# Constructing the $J^{P(C)}=1^{-(+)}$ light flavor hybrid nonet with the
newly observed $\eta_{1}(1855)$

Bing Chen1,3<EMAIL_ADDRESS>Si-Qiang Luo2,3,4,6<EMAIL_ADDRESS>Xiang Liu2,3,4,5,6<EMAIL_ADDRESS>1School of Electrical and Electronic
Engineering, Anhui Science and Technology University, Bengbu 233000, China
2School of Physical Science and Technology, Lanzhou University, Lanzhou
730000, China
3Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
4Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou University,
Lanzhou 730000, China
5MoE Frontiers Science Center for Rare Isotopes, Lanzhou University, Lanzhou
730000, China
6Research Center for Hadron and CSR Physics, Lanzhou University and Institute
of Modern Physics of CAS, Lanzhou 730000, China

###### Abstract

The recently discovered $\eta_{1}(1855)$ and the previously observed
$\pi_{1}(1600)$ state present a valuable opportunity for the investigation of
the $J^{P(C)}=1^{-(+)}$ light hybrid nonet. In this study, we employ a
semirelativistic quark potential model to examine the masses of the
$J^{P(C)}=1^{-(+)}$ light hybrid states. The static potential, which portrays
the confinement force between the quark-antiquark pair in a hybrid system, is
borrowed from the SU$(3)$ lattice gauge theory. Additionally, we utilize a
constituent gluon model to analyze the strong decay characteristics of these
light $1^{-+}$ hybrids. Our findings suggest that the $\pi_{1}(1600)$ and
$\eta_{1}(1855)$ states could be potential candidates for $1^{-+}$
$(u\bar{u}-d\bar{d})g/\sqrt{2}$ and $s\bar{s}g$ hybrids, respectively. To
ensure comprehensiveness, we also investigate the isospin partners of the
$\pi_{1}(1600)$ and $\eta_{1}(1855)$ states within the $1^{-(+)}$
nonet$-$specifically, the $(u\bar{u}+d\bar{d})g/\sqrt{2}$ and $s\bar{q}g$
($q=u$ and $d$ quarks) states. We propose some potential decay channels which
could be explored in experimental settings to detect these undiscovered
states.

###### pacs:

12.39.Jh, 14.40.Rt

##  I Introduction

Nearly half a century has elapsed since the establishment of quantum
chromodynamics (QCD) tHooft:1972tcz ; Fritzsch:1973pi ; Gross:1973id ;
Politzer:1973fx . However, physicists are still confronted with the challenge
of describing the fundamental properties of hadrons directly from first
principles. This difficulty arises primarily from the lack of a clear
understanding of the role played by the gluonic field in the low-energy regime
of strong interactions (for a comprehensive review of QCD, refer to Ref.
Gross:2022hyw ). Consequently, various phenomenological models, incorporating
key aspects of QCD, have been employed to gain insight into the properties of
meson and baryon states Olsen:2017bmm .

Within the framework of quark potential models, the gluonic field in
conventional hadron systems, responsible for mediating the interaction between
valence quarks, is typically approximated by an effective adiabatic potential
Quigg:1979vr ; Lucha:1991vn ; Mukherjee:1993hb . By employing this effective
potential, the mass spectra of meson and baryon states can be successfully
reproduced. However, it is important to note that the quark potential model
has not been rigorously examined in the context of hadrons containing an
excited gluonic field. These particular types of mesons, which possess an
excited gluonic field, are referred to as the hybrid states and are considered
to be exotic states. Undoubtedly, conducting a comprehensive study of hybrid
states could significantly contribute to our understanding of the role played
by the gluonic field in low-energy strong interactions. Unfortunately, as of
now, none of the hybrid states have been definitively established, which
underscores the need for further investigation in this area.

According to the results of lattice QCD Dudek:2011bn , the quantum numbers
$J^{PC}$ of the lightest hybrid mesons have been proposed to be $0^{-+}$,
$1^{-+}$, $2^{-+}$, and $1^{--}$. However, it should be noted that identifying
the $0^{-+}$, $2^{-+}$, and $1^{--}$ hybrid states in experiments can be
challenging, as they may mix with the conventional $q\bar{q}$ mesons in the
nearby mass region. On the other hand, the $1^{-+}$ states have the potential
to serve as a distinctive “smoking gun” signature for the presence of spin-
exotic states, provided that their existence is confirmed experimentally. To
date, the Particle Data Group (PDG) has cataloged three $1^{-+}$ states,
specifically the $\pi_{1}(1400)$, $\pi_{1}(1600)$, and $\pi_{1}(2015)$
ParticleDataGroup:2022pth . Among these states, the $\pi_{1}(2015)$ has solely
been observed by the E852 experiment in the decay modes $f_{1}(1285)\pi$
E852:2004gpn  and $b_{1}(1235)\pi$ E852:2004rfa . Nevertheless, further
substantial experimental evidence is required to establish its existence
definitively.

The existence of the $\pi_{1}(1400)$ state remains a subject of controversy.
In various processes such as $\pi^{-}$ diffraction IHEP-Brussels-LosAlamos-
AnnecyLAPP:1988iqi ; Aoyagi:1993kn ; E852:1997gvf ; E852:1999xev ; VES:2001rwn
; Dzierba:2003fw ; E862:2006cfp  and $p\bar{p}/n\bar{p}$ annihilation
CrystalBarrel:1998cfz ; CrystalBarrel:1999reg ; CrystalBarrel:2019zqh , the
$\pi_{1}(1400)$ state has been observed with decay modes including $\eta\pi$
and $\rho\pi$. However, in the cascade process
$\psi(3686)\to\gamma\chi_{c1}\to\gamma\pi_{1}(1400)^{\pm}\pi^{\mp}\to\gamma\eta\pi^{\pm}\pi^{\mp}$
involving approximately $4.48\times 10^{8}$ $\psi(3686)$ events, the BESIII
Collaboration did not detect a clear signal for the $\pi_{1}(1400)$
BESIII:2016tqo . Similarly, the COMPASS Collaboration conducted a
comprehensive resonance-model analysis of the $\pi^{-}\pi^{-}\pi^{+}$
invariant mass spectrum in the reaction
$\pi^{-}+p\to\pi^{-}\pi^{-}\pi^{+}+p_{\textup{spectator}}$ and found no
distinct resonance signal for the $\pi_{1}(1400)$ stateCOMPASS:2018uzl .
Moreover, the JPAC Collaboration performed a coupled-channel analysis of
COMPASS data for diffractively produced $\eta^{(\prime)}\pi^{-}$ final states
JPAC:2018zyd , revealing only one clear resonance pole corresponding to the
$\pi_{1}(1600)$ state in the spin-exotic wave. Consequently, it is plausible
to suggest that the structure attributed to the $\pi_{1}(1400)$ might be an
artifact resulting from imperfections in the analysis methodology. This
conclusion finds support in the work of Kopf _et al_. Kopf:2020yoa , where a
sophisticated coupled-channel analysis of data from the COMPASS and Crystal
Barrel experiments demonstrated that a single $\pi_{1}(1600)$ pole effectively
describes the amplitude of the $1^{-}(1^{-+})$ wave.

Among the spin-exotic states, the $\pi_{1}(1600)$ stands as the sole
experimental finding thus far. It has been detected in various decay modes,
including $\eta^{\prime}\pi$ VES:1993scg ; E852:2001ikk ; Amelin:2005ry ,
$\rho\pi$ E852:1998mbq ; Zaitsev:2000rc ; COMPASS:2009xrl ; COMPASS:2021ogp ,
$f_{1}(1285)\pi$ E852:2004gpn , and $b_{1}(1235)\pi$ E852:2004rfa ;
Baker:2003jh . However, precise measurements of the Breit-Wigner parameters
and important branching fractions for the $\pi_{1}(1600)$ state have been
challenging due to substantial background contributions from nonresonant
processes. Detailed information regarding the $\pi_{1}(1400)$,
$\pi_{1}(1600)$, and $\pi_{1}(2015)$ states can be found in the comprehensive
reviews Meyer:2010ku ; Meyer:2015eta ; Ketzer:2019wmd ; Chen:2022asf ;
Gross:2022hyw . The available data on the $\pi_{1}(1600)$ state provide
support for its candidacy as a hybrid meson. Assuming the $\pi_{1}(1600)$ to
be an isovector hybrid within the $J^{PC}=1^{-+}$ nonet, one can anticipate
the potential experimental discovery of its isospin partners.

A significant advancement was recently achieved by the BESIII Collaboration,
as they discovered the first isoscalar $1^{-+}$ state, named the
$\eta_{1}(1855)$, through a partial wave analysis of
$J/\psi\to\gamma\eta_{1}(1855)\to\gamma\eta\eta^{\prime}$ BESIII:2022iwi ;
BESIII:2022riz . The observation of the $\eta_{1}(1855)$ state has spurred
various theoretical interpretations. It has been proposed as a molecular state
of $K\bar{K}_{1}(1400)$ or a dynamically generated state in Refs. Dong:2022cuw
; Yang:2022rck ; Yan:2023vbh . In the framework of QCD sum rules, the
$\eta_{1}(1855)$ has been suggested as a tetraquark state with the
configuration $[1_{c}]{s\bar{s}}\otimes[1_{c}]_{q\bar{q}}$ Wan:2022xkx .
Furthermore, investigations into the $\eta_{1}(1855)$ as a hybrid candidate
have been conducted using the flux tube model Qiu:2022ktc , QCD sum rules
Chen:2022qpd , and the effective Lagrangian method Shastry:2022mhk . In
addition, the production cross sections of the $\eta_{1}(1855)$ in reactions
such as $K^{-}p\to\eta(1855)\Lambda$ Wang:2022sib  and $\gamma
p\to\eta_{1}(1855)p$ reactions Huang:2022tpq , as well as the branching
fractions of processes like $J/\psi\to\gamma\eta_{1}(1855)$ Shastry:2023ths ;
Chen:2022isv  and $J/\psi\to\eta^{(\prime)}\eta_{1}(1855)$ Yu:2022wtu , have
been extensively studied.

As emphasized in Ref. Gross:2022hyw , establishing the complete
SU(3)${}_{\textup{flavor}}$ multiplet for the $1^{-+}$ states is an essential
endeavor. The observations of the $\pi_{1}(1600)$ and $\eta_{1}(1855)$ states
provide an excellent opportunity to pursue this research. In this study, we
investigate the masses of $1^{-+}$ hybrid states in Sec. II, utilizing a
hybrid static potential simulated by lattice gauge theory. Furthermore, we
analyze the strong decays of the $\eta_{1}(1855)$, $\pi_{1}(1600)$, and their
isospin partners within the $1^{-+}$ nonet. The decay model is introduced in
Sec. III, and the obtained results are presented in Sec. IV. Finally, we end
the paper with discussion and conclusion in Sec. V.

##  II Masses of $1^{-+}$ hybrid nonet

It is worth noting that the mass difference between the $\eta_{1}(1855)$ and
$\pi_{1}(1600)$ states is remarkably similar to that between the $\phi(1020)$
and $\rho(770)$ mesons (as shown in Fig. 1). This observation can be explained
as follows: If we consider the $\pi_{1}(1600)$ and $\eta_{1}(1855)$ as
$1^{-+}$ hybrids, the spin of the constituent quark-antiquark pair, denoted as
$S_{q\bar{q}}$, is equal to 1, which is the same as for the $\rho(770)$ and
$\phi(1020)$ mesons. The primary distinction lies in the fact that the
$q\bar{q}$ pair in the $\rho(770)$ and $\phi(1020)$ systems is associated with
the lowest-lying static potential, while the $q\bar{q}$ pair in the
$\pi_{1}(1600)$ and $\eta_{1}(1855)$ systems is formed from the lowest excited
configurations of the gluon field. Under this assumption, the mass gaps
observed between $\phi(1020)/\rho(770)$ and $\eta_{1}(1855)/\pi_{1}(1600)$ can
be predominantly attributed to the inherent mass difference between the $s$
and $u/d$ quarks.

Figure 1: Mass differences of $\phi(1020)/\rho(770)$ and
$\eta_{1}^{\prime}(1855)/\pi_{1}(1600)$. For the light mesons, $\Delta{m}$ is
defined as $m_{X}-m_{\rho}$ where $X$ denotes a state in the $1^{3}S_{1}$
nonet. For the light hybrids, $\Delta{m}$ is defined as $m_{X}-m_{\pi_{1}}$
where $X$ denotes a state in the $1^{-+}$ nonet. The $\omega(783)$,
$K^{\ast}(893)$, and the unknown $\eta_{1}(1600)/K_{1}(1720)$ are also
presented. The masses of the $\rho(770)$, $\omega(783)$, $K^{\ast}(893)$, and
$\phi(1020)$ states are taken from Ref. ParticleDataGroup:2022pth . The masses
of the $\pi_{1}(1600)$ and $\eta_{1}^{\prime}(1855)$ states are taken from
Refs. COMPASS:2018uzl ; BESIII:2022iwi ; BESIII:2022riz .

To further investigate the hybrid nature of the $\pi_{1}(1600)$ and
$\eta_{1}^{\prime}(1855)$ states,111Since the $1^{-+}$ isoscalar state
observed by BESIII BESIII:2022iwi ; BESIII:2022riz  is proposed to be the
strange isoscalar partner of the $\pi_{1}(1600)$ state, we henceforth refer to
it as the $\eta_{1}^{\prime}(1855)$ state. we employ a quark potential model
to calculate the masses of these lowest hybrid states. The quark potential
model, which has been validated by lattice QCD simulations Barkai:1984ca ;
Itoh:1985wk , has exhibited considerable success in describing the mass
spectrum of meson and baryon systems. Consequently, it can be extended to
incorporate hybrid masses. Notably, the interaction between quarks, mediated
by the excited gluon field, has been simulated as “effective potentials” in
lattice QCD studies UKQCD:1998zbe ; Karbstein:2018mzo ; Capitani:2018rox ;
Schlosser:2021wnr .

In the subsequent analysis, we utilize a quark potential model to compute the
masses of $1^{-+}$ light hybrid states, incorporating the hybrid static
potential extracted from lattice simulations Capitani:2018rox .222Reference
Capitani:2018rox  provides more precise hybrid static potentials with smaller
statistical errors and finer spatial resolution for different gluon field
configurations. It is worth noting that lattice QCD not only obtains effective
potentials for hybrids but also successfully reproduces the well-established
Cornell potential for conventional meson states. For the $q\bar{q}$ mesons,
the effective potential is

$V_{\Sigma_{g}^{+}}(r)=-\frac{4}{3}\frac{\alpha_{s}}{r}+br+V_{0}.$ (1)

The parameters $\alpha$, $b$, and $C$ stand for the strength of the color
Coulomb potential, the strength of linear confinement, and a mass-renormalized
constant, respectively. For the lowest-energy hybrids, the effective potential
is given as

$V_{\Pi_{u}}(r)=\frac{A_{1}}{r}+A_{2}r^{2}+V_{1}.$ (2)

On the other hand, the short-distance behavior of the gluonic field $\Pi_{u}$
can be accurately described by a $1^{+-}$ gluelump, as discussed in the Ref.
Braaten:2014qka . This understanding has received further confirmation through
lattice QCD computations, which have shown that the gluonic field in the
lowest-energy hybrids exhibits a significant overlap with the chromomagnetic
structure characterized by $J^{PC}_{g}=1^{+-}$ Dudek:2011bn . When considering
a $q\bar{q}$ pair in an internal $S$-wave, the permitted $J^{PC}$ values for
the lightest hybrid mesons are $0^{-+}$, $1^{-+}$, $2^{-+}$, and $1^{--}$.
These conclusions align with the findings of lattice QCD calculations
Dudek:2011bn , where the masses of the $0^{-+}$, $1^{-+}$, $2^{-+}$, and
$1^{--}$ hybrid mesons were expected to be of the order of

$M_{0^{-+}}<M_{1^{-+}}<M_{1^{--}}<M_{2^{-+}}.$ (3)

Here, we are only concerned about the masses of $1^{-+}$ hybrids. In our
calculations, the following spinless Salpeter equation333The spinless Salpeter
equation could be regarded as an approximate version of the relativistic
Bethe-Salpeter equation Salpeter:1951sz ; Salpeter:1952ib . More details can
be found in Ref. Lucha:1998ix . is solved for the masses of the light mesons
and the lowest hybrid states:

$\left[\sqrt{m_{1}^{2}+p^{2}}+\sqrt{m_{2}^{2}+p^{2}}+V(r)\right]\psi_{nL}=E\psi_{nL}.$
(4)

For mesons, the potential $V(r)$ in Eq. (2) includes not only the
$V_{\Sigma_{g}^{+}}(r)$ in Eq. (1), but also the spin-spin contact hyperfine
interaction,

$V_{q_{1}\bar{q}_{2}}^{\textup{cont.}}(r)=\frac{32\alpha\sigma^{3}e^{-\sigma^{2}r^{2}}}{9\sqrt{\pi}m_{1}m_{2}}\textbf{s}_{q_{1}}\cdot\textbf{s}_{\bar{q}_{2}}.$
(5)

The parameters in Eqs.(1) and (5) can be determined by the measured masses of
well-established $q\bar{q}$ states. Then the parameters of the potential
$V_{\Pi_{u}}(r)$ for the lowest hybrid states can be constrained by the
following relationships: $A_{1}=0.0958$, $A_{2}=\xi_{2}(b/\xi_{1})^{3/2}$, and
$V_{1}=V_{0}+\xi_{3}\sqrt{b/\xi_{1}}$. Here, we have $\xi_{1}=0.04749$,
$\xi_{2}=0.001599$, and $\xi_{3}=0.5385$. In principle, the potential of
hybrid mesons should include the spin-spin interaction. However, due to the
current limitations in experimental data, the scarcity of measurements
restricts our ability to thoroughly discuss the spin-spin effects of these
hybrids. As a result, our primary focus in this work centers aon determining
the central masses of $0^{-+}$, $1^{-+}$, $2^{-+}$, and $1^{--}$ hybrids.

Table 1: Predicted masses of $n^{2S+1}L_{J}$ light mesons with the corresponding states (GeV).  $n^{2S+1}L_{(J)}$ |  $q\bar{q}$ meson |  $q\bar{s}$ meson |  $s\bar{s}$ meson
---|---|---|---
Pred. | Exp. ParticleDataGroup:2022pth  | Pred. | Exp. ParticleDataGroup:2022pth  | Pred. | Exp. ParticleDataGroup:2022pth
$1^{1}S_{0}$ | 0.139 | $\pi(140)$ | 0.490 | $K(496)$ | 0.754 | $\eta^{\prime}(958)$
$1^{3}S_{1}$ | 0.772 | $\rho(775)$ | 0.897 | $K^{\ast}(892)$ | 1.018 | $\phi(1020)$
$2^{1}S_{0}$ | 1.107 | $\pi(1300)$ | 1.342 | $K(1460)$ | 1.540 | $\eta^{\prime}(1475)$
$2^{3}S_{1}$ | 1.397 | $\rho(1450)$ | 1.529 | $K^{\ast}(1410)$ | 1.659 | $\phi(1680)$
$1^{1}P_{1}$ | 1.072 | $h_{1}(1170)$ | 1.286 | $K_{1}(1270)$ | 1.465 | $h_{1}(1415)$
$1^{3}P_{J}$ | 1.262 | $a_{1}(1260)$ | 1.413 | $K_{1}(1400)$ | 1.553 | $f_{2}^{\prime}(1525)$
$1^{1}D_{2}$ | 1.559 | $\eta_{2}(1645)$ | 1.722 | $K_{2}(1770)$ | 1.871 |
$1^{3}D_{J}$ | 1.611 | $\rho_{3}(1690)$ | 1.761 | $K_{3}(1780)$ | 1.900 | $\phi_{3}(1850)$

By reproducing the masses of these well-determined $q\bar{q}$ mesons, as shown
in Table 1, we fix the parameters of the potential model as follows: The
masses of the $u/d$ and $s$ quarks are taken to be 0.32 GeV and 0.45 GeV,
respectively. The parameters $\alpha_{s}$ and $b$ are set to 0.64 and 0.165
GeV2 for all light mesons. The values for $\sigma$ and $V_{0}$ are assigned as
follows: $\sigma_{q\bar{q}}=0.47$ GeV, $\sigma_{q\bar{s}}=0.45$ GeV,
$\sigma_{s\bar{s}}=0.43$ GeV, $V_{0}^{(q\bar{q})}=-0.48$ GeV,
$V_{0}^{(q\bar{s})}=-0.40$ GeV, and $V_{0}^{(s\bar{s})}=-0.33$ GeV. With these
parameter values, the average mass of the $\Pi_{u}$ hybrid multiplet is
summarized in the bottom row of Table 2, which also includes the newly
measured masses of the $\pi_{1}(1600)$ and $\eta_{1}^{\prime}(1855)$ states
for comparison.

Table 2: A comparison of the predicted average masses of the $\Pi_{u}$ hybrid multiplet with the measured masses of the $\pi_{1}(1600)$ and $\eta_{1}^{\prime}(1855)$ states (MeV). $\pi_{1}(1600)$ | $K_{1}$ | $\eta_{1}^{\prime}(1855)$
---|---|---
$1600^{+110}_{-60}$ COMPASS:2018uzl  |  |  $1855\pm 9^{+6}_{-1}$ BESIII:2022riz
$1564\pm 24\pm 86$ JPAC:2018zyd  |  |
$1623\pm 47^{+24}_{-75}$ Kopf:2020yoa  |  |
1669 | 1852 | 2023

As presented in Table 2, the measured mass of the $\pi_{1}(1600)$ state is
comparable to the predicted average mass of the lowest $q\bar{q}g$ hybrids,
suggesting that the $\pi_{1}(1600)$ could indeed be a viable $1^{-+}$ hybrid
state. In fact, our prediction for the lowest $q\bar{q}g$ hybrids aligns well
with most theoretical predictions for the $1^{-+}$ $q\bar{q}g$ state (see Fig.
78 of Ref. Chen:2022asf  for a comparison). Furthermore, several lattice QCD
calculations also support the hybrid assignment for the $\pi_{1}(1600)$ state,
particularly when extrapolating the pion mass in the lattice QCD simulations
to the physical value Chen:2022asf . Thus, the mass of the $\pi_{1}(1600)$
state provides a compelling evidence in favor of its hybrid nature.

Regarding the predicted average mass for the $s\bar{s}g$ states, it is
consistent with the findings of Refs. Isgur:1984bm ; Close:2003ae ;
Eshraim:2020ucw , but approximately 150 MeV higher than the measured mass of
the $\eta_{1}^{\prime}(1855)$ state. However, it is important to note that our
calculations do not account for spin-dependent interactions. Thus, the hybrid
assignment for the $\eta_{1}^{\prime}(1855)$ remains a plausible possibility.
Swanson recently studied the flavor mixing of hybrid states and also proposed
the $\eta_{1}^{\prime}(1855)$ state as the probable $s\bar{s}g$ partner of the
$\pi_{1}(1600)$ state Swanson:2023zlm .

Certainly, further investigation is warranted to explore the possibility of
the $\pi_{1}(1600)$ and $\eta_{1}^{\prime}(1855)$ states as two hybrid members
of the SU(3)${}_{\textup{flavor}}$ $1^{-+}$ multiplet. In the following
analysis, we will examine their strong decay behavior under the assumption of
the $1^{-+}$ hybrid assignment. To calculate the strong decays of the
$\pi_{1}(1600)$ and $\eta_{1}^{\prime}(1855)$ states, we will employ a
constituent gluon model proposed in Ref. Farina:2020slb . For the spatial wave
functions of the initial hybrid and final meson states, we will approximate
them using the simple harmonic oscillator (SHO) wave function. By adopting
this approach, we can obtain analytical expressions for the decay amplitudes.
In our calculations, we will determine the scale parameter $\beta$ of the SHO
wave function by solving the Salpeter equation. The specific values of $\beta$
for the final meson states are collected in Table 3. Additionally, we present
the $\beta^{H}_{q_{1}\bar{q}_{2}}$ values (where $q_{1}$ and $q_{2}$ represent
the $u$, $d$, and $s$ quarks) for the lowest hybrid states in the $\Pi_{u}$
multiplet as follows:

$\displaystyle\beta^{H}_{q\bar{q}}=0.264~{}\textup{GeV},~{}~{}~{}\beta^{H}_{q\bar{s}}=0.271~{}\textup{GeV},~{}~{}~{}\beta^{H}_{s\bar{s}}=0.277~{}\textup{GeV}.$

The predicted values of $\beta^{H}_{q\bar{q}}$ and $\beta^{H}_{q\bar{s}}$ are
consistent with the findings of the flux-tube model Close:2003ae , indicating
agreement between the two approaches. However, the predicted value of
$\beta^{H}_{s\bar{s}}$ is slightly smaller than the corresponding result from
the flux-tube model.

Table 3: Values of the SHO wave function scale $\beta$ (GeV).  System | $1^{1}S_{0}$ | $1^{3}S_{1}$ | $2^{1}S_{0}$ | $2^{3}S_{1}$ | $1^{1}P_{1}$ | $1^{3}P_{J}$
---|---|---|---|---|---|---
$q\bar{q}$ | 0.669 | 0.396 | 0.435 | 0.356 | 0.446 | 0.339
$q\bar{s}$ | 0.609 | 0.432 | 0.421 | 0.371 | 0.421 | 0.353
$s\bar{s}$ | 0.579 | 0.466 | 0.417 | 0.384 | 0.412 | 0.367

##  III Decay model of the hybrid state

Numerous models and methods have been proposed to investigate the strong
decays of hybrid states, including the constituent gluon model Tanimoto:1982eh
; Close:2003ae ; Tanimoto:1982wy ; LeYaouanc:1984gh ; Iddir:1988jd ;
Ishida:1991mx ; Kalashnikova:1993xb ; Swanson:1997wy ; Iddir:2000yb ;
Ding:2006ya ; Iddir:2007dq ; Benhamida:2019nfx ; Farina:2020slb , the flux-
tube model Isgur:1985vy ; Close:1994hc ; Barnes:1995hc ; Page:1998gz , QCD sum
rules DeViron:1984svx ; Zhu:1998sv ; Zhang:2002id ; Chen:2010ic ; Huang:2010dc
; Huang:2016upt , lattice QCD McNeile:2002az ; McNeile:2006bz ; Woss:2020ayi ,
and other approaches Page:2003qn ; Shastry:2022mhk . In this study, we employ
the constituent gluon model, recently developed in Ref. Farina:2020slb , to
calculate the strong decays of $1^{-+}$ light hybrid states. This model has
successfully reproduced reasonable spin-averaged $c\bar{c}g$ spectra, which
are in good agreement with lattice QCD results HadronSpectrum:2012gic . In the
following, we will demonstrate that the constituent gluon model can also
capture the main decay characteristics of the $\pi_{1}(1600)$ state predicted
by lattice QCD Woss:2020ayi .

In the constituent gluon model, the gluon dynamics in the lowest hybrid states
is approximated by an axial gluon with quantum numbers $J^{PC}=1^{+-}$. The
foundation of this model can be understood as follows: in these lowest hybrid
states, the close proximity of the quark-antiquark pair allows the gluon field
configuration to be effectively described by the corresponding gluelumps
Braaten:2014qka . Consequently, a $1^{-+}$ hybrid state can be simplified as a
three-body system, with the quark-antiquark pair and a transverse electric
(TE) gluon $(J_{g}^{PC}=1^{+-})$ as its essential degrees of freedom.

Figure 2: A connected diagram for a hybrid state $H$ decaying into the $B$ and
$C$ mesons.

A matrix element of the interaction Hamiltonian in the QCD theory, which
describes the annihilation of a gluon and the creation of a quark-antiquark
pair, is taken as the coupling vertex to describe a decay process of a hybrid
state into two normal mesons. As shown in Fig. 2, the corresponding
Hamiltionian is given by

$\hat{H}_{I}=g_{s}\int\textup{d}^{3}\vec{x}~{}\bar{\psi}(\vec{x})\gamma_{\mu}\frac{\lambda^{a}}{2}\psi(\vec{x})A_{a}^{\mu}(\vec{x})$
(6)

in the lowest order LeYaouanc:1984gh . The transition operator
$\hat{\mathcal{T}}$ could be derived from the above matrix element as

$\begin{split}\hat{\mathcal{T}}=&g_{s}\sum_{s,s^{\prime},\lambda}\int\frac{\textup{d}^{3}\vec{p}_{3}\textup{d}^{3}\vec{p}_{4}\textup{d}^{3}\vec{k}}{\sqrt{2\omega_{g}}(2\pi)^{6}}\delta^{3}(\vec{p}_{3}+\vec{p}_{4}-\vec{k})\\\
&\times\frac{\lambda_{c_{g}}^{c,c^{\prime}}}{2}\omega_{g}^{(34)}\chi_{s}^{\dagger}\sigma\tilde{\chi}_{s^{\prime}}\epsilon^{\mu}(\hat{\textbf{k}},\lambda)b_{3}^{\dagger}(\vec{p}_{3})d_{4}^{\dagger}(\vec{p}_{4})a_{\textbf{k}\lambda}^{c_{g}}(\vec{k})\end{split}$
(7)

in the nonrelativistic limit. The parameter $\omega_{g}$ could be regarded as
the effective mass of the constituent gluon Swanson:1997wy , which is taken to
be 0.80 GeV in our calculations. The strong coupling constant $g_{s}$ is
constrained by the decay width of $\pi_{1}(1600)$ with the $1^{-+}$ hybrid
assignment. In practical application, the mock state is adopted to describe
the spatial wave function of initial and final hadrons Hayne:1981zy . For the
hybrid state, the wave function could be written as

$\begin{split}|\textup{H}(J_{H}m_{H}~{}[&L_{H}(L_{q_{1}\bar{q}_{2}}J_{g})S_{H}](\vec{P}_{H})\rangle\equiv\\\
&\omega_{H}\phi_{H}\mathcal{W}_{\textup{CG}}\int\textup{d}^{3}\vec{p}_{1}\textup{d}^{3}\vec{p}_{2}\textup{d}^{3}\vec{k}\delta^{3}(\vec{p}_{1}+\vec{p}_{2}+\vec{k}-\vec{P}_{H})\\\
&\times\Psi_{n_{q_{1}\bar{q}_{2}},n_{g}}^{L_{q\bar{q}}m_{q\bar{q}},L_{g}m_{g}}(\vec{p}_{1},\vec{p}_{2},\vec{k})|q_{1}(\vec{p}_{1})\bar{q}_{2}(\vec{p}_{2})g(\vec{k})\rangle.\end{split}$
(8)

where $\omega_{H}$ and $\phi_{H}$ denote the color and flavor wave functions,
respectively, of an initial hybrid state. As a color singlet state, the color
wave functions of a hybrid state could be represented as
$\omega_{H}=|(q\bar{q})_{8}\otimes{g_{8}}\rangle^{0}$. The flavor wave
function of a hybrid state is the same as that of a meson state, since the
gluon is a flavor singlet. The value $\mathcal{W}_{\textup{CG}}$ in Eq. (4)
involves the Clebsch-Gordan coefficients, which denote the spins and angular
momentums of the quarks and gluon coupling to the total spin ($J_{H}$) of a
hybrid state. Specifically, it is given as

$\begin{split}\mathcal{W}_{\textup{CG}}=&\sqrt{\frac{2J_{g}+1}{4\pi}}\mathcal{D}^{J_{g}\ast}_{m_{g},\mu}\chi_{\mu,\lambda}\langle{L_{q_{1}\bar{q}_{2}}m_{q_{1}\bar{q}_{2}},J_{g}m_{g}}|L_{H}m_{L}\rangle\\\
&\times\langle\frac{1}{2}m_{1},\frac{1}{2}m_{2}|S_{H}s_{H}\rangle\langle{S_{H}s_{H},L_{H}m_{L}}|J_{H}m_{H}\rangle.\end{split}$
(9)

The value $\chi_{\mu,\lambda}$ denotes the spin of a transverse gluon in the
gluon helicity basis$-$i.e.
$\chi_{\mu,\lambda}\equiv\langle{1\lambda,L_{g}0}|J_{g}\mu\rangle\delta_{\mu\lambda}$.
For the $1^{-+}$ hybrid states containing a TE gluon, the factor
$\chi_{\mu,\lambda}$ is always $1/\sqrt{2}$. The role of the rotation matrix
$\mathcal{D}^{J_{g}\ast}_{m_{g},\mu}$ in Eq. (5) is to convert the angular
momentum projection of a gluon to the basis of the hybrid system. As done in
Refs. LeYaouanc:1984gh ; Iddir:1988jd ; Farina:2020slb , a simple product
ansatz could be made for the spatial wave function of the hybrid state.
Namely, the
$\Psi_{n_{q\bar{q}}}^{L_{q\bar{q}},L_{g}}(\vec{p}_{1},\vec{p}_{2},\vec{k})$
could be written as

$\Psi_{n_{q\bar{q}},n_{g}}^{L_{q\bar{q}},L_{g}}(\vec{p}_{1},\vec{p}_{2},\vec{k})=\psi_{n_{q_{1}\bar{q}_{2}}}^{L_{q\bar{q}}m_{q\bar{q}}}(\vec{p}_{\rho})\psi_{n_{g}}^{L_{g}m_{g}}(\vec{p}_{\lambda}).$
(10)

With the Jacobian coordinates, $\vec{p}_{\rho}$ and $\vec{p}_{\lambda}$ are
represented as follows:

$\vec{p}_{\rho}=\frac{1}{2}\vec{p}-\vec{p}_{B}-\frac{1}{2}\frac{m_{1}-m_{2}}{m_{1}+m_{2}}\vec{k};~{}~{}~{}~{}\vec{p}_{\lambda}=\vec{k}$
(11)

in the rest frame of the hybrid system. The values $m_{1}$ and $m_{2}$ refer
to the masses of $q_{1}$ and $\bar{q}_{2}$ in the hybrid state. As mentioned
before, the SHO wave function could be used for the spatial wave function
$\psi_{n}^{Lm}(\vec{p})$. When the wave functions of the final meson states in
the decay process are constructed in the same way, the momentum $\vec{p}$ and
$\vec{k}$ can be integrated by performing the partial wave amplitude
calculation. In this way, the helicity amplitude of a decay process
$H\to{B+C}$ (see Fig. 2) can be obtained analytically by the relation
$\langle{BC}|\hat{\mathcal{T}}|H\rangle=\delta^{3}(\textbf{K}_{B}+\textbf{K}_{C})\mathcal{M}^{j_{H},j_{B},j_{C}}(\textbf{p})$.
The concrete expression of $\mathcal{M}^{j_{H},j_{B},j_{C}}(\textbf{p})$ can
be found in Ref. Farina:2020slb .

Finally, the partial wave amplitude is given as

$\displaystyle\mathcal{M}^{H\to BC}_{LS}=\frac{\sqrt{2L+1}}{2J_{H}+1}$
$\displaystyle\sum_{\text{$j_{B}$,$j_{C}$}}\langle L0Jj_{H}|J_{H}j_{A}\rangle$
(12) $\displaystyle\times\langle
J_{B}j_{B},J_{C}j_{C}|Jj_{H}\rangle\mathcal{M}^{j_{H},j_{B},j_{C}}(\textbf{p}),$

while the partial width of the process $H\to BC$ in the $H$ rest frame is
given by

$\displaystyle\Gamma(H\rightarrow
BC)=2\pi\frac{E_{B}E_{C}}{M_{H}}p\sum_{L,S}|\mathcal{M}^{H\to
BC}_{LS}(p)|^{2}.$ (13)

##  IV Strong decays of the $1^{-+}$ nonet

###  IV.1 The $\pi_{1}(1^{-+})$ state

Table 4: The partial and total widths of the strong decays of the $\pi_{1}(1^{-+})$ state (MeV). Here, the final mesons refer to all allowed charge conjugate pairs. For example, when we consider $\pi_{1}(1600)^{-}$ decaying into the $K^{\ast}(892)$ and $K$ states, $K^{\ast}{K}$ denotes the $K^{\ast}(892)^{-}+K^{0}$ and $K^{\ast}(892)^{0}+K^{-}$ channels. If a partial width is predicted to be smaller than 1 MeV, we denote it as zero. $\rho\pi$ | $K^{\ast}{K}$ | $b_{1}(1235)\pi$ | $f_{1}(1285)\pi$ | $f_{2}(1270)\pi$
---|---|---|---|---
2 | $\approx 0$ | 244 | 15 | $\approx 0$
|  |  | Total | Exp. ParticleDataGroup:2022pth
|  |  | 261 | 240$\pm$50

By assigning $\pi_{1}(1600)$ as the isovector member in the $J^{PC}=1^{-+}$
nonet, we give a total decay width of 261 MeV,444By comparing with the average
decay width of $\pi_{1}(1600)$ listed in the PDG table, we may find that the
uncertainty of our result is about $10\%\sim 30\%$. The same uncertainty also
exists in the results of Tables 5, 6, and 7. which is comparable to the
average value reported in the PDG table ParticleDataGroup:2022pth .555It
should be noted that the pole position (or the Breit-Wigner parameter) of the
$\pi_{1}(1600)$ state currently has a large uncertainty, and more precise
measurements are required in the future. The dominant decay modes
$b_{1}(1260)\pi$ and $f_{1}(1280)\pi$ are predicted. This finding is
consistent with the results of the flux-tube model Close:1994hc  and lattice
QCD Woss:2020ayi , where the $b_{1}(1260)\pi$ channel was also identified to
have the largest partial decay width for the $\pi_{1}(1600)$ state. The decays
of $\pi_{1}(1600)$ into the $b_{1}(1260)\pi$ and $f_{1}(1280)\pi$ channels can
occur via both $s$-wave and $d$-wave processes. However, our calculations show
that the $d$-wave partial widths are quite negligible ($<$ 1 MeV) due to the
limited phase space available. This conclusion is in agreement with the
lattice QCD results Woss:2020ayi , but it contradicts the predictions of the
flux-tube model Close:1994hc .

In addition to the $b_{1}(1260)\pi$ and $f_{1}(1280)\pi$ channels, the
$\pi_{1}(1600)$ state has also been observed in the $\rho(770)\pi$
E852:1998mbq ; Zaitsev:2000rc ; COMPASS:2009xrl ; COMPASS:2021ogp  and
$\eta^{\prime}(958)\pi$ VES:1993scg ; E852:2001ikk ; Amelin:2005ry  channels.
According to the results presented in Table 4, the partial decay width for the
$\rho\pi$ channel is estimated to be approximately 2 MeV. This suggests that
the $\pi_{1}(1600)$ state could be observed in the $\pi_{1}(1600)\to\rho\pi$
decay process, given a sufficiently large data sample. However, the isospin
symmetry Close:1987aw  prohibits the decays of $\pi_{1}(1600)$ into $\eta\pi$
and $\eta^{\prime}\pi$ via the mechanism shown in Fig. 2. Therefore, this
conclusion contradicts the observation of $\pi_{1}(1600)\to\eta^{\prime}\pi$
reported in previous studies VES:1993scg ; E852:2001ikk ; Amelin:2005ry . One
possible explanation for this puzzle is that the $\eta^{\prime}$ meson may
have a significant gluonium component666The KLOE Collaboration determined the
gluonium content of the $\eta^{\prime}$ meson through a global fit to the
radiative decays of pseudoscalar and vector mesons Ambrosino:2009sc . Their
analysis indicated a substantial gluonium component in the wave function of
the $\eta^{\prime}$ meson.. Consequently, the decay mechanism of
$\pi_{1}(1600)\to\eta^{\prime}\pi$ would differ from the other decay modes
listed in Table 4. Further discussion on this point will be provided in the
next subsection.

We would like to emphasize the significance of the branching ratio
$\mathcal{B}=\frac{\Gamma(\pi_{1}(1600)\to
b_{1}(1235)\pi)}{\Gamma(\pi_{1}(1600)\to f_{1}(1285)\pi)}$ in providing
insights into the nature of the $\pi_{1}(1600)$ state. When considering the
mass and SHO wave function scale parameter $\beta$ of the $b_{1}(1235)$ and
$f_{1}(1285)$ to be equal, the ratio $\mathcal{B}$ is determined to be 4, a
result consistent with the findings of the flux-tube model Burns:2006wz .
However, by incorporating the differences between the $b_{1}(1235)$ and
$f_{1}(1285)$ in the calculation, the value of $\mathcal{B}$ increases to
approximately 16.3. This suggests that if the $\pi_{1}(1600)$ is indeed a
$1^{-+}$ hybrid state, its partial width for $\pi_{1}(1600)\to b_{1}(1235)\pi$
is expected to be significantly larger than that for $\pi_{1}(1600)\to
f_{1}(1285)\pi$. Currently, only the VES experiment has evaluated the ratio
$\mathcal{B}$ with a large uncertainty Amelin:2005ry . Therefore, further
experimental efforts are needed to measure this branching ratio accurately.

###  IV.2 The $\eta_{1}(1^{-+})$ state

As the isoscalar counterpart to the $\pi_{1}(1600)$ state, it is anticipated
that the $\eta_{1}(1^{-+})$ state will have a mass approximately equal to that
of the $\pi_{1}(1600)$ state Eshraim:2020ucw . However, the decay width of the
$\eta_{1}(1^{-+})$ state is predicted to be narrow (see Table 5), which is in
stark contrast to the broader decay width of the $\pi_{1}(1600)$. The same
result has also been presented in Ref. Shastry:2023ths  where the lighter
isoscalar hybrid was predicted to be the narrowest state in the $1^{-+}$
multiplet.

Table 5 reveals that the dominant decay channel for the $\eta_{1}(1^{-+})$
state is $a_{1}(1260)\pi$. Therefore, a promising avenue for detecting the
$\eta_{1}(1^{-+})$ signal is through the
$J/\psi\to\gamma\eta_{1}(1^{-+})\to\gamma+a_{1}(1260)\pi$ process, as $J/\psi$
radiative decays are known to be rich in hybrid production. However, the
situation is different for the $\pi p$ diffraction process at high beam
energy, as this process is primarily governed by Pomeron exchange
Ketzer:2019wmd . Consequently, the production of $\eta_{1}(1^{-+})$ in $\pi p$
reactions is suppressed.

Table 5: The partial and total widths of the strong decays of the $\eta_{1}(1^{-+})$ state (MeV). $\eta\eta^{\prime}$ | $K^{\ast}{K}$ | $a_{1}(1260)\pi$ | $a_{2}(1320)\pi$ | $\pi(1300)\pi$
---|---|---|---|---
$\approx 0$ | $\approx 0$ | 55 | $\approx 0$ | 5
|  |  | Total | Exp.
|  |  | 60 | $-$

In the scenario where the $\eta_{1}(1^{-+})$ hybrid contains a small
$s\bar{s}g$ component,777The mixing between two isoscalar states in $1^{-+}$
nonet is expected to be small due to the homochiral nature Shastry:2023ths . a
visible mass difference between the $\eta_{1}(1^{-+})$ and $\pi_{1}(1600)$
states could be expected. However, considering the decay properties of a pure
$1^{-+}$ $s\bar{s}g$ state discussed in the next subsection, it is evident
that the $a_{1}(1260)\pi$ channel remains the dominant decay mode for the
$\eta_{1}(1^{-+})$ state when its mass is below 1.76 GeV. Consequently, the
main partial widths predicted and listed in Table 5 would not undergo
significant changes.

###  IV.3 The $\eta^{\prime}_{1}(1^{-+})$ state

As discussed earlier, the mass difference between the
$\eta_{1}^{\prime}(1855)$ and $\pi_{1}(1600)$ states can be naturally
explained by treating the $\eta_{1}^{\prime}(1855)$ state as the $s\bar{s}g$
partner of the $\pi_{1}(1600)$. In this section, we will initially consider
the $\eta_{1}^{\prime}(1855)$ as a pure $s\bar{s}g$ hybrid state and examine
its strong decays. Subsequently, we will discuss the possibility of the
$\eta_{1}^{\prime}(1855)$ state being a mixture of $s\bar{s}g$ and
$(u\bar{u}+d\bar{d})g/\sqrt{2}$ components.

Treating the $\eta_{1}^{\prime}(1855)$ state as a $s\bar{s}g$ hybrid, the
total decay width is predicted to be approximately 160 MeV, which agrees with
experimental measurements BESIII:2022iwi ; BESIII:2022riz . However, the
partial decay width of $\eta\eta^{\prime}$ is estimated to be no more than 1
MeV, which may appear too small to be detected in experiments. This result is
analogous to the case of $\pi_{1}(1600)\to\eta^{\prime}\pi$. The presence of
gluonium content in the $\eta^{\prime}$ meson can significantly influence the
$\eta^{\prime}(1855)\to\eta\eta^{\prime}$ process. In other words, the decay
mechanism governing the $\eta^{\prime}(1855)\to\eta\eta^{\prime}$ channel
differs from that of other channels listed in Table 6. If the $\pi_{1}(1600)$
and $\eta_{1}^{\prime}(1855)$ are $1^{-+}$ hybrids, there should be a nearby
$0^{-+}$ isoscalar hybrid state. This $0^{-+}$ isoscalar hybrid state can mix
with the $\eta^{\prime}$ meson, resulting in the gluonic content in the
$\eta^{\prime}$. Experimental measurements of the branching ratio
$\mathcal{B}(\eta_{1}^{\prime}(1855)\to\eta\eta^{\prime})/\mathcal{B}(\eta_{1}^{\prime}(1855)\to
K^{\ast}K)$ can test this speculation in the future. Taking into account the
gluonic content of the $\eta^{\prime}$, the partial width of
$\eta_{1}^{\prime}(1855)\to\eta\eta^{\prime}$ is comparable to or even larger
than that of $\eta_{1}^{\prime}(1855)\to K^{\ast}K$.

Table 6: The partial and total widths of the strong decays of the $\eta_{1}^{\prime}(1^{-+})$ state (MeV). $\eta\eta^{\prime}$ | $K^{\ast}{K}$ | $K_{1}(1270)K$ | Total | Exp. BESIII:2022iwi ; BESIII:2022riz
---|---|---|---|---
$\approx 0$ | 2 | 157 | 159 | 188$\pm$18${}^{+3}_{-8}$

It is also crucial for future experiments to search for the
$\eta_{1}^{\prime}(1855)$ state in the $\eta_{1}^{\prime}(1855)\to
K_{1}(1270)K$ process since $K_{1}(1270)K$ is predicted to be a dominant decay
channel for the $\eta_{1}^{\prime}(1855)$ state. For the decay channels
associated with $K_{1}(1270)$ and $K_{1}(1400)$, it is necessary to regard
them as a combination of the $K_{1}(^{1}P_{1})$ and $K_{1}(^{3}P_{1})$ states.
The mixing scheme reads as Suzuki:1993yc

$\left\\{\begin{array}[]{c}|K_{1}(1270)\rangle=\cos\theta~{}|K_{1}(^{1}P_{1})\rangle-\sin\theta~{}|K_{1}(^{3}P_{1})\rangle\\\
~{}|K_{1}(1400)\rangle=\sin\theta~{}|K_{1}(^{1}P_{1})\rangle+\cos\theta~{}|K_{1}(^{3}P_{1})\rangle.\end{array}\right.$
(14)

The mixing angle $\theta$ is adopted as $60^{\circ}$, a value established
through an examination of the $K_{1}\to K\pi\pi$ strong decays
Tayduganov:2011ui .

As an isoscalar meson, $\eta_{1}^{\prime}(1855)$ in principle could contain
some $(u\bar{u}+d\bar{d})g/\sqrt{2}$ component Swanson:2023zlm . In future,
experiments can check the $(u\bar{u}+d\bar{d})g/\sqrt{2}$ component of the
$\eta_{1}^{\prime}(1855)$ state by searching for it in the $a_{1}(1260)\pi$
channel.

###  IV.4 The $K_{1}(1^{-})$ state

Given the predicted decay width of 278 MeV, the $K_{1}(1^{-})$ state, as the
strange partner of the $\pi_{1}(1600)$ and $\eta^{\prime}(1855)$ states, is
anticipated to be broad. Consequently, it may pose a challenge for future
experimental searches, as detecting such a broad state can be more challenging
compared to the narrower resonances. Nonetheless, further investigations and
experimental efforts are crucial to exploring the properties and existence of
the $K_{1}(1^{-})$ state.

Table 7: The partial and total widths of the strong decays of the $K_{1}(1^{-})$ state where the mass of the $K_{1}(1^{-})$ state is taken as 1.72 GeV (MeV). $K\pi$ | $K\eta$ | $K\eta^{\prime}$ | $K\rho$ | $K\omega$
---|---|---|---|---
1 | $\approx 0$ | $\approx 0$ | $\approx 0$ | $\approx 0$
$K^{\ast}\pi$ | $K^{\ast}\eta$ | $K^{\ast}\rho$ | $K^{\ast}\omega$ | $K_{1}(1270)\pi$
3 | 1 | $\approx 0$ | $\approx 0$ | 106
$K_{1}(1400)\pi$ | $K_{2}(1430)\pi$ | $h_{1}(1116)K$ | $K(1460)\pi$ | $K^{\ast}(1410)\pi$
146 | $\approx 0$ | 16 | 2 | 3
|  |  | Total | Exp.
|  |  | 278 | $-$

The $K_{1}(1^{-})$ state is predicted to exhibit significant decays into
$K_{1}(1400)\pi$ and $K_{1}(1270)\pi$, making them the two dominant decay
channels (see Table 7). Furthermore, the decays $h_{1}(1170)K$,
$K^{\ast}(1410)\pi$, and $K^{\ast}\pi$ are also expected to be observable for
the $K_{1}(1^{-})$ state. Given that the $\pi_{1}(1600)$ state has been
observed in the $\eta^{\prime}\pi$ channel and the $\eta_{1}^{\prime}(1855)$
state in the $\eta^{\prime}\eta$ channel, we propose that future experimental
investigations also target the $K_{1}(1^{-})$ state in the $K\eta^{\prime}$
channel. These suggested explorations would provide valuable insights into the
existence and properties of the $K_{1}(1^{-})$ state.

##  V Discussion and conclusion

In the field of hadron physics, the study of exotic states has gained
significant attention. While flavor-exotic states like the $Z_{c}^{\pm}$ and
$Z_{b}^{\pm}$ have been extensively investigated, spin-exotic states hold a
special significance in understanding the nonperturbative nature of strong
interactions at low energies. Currently, there have been reports of three
$1^{-+}$ states: the $\pi_{1}(1400)$, $\pi_{1}(1600)$, and $\pi_{1}(2015)$.
However, the existence of the $\pi_{1}(1400)$ and $\pi_{1}(2015)$ states is
still subject to debate, while the existence of the $\pi_{1}(1600)$ is
relatively more certain. Even so, crucial properties such as the pole position
and important decay ratios of the $\pi_{1}(1600)$ state are yet to be
precisely measured. Consequently, the establishment of the light
$J^{P(C)}=1^{-(+)}$ hybrid nonet is still a work in progress.

Recently, the discovery of the $\eta^{\prime}_{1}(1855)$ state by the BESIII
Collaboration in the $\eta\eta^{\prime}$ decay channel has significantly
impacted the situation BESIII:2022iwi ; BESIII:2022riz . As both the
$\eta^{\prime}(1855)$ and $\pi_{1}(1600)$ are $1^{-+}$ states, their mass
difference closely matches that of the $\phi$ and $\rho$ mesons. This
observation suggests that the $\eta^{\prime}(1855)$ and $\pi_{1}(1600)$ could
be two members of a $J^{P(C)}=1^{-(+)}$ hybrid nonet. By assuming the
$\eta^{\prime}(1855)$ to be the $s\bar{s}g$ partner of the $\pi_{1}(1600)$, we
conducted a study on their masses and strong decays. The results indicate that
the $\pi_{1}(1600)$ and $\eta^{\prime}(1855)$ are plausible candidates to be
considered as the members of $1^{-+}$ hybrid nonet.

The predicted dominant decay modes of the $\pi_{1}(1600)$ state include
$b_{1}(1260)\pi$ and $f_{1}(1280)\pi$, which is consistent with experimental
observations since the $\pi_{1}(1600)$ has been detected in these two decay
channels. To further determine the nature of the $\pi_{1}(1600)$ state, future
experiments could focus on measuring the ratio of $\Gamma(\pi_{1}(1600)\to
b_{1}(1235)\pi)$ to $\Gamma(\pi_{1}(1600)\to f_{1}(1285)\pi)$. The predicted
partial width for the $\rho\pi$ channel is around 2 MeV, in agreement with
lattice QCD results Woss:2020ayi . However, the $\eta^{\prime}\pi$ channel is
forbidden for the $\pi_{1}(1600)$ if the $\eta^{\prime}$ is considered a
conventional $q\bar{q}$ meson. This discrepancy seems to contradict
experimental observations where the $\pi_{1}(1600)$ state has been found in
the $\eta^{\prime}\pi$ channel VES:1993scg ; E852:2001ikk ; Amelin:2005ry . We
have proposed that the decay mechanism of $\pi_{1}(1600)\to\eta^{\prime}\pi$
might be different from other observed decay processes due to the significant
gluonic component of the $\eta^{\prime}$ meson. This assumption also provides
an explanation for the observation of
$\eta^{\prime}_{1}(1855)\to\eta^{\prime}\eta$. In fact, decay processes
involving the $\eta^{\prime}$ meson in the final states are considered as
promising channels to search for hybrid states Bibrzycki:2021rwh . Therefore,
it is crucial to measure the partial widths of
$\pi_{1}(1600)\to\eta^{\prime}\pi$ and
$\eta^{\prime}_{1}(1855)\to\eta^{\prime}\eta$ more precisely in future
experiments.

Being isospin partners of the $\pi_{1}(1600)$ and $\eta_{1}^{\prime}(1855)$ in
the $1^{-+}$ hybrid multiplet, the $K_{1}$ state is expected to have a broad
decay width, which presents a challenge for experimental searches. In
contrast, the $\eta_{1}$ state could potentially be observed in the cascade
process $J/\psi\to\gamma\eta_{1}(1^{-+})\to\gamma+a_{1}(1260)\pi$. In the
future, experiments such as BESIII BESIII:2020nme , Belle II Belle-II:2018jsg
, GlueX Dudek:2012vr ; Burkert:2018nvj , PANDA PANDA:2009yku , and COMPASS
Ketzer:2019wmd  will provide more valuable data to enhance our understanding
of hybrid states. Therefore, the field of hybrids warrants increased attention
from both experimentalists and theorists.

## Acknowledgements

This project is supported by the National Natural Science Foundation of China
under Grants No. 11305003, No. 12047501, and No. 12247101. X. L. is also
supported by China National Funds for Distinguished Young Scientists under
Grant No. 11825503, the National Key Research, the Development Program of
China under Contract No. 2020YFA0406400, the Fundamental Research Funds for
the Central Universities, and the Project for Top-notch Innovative Talents of
Gansu Province.

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