# Study of the $\omega$ and $\omega_{3}$, $\rho$ and $\rho_{3}$ and the newly
observed $\omega$-like state $X(2220)$

Ya-Rong Wang1,2<EMAIL_ADDRESS>Ting-Yan Li1,2<EMAIL_ADDRESS>Zheng-Yuan Fang1,2<EMAIL_ADDRESS>Hao Chen1,2,3<EMAIL_ADDRESS>Cheng-Qun Pang1,2,3111Corresponding author<EMAIL_ADDRESS>1College of
Physics and Electronic Information Engineering, Qinghai Normal University,
Xining 810000, China
2Joint Research Center for Physics, Lanzhou University and Qinghai Normal
University, Xining 810000, China
3Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China

###### Abstract

We study the excited states of $\omega$ and $\omega_{3}$ by comparison with
the $\rho$ and $\rho_{3}$ families, and discuss the possibility of $X(2220)$
as $\omega$ excitation by analyzing the mass spectra and strong decay
behaviors. In addition, we predict the masses and widths of $\omega(2D)$ and
$\omega_{3}$and $\rho_{3}(4D)$, $\rho_{3}(1G)$, $\omega_{3}$ and
$\rho_{3}(2G)$ and $\omega_{3}(3G)$ and $\rho_{3}(3G)$. The abundant
information of their two-body strong decays predicted in this work will be
helpful to further study of these $\omega$ and $\omega_{3}$ and $\rho$ and
$\rho_{3}$ states in experiment and theory.

###### pacs:

14.40.Be, 12.38.Lg, 13.25.Jx

##  I introduction

Very recently, the BESIII Collaboration announced the observation of the
$X(2220)$ state in the $e^{+}e^{-}\to\omega\pi^{0}\pi^{0}$ process Ablikim
_et al._ (2022), which has a statistical significance larger than $5\sigma$.
The measurement shows that the $X(2220)$ has the mass of $M=2222\pm 7\pm 2$
MeV and the width of $\Gamma=59\pm 30\pm 6$ MeV. It could be an $\omega$
excited state from its report. The Lanzhou group indicates that the
enhancement structures around 2.2 GeV existing in $e^{+}e^{-}\to\omega\eta$
and $e^{+}e^{-}\to\omega\pi^{0}\pi^{0}$ contain the $\omega(4S)$ and
$\omega(3D)$ signals Zhou  _et al._ (2022). The BESIII Collaboration announced
$X(2220)$ could be an excited $\omega$ resonance Li  _et al._ (2022). Present
studies show that it is reasonable to regard $X(2220)$ as an $\omega$ state.
When further checking the experimental status of light mesons, we notice an
interesting phenomenon: the states of unflavored light meson families with
$3^{--}$ are not established well. In our previous work Pang  _et al._ (2020),
we carried out a system investigation of the $\omega$ mesons and studied the
$\omega$-like $X(2240)$ state observed by BESIII Collaboration Ablikim  _et
al._ (2019a). Now, we reconsider the newly observed $X(2220)$ state and the
$\omega$ and $\omega_{3}$, $\rho$ and $\rho_{3}$ families.

As an important part of the meson family, the $\omega$ and $\rho$ and
$\omega_{3}$ and $\rho_{3}$ families have become more and more abundant in
experiment. BESIII Collaboration reported a result that in the
$e^{+}e^{-}\to\omega\pi^{0}$ cross section, a resonance denoted by $Y(2040)$
was observed recently. Its mass and width are determined to be $2034\pm 13\pm
9$ and $234\pm 30\pm 25$ MeV Ablikim  _et al._ (2021), respectively; Ref. Li
_et al._ (2021) also treated it as $\rho(2000)$. BESIII Collaboration also
reported a state []we call it $X(2239)$] with the mass of $M=2239.2\pm 7.1\pm
11.3$ MeV and the width of $\Gamma=139.8\pm 12.3\pm 20.6$ MeV Ablikim  _et
al._ (2019b) by studying the cross section of the $e^{+}e^{-}\to K^{+}K^{-}$
process. $X(2239)$ is treated as $Y(2175)$ in the Ref. Chen  _et al._ (2020).
Reference Pang  _et al._ (2020) assigned $X(2239)$ as the $\omega(4^{3}S_{1})$
state and it treated $X(2239)$ as the candidate of tetraquark states in Refs.
Lü  _et al._ (2020); Azizi  _et al._ (2019). Barnes, Godfrey, and Swanson
suggest that further investigation is expected to understand the structures
near the $\Lambda\bar{\Lambda}$ threshold, such as $X(2239)$, $\eta(2225)$,
and $X(2175)$. However, the properties of $X(2240)$ still remain unclear Zhu
_et al._ (2020).

Afonin classified light nonstrange mesons according to the values of $(L,n)$
and indicated that $\omega_{3}(1670)$, $\omega_{3}(2285)$, $\rho_{3}(1690)$,
are $D$-wave mesons Afonin (2007). In 2009, Ebert calculated masses of the
ground state as well as the orbital and radial excited states of quark-
antiquark mesons within the relativistic quark model, which included the
masses of $\omega$, $\omega_{3}$, $\rho$ and $\rho_{3}$ meson families Ebert
_et al._ (2009). Anisovish $et~{}al$. consider the decays of $\omega$ and
$\rho$ mensons, the ${}^{3}S_{1}~{}q\bar{q}$ states of constituent quark
model: $\rho\to\gamma\pi$, $\omega\to\gamma\pi$, and $\rho\to\pi\pi$ and give
correct values for the partial widths of radiative and hadronic decays of
confined $q\bar{q}$ states Anisovich  _et al._ (2010). In 2013, the Lanzhou
group studied $\rho$, $\rho_{3}$ family He  _et al._ (2013), which helps to
construct the whole $\rho$ family. Later, Wang $et~{}al$. predicted the
spectroscopy behavior of $\omega$, $\rho$, and $\phi$ mesons at the range of
$2.4\sim 3$ GeV Wang  _et al._ (2021). Feng $et~{}al$. analyze the masses and
two-body strong decay widths of ${}^{3}S_{1}$ vector mesons Feng  _et al._
(2021). The masses and strong decay widths of the higher excited $\rho$ mesons
are calculated in Ref. Feng  _et al._ (2022). All of the above mentioned works
refer to $\omega$ and $\omega_{3}$ and $\rho$ and $\rho_{3}$ mesons.

Not only are the lightest vector meson resonances, $\rho$, $\omega$, and
$\phi$ benchmark states in understanding of the quark substructure of hadrons
Johnson and Dudek (2021), but also they are important parts of the light meson
family. A systematic study for the $\omega$, $\rho$, and $\omega_{3}$,
$\rho_{3}$ meson family becomes very necessary and urgent. $\omega$ and
$\omega_{3}$ and $\rho$ and $\rho_{3}$ mesons have same quantum numbers ($P$
and $C$), for which reason the study of them can be borrowed from each other.
We estimate the mass of the missing states in these meson families. Similarly,
we study the strong decays of the $\omega$ and $\omega_{3}$ meson family
comparing with that of the $\rho$ and $\rho_{3}$ meson family.

In this work, we will study the excited states of $\omega$, $\rho$, and
$\omega_{3}$, $\rho_{3}$ mesons. By using the modified Godfrey-Isgur quark
(MGI) model and quark pair creation (QPC) model, the mass spectra and strong
decay behavior of the excited states of these mesons are analyzed, which
indicates that $X(2220)$ is the candidate of $\omega(3D)$ meson with
$I(J^{P})=0(1^{-})$. The spectra of the $\omega$, $\rho$ and $\omega_{3}$,
$\rho_{3}$ meson families are studied.

This paper is organized as follows. We first introduce the models applied in
this paper in section II and analyze the mass spectra and decay behaviors of
the $\omega,\rho$ and $\omega_{3},\rho_{3}$ mesons in section III. The paper
ends with a summary.

##  II Models employed in the work

The modified GI quark model, Regge trajectory, and the QPC model can be used
to calculate the mass spectra and the two-body strong decays of the meson
family, respectively. Before discussing the results, the above models will be
introduced briefly here.

###  II.1 Regge trajectory

It is efficient to investigate a light-meson spectrum by determining the Regge
trajectory Chew and Frautschi (1962); Anisovich  _et al._ (2000a). The
following relation is satisfied by the masses and radial quantum numbers of
the light mesons that belong to the same meson family:

$\displaystyle M^{2}=M_{0}^{2}+(n-1)\mu^{2},$ (2.1)

where $M_{0}$ represents the mass of the ground state, $n$ is the radial
quantum number of the corresponding meson with the mass $M$, and $\mu^{2}$ is
the trajectory slope. The Regge trajectories are a great method to help us
decide the approximate mass range of a particle and it lays a good foundation
for our next step. This method can approximately hold even if it is not as
strict as the Regge trajectory as a function of the particle spin, it can
still approximately hold since at most one or two excitations are known for
each quantum number. And we will calculate the mass spectrum using the MGI
model, which obviously takes into account spin breaking.

Figure 1: The $S$ and $D$ waves’ Regge trajectories of $\omega$, $\omega_{3}$
and $\rho$, $\rho_{3}$ families. Here $J$ is the angular momentum of the
meson. The open circle and the filled geometry are the theoretical and
experimental values, respectively.

###  II.2 The modified GI model

Godfrey and Isgur successfully proposed the GI model in 1985 for
characterizing relativistic meson spectra, particularly for low-lying mesons
Godfrey and Isgur (1985). In order to better describe the spectrum of the
highly excited states, Song $et~{}al$. Song  _et al._ (2015a, b) introduced
the color screening effect on the basis of the Cornell potential to modify the
GI model (MGI model).

The Hamiltonian of the potential model reads

$\tilde{H}=\sqrt{m_{1}^{2}+\mathbf{p}^{2}}+\sqrt{m_{2}^{2}+\mathbf{p}^{2}}+\tilde{V}_{\mathrm{eff}}(\mathbf{p,r}),$
(2.2)

which represents the internal interactions of mesons. $m_{1}$ and $m_{2}$ are
the mass of quark and antiquark, respectively, the effective potential has a
familiar format in the nonrelativistic limit before smeared Godfrey and Isgur
(1985); Lucha  _et al._ (1991)

$\displaystyle
V_{\mathrm{eff}}(r)=H^{\mathrm{conf}}+H^{\mathrm{hyp}}+H^{\mathrm{so}},$ (2.3)

with

$\displaystyle H^{\mathrm{conf}}$
$\displaystyle=\Big{[}-\frac{3}{4}(\frac{b(1-e^{-\mu
r})}{\mu}+c)+\frac{\alpha_{s}(r)}{r}\Big{]}(\bm{F}_{1}\cdot\bm{F}_{2})$
$\displaystyle=S(r)+G(r),$ (2.4) $\displaystyle H^{\mathrm{hyp}}$
$\displaystyle=-\frac{\alpha_{s}(r)}{m_{1}m_{2}}\Bigg{[}\frac{8\pi}{3}\bm{S}_{1}\cdot\bm{S}_{2}\delta^{3}(\bm{r})+\frac{1}{r^{3}}\Big{(}\frac{3\bm{S}_{1}\cdot\bm{r}\bm{S}_{2}\cdot\bm{r}}{r^{2}}$
$\displaystyle\quad-\bm{S}_{1}\cdot\bm{S}_{2}\Big{)}\Bigg{]}(\bm{F}_{1}\cdot\bm{F}_{2}),$
(2.5) $\displaystyle H^{\mathrm{so}}=$ $\displaystyle
H^{\mathrm{so(cm)}}+H^{\mathrm{so(tp)}},$ (2.6)

where $\bm{S}$ indicates the spin of quark or antiquark and $\bm{L}$ is the
orbital momentum. For a meson, $\langle\bm{F}_{1}\cdot\bm{F}_{2}\rangle=-4/3$,
the running coupling constant $\alpha_{s}(r)$ has the following form:

$\displaystyle\alpha_{s}(r)=\sum_{k=1}^{3}\frac{2\alpha_{k}}{\sqrt{\pi}}\int_{0}^{\gamma_{k}r}e^{-x^{2}}dx,$
(2.7)

where $\alpha_{1,2,3}=0.25,0.15,0.2$ and
$\gamma_{1.2.3}=\frac{1}{2},\frac{\sqrt{10}}{2},\frac{\sqrt{1000}}{2}$ Godfrey
and Isgur (1985). $H^{\mathrm{conf}}$ reflects the spin-independent
interaction, and it can divided into two parts: $S(r)$ and $G(r)$.
$H^{\mathrm{hyp}}$ represents the color-hyperfine interaction.
$H^{\mathrm{so}}$ is the spin-orbit interaction that contains the color
magnetic term and the Thomas precession term. $H^{\mathrm{so}}$ can be written
as

$\displaystyle
H^{\mathrm{so(cm)}}=\frac{-\alpha_{s}(r)}{r^{3}}\left(\frac{1}{m_{1}}+\frac{1}{m_{2}}\right)\left(\frac{\bm{S}_{1}}{m_{1}}+\frac{\bm{S}_{2}}{m_{2}}\right)\cdot\bm{L}(\bm{F}_{1}\cdot\bm{F}_{2}),$
(2.8) $\displaystyle H^{\mathrm{so(tp)}}=-\frac{1}{2r}\frac{\partial
H^{\mathrm{conf}}}{\partial
r}\Bigg{(}\frac{\bm{S}_{1}}{m^{2}_{1}}+\frac{\bm{S}_{2}}{m^{2}_{2}}\Bigg{)}\cdot\bm{L}.$
(2.9)

The parameters of the MGI model are defined in Table 1 Pang  _et al._ (2019).

Table 1: Parameters and their values in this work, which are given by Ref. Pang  _et al._ (2019).  Parameter | value Pang  _et al._ (2019) | Parameter | value Pang  _et al._ (2019)
---|---|---|---
$m_{u}$ (GeV) | 0.163 |  $\sigma_{0}$ (GeV) | 1.799
$m_{d}$ (GeV) | 0.163 |  $s$ (GeV) | 1.497
$m_{s}$ (GeV) | 0.387 |  $\mu$ (GeV) | 0.0635
$b$ (GeV2) | 0.221 |  $c$ (GeV) | -0.240
$\epsilon_{c}$ | -0.138 | $\epsilon_{sov}$ | 0.157
$\epsilon_{sos}$ | 0.9726 | $\epsilon_{t}$ | 0.893

Through diagonalizing and solving the Hamiltonian in Eq. (2.2) by exploiting a
simple harmonic oscillator (SHO) basis, the mass spectra and wave functions
can be obtained. The mass and wave function of the meson are applied to the
strong decay process.

###  II.3 A brief review of QPC mdoel

The QPC model is used to calculate the hadronic strong decays allowed by the
Okubo-Zweig-Iizuka(OZI) rule. This model was initially put up by Micu Micu
(1969) and further developed by the Orsay group Le Yaouanc  _et al._ (1973,
1974, 1975, 1977a, 1977b). The QPC model is widely applied to the OZI-allowed
two-body strong decays of mesons in Refs. van Beveren  _et al._ (1980, 1983);
Capstick and Roberts (1994); Page (1995); Titov  _et al._ (1996); Ackleh  _et
al._ (1996); Blundell (1996); Bonnaz  _et al._ (2002); Zhou  _et al._ (2005);
Lu  _et al._ (2006); Zhang  _et al._ (2007); Luo  _et al._ (2009); Sun and Liu
(2009); Liu  _et al._ (2010); Sun  _et al._ (2010); Rijken  _et al._ (2010);
Ye  _et al._ (2012); Wang  _et al._ (2012); He  _et al._ (2013); Sun  _et al._
(2013); Pang  _et al._ (2014); Wang  _et al._ (2015); Chen  _et al._ (2015);
Pang  _et al._ (2017, 2019). The transition operator $\mathcal{T}$ describes a
quark-antiquark pair (denoted by indices $3$ and $4$) creation from vacuum
that $J^{PC}=0^{++}$. For the process $A\to B+C$, $\mathcal{T}$ can be written
as Pang  _et al._ (2020)

$\displaystyle\mathcal{T}=$ $\displaystyle-3\gamma\sum_{m}\langle
1m;1~{}-m|00\rangle\int
d\mathbf{p}_{3}d\mathbf{p}_{4}\delta^{3}(\mathbf{p}_{3}+\mathbf{p}_{4})$
$\displaystyle~{}\times\mathcal{Y}_{1m}\left(\frac{\textbf{p}_{3}-\mathbf{p}_{4}}{2}\right)\chi_{1,-m}^{34}\phi_{0}^{34}\left(\omega_{0}^{34}\right)_{ij}b_{3i}^{{\dagger}}(\mathbf{p}_{3})d_{4j}^{{\dagger}}(\mathbf{p}_{4}),$
(2.10)

where $\mathcal{Y}_{\ell m}(\mathbf{p})={|\mathbf{p}|^{\ell}}Y_{\ell
m}(\mathbf{p})$ represents the solid harmonics. $\chi$, $\phi$, and $\omega$
denote the spin, flavor, and color wave functions respectively. Subindices $i$
and $j$ are the color index of a $q\bar{q}$ pair. The amplitude
${M}^{{M}_{J_{A}}M_{J_{B}}M_{J_{C}}}$ of the decay process is defined with the
transition operator $T$,

$\displaystyle\langle
BC|\mathcal{T}|A\rangle=\delta^{3}(\mathbf{P}_{B}+\mathbf{P}_{C})\mathcal{M}^{{M}_{J_{A}}M_{J_{B}}M_{J_{C}}},$
(2.11)

$\mathbf{P}$ is the three-momentum in the rest frame of a meson A. Finally,
the total width can be expressed in terms of the partial wave amplitude
squared,

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\frac{\pi}{4}\frac{|\mathbf{P}|}{m_{A}^{2}}\sum_{J,L}|\mathcal{M}^{JL}(\mathbf{P})|^{2},$
(2.12)

where $m_{A}$ is the mass of an initial state $A$, and the two decay
amplitudes can be related by the Jacob-Wick formula Jacob and Wick (1959) as

$\displaystyle\mathcal{M}^{JL}(\mathbf{P})=$
$\displaystyle\frac{\sqrt{4\pi(2L+1)}}{2J_{A}+1}\sum_{M_{J_{B}}M_{J_{C}}}\langle
L0;JM_{J_{A}}|J_{A}M_{J_{A}}\rangle$ (2.13) $\displaystyle\times\langle
J_{B}M_{J_{B}};J_{C}M_{J_{C}}|{J_{A}}M_{J_{A}}\rangle\mathcal{M}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}.$

##  III Numerical results and phenomenological analysis

We adopt the Regge trajectory and the MGI model to calculate the mass
spectrums of $\omega$ and $\omega_{3}$ and $\rho$ and $\rho_{3}$ mesons and
employ the QPC model with the meson wave functions obtained from the MGI model
to evaluate the meson’ decay widths. Following, we will give the numerical
results and phenomenological analysis.

###  III.1 Mass spectra analysis

We plot the Regge trajectory of $\omega$ and $\omega_{3}$, $\rho$ and
$\rho_{3}$ meson families in the Fig. 1. By analyzing the Regge trajectory,
the results demonstrate that $\omega(782)$, $\omega(1420)$, and $\omega(1960)$
are $1S$, $2S$, and $3S$ states, respectively. $\rho(770)$, $\rho(1450)$,
$\rho(1900)$, and $\rho(2150)$ are $1S$, $2S$, $3S$, and $4S$ states. However,
the situation of $\omega(4S)$ is not so distinct. $\omega(2330)$,
$\omega(2290)$, and $X(2240)$ can be the candidates of $\omega(4S)$ because of
their approximate masses. The widths of those states are different. The mass
of $\omega(2330)$ is larger than that of others, more details can be seen in
the Fig. 1 and the numerical result in Table 2. As for the $D$ wave, the
states $\rho(1700)$, $\rho(2000)$, and $\rho(2270)$ are on the ($n+I+J,M^{2}$)
planes. We assign $X(2220)$ and $\omega(2205)$ as the $3^{3}D_{1}$ states.
$\omega_{3}(1945)$ and $\omega_{3}(2285)$ are the excited states of
$\omega_{3}(1670)$. $\rho_{3}(1990)$ and $\rho_{3}(2250)$ are the $2D$ and
$3D$ states, which agrees well with the results in Ref. He  _et al._ (2013).
We can see the masses of $\omega(1650)$ and $\rho(1700)$, $\omega_{3}(1670)$
and $\rho_{3}(1670)$, $\omega_{3}(1945)$ and $\rho_{3}(1990)$ are very
similar.

Applying the MGI model and the parameters in Table 1, the mass spectra of $S$
wave and $D$ wave of $\omega$ and $\rho$ mesons as well as $\omega_{3}$ and
$\rho_{3}$ mesons are shown in the Table 2. Jia $et~{}al$. revisited the
orbital Regge spectra of the light unflavored mesons Jia  _et al._ (2017). And
as we reviewed in section I, Anisovich $et~{}al$. investigated the Regge
trajectory Anisovich  _et al._ (2005), Ebert $et~{}al$. gave the mass spectrum
of the $\omega$ and $\omega_{3}$ and $\rho$ and $\rho_{3}$ mesons Ebert  _et
al._ (2009). Our numerical results are compared with experimental masses and
their theoretical results in Table 2. As shown in Table 2, Ref. Anisovich  _et
al._ (2005) estimated the mass of $\omega_{3}(1D)$ to be 1671 MeV, which is
smaller than the mass obtained with the MGI model and Ref. Ebert  _et al._
(2009). The mass of $\omega(1^{3}S_{1})$ calculated in the MGI model is
smaller than that in Refs. Jia  _et al._ (2017); Anisovich  _et al._ (2005);
Ebert  _et al._ (2009). The mass spectrum of Ref. Anisovich  _et al._ (2005)
and MGI model all indicate that the newly observed state $X(2220)$ may be an
$\omega(3D)$ state. Furthermore, we find that the lower energy excited states
such as $1S$, $2S$ and $1D$, $2D$ are consistent with experimental values Zyla
_et al._ (2020); Bugg (2004a).

As for $\omega_{3}$ and $\rho_{3}$ meson families, there is too little
information to help us determine their structures. For the states of $D$ wave,
$\omega_{3}(1670)$ and $\rho_{3}(1690)$ are the $1D$ states which has been
certified by many researchers Buisseret and Semay (2005); Ebert  _et al._
(2009); Afonin (2007); Pang  _et al._ (2015). With respect to $2D$ states, we
predict the masses of $\omega(2D)$ and $\rho(2D)$ are both $2074$ MeV. At the
same time, $\rho(2000)$ is a $\rho(2D)$ state He  _et al._ (2013); Afonin
(2007), which is also close to our theoretical prediction. With regard to the
$3D$ states of $\omega_{3}$ and $\rho_{3}$, the value of the mass that we
calculated is $2376.5$ MeV. Moreover, we consider $\rho_{3}(2250)$ as the
$\rho_{3}(3D)$ state, which agrees well with the result in Ref. He  _et al._
(2013).

Besides, we have also calculated the mass spectra of $\omega_{3}$ and
$\rho_{3}$ of $G$ wave. The masses of $1G$, $2G$, and $3G$ are $2254.8$,
$2522.3$ MeV, and $2748.5$, respectively. Further discussion based on the
decay behaviors will be given in the next section. We hope that this
information can help find these resonances in experiment.

Table 2: The mass spectra of $\omega$ and $\omega_{3}$ and $\rho$ and
$\rho_{3}$ mesons. The unit is MeV.
$\begin{array}[]{cccccc}\hline\cr\hline\cr
n^{2s+1}L_{J}&\text{Jia}$\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Jia_2017}{\@@citephrase{(}}{\@@citephrase{)}}}$&\text{Anisovish}$\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{2005Quark}{\@@citephrase{(}}{\@@citephrase{)}}}$&\text{Ebert}$\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Ebert:2009ub}{\@@citephrase{(}}{\@@citephrase{)}}}$&\text{This~{}work}&\text{Experimental
mass}\\\ \omega(1^{3}S_{1})&{846}&\lx@intercol\hfil\hbox{\multirowsetup
778}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
776}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
773.8}\hfil\lx@intercol&\lx@intercol\hfil$
782.65$\pm$0.12~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\rho(1^{3}S_{1})&775&\lx@intercol\hfil\lx@intercol&\lx@intercol\hfil\lx@intercol&\lx@intercol\hfil\lx<EMAIL_ADDRESS>\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\omega(2^{3}S_{1})&{}\hfil&\lx@intercol\hfil\hbox{\multirowsetup
1473}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
1486}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
1424.3}\hfil\lx@intercol&\lx@intercol\hfil$1410$\pm$ 60
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\rho(2^{3}S_{1})&&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil$1465$\pm$25~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\omega(3^{3}S_{1})&{}\hfil&\lx@intercol\hfil\hbox{\multirowsetup
1763}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
1921}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
1906.1}\hfil\lx@intercol&\lx@intercol\hfil$1960$\pm$25
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\rho(3^{3}S_{1})&&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil$1909
$\pm$ 30.2 \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{aubert2008measurements}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\omega(4^{3}S_{1})&&\lx@intercol\hfil\hbox{\multirowsetup
2158}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
2195}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
2259.1}\hfil\lx@intercol&\lx@intercol\hfil$2330 $\pm$ 30
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$,$2290
$\pm$ 20 \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$,$2239.2
$\pm$ 7.1 $\pm$ 11.3 \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Ablikim:2018iyx}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\rho(4^{3}S_{1})&&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil
2201\pm$19 \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{PhysRevD.101.012011}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\omega(5^{3}S_{1})&&\lx@intercol\hfil\hbox{\multirowsetup
2363}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
2542.1}\hfil\lx@intercol&\lx@intercol\hfil--\hfil\lx@intercol\\\
\rho(5^{3}S_{1})&&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil--\hfil\lx@intercol\\\
\hline\cr\omega(1^{3}D_{1})&&\lx@intercol\hfil\hbox{\multirowsetup
1701}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
1557}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
1646.2}\hfil\lx@intercol&\lx@intercol\hfil$1670 $\pm$ 30
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\rho(1^{3}D_{1})&&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil$1720
$\pm$ 20 \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\omega(2^{3}D_{1})&&\lx@intercol\hfil\hbox{\multirowsetup
1992}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
1895}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
2047.6}\hfil\lx@intercol&\lx@intercol\hfil--\hfil\lx@intercol\\\
\rho(2^{3}D_{1})&&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil$2000
$\pm$ 30 \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{bugg2004four}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\omega(3^{3}D_{1})&&\lx@intercol\hfil\hbox{\multirowsetup
2212}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
2168}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
2365.0}\hfil\lx@intercol&\lx@intercol\hfil$2205 $\pm$ 30
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$,$2222
$\pm$ 7.3 \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{BESIII:2021uni}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\rho(3^{3}D_{1})&&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil$2265
$\pm$ 40 \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\omega(4^{3}D_{1})&&\lx@intercol\hfil\hbox{\multirowsetup
}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
2624.0}\hfil\lx@intercol&\lx@intercol\hfil--\hfil\lx@intercol\\\
\rho(4^{3}D_{1})&&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil--\hfil\lx@intercol\\\
\hline\cr\omega_{3}(1^{3}D_{3})&1661&\lx@intercol\hfil\hbox{\multirowsetup
1671}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
1714}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
1708.2}\hfil\lx@intercol&\lx@intercol\hfil$1667$\pm$4
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\rho_{3}(1^{3}D_{3})&1675&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx<EMAIL_ADDRESS>\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\omega_{3}(2^{3}D_{3})&&\lx@intercol\hfil\hbox{\multirowsetup
1987}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
2066}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
2074.0}\hfil\lx@intercol&\lx@intercol\hfil$1945$\pm$20
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\rho_{3}(2^{3}D_{3})&&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil$1982$\pm$14
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\omega_{3}(3^{3}D_{3})&&\lx@intercol\hfil\hbox{\multirowsetup
2376}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
2309}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
2374.0}\hfil\lx@intercol&\lx@intercol\hfil$2285$\pm$60
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\rho_{3}(3^{3}D_{3})&&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil$2232
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\omega_{3}(4^{3}D_{3})&&\lx@intercol\hfil\hbox{\multirowsetup
2705}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
2628.8}\hfil\lx@intercol&\lx@intercol\hfil--\hfil\lx@intercol\\\
\rho_{3}(4^{3}D_{3})&&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil--\hfil\lx@intercol\\\
\hline\cr\omega_{3}(1^{3}G_{3})&&\lx@intercol\hfil\hbox{\multirowsetup
2252}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
2002}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
2254.8}\hfil\lx@intercol&\lx@intercol\hfil$2255$\pm$15
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\rho_{3}(1^{3}G_{3})&&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil--\hfil\lx@intercol\\\
\omega_{3}(2^{3}G_{3})&&\lx@intercol\hfil\hbox{\multirowsetup
2482}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
2267}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
2522.3}\hfil\lx@intercol&\lx@intercol\hfil--\hfil\lx@intercol\\\
\rho_{3}(2^{3}G_{3})&&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil--\hfil\lx@intercol\\\
\omega_{3}(3^{3}G_{3})&&\lx@intercol\hfil\hbox{\multirowsetup
2746}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
}\hfil\lx@intercol&\lx@intercol\hfil\hbox{\multirowsetup
2748.5}\hfil\lx@intercol&\lx@intercol\hfil--\hfil\lx@intercol\\\
\rho_{3}(3^{3}G_{3})&&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil\hfil\lx@intercol&\lx@intercol\hfil--\hfil\lx@intercol\\\
\hline\cr\hline\cr\end{array}$

###  III.2 Decay modes and width

$\omega$ and $\omega_{3}$ and $\rho$ and $\rho_{3}$ mesons have the same
quantum numbers ($P$ and $C$), for which reason the study of them can be
learned from each other. We estimate the masses of the missing states in these
meson families. Similarly, we study the strong decays of $\omega$ and
$\omega_{3}$ meson family comparing with that of $\rho$ and $\rho_{3}$ meson
family. The $\gamma$ value in Eq. (II.3) is taken by the following method.
When fitting the $\rho$ and $\rho_{3}$ meson experimental width value (with
error) using the theoretical total width, the range of $\gamma$ value can be
fixed. Then we use this $\gamma$ value to calculate the width of $\omega$ and
$\omega_{3}$ meson. Some decay channels which less than 1 MeV are omitted.

####  III.2.1 $S$wave $\omega$ and $\rho$ mesons

Table 3: The total and partial decay widths of the $\omega(2S)$ and
$\rho(2S)$, the unit of width is MeV. The $\gamma$ value is 11.5$-$13.4.
$\begin{array}[]{ccccccc}\hline\cr\hline\cr\lx@intercol\hfil\omega(1420),~{}{\Gamma_{exp.}=880\pm
170}$~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Aulchenko:2015mwt}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol&\lx@intercol\hfil{\rho(1450)},~{}{\Gamma_{exp.}=400\pm
60}$~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Tanabashi:2018oca}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\hline\cr\text{Channel}&\text{Value}&\text{Ref.~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Barnes:1996ff}{\@@citephrase{(}}{\@@citephrase{)}}}}&\text{Channel}&\text{Value}&\text{Ref.~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Barnes:1996ff}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
\hline\cr\text{Total}&690\pm 105&{378}&\text{Total}&400&{279}\\\
\pi\rho&{608}&{328}&\pi\omega&{220}&{122}\\\
\eta\omega&{33.3}&{}\hfil&\eta\rho&{58.0}&{}\hfil\\\
KK&{28.3}&{31}&KK&{31.9}&{}\hfil\\\
{b_{1}}\pi&{12.1}&{1}&\pi\pi&{30.9}&{74}\\\
KK^{*}&{9.50}&{5}&KK^{*}&{30.3}&{}\hfil\\\
{}\hfil&&{}\hfil&{a_{1}}\pi&{14.9}&{3}\\\
{}\hfil&&{}\hfil&{h_{1}}\pi&{9.05}&{1}\\\ \hline\cr\hline\cr\end{array}$

By analyzing the above masses of $\omega$ and $\rho$ states, we know that
$\omega(1420)$, $\omega(1960)$, $\omega(2330)$, $\omega(2290)$, and $X(2240)$
are the radial excitations of $\omega(782)$. $\rho(1450)$, $\rho(1900)$, and
$\rho(2150)$ are the $\rho(2S)$, $\rho(3S)$, and $\rho(4S)$ states
respectively. We will then discuss these states.

From Table 3, $\omega(1420)$ dominantly decay into $\pi\rho$ which is
consistent with experiment Tanabashi  _et al._ (2018) and Ref. Barnes  _et
al._ (1997). $\eta\omega$, $KK$, ${b_{1}}\pi$ and $KK^{*}$ are the important
decay modes in which ${b_{1}}\pi$ was observed in experiment Tanabashi  _et
al._ (2018). The total width of $\omega(1420)$ in our calculation has a
overlap with the experiment value Aul’chenko  _et al._ (2015). $\omega(1420)$
is a good candidate of $\omega(2S)$ which is consistent with Ref. Wang  _et
al._ (2012).

Table 4: The total and partial decay width of the $\omega(3S)$ and $\rho(3S)$,
the unit of width is MeV. The $\gamma$ value is 6.3$-$7.9.
$\begin{array}[]{cccc}\hline\cr\hline\cr\lx@intercol\hfil\omega(1960),~{}{\Gamma_{exp.}=195\pm
60}$~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Anisovich:2011sva}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol&\lx@intercol\hfil\rho(1900),~{}{\Gamma_{exp.}=130\pm
30}$~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Tanabashi:2018oca}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\hline\cr\text{Channel}&\text{Value}&\text{Channel}&\text{Value}\\\
\hline\cr\text{Total}&{191\pm 42.5}&\text{Total}&130\\\
\rho\pi&{108}&{a_{2}}\pi&{31.1}\\\
{b_{1}}\pi&{32.0}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>K{K_{1}}&{14.8}&K{K_{1}}&{18.7}\\\ \eta\omega&{9.90}&\pi\pi(1300)&{16.1}\\\
KK^{*}(1410)&{9.40}&\pi\pi&{13.9}\\\ {h_{1}}\eta&{4.74}&{a_{1}}\pi&{13.0}\\\
KK^{*}&{3.77}&KK&{2.60}\\\ KK&{3.14}&{b_{1}}\eta&{2.55}\\\
K{K_{1}^{\prime}}&{2.35}&KK^{*}&{2.46}\\\ \hline\cr\hline\cr\end{array}$ Table
5: The total and partial decay widths of the $\omega(4S)$ and $\rho(4S)$, the
unit of width is MeV. The $\gamma$ value is 6.9$-$11.3.
$\begin{array}[]{cccccc}\hline\cr\hline\cr&{\omega(2330)}&{\omega(2290)}&{X(2240)}&\lx@intercol\hfil\rho(2150)\hfil\lx@intercol\\\
\hline\cr\text{Channel}&\text{Value}&\text{Value}&\text{Value}&\text{Channel}&\text{Value}\\\
\hline\cr\Gamma_{exp.}&$435$\pm$75~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$&$275$\pm$35~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$&$139.8$\pm$23.99~{}
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Ablikim:2018iyx}{\@@citephrase{(}}{\@@citephrase{)}}}$&\Gamma_{exp.}&$310$\pm$140~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$\\\
\text{Total}&{372\pm 170}&346\pm 158&306\pm 140&\text{Total}&{310}\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi\rho(1450)}&{134}&122&95.8&\pi\omega_{3}&{58.6}\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi\rho}&{62.6}&68.0&69.0&{a_{2}}\pi(1700)&{40.7}\\\
{b_{1}}\pi&{61.6}&54.4&42.4&\pi\pi(1300)&{32.1}\\\
{a_{1}}\rho&{37.2}&39.0&37.3&\pi\omega&{31.5}\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>K{K_{1}}&{8.39}&8.33&6.69&h_{1}\pi&{14.7}\\\
f_{1}\omega&{7.84}&6.28&4.28&a_{1}\pi&{13.6}\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>K^{*}K^{*}&{5.91}&5.65&4.88&\pi\pi&{11.4}\\\
\eta\omega&{4.81}&3.28&1.70&a_{1}\omega&{10.5}\\\
KK(1460)&{4.81}&3.45&2.00&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>{a_{2}}\rho&{2.65}&11.5&19.0&\rho\rho&{7.74}\\\
KK&{0.599}&0.503&0.379&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>KK_{1}^{\prime}&{0.421}&0.224&0.0783&f_{2}\rho&{6.19}\\\
K^{*}{K_{1}}&0.164&0.152&0.0761&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>KK^{*}(1430)&0.121&0.0366&0.00138&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>{f_{2}}\omega&{0.0228}&1.39&4.97&a_{2}\omega&{5.34}\\\ &&&&f_{1}\rho&{4.99}\\\
&&&&K^{*}K^{*}&{4.20}\\\
&&&&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}KK^{*}(1410)}&{2.92}\\\
&&&&b_{1}\eta&{1.74}\\\ &&&&KK(1460)&{1.20}\\\
&&&&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>&&&&\eta\rho(1450)&0.903\\\ \hline\cr\hline\cr\end{array}$

$\omega(1960)$ is observed in the $p\bar{p}\to\omega\eta$, and $\omega\pi\pi$
process Anisovich  _et al._ (2002a). As shown in Table 4, $\omega(1960)$
dominantly decay into $\rho\pi$ under $\omega(3S)$ assignment. The decay modes
${b_{1}}\pi$, $KK_{1}$ and $\eta\omega$ make some contribution to the total
width, and ${b_{1}}\pi$ can decay to $\omega\pi\pi$ which is the final channel
observed in experiment Anisovich  _et al._ (2002a). According to our
calculation, the total width of $\omega(1960)$ is $191.1\pm 42.5$ MeV and this
is consistent with the experimental data Anisovich  _et al._ (2002a). In
addition, $\eta\omega$ is a sizable final channel and has been observed in
experiment Anisovich  _et al._ (2002a). Other detailed information is
demonstrated in Table 4. Our calculation indicates that $\omega(1960)$ can be
assigned as $\omega(3S)$ state, which is consistent with Ref. Feng  _et al._
(2021).

$X(2240)$ is observed in the $e^{+}e^{-}\to K^{+}K^{-}$ process by the BESIII
Collaboration, which has the mass of $2239.2\pm 7.1\pm 11.3\,\rm{MeV}$ and the
width of 139.8$\pm$12.3$\pm$20.6 MeV Ablikim  _et al._ (2019b). The quantum
number of this resonant structure can be assigned as $J^{PC}=1^{--}$. Assuming
that it is a isospin scalar state, it may be a $\omega(4S)$ candidate from our
previous mass analysis. The state $X(2240)$ needs more theoretical and
experimental research to recover its structure. $\omega(2330)$ is firstly
observed in the $\gamma{p}\to\rho^{\pm}\rho^{0}\pi^{\mp}$ process Valencia
(2006). It has the mass of $2330\pm 30$ MeV and the width of $435\pm 75$ MeV.
$\omega(2290)$ is reported in the partial wave analysis of the data on
$p\bar{p}\to\bar{\Lambda}\Lambda$ Bugg (2004b), which has the mass of $2290\pm
20$ MeV and the width of $275\pm 35$ MeV. $\omega(2330)$, $\omega(2290)$ and
$X(2240)$ are the candidates of the $\omega(4S)$ according to our calculation
and the obtained widths are in good agreement with experimental values.
$\pi\rho(1450)$ is the primary decay mode of $\omega(2330)$, $X(2240)$ and
$\omega(2290)$ if assign them as $\omega(4S)$ states. Besides, $\pi\rho$,
$b_{1}\pi$ and $a_{1}\rho$ make needful contribution to the total width of
$\omega(4S)$, other decay modes contribute less to the total width as shown in
Table 5.

As for $S$ wave states of $\rho$ family, we assign $\rho(1450)$, $\rho(1900)$
and $\rho(2150)$ as $\rho(2S)$, $\rho(3S)$ and $\rho(4S)$ states, which
consists with the results of Ref. Ebert  _et al._ (2009); Afonin (2007); Wang
_et al._ (2021); He  _et al._ (2013). When $\rho(1450)$ is treated as the
$\rho(2S)$, $\pi\omega$ and $\eta\rho$ are predicted to be its important decay
channels. $KK$, $\pi\pi$ and $KK^{*}$ are the sizable decay modes as well.
$\rho(1900)$ is a good candidate of $\rho(3S)$. As shown in the second column
of Table 4, the decay modes of $\rho(1900)$ are predicted. $a_{2}\pi$,
$\pi\omega$ and $KK_{1}$ are its important decay modes. $\pi\pi(1300)$,
$\pi\pi$ and $a_{1}\pi$ are the sizable final states, which is consistent with
Ref. Wang  _et al._ (2020). We assign $\rho(2150)$ as $\rho(4S)$ based on our
calculation. The channels $\pi\omega_{3}$, $a_{2}(1700)\pi$, $\pi\pi(1300)$
and $\pi\omega$ are the main channels. Otherwise, $a_{2}\pi$, $h_{1}\pi$,
$a_{1}\pi$ and ${b_{1}}\rho$ have sizable contributions to the total width of
$\rho(4S)$.

####  III.2.2 D-wave $\omega$ and $\rho$ mesons

In this section, we will give an analysis for the D-wave $\omega$ and $\rho$
mesons by the means of their two-body strong decay behaviors.

Table 6: The partial decay widths of the $\omega(1D)$, $\rho(1D)$, the unit of
width is MeV. The $\gamma$ value is 4.14$-$6.3.
$\begin{array}[]{cccccc}\hline\cr\hline\cr\lx@intercol\hfil\omega(1650),~{}{\Gamma_{exp.}=315\pm
35}$~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Tanabashi:2018oca}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol&\lx@intercol\hfil\rho(1700),~{}{\Gamma_{exp.}=250\pm
100}$\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Tanabashi:2018oca}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\hline\cr\text{Channel}&\text{Value}&\text{Ref.~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Barnes:1996ff}{\@@citephrase{(}}{\@@citephrase{)}}}}&\text{Channel}&\text{Value}&\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Barnes:1996ff}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\
\hline\cr\text{Total }&284\pm 113&{542}&\text{Total}&{250}&{435}\\\
{b_{1}}\pi&{272}&{371}&{a_{1}}\pi&{93.9}&{134}\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>KK&{9.00}&{35}&\rho\rho&{26.4}&{14}\\\
\eta\omega&{7.84}&{}\hfil&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>KK^{*}&{5.29}&{21}&KK&{9.84}&{36}\\\
{}\hfil&{}\hfil&{}\hfil&\eta\rho&{8.76}&{}\hfil\\\
{}\hfil&{}\hfil&{}\hfil&KK^{*}&{6.69}&{}\hfil\\\
{}\hfil&{}\hfil&{}\hfil&{a_{2}}\pi&{2.64}&{}\hfil\\\
\hline\cr\hline\cr\end{array}$

$\omega(1650)$ is well established as a $1D$ state in the $\omega$ family in
theory Wang  _et al._ (2012); Tanabashi  _et al._ (2018); Pang  _et al._
(2020), and its dominant decay channel is $b_{1}\pi$, whose branching ratio is
about 0.8. $\pi\rho$, $KK$, $\eta\omega$ and $KK^{*}$ are the important modes.
$\rho(1700)$ is a good candidate of the $1^{3}D_{1}$ $\rho$ meson. The
analysis of $\rho(1700)\to 2\pi,4\pi$ Collaboration and Al. (2001) and the
study of $e^{+}e^{-}\to\pi\omega$ via the nonrelativistic ${}^{3}P_{0}$ quark
model Kittimanapun  _et al._ (2009) both indicate that $\rho(1700)$ is a
$1^{3}D_{1}$ state. $\rho(1700)$ mainly decays into $a_{1}\pi$, $h_{1}\pi$ and
$\rho\rho$ with branching ratios being about 0.38, 0.34 and 0.11,
respectively.

Table 7: The total and partial decay widths of the $\omega(2D)$ and
$\rho(2D)$, the unit of width is MeV. The $\gamma$ value is 5.4$-$6.4.
$\begin{array}[]{cccc}\hline\cr\hline\cr\lx@intercol\hfil\omega(2D)\hfil\lx@intercol&\lx@intercol\hfil\rho(2000),~{}{\Gamma=260\pm
45}$~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Tanabashi:2018oca}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\hline\cr\text{Channel}&\text{ Value}&\text{Channel}&\text{ Value}\\\
\hline\cr\text{Total}&{212\pm 36.1}&\text{Total}&{260}\\\
{b_{1}}\pi&{110}&{a_{1}(1640)}\pi&58.4\\\
\pi\rho(1450)&{24.7}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\pi\rho&{22.7}&\rho\rho&{35.2}\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{h_{1}}\eta}&{13.2}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>KK^{*}(1410)&{4.29}&\pi\pi&{21.9}\\\
\eta\omega(1420)&{3.26}&\pi\pi(1300)&{19.8}\\\
\eta\omega&{2.60}&\pi\omega(1420)&{8.22}\\\ KK&{2.18}&\pi\omega&{6.39}\\\
KK_{1}&1.83&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>KK^{*}&1.15&KK^{*}(1410)&2.54\\\ &&KK&{1.81}\\\ &&\eta\rho&1.70\\\
\hline\cr\hline\cr\end{array}$

As is shown in Fig. 1 and Table 2, the mass of $\omega(2D)$ is predicted,
which is still unobserved. $\omega(2D)$ has the total width of about $212\pm
36.1$ MeV. $b_{1}\pi$ is its dominant decay channel, whose branching ratio is
0.5. $\pi\rho(1450)$, $\pi\rho$ and ${a_{1}}\rho$ are its important decay
channels and more details can be seen in Table 7. We suggest that
experimentalists focus on these final channels. In 1994, a resonance around
1988 MeV was found in $\bar{p}p\to\pi\pi$ from 0.36 to 2.5 GeV reaction Hasan
and Bugg (1994). Later in 2011, Anisovich $et~{}al$. found a resonance that
$J^{PC}=1^{--}$ and its mass is of $2000\pm 30$ MeV  Anisovich  _et al._
(2000b). $\rho(2000)$ is treated as $\rho(2D)$ state  He  _et al._ (2013);
Wang  _et al._ (2020). $a_{1}(1640)\pi$, ${h_{1}}\pi$, $\rho\rho$, and
$\pi\pi_{2}(1670)$ have sizable contributions to the total width of
$\rho(2D)$.

Table 8: The partial decay widths of the $\omega(3D)$, $\rho(3D)$,
$\Gamma_{e}xp.$ means the experimental total width, the unit of width is MeV.
The $\gamma$ value is 11.3$-$14.6.
$\begin{array}[]{ccccc}\hline\cr\hline\cr&\omega(2205)&X(2220)&\lx@intercol\hfil\rho(2270)\hfil\lx@intercol\\\
\hline\cr\text{Channel}&\text{Value}&\text{ Value}&\text{Channel
}&\text{Value}\\\
\hline\cr\Gamma_{exp.}&$350$\pm$90~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$&$59$\pm$30.6~{}
\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{BESIII:2021uni}{\@@citephrase{(}}{\@@citephrase{)}}}$&\Gamma_{exp.}&$325$\pm$80~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Anisovich:2011sva}{\@@citephrase{(}}{\@@citephrase{)}}}$\\\
\text{Total}&147\pm 36.1&164\pm 40.4&\text{Total}&325\\\
{b_{1}}\pi&{56.7}&66.0&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>{\pi}\rho(1450)&{21.0}&23.7&\pi\pi(1300)&64.3\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{a_{1}}\rho}&{23.7}&22.0&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi\rho}&{8.32}&9.78&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\pi\rho_{3}&{6.09}&6.97&\pi\pi_{2}&23.2\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{f_{1}}\omega}&{5.24}&5.54&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>KK_{1}&{4.78}&4.58&{a_{1}}\omega&5.81\\\
\pi\rho(1700)&{2.65}&2.90&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi(1300)\rho}&{1.34}&0.495&KK_{1}&4.00\\\
&&&\rho\rho(1450)&4.00\\\ &&&\pi\omega_{3}&{3.41}\\\
&&&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>&&&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi\omega(1650)}&{1.5}\\\
\hline\cr\hline\cr\end{array}$

$X(2220)$ state is found in the $e^{+}e^{-}\to\omega\pi^{0}\pi^{0}$ process
Ablikim  _et al._ (2022), which has the mass of $M=2222\pm 7\pm 2$ MeV and the
width of $\Gamma=59\pm 30\pm 6$ MeV. $X(2220)$ is an excellent candidate of
$\omega(3D)$ by Regge trajectory analysis and according to Table 8. When
$X(2220)$ is treated as the $\omega(3D)$ state, its width is close to the
experiment value. $\omega(2205)$ is also assigned as $\omega(3D)$ state based
on mass spectra analysis. $b_{1}\pi$ is the main decay mode of $\omega(3D)$.
$\pi\rho(1450)$ and ${a_{1}}\rho$ are the important decay channels of
$\omega(3D)$, other modes like ${a_{2}}\rho$, $\pi\rho$ and $\pi\rho_{3}$ and
so on also have contribution to the total width of $\omega(3D)$. We assign
$\rho(2270)$ as $\rho(3D)$ state, which is consistent with Ref. He  _et al._
(2013). $\rho(2270)$ primarily decays into ${a_{2}}\omega$, $\pi\pi(1300)$ and
${h_{1}}\pi$. More details can be seen in Table 8.

####  III.2.3 D-wave $\omega_{3}$ and $\rho_{3}$ mesons

Referring to Particle Data Group (PDG), we can find some experimental results
about $\omega_{3}$ and $\rho_{3}$, which are $\rho_{3}(1690)$,
$\rho_{3}(1990)$, $\rho_{3}(2250)$, $\omega_{3}(1670)$, $\omega_{3}(1945)$,
$\omega_{3}(2285)$, and $\omega_{3}(2255)$. We carry out an investigation of
the Okubo-Zweig-Iizuka allowed two-body strong decay of the $D$-wave and
$G$-wave $\omega_{3}$ and $\rho_{3}$ mesons.

Table 9: The partial decay widths of the $\omega_{3}(1D)$, $\rho_{3}(1D)$, the
unit of width is MeV. The $\gamma$ value is 7.5$-$9.2.
$\begin{array}[]{cccc}\hline\cr\hline\cr\lx@intercol\hfil\omega_{3}(1670),~{}{\Gamma_{exp.}=90\pm
20}~{}$\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{1969Evidence}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol&\lx@intercol\hfil\rho_{3}(1690),~{}{\Gamma_{exp.}=190\pm
40}~{}$\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\hline\cr\text{Channel}&\text{Value}&\text{Channel}&\text{Value}\\\
\hline\cr\text{Total }&{81.8\pm 16.5}&\text{Total}&{190}\\\
\pi\rho&{62.8}&\rho\rho&{105}\\\ b_{1}\pi&{14.7}&\pi\pi&{32.0}\\\
\eta\omega&{2.76}&\pi\omega&{23.1}\\\ KK&{1.16}&\pi{h_{1}}&{10.6}\\\
&&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>&&\eta\rho&{3.68}\\\
&{}\hfil&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>&{}\hfil&KK&{2.66}\\\ \hline\cr\hline\cr\end{array}$

There exists a lot of data for $\omega_{3}(1670)$ from PDG. Amelin $et~{}al$.
first found it in the $\pi^{-}p\to\pi^{+}\pi^{-}\pi^{0}n$ reactions, which has
the mass of $1665.3\pm 5.2\pm 4.5$ MeV Amelin  _et al._ (1996). And then
Baltay $et~{}al$. reported the existence of a high-mass, $I=0$ resonant state
with odd $G$ parity, whose mass and width are $M=1.63\pm 0.012$ GeV and
$\Gamma=0.173\pm 0.016$ GeV  Baltay  _et al._ (1978), respectively. At
present, $\omega_{3}(1670)$ is well established to be a $1^{3}D_{3}$ state
Buisseret and Semay (2005). $\pi\rho$ is the dominant decay modes of
$\omega_{3}(1D)$, which has the branching ratios of 0.77. Additionally,
$b_{1}\pi$, $\eta\omega$, and $KK$ also make contribution to the total width
of $\omega_{3}(1D)$.  $\rho_{3}(1690)$ is well established to be a
$1^{3}D_{3}$ state, which decays into $\rho\rho$, $\pi\pi$, $\pi\omega$, and
$\pi{h_{1}}$ as shown in Table 9 and the branching ratios are about 0.55,
0.17, 0.12, and 0.056, respectively. These results are in good agreement with
that in Ref. He  _et al._ (2013).

$\omega_{3}(1945)$ was firstly observed in the
$p\bar{p}\to\omega\eta,\omega\pi^{0}\pi^{0}$ Anisovich  _et al._ (2002b) . It
has the mass of $1945\pm 20$ MeV and the width of $115\pm 22$ MeV. We assign
it as $\omega_{3}(2D)$ state and calculated the two-body strong decay as shown
in Table 10. We see from Table 10 that $b_{1}\pi$, $\pi\rho$, $\pi\rho(1450)$
and $\pi\rho_{3}$ are major decay modes of $\omega_{3}(2D)$. $\eta\omega$,
$h_{1}\eta$, $K^{*}K^{*}$ and $KK^{*}$ are important channels. When treated as
$\rho_{3}(2D)$, $\rho_{3}(1990)$ mainly decay into $\pi\pi(1300)$,
${a_{2}}\pi$, $\rho\rho$, which branching ratios are 0.22, 0.18 and 0.14,
respectively and this result was certified in Ref. He  _et al._ (2013).
Besides, $\rho_{3}(1990)\to\pi{a_{1}},\pi{h_{1}},\pi\omega(1420)$ are
sizeable. More details can be seen in Table 9.

As for $\omega_{3}(3D)$, we consider $\omega_{3}(2285)$ is the best candidate
according to the mass spectra analysis and we give its decay modes and values
in Table 11. The total width of $\omega_{3}(2285)$ is $226\pm 126$ MeV if we
treat $\omega_{3}(2285)$ as a $\omega_{3}(3D)$ state, which is in great
agreement with experimental width Bugg (2004b). $a_{2}\rho$, $\pi\rho$,
$f_{2}\omega$ and $b_{1}\pi$ make great contribution to its total width and
$\pi\rho_{3}$, $\pi(1300)\rho$, $\pi\rho(1450)$ and so forth are also
important decay channels. Meanwhile, $\rho_{3}(2250)$ is established to be a
$\rho_{3}(3D)$ state, which can decay into ${a_{2}}\omega$, ${f_{2}}\rho$,
$\pi\omega$, $\rho\rho$ as shown in Table 11.

$\omega_{3}(4D)$ and $\rho_{3}(4D)$ are still missing in experiment. We
calculated the width of $\omega_{3}(4D)$ is about 290 MeV and $\rho_{3}(4D)$
is approximately 420 MeV. According to Table 12, the main decay modes of
$\omega_{3}(4D)$ are $a_{2}\rho$, $\pi(1300)\rho$, $b_{1}\pi$ and $\pi\rho$.
$\rho_{3}(4D)$ mainly decay into ${a_{4}(2040)\pi}$, ${a_{2}(1700)\omega}$,
${a_{2}(1700)\pi}$ and so on. These predictions can help us search for
$\omega_{3}(4D)$ and $\rho_{3}(4D)$ states.

Table 10: The partial decay widths of the $\omega_{3}(2D)$, $\rho_{3}(2D)$,
the unit of width is MeV. The $\gamma$ value is 10.4$-$11.8.
$\begin{array}[]{cccc}\hline\cr\hline\cr\lx@intercol\hfil\omega_{3}(1945),~{}{\Gamma_{exp.}=115\pm
22}~{}$\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol&\lx@intercol\hfil\rho_{3}(1990),~{}{\Gamma_{exp.}=188\pm
24}~{}$\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Anisovich:2011sva}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\hline\cr\text{Channel}&\text{Value }&\text{Channel}&\text{Value}\\\
\hline\cr\text{Total }&{105\pm 13.2}&\text{Total}&{188}\\\
{b_{1}}\pi&{38.5}&\pi\pi(1300)&{40.6}\\\
\pi\rho&{22.1}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\pi\rho(1450)&{18.7}&\rho\rho&{26.9}\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi\rho_{3}}&{11.5}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\eta\omega&{4.51}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>h_{1}\eta&{4.44}&\pi\omega(1420)&{15.5}\\\
K^{*}K^{*}&{2.31}&\pi\pi_{2}&{9.19}\\\ KK^{*}&{1.28}&\pi\omega_{3}&{7.80}\\\
&{}\hfil&\pi\omega&{7.14}\\\ &{}\hfil&\eta\rho&{4.58}\\\
&{}\hfil&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>&{}\hfil&K^{*}K^{*}&{2.54}\\\ &{}\hfil&KK^{*}&{1.61}\\\
\hline\cr\hline\cr\end{array}$ Table 11: The partial decay widths of the
$\omega_{3}(3D)$, $\rho_{3}(3D)$, the unit of width is MeV. The $\gamma$ value
is 6.7$-$12.5.
$\begin{array}[]{cc|cc}\hline\cr\hline\cr\lx@intercol\hfil\omega_{3}(2285),~{}{\Gamma_{exp.}=224\pm
50}~{}$\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Bugg:2004rj}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol&\lx@intercol\hfil\rho_{3}(2250),~{}{\Gamma_{exp.}=135\pm
75}~{}$\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{1977Antiproton}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol\\\
\hline\cr\text{Channel}&\text{Value }&\text{Channel}&\text{Value}\\\
\hline\cr\text{Total }&{226\pm 126}&\text{Total}&{135}\\\
a_{2}\rho&{76.3}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\pi\rho&{32.6}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>f_{2}\omega&{25.6}&\pi\omega&{12.3}\\\ b_{1}\pi&{24.5}&\rho\rho&{11.6}\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi\rho_{3}}&{12.9}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\pi(1300)\rho&{10.8}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\pi\rho(1450)&{5.22}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>a_{1}\rho&{4.37}&\pi\omega_{3}&{4.32}\\\ \eta\omega&{3.78}&\pi\pi&{3.68}\\\
\eta\omega(1420)&{3.68}&\pi\pi_{2}&{2.95}\\\
h_{1}\eta&{3.28}&\pi\omega(1420)&{2.70}\\\
\eta(1295)\omega&2.14&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>K^{*}K^{*}&{1.45}&KK^{*}(1410)&{2.06}\\\
&&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>&&\eta\rho&{1.82}\\\
&&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\eta(1295)\rho}&{1.81}\\\
&&\eta\rho(1450)&{1.70}\\\ &&K^{*}K^{*}&{1.69}\\\
\hline\cr\hline\cr\end{array}$ Table 12: The partial decay widths of the
$\omega_{3}(4D)$, $\rho_{3}(4D)$, the unit of width is MeV. The $\gamma$ value
is 11.6.
$\begin{array}[]{cccc}\hline\cr\hline\cr\lx@intercol\hfil\omega_{3}(2630)\hfil\lx@intercol&\lx@intercol\hfil\rho_{3}(2630)\hfil\lx@intercol\\\
\hline\cr\text{Channel}&\text{Value }&\text{Channel}&\text{Value}\\\
\hline\cr\text{Total }&{288}&\text{Total}&{420}\\\
a_{2}\rho&57.2&a_{4}(2040)\pi&184\\\
\pi(1300)\rho&40.3&a_{2}(1700)\omega&38.7\\\
b_{1}\pi&37.3&a_{2}(1700)\pi&35.4\\\ \pi\rho&33.3&a_{2}\pi&25.3\\\
\pi\rho(1450)&32.8&\rho\rho(1450)&22.4\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>f_{2}\omega&17.2&h_{1}\pi&14.3\\\ \eta(1295)\omega&8.03&b_{1}\rho&11.8\\\
\pi_{2}\rho&6.85&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi\omega(1420)}&10.2\\\
\eta\omega(1420)&6.42&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>K^{*}K^{*}(1410)&5.65&\rho\rho&6.76\\\
h_{1}\eta&5.30&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\eta\omega&3.46&K^{*}K^{*}(1410)&5.65\\\
K^{*}K_{2}^{*}&2.36&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\eta\rho(1450)}&5.51\\\
f_{2}h_{1}&1.94&\pi\pi&4.79\\\ \eta_{2}\omega&1.86&b_{1}\eta&4.01\\\
f_{1}\omega&1.66&a_{1}\omega&3.27\\\
a_{2}{b_{1}}&1.50&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>f_{1}h_{1}&1.09&\pi_{2}\omega&2.14\\\ \pi\rho(1700)&1.01&h_{1}{a_{1}}&1.94\\\
&&b_{1}{b_{1}}&1.83\\\
&&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}KK^{*}(1410)}&1.71\\\
&&a_{2}h_{1}&1.51\\\
&&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\eta^{\prime}\rho(1450)}&1.39\\\
&&\pi\pi(1300)&1.21\\\ &&h_{1}\pi(1300)&1.04\\\ \hline\cr\hline\cr\end{array}$

####  III.2.4 G-wave $\rho_{3}$ and $\omega_{3}$ mesons

$\omega_{3}(2285)$ is regarded as $\omega_{3}(1G)$ in Ref. Afonin (2007). The
reference Anisovich  _et al._ (2002a) observed $\omega_{3}(2255)$ and assigned
it as $\omega_{3}(1G)$ state. According to the mass spectra analysis and two-
body decay modes, we agree that $\omega_{3}(2255)$ may be $\omega_{3}(1G)$
state. According to Table 13, $b_{1}\pi$, $\pi\rho_{3}$ and $h_{1}\eta$ are
predicted to be $\omega_{3}(1G)$’s important decay channels, which have the
ratios of 0.45, 0.14 and 0.1, respectively. We suggest that experimentalists
focus on these final channels. The total width of $\omega_{3}(1G)$ is 325 MeV.
The total width of $\rho_{3}(1G)$ is approximately 721 MeV. $\pi\pi_{2}$,
$b_{1}\rho$, $\rho\rho$ and $a_{1}\pi$ are its important decay channels.
Considering the final decay channels, $\pi\pi_{2}$ will be the most important
final channels in searching for $\rho_{3}(1G)$ state experimentally. More
details can be seen in Tale 13. These predictions can help us search for and
establish $\omega_{3}(1G)$ and $\rho_{3}(1G)$ state.

$\omega_{3}(2G)$ and $\rho_{3}(2G)$ and $\omega_{3}(3G)$ and $\rho_{3}(3G)$
are still missing, We make predictions about their widths and decay modes. The
decay information of the $\omega_{3}(2G)$ and $\rho_{3}(2G)$ is listed in
Table 14. As shown in the first column of Table 14, the strong decay of
$\omega_{3}(2G)$ is predicted, which is still unobserved. $\omega_{3}(2G)$ has
the total width of 312 MeV. $b_{1}\pi$ is its dominant decay channel, which
ratios is 0.34. $a_{2}\rho$, $\pi_{2}\rho$, $\pi\rho_{3}$, $h_{1}\eta$ and
$\pi\rho$ are the important final states. $\eta^{\prime}\omega$, $a_{1}\rho$
and $h_{1}\eta^{\prime}$ are small. The total width of $\rho_{3}(2G)$ is 397
MeV. $\rho_{3}(2G)\to\pi\pi_{2}$ will be the dominant decay mode. In the
calculation, $h_{1}\pi$, $a_{1}\pi$ and $\rho\rho(1450)$ are the important
decay channels. Other decay information can be seen in Table 14.

The decay information of the $\omega_{3}(3G)$ and $\rho_{3}(3G)$ are also
predicted in this work in Table 15. The total width of $\omega_{3}(3G)$ is
approximately 181 MeV. The channels $b_{1}\pi$, $\pi\rho$, $\pi\rho_{3}$ and
$f_{1}\omega$ have the branching ratios of 0.35, 0.07, 0.07 and 0.06,
respectively, which are the main decay modes. $\pi_{2}\rho$, $h_{1}\eta$,
$b_{1}\pi(1300)$ and $f_{2}\omega$ are its important decay channels. This work
suggests that experimentlist should search for this missing state in
$b_{1}\pi$ final state. Otherwise, $\pi(1300)\rho$, $a_{2}\rho$,
$\pi\rho(1450)$, $a_{0}\rho$ and $h_{1}\eta(1295)$ have sizeable contributions
to the total width of $\omega_{3}(3G)$. $\rho_{3}(3G)$ has the total width of
about 270 MeV from our calculation. The main decay modes are $\pi\pi_{2}$,
$\rho\rho$, $h_{1}\pi$, $a_{1}\omega$ and $a_{1}\pi$, which branching ratios
are 0.17, 0.090, 0.089, 0.086 and 0.078. $a_{2}\pi$, $a_{2}(1700)\pi$,
$f_{2}\rho$, $\eta(1295)\rho$ and $\rho\rho(1450)$ are the important decay
modes of $\rho_{3}(3G)$.

Table 13: The partial decay widths of the $\omega_{3}(1G)$, $\rho_{3}(1G)$,
the unit of width is MeV. The $\gamma$ value is 11.6.
$\begin{array}[]{cccc}\hline\cr\hline\cr\lx@intercol\hfil\omega_{3}(2255),\Gamma_{exp.}=175\pm$30~{}\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Zyla:2020zbs}{\@@citephrase{(}}{\@@citephrase{)}}}$\hfil\lx@intercol&\lx@intercol\hfil\rho_{3}(2255)\hfil\lx@intercol\\\
\hline\cr\text{Channel}&\text{Value }&\text{Channel}&\text{Value}\\\
\hline\cr\text{Total}&$325$&\text{Total}&$721$\\\
b_{1}\pi&$147$&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi\pi_{2}}&$296$\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>h_{1}\eta&$32.9$&\rho\rho&$62.6$\\\ a_{2}\rho&$23.2$&a_{1}\pi&$57.4$\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>KK_{1}^{\prime}&$12.5$&a_{2}\pi&$37.1$\\\
f_{2}\omega&$11.3$&b_{1}\eta&$31.5$\\\
\pi\rho(1450)&$8.69$&a_{1}\omega&$28.9$\\\
\pi(1300)\rho&$5.32$&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\eta\omega&$4.00$&\pi\pi(1300)&$14.2$\\\
\pi\rho(1700)&$2.57$&f_{2}\rho&$12.2$\\\
KK_{1}&$2.53$&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>KK_{2}^{*}&$2.20$&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\eta^{\prime}\omega&$1.75$&a_{2}(1700)\pi&$5.02$\\\
a_{1}{\rho}&$1.75$&\omega(1420)\pi&$3.01$\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\eta\omega(1420)&$1.05$&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\eta(1295)\omega&$1.04$&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>KK^{*}&$1.03$&f_{1}\rho&$2.201$\\\
&&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>&&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi\omega(1650)}&$1.54$\\\
&&b_{1}\eta^{\prime}&$1.27$\\\
&&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\eta(1295)\rho}&$1.15$\\\
&&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\hline\cr\hline\cr\end{array}$ Table 14: The partial decay widths of the
$\omega_{3}(2G)$, $\rho_{3}(2G)$, the unit of width is MeV. The $\gamma$ value
is 11.6.
$\begin{array}[]{cccc}\hline\cr\hline\cr\lx@intercol\hfil\omega_{3}(2520)\hfil\lx@intercol&\lx@intercol\hfil\rho_{3}(2520)\hfil\lx@intercol\\\
\hline\cr\text{Channel}&\text{Value }&\text{Channel}&\text{Value}\\\
\hline\cr\text{Total }&$312$&\text{Total}&$397$\\\
{b_{1}}\pi&$107$&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi\pi_{2}}&$107$\\\
a_{2}\rho&$40.9$&h_{1}\pi&$37.1$\\\ \pi_{2}\rho&$21.4$&a_{1}\pi&$36.0$\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>$&\rho\rho(1450)&$31.4$\\\ h_{1}\eta&$18.9 $&a_{1}\omega&$22.6$\\\
\pi\rho&$18.6$&b_{1}\rho&$21.0$\\\ f_{2}\omega&$17.5 $&f_{2}\rho&$18.4$\\\
KK_{2}&$15.3$&b_{1}\eta&$16.4$\\\ f_{1}\omega&$13.2$&a_{2}(1700)\pi&$16.4$\\\
a_{0}{\rho}&$9.88$&a_{2}\pi&$16.04$\\\ \pi(1300)\rho&$4.88$&\rho\rho&$9.21$\\\
\eta_{2}\omega&$3.57$&\pi_{2}\omega&$7.03$\\\
f_{1}{h_{1}}&$3.03$&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>h_{1}\eta(1295)&$3.16$&b_{1}{b_{1}}&$6.46$\\\
f_{2}{h_{1}}&$3.11$&a_{4}(2040)\pi&$6.16$\\\ \eta\omega&$3.08
$&\pi\pi&$6.15$\\\ f_{1}(1420)\omega&$3.03$&f_{1}\rho&$4.72$\\\
h_{1}\eta^{\prime}&$2.63$&h_{1}\pi(1300)&$4.35$\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\eta\omega_{3}}&$1.81$&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>a_{1}\rho&$1.31$&a_{2}\omega&$3.50$\\\
\eta^{\prime}\omega&$1.06$&f_{1}(1420)\rho&$3.20$\\\ &&\omega{a_{0}}&$3.08$\\\
&&h_{1}{a_{1}}&$2.52$\\\ &&a_{2}h_{1}&$2.18$\\\
&&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>&&KK(1630)&$1.42$\\\ &&\pi\pi(1300)&$1.38$\\\ &&b_{1}\eta^{\prime}&$1.31$\\\
\hline\cr\hline\cr\end{array}$ Table 15: The partial decay widths of the
$\omega_{3}(3G)$, $\rho_{3}(3G)$, the unit of width is MeV. The $\gamma$ value
is 11.6.
$\begin{array}[]{cccc}\hline\cr\hline\cr\lx@intercol\hfil\omega_{3}(2750)\hfil\lx@intercol&\lx@intercol\hfil\rho_{3}(2750)\hfil\lx@intercol\\\
\hline\cr\text{Channel}&\text{Value }&\text{Channel}&\text{Value}\\\
\hline\cr\text{Total }&181&\text{Total}&270\\\
b_{1}\pi&63.8&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\pi\rho&12.3&\rho\rho&24.3\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>f_{1}\omega&11.7&a_{1}\omega&23.6\\\ \pi_{2}\rho&10.4&a_{1}\pi&21.0\\\
h_{1}\eta&9.84&a_{2}\pi&10.5\\\ b_{1}\pi(1300)&8.28&a_{2}(1700)\pi&9.40\\\
f_{2}\omega&8.25&f_{2}\rho&8.45\\\
\pi(1300)\rho&7.77&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\eta(1295)\rho}&8.17\\\
{a_{2}}\rho&5.59&\rho\rho(1450)&7.91\\\ \pi\rho(1450)&5.32&b_{1}\eta&7.90\\\
{a_{0}}\rho&4.89&b_{1}\rho&7.84\\\
h_{1}\eta(1295)&4.21&a_{2}(1700)\omega&7.05\\\
{\eta_{2}}\omega&1.96&{h_{1}}{\pi(1300)}&6.56\\\
{f_{1}(1420)}\omega&1.85&{\pi}{\pi}&6.40\\\
\eta\omega&1.59&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>{f_{2}}{h_{1}}&1.59&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>\eta(1295)\omega&1.55&a_{4}(2040)\pi&3.94\\\
K_{1}K_{1}&1.50&\pi_{2}\omega&3.55\\\
\eta\omega(1420)&1.50&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pi(1300)\omega}&2.62\\\
{a_{1}}\rho&1.37&a_{1}\pi(1300)&2.17\\\
K{K_{1}^{\prime}}&1.34&f_{1}(1420)\rho&1.96\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\eta\omega_{3}}&1.07&a_{1}\omega(1420)&1.86\\\
h_{1}{\eta^{\prime}}&1.01&b_{1}\eta(1295)&1.84\\\ &&a_{2}\omega&1.69\\\
&&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\eta(1475)\rho}&1.64\\\
&&a_{2}(1320){h_{1}}&1.55\\\ &&b_{1}{b_{1}}&1.54\\\ &&b_{1}{f_{1}}&1.51\\\
&&K_{1}K_{1}&1.50\\\
&&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\eta\rho(1450)}&1.48\\\
&&\omega{a_{0}}&1.48\\\ &&\omega(1420)\pi&1.43\\\ &&h_{1}{a_{1}}&1.39\\\
&&\pi(1300)\pi(1300)&1.19\\\ &&a_{2}{a_{2}}&1.03\\\
\hline\cr\hline\cr\end{array}$

##  IV CONCLUSION

In this work, we have systematically studied the mass spectra and the OZI-
allowed two-body strong decay behaviors of the newly observed $X(2220)$ state
as well as $\omega$ and $\omega_{3}$ and $\rho$ and $\rho_{3}$ states. We
adopt a new way to get the value of $\gamma$ in $\omega$ and $\omega_{3}$
system. Following is the main work.

  1. 1.

The newly observed state $X(2220)$ should be a $3^{3}D_{1}$ $\omega$ state.

  2. 2.

The $\omega(782)$, $\omega(1420)$ and $\omega(1960)$ states can be interpreted
as the $\omega(1^{3}S_{1})$, $\omega(2^{3}S_{1})$ and $\omega(3^{3}S_{1})$.
$\omega(2240)$, $\omega(2330)$ and $\omega(2290)$ can be the candidates of
$\omega(4^{3}S_{1})$. $\rho(1450)$, $\rho(1900)$ and $\rho(2150)$ are the
first, second and third excited states of $\rho(770)$ as shown in Fig. 1.

  3. 3.

The mass and width of $\omega(2D)$, by our prediction, are $2047.6$ MeV and
$212\pm 36.1$ MeV. $\omega(2205)$ and $X(2220)$ are the $\omega(3D)$ states.

  4. 4.

$\omega_{3}(1945)$, $\omega_{3}(2285)$ and $\rho_{3}(1990)$, $\rho_{3}(2250)$
are the first and second excited states of $\omega_{3}(1670)$ and
$\rho_{3}(1690)$.

  5. 5.

The masses of $\omega_{3}(4D)$ and $\rho_{3}(4D)$ are both predicted to be
2.63 GeV, the width of them are 288 MeV and 420 MeV. We also predict the mass
spectra and decay behaviors of $\omega_{3}(1G)$ and $\rho_{3}(1G)$,
$\omega_{3}(2G)$ and $\rho_{3}(2G)$ and $\omega_{3}(3G)$ and $\rho_{3}(3G)$.

In addition, it is our hope and belief that more experimental measurements
will be released in the future.

##  V ACKNOWLEDGE

This work is supported by the National Natural Science Foundation of China
under Grants No. 11965016 and the Natural Science Foundation of Qinghai
Province (2020-ZJ-728). Y. -R. W. and T.-Y. L. contributed equally to this
work.

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