Towards the establishment of the light $J^{P(C)}=1^{-(+)}$ hybrid nonet

On the SU(3) flavor basis the light hybrid mesons are described by a pair of $q\bar{q}$ associated by gluonic quasiparticle excitations. Taking the flux tube model picture, the $q\bar{q}$ inside hybrid mesons are separated static color sources and they are connected by the gluonic flux tube to form an overall color singlet. The transverse oscillations of the flux tube that manifests the explicit effective gluonic degrees of freedom, will give rise to energy spectrum of the hybrid mesons. As studied in the literature, the lowest energy flux tube motion has $J_{g}^{PC}=1^{+-}$. Namely, the lightest hybrid multiplet can be formed by the relative $S$-wave coupling between a gluonic lump of $J_{g}^{PC}=1^{+-}$ and a $S$-wave $q\bar{q}$ pair. With the total gluon spin $J_{g}^{PC}=1^{+-}$, the lowest hybrid multiplets can be obtained: $(0,\ 1,\ 2)^{-+},\ 1^{--}$ Bali:2003jq ; Dudek:2011bn . Alternatively, in the constituent gluon picture the lowest energy flux tube excitation can be described by the motion of quasigluon in a $P$ wave with respect to the $S$-wave $q\bar{q}$.

The gluonic excitations additive to the $S$-wave constituent $q\bar{q}$ configuration suggests that for each $S$-wave $q\bar{q}$ pair, there should exist an SU(3) flavor nonet as the eigenstates of the corresponding Hamiltonian. For the same coupling mode involving the gluonic lump, these states can be related to each other by the Gell-Mann-Okubo mass relation similar to that for the ground states in the $q\bar{q}$ scenario. This conjecture may have a caveat when the strange multiplets are included. Since the charged and strange states do not have the fixed $C$ parity, it may raise the question whether a nonet scheme makes sense or not. Note that signals for charged $\pi_{1}(1600)$ have been seen in the decay channels of $\rho^{0}\pi^{-}$ COMPASS:2009xrl and $\eta^{\prime}\pi^{-}$  E852:2001ikk ; COMPASS:2014vkj . Similar dynamics should appear in the strange sector and a nonet structure among the $1^{-(+)}$ multiplets should be a good guidance for a better understanding of the underlying dynamics.

Taking the $1^{-+}$ hybrid as an example, it should contain flavor multiplets as follows:

	$\displaystyle\pi_{1}^{+},\ \pi_{1}^{-},\ \pi_{1}^{0}$	$\displaystyle:$	$\displaystyle u\bar{d}\tilde{g},\ d\bar{u}\tilde{g},\ \frac{1}{\sqrt{2}}(u\bar{u}-d\bar{d})\tilde{g}\ ,$		(1)
	$\displaystyle\eta_{1}^{(8)}$	$\displaystyle:$	$\displaystyle\frac{1}{\sqrt{6}}(u\bar{u}+d\bar{d}-2s\bar{s})\tilde{g}\ ,$		(2)
	$\displaystyle\eta_{1}^{(1)}$	$\displaystyle:$	$\displaystyle\frac{1}{\sqrt{3}}(u\bar{u}+d\bar{d}+s\bar{s})\tilde{g}\ ,$		(3)
	$\displaystyle K^{*+},\ K^{*0},\ K^{*-},\ \bar{K}^{*0}$	$\displaystyle:$	$\displaystyle u\bar{s}\tilde{g},\ d\bar{s}\tilde{g},\ s\bar{u}\tilde{g},\ s\bar{d}\tilde{g}\ ,$		(4)

where $\tilde{g}$ represents the gluonic lump with $J_{g}^{PC}=1^{+-}$. For the flavor octet $\eta_{1}^{(8)}$ and singlet $\eta_{1}^{(1)}$ with isospin $I=0$, they may mix with each other to form the corresponding physical states similar to the familiar $\eta$-$\eta^{\prime}$ mixing.

Considering the mixing between the hybrid flavor singlet and octet, the physical states can be expressed as

$\displaystyle\left(\begin{array}[]{c}\eta_{1L}\\ \eta_{1H}\end{array}\right)$

$\displaystyle=$

$\displaystyle\left(\begin{array}[]{cc}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{array}\right)\left(\begin{array}[]{c}\eta_{1}^{(8)}\\ \eta_{1}^{(1)}\end{array}\right)=\left(\begin{array}[]{cc}\cos\alpha&-\sin\alpha\\ \sin\alpha&\cos\alpha\end{array}\right)\left(\begin{array}[]{c}n\bar{n}\tilde{g}\\ s\bar{s}\tilde{g}\end{array}\right)\ ,$

(15)

where $\theta$ is the mixing angle between the flavor octet and singlet, and $\alpha$ is the mixing angle defined on the flavor basis $n\bar{n}\tilde{g}$ (with $n\bar{n}\equiv(u\bar{u}+d\bar{d})/\sqrt{2}$) and $s\bar{s}\tilde{g}$.

The Gell-Mann-Okubo relation can provide a constraint on the mixing angle $\theta$ via the following equation:

$\displaystyle\tan\theta$

$\displaystyle=$

$\displaystyle\frac{4m_{K^{*}}-m_{\pi_{1}}-3m_{\eta_{1L}}}{2\sqrt{2}(m_{\pi_{1}}-m_{K^{*}})}\ ,$

(16)

where $\eta_{1L}$ is the lower mass state in Eq. (15) and the sign of $\theta$ can be determined here. We note that the same relation is also satisfied for the quadratic masses. The Gell-Mann-Okubo relation also leads to the following mass relation,

$\displaystyle(m_{\eta_{1H}}+m_{\eta_{1L}})(4m_{K^{*}}-m_{\pi_{1}})-3m_{\eta_{1H}}m_{\eta_{1L}}$

$\displaystyle=$

$\displaystyle 8m_{K^{*}}^{2}-8m_{K^{*}}m_{\pi_{1}}+3m_{\pi_{1}}^{2}\ ,$

(17)

which is symmetric for $\eta_{1L}$ and $\eta_{1H}$ although it is the lower mass state $\eta_{1L}$ defined in Eq. (15) to appear in Eq. (16). With the masses of $\pi_{1}(1600)$ and $\eta_{1}(1855)$ as the input for Eqs. (16) and (17), we are still unable to determine these three quantities, i.e. $\theta$, $m_{\eta_{1H}}/m_{\eta_{1L}}$ and $m_{K^{*}}$. Also, it is unclear whether $\eta_{1}(1855)$ is the lower or higher mass state in Eq. (15). However, we will show later that the $\eta\eta^{\prime}$ channel is informative to impose constraint on the determination of the $1^{-(+)}$ nonet.

The flavor-blindness of the strong interactions also allows us to relate the SU(3) decay channels together Close:2000yk ; Close:2005vf ; Zhao:2007ze ; Zhao:2006cx . Considering the two-body decay of $\eta_{1L}$ and $\eta_{1H}$ into pseudoscalar meson pair $PP^{\prime}$ ³³3Due to Bose symmetry, the $1^{-+}$ hybrid cannot decay into two identical mesons. Namely, the decays into $\pi^{0}\pi^{0}$, $\eta\eta$, and $\eta^{\prime}\eta^{\prime}$ are forbidden. Also, the $G$-parity conservation will forbid the decays of $\pi_{1}$ and $\eta_{1}$ into $\pi\pi$ and $K\bar{K}$. , two independent transition mechanisms can be identified and they are illustrated by Fig. 1 (a) and (b). The transition of Fig. 1 (a) represents the flux tube string breaking with the quark pair creation. It is similar to the decay of a conventional $q\bar{q}$ state into two mesons by the quark pair creation (QPC) mechanism. In the flux tube scenario it corresponds to the flux excitation mode along the displacement between the quark and anti-quark, for which the potential is denoted as $\hat{V}_{L}$. The transition of Fig. 1 (b) corresponds to the flux excitation mode transverse to the displacement between the quark and anti-quark. The quark pair created from this mode will recoil the initial color-octet $q\bar{q}$ via the transverse flux motion. For a conventional $q\bar{q}$ decay via the ${}^{3}P_{0}$ QPC mechanism, the kinematic regime as Fig. 1 (b) will be relatively suppressed with respect to Fig. 1 (a). Since in such a case, in order to balance the color, an additional relatively-hard gluon will be exchanged between the recoiled $q\bar{q}$ and the created $q\bar{q}$. In contrast, such a transition in the hybrid decay can naturally occur via the transverse mode of the flux tube oscillations Kokoski:1985is . Namely, the created $q\bar{q}$ can easily get the color balanced by soft gluon exchanges which can be absorbed into the effective potential without suppression. Such a transition through the transverse mode of the flux tube motions can be parametrized by the effective potential $\hat{V}_{T}$.

The transition amplitude for a $1^{-+}$ hybrid of $q\bar{q}\tilde{g}$ decaying into two pseudoscalar mesons can then be expressed as

$\displaystyle{\cal M}_{a}$

$\displaystyle=$

$\displaystyle\langle(q_{1}\bar{q}_{4})_{M_{1}}(q_{3}\bar{q}_{2})_{M_{2}}|\hat{V}_{L}|q_{1}\bar{q}_{2}\tilde{g}\rangle\equiv g_{1}|{\bf k}|\ ,$

(18)

and

$\displaystyle{\cal M}_{b}$

$\displaystyle=$

$\displaystyle\langle(q_{1}\bar{q}_{2})_{M_{1}}(q_{3}\bar{q}_{4})_{M_{2}}|\hat{V}_{T}|q_{1}\bar{q}_{2}\tilde{g}\rangle\equiv g_{2}|{\bf k}|\ ,$

(19)

for these two decay modes, respectively. In the above two equations, ${\bf k}$ is the three-vector momentum of the final-state meson in the c.m. frame of the hybrid, and the quarks (anti-quarks) are the non-strange quarks (anti-quarks). Note that the QPC only contributes to a flavor singlet $\tilde{g}\to(u\bar{u}+d\bar{d}+s\bar{s})/\sqrt{3}$. We mention that when the $s\bar{s}$ pair is created, an SU(3) flavor symmetry breaking parameter will be included. Also, in the above two amplitudes the interchanges of the final-state hadron indices are implied.

This parametrization leads to a connection among the couplings of an initial hybrid state to different SU(3) channels, and they are collected in Table 1. Interesting features with the hybrid nonet decays can be learned as follows:

• •

  It is rather clear that if the final states do not contain isoscalar mesons, the transitions will be via the string breaking potential $\hat{V}_{L}$ along the displacement between the quark and anti-quark. Namely, the transitions are similar to the conventional ${}^{3}P_{0}$ process. For $K^{*}$ decays into $K\pi$, it will be difficult to distinguish it from the conventional $q\bar{q}$ vector mesons.

• •

  For the $\pi_{1}$ and $K^{*}$ decays into $\eta$ or $\eta^{\prime}$ plus a $I\neq 0$ state, such as $\pi_{1}^{0}\to\eta\pi^{0}$ and $\eta^{\prime}\pi^{0}$, the couplings involve interferences between processes of Fig. 1 (a) and (b). Since the mixing angle between $\eta$ and $\eta^{\prime}$ is $\alpha_{P}\simeq 42^{\circ}$, the couplings for the channels between $\eta$ and $\eta^{\prime}$ would be very different.

• •

  $\eta_{1L}$ and $\eta_{1H}$ decays into $\pi\pi$ and $K\bar{K}$ are forbidden by the Bose symmetry and $G$-parity conservation. They can only access $\eta\eta^{\prime}$ via the octet and singlet mixing. The coupling strengths have non-trivial dependence of the mixing angle $\alpha$. One can see that the decay pattern for these channels in a combined analysis should be sensitive to the value of $\alpha$.

A typical process for the production of a $J^{PC}=1^{-+}$ hybrid in the $J/\psi$ radiative decays is illustrated by Fig. 2. It shows that the annihilations of the charm and anti-charm quark can create a pair of light $S$-wave $q\bar{q}$ associated by a constituent gluon in a relative $P$-wave to the $q\bar{q}$. At the hadronic level, the Lagrangian for a general vector-vector-vector field interaction at the leading-order can be described by

$$\mathcal{L}_{VVV}=ig_{VVV}(V_{1,\nu}\overleftrightarrow{\partial^{\mu}}V_{2}^{\nu}V_{3,\mu}+V_{1,\mu}V_{2}^{\nu}\overleftrightarrow{\partial^{\mu}}V_{3,\nu}+V_{2,\mu}V_{3}^{\nu}\overleftrightarrow{\partial^{\mu}}V_{1,\nu})\ ,$$

(20)

where $V_{1}$, $V_{2}$, $V_{3}$ denotes the vector fields. For the radiative decay of $J/\psi\to\gamma\eta_{1}$, since the photon is transversely polarized, the above Lagrangian will reduce to the following form:

$${\cal L}_{J/\psi\to\gamma\eta_{1}}=ig_{J/\psi\eta_{1}\gamma}F_{\mu\nu}V_{J/\psi}^{\mu}V_{\eta_{1}}^{\nu}\ ,$$

(21)

where $F_{\mu\nu}\equiv\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, and the vector fields $V_{J/\psi}$, $A$, and $V_{\eta_{1}}$ stand for the initial $J/\psi$, final-state photon, and hybrid $\eta_{1}$ fields, respectively; $g_{J/\psi\eta_{1}\gamma}$ is the coupling constant. Note that the leading transition of $J/\psi\to\gamma\eta_{1}$ is via a $P$ wave. In the center of mass (c.m.) frame of $J/\psi$ the squared transition amplitudes for the two $I=0$ states can be expressed like below:

	$\displaystyle|i{\cal M}(J/\psi\to\gamma\eta_{1L})|^{2}$	$\displaystyle\propto$	$\displaystyle g_{J/\psi\eta_{1L}\gamma}^{2}|{\bf q}_{L}|^{2}(1+m_{J/\psi}^{2}/m_{\eta_{1L}}^{2})\ ,$		(22)
	$\displaystyle|i{\cal M}(J/\psi\to\gamma\eta_{1H})|^{2}$	$\displaystyle\propto$	$\displaystyle g_{J/\psi\eta_{1H}\gamma}^{2}|{\bf q}_{H}|^{2}(1+m_{J/\psi}^{2}/m_{\eta_{1H}}^{2})\ ,$		(23)

where ${\bf q}_{L}$ and ${\bf q}_{H}$ are the three-vector momenta of $\eta_{1L}$ and $\eta_{1H}$ in the $J/\psi$ rest frame, respectively. The subscripts, “$L$” and “$H$”, stand for the low and high mass states, respectively. The two coupling constants, $g_{J/\psi\eta_{1L}\gamma}$ and $g_{J/\psi\eta_{1H}\gamma}$, which account for the production mechanism for these two isoscalars, can be parametrized out:

	$\displaystyle g_{J/\psi\eta_{1L}\gamma}$	$\displaystyle=$	$\displaystyle g_{0}(\sqrt{2}\cos\alpha-R\sin\alpha)\ ,$		(25)
	$\displaystyle g_{J/\psi\eta_{1H}\gamma}$	$\displaystyle=$	$\displaystyle g_{0}(\sqrt{2}\sin\alpha+R\cos\alpha)\ ,$		(26)

where $R\simeq f_{\pi}/f_{K}\simeq 0.93$ indicates the SU(3) flavor symmetry breaking effects in the production of the $s\bar{s}$ pair in comparison with the non-strange $q\bar{q}$ pairs, and $g_{0}$ describes the coupling strength for the production of a light hybrid configuration $q\bar{q}\tilde{g}$ of $J^{PC}=1^{-+}$ in the $J/\psi$ radiative decays. It can be expressed as

$$g_{0}\equiv\langle(q\bar{q}\tilde{g})_{1^{-+}}|\hat{H}_{em}|J/\psi\rangle\ ,$$

(27)

where $\hat{H}_{em}$ contains the dynamics for the transition of Fig. 2.

The coupling relation in Eq. (25) leads to the relative production rate for $\eta_{1L}$ and $\eta_{1H}$ as follows:

$\displaystyle r_{L/H}$

$\displaystyle\equiv$

$\displaystyle\frac{BR(J/\psi\to\gamma\eta_{1L})}{BR(J/\psi\to\gamma\eta_{1H})}=\left(\frac{|{\bf q}_{L}|}{|{\bf q}_{H}|}\right)^{3}\frac{(\sqrt{2}\cos\alpha-R\sin\alpha)^{2}}{(\sqrt{2}\sin\alpha+R\cos\alpha)^{2}}\frac{m_{\eta_{1H}}^{2}(m_{J/\psi}^{2}+m_{\eta_{1L}}^{2})}{m_{\eta_{1L}}^{2}(m_{J/\psi}^{2}+m_{\eta_{1H}}^{2})}\ ,$

(28)

which seems to be sensitive to the mixing angle $\alpha$. Note that, in Ref. besiii-hybrid ; besiii-hybrid-pwa the $\eta_{1}(1855)$ signal is actually observed in its decays into $\eta\eta^{\prime}$. Moreover, the PWA results suggest that only one $I=0$ hybrid state has been clearly seen in the $1^{-+}$ partial wave amplitude. As shown in Subsection II.2, the decays of $\eta_{1L}$ and $\eta_{1H}$ into $\eta\eta^{\prime}$ are strongly correlated with the mixing angle $\alpha$ and mechanisms for the flux tube breaking. It means that the following branching ratio fractions can serve as constraints on the mixing angle:

Scheme-I

$\displaystyle:$

$\displaystyle R_{\eta_{1L}/\eta_{1}(1855)}\equiv\frac{BR(J/\psi\to\gamma\eta_{1L}\to\gamma\eta\eta^{\prime})}{BR(J/\psi\to\gamma\eta_{1}(1855)\to\gamma\eta\eta^{\prime})}<10\%\ ,$

(29)

and

Scheme-II

$\displaystyle:$

$\displaystyle R_{\eta_{1H}/\eta_{1}(1855)}\equiv\frac{BR(J/\psi\to\gamma\eta_{1H}\to\gamma\eta\eta^{\prime})}{BR(J/\psi\to\gamma\eta_{1}(1855)\to\gamma\eta\eta^{\prime})}<10\%\ ,$

(30)

where we have assigned $\eta_{1}(1855)$ as either the higher mass state (Scheme-I) or the lower mass state (Scheme-II). The relative rate $10\%$ is the production upper limit for the partner of $\eta_{1}(1855)$ from the experimental measurement besiii-hybrid ; besiii-hybrid-pwa .

With the production and decay couplings extracted earlier, the general form for the joint branching ratio fraction can be expressed as

	$\displaystyle R_{\eta_{1L}/\eta_{1H}}$	$\displaystyle=$	$\displaystyle\left(\frac{|{\bf q}_{L}|}{|{\bf q}_{H}|}\right)^{3}\frac{(\sqrt{2}\cos\alpha-R\sin\alpha)^{2}}{(\sqrt{2}\sin\alpha+R\cos\alpha)^{2}}\frac{m_{\eta_{1H}}^{2}(m_{J/\psi}^{2}+m_{\eta_{1L}}^{2})}{m_{\eta_{1L}}^{2}(m_{J/\psi}^{2}+m_{\eta_{1H}}^{2})}\left(\frac{|{\bf k}_{L}|}{|{\bf k}_{H}|}\right)^{3}\left(\frac{\Gamma_{H}m_{\eta_{1H}}}{\Gamma_{L}m_{\eta_{1L}}}\right)^{2}$		(31)
		$\displaystyle\times$	$\displaystyle\frac{[(1+\delta)\tan 2\alpha_{P}(\cos\alpha+R\sin\alpha)+2\delta(R\cos\alpha-\sin\alpha)]^{2}}{[(1+\delta)\tan 2\alpha_{P}(\sin\alpha-R\cos\alpha)+2\delta(R\sin\alpha+\cos\alpha)]^{2}}\ ,$

where $\Gamma_{L}$ and $\Gamma_{H}$ are the total widths of the lower and higher mass states, respectively; ${\bf q}_{L,H}$ and ${\bf k}_{L,H}$ are the three-vector momenta of the photon and pseudoscalar meson in the rest frames of $J/\psi$ and $\eta_{L,H}$, respectively; $\delta\equiv g_{2}/g_{1}$ indicates the relative strength between the two decay mechanisms for the flux tube breaking. As discussed earlier, $|\delta|\simeq 1$ is for the hybrid decays, while $|\delta|<<1$ for conventional $q\bar{q}$ decays. In Eq. (31) if we approximate ${\Gamma_{H}}/{\Gamma_{L}}\simeq 1$, ratio $R_{\eta_{1L}/\eta_{1H}}$ will strongly depend on $\alpha$ and $\delta$.

Before we go to the detailed studies of the two schemes, we briefly summarize the present experimental information on the strange vector mesons. As listed by the Particle Data Group (PDG) ParticleDataGroup:2020ssz , two excited $K^{*}$ states are observed in experiment, i.e. $K^{*}(1410)$ and $K^{*}(1680)$. While $K^{*}(1410)$ can be well accommodated by the first radial excitations of the vector meson nonet, the property of $K^{*}(1680)$ is far from well explored. Note that the second radial excitations of the isoscalar pseudoscalar mesons can be occupied by $\eta(1760)$ and $\eta(1860)$ in the Regge trajectory Yu:2011ta , the mass of $K^{*}(1680)$ as the second radial excitation in the conventional $q\bar{q}$ vector nonet seems to be too small. We also note that the strange pseudoscalar partner in the second radial excitation nonet has not yet been established in experiment though $K(1630)$ could be a candidate ParticleDataGroup:2020ssz . In the following analysis we first treat $K^{*}(1680)$ as the strange partner of the $1^{-(+)}$ nonet and examine whether it fits the constraint. If not, we then investigate the mass correlation of $K^{*}$ with the mixing angle and other multiplets as required by the Gell-Mann-Okubo relation.

In order to further investigate the characters arising from the nonet structure of the $1^{-(+)}$ hybrid states, we analyse the hadronic decays of $J/\psi\to VH$ and look for signals for the $I=0$ partner of $\eta_{1}(1855)$. Here, $V$ stands for the vector mesons $\rho$, $\omega$, and $\phi$, while $H$ stands for the light hybrid multiplets. This process is illustrated by Fig. 9 and the leading-order Lagrangian has been given in Eq. (20). The coupling for $J/\psi\rightarrow[q\bar{q}]_{1^{--}}[q\bar{q}\tilde{g}]_{1^{-+}}$ can be parametrized as,

$\displaystyle g_{P}\equiv\langle[q\bar{q}]_{1^{--}}[q\bar{q}\tilde{g}]_{1^{-+}}|\hat{V}_{P}|J/\psi\rangle\ ,$

(33)

where $\hat{V}_{P}$ represents the potential for the hadronic decays of $J/\psi\to VH$. And the detailed coupling constants for different decay channels list as following:

	$\displaystyle g_{J/\psi\rho^{+}\pi_{1}^{-}}$	$\displaystyle=$	$\displaystyle g_{P},$
	$\displaystyle g_{J/\psi\omega\eta_{1L}}$	$\displaystyle=$	$\displaystyle g_{P}\cos\alpha\ ,$
	$\displaystyle g_{J/\psi\omega\eta_{1H}}$	$\displaystyle=$	$\displaystyle g_{P}\sin\alpha\ ,$
	$\displaystyle g_{J/\psi\phi\eta_{1L}}$	$\displaystyle=$	$\displaystyle-g_{P}R^{2}\sin\alpha\ ,$
	$\displaystyle g_{J/\psi\phi\eta_{1H}}$	$\displaystyle=$	$\displaystyle g_{P}R^{2}\cos\alpha\ ,$
	$\displaystyle g_{J/\psi K^{*+}K_{H}^{*-}}$	$\displaystyle=$	$\displaystyle g_{P}R\ ,$		(34)

where $R$ is the SU(3) flavor symmetry breaking factor defined earlier. In this Section to distinguish the hybrid $K^{*}$ from $K^{*}(892)$, we denote it as $K^{*}_{H}$. Apart from the partial wave factor ($\propto|{\bf q}|^{3}$) which should be included for each channel and a mass function which has the same form for each channel, the branching ratio fractions among all the $VH$ decay channels will be driven by the following relative strengths:

			$\displaystyle\rho^{+}\pi_{1}^{-}:\omega\eta_{1L}:\omega\eta_{1H}:\phi\eta_{1L}:\phi\eta_{1H}:K^{*+}K_{H}^{*-}$		(35)
		$\displaystyle=$	$\displaystyle 1:\cos^{2}\alpha:\sin^{2}\alpha:R^{4}\sin^{2}\alpha:R^{4}\cos^{2}\alpha:R^{2}\ .$

Note that for the total of $\rho\pi$, a factor of 3 should be multiplied to the $\rho^{+}\pi_{1}^{-}$ channel, while the total of $K^{*}\bar{K}_{H}+c.c.$, a factor of 4 should be multiplies to the $K^{*+}K_{H}^{*-}$ channel.

If we take into account the effects from the partial wave factor and the SU(3) flavor symmetry breaking factor $R$, we find that the $\rho\pi_{1}$ channel has the largest branching ratio, while the production strengths for most of the other channels are actually comparable with $\alpha\sim 30^{\circ}$. In particular, it suggests that the production of $\eta_{1L}$ and $\eta_{1H}$ are accessible in the same channel such as $J/\psi\to\omega\eta\eta^{\prime}$. This is different from the case of $J/\psi\to\gamma\eta\eta^{\prime}$.

It should also be interesting to study the $J/\psi\to K^{*}\bar{K}_{H}^{*}+c.c.$ channel. Since $K_{H}^{*}$ has $J^{P}=1^{-}$ which are the same as the radial excitation states of the vector $K^{*}(892)$, it is difficult to identify the hybrid-like state. But as discussed earlier, the isolated $K^{*}(1680)$, either as a radial excitation state of $K^{*}(892)$ or a hybrid state, will bring crucial understandings of the $K^{*}$ spectrum. In the hybrid scenario $K_{H}^{*}$ will favor to decay into $K_{1}\pi\to K^{*}\pi\pi$. It means that $J/\psi\to K\bar{K}\pi\pi\pi$ will be ideal for the search of $K_{H}^{*}$.

In Fig. 10, we plot the branching ratio fractions in Scheme-I for each channels in respect with $J/\psi\to\rho^{+}\pi_{1}^{-}$ for which we set it as unity, and the other channels are normalized to the $\rho^{+}\pi_{1}^{-}$ channel with $R=0.93$ adopted. The favored range of the mixing angle is about $\alpha\in(17^{\circ},43^{\circ})$. But a relatively broad range is plotted in Fig. 10 as an illustration. The dependence of the mixing angle produces certain patterns which makes the combined study of the $VH$ channel useful for further confirmation of the $1^{-(+)}$ nonet.

For the results of Scheme-II, the $\omega\eta_{1H}$ and $\phi\eta_{1H}$ thresholds are higher than the $J/\psi$ mass. A combined study can be done in other higher heavy quarkonium decays such as $\Upsilon\to VH$.