bar{s}(g)]$ |  0.00 |  0.00 |  0.72 |  1.08 |  0.00
Table 5: .The coefficients $\zeta^{2}$ of $\xi$ and $\xi^{\prime}$ in the
ideal mixing.

Here the mixing of $\eta(548)$ and $\eta^{\prime}(958)$ has been considered.
It is sure that the $\zeta^{2}$ are zero for the processes,
$\xi^{\prime}\rightarrow a_{0}\pi$, $\xi^{\prime}\rightarrow\sigma\eta$ and
$\xi^{\prime}\rightarrow\sigma\eta^{\prime}$, since they are OZI-forbidden.
$\zeta^{2}$ of $\xi^{\prime}\rightarrow f_{0}\eta^{\prime}$ hasn't been
considered in Table 5 since $X(1870)$ lies below the threshold of
$f_{0}\eta^{\prime}$.

As illustrated in the Fig.3 and Fig.4, the primary decay channels of a
$s\bar{s}$ or $s\bar{s}g$ predominant excitation are $f_{0}\eta$ and $\kappa
K$. If the deviation of $\theta$ from the ideal mixing angle is not large,
$X(1870)$ should be a $n\bar{n}$ or $n\bar{n}g$ predominant state since
$X(1870)$ primarily decay via the $a_{0}(980)\pi$ channel. At present, only
the ground $0^{-+}$ and the $0^{++}$ isoscalar mesons deviate from the ideal
mixing distinctly. In addition, if the $X(1870)$ is produced via a diagram of
Fig.1 [B], its should also be $n\bar{n}$ or $n\bar{n}g$ predominant state.

Of course, the SU(3)f symmetry breaking will effect the ratios of these
channels listed in Table 5, because the three-momentum of the these products
are different. However, the coefficients $\zeta^{2}$ have presented the
valuable information for these specific decay channels. When $\eta_{2}(1870)$
occupies the $2^{1}D_{2}$ $n\bar{n}$ state, $X(1870)$ becomes a good
$n\bar{n}g$ candidate. In the following subsection, we will explore the two-
body strong decays of $X(1870)$ within the ${}^{3}P_{0}$ model and the flux-
tube model. Of course, the analysis of $X(1870)$ also suit $\eta_{2}(1870)$
for their nearly equal masses.

###  III.2 The strong decays of $\eta_{2}(1870)$ and $X(1870)$

In Ref. Li3 , the ${}^{3}P_{0}$ model Micu ; Oliver1 ; Oliver2  and the flux-
tube model Kokoski  were employed to study the two-body strong decays of
$\eta_{2}(1870)$. There, the pair production (creation) strength $\gamma$ and
the simple harmonic oscillator (SHO) wave function scale parameter, $\beta$s,
were taken as constants.

However, a series of studies indicate that the strength $\gamma$ may depend on
both the flavor and the relative momentum of the produced quarks Ackleh ;
Bonnaz . $\gamma$ may also depend on the reduced mass of quark-antiquark pair
of the decaying meson Segovia . Firstly, the relations of the ${}^{3}P_{0}$
model to ``microscopic'' QCD decay mechanisms have been studied in Ref. Ackleh
. There, the authors found that the constant $\gamma$ corresponds
approximately to the dimensionless combination, $\sigma/m_{q}\beta$, where
$m_{q}$ is the mass of produced quark, $\beta$ means the meson wave function
scale and $\sigma$ is the string tension. Secondly, the momentum dependent
manner of $\gamma$ has been studied in Ref. Bonnaz . It was found that
$\gamma$ is dependent on the relative momentum of the created $q\bar{q}$ pair,
and the form of $\gamma(k)=A+B\exp(-Ck^{2})$ with
$k=|\vec{k}_{3}-\vec{k}_{4}|$ was suggested. Thirdly, J.Segovia, _et al_.,
proposed that $\gamma$ is a function of the reduced mass of quark-antiquark
pair of the decaying meson Segovia . Based on the first and third points
above, $\gamma$ will depend on the flavors of both the decaying meson and
produced pairs. In our calculations, we will treat the $\gamma$ as a free
parameter and fix it by the well-measured partial decay widths.

In addition, the amplitudes given by the ${}^{3}P_{0}$ model and the flux-tube
model often contain the nodal-type Gaussian form factors which can lead to a
dynamic suppression for some channels. Then the values of $\beta$ are
important to exact the decay width for the higher excited mesons in these two
strong decay models.

In the following, the two-body strong decay of $X(1870)$ will be investigated
in the ${}^{3}P_{0}$ model where the strength $\gamma$ will be extracted by
fitting the experimental data. The SHO wave function scale parameter,
$\beta$s, will be borrowed from the 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 non relativistic limit, the transition operator $\mathcal{\hat{T}}$ of
the ${}^{3}P_{0}$ model is depicted as

$\displaystyle\mathcal{\hat{T}}$ $\displaystyle=$
$\displaystyle-3\gamma\sum_{\text{\emph{m}}}\langle 1,m;1,-m|0,0\rangle\iint
d^{3}\vec{k}_{3}d^{3}\vec{k}_{4}\delta^{3}(\vec{k}_{3}+\vec{k}_{4})\mathcal{Y}_{1}^{m}(\frac{\vec{k}_{3}-\vec{k}_{4}}{2})\omega_{0}^{(3,4)}\varphi^{(3,4)}_{0}\chi^{(3,4)}_{1,-m}d^{\dagger}_{3i}(\vec{k}_{3})d^{\dagger}_{4j}(\vec{k}_{4})$
(12)

Where the $\omega_{0}^{(3,4)}$ and $\varphi^{(3,4)}_{0}$ are the color and
flavor wave functions of the $q_{3}\bar{q}_{4}$ pair created from vacuum.
Thus, $\omega_{0}^{(3,4)}=(R\bar{R}+G\bar{G}+B\bar{B})/\sqrt{3}$,
$\varphi^{(3,4)}_{0}=(u\bar{u}+d\bar{d}+s\bar{s})/\sqrt{3}$ are color and
flavor singlets. The pair is also assumed to carry the quantum numbers of
$0^{++}$, suggesting that they are in a ${}^{3}P_{0}$ state. Then
$\chi^{(3,4)}_{1,-m}$ represents the pair production in a spin triplet state.
The solid harmonic polynomial
$\mathcal{Y}_{1}^{m}(\vec{k})\equiv|\vec{k}|\mathcal{Y}_{1}^{m}(\theta_{k},\phi_{k})$
reflects the momentum-space distribution of the $q_{3}\bar{q}_{4}$.

The helicity amplitude $\mathcal{M}^{M_{J_{A}},M_{J_{B}},M_{J_{C}}}(p)$ of
$A\rightarrow B+C$ is given by

$\displaystyle\langle
BC|\mathcal{\hat{T}}|A\rangle=\delta^{3}(\vec{P}_{A}-\vec{P}_{B}-\vec{P}_{C})\mathcal{M}^{M_{J_{A}},M_{J_{B}},M_{J_{C}}}(p),$
(13)

where _p_ represents the momentum of the outgoing meson in the rest frame of
the meson _A_. When the mock state Hayne  is adopted to describe the spatial
wave function of a meson, the helicity amplitude
$\mathcal{M}^{M_{J_{A}},M_{J_{B}},M_{J_{C}}}(p)$ can be constructed in the
$L-S$ basis easily Oliver1 ; Oliver2 . The mock state for _A_ meson is

$\displaystyle|A({n_{A}}$
$\displaystyle{}^{2S_{A}+1}L_{A}^{J_{A}M_{J_{A}}}(\vec{P}_{A})\rangle$ (14)
$\displaystyle\equiv$
$\displaystyle\sqrt{2E_{A}}\sum_{{M_{L_{A}}}{M_{S_{A}}}}\langle
L_{A}M_{L_{A}}S_{A}M_{S_{A}}|J_{A}M_{J_{A}}\rangle\omega_{A}^{12}\phi_{A}^{12}\chi_{S_{A}M_{S_{A}}}^{12}$
$\displaystyle\times\int
d\vec{P}_{A}\psi_{n_{A}}^{L_{A}M_{L_{A}}}(\vec{k}_{1},\vec{k}_{2})|q_{1}(\vec{k}_{1})q_{2}(\vec{k}_{2})\rangle.$

To obtain the analytical amplitudes, the SHO wave functions are usually
employed for $\psi_{n_{A}}^{L_{A}M_{L_{A}}}(\vec{k}_{1},\vec{k}_{2})$. For
comparison with experiments, one obtains the partial decay width
$\mathcal{M}^{JL}(p)$ via the Jacob-Wick formula Jacob

$\displaystyle\mathcal{M}_{LS}(p)=$
$\displaystyle\frac{\sqrt{2L+1}}{2J_{A}+1}\sum_{\text{$M_{J_{B}}$,$M_{J_{C}}$}}\langle
L0JM_{J_{A}}|J_{A}M_{J_{A}}\rangle$ (15) $\displaystyle\times\langle
J_{B},M_{J_{B}}J_{C},M_{J_{C}}|JM_{J_{A}}\rangle\mathcal{M}^{M_{J_{A}},M_{J_{B}},M_{J_{C}}}(p).$

Finally, the decay width $\Gamma(A\rightarrow BC)$ is derived analytically in
terms of the partial wave amplitudes

$\displaystyle\Gamma(A\rightarrow
BC)=2\pi\frac{E_{B}E_{C}}{M_{A}}p\sum_{LS}|\mathcal{M}_{LS}(p)|^{2}.$ (16)

More technical details of the ${}^{3}P_{0}$ model can be found in Ref. Oliver2
. The inherent uncertainties of the ${}^{3}P_{0}$ decay model itself have been
discussed in the Refs. Bonnaz ; model1 ; model2 .

The dimensionless parameter $\gamma$ will be fixed by the 8 well-measured
partial decay widths which are listed in Table6. The $\mathcal{M}_{LS}$
amplitudes of these decay channels are presented explicitly in the Appendix A.

Decay channels |  p (GeV) |  $\gamma(10^{3})$ |  $\gamma$Bonnaz  |  Decay channels |  p (GeV) |  $\gamma(10^{3})$ |  $\gamma$Bonnaz
---|---|---|---|---|---|---|---
$\rho\rightarrow\pi\pi$ |  0.362 |  17.8 |  9.18 |  $f^{\prime}_{1}\rightarrow K^{*}\bar{K}$ |  0.158 |  4.9 |  -
$a_{2}\rightarrow\eta\pi$ |  0.535 |  11.5 |  - |  $f_{2}\rightarrow K\bar{K}$ |  0.401 |  2.9 |  6.11
$f_{2}\rightarrow\pi\pi$ |  0.622 |  7.8 |  7.13 |  $a_{2}\rightarrow K\bar{K}$ |  0.434 |  2.3 |  3.91
$\rho_{3}\rightarrow\pi\pi$ |  0.833 |  4.2 |  - |  $f^{\prime}_{2}\rightarrow K\bar{K}$ |  0.579 |  2.0 |  5.66
Table 6: . Values of $\gamma$ in different channels and comparison with the
results given in Ref.Bonnaz . Here, $\rho(770)$, $a_{2}(1320)$,
$f^{\prime}_{1}(1420)$, $f_{2}(1270)$, $f^{\prime}_{2}(1525)$ and
$\rho_{3}(1690)$ have been studied.

As mentioned before, $\gamma$ 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 $n\bar{n}\rightarrow n\bar{n}+n\bar{n}$, the other includes
$s\bar{s}\rightarrow n\bar{s}+s\bar{n}$ and $n\bar{n}\rightarrow
n\bar{s}+s\bar{n}$. The values of $\gamma$ here are a little different from
these given in Ref.Bonnaz  where an _AL_ 1 potential (for details of _AL_ 1
potential, see Ref.Roberts ) was selected to determine the meson wave
functions. Of course, the meson wave function given by different potentials
will influence the values of $\gamma$.

It is clear in Table 6 that $\gamma$ decrease with _p_ increase. In addition,
our calculation indicates that $\gamma$ depend on flavors of both the decaying
meson and the produced quark pairs. For example, values of $\gamma$ fixed by
$a_{2}\rightarrow K\bar{K}$ and $f^{\prime}_{2}\rightarrow K\bar{K}$ are
roughly equal.

In the following calculations, we assume that the values of $\gamma$
corresponding to the processes of $s\bar{s}\rightarrow n\bar{s}+s\bar{n}$ and
$n\bar{n}\rightarrow n\bar{s}+s\bar{n}$ are determined by one function.
Similarly, we take the function, $\gamma(p)=A+B\exp(-Cp^{2})$, for the
creation vertex. _The function of the creation vertex here is different with
the one used in the RefBonnaz _. With the four decay channels listed in fifth
column of Table 6, we fix the function as $\gamma(p)=1.8+4\exp(-10p^{2})$. For
the processes of $n\bar{n}\rightarrow n\bar{n}+n\bar{n}$ (the first column of
Table 6), we fix the creation vertex function as
$\gamma(p)=3.0+25\exp(-4p^{2})$. The dependence of $\gamma$ on the momentum
_p_ are plotted in the Fig. 5. Obviously the functions can describe the
dependence of $\gamma$ and _p_ well. The functions of creation vertex given
here need further test.

Figure 5: The functions of $\gamma(p)=A+B\exp(-Cp^{2})$ in different decay
processes. The symbols of red ``⚫'' and black ``◼'' denote $\gamma$ 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 $\mathcal{F}$ (Table 7). This $\mathcal{F}$ factor
also incorporates the statistical factor 1/2 if the final state mesons _B_ and
_C_ are identical (as illustrated in Fig.6). More details of $\mathcal{F}$ can
be found in the Appendix A of Ref.Barnes .

Figure 6: Two topological diagrams for a $q\bar{q}$ meson decay in the ${}^{3}P_{0}$ decay model. We refer to the left one as _d_ 1 where the produced quark goes into meson _C_ , and _d_ 2 where it goes into _B_. Decay |  $\mathcal{I}_{flavor}(d1)$ |  $\mathcal{I}_{flavor}(d1)$ |  $\mathcal{F}$
---|---|---|---
channels
$\rho\rightarrow\pi\pi$ |  $+1/\sqrt{2}$ |  $-1/\sqrt{2}$ |  1
$f_{2}\rightarrow\pi\pi$ |  $-1/\sqrt{2}$ |  $-1/\sqrt{2}$ |  3/2
$f_{2}\rightarrow KK$ |  0 |  $-1/\sqrt{2}$ |  2
$f^{\prime}_{1}\rightarrow K^{*}K$ |  +1 |  0 |  4
$f^{\prime}_{2}\rightarrow KK$ |  +1 |  0 |  2
$a_{2}\rightarrow KK$ |  0 |  -1 |  1
$a_{2}\rightarrow\eta\pi$ |  $+1/2$ |  $+1/2$ |  1
$\eta_{2}\rightarrow\omega\omega$ |  $-1/\sqrt{2}$ |  $-1/\sqrt{2}$ |  1/2
$\eta_{2}\rightarrow a_{i}\pi$ |  $-1/\sqrt{2}$ |  $-1/\sqrt{2}$ |  3
$\eta_{2}\rightarrow f_{i}\eta$ |  $+1/2$ |  $+1/2$ |  1
Table 7: The second and third columns for the flavor weight factors corresponding to two topological diagrams shown in Fig.6. The last column for the the flavor multiplicity factor $\mathcal{F}$. Here, $|\eta\rangle=(|n\bar{n}\rangle-|s\bar{s}\rangle)/\sqrt{2}$ and $|\eta^{\prime}\rangle=(|n\bar{n}\rangle+|s\bar{s}\rangle)/\sqrt{2}$ have been taken for simplicity. Decay |  $\eta_{2}(2^{1}D_{2})$ |  |  | $\eta_{H}(0^{+}0^{-+})$ | $f_{H}(0^{+}1^{++})$ | $\eta_{H}(0^{+}2^{-+})$
---|---|---|---|---|---|---
channels |  Our | Ref. Li3  |  Our |  Ref. Swanson  |  Our |  Ref. Swanson  |  Our |  Ref. Swanson
$K^{*}K$ |  0.5 |  17.7 |  19.3 |  12.6 |  10 5 |  4.9 |  24.1 18.0 |  3.2 |  2.0 1.0
$\rho\rho$ |  12.9 |  52.2 |  56.8 |  $\times$ |  $\times$ $\times$ |  $\times$ |  $\times$ $\times$ |  $\times$ |  $\times$ $\times$
$\omega\omega$ |  4.2 |  16.9 |  18.4 |  $\times$ |  $\times$ $\times$ |  $\times$ |  $\times$ $\times$ |  $\times$ |  $\times$ $\times$
$K^{*}K^{*}$ |  0.2 |  2.1 |  2.3 |  $\times$ |  $\times$ $\times$ |  $\times$ |  $\times$ $\times$ |  $\times$ |  $\times$ $\times$
$a_{0}(1450)\pi$ |  16.0 |  2.4 |  2.6 |  56.3 |  70 175 |  0.5 |  $\times$ 6 |  0.5 |  0.0 0.6
$a_{1}(1260)\pi$ |  0.0 |  15.2 |  16.6 |  $\times$ |  $\times$ $\times$ |  57.3 |  14 232 |  $\times$ |  0.3 $\times$
$f_{1}(1280)\eta$ |  0.0 |  0.0 |  0.0 |  $\times$ |  $\times$ $\times$ |  2.5 |  $-$ $-$ |  $\times$ |  0.0 $\times$
$a_{2}(1320)\pi$ |  54.2 |  102.5 |  111.6 |  8.8 |  1 16 |  35.1 |  5.0 179.4 |  26.7 |  25.1 67
$f_{2}(1275)\eta$ |  15.1 |  17.5 |  19.0 |  0.0 |  $\times$ $\times$ |  1.0 |  $-$ $-$ |  4.6 |  0.0 0.0
$\sum\Gamma_{i}$ |  103.3 |  226.5 |  246.7 |  77.7 |  81 196 |  101.3 |  43.1 435.4 |  35.0 |  27.4 68.6
Expt (MeV) | 225$\pm$14 PDG  |  |  |  |  | 57$\pm$12${}^{+19}_{-4}$ Bes6
Table 8: The partial widths of $X(1870)$ and compared with results from Refs.
Li3 ; Swanson . The symbol ``$\times$'' indicates that the decay modes are
forbidden and ``$-$'' denotes that the decay channels can be ignored. Here, we
collected the results given by the ${}^{3}P_{0}$ model from Ref. Li3  in the
left column, the right column by the flux-tube model. In Ref. Swanson , the
masses are taken as 1.8 GeV for the $0^{-+}$, $1^{++}$ and $2^{-+}$ for the
hybrid states.

The partial decay widths of $X(1870)$ are shown in Table 8 except the channels
of $\mathcal{S}$ $+$ $P$ mesons. $a_{2}(1320)\pi$ and $f_{2}(1275)\eta$ are
large channels for the $\eta_{2}(2^{1}D_{2})$ $n\bar{n}$ state in our work and
the Ref. Li3 , which are consistent with the experimental observations of the
$\eta_{2}(1870)$. The partial widths of $K^{*}K$, $\rho\rho$ and
$\omega\omega$ are narrower in our work than the expectations from Ref. Li3 .
$\eta_{2}(1870)$ has been observed by the BES Collaboration in the radiative
decay channel of $J/\psi\rightarrow\gamma\eta\pi\pi$ Li3 . However, no
apparent $\eta_{2}(1870)$ signals were detected in the channels of
$J/\psi\rightarrow\gamma\rho\rho$ Mark1
and$J/\psi\rightarrow\gamma\omega\omega$ Mark2 ; Bes4 . Therefore, improved
experimental measurements of the radiative $J/\psi$ decay channels are needed
for the $\eta_{2}(1870)$ in future.

Figure 7: The diagram for the `` $\mathcal{S}$ $+$ P'' channels through a
virtual intermediate $1^{3}P_{0}$ $q\bar{q}$ meson.

Nextly, we shall evaluate the partial widths of ``$\mathcal{S}+P$'' channels
which have not been listed in the table8. _The scheme is proposed as
following._ As illustrated in the Fig.7, we assume $X(1870)$ decay into
$a_{0}(980)\pi$ via a virtual intermediate $1^{3}P_{0}$ $q\bar{q}$ meson . We
notice the $\eta(1295)$ also dominantly decay into the $\eta\pi\pi$ Bes6 . Its
three-body decay can occur via three intermediate processes:
$\eta(1295)\rightarrow\eta\sigma/a_{0}(980)\pi/\eta(\pi\pi)_{S-wave}\rightarrow\eta\pi\pi$PDG
. With the ratio $\Gamma(a_{0}(980)\pi)/\Gamma(\eta\pi\pi)=0.65\pm 0.10$ and
$\Gamma(\eta(1295))=55\pm 5$MeV, the partial width of $\eta(1295)$ decaying
into $a_{0}(980)\pi$ is estimated no more than 45MeV. By the ${}^{3}P_{0}$
model, the ratio of $\frac{\Gamma(X(1870)\rightarrow
a_{0}(1^{3}P_{0})\pi)}{\Gamma(\eta(1295)\rightarrow a_{0}(1^{3}P_{0})\pi)}$
can be reached easily. If the uncertainty of the coupling vertex of
$\varepsilon(1^{3}P_{0}(q\bar{q})\rightarrow a_{0}(980))$ (see in Fig.7) is
assumed to be canceled in the ratio of $\frac{\Gamma(X(1870)\rightarrow
a_{0}(1^{3}P_{0})\pi)\cdot\varepsilon(1^{3}P_{0}(q\bar{q})\rightarrow
a_{0}(980))}{\Gamma(\eta(1295)\rightarrow
a_{0}(1^{3}P_{0})\pi)\cdot\varepsilon(1^{3}P_{0}(q\bar{q})\rightarrow
a_{0}(980))}$, the value of $\frac{\Gamma(X(1870)\rightarrow
a_{0}(980)\pi)}{\Gamma(\eta(1295)\rightarrow a_{0}(980)\pi)}$ can be extracted
roughly. Although the assumption above seems a little rough, we just need to
evaluate magnitudes of these decay channels.

$\eta(1295)$ is proposed to be the first radial excited state of $\eta(550)$.
Then the total decay widths of $\Gamma(X(1870)\rightarrow\mathcal{S}+P)$ is
evaluated no more than 12.6MeV and $\Gamma(X(1870)\rightarrow
a_{0}(980)\pi)\leq 3.8$MeV. The BESIII Collaboration claimed that $X(1870)$
primarily decay via $a_{0}(980)\pi$ Bes6 . The small partial width of
$\Gamma(X(1870)\rightarrow a_{0}(980)\pi)$ also indicates that the $X(1870)$
can't be interpreted as the $2^{1}D_{2}$ $q\bar{q}$ state.

In addition, our results do not support $X(1870)$ as the
$\eta_{2}(2^{1}D_{2})$ $n\bar{n}$ state since its observed decay width is much
smaller than the theoretical estimate. The $a_{2}(1320)\pi$ is the largest
decay channel in our numerical results and in Ref. Li3  for the
$\eta_{2}(2^{1}D_{2})$ $n\bar{n}$ state (Table 8). If the partial width of
$a_{0}(980)\pi$ channel is as large as $a_{2}(1320)\pi$, 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 the Appendix B.

Two groups of the partial widths predicted in the Ref. Swanson  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'' IKP . The right
column was by the developed flux tube decay model of Swanson-Szczepaniak (SS).
In Ref. Swanson , the masses are taken as 1.8 GeV for the $0^{-+}$, $1^{++}$
and $2^{-+}$ for the hybrid states.

For a hybrid meson, $X(1870)$ seems most possible to be the
$\eta_{H}(0^{+}2^{-+})$ state because the total widths exclude the channels of
$\mathcal{S}+P$ are much narrow in our work and in Ref.Swanson . It is
consistent with the narrow width of $X(1870)$.

As shown in Table 8, $X(1870)$ is impossible to be the $\eta_{H}(0^{+}0^{-+})$
hybrid state. The predicted width in both our work and in Ref.Swanson  are
broader. In addition, $\eta\pi$ is a visible channel for both $a_{0}(1450)$. A
week signal was found in the region of 1200$\sim$1400MeV in the analysis of
$\eta\pi^{\pm}$ (Fig.2(b) of Ref.Bes6 ), which contradicts the large
$a_{0}(1450)\pi$ channel of the $\eta_{H}(0^{+}0^{-+})$ state. We can exclude
the possibility of $X(1870)$ as the $\eta_{H}(0^{+}0^{-+})$ hybrid state
preliminarily.

The assignment for $X(1870)$ as the $f_{H}(0^{+}1^{++})$ hybrid seems
impossible since the theoretical width of $a_{1}(1260)\pi$ is rather broad in
our results and in the IKP model. If the partial width of $a_{0}(980)\pi$
channel is as large as $a_{1}(1260)\pi$, the total widths of $X(1870)$ will be
much broader than the experimental value. But the width given by the SS flux
tube decay model for the $f_{H}(0^{+}1^{++})$ hybrid is much small. So the
possibility of $X(1870)$ as a $f_{H}(0^{+}1^{++})$ hybrid can not be excluded.
We suggest to detect the decay channel of $a_{1}(1260)\pi$ because this
channel is forbidden for the $\eta_{H}(0^{+}2^{-+})$ state in the IKP flux
tube decay model and very small in the SS flux tube decay model (see Table8).
Then the channel of $a_{1}(1260)\pi$ can discriminate the state
$f_{H}(0^{+}1^{++})$ and $\eta_{H}(0^{+}2^{-+})$ for $X(1870)$.

Finally, if $\eta_{2}(1870)$ is the $\eta_{2}(2^{1}D_{2})$ state, its decay
width is predicted about 100MeV which is much smaller than the experiments.
However, the difference can be explained by the remedy of mixing effect. If
$X(1870)$ and $\eta_{2}(1870)$ have the same quantum numbers, $0^{+}2^{-+}$,
they should mix with each other with a visible mixing angle. Then the
interference enhancement will enlarge the width of $\eta_{2}(1870)$. The broad
decay width of $\eta_{2}(1870)$ could be explained naturally. On the other
hand, $\eta_{2}(1870)$ has been observed in the channel of $a_{0}(980)\pi$.
However, this channel seems much small if $\eta_{2}(1870)$ is a pure
$2^{1}D_{2}$ $n\bar{n}$ meson. The mixing effect will also enlarge this
partial width. Here, we don't plan to discuss the mixing of $X(1870)$ and
$\eta_{2}(1870)$ further for the complex mechanism.

##  IV  DISCUSSIONS AND CONCLUSIONS

A isoscalar resonant structure of $X(1870)$ was observed by BESIII in the
channels $J/\psi\rightarrow\omega X(1870)\rightarrow\omega\eta\pi^{+}\pi^{-}$
recently. Although the mass of $X(1870)$ is consistent with the
$\eta_{2}(1870)$, the production, decay width and decay properties are much
different. In this paper, the mass spectrum and strong decays of the $X(1870)$
and $\eta_{2}(1870)$ are analyzed.

Firstly, the mass spectrum are studied in the GI potential model and the RTs
framework. In the GI potential model, both $X(1870)$ and $\eta_{2}(1870)$
could be the $\eta^{\prime}_{2}(1^{1}D_{2})$, $f^{\prime}_{1}(2^{3}P_{1})$ and
$\eta_{2}(2^{1}D_{2})$ states. In RTs, the possible assignments are the
$\eta(3^{1}S_{0})$, $f^{\prime}_{1}(2^{3}P_{1})$ and $\eta_{2}(2^{1}D_{2})$
states. For the mass spectrum, they are also good hybrid candidates since the
masses overlap the predictions given by different models (see Table4).

Secondly, the processes of a $n\bar{n}$ quarkonium or a $n\bar{n}g$ hybrid
meson decaying into the ``$\mathcal{S}+P$'' mesons are studied under the
SU(3)f symmetry and the diquark-antidiquark description of the $\mathcal{S}$
mesons. We assumed the processes obey the OZI rule. We find that the channels
of $a_{0}\pi$, $\sigma\eta$ and $f_{0}\eta$ are the dominant when a $n\bar{n}$
quarkonium or a $n\bar{n}g$ hybrid meson decays primarily through this kind of
processes. This result can explain why $X(1870)$ has been first observed in
the $\eta\pi\pi$ channel.

Thirdly, the two-body strong decay of $X(1870)$ is computed in the
${}^{3}P_{0}$ model. As the $\eta_{2}(2^{1}D_{2})$ quarkonium, the predicted
width of $X(1870)$ looks much larger than the observations. The broad
resonance, $\eta_{2}(1870)$, can be a natural candidate for the $2^{1}D_{2}$
$n\bar{n}$ meson. There, we fix the creation strength, $\gamma$, in two kinds
of processes: ①.$n\bar{n}\rightarrow n\bar{n}+n\bar{n}$; ②.
$n\bar{n}\rightarrow n\bar{s}+s\bar{n}$ and $s\bar{s}\rightarrow
n\bar{s}+s\bar{n}$. The functions of creation vertex are determined as
$\gamma(p)=3.0+25\exp(-4p^{2})$ and $\gamma(p)=1.8+4\exp(-10p^{2})$
respectively. Meanwhile, the SHO wave function scale, $\beta$s, are obtained
by the GI potential model.

We have evaluated the magnitude of the partial widths of ``$\mathcal{S}+P$''
channels by the ratio, $\frac{\Gamma(X(1870)\rightarrow
a_{0}(980)\pi)}{\Gamma(\eta(1295)\rightarrow a_{0}(980)\pi)}$, under a rather
crude assumption that $\eta(1295)/X(1870)\rightarrow a_{0}(980)\pi$ through a
virtual intermediate $1^{3}P_{0}$ $q\bar{q}$ meson (see Fig.7). Then the
uncertainties of the coupling vertex for $1^{3}P_{0}(q\bar{q})\rightarrow
a_{0}(980)$ are assumed to be canceled in the ratio. The total widths of
``$\mathcal{S}+P$'' are evaluated no more than 12.6MeV and
$\Gamma(X(1870)\rightarrow a_{0}(980)\pi)\leq 3.8$MeV. Since $X(1870)$
primarily decay via $a_{0}(980)\pi$, it also indicated that the $X(1870)$
can't be interpreted as the $2^{1}D_{2}$ $n\bar{n}$ 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 Ref. Swanson . $X(1870)$
looks most like the $\eta_{H}(0^{+}2^{-+})$ state for the narrow predicted
width, which is consistent with the experiments. But we can't exclude the
possibility of $0^{+}1^{++}$. A precise measurement of $a_{1}(1260)\pi$ is
suggested to pin down this uncertainty.

Finally, some important arguments and useful suggestions are given as follows.
➀.If $\eta_{2}(1870)$ is the $\eta_{2}(2^{1}D_{2})$ state, the broad
$\pi_{2}(1880)$ should be isovector partner of $\eta_{2}(1870)$.
$\pi_{2}(1880)$ has been interpreted as the conventional $2^{1}D_{2}$
$q\bar{q}$ meson in Ref. Li4 . In deed, the decay channel of $\omega\rho$ is
large enough for $\pi_{2}(1880)$ E852 . This observation disfavors the
$\pi_{2}(1880)$ as a $2^{-+}$ hybrid candidate for the selection rule that a
hybrid meson decaying into two S-wave mesons is strongly suppressed page .
➁.If $X(1870)$ is a hybrid meson, we suggest to search its isospin partner in
the decay channels of $J/\psi\rightarrow\rho f_{0}(980)\pi$ and
$J/\psi\rightarrow\rho b_{1}(1235)\pi$, which are accessible at BESIII, Belle
and BABAR Collaborations. The decay channel of $b_{1}(1235)\pi$ is forbidden
for the $\pi_{2}(2^{1}D_{2})$ quarkonium due to the ``spin selection rule''
Burns ; flux1 . We also suggest to search the $\eta_{2}(1870)$ in the decay
channels of $J/\psi\rightarrow\gamma\rho\rho$ and
$J/\psi\rightarrow\gamma\omega\omega$ since these channels are forbidden for
the hybrid production.

###### Acknowledgements.

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 (No.13A140014), the National Natural Science Foundation of China
under grant No. 11305003, No. 11075102, No. 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).

##  Appendix A The expressions of amplitudes

We have omitted a exponential factor in following decay amplitudes
$\mathcal{M}_{LS}$ for compactness,

$\exp(-\frac{2\lambda\mu-\nu^{2}}{4\mu}p^{2}).$ (17)

where we defined

$\displaystyle\mu=\frac{1}{2}(\frac{1}{\beta_{A}^{2}}+\frac{1}{\beta_{B}^{2}}+\frac{1}{\beta_{C}^{2}});\hskip
3.41418pt\nu=\frac{m_{1}}{(m+m_{1})\beta_{B}^{2}}+\frac{m_{2}}{(m+m_{2})\beta_{C}^{2}}.$

and

$\displaystyle\lambda=\frac{m_{1}^{2}}{(m+m_{1})^{2}\beta_{B}^{2}}+\frac{m_{2}^{2}}{(m+m_{2})^{2}\beta_{C}^{2}};\hskip
10.243pt\eta=\frac{m_{1}}{m+m_{1}}.$

For $1^{3}S_{1}\rightarrow 1^{1}S_{0}+1^{1}S_{0}$,

$\mathcal{M}_{10}=\frac{2\mu-\nu}{8\sqrt{3}\pi^{5/4}\mu^{5/2}(\beta_{A}\beta_{B}\beta_{C})^{3/2}}p$
(18)

For $1^{3}P_{2}\rightarrow 1^{1}S_{0}+1^{1}S_{0}$,

$\mathcal{M}_{20}=\frac{2\mu\beta_{B}^{3/2}-(p^{2}\nu^{2}+2\mu(1-p^{2}\nu))\beta_{C}^{3/2}}{8\sqrt{15}\pi^{5/4}\mu^{7/2}\beta_{A}^{5/2}\beta_{B}^{3/2}\beta_{C}^{3}}$
(19)

For $1^{3}P_{2}\rightarrow 1^{3}S_{1}+1^{1}S_{0}$,
$\mathcal{M}_{21}=-\sqrt{3/2}\mathcal{M}_{20}$.

For $1^{3}P_{1}\rightarrow 1^{3}S_{1}+1^{1}S_{0}$,

$\mathcal{M}_{01}=\frac{4\mu\beta_{B}^{3/2}+(p^{2}\nu^{2}+2\mu(1-p^{2}\nu))\beta_{C}^{3/2}}{24\pi^{5/4}\mu^{7/2}\beta_{A}^{5/2}\beta_{B}^{3/2}\beta_{C}^{3}}$
(20)
$\mathcal{M}_{21}=\frac{(2\mu-\nu)\nu}{24\sqrt{2}\pi^{5/4}\mu^{7/2}\beta_{A}^{5/2}\beta_{B}^{3/2}\beta_{C}^{3}}p^{2}$
(21)

For $1^{3}D_{3}\rightarrow 1^{1}S_{0}+1^{1}S_{0}$,

$\mathcal{M}_{30}=-\frac{2\mu-\nu}{16\sqrt{35}\pi^{5/4}\mu^{9/2}\beta_{A}^{7/2}\beta_{B}^{3/2}\beta_{C}^{3/2}}\nu^{2}p^{3}$
(22)

For $2^{1}S_{0}\rightarrow 1^{3}P_{0}+1^{1}S_{0}$,

$\displaystyle\mathcal{M}_{11}=$
$\displaystyle\frac{1}{96\pi^{5/4}\mu^{11/2}\beta_{A}^{7/2}\beta_{B}^{5/2}\beta_{C}^{3/2}}$
(23)
$\displaystyle\times(-24p^{2}\eta\mu^{3}-p^{4}\nu^{4}+2p^{2}\mu\nu^{2}(-10+p^{2}(1+\eta)\nu)-4\mu^{2}(15-5p^{2}(1+\eta)\nu+p^{4}\eta\nu^{2})$
$\displaystyle+6\mu^{2}(4p^{2}\eta\mu^{2}+p^{2}\nu^{2}-2\mu(-3+p^{2}(1+\eta)\nu))\beta_{A}^{2})$

For $2^{1}D_{2}\rightarrow 1^{3}S_{1}+1^{1}S_{0}$,

$\mathcal{M}_{11}=\frac{-p^{4}\nu^{4}+2p^{2}\nu^{2}\mu(\nu
p^{2}-14)+28\mu^{2}(\nu p^{2}-5)+14\mu^{2}(p^{2}\nu^{2}-2(\nu
p^{2}-5)\mu\beta_{A}^{2})}{160\sqrt{21}\pi^{5/4}\mu^{13/2}\beta_{A}^{11/2}\beta_{B}^{3/2}\beta_{C}^{3/2}}\nu
p$ (24) $\mathcal{M}_{31}=\frac{-28\mu^{2}+\nu^{3}p^{2}-2\mu\nu(\nu
p^{2}-9)+14(2\mu-\nu)\mu^{2}\beta_{A}^{2}}{160\sqrt{14}\pi^{5/4}\mu^{13/2}\beta_{A}^{11/2}\beta_{B}^{3/2}\beta_{C}^{3/2}}\nu^{2}p^{3}$
(25)

For $2^{1}D_{2}\rightarrow 1^{3}S_{1}+1^{3}S_{1}$,
$\mathcal{M}^{\prime}_{11}=\sqrt{2}\mathcal{M}_{11}$ and
$\mathcal{M}^{\prime}_{31}=\sqrt{2}\mathcal{M}_{31}$.

For $2^{1}D_{2}\rightarrow 1^{3}P_{0}+1^{1}S_{0}$

$\displaystyle\mathcal{M}_{20}=$
$\displaystyle-\frac{1}{192\sqrt{35}\pi^{5/4}\mu^{15/2}\beta_{A}^{11/2}\beta_{B}^{5/2}\beta_{C}^{3/2}}$
(26)
$\displaystyle\times(p^{4}\nu^{5}-2p^{2}\mu\nu^{3}(-18+p^{2}(1+\eta)\nu)+4\mu^{2}\nu(63-11p^{2}(1+\eta)\nu+p^{4}\eta\nu^{2})+56\mu^{3}(-2+\eta(-2+p^{2}\nu))$
$\displaystyle-14\mu^{2}(p^{2}\nu^{3}-2\mu\nu(-7+p^{2}(1+\eta)\nu)+4\mu^{2}(-2+\eta(-2+p^{2}\nu)))\beta_{A}^{2})\nu
p^{2}.$

For $2^{1}D_{2}\rightarrow 1^{3}P_{1}+1^{1}S_{0}$

$\displaystyle\mathcal{M}_{21}=-\frac{p^{2}\nu^{2}-14\mu^{2}\beta^{2}_{A}+14\mu}{16\sqrt{35}\pi^{5/4}\mu^{11/2}\beta_{A}^{11/2}\beta_{B}^{5/2}\beta_{C}^{3/2}}(\eta-1)\nu
p^{2}.$ (27)

For $2^{1}D_{2}\rightarrow 1^{3}P_{2}+1^{1}S_{0}$

$\displaystyle\mathcal{M}_{02}=$
$\displaystyle\frac{1}{480\sqrt{14}\pi^{5/4}\mu^{15/2}\beta_{A}^{11/2}\beta_{B}^{5/2}\beta_{C}^{3/2}}$
(28)
$\displaystyle\times(p^{6}\mu^{6}-2p^{4}\mu\nu^{4}(p^{2}(1+\eta)\nu-21)+4p^{2}\mu^{2}\nu^{2}(105-14p^{2}(1+\eta)\nu+p^{4}\eta\nu^{2})+56\mu^{3}(15-5p^{2}(1+\eta)\nu$
$\displaystyle+p^{4}\eta\nu^{2})-14\mu^{2}(p^{4}\nu^{4}-2p^{2}\mu\nu^{2}(-10+p^{2}(1+\eta)\nu)+4\mu^{2}(15-5p^{2}(1+\eta)\nu+p^{4}\eta\nu^{2}))\beta_{A}^{2}).$

$\displaystyle\mathcal{M}_{22}=$
$\displaystyle-\frac{1}{672\sqrt{5}\pi^{5/4}\mu^{15/2}\beta_{A}^{15/2}\beta_{B}^{5/2}\beta_{C}^{3/2}}$
(29)
$\displaystyle\times(p^{4}\nu^{5}-2p^{2}\mu\nu^{3}(-18+p^{2}(1+\eta)\nu)+2\mu^{2}\nu(126-25p^{2}(1+\eta)\nu+2p^{4}\eta\nu^{2})+28\mu^{3}(-7+\eta(-7+2p^{2}\nu))$
$\displaystyle-14(\mu^{2})(p^{2}\nu^{3}-2\mu\nu(-7+p^{2}(1+\eta)\nu)+2\mu^{2}(-7+\eta(-7+2p^{2}\nu)))\beta_{A}^{2})\nu
p^{2}.$

$\displaystyle\mathcal{M}_{42}=$
$\displaystyle-\frac{1}{1120\pi^{5/4}\mu^{15/2}\beta_{A}^{11/2}\beta_{B}^{5/2}\beta_{C}^{3/2}}$
(30)
$\displaystyle\times(\nu(p^{2}\nu^{3}-36\mu^{2}+2\mu\nu(11-p^{2}\nu))+2\eta\mu(28\mu^{2}-p^{2}\nu^{3}+2\mu\nu(p^{2}\nu-9))-14\mu^{2}(2\mu-\nu)(2\eta\mu-\nu)\beta_{A}^{2})\nu^{2}p^{4}.$

For $2^{1}D_{2}\rightarrow 2^{3}P_{1}+1^{1}S_{0}$

$\displaystyle\mathcal{M}_{21}=$
$\displaystyle\frac{1}{160\sqrt{14}\pi^{5/4}\mu^{15/2}\beta_{A}^{11/2}\beta_{B}^{9/2}\beta_{C}^{3/2}}$
(31)
$\displaystyle\times(-112\eta\mu^{3}+252\mu^{2}\nu+56p^{2}\eta^{2}\mu^{3}\nu-80p^{2}\eta\mu^{2}\nu^{2}+36p^{2}\mu\nu^{3}+4p^{4}\eta^{2}\mu^{2}\nu^{3}-4p^{4}\eta\mu\nu^{4}+p^{4}\nu^{5}$
$\displaystyle-10\mu^{2}\nu(14\mu+p^{2}\nu^{2})\beta_{B}^{2}-14\mu^{2}\beta_{A}^{2}(14\mu\nu+4p^{2}\eta^{2}\mu^{2}\nu+p^{2}\nu^{3}-4\eta\mu(2\mu+p^{2}\nu^{2})-10\mu^{2}\nu\beta_{B}^{2}))(\eta-1)p^{2}.$

$m_{1}$ and $m_{2}$ are the masses of quarks in the decaying meson _A_. _m_ is
the mass of the created quark from the vacuum. For calculating the decay
widths, the masses of quarks are taken as: $m_{u}$ = $m_{d}$ = 0.220 GeV,
$m_{s}=$ 0.428 GeV, which are as same as these in the Section II. The above
amplitudes, $\mathcal{M}_{LS}$, can be reduced further in the approximation of
$m_{1}=m_{2}=m$ and $\beta_{A}=\beta_{B}=\beta_{C}=\beta$. The reduced
$\mathcal{M}_{LS}$ are consistent with these given by Ref. Barnes  except for
an unimportant factor, $-2^{9/2}$, since this factor can be absorbed into the
coefficient $\gamma$.

##  Appendix B Hybrid decay in the flux tube model

The flux tube model was motivated by the strong coupling expansion of the
lattice QCD. In this model, decay occurs when the flux-tube breaks at any
point along its length, with a $q\bar{q}$ pair production in a relative
$J^{PC}=0^{++}$ state. It is similar to the ${}^{3}P_{0}$ decay model but with
an essential difference. The flux tube model extend the nonrelativistic
constituent quark model to include gluonic degrees of freedom in a very simple
and intuitive way, where the gluonic field is regarded as tubes of color flux.
Then it can be extended to the hybrid research. When the hybrid mesons are
assumed to be narrow, and the threshold effects aren't taken into account, the
partial decay width $\Gamma_{LS}(H\rightarrow BC)$ is given by the flux model
as flux1

$\Gamma_{LS}(H\rightarrow
BC)=\frac{p}{(2J_{A}+1)\pi}\frac{\tilde{M}_{B}\tilde{M}_{C}}{\tilde{M}_{A}}|\mathcal{M}_{LS}(H\rightarrow
BC)|^{2}$ (32)

where $\tilde{M}_{A}$, $\tilde{M}_{B}$, $\tilde{M}_{C}$ are the ``mock-meson''
masses of A, B, C Kokoski . When a hybrid meson decay into _P_ -wave and
pseudoscalar mesons, the partial wave amplitude $\mathcal{M}_{L}(H\rightarrow
BC)$ (with $S=S_{B}$) is given as the following form

$\mathcal{M}_{L}(H\rightarrow
BC)=\langle\phi_{B}\phi_{C}|\phi_{A}\phi_{0}\rangle(\frac{a\tilde{c}}{9\sqrt{3}}\frac{1}{2}A^{0}_{00}\sqrt{\frac{fb}{\pi}})\frac{\kappa\sqrt{b}}{(1+fb/(2\tilde{\beta}^{2}))^{2}}\sqrt{\frac{2\pi}{3\Gamma(3/2+\delta)}}\frac{\beta_{A}^{3/2+\delta}}{\tilde{\beta}}\tilde{\mathcal{M}}_{L}(H\rightarrow
BC)$ (33)

The flavor matrix element $\langle\phi_{B}\phi_{C}|\phi_{A}\phi_{0}\rangle$
have been discussed before. $\tilde{\mathcal{M}}_{L}(H\rightarrow BC)$ are
listed in Table 9 for the states of $\eta_{H}(0^{+}0^{-+})$,
$f_{H}(0^{+}1^{++})$ and $\eta_{H}(0^{+}2^{-+})$.

_B_ |  _H_($0^{+}0^{-+}$) |  _H_($0^{+}1^{++}$) |  _H_($0^{+}2^{-+}$)
---|---|---|---
$0^{++}$ | $+\sqrt{2}\mathcal{M}_{S}/3$ | $-\sqrt{2}\mathcal{M}_{P_{2}}/\sqrt{3}$ | $+\mathcal{M}_{D}/3$
$1^{++}$ | $-$ | $-\mathcal{M}_{P_{1}}/\sqrt{2}$ | -
$2^{++}$ | $+\mathcal{M}_{D}/3$ | $-\mathcal{M}_{P_{4}}/\sqrt{30}$ | $-\sqrt{5}\mathcal{M}_{S}/\sqrt{18}$
|  | $+\mathcal{M}_{F}/\sqrt{5}$ | $-\sqrt{7}\mathcal{M}_{D}/3$
Table 9: Partial wave amplitudes $\tilde{\mathcal{M}}_{L}(H\rightarrow BC)$
for an initial hybrid _H_ decaying into a _P_ -wave and pseudoscalar mesons.

Here the $\mathcal{M}_{S}$, $\mathcal{M}_{D}$, $\mathcal{M}_{P_{i}}$ and
$\mathcal{M}_{F}$ are defined as
$\mathcal{M}_{S}=-(3\tilde{h}_{0}-\tilde{g}_{1}+4\tilde{h}_{2})$,
$\mathcal{M}_{D}=(\tilde{g}_{1}+5\tilde{h}_{2})$,
$\mathcal{M}_{P_{1}}=-i(2\tilde{g}_{0}+3\tilde{h}_{1}-\tilde{g}_{2})$,
$\mathcal{M}_{P_{2}}=-i(\tilde{g}_{0}+\tilde{g}_{2})$,
$\mathcal{M}_{P_{4}}=-i(10\tilde{g}_{0}+9\tilde{h}_{1}+\tilde{g}_{2})$ and
$\mathcal{M}_{F}=-3i(\tilde{g}_{2}+\tilde{h}_{3})$. The analytical expressions
of $\tilde{g}_{i}$ and $\tilde{h}_{i}$ are given as

$\tilde{g}_{n}=2^{3+\delta}\frac{M^{n}m}{(M+m)^{n+1}}(2\beta_{A}^{2}+\tilde{\beta}^{2})^{-\frac{n+\delta+3}{2}}\Gamma(\frac{n+\delta+3}{2})_{1}F_{1}[\frac{n+\delta+3}{2},n+1,-(\frac{M}{M+m})^{2}\frac{p^{2}}{2\beta_{A}^{2}+\tilde{\beta}^{2}}]p^{n+1}$
(34)
$\tilde{h}_{n}=2^{3+\delta}\tilde{\beta}^{2}(\frac{M}{M+m})^{n}(2\beta_{A}^{2}+\tilde{\beta}^{2})^{-\frac{n+\delta+4}{2}}\Gamma(\frac{n+\delta+4}{2})_{1}F_{1}[\frac{n+\delta+4}{2},n+1,-(\frac{M}{M+m})^{2}\frac{p^{2}}{2\beta_{A}^{2}+\tilde{\beta}^{2}}]p^{n}$
(35)

where ${}_{1}F_{1}[\cdots]$ are the confluent hypergeometric functions. Here
we don't take account of the decay channels of $H\rightarrow 2S+1S$ because
they are forbidden by the conservation laws, or the ``spin selection rule'',
or the phase space, _e.g._ , the decay channel of $\pi(1300)+\pi$ is forbidden
for the $f_{H}(0^{+}1^{++})$ state by the ``spin selection rule''. In this
work, we choose to follow the Refs. flux1  and take the combination
$(a\tilde{c}/9\sqrt{3})\frac{1}{2}A^{0}_{00}\sqrt{\frac{fb}{\pi}}$ as 0.64
which was fixed by the conventional mesons Kokoski , $M=m=m_{u,d}=330$MeV,
$\tilde{M}^{I=0}_{B}$ $=\tilde{M}^{I=1}_{B}=1250$MeV,
$\tilde{M}^{I=0}_{C}=720$MeV, $\tilde{M}^{I=1}_{C}=850$MeV,
$\beta_{A}=$0.27GeV, $\delta=$0.62, $b=$0.18GeV2 and $\kappa=$0.9. Final
states containing $\pi$ have $\tilde{\beta}=$0.36GeV, otherwise
$\tilde{\beta}=$0.40GeV.

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