# Light unflavored vector meson spectroscopy around the mass range of $2.4\sim
3$ GeV and possible experimental evidence

Li-Ming Wang1,3<EMAIL_ADDRESS>Si-Qiang Luo2,3,4<EMAIL_ADDRESS>Xiang
Liu2,3,4,5111Corresponding author<EMAIL_ADDRESS>1Key Laboratory for
Microstructural Material Physics of Hebei Province, School of Science, Yanshan
University, Qinhuangdao 066004, 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 and Frontiers Science Center for Rare Isotopes, Lanzhou
University, Lanzhou 730000, China
4Research Center for Hadron and CSR Physics, Lanzhou University and Institute
of Modern Physics of CAS, Lanzhou 730000, China
5Joint Research Center for Physics, Lanzhou University and Qinghai Normal
University, Xining 810000, China

###### Abstract

In this work, we predict the spectroscopy behavior of these light unflavored
vector mesons with masses at the range of $2.4\sim 3$ GeV, which are still
missing in the experiment. By presenting their mass spectrum and studying
their two-body Okubo-Zweig-lizuka allowed decay widths, we discuss the
possible experimental evidences of these discussed states combining with the
present experimental data. Especially, we strongly suggest that our
experimental colleagues carry out the exploration of these higher states via
the $e^{+}e^{-}$ annihilation into light mesons. It is obvious that BESIII and
Belle II will be potential experiments to achieve this target.

##  I Introduction

As a main body of the hadron family, light hadron spectroscopy has special
status in the whole hadron family, which has inspired the classification of
hadron based on SU(3) symmetry in 1964 Gell-Mann:1964ewy ; Zweig:1964ruk , and
has aroused the interest from both theorists and experimentalists in exploring
exotic states like glueball, hybrid, and multiquark states Teper:1998kw ;
Meyer:2015eta ; Chen:2016qju ; Liu:2019zoy . With the running of BESIII and
Belle II, it is a good chance to launch a profound research on light flavor
mesons, which can deepen our understanding of how quarks and gluons interact
with each other to form hadrons.

Focusing on the study of light unflavored vector meson spectroscopy, we find
big progress being made by experiments in the past years Ablikim:2005um ;
Ablikim:2010au ; Ablikim:2016itz . A typical example is the observation of the
$Y(2175)$ Ablikim:2007ab  and the following measurement Ablikim:2014pfc ;
Lees:2011zi ; Aubert:2007ym ; Shen:2009zze ; Aubert:2007ur ; Aubert:2006bu
around the $Y(2175)$, which have stimulated extensive discussions of decoding
the property of the $Y(2175)$ Wang:2021gle ; Wang:2006ri ; Ding:2007pc ;
MartinezTorres:2008gy ; Chen:2008ej ; Klempt:2007cp . Among these possible
explanations to the $Y(2175)$, categorizing the $Y(2175)$ into the $\phi$
meson family is a popular one Wang:2012wa ; Ding:2007pc ; Barnes:2002mu ,
which should still be tested in an experiment. In fact, the light unflavored
vector mesons also include these states in the $\rho$ and $\omega$ meson
families. With the accumulation of data with unprecedented statistical
accuracy, a series of $\rho$ and $\omega$ states around 2 GeV were observed in
experiments Antonelli:1996xn ; Clegg:1989mp ; Hasan:1994he ; Bugg:2004xu ;
Atkinson:1985yx ; Chung:2002pu ; Anisovich:2011sva ; Bugg:2004rj ;
Atkinson:1988xa . By the joint effort from both theorists and
experimentalists, our knowledge of light unflavored vector meson spectroscopy
below 2.2 GeV has been promoted, which is the situation of the study of
unflavored vector meson spectroscopy. But, it is not the whole aspect of light
unflavored vector meson spectroscopy.

The experiments have released the data of $e^{+}e^{-}$ annihilation into light
mesons BaBar:2006vzy ; BESIII:2019gjz ; BaBar:2011btv ; Belle:2008kuo , where
the collision energy reaches up to $\sim 3$ GeV. When checking these data, we
find possible enhancement structures existing in the corresponding invariant
mass spectra with mass range from 2.4 GeV to 3 GeV. For example, in the
$e^{+}e^{-}\to\omega\pi^{+}\pi^{-}\pi^{0}$ process BaBar:2006vzy , several
enhancement structures existing in the energy range of 2.4 GeV are obvious.
However, in the process of $e^{+}e^{-}\to\pi^{+}\pi^{-}\pi^{0}$, we cannot
find conspicuous enhancement structure BESIII:2019gjz . Thus, we still need
more precise data. In the following, we can find many narrow structures above
2.4 MeV in the process of $e^{+}e^{-}\to K_{2}^{*0}(1430)K^{-}\pi^{+}$
reported by the BaBar Collaboration BaBar:2011btv . The cross section of
$e^{+}e^{-}\to\phi f_{2}^{\prime}(1525)$ was also measured by the BaBar
Collaboration. Here, there exists possible enhancement clusters around 2.7 GeV
and a clear $J/\psi$ signal BaBar:2011btv . And then, the Belle Collaboration
studied the process of $e^{+}e^{-}\to\phi\pi^{+}\pi^{-}$ and
$e^{+}e^{-}\to\phi f_{0}(980)$ Belle:2008kuo , where the event accumulation
around 2.4 GeV and 2.6 GeV can be found. We conjecture that these enhancement
structures with low significance can be due to these unknown light unflavored
vector mesons.

At present, our knowledge of higher states of the light unflavored vector
meson family, which have the masses above 2.4 GeV, is obviously absent. This
research status causes us to reinforce the investigation of these higher
states of the light unflavored vector meson family, which becomes a central
task of this work.

In this work, we apply the modified Godfrey-Isgur model Song:2015nia  to
present the masses distribution of the discussed higher states of the light
unflavored vector meson family, where their spatial wave functions can be
gotten numerically, which can be as the input of the following calculation of
their two-body Okubo-Zweig-lizuka (OZI) allowed decays. For performing the
realistic estimate of these OZI allowed decays, the quark pair creation (QPC)
model is adopted here Anisovich:2005wf ; Roberts:1992js ; Blundell:1996as ,
which was applied to study the strong decay of different kinds of hadrons
Yu:2011ta ; Wang:2012wa ; He:2013ttg ; Ye:2012gu ; Wang:2014sea ; Chen:2015iqa
; Guo:2019wpx ; Pang:2017dlw ; Pang:2018gcn ; Wang:2020due . We hope that our
study on mass spectrum and decay width of the $\rho$, $\omega$, and $\phi$
mesons in $2.4\sim 3$ GeV can provide valuable information to the future
experimental explorations.

As emphasized in Ref. BESIII:2020nme , the BESIII experiment at the BEPCII is
still an ideal platform to hunt for light hadrons. We have reason to believe
that the present work can attract the BESIII interest of studying this topic
in an experiment. In addition, the running of Belle II with the initial state
radiation method will be a potential experiment to finding these discussed
light unflavored vector mesons.

This paper is organized as follows. After the Introduction in Sec. I, we
present the spectroscopy behavior by the modified Godfrey-Isgur (MGI) model
and the QPC model, where the mass spectrum and two-body OZI allowed strong
decays of these discussed higher states of light unflavored vector meson are
given in Sec. II. Finally, the paper ends with a short summary in Sec. IV.

##  II Spectroscopy behavior

There were different approaches to study the meson spectroscopy, which include
lattice QCD Dudek:2010wm ; Dudek:2011tt , Regge trajectories Pang:2015eha ;
He:2013ttg ; Badalian:2002xy , nonrelativistic potential models Eichten:1974af
; Eichten:1978tg ; Close:2005se , QCD string approaches Badalian:2006sg ;
Badalian:2008sv , relativistic potential models Ebert:2009ub ; Ebert:2009ua ,
and so on. In this work, we adopt the MGI model Song:2015nia  by replacing the
linear potential with the screened potential in the Godfrey-Isgur (GI) model
Godfrey:1985xj . Here, the screened potential indicated by the lattice studies
Laermann:1986pu ; Born:1989iv  was employed in Refs. Chao:1992et ; Ding:1993uy
; Ding:1995he , which is also inspired by the the coupled-channel effects
Li:2009zu ; Duan:2021alw . A successful application of the MGI model is to
depict the mass spectrum of these observed charmed-strange mesons Song:2015nia
, where the masses of the most observed $D_{s}$ mesons can be understood well.
Later, the MGI model was further adopted to perform the mass spectrum analysis
of charmed mesons Song:2015fha , charmonia Wang:2019mhs ; Wang:2020prx , and
bottomonia Wang:2018rjg . Along this line, in this work we still apply the MGI
model to discuss these higher states of the light unflavored vector meson
family. Besides the mass spectrum, the spatial wave functions of these higher
unflavored mesons could be extracted for getting the two-body OZI allowed
strong decay widths via the QPC model. In Fig. 1, we show the procedure of how
to obtain the whole aspect of these higher light unflavored vector meson
spectroscopies.

Figure 1: The procedure of presenting the aspect of light unflavored vector
meson spectroscopy.

###  II.1 Mass spectrum

The Hamiltonian depicting the interaction between quark and antiquark in the
MGI model is

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

where $m_{1}$ and $m_{2}$ denote the mass of quark and antiquark,
respectively. The effective potential of the $q\bar{q}$ interaction has the
form Song:2015nia

$\widetilde{V}_{\mathrm{eff}}(\mathbf{p},\mathbf{r})=\widetilde{H}^{\mathrm{OGE}}+\widetilde{H}^{\mathrm{conf}},$
(2)

where the first term $\widetilde{H}^{\mathrm{OGE}}$ is a
$\gamma^{\mu}\otimes\gamma_{\mu}$ one-gluon-exchange potential, which is a
short distance interaction, and the $\widetilde{H}^{\mathrm{conf}}$ in Eq. (2)
is a long distance $1\otimes 1$ confining interaction. The model includes
smearing transformations and momentum-dependent terms. Firstly, the smearing
transformations could be expressed as

$\begin{split}\widetilde{V}_{\rm eff}({\bf p},{\bf r})=&\int d^{3}{\bf
r}^{\prime}\rho_{ij}({\bf r}-{\bf r}^{\prime})V_{\rm eff}({\bf p},{\bf
r}^{\prime}),\\\ \end{split}$ (3)

where the $\rho_{ij}({\bf r}-{\bf r}^{\prime})$ is the Gaussian smearing
function with

$\rho_{ij}({\bf r}-{\bf
r}^{\prime})=\frac{\sigma_{ij}^{3}}{\pi^{3/2}}e^{-\sigma_{ij}^{2}({\bf r}-{\bf
r}^{\prime})^{2}},$ (4)
$\sigma_{ij}^{2}=\sigma_{0}^{2}\left(\frac{1}{2}+\frac{1}{2}\left(\frac{4m_{i}m_{j}}{(m_{i}+m_{j})^{2}}\right)^{4}\right)+s^{2}\left(\frac{2m_{i}m_{j}}{m_{i}+m_{j}}\right)^{2}.$
(5)

The parameters $\sigma_{0}$ and $s$ in Eq. (5) could be determined by the spin
splits of the mesons.

The $\widetilde{H}^{\mathrm{OGE}}$ in Eq. (2) contains Coulomb, contact,
tensor, and vector spin-orbit terms, i.e.,

$\widetilde{H}^{\mathrm{OGE}}=\widetilde{G}_{12}(r)+\widetilde{V}^{\mathrm{cont}}(r)+\widetilde{V}^{\mathrm{tens}}(r)+\widetilde{V}^{\mathrm{sov}}(r).$
(6)

The Coulomb term $\widetilde{G}_{12}(r)$ is spin independent with definition

$\begin{split}\widetilde{G}_{ij}(r)=&\int d^{3}{\bf r}^{\prime}\rho_{ij}({\bf
r}-{\bf r}^{\prime})\left(\sum\limits_{k}-\frac{4\alpha_{k}}{3r}{\rm
erf}(\gamma_{k}r)\right)\\\ =&\sum\limits_{k}-\frac{4\alpha_{k}}{3r}{\rm
erf}(\tau_{k}r),\end{split}$ (7)

where $\alpha_{k}=(0.25,0.15,0.2)$ and
$\gamma_{k}=(1/2,\sqrt{10}/2,\sqrt{1000}/2)$ with $k=1,2,3$ Godfrey:1985xj ,
and the $\tau_{k}$ could be obtained by

$\tau_{k}=\frac{1}{\sqrt{\frac{1}{\sigma_{ij}^{2}}+\frac{1}{\gamma_{k}^{2}}}}.$
(8)

Additionally, the semirelativistic correction with momentum-dependent factors
is introduced. For the Coulomb term, the correction is

$G_{12}(r)\to\left(1+\frac{p^{2}}{E\bar{E}}\right)^{1/2}G_{12}(r)\left(1+\frac{p^{2}}{E\bar{E}}\right)^{1/2}.$
(9)

For the spin-dependent terms, the semi-relativistic correction could be
written as

$V^{i}_{\alpha\beta}\to\left(\frac{m_{\alpha}m_{\beta}}{E_{\alpha}E_{\beta}}\right)^{1/2+\epsilon_{i}}V_{\alpha\beta}\left(\frac{m_{\alpha}m_{\beta}}{E_{\alpha}E_{\beta}}\right)^{1/2+\epsilon_{i}}.$
(10)

The remaining terms in Eq. (6) are spin-dependent potentials, which are
defined as

$\begin{split}\widetilde{V}^{\mathrm{cont}}(r)=\frac{2{\bf S}_{1}\cdot{\bf
S}_{2}}{3m_{1}m_{2}}\nabla^{2}\widetilde{G}_{12}^{c}(r),\end{split}$ (11)
$\begin{split}\widetilde{V}^{\mathrm{tens}}(r)=&-\left(\frac{3{\bf
S}_{1}\cdot{\bf r}{\bf S}_{2}\cdot{\bf r}/r^{2}-{\bf S}_{1}\cdot{\bf
S}_{2}}{3m_{1}m_{2}}\right)\left(\frac{\partial^{2}}{\partial
r^{2}}-\frac{1}{r}\frac{1}{\partial
r}\right)\widetilde{G}_{12}^{t}(r),\end{split}$ (12)
$\begin{split}\widetilde{V}^{\mathrm{sov}}(r)=&\frac{{\bf S}_{1}\cdot{\bf
L}}{2m_{1}^{2}}\frac{1}{r}\frac{\partial\widetilde{G}_{11}^{\rm
sov}(r)}{\partial r}+\frac{{\bf S}_{2}\cdot{\bf
L}}{2m_{2}^{2}}\frac{1}{r}\frac{\partial\widetilde{G}_{22}^{\rm
sov}(r)}{\partial r}\\\ &+\frac{({\bf S}_{1}+{\bf S}_{2})\cdot{\bf
L}}{m_{1}m_{2}}\frac{1}{r}\frac{\partial\widetilde{G}_{12}^{\rm
sov}(r)}{\partial r}.\\\ \end{split}$ (13)

In Eqs. (11)-(13), the semirelativistic correction factors in Eq. (10) with
$\epsilon_{i}=\epsilon_{c}$, $\epsilon_{t}$, and $\epsilon_{\rm sov}$ impacts
the potentials $\widetilde{V}^{\mathrm{cont}}(r)$,
$\widetilde{V}^{\mathrm{tens}}(r)$, and $\widetilde{V}^{\mathrm{sov}}(r)$,
respectively222The spin-orbit and tensor interactions without smearing
transformations may contain the $1/r^{3}$ term. But for $S$-wave states, the
spin-orbit and tensor operators are zero, thus the spin-orbit and tensor
potentials do not lead to singularities..

The second part of Eq. (2) contains a screened confinement potential and a
scalar spin-orbit interaction, i.e.,

$\widetilde{H}^{\mathrm{conf}}=\widetilde{S}_{12}+\widetilde{V}^{\mathrm{sos}}.$
(14)

In the MGI model, the screen effect Song:2015nia  is introduced, where we
should make a replacement

$br\to\frac{b(1-e^{-\mu r})}{\mu}.$ (15)

Then the $\widetilde{S}_{12}$ in Eq. (14) is defined as

$\begin{split}\widetilde{S}_{ij}(r)=&\int d^{3}{\bf r}^{\prime}\rho_{ij}({\bf
r}-{\bf r}^{\prime})\frac{b(1-e^{-\mu r})}{\mu}+c,\end{split}$ (16)

where $b$ is the strength of the confinement, and the $\mu$ is a parameter to
scale the screening effect. Then the scalar spin-orbit interaction could be
obtained by

$\begin{split}\widetilde{V}^{\mathrm{sos}}(r)=&-\frac{{\bf S}_{1}\cdot{\bf
L}}{2m_{1}^{2}}\frac{1}{r}\frac{\partial\widetilde{S}_{11}^{\rm
sos}(r)}{\partial r}-\frac{{\bf S}_{2}\cdot{\bf
L}}{2m_{2}^{2}}\frac{1}{r}\frac{\partial\widetilde{S}_{22}^{\rm
sos}(r)}{\partial r},\\\ \end{split}$ (17)

where the $\widetilde{S}_{11}^{\rm sos}(r)$ and $\widetilde{S}_{11}^{\rm
sos}(r)$ contain semirelativistic correction factors in Eq. (10) with
$\epsilon_{i}=\epsilon_{\rm sos}$.

In the above formula, $\mathbf{S_{1}}$ and $\mathbf{S_{2}}$ denote the spin of
quark and antiquark, respectively. $\bm{L}$ indicates the orbital momentum
between two quarks. More details of the MGI model can be found in Ref.
Song:2015nia .

Table 1: The value of these parameters involved in the MGI model. Parameter | Value | Parameter | Value
---|---|---|---
$m_{u}$ (GeV) | 0.22 |  $m_{d}$ (GeV) | 0.22
$m_{s}$ (GeV) | 0.424 |  $b$ (GeV2) | 0.229
$\epsilon_{c}$ | -0.164 | $\epsilon_{\mathrm{sos}}$ | 0.9728
$\sigma_{0}$ (GeV) | 1.8 |  $s$ (GeV) | 3.88
$\mu$ (GeV) | 0.081 |  $c$ (GeV) | -0.30
$\epsilon_{\mathrm{sov}}$ | 0.262 | $\epsilon_{t}$ | 1.993

The values of all parameters used in the MGI model are collected in Table. 1.
With these preparations, we further present the mass spectrum of light
unflavored vector mesons in Fig. 2. In Refs. Wang:2012wa ; He:2013ttg ;
Wang:2020kte , the Lanzhou group performed the study of mass spectrum and two-
body OZI allowed decay of the mesons in the $\rho$, $\omega$, and $\phi$
family and discussed how to categorize these observed light vector $\rho$,
$\omega$, and $\phi$ states below 2.4 GeV collected in Particle Data Group
(PDG) PDG:2020ssz  into the $\rho$, $\omega$, and $\phi$ family. Here, the MGI
model was adopted, which was applied to study various light flavor systems,
including kaon family Pang:2017dlw , $\phi$ states below 2.6 GeV Pang:2019ttv
, and high spin light flavor mesons Pang:2018gcn . When further exploring the
property of these higher states of the $\rho$, $\omega$, and $\phi$ family, we
still use the MGI model for presenting the mass spectrum analysis. Here,
besides reproducing the masses of these low lying states, the masses of higher
states are also obtained. Our calculated result explicitly shows that there
exists accumulation of these $5S$, $6S$, and $7S$ states of $\rho$, $\omega$
and $\phi$ mesons in the mass range from 2.4 GeV to 3 GeV.

In this work, we also calculate the dilepton decay widths of these discussed
light unflavored vector mesons. The dilepton decay width of a unflavored meson
can be expressed as Godfrey:1985xj

$\Gamma_{e^{+}e^{-}}=\frac{4\pi}{3}\alpha^{2}m|{\cal M}|^{2},$ (18)

where the $m$ is the mass of the vector meson. The ${\cal M}$ in Eq. (18) is
the decay amplitude. For $S$-wave vector mesons, we employ ${\cal
M}_{\rho}=\sqrt{6}V_{\rho}$, ${\cal M}_{\omega}=\sqrt{2/3}V_{\omega}$, and
${\cal M}_{\phi}=-\sqrt{4/3}V_{\phi}$ to express the decay amplitudes of the
$\rho$, $\omega$, and $\phi$ states, respectively. In potential models, the
factors $V_{\rho}$, $V_{\omega}$, and $V_{\phi}$ can be written as

$V_{\rho/\omega/\phi}=m^{-2}\widetilde{m}^{1/2}(2\pi)^{-3/2}\int{\rm
d}^{3}{\bf
p}(4\pi)^{-1/2}\Phi(p)\left(\frac{m_{1}m_{2}}{E_{1}E_{2}}\right)^{1/2},$ (19)

where $m_{1}$ and $m_{2}$ are consistent quark masses in the meson, and
$E_{1}$ and $E_{2}$ are energies of the quarks. The $\Phi(p)$ in Eq. (19) is
the radial part of the spatial wave function in the momentum space, which is
extracted from the potential model. And the $\widetilde{m}$ is calculated by
$\widetilde{m}=2\int{\rm d}^{3}{\bf p}E|\phi({\bf p})|^{2}$, where $\phi({\bf
p})=\Phi(p)Y_{lm}(\Omega_{\bf p})$. Besides, we use
$M_{\rho}^{\prime}=\sqrt{4/3}V_{\rho}^{\prime}$,
$M_{\omega}^{\prime}=\sqrt{4/27}V_{\omega}^{\prime}$, and
$M_{\phi}^{\prime}=-\sqrt{8/27}V_{\phi}^{\prime}$ to denote the decay
amplitudes of the $D$-wave vector $\rho$, $\omega$, and $\phi$ mesons,
respectively. The $V_{\rho}^{\prime}$, $V_{\omega}^{\prime}$, and
$V_{\phi}^{\prime}$ could be expressed by

$\begin{split}V_{\rho/\omega/\phi}^{\prime}=&m^{-2}\widetilde{m}^{1/2}(2\pi)^{-3/2}\int{\rm
d}^{3}{\bf p}(4\pi)^{-1/2}\phi(p)\\\
&\times\left(\frac{m_{1}m_{2}}{E_{1}E_{2}}\right)^{1/2}\left(\frac{p}{E_{1}}\right)^{2}.\end{split}$
(20)

With preparations above, the dilepton decay widths of these discussed light
unflavored vector mesons are calculated and presented in Table 2.

Table 2: The dilepton decay widths of the discussed light unflavored vector mesons in units of keV. States  | $\rho(5S)$ | $\rho(6S)$ | $\rho(7S)$ | $\rho(4D)$ | $\rho(5D)$ | $\rho(6D)$
---|---|---|---|---|---|---
$\Gamma_{e^{+}e^{-}}$ | 0.0377 | 0.0204 | 0.0195 | 0.0070 | 0.0052 | 0.0029
States  | $\omega(5S)$ | $\omega(6S)$ | $\omega(7S)$ | $\omega(4D)$ | $\omega(5D)$ | $\omega(6D)$
$\Gamma_{e^{+}e^{-}}$ | 0.0042 | 0.0023 | 0.0022 | 0.0008 | 0.0006 | 0.0003
States  | $\phi(4S)$ | $\phi(5S)$ | $\phi(6S)$ | $\phi(3D)$ | $\phi(4D)$ | $\phi(5D)$
$\Gamma_{e^{+}e^{-}}$ | 0.0492 | 0.0291 | 0.0162 | 0.0096 | 0.0058 | 0.0037
Figure 2: The comparison of calculated results calculated in this work and the
experimental data of mass of higher $\rho$, $\omega$ and $\phi$ states. Here,
the red dots denote the experimental data from Refs. PDG:2020ssz ;
BaBar:2006vzy ; Anisovich:2002su ; Bugg:2004xu .

In the following section, we pay more attention to their two-body OZI allowed
strong decays, which can give important hints to the experimental search for
them. Here, we need to emphasize that the numerical forms of the spatial wave
function of these discussed light vector unflavored mesons can be obtained,
which are as the input of the following discussion, by their decay behavior.

###  II.2 Two-body OZI allowed strong decay

For calculating the two-body OZI allowed strong decay of these discussed
higher unflavored vector meson states, we apply the QPC model, which is an
effective approach to quantitatively estimate such physical quantity
Anisovich:2005wf ; Roberts:1992js ; Blundell:1996as .

In the QPC model, when a meson decay occurs, a quark-antiquark pair is created
from the vacuum with the quantum number $J^{PC}=0^{++}$ and combines with the
corresponding antiquark and quark in the initial meson to produce two final
mesons. In Ref. Hayne:1981zy , the authors introduced the wave functions of
the mock states when calculating the transition amplitudes. In the QPC model,
the wave function of a mock state is also adopted. In the decay process $A\to
BC$, the wave function of the mock state $A$ is defined as Hayne:1981zy ;
Blundell:1996as

$\begin{split}&\left|A\left(n_{A}^{2S_{A}+1}L_{AJ_{A}M_{J_{A}}}\right)\left(\bm{P}_{A}\right)\right\rangle\\\
&\equiv\sqrt{2E_{A}}\sum\limits_{M_{L_{A}},M_{S_{A}}}\left\langle
L_{A}M_{L_{A}}S_{A}M_{S_{A}}|J_{A}M_{J_{A}}\right\rangle\\\ &\quad\times\int
d^{3}\bm{p}_{A}\psi_{n_{A}L_{A}M_{L_{A}}}(\bm{p}_{A})\chi_{S_{A}M_{S_{A}}}^{12}\phi_{A}^{12}\omega_{A}^{12}\\\
&\quad\times\left|q_{1}\left(\frac{m_{1}}{m_{1}+m_{2}}\bm{P}_{A}+\bm{p}_{A}\right)\bar{q}_{2}\left(\frac{m_{2}}{m_{1}+m_{2}}\bm{P}_{A}-\bm{p}_{A}\right)\right\rangle,\end{split}$
(21)

where the $m_{1}$ and $m_{2}$ are masses of quark $q_{1}$ and antiquark
$\bar{q_{2}}$, respectively. For the mock state wave function of the meson $B$
($C$), we simply replace ‘$A$’ by ‘$B$’ (‘$C$’) in Eq. (21). The $n_{A}$ in
Eq. (21) is the radial quantum number of the meson $A$. The $S_{A}$, $L_{A}$,
and $J_{A}$ are spin, orbital angular momentum, and total angular momentum,
respectively. Note that
$\bm{p}_{A}=\frac{m_{1}\bm{p_{1}}-m_{2}\bm{p_{2}}}{m_{1}+m_{2}}$ is the
relative momentum of the $q_{1}\bar{q}_{2}$ pair, where $\bm{p}_{1}$ and
$\bm{p}_{2}$ are the momentum of $q_{1}$ and $\bar{q}_{2}$, respectively.
Since the particles here are on shell, the integral variable ${\bm{p}}_{A}$ in
Eq. (21) is three dimensional. Note that $\bm{P}_{A}=\bm{p}_{1}+\bm{p}_{2}$
and $E_{A}$ are total momentum and total energy of the meson $A$,
respectively. The $\psi_{n_{A}L_{A}M_{L_{A}}}(\bm{p}_{A})$ is the spatial wave
function in momentum space. The $\chi_{S_{A}M_{S_{A}}}^{12}$, $\phi_{A}^{12}$,
and $\omega_{A}^{12}$ are the spin, flavor, and color wave functions,
respectively.

The total decay width in the center-of-mass (CM) frame is given by

$\displaystyle\Gamma=\frac{\pi}{4}\frac{|\mathbf{P}|}{M_{A}^{2}}\sum\limits_{LS}\left|M^{LS}\right|^{2}.$
(22)

Here, $\mathbf{P}$ is the momentum outgoing meson (meson $B$ or meson $C$).
$M_{A}$ is the mass of meson $A$. $\bm{L}$ and $\bm{S}$ denote the relative
orbital angular and total spin momentum between meson $B$ and $C$,
respectively. $M^{LS}$ denotes the partial wave amplitude and is related to
the helicity amplitude $M^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}$ according to Jacob-
Wick formula Jacob:1959at . In the CM frame, the specific form of
$M^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}$ can be written as

$\displaystyle\begin{split}&M^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}(\bm{P})\\\
&=\gamma\sqrt{8E_{A}E_{B}E_{C}}\sum\limits_{\mbox{${\begin{split}M_{L_{A}},M_{S_{A}},M_{L_{B}},\\\
M_{S_{B}},M_{L_{C}},M_{S_{C}},m\end{split}}$}}\langle
L_{A}M_{L_{A}}S_{A}M_{S_{A}}|J_{A}M_{J_{A}}\rangle\;\\\ &\quad\times\langle
L_{B}M_{L_{B}}S_{B}M_{S_{B}}|J_{B}M_{J_{B}}\rangle\langle
L_{C}M_{L_{C}}S_{C}M_{S_{C}}|J_{C}M_{J_{C}}\rangle\\\ &\quad\times\langle
1m1-m|00\rangle\;\langle\chi_{S_{B}M_{S_{B}}}^{14}\chi_{S_{C}M_{S_{C}}}^{32}|\chi_{S_{A}M_{S_{A}}}^{12}\chi_{1-m}^{34}\rangle\\\
&\quad\times\bigg{[}\langle\phi_{B}^{14}\phi_{C}^{32}|\phi_{A}^{12}\phi_{0}^{34}\rangle
I(\bm{P},m_{2},m_{1},m_{3})\\\
&\quad+(-1)^{1+S_{A}+S_{B}+S_{C}}\langle\phi_{B}^{32}\phi_{C}^{14}|\phi_{A}^{12}\phi_{0}^{34}\rangle
I(\bm{-P},m_{2},m_{1},m_{3})\bigg{]},\end{split}$ (23)

where $\gamma$ as the strength of the quark-antiquark pair created from the
vacuum is fixed to be 6.57 Pang:2019ttv . $\phi_{0}$ denotes the flavor wave
function of this quark-antiquark pair. Note that $I(\bm{P},m_{2},m_{1},m_{3})$
is the momentum space integral

$\displaystyle\begin{split}I(\bm{P},m_{2},m_{1},m_{3})=&\int
d^{3}\bm{p}\psi_{n_{B}L_{B}M_{L_{B}}}^{*}\left(\frac{m_{3}}{m_{1}+m_{3}}\bm{P}+\bm{p}\right)\\\
&\times\psi_{n_{C}L_{C}M_{L_{C}}}^{*}\left(\frac{m_{3}}{m_{2}+m_{3}}\bm{P}+\bm{p}\right)\\\
&\times\psi_{n_{A}L_{A}M_{L_{A}}}(\bm{P}+\bm{p})\mathcal{Y}_{1}^{m}(\bm{p}),\end{split}$
(24)

where $\mathcal{Y}_{1}^{m}(\bm{p})$ is a solid harmonic that gives the
momentum space distribution of the created quark-antiquark pair.

In our previous works, the QPC model was employed to deal with the two-body
OZI allowed strong decays of light unflavored vector mesons, which include
$\omega(2S)$, $\omega(1D)$, $\phi(2S)$, and $\phi(2D)$ Wang:2012wa  and
$\rho(2S)$, $\rho(3S)$, $\rho(4S)$, $\rho(1D)$, $\rho(2D)$, and $\rho(3D)$
He:2013ttg , where the experimental widths have been well reproduced. In
addition, we predicted the partial and total widths of the $\phi(1D)$,
$\omega(2D)$, $\phi(3S)$, and $\omega(3S)$ states Wang:2012wa . Until now,
these excited light unflavored vector mesons below 2.4 GeV have been
systemically studied. Thus, in the present work, we do not list the results of
these low-lying states.

Focusing on the light unflavored vector mesons existing in the range of 2.4
$\sim$ 3 GeV, we continue to adopt the QPC model to estimate the strong decay
widths, which is a main task of the following subsections.

Figure 3: The two-body strong decay behaviors of these discussed light vector
mesons with the mass range from 2.4 GeV to 3 GeV. Here, we show their
branching ratio of main channels and the total decay width.

####  II.2.1 $\rho$ states

As shown in Fig. 2, there are six $\rho$ states [$\rho(5S)$, $\rho(6S)$,
$\rho(7S)$, $\rho(4D)$, $\rho(5D)$, and $\rho(6D)$] existing in the mass range
of $2.4\sim 3$ GeV.

The obtained total and main partial decay widths of $\rho(5S)$ are shown in
Fig. 3, which show that $\pi a_{2}(1700)$ is the dominant decay channel and
the total width of $\rho(5S)$ can reach up to 81.85 MeV. In Fig. 3, we also
list their several typical decay modes like $\pi\pi$ and $\pi\omega$ easily
accessible at experiment. However, the branching ratios of these typical
decays are not obvious. Of course, the $\rho(5S)$ state can also decay into an
open-strange meson pair. However, our result shows that their contribution to
the total width is not significant. For example, $\rho(5S)\to KK_{1}(1270)$
and $K^{*}K_{1}(1270)$, have branching ratios of $0.43\%$ and $0.41\%$,
respectively. Thus, in this work, we do not collect the decay channels
relevant to the open-strange meson pair in Fig. 3.

The predicted decay behavior of $\rho(6S)$ is given in Fig. 4. Similar to the
$\rho(5S)$ state, its dominant decay channel is also $\pi a_{2}(1700)$, while
the $\rho\rho(1450)$, $\pi\pi_{2}(1880)$, $\pi\pi(1800)$ and $\pi\pi(1300)$
modes have sizable contributions to the total decay width, which is predicted
to be around 57.25 MeV. Additionally, the particle widths of $KK_{1}(1270)$
and $K^{*}K_{1}(1270)$ are 0.18 MeV and 0.16 MeV, respectively, which are not
listed in Fig. 3.

The calculation results for the $\rho(7S)$ are presented in Fig. 3, where
$\rho\rho(1450)$ and $\pi\pi_{2}(1880)$ are its dominant decay modes. In
addition, the contribution of the $\pi\pi(1800)$, $\pi a_{2}(1700)$,
$\pi\pi(1300)$ and $\rho\rho$ decay modes to total width cannot ignored. As
the typical channels, $\pi\pi$ and $\pi\omega$ have the branching ratios of
$1.09\%$ and $0.10\%$, respectively.

Accompanying with these three $S$-wave states, three corresponding $D$-wave
states exist in the same mass range. In Fig. 3, the two-body OZI allowed
strong decay behavior of $\rho(4D)$, $\rho(5D)$, and $\rho(6D)$ are given by
presenting the total decay width and the branching ratios of their partial
decays. The total decay widths of $\rho(4D)$, $\rho(5D)$, and $\rho(6D)$ are
92.33 MeV, 55.69 MeV, and 32.96 MeV, respectively, where their main decay
modes are $\pi\pi_{2}(1880)$, $\pi\pi(1300)$ and $\pi\pi(1800)$. Since these
three $D$-wave $\rho$ states have sizable $\pi\pi$ decay rates, the
$e^{+}e^{-}\to\pi\pi$ process can be as the ideal channel to identify these
$D$-wave $\rho$ states, which should stimulate experimentalists’ interest in
measuring the cross section of $e^{+}e^{-}\to\pi\pi$ with energy of collision
up to 3 GeV.

####  II.2.2 $\omega$ states

In the following, we check the two-body OZI allowed strong decay of six higher
$\omega$ mesonic states, which include $\omega(5S)$, $\omega(6S)$,
$\omega(7S)$, $\omega(4D)$, $\omega(5D)$, and $\omega(6D)$, where their total
decay widths and partial decay widths estimated by the QPC model are shown in
Fig. 3.

The total widths of $\omega(5S)$ and $\omega(6S)$, which are contributed by
the main decay channels like $\rho a_{0}(1450)$, $\pi\rho(1450)$, $\rho
a_{1}(1260)$, and $\rho a_{2}(1320)$, are 69.98 MeV and $45.10$ MeV,
respectively. It is worth noting that the typical mode $\pi\rho$ has sizable
contribution to the total width. Thus, $e^{+}e^{-}$ annihilation into
$\pi\rho$ is a perfect process to experimentally search for these $\omega$
states. Similar to the situation of the discussed $\rho$ states, $\omega(5S)$
and $\omega(6S)$ decaying into an open-strange meson pair are not significant.

The decay properties of $\omega(7S)$ we predicted are presented in Fig. 3,
where its dominant decay mode is the $\rho a_{2}(1700)$ channel. Other main
decay modes of $\omega(7S)$ include $\rho a_{2}(1320)$,
$a_{0}(980)\rho(1450)$, $\rho a_{0}(1450)$, and $\rho a_{1}(1260)$. These
partial decay widths of $\omega(7S)$ into an open-strange meson pair are tiny
in our calculation, which are not shown here.

Additionally, we also study three $D$-wave $\omega$ states existing in the
mass range $2.4\sim 3$ GeV. The estimated total decay width and branching
ratios of partial decay of $\omega(4D)$, $\omega(5D)$, and $\omega(6D)$ are
listed in Fig. 3, where the obtained total width of $\omega(4D)$,
$\omega(5D)$, and $\omega(6D)$ are 49.84 MeV, 27.25 MeV, and 15.26 MeV,
respectively. These $D$-wave $\omega$ states mainly decay into $\pi
b_{1}(1235)$, and $\rho a_{0}(1450)$. Other decay channels like
$\pi\rho(1450)$ and $\rho a_{0}(980)$ also have a non-negligible effect on
their total decay widths. Additionally, $\omega(4D)$ and $\omega(5D)$ all have
sizable $\pi\rho$ decay rates, which are more significant than $\omega(6D)$.

####  II.2.3 $\phi$ states

In this subsection, we discuss two-body strong decays of $\phi(4S)$,
$\phi(5S)$, $\phi(6S)$, $\phi(3D)$, $\phi(4D)$, and $\phi(5D)$, which are also
listed in Fig. 3.

From Fig. 3, we can notice that the main decay channels of $\phi(4S)$,
$\phi(5S)$ and $\phi(6S)$ are $KK_{1}(1270)$, $KK^{*}$, $KK^{*}(1410)$, and
$K^{*}K_{1}(1400)$, where the total decay widths of them are 140.87 MeV,
103.79 MeV, and 71.24 MeV, respectively. Due to the sizable contribution of
the channel $KK^{*}$ to the total decay widths of these discussed $\phi$
states, exploring $S$-wave $\phi$ mesonic states via the $e^{+}e^{-}\to
KK^{*}$ process is suggested.

In the same mass range, there are three $D$-wave $\phi$ mesons. In this work,
we also estimate their decay behaviors. $\phi(3D)$ has the total width of
171.56 MeV, where the branching ratio of its $KK_{1}(1270)$ decay channel is
$29.20\%$, which is the main decay channel. Other modes including
$K^{*}K^{*}$, $KK(1460)$, $K^{*}K_{1}(1270)$, and $KK$ are also significant.
The predicted total width of $\phi(4D)$ is 128.95 MeV. The channels of
$KK_{1}(1270)$, $K^{*}K^{*}$, $K^{*}K^{*}(1410)$, and $K^{*}K_{1}(1270)$ have
the branching ratios of $24.18\%$, $12.90\%$, $9.81\%$, and $9.64\%$,
respectively. And then, the obtained total decay width of $\phi(5D)$ is 95.12
MeV in this work. The results presented in Fig. 3 further indicate that the
branching ratio of the $\phi(5D)$ decays into $KK_{1}(1270)$,
$K^{*}K^{*}(1410)$, $K^{*}K^{*}$, and $K^{*}K_{1}(1270)$ can reach up to
$20.54\%$, $12.50\%$, $10.64\%$, and $8.11\%$, respectively.

Through the theoretical calculation given in this work, we find that the decay
mode $K^{*}K_{2}^{*}(1430)$ of the six $\phi$ states discussed above has a
significant contribution to their total decay widths. Thus, measuring the
cross section of $e^{+}e^{-}\to K_{2}^{*}(1430)K\pi$ with energy of collision
up to 3 GeV will be a promising approach to identify these highly excited
$\phi$ states. As shown in Fig. 4 (c), indeed there are many obvious
enhancement structures existing in the collision energy range of 2.4$\sim$2.7
GeV for $e^{+}e^{-}\to K_{2}^{*0}(1430)K^{-}\pi^{+}$, which may correspond to
these discussed $\phi$ mesons. More precise data is expected in future
experiments.

##  III Experimental evidence

When having the theoretical results of masses of these higher states of light
unflavored vector mesons, we can make comparison of our results with the
experimental data as shown in Fig. 4. Firstly, focusing on the data of
$e^{+}e^{-}\to\omega\pi^{+}\pi^{-}\pi^{0}$ BaBar:2006vzy , which has a close
relation with higher $\rho$ mesons due to the $G$-parity conservation, we find
that the invariant mass distribution shows obvious enhancement structures like
the event accumulation around 2.65 GeV. This broad structure overlaps with the
predicted $\rho(6S)$ and $\rho(5D)$ states. There exists a jump point around
2.76 GeV, which may corresponds to $\rho(7S)$. And then, an unnotable
fluctuation around 2.81 GeV just locates at the position of $\rho(6D)$.
Additionally, a bump structure around 2.4 GeV should have relation with
$\rho(5S)$, while around the $\rho(4D)$ state we can also find an obvious jump
point.

The data of $e^{+}e^{-}\to\pi^{+}\pi^{-}\pi^{0}$ BESIII:2019gjz  is an ideal
process to identify $\omega$ states. Although we mark these predicted $\omega$
states in the corresponding cross section data, we cannot identify the obvious
bump structure corresponding to these states since the precision of the
measured data is not enough. Thus, we have to wait for further precise
measurements of the cross section of $e^{+}e^{-}\to\pi^{+}\pi^{-}\pi^{0}$.

In the following, we check four measured processes $e^{+}e^{-}\to
K_{2}^{*0}(1430)K^{-}\pi^{+}$ BaBar:2011btv ,
$e^{+}e^{-}\to\phi\pi^{+}\pi^{-}$ Belle:2008kuo , $e^{+}e^{-}\to\phi
f_{0}(980)$ Belle:2008kuo , and $e^{+}e^{-}\to\phi f_{2}(1525)$ BaBar:2011btv
. For $e^{+}e^{-}\to K_{2}^{*0}(1430)K^{-}\pi^{+}$, there exists abundant
event clusters. Since these predicted higher $\rho$, $\omega$ and $\phi$
states can decay into open-strange meson pairs, it is difficult to distinguish
$\rho$, $\omega$ and $\phi$ states via the $e^{+}e^{-}\to
K_{2}^{*0}(1430)K^{-}\pi^{+}$ channel. Further data analysis of $e^{+}e^{-}\to
K_{2}^{*0}(1430)K^{-}\pi^{+}$ is an interesting research issue similar to the
way adopted in Refs. Wang:2020kte ; Wang:2021gle .

Figure 4: A comparison of theoretical masses of these discussed light vector
$\rho$, $\omega$ and $\phi$ mesons and the experimental data of
$e^{+}e^{-}\to\omega\pi^{+}\pi^{-}\pi^{0}$ BaBar:2006vzy ,
$e^{+}e^{-}\to\pi^{+}\pi^{-}\pi^{0}$ BESIII:2019gjz , $e^{+}e^{-}\to
K_{2}^{*0}(1430)K^{-}\pi^{+}$ BaBar:2011btv ,
$e^{+}e^{-}\to\phi\pi^{+}\pi^{-}$ BaBar:2011btv , $e^{+}e^{-}\to\phi
f_{0}(980)$ BaBar:2011btv , and $e^{+}e^{-}\to\phi f_{2}(1525)$ BaBar:2011btv
. These observed light vector $\rho$, $\omega$, and $\phi$ states below 2.4
GeV collected in PDG PDG:2020ssz  have been studied in our previous work
Wang:2012wa ; He:2013ttg ; Wang:2020kte . Thus, here we do not list these
light vector $\rho$, $\omega$, and $\phi$ states below 2.4 GeV.

Different from $e^{+}e^{-}\to K_{2}^{*0}(1430)K^{-}\pi^{+}$,
$e^{+}e^{-}\to\phi f_{0}(980)$ can be the ideal process to identify these
higher $\phi$ states. Two obvious structures around 2.4 GeV and 2.7 GeV may
correspond to $\phi(4S)$ and $\phi(5S)$, respectively. Of course, we may also
find an event accumulation around 2.5 GeV corresponding to $\phi(3D)$. For
$e^{+}e^{-}\to\phi\pi^{+}\pi^{-}$, the structure at 2.41 GeV and the
enhancement around 2.7 GeV may have relation to the $\phi(4S)$,
$\phi(5S)/\phi(4D)$. For $e^{+}e^{-}\to\phi\pi^{+}\pi^{-}$, the broad
structure around 2.65 GeV contains the information of the predicted
$\phi(5S)/\phi(4D)$.

In fact, this simple comparison of the experimental data and masses of these
discussed vector states only reflects some possible evidences of these
predicted higher states of light unflavored vector meson. For further
illustrating it, in the following, we take two processes
$e^{+}e^{-}\to\omega\pi^{+}\pi^{-}\pi^{0}$ and $e^{+}e^{-}\to\phi f_{0}(980)$
as examples to make further analysis.

In general, the cross section can be parametrized as the coherent sum of an
$s$-dependent continuum amplitude and a resonant amplitude described by a
Breit-Wigner function BESIII:2020kpr , i.e.,

$\displaystyle\sigma(s)=\left|a\,e^{-b(\sqrt{s}-M_{th})}+\sum_{R}BW_{R}\left(\sqrt{s}\right)\,e^{i\phi_{R}}\right|^{2},$
(25)

where $a$ and $b$ are the continuum parameters, $M_{th}$ is the sum of the
masses of the final state particles, and $\phi_{R}$ is the phase angle between
the amplitudes. The Breit-Wigner function is given by

$\displaystyle
BW_{R}(\sqrt{s})=\frac{\sqrt{12\pi\Gamma^{e^{+}e^{-}}_{R}\mathcal{B}_{R}\Gamma^{\rm{tot}}_{R}}}{s-M_{R}^{2}+iM_{R}\Gamma^{\rm{tot}}_{R}},$
(26)

where $M_{R}$, $\Gamma^{e^{+}e^{-}}_{R}$, and $\Gamma^{\rm{tot}}_{R}$ denote
the mass, partial width to $e^{+}e^{-}$, and total width of the assumed
intermediate resonance $R$. $\mathcal{B}_{R}$ is the branching ratio for
$R\to\rm{(final}$ $\rm{states)}$.

Table 3: The fitting parameters of depicting the Born cross section of $e^{+}e^{-}\to\omega\pi^{+}\pi^{-}\pi^{0}$. Parameters |  | Solution
---|---|---
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(5S)\to\omega\pi^{+}\pi^{-}\pi^{0})$ (eV) |  | $3.38\pm 0.96$
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(6S)\to\omega\pi^{+}\pi^{-}\pi^{0})$ (eV) |  | $0.62\pm 0.44$
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(5D)\to\omega\pi^{+}\pi^{-}\pi^{0})$ (eV) |  | $0.86\pm 0.52$
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\rho(6D)\to\omega\pi^{+}\pi^{-}\pi^{0})$ (eV) |  | $0.96\pm 0.52$
$\phi_{\rho(5S)}$ (rad) |  | $4.76\pm 0.21$
$\phi_{\rho(6S)}$ (rad) |  | $3.92\pm 0.44$
$\phi_{\rho(5D)}$ (rad) |  | $5.52\pm 0.35$
$\phi_{\rho(6D)}$ (rad) |  | $5.20\pm 0.38$
$a$ ($\times 10^{-3}$ $\rm{GeV}^{-2}$) |  | $-4.08\pm 0.05$
$b$ |  | $0.54\pm 0.01$
$\chi^{2}/\rm{d.o.f}$ |  | $0.60$
Table 4: The fitting parameters of depicting the Born cross section of $e^{+}e^{-}\to\phi f_{0}(980)$. Parameters |  | Solution
---|---|---
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\phi(5S)\to\phi f_{0}(980))$ (eV) |  | $1.36\pm 0.0.31$
$\Gamma_{e^{+}e^{-}}\mathcal{B}(\phi(6S)\to\phi f_{0}(980))$ (eV) |  | $0.94\pm 0.32$
$\phi_{\phi(5S)}$ (rad) |  | $4.32\pm 0.21$
$\phi_{\phi(6S)}$ (rad) |  | $3.86\pm 0.17$
$a$ ($\times 10^{-3}$ $\rm{GeV}^{-2}$) |  | $-0.92\pm 0.004$
$b$ |  | $1.92\pm 0.09$
$\chi^{2}/\rm{d.o.f}$ |  | $0.49$

With this preparation, we first fit the measured cross section of
$e^{+}e^{-}\to\omega\pi^{+}\pi^{-}\pi^{0}$ BaBar:2006vzy . The comparison in
Fig. 4 (a) indicates that $\rho(5S)$, $\rho(6S)$, $\rho(5D)$ and $\rho(6D)$
should be considered in our fit, where the resonance parameters for these
vector states are taken from our calculation given in Sec. II and the fitted
parameters are listed in Table 3. As shown in Fig. 5 (a), we can depict the
data of the cross section of $e^{+}e^{-}\to\omega\pi^{+}\pi^{-}\pi^{0}$
BaBar:2006vzy , by which the values of
$\Gamma_{R}^{e^{+}e^{-}}\mathcal{B}_{R}$ for these considered light unflavored
vector mesons can be obtained. This information is valuable to further study
around these light unflavored vector mesons.

Along this line, we may study the $e^{+}e^{-}\to\phi f_{0}(980)$ process
BaBar:2011btv . The fitted result can be found in Fig. 5 (b). When considering
the contributions from $\phi(5S)$ and $\phi(6S)$, we can describe the
experimental data well, where the fitting parameters are listed in Table 4.

We expect more precise data, which can be applied to identify these discussed
higher light unflavored vector mesons.

Figure 5: The fitted result of the cross section of (a)
$e^{+}e^{-}\to\omega\pi^{+}\pi^{-}\pi^{0}$ and (b) $e^{+}e^{-}\to\phi
f_{0}(980)$. The black dots with the error bar are experimental data
BaBar:2006vzy ; BaBar:2011btv .

##  IV summary

In the past decades, the process of $e^{+}e^{-}$ annihilation into light
mesons can be as an ideal platform to probe the light vector mesonic state. A
typical example is the observation of $Y(2175)$, which had stimulated
theorists’ interest in revealing its inner structure Wang:2021gle ;
Wang:2006ri ; Ding:2007pc ; MartinezTorres:2008gy ; Chen:2008ej ;
Klempt:2007cp . In recent years, BESIII, as the main force of studying light
hadron spectroscopy, has accumulated more and more data of $e^{+}e^{-}$
annihilation into light mesons BESIII:2019gjz ; BESIII:2021bjn ;
BESIII:2021yam ; BESIII:2018ldc ; BESIII:2019ebn ; BESIII:2020gnc ;
BESIII:2020kpr ; BESIII:2020vtu ; BESIII:2020xmw . Especially, these
experimental measurements from BESIII make that constructing light unflavored
vector mesons around 2 GeV become possible BESIII:2018ldc ; BESIII:2020vtu ;
Wang:2020kte ; Wang:2021gle ; BESIII:2020xmw .

Although big progress has been made by both experimentalists and theorists,
our knowledge of these light unflavored vector mesons existing in the mass
range of $2.4\sim 3$ GeV is not abundant, which inspires our attention.

In this work, we obtain the masses and decay widths of these discussed higher
$\rho$, $\omega$, and $\phi$ states in the $2.4\sim 3$ GeV mass range and make
a rough comparison with the present experimental data as shown in Fig. 4.
Although the present precision of experimental data is not enough to conclude
whether these event clusters can be identified as these predicted states, we
still want to show this possible evidence. With the improvement of
experimental precision, we suggest our experimental colleagues pay attention
to these predicted higher vector states.

As emphasized in the recent white paper released by BESIII BESIII:2020nme ,
the study around light hadrons will still be one of the tasks of the BESIII
experiment in the next ten years. According to our study, we strongly suggest
our experimental colleagues pay attention to the $2.4\sim 3$ GeV energy range
of collision, which has a close relation to these predicted light unflavored
vector meson states.

## Acknowledgments

L.-M.W. would like to thank Lanzhou Center for Theoretical Physics to support
his stay at Lanzhou University, where this work was finished. Additionally,
L.-M. W. would like to thank Qin-Song Zhou for his help of fitting the
experimental data. This work is supported by the China National Funds for
Distinguished Young Scientists under Grant No. 11825503, National Key Research
and Development Program of China under Contract No. 2020YFA0406400, the 111
Project under Grant No. B20063, the National Natural Science Foundation of
China under Grant No. 12047501, and the projects funded by Science and
Technology Department of Qinghai Province Project No. 2020-ZJ-728.

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