# Deciphering the light vector meson contribution to the cross sections of
$e^{+}e^{-}$ annihilations into the open-strange channels through a combined
analysis

Jun-Zhang Wang1,2<EMAIL_ADDRESS>Li-Ming Wang3<EMAIL_ADDRESS>Xiang
Liu1,2,4111Corresponding author<EMAIL_ADDRESS>Takayuki Matsuki5
<EMAIL_ADDRESS>1School of Physical Science and Technology, Lanzhou
University, Lanzhou 730000, China
2Research Center for Hadron and CSR Physics, Lanzhou University $\&$ Institute
of Modern Physics of CAS, Lanzhou 730000, China
3Key Laboratory for Microstructural Material Physics of Hebei Province, School
of Science, Yanshan University, Qinhuangdao 066004, China
4Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics
of Gansu Province, and Frontiers Science Center for Rare Isotopes, Lanzhou
University, Lanzhou 730000, China
5Tokyo Kasei University, 1-18-1 Kaga, Itabashi, Tokyo 173-8602, Japan

###### Abstract

In this work, we perform a combined analysis to the measured data of the cross
section of open-strange processes $e^{+}e^{-}\to K^{+}K^{-}$, $e^{+}e^{-}\to
K\bar{K}^{*}+c.c.$, $e^{+}e^{-}\to K^{*+}K^{*-}$, $e^{+}e^{-}\to
K_{1}(1270)^{+}K^{-}$, $e^{+}e^{-}\to K_{1}(1400)^{+}K^{-}$, $e^{+}e^{-}\to
K_{2}^{*}(1430)\bar{K}+c.c.$ and $e^{+}e^{-}\to K(1460)^{+}K^{-}$ with the
support of study of hadron spectroscopy. We reveal the contribution of the
possible light vector mesons around 2 GeV to reproduce the cross section data
of the reported open-strange processes from $e^{+}e^{-}$ annihilation which
may provide a new perspective to construct the light vector meson family and
understand the $Y(2175)$.

##  I Introduction

Recently, the BESIII Collaboration released the data of the cross section for
the open-strange process $e^{+}e^{-}\to K^{+}K^{-}$, where an obvious
enhancement at 2.2 GeV was observed and its Breit-Wigner resonance parameters
were measured to be $M=2239.2\pm 7.1\pm 11.3$ MeV and $\Gamma=139.8\pm 12.3\pm
20.6$ MeV Ablikim:2018iyx . This is the first evidence showing the observation
of the vector structure around 2.2 GeV in the open-strange channel. After
that, BESIII reported the partial-wave analysis result of a new open-strange
process $e^{+}e^{-}\to K^{+}K^{-}\pi^{0}\pi^{0}$, where the Born cross
sections for the subprocesses $e^{+}e^{-}\to
K^{+}(1460)K^{-},K^{+}_{1}(1400)K^{-},K^{+}_{1}(1270)K^{-}$, and
$K^{*+}K^{*-}$ were measured Ablikim:2020pgw . By performing an overall fit to
the above four processes, a structure with the mass of $2126.5\pm 16.8\pm
12.4$ MeV and the width of $106.9\pm 32.1\pm 28.1$ MeV was found, although a
limited significance appears in the process $e^{+}e^{-}\to
K^{+}_{1}(1270)K^{-}$ and $K^{*+}K^{*-}$ Ablikim:2020pgw . These new open-
strange processes again indicate the existence of a vector structure around
2.2 GeV. When checking the data collected by the Particle Data Group (PDG)
Zyla:2020zbs , one finds that the resonance parameter of this structure
observed in these open-strange processes is close to the corresponding average
value of that of the $Y(2175)$. Therefore, such simple comparison may suggest
that this newly observed vector structure around 2.2 GeV and the $Y(2175)$ are
the same state.

As we all know, the $Y(2175)$ was first reported in the process
$e^{+}e^{-}\to\phi f_{0}(980)$ by the BaBar Collaboration based on the
initial-state-radiation (ISR) method Aubert:2006bu , which was later confirmed
in the same process by Belle Shen:2009zze  and in the process
$J/\psi\to\eta\phi f_{0}(980)$ by BES Ablikim:2007ab  and BESIII
Ablikim:2014pfc . Different from most of the observed charmoniumlike $Y$
states, the $Y(2175)$ is the only light-flavor $Y$ particle, which was
reported in the hidden-strange final states and still remains a mystery even
though more than a decade have passed since its discovery. Thus, it has
inspired theorist’s great interest in exploring its inner structure. In the
past years, the theoretical explanations for the $Y(2175)$ include hybrid
$s\bar{s}g$ Ding:2006ya , vector strangeonium state $\phi(3S)$ Barnes:2002mu ;
Pang:2019ttv  and $\phi(2D)$ Wang:2012wa ; Ding:2007pc ; Pang:2019ttv , which
should favor the open-strange decay modes. However, the recent BESIII
measurement of the $e^{+}e^{-}\to K^{+}K^{-}$ Ablikim:2018iyx  and
$e^{+}e^{-}\to K^{+}K^{-}\pi^{0}\pi^{0}$ Ablikim:2020pgw  reactions indicates
that it is hard to understand these open-strange experimental data under the
hybrid and strangeonium assignments to the $Y(2175)$.

If the $Y(2175)$ is the $s\bar{s}g$ hybrid state, the dominant decay modes
should be $K_{1}(1270)^{+}K^{-}$ and $K_{1}(1400)^{+}K^{-}$ Ding:2006ya ,
while two open-strange decay channels like $K^{+}K^{-}$ and $K^{*+}K^{*-}$
should be strongly suppressed according to the flux tube model analysis
Close:1994hc ; Close:2003mb  and the QCD sum rule calculation Zhu:1998sv ;
Zhu:1999wg . On the other hand, different predictions for the branching ratios
of the open-strange decay modes of strangeonium states $\phi(3S)$ and
$\phi(2D)$ were given in former theoretical studies. For example, the
branching ratio of $\phi(3S)\to K^{+}K^{-}$ was calculated to be almost zero
in Ref. Barnes:2002mu , and instead, $K_{1}(1270)^{+}K^{-}$ and
$K^{*+}{K}^{*-}$ were predicted to be dominant decay channels of $\phi(3S)$
and $\phi(2D)$ in Refs. Barnes:2002mu ; Ding:2007pc ; Pang:2019ttv .
Furthermore, the theoretical total widths of $\phi(3S)$ and $\phi(2D)$ are
generally estimated to be $200\sim 400$ MeV Barnes:2002mu ; Pang:2019ttv ;
Wang:2012wa ; Ding:2007pc , which obviously deviates from the present
experimental data of the $Y(2175)$ Zyla:2020zbs . Therefore, the BESIII data
Ablikim:2018iyx ; Ablikim:2020pgw  does not support the above theoretical
predictions, which becomes a big challenge for the $s\bar{s}$ or $s\bar{s}g$
assignment to the $Y(2175)$. In addition to the above difficulties in decoding
the $Y(2175)$ structure, there are still two confusing problems, which should
be mentioned here. Through a comparison among the cross sections of the
reported open-strange processes from $e^{+}e^{-}$ annihilation, we can see
that although the resonance parameter of the observed vector structure around
2.2 GeV in $e^{+}e^{-}\to K^{+}K^{-}$ Ablikim:2018iyx  is similar to that in
$e^{+}e^{-}\to K^{+}K^{-}\pi^{0}\pi^{0}$ Ablikim:2020pgw , there is still
about 100 MeV deviation on these measured masses. Hence, this measured mass
discrepancy problem should be clarified. In addition, the BESIII measurement
shows no observation of the structure around 2.2 GeV in a typical open-strange
$K^{*}K^{*}$ mode Ablikim:2020pgw , which also should be appropriately
understood.

When facing this puzzling situation, let us firstly return to the open-strange
reaction itself $e^{+}e^{-}\to K^{(*/\prime)}_{(J)}\bar{K}^{(*/\prime)}_{(J)}$
($e^{+}e^{-}\to K^{+}K^{-}$, $e^{+}e^{-}\to K\bar{K}^{*}+c.c.$, $e^{+}e^{-}\to
K^{*+}K^{*-}$, $e^{+}e^{-}\to K_{1}(1270)^{+}K^{-}$, $e^{+}e^{-}\to
K_{1}(1400)^{+}K^{-}$, $e^{+}e^{-}\to K_{2}^{*}(1430)\bar{K}+c.c.$ and
$e^{+}e^{-}\to K(1460)^{+}K^{-}$). In fact, the open-strange process from
$e^{+}e^{-}$ annihilation at center-of-mass energy $\sqrt{s}\sim 2$ GeV is
mediated by different intermediate vector mesons. Since there is no isospin
restriction, the highly excited $\rho$, $\omega$, and $\phi$ meson states
around 2 GeV may have contributions to the cross sections of the discussed
open-strange processes, which makes the whole analysis quite complicated.
Therefore, when facing the enhancement or dip structure around 2.2 GeV
observed in both $e^{+}e^{-}\to K^{+}K^{-}$ and $e^{+}e^{-}\to
K^{+}K^{-}\pi^{0}\pi^{0}$ Ablikim:2018iyx ; Ablikim:2020pgw , we see that it
is a rough treatment to consider only a simple Breit-Wigner fit to the cross
section data of the discussed open-strange processes from $e^{+}e^{-}$
annihilation Ablikim:2018iyx ; Ablikim:2020pgw ; BaBar:2013jqz ; Aubert:2007ym
, which may inevitably lead to a puzzling situation mentioned above.

Obviously, we need to carry out a combined analysis to the present measured
data of cross sections producing open-strange channels via $e^{+}e^{-}$
annihilation Ablikim:2018iyx ; Ablikim:2020pgw ; BaBar:2013jqz ; Aubert:2007ym
, which is supported by the study of hadron spectroscopy. In fact, this
approach was applied to construct higher charmonia above 4 GeV Wang:2019mhs ;
Wang:2020prx ; Wang:2018rjg . In this work, the hadron spectrum analysis is
based on an unquenched potential model Pang:2017dlw ; Song:2015nia ;
Song:2015fha ; Wang:2016krl ; Wang:2019mhs ; Wang:2020prx ; Wang:2018rjg , by
which we select suitable light vector mesons with mass around 2 GeV involved
in the discussed processes. Of course, this mass spectrum analysis
simultaneously provides the numerical spatial wave functions of the selected
light vector mesons, which are applied to calculate their two-body open-
strange decay widths and di-lepton widths. With this preparation supported by
the meson spectroscopy, the main task of the present work is to perform a
combined analysis to the experimental cross section data of $e^{+}e^{-}$
annihilations into open-strange channels, which can provide valuable
information of light vector meson contribution to depict the experimental
cross section data of $e^{+}e^{-}$ annihilation into open-strange channels.
Finally, we show later that the above puzzling situation can be clarified,
which also provides a new perspective to construct the light vector meson
family and understand the $Y(2175)$.

This paper is organized as follows. After Introduction, we discuss the cross
section of open-strange processes $e^{+}e^{-}\to
K^{(*/\prime)}_{(J)}\bar{K}^{(*/\prime)}_{(J)}$ by considering the direct
production and light vector meson contributions in Sec. II. Furthermore, we
illustrate how to obtain the mass spectrum and decay behaviors of light vector
mesons in Sec. III. In Sec. IV, based on our theoretical predictions for the
branching ratios of the open-strange strong decay modes and di-lepton widths
of higher light vector mesons above 2 GeV, we find that higher light vector
meson states play an important role to perform a combined analysis of the
cross sections of $e^{+}e^{-}$ annihilations into the open-strange processes.
Finally, this work ends with a summary in Sec. V.

##  II  cross sections of open-strange processes $e^{+}e^{-}\to
K^{(*/\prime)}_{(J)}\bar{K}^{(*/\prime)}_{(J)}$ around 2.0 GeV

Figure 1:  The schematic diagrams for depicting the open-strange process
$e^{+}e^{-}\to K^{(*/\prime)}_{(J)}\bar{K}^{(*/\prime)}_{(J)}$. Diagram (a)
corresponds to a direct annihilation process, while diagram (b) is the
resonance contributions from the intermediate $\rho$, $\omega$, $\phi$ state.

In this section, we focus on the open-strange reactions $e^{+}e^{-}\to
K^{+}{K}^{-}$ Ablikim:2018iyx ; BaBar:2013jqz , $e^{+}e^{-}\to
K\bar{K}^{*}+c.c.$ Aubert:2007ym , $e^{+}e^{-}\to K^{*+}K^{*-}$
Ablikim:2020pgw , $e^{+}e^{-}\to K_{1}(1270)^{+}K^{-}$ Ablikim:2020pgw ,
$e^{+}e^{-}\to K_{1}(1400)^{+}K^{-}$ Ablikim:2020pgw , $e^{+}e^{-}\to
K_{2}^{*}(1430)\bar{K}+c.c.$ Aubert:2007ym  and $e^{+}e^{-}\to
K(1460)^{+}K^{-}$ Ablikim:2020pgw  with center-of-mass (CM) energy around 2.0
GeV and illustrate how to prensent the cross section. In addition to the
$K^{*}K^{*}$, $K_{1}(1270)K$, $K_{1}(1400)K$ and $K(1460)K$ channels measured
by BESIII for the first time, there also exist the experimental results of
several other open-strange channels, such as $KK^{*}$ and $K_{2}(1430)^{*}K$,
where their measured results were reported by the BaBar Collaboration in 2008
Aubert:2007ym . As shown in Fig. 1, there exist two mechanisms working
together for the $e^{+}e^{-}\to
K^{(*/\prime)}_{(J)}\bar{K}^{(*/\prime)}_{(J)}$ process. The first one is the
direct annihilation of $e^{+}e^{-}$ into open-strange channel, where the
virtual photon directly couples with the final states
$K^{(*/\prime)}_{(J)}\bar{K}^{(*/\prime)}_{(J)}$, which provides the
background contribution. The second one occurs via the intermediate light
vector meson states, which include the $\rho,\omega$, and $\phi$-like
resonances.

In this work, the effective Lagrangian approach is adopted to calculate the
discussed open-strange process $e^{+}e^{-}\to
K^{(*/\prime)}_{(J)}\bar{K}^{(*/\prime)}_{(J)}$ as shown in Fig. 1. The
effective Lagrangian densities involved in the concrete works include
Bauer:1975bv ; Bauer:1975bw ; Chen:2011cj ; Huang:2013jda ; Xie:2015wja

$\displaystyle\mathcal{L}_{KK\gamma}$ $\displaystyle=$ $\displaystyle
ieA^{\mu}(\bar{K}\partial_{\mu}K-\partial_{\mu}\bar{K}K),$
$\displaystyle\mathcal{L}_{KK^{*}\gamma}$ $\displaystyle=$ $\displaystyle
e\varepsilon^{\mu\nu\rho\sigma}\partial_{\mu}A_{\nu}(\bar{K}\partial_{\rho}K^{*}_{\sigma}+\partial_{\rho}\bar{K}^{*}_{\sigma}K),$
$\displaystyle\mathcal{L}_{K^{*}K^{*}\gamma}$ $\displaystyle=$ $\displaystyle
ie(A^{\mu}(\bar{K}^{*}_{\nu}\overleftrightarrow{\partial}_{\mu}K^{*\nu})+\bar{K}^{*\mu}(K^{*}_{\nu}\overleftrightarrow{\partial}_{\mu}A^{\nu})$
$\displaystyle+(A^{\nu}\overleftrightarrow{\partial}_{\mu}\bar{K}^{*}_{\nu})K^{*\mu}),$
$\displaystyle\mathcal{L}_{KK_{1}\gamma}$ $\displaystyle=$ $\displaystyle
ieA_{\mu}(\bar{K}K_{1}^{\mu}-\bar{K}_{1}^{\mu}K),$
$\displaystyle\mathcal{L}_{KK_{2}\gamma}$ $\displaystyle=$ $\displaystyle
e\varepsilon_{\mu\nu\rho\sigma}\partial^{\rho}A^{\sigma}(\partial_{\delta}\bar{K}\partial^{\mu}K_{2}^{\nu\delta}+\partial^{\mu}\bar{K}_{2}^{\nu\delta}\partial_{\delta}K),$
$\displaystyle\mathcal{L}_{KK^{\prime}\gamma}$ $\displaystyle=$ $\displaystyle
ieA^{\mu}(\bar{K}\overleftrightarrow{\partial}_{\mu}K^{\prime}+c.c.),$ (1)
$\displaystyle\mathcal{L}_{\gamma\mathcal{V}}$ $\displaystyle=$
$\displaystyle\frac{-em_{\mathcal{V}}^{2}}{f_{\mathcal{V}}}\mathcal{V}_{\mu}A^{\mu},$
$\displaystyle\mathcal{L}_{\mathcal{V}KK}$ $\displaystyle=$ $\displaystyle
ig_{\mathcal{V}KK}(\bar{K}\partial_{\mu}K-\partial_{\mu}\bar{K}K)\mathcal{V}^{\mu},$
$\displaystyle\mathcal{L}_{\mathcal{V}KK^{*}}$ $\displaystyle=$ $\displaystyle
g_{\mathcal{V}KK^{*}}\varepsilon^{\mu\nu\rho\sigma}(\bar{K}\partial_{\rho}K^{*}_{\sigma}+\partial_{\rho}\bar{K}^{*}_{\sigma}K)\partial_{\mu}\mathcal{V}_{\nu},$
$\displaystyle\mathcal{L}_{\mathcal{V}K^{*}K^{*}}$ $\displaystyle=$
$\displaystyle
ig_{\mathcal{V}K^{*}K^{*}}((\bar{K}^{*}_{\nu}\overleftrightarrow{\partial}_{\mu}K^{*\nu})\mathcal{V}^{\mu}+\bar{K}^{*\mu}(K^{*}_{\nu}\overleftrightarrow{\partial}_{\mu}\mathcal{V}^{\nu})$
$\displaystyle+(\mathcal{V}^{\nu}\overleftrightarrow{\partial}_{\mu}\bar{K}^{*}_{\nu})K^{*\mu}),$
$\displaystyle\mathcal{L}_{\mathcal{V}KK_{1}}$ $\displaystyle=$ $\displaystyle
ig_{\mathcal{V}KK_{1}}(\bar{K}K_{1}^{\mu}-\bar{K}_{1}^{\mu}K)\mathcal{V}_{\mu},$
$\displaystyle\mathcal{L}_{\mathcal{V}KK_{2}}$ $\displaystyle=$ $\displaystyle
g_{\mathcal{V}KK_{2}}\varepsilon_{\mu\nu\rho\sigma}(\partial_{\delta}\bar{K}\partial^{\mu}K_{2}^{\nu\delta}+\partial^{\mu}\bar{K}_{2}^{\nu\delta}\partial_{\delta}K)\partial^{\rho}\mathcal{V}^{\sigma},$
$\displaystyle\mathcal{L}_{\mathcal{V}KK^{\prime}}$ $\displaystyle=$
$\displaystyle
ig_{\mathcal{V}KK^{\prime}}(\bar{K}\overleftrightarrow{\partial}_{\mu}K^{\prime}+c.c.)\mathcal{V}^{\mu},$
(2)

where $K_{1}$, $K_{2}$, and $K^{\prime}$ stand for the kaon meson fields of
$K_{1}(1270)/K_{1}(1400)$, $K_{2}^{*}(1430)$ and $K(1460)$, respectively. And,
$\mathcal{V}$ is the intermediate light vector meson field.
$g_{\mathcal{V}K^{(*/\prime)}_{(J)}K^{(*/\prime)}_{(J)}}$ is the corresponding
coupling constants involved in the $\mathcal{V}$ state and the open-strange
channel.

The scattering amplitudes of seven open-strange processes $e^{+}e^{-}\to KK$,
$e^{+}e^{-}\to KK^{*}$, $e^{+}e^{-}\to K^{*}K^{*}$, $e^{+}e^{-}\to
K_{1}(1270)K$, $e^{+}e^{-}\to K_{1}(1400)K$, $e^{+}e^{-}\to K_{2}^{*}(1430)K$,
$e^{+}e^{-}\to K(1460)K$ generally depicted in in Fig. 1 can be written as

$\displaystyle\mathcal{M}_{\mathrm{Dir}}^{KK^{(\prime)}}$ $\displaystyle=$
$\displaystyle[\bar{v}(k_{2})ie\gamma^{\mu}u(k_{1})]\frac{-g_{\mu\nu}}{q^{2}}[-e(p_{4}^{\nu}-p_{3}^{\nu})F_{KK^{(\prime)}}(q^{2})],$
$\displaystyle\mathcal{M}_{\mathrm{\mathcal{V}}}^{KK^{(\prime)}}$
$\displaystyle=$
$\displaystyle[\bar{v}(k_{2})ie\gamma_{\mu}u(k_{1})]\frac{-g^{\mu\rho}}{q^{2}}\frac{-em_{\mathcal{V}}^{2}}{f_{\mathcal{V}}}\frac{-g_{\rho\nu}+q_{\rho}q_{\nu}/m_{\mathcal{V}}^{2}}{q^{2}-m_{\mathcal{V}}^{2}+im_{\mathcal{V}}\Gamma_{\mathcal{V}}}$
(3)
$\displaystyle\times[-g_{\mathcal{V}KK^{(\prime)}}(p_{4}^{\nu}-p_{3}^{\nu})],$
$\displaystyle\mathcal{M}_{\mathrm{Dir}}^{KK^{*}}$ $\displaystyle=$
$\displaystyle[\bar{v}(k_{2})ie\gamma^{\mu}u(k_{1})]\frac{-g_{\mu\nu}}{q^{2}}[e\varepsilon^{\alpha\nu\rho\sigma}q_{\alpha}p_{4\rho}\epsilon_{K^{*}\sigma}^{*}F_{KK^{*}}(q^{2})],$
$\displaystyle\mathcal{M}_{\mathrm{\mathcal{V}}}^{KK^{*}}$ $\displaystyle=$
$\displaystyle[\bar{v}(k_{2})ie\gamma_{\mu}u(k_{1})]\frac{-g^{\mu\rho}}{q^{2}}\frac{-em_{\mathcal{V}}^{2}}{f_{\mathcal{V}}}\frac{-g_{\rho\nu}+q_{\rho}q_{\nu}/m_{\mathcal{V}}^{2}}{q^{2}-m_{\mathcal{V}}^{2}+im_{\mathcal{V}}\Gamma_{\mathcal{V}}}$
(4)
$\displaystyle\times[g_{\mathcal{V}KK^{*}}\varepsilon^{\alpha\nu\omega\sigma}q_{\alpha}p_{4\omega}\epsilon_{K^{*}\sigma}^{*}],$
$\displaystyle\mathcal{M}_{\mathrm{Dir}}^{K^{*}K^{*}}$ $\displaystyle=$
$\displaystyle[\bar{v}(k_{2})ie\gamma^{\mu}u(k_{1})]\frac{-g_{\mu\nu}}{q^{2}}[-e\Big{(}g^{\alpha\beta}(p_{4}^{\nu}-p_{3}^{\nu})-g^{\nu\beta}q^{\alpha}$
$\displaystyle+g^{\nu\alpha}q^{\beta}+g^{\nu\alpha}p_{3}^{\beta}-g^{\nu\beta}p_{4}^{\alpha}\Big{)}\epsilon_{K^{*}\alpha}^{*}\epsilon_{K^{*}\beta}^{*}F_{K^{*}K^{*}}(q^{2})],$
$\displaystyle\mathcal{M}_{\mathrm{\mathcal{V}}}^{K^{*}K^{*}}$
$\displaystyle=$
$\displaystyle[\bar{v}(k_{2})ie\gamma_{\mu}u(k_{1})]\frac{-g^{\mu\rho}}{q^{2}}\frac{-em_{\mathcal{V}}^{2}}{f_{\mathcal{V}}}\frac{-g_{\rho\nu}+q_{\rho}q_{\nu}/m_{\mathcal{V}}^{2}}{q^{2}-m_{\mathcal{V}}^{2}+im_{\mathcal{V}}\Gamma_{\mathcal{V}}}$
(5)
$\displaystyle\times[-g_{\mathcal{V}K^{*}K^{*}}\Big{(}g^{\alpha\beta}(p_{4}^{\nu}-p_{3}^{\nu})-g^{\nu\beta}q^{\alpha}+g^{\nu\alpha}q^{\beta}$
$\displaystyle+g^{\nu\alpha}p_{3}^{\beta}-g^{\nu\beta}p_{4}^{\alpha}\Big{)}\epsilon_{K^{*}\alpha}^{*}\epsilon_{K^{*}\beta}^{*}],$
$\displaystyle\mathcal{M}_{\mathrm{Dir}}^{KK_{1}}$ $\displaystyle=$
$\displaystyle[\bar{v}(k_{2})ie\gamma^{\mu}u(k_{1})]\frac{-g_{\mu\nu}}{q^{2}}[ie\epsilon_{K_{1}}^{\nu*}F_{KK_{1}}(q^{2})],$
$\displaystyle\mathcal{M}_{\mathrm{\mathcal{V}}}^{KK_{1}}$ $\displaystyle=$
$\displaystyle[\bar{v}(k_{2})ie\gamma_{\mu}u(k_{1})]\frac{-g^{\mu\rho}}{q^{2}}\frac{-em_{\mathcal{V}}^{2}}{f_{\mathcal{V}}}\frac{-g_{\rho\nu}+q_{\rho}q_{\nu}/m_{\mathcal{V}}^{2}}{q^{2}-m_{\mathcal{V}}^{2}+im_{\mathcal{V}}\Gamma_{\mathcal{V}}}$
(6) $\displaystyle\times[ig_{\mathcal{V}KK_{1}}\epsilon_{K_{1}}^{\nu*}],$
$\displaystyle\mathcal{M}_{\mathrm{Dir}}^{KK_{2}}$ $\displaystyle=$
$\displaystyle[\bar{v}(k_{2})ie\gamma^{\mu}u(k_{1})]\frac{-g_{\mu\nu}}{q^{2}}[ie\varepsilon^{\alpha\beta\rho\nu}p_{3}^{\delta}q_{\rho}p_{4\alpha}\epsilon_{K_{2}\beta\delta}^{*}$
$\displaystyle\times F_{KK_{2}}(q^{2})],$
$\displaystyle\mathcal{M}_{\mathrm{\mathcal{V}}}^{KK_{2}}$ $\displaystyle=$
$\displaystyle[\bar{v}(k_{2})ie\gamma_{\mu}u(k_{1})]\frac{-g^{\mu\rho}}{q^{2}}\frac{-em_{\mathcal{V}}^{2}}{f_{\mathcal{V}}}\frac{-g_{\rho\nu}+q_{\rho}q_{\nu}/m_{\mathcal{V}}^{2}}{q^{2}-m_{\mathcal{V}}^{2}+im_{\mathcal{V}}\Gamma_{\mathcal{V}}}$
(7)
$\displaystyle\times[ig_{\mathcal{V}KK_{2}}\varepsilon^{\alpha\beta\omega\nu}p_{3}^{\delta}q_{\omega}p_{4\alpha}\epsilon_{K_{2}\beta\delta}^{*}],$

where $q^{\mu}=(\sqrt{s},0,0,0)$ and $p_{3}$ and $p_{4}$ are the four-momenta
of final states $K^{(*/\prime)}_{(J)}$/$\bar{K}^{(*/\prime)}_{(J)}$.
Additionally, the time-like form factor
$F_{K^{(*/\prime)}_{(J)}K^{(*/\prime)}_{(J)}}(q^{2})$ is introduced when
depicting the direct production process. Generally speaking, the expression of
the form factor in the time-like region is relatively complicated. For
simplicity, we adopt a universal form factor
$F_{K^{(*/\prime)}_{(J)}K^{(*/\prime)}_{(J)}}(q^{2})=f_{\mathrm{Dir}}\,e^{-aq^{2}}$
with free parameters $f_{\mathrm{Dir}}$ and $a$ Chen:2020xho , which is an
approximate description for the cross section of a direct production process
in a narrow energy region between $\sqrt{s}=2.0\sim 2.6$ GeV. In addition,
$m_{\mathcal{V}}$ and $\Gamma_{\mathcal{V}}$ are resonance parameters of the
selected intermediate light vector meson states, which can be fixed by the
corresponding experimental values or theoretical predictions. The total
amplitude of $e^{+}e^{-}\to K^{(*/\prime)}_{(J)}\bar{K}^{(*/\prime)}_{(J)}$
can be written as the sum of a direct amplitude and different resonance
contributions, i.e.,

$\displaystyle\mathcal{M}_{\mathrm{Total}}^{K^{(*/\prime)}_{(J)}\bar{K}^{(*/\prime)}_{(J)}}=\mathcal{M}_{\mathrm{Direct}}^{K^{(*/\prime)}_{(J)}\bar{K}^{(*/\prime)}_{(J)}}+e^{i\theta_{n}}\sum_{\mathcal{V}_{n}}\mathcal{M}_{\mathcal{V}_{n}}^{K^{(*/\prime)}_{(J)}\bar{K}^{(*/\prime)}_{(J)}},$
(8)

where $\mathcal{V}_{n}$ stands for the selected intermediate vector
resonances, and $\theta_{n}$ is the phase angle between the direct
annihilation amplitude and the intermediate light vector meson contribution.
With the above total amplitude, the cross section of $e^{+}e^{-}\to
K^{(*/\prime)}_{(J)}\bar{K}^{(*/\prime)}_{(J)}$ can be calculated directly by

$\displaystyle\sigma(e^{+}e^{-}\to
K^{(*/\prime)}_{(J)}\bar{K}^{(*/\prime)}_{(J)})=\int\frac{1}{64\pi
s}\frac{1}{\left|p_{3\mathrm{cm}}\right|^{2}}\overline{\left|\mathcal{M}_{\mathrm{Total}}^{K^{(*/\prime)}_{(J)}\bar{K}^{(*/\prime)}_{(J)}}\right|^{2}}dt,$

where $p_{3\mathrm{cm}}$ stands for the corresponding momentum of $p_{3}$ in
the CM frame of a reaction.

Before depicting the cross section of our discussed open-strange processes, we
need to select the suitable intermediate light vector meson states and
investigate their resonance contributions, which can directly determine the
coupling constants $g_{\mathcal{V}K^{(*/\prime)}_{(J)}K^{(*/\prime)}_{(J)}}$
and $f_{\mathcal{V}}$. In this work, we mainly concern with the CM energy
region between 2.0 and 2.6 GeV. Consequently, how to obtain a mass spectrum of
light vector mesons above 2.0 GeV is a crucial step before carrying out a
theoretical analysis of the open-strange processes $e^{+}e^{-}\to
K^{(*/\prime)}_{(J)}\bar{K}^{(*/\prime)}_{(J)}$. In the next section, we
present the mass spectrum of light vector mesons by an unquenched quark model
and discuss their decay behaviors.

##  III  Mass spectrum and decay behaviors of light vector mesons

The mass spectrum of light vector mesons around 2.0 GeV has been studied by
various potential models in the past several decades Godfrey:1985xj ;
Barnes:1996ff ; Stanley:1980zm ; Ebert:2009ub ; Ebert:2005ha ; Ishida:1986vn .
However, the results of the different models obviously differ from each other.
Focusing on the excited strangeonium $\phi(3S)$ state that is considered to be
a potential candidate of the $Y(2175)$, the analysis of the Regge trajectories
indicates that its mass should be around 1.92 GeV Wang:2012wa . However,
Barnes et al. predicted a corresponding mass as 2.05 GeV Barnes:2002mu . In
Ref. Ishida:1986vn , the authors employed a covariant oscillator quark model
with one gluon exchange effect, where the mass of $3^{3}S_{1}$ $s\bar{s}$
vector state is predicted to be 2.25 GeV. Therefore, obviously, there exists
room for theorists to develop the phenomenological model to obtain a
relatively reliable spectroscopy description of light vector mesons above 2.0
GeV.

For describing the highly excited light-flavor hadronic states, the
relativistic effect and unquenched correction should be considered in the
calculation. In this work, we adopt an unquenched relativized potential model
to study the mass spectrum of light vector mesons, which has been widely
applied to study other meson families from a kaon sector to a heavy quarkonium
sector Pang:2017dlw ; Song:2015nia ; Song:2015fha ; Wang:2016krl ;
Wang:2019mhs ; Wang:2020prx ; Wang:2018rjg . Here, we propose that the
unquenched correction for the light vector meson family can be determined by
other experimentally established $S$-wave and $D$-wave light meson families
involving pion, $\rho_{2}$, and $\rho_{3}$, which could provide a relatively
reliable scaling point for the energy region above 2.0 GeV in our theoretical
model. Furthermore, the Okubo-Zweig-Iizuka ($\mathrm{OZI}$)-allowed strong
decay behaviors of these vector mesons can be obtained by the quark pair
creation (QPC) model without $\beta$ parameter dependence involved in the
simple harmonic oscillator (SHO) wave function necessary for calculating the
transition matrix element, where we take the corresponding numerical meson
wave function directly from our unquenched relativized potential model as
input. For the convenience of the readers, we give a concise introduction to
the method adopted in this work in Appendices A and B.

Table 1: The mass spectra of the observed $S$-wave and $D$-wave light mesons calculated by the unquenched relativized potential model. Here, the comparison of the theoretical results and experimental data is given. All masses are in units of MeV. Predicted states | Mass (The.) | Observed states | Mass (Exp.)
---|---|---|---
$\pi(1S)$ | 131 | $\pi$ | 135 Zyla:2020zbs
$\pi(2S)$ | 1265 | $\pi(1300)$ |  $1300\pm 100$ Zyla:2020zbs
$\pi(3S)$ | 1759 | $\pi(1800)$ |  $1810^{+9}_{-11}$ Zyla:2020zbs
$\pi(4S)$ | 2115 | $\pi(2070)$ |  $2070\pm 35$ Anisovich:2001pn
$\pi(5S)$ | 2372 | $\pi(2360)$ |  $2360\pm 25$ Anisovich:2001pn
$\rho(1S)$ | 775 | $\rho(770)$ | 775 Zyla:2020zbs
$\rho(2S)$ | 1413 | $\rho(1450)$ |  $1465\pm 25$ Zyla:2020zbs
$\rho(3S)$ | 1862 | $\rho(1900)$ |  $1880\pm 30$ Aubert:2006jq
$\omega(1S)$ | 775 | $\omega(782)$ | 783 Zyla:2020zbs
$\omega(2S)$ | 1413 | $\omega(1420)$ |  $1410\pm 60$ Zyla:2020zbs
$\pi_{2}(1D)$ | 1650 | $\pi_{2}(1670)$ |  $1670.6^{+2.9}_{-1.6}$ Zyla:2020zbs
$\pi_{2}(2D)$ | 2003 | $\pi_{2}(2005)$ |  $1963.6^{+17}_{-27}$ Zyla:2020zbs
$\pi_{2}(3D)$ | 2278 | $\pi_{2}(2285)$ |  $2285\pm 20\pm 25$ Anisovich:2010nh
$\omega(1D)$ | 1633 | $\omega(1650)$ |  $1670\pm 30$ Zyla:2020zbs
$\rho(1D)$ | 1633 | $\rho(1700)$ |  $1720\pm 20$ Zyla:2020zbs
$\rho(2D)$ | 2003 | $\rho(2000)$ |  $2000\pm 30$ Bugg:2004xu
$\rho_{2}(2D)$ | 2007 | $\rho_{2}(1940)$ |  $1940\pm 40$ Anisovich:2002su
$\rho_{2}(3D)$ | 2284 | $\rho_{2}(2225)$ |  $2225\pm 35$ Anisovich:2002su
$\rho_{3}(1D)$ | 1672 | $\rho_{3}(1690)$ |  $1688.8\pm 2.1$ Zyla:2020zbs
$\rho_{3}(2D)$ | 2013 | $\rho_{3}(1990)$ |  $1982\pm 14$ Anisovich:2002su
$\rho_{3}(3D)$ | 2284 | $\rho_{3}(2250)$ |  $2290\pm 20\pm 30$ Amelin:2000nm
$\phi(1S)$ | 1020 | $\phi(1020)$ | 1019 Zyla:2020zbs
$\phi(2S)$ | 1665 | $\phi(1680)$ |  $1680\pm 20$ Zyla:2020zbs
Table 2:  Parameters in the unquenched relativized potential model adopted in this work. Parameter | Value | Parameter | Value
---|---|---|---
$b$ (GeV2) | 0.229 |  $c$ (GeV) | -0.300
$\epsilon_{\mathrm{sos}}$ | 0.973 | $\epsilon_{c}$ | -0.164
$\epsilon_{\mathrm{sov}}$ | 0.262 | $\epsilon_{t}$ | 1.993
$\mu$ (GeV) | 0.081 |  |
Table 3: A comparison of the decay behavior of some well-established light vector mesons of the experimental average values and our theoretical estimate. Here, $\mathcal{B}_{e^{+}e^{-}}(\mathcal{V})$ and $\mathcal{B}_{h_{1}h_{2}}(\mathcal{V})$ represent the branching ratio $\Gamma(\mathcal{V}\to e^{+}e^{-})/\Gamma^{Total}_{\mathcal{V}}$ and $\Gamma(\mathcal{V}\to h_{1}h_{2})/\Gamma^{Total}_{\mathcal{V}}$, respectively.  Decay | Our | Exp. (Ave.) Zyla:2020zbs
---|---|---
$\Gamma_{e^{+}e^{-}}(\rho(770))$ (keV) | 6.98 | $7.04\pm 0.06$
$\mathcal{B}_{e^{+}e^{-}}(\rho(1450))\mathcal{B}_{\omega\pi}(\rho(1450))$ ($\times 10^{-6}$) | 4.4 | $3.7\pm 0.4$
$\mathcal{B}_{e^{+}e^{-}}(\rho(1450))\mathcal{B}_{\rho\eta}(\rho(1450))$ ($\times 10^{-6}$) | 0.60 | $0.58\pm 0.07$
$\Gamma_{e^{+}e^{-}}(\omega(782))$ (keV) | 0.78 | $0.60\pm 0.02$
$\mathcal{B}_{e^{+}e^{-}}(\omega(1420))\mathcal{B}_{\omega\eta}(\omega(1420))$ ($\times 10^{-7}$) | 0.11 | $0.29\pm 0.15$
$\mathcal{B}_{e^{+}e^{-}}(\omega(1420))\mathcal{B}_{\rho\pi}(\omega(1420))$ ($\times 10^{-7}$) | 2.74 | $6.58\pm 1.49$
$\Gamma_{e^{+}e^{-}}(\phi(1020))$ (keV) | 3.19 | $1.27\pm 0.04$
$\mathcal{B}_{e^{+}e^{-}}(\phi(1680))\mathcal{B}_{KK^{*}}(\phi(1680))$ ($\times 10^{-7}$) | 20.4 | $22.2\pm 8.7$
$\mathcal{B}_{e^{+}e^{-}}(\phi(1680))\mathcal{B}_{K^{0}\bar{K}^{0}}(\phi(1680))$ ($\times 10^{-7}$) | 1.37 | $1.31\pm 0.59$

As pointed out in Ref. Wang:2018rjg , the color screened effect is mainly
reflected on the highly excited hadronic states and so the key point of
obtaining the reliable mass spectrum of light vector mesons above 2.0 GeV is
to determine the screened parameter $\mu$ in the unquenched relativized
potential model. In fact, the unquenched effect from the color screened
interaction is a kind of nonperturbative behavior of QCD and is difficult to
be solved from a theoretical perspective, so its quantification needs the
input of available experimental hints. At present, within the messy research
situation of the light-flavor vector mesons, we believe that other light meson
families could provide some valuable hints on our theoretical model.

In the quark model, in addition to an $S$-wave vector meson state, there still
exist the corresponding vector $D$-wave partners, i.e.,
$n^{2S+1}L_{J}=n^{3}D_{1}$. Thus, as their spin singlet or triplet states, the
experimental information of an $S$-wave pseudoscalar pion family and $D$-wave
$\pi_{2}$, $\rho_{2}$, and $\rho_{3}$ states can help us determine the
potential model parameters, where we notice that the masses of some highly
excited states can reach around 2.3 GeV, such as $\pi(2360)$ Anisovich:2001pn
, $\pi_{2}(2285)$ Anisovich:2010nh , $\rho_{2}(2225)$ Anisovich:2002su  and
$\rho_{3}(2250)$ Amelin:2000nm . Thus, these highly excited states play the
role of the important scaling point of color screened effect. The measured
masses of these light-flavor mesons are summarized in Table 1, which can be
used to constrain three main screened confinement parameters and four
relativistic correction factors $\epsilon_{i}$. Based on a set of parameters
listed in Table 2, the global aspect of the light-flavor $S$-wave and $D$-wave
meson mass spectra is well consistent with those of experiments, which can be
clearly shown by the comparison between experimental values and our
theoretical estimates in Table 1. The screening parameter $\mu=0.81$ indicates
that the unquenched effect is indeed significant for describing the mass
spectrum of light vector mesons.

With the above preparation, we can directly predict the mass spectrum and
decay behavior of higher $\rho$, $\omega$, and $\phi$ mesonic states above 2.0
GeV. In the QPC model, since the meson wave functions of initial and final
states can be fixed by the corresponding eigenvectors solved from the above
unquenched potential model, the partial width of each decay channel is only
dependent on the parameter $\gamma$. However, it is worth noting that the
corresponding branching ratio is independent on any parameter because the
$\gamma$ in the numerator and the denominator cancel out each other. In
addition, the di-lepton widths of the higher vector states can also be related
to the zero-point behavior of their radial wave functions, whose concrete
width formula can be found in Refs. Wang:2020kte ; Godfrey:1985xj . Thus, the
production rates of intermediate light vector mesons in $e^{+}e^{-}$
annihilations can also be estimated from a theoretical perspective. In Table
3, we show the decay behavior comparison of some well-established light vector
mesons between the experimental values and our theoretical results, which
involve $\rho(770)$, $\rho(1450)$, $\omega(782)$, $\omega(1420)$,
$\phi(1020)$, and $\phi(1680)$. It can be seen that our theoretical
predictions are well consistent with measured results, in a sense, which can
prove the reliability of our models in estimating the decay properties of
higher light vector meson states above 2.0 GeV.

The calculated mass spectrum and branching ratios of two-body open-strange
strong decays as well as di-lepton decay widths of $\rho$($\omega$) and $\phi$
meson states above 2.0 GeV are listed in Tables 4 and 5, respectively. There
are eight light vector mesons existing near the energy region of $Y(2175)$,
which include $\rho(2D)/\omega(2D)$, $\rho(4S)/\omega(4S)$,
$\rho(3D)/\omega(3D)$, $\phi(3S)$ and $\phi(2D)$. However, it is almost
impossible to study the present experimental data of open-strange channels by
adding so many resonance contributions. Thus, we have to select the main
intermediate resonances in our practical analysis. From the calculated
numerical results, the di-lepton widths of the isoscalar $\omega$ states are
one order smaller than those of the isovector $\rho$ partners, and their
branching ratios to open-strange decay channels are almost the same.
Furthermore, one can see that the branching ratios of $\phi(3S)/\phi(2D)\to
K^{(*/\prime)}_{(J)}\bar{K}^{(*/\prime)}_{(J)}$ are far larger than those of
$\rho(2D)$, $\rho(4S)$, and $\rho(3D)$ although there is no obvious difference
among their production rates via electron-positron collisions. Therefore, we
can conclude that the resonance contributions from the highly excited $\rho$
and $\omega$ states can be safely ignored in studying $e^{+}e^{-}\to
K^{(*/\prime)}_{(J)}\bar{K}^{(*/\prime)}_{(J)}$ reaction above 2.0 GeV.
Additionally, it is worth mentioning that the recently observed structure near
2.2 GeV in $e^{+}e^{-}\to K^{+}K^{-}$ by BESIII has been collected into
$``\rho(2170)"$ in the 2020 version of PDG Zyla:2020zbs , which should need a
careful judgement and is not supported by our theoretical results here.

In Tables 4 and 5, we also give the theoretical total widths of the light
vector mesons by taking a typical value of $\gamma=6.57$ Pang:2019ttv , which
can be used as a reference. One can see that $\phi$ states play an absolutely
dominant role in the open-strange processes from $e^{+}e^{-}$ collisions, so
three strangeonium states $\phi(3S)$, $\phi(2D)$, and $\phi(4S)$ will be
considered in our analysis. Utilizing the calculated decay behaviors of the
higher strange quarkonium states presented in Table 5, in the next section, we
can directly decipher the cross sections of the present reported open-strange
processes based on $e^{+}e^{-}$ collisions.

Table 4: The branching ratios of two-body open-strange strong decay and di-lepton widths of $\rho$ and $\omega$ meson states above 2.0 GeV. Here, the tiny branching ratio is marked as “$\cdots$”.  States | $\rho(2D)$ | $\rho(4S)$ | $\rho(3D)$ | $\rho(5S)$ | $\omega(2D)$ | $\omega(4S)$ | $\omega(3D)$ | $\omega(5S)$
---|---|---|---|---|---|---|---|---
Mass (GeV) | 2.003 | 2.180 | 2.283 | 2.422 | 2.003 | 2.180 | 2.283 | 2.422
$\Gamma_{e^{+}e^{-}}$ (keV) | 0.020 | 0.063 | 0.016 | 0.036 | 0.0022 | 0.007 | 0.0018 | 0.004
$\Gamma_{\mathrm{Total}}$ (MeV) | 179 | 102 | 158 | 80 | 181 | 104 | 94 | 69
$\mathcal{B}(KK)$ | 0.006 | 0.002 | 0.002 | $\cdots$ | 0.006 | 0.002 | 0.003 | $\cdots$
$\mathcal{B}(KK^{*})$ | 0.001 | $\cdots$ | $\cdots$ | $\cdots$ | 0.001 | $\cdots$ | $\cdots$ | $\cdots$
$\mathcal{B}(K^{*}K^{*})$ | $\cdots$ | 0.003 | $\cdots$ | 0.003 | $\cdots$ | 0.003 | 0.001 | 0.003
$\mathcal{B}(KK_{1}(1270))$ | $\cdots$ | 0.010 | $\cdots$ | 0.004 | $\cdots$ | 0.010 | $\cdots$ | 0.005
$\mathcal{B}(KK^{*}(1410))$ | $\cdots$ | 0.001 | $\cdots$ | $\cdots$ | $\cdots$ | 0.001 | $\cdots$ | $\cdots$
$\mathcal{B}(K^{*}K_{1}(1270))$ | $\cdots$ | 0.004 | $\cdots$ | 0.004 | $\cdots$ | 0.004 | $\cdots$ | 0.005
$\mathcal{B}(KK(1460))$ | 0.002 | 0.003 | 0.003 | 0.001 | 0.002 | 0.002 | 0.005 | 0.001
Table 5: The branching ratios of two-body open-strange strong decay and di-lepton widths of $\phi$ mesons above 2.0 GeV. Additionally, the mass and decay behaviors of missing $\phi(1D)$ in experiments are also given. Here, the tiny branching ratio is marked as “$\cdots$”. States | $\phi(1D)$ | $\phi(3S)$ | $\phi(2D)$ | $\phi(4S)$ | $\phi(3D)$
---|---|---|---|---|---
Mass (GeV) | 1.860 | 2.103 | 2.236 | 2.423 | 2.519
$\Gamma_{e^{+}e^{-}}$ (keV) | 0.063 | 0.106 | 0.017 | 0.050 | 0.010
$\Gamma_{\mathrm{Total}}$ (MeV) | 515 | 156 | 265 | 140 | 171
$\mathcal{B}(KK)$ | 0.076 | 0.059 | 0.087 | 0.047 | 0.084
$\mathcal{B}(KK^{*})$ | 0.100 | 0.242 | 0.067 | 0.145 | 0.050
$\mathcal{B}(K^{*}K^{*})$ | 0.016 | 0.007 | 0.105 | 0.014 | 0.139
$\mathcal{B}(KK^{*}(1410))$ | $\cdots$ | 0.180 | 0.080 | 0.142 | 0.052
$\mathcal{B}(KK^{*}(1680))$ | $\cdots$ | $\cdots$ | $\cdots$ | 0.004 | 0.003
$\mathcal{B}(K^{*}K^{*}(1410))$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 0.045
$\mathcal{B}(KK_{2}^{*}(1430))$ | $\cdots$ | 0.115 | 0.043 | 0.091 | 0.043
$\mathcal{B}(K^{*}K_{2}^{*}(1430))$ | $\cdots$ | $\cdots$ | $\cdots$ | 0.057 | 0.047
$\mathcal{B}(KK_{1}(1270))$ | 0.799 | 0.185 | 0.356 | 0.154 | 0.293
$\mathcal{B}(K^{*}K_{1}(1270))$ | $\cdots$ | $\cdots$ | 0.110 | 0.018 | 0.095
$\mathcal{B}(KK_{1}(1400))$ | $\cdots$ | 0.064 | 0.026 | 0.027 | 0.015
$\mathcal{B}(K^{*}K_{1}(1400))$ | $\cdots$ | $\cdots$ | $\cdots$ | 0.126 | 0.003
$\mathcal{B}(KK(1460))$ | $\cdots$ | 0.127 | 0.112 | 0.073 | 0.101
$\mathcal{B}(K^{*}K(1460))$ | $\cdots$ | $\cdots$ | $\cdots$ | 0.013 | 0.005
$\mathcal{B}(K^{*}K_{0}(1430))$ | $\cdots$ | $\cdots$ | $\cdots$ | 0.049 | $\cdots$
$\mathcal{B}(KK_{3}^{*}(1780))$ | $\cdots$ | $\cdots$ | $\cdots$ | 0.021 | 0.014

##  IV  results of numerical analysis

Figure 2:  The combined analysis to seven open-strange processes from
$e^{+}e^{-}$ collisions Ablikim:2018iyx ; Ablikim:2020pgw ; BaBar:2013jqz ;
Aubert:2007ym , which are shown in figure (a)-(g), successively. Here,
$\sigma_{I=0}$ in figure (b) and (c) means that the measured cross section of
reaction process corresponds to the isoscalar component.

In the following, we fit the cross sections for seven open-strange reactions
of $e^{+}e^{-}\to K^{+}{K}^{-}$ Ablikim:2018iyx ; BaBar:2013jqz ,
$e^{+}e^{-}\to K\bar{K}^{*}+c.c.$ Aubert:2007ym , $e^{+}e^{-}\to K^{*+}K^{*-}$
Ablikim:2020pgw , $e^{+}e^{-}\to K_{1}(1270)^{+}K^{-}$ Ablikim:2020pgw ,
$e^{+}e^{-}\to K_{1}(1400)^{+}K^{-}$ Ablikim:2020pgw , $e^{+}e^{-}\to
K_{2}^{*}(1430)\bar{K}+c.c.$ Aubert:2007ym  and $e^{+}e^{-}\to
K(1460)^{+}K^{-}$ Ablikim:2020pgw , which provide us a direct evidence to
demonstrate the nature of the vector structure around 2.2 GeV observed in
these processes. It is worth mentioning that although the contributions from
resonances $\phi(1020)$, $\phi(1680)$, and $\phi(1D)$ in the energy region
above 2.0 GeV are obviously suppressed compared with those at their resonant
peak positions, they still may play a significant background role in some
specific open-strange channels, which have to be included in our analysis.
According to the corresponding decay behaviors presented in Tables 3 and 5, we
consider the contributions of lower $\phi(1020)$, $\phi(1680)$, and $\phi(1D)$
in reaction channels $KK$, $KK^{(*)}$, and $KK_{1}(1270)$, respectively. The
resonance masses and total widths of $\phi(1020)$ and $\phi(1680)$ can be
fixed by the experimental average values in PDG Zyla:2020zbs . All coupling
constants $g_{\phi K^{(*/\prime)}_{(J)}K^{(*/\prime)}_{(J)}}$/$f_{\phi}$ in
Eqs. (3)-(7) can be related to the corresponding products of $\Gamma(\phi\to
e^{+}e^{-})\times\mathcal{B}(\phi\to
K^{(*/\prime)}_{(J)}\bar{K}^{(*/\prime)}_{(J)})$ by the following expressions

$\displaystyle\frac{g_{\phi KK}}{f_{\phi}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{288\pi^{2}\Gamma_{\phi}m_{\phi}\Gamma(\phi\to
e^{+}e^{-})\mathcal{B}(\phi\to
K\bar{K})}{e^{4}\lambda(m_{\phi},m_{K},m_{K})^{\frac{3}{2}}}},$
$\displaystyle\frac{g_{\phi KK^{*}}}{f_{\phi}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{144\pi^{2}\Gamma_{\phi}\Gamma(\phi\to
e^{+}e^{-})\mathcal{B}(\phi\to
K\bar{K}^{*}+c.c.)}{e^{4}\lambda(m_{\phi},m_{K},m_{K^{*}})^{\frac{3}{2}}m_{\phi}}},$
$\displaystyle\frac{g_{\phi K^{*}K^{*}}}{f_{\phi}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{288\pi^{2}\Gamma_{\phi}m_{\phi}\Gamma(\phi\to
e^{+}e^{-})\mathcal{B}(\phi\to
K^{*}\bar{K}^{*})}{e^{4}\lambda(m_{\phi},m_{K^{*}},m_{K^{*}})^{\frac{3}{2}}(m_{\phi}^{4}-4m_{\phi}^{2}m_{K^{*}}^{2}+12m_{K^{*}}^{4})}},$
$\displaystyle\frac{g_{\phi KK_{1}}}{f_{\phi}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{144\pi^{2}\Gamma_{\phi}m_{\phi}\Gamma(\phi\to
e^{+}e^{-})\mathcal{B}(\phi\to
K\bar{K}_{1}+c.c.)}{e^{4}\lambda(m_{\phi},m_{K},m_{K_{1}})^{\frac{1}{2}}}},$
$\displaystyle\frac{g_{\phi KK_{2}}}{f_{\phi}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{144\pi^{2}\Gamma_{\phi}\Gamma(\phi\to
e^{+}e^{-})\mathcal{B}(\phi\to
K\bar{K_{2}}+c.c.)}{e^{4}\lambda(m_{\phi},m_{K},m_{K_{2}})^{\frac{5}{2}}m_{\phi}^{3}m_{\phi}^{2}}},$
$\displaystyle\frac{g_{\phi KK^{\prime}}}{f_{\phi}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{144\pi^{2}\Gamma_{\phi}m_{\phi}\Gamma(\phi\to
e^{+}e^{-})\mathcal{B}(\phi\to
K\bar{K}^{\prime}+c.c.)}{e^{4}\lambda(m_{\phi},m_{K},m_{K}^{\prime})^{\frac{3}{2}}}},$

where $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2xz-2yz$ is the
K$\ddot{\mathrm{a}}$llen function. Thus, the only fitting parameters are
relative phase angles and $f_{\mathrm{Dir}}$ and $a$ in a direct production
amplitude. However, in a practical analysis, we find that the fitted
$\chi^{2}$ value is obviously dependent on resonance parameters of three
strange quarkonium states $\phi(3S)$, $\phi(2D)$, and $\phi(4S)$. At the same
time, combining the fact that the measured errors of experimental resonance
parameters of most of light-flavor mesons are generally large as seen in Table
1, the masses of $m_{\phi(3S)}$, $m_{\phi(2D)}$, $m_{\phi(4S)}$, and strong
decay parameter $\gamma$ can be set as free parameters, which should vary
around their fixed theoretical values.

The combined fit to seven open-strange processes from $e^{+}e^{-}$ collisions
mentioned above is presented in Fig. 2. Benefiting from the calculated decay
behaviors of light vector mesons in Sec. III, the contributions of each
individual light vector meson to the cross sections of different open-strange
processes can be directly obtained, which are also shown in Fig. 2. One can
see that experimental cross sections of the measured open-strange processes
can be described well by introducing the resonance contributions from highly
excited $3S$, $2D$, and $4S$ strange quarkonium states and a combined
$\chi^{2}/d.o.f=2.29$ is obtained. Especially, the apparent enhancement or dip
around 2.2 GeV seen in $e^{+}e^{-}\to
K^{+}K^{-},K(1460)^{+}K^{-},K_{1}(1270)^{+}K^{-}$, $K_{1}(1400)^{+}K^{-}$, and
$K_{2}^{*}(1430)\bar{K}+c.c.$ can be completely reproduced via the
interference effect from the resonance contributions of $\phi(3S)$ and
$\phi(2D)$ and the continuum background, where an $S$-wave state plays a
dominant role. Here, it is worth mentioning that the $e^{+}e^{-}\to
K^{*}\bar{K}_{1}(1270)+c.c.$ could be an excellent reaction process to detect
the dominant signal of a $D$-wave state, where the contribution from a $3S$
state is suppressed as shown in Table 5. Based on the interference effect, we
can naturally explain why there exists an inconsistent peak position of the
observed vector structure around 2.2 GeV among different reaction channels,
such as $KK$ channel in Fig. 2(a) and other channels in Figs. 2(c)-(f). Here,
we can see that a simple Breit-Wigner fit to the corresponding cross section
is indeed a very rough treatment.

The obtained phase angles and direct production parameters of each process in
the combined fit are summarized in Table 6. In addition, the resonance masses
and widths of $\phi(3S)$, $\phi(2D)$, and $\phi(4S)$ are fitted to be

$\displaystyle m_{\phi(3S)}=2183\pm 1~{}\mathrm{MeV},\Gamma_{\phi(3S)}=185\pm
4~{}\mathrm{MeV},$ $\displaystyle m_{\phi(2D)}=2290\pm
3~{}\mathrm{MeV},\Gamma_{\phi(2D)}=312\pm 6~{}\mathrm{MeV},$ $\displaystyle
m_{\phi(4S)}=2485\pm 5~{}\mathrm{MeV},\Gamma_{\phi(4S)}=165\pm
3~{}\mathrm{MeV},$

respectively. The above results indicate that both $\phi(3S)$ and $\phi(2D)$
are broad states, which are consistent with previous theoretical calculations
Barnes:2002mu ; Pang:2019ttv ; Wang:2012wa ; Ding:2007pc . As mentioned in
Introduction, the measured resonance width of $Y(2175)$ is narrower than
predictions for $\phi(3S)$ or $\phi(2D)$. From Fig. 2, it can be seen that
this width problem can also be explained in our theoretical analysis of
considering the interference effect among two light vector meson contributions
as well as a background term. Here, if we fit the cross sections in Fig. 2(a)
or 2(c)-(f) by a simple Breit-Wigner formula, we would obtain the narrow width
around 100 MeV.

Additionally, we notice that the experimental data of reaction $e^{+}e^{-}\to
K^{*+}K^{*-}$ in Fig. 2(g) does not show an obvious signal for the existence
of a vector structure around 2.2 GeV. Here, we want to emphasize that this
interesting phenomenon can also be understood in our theoretical framework.
From Table 5, we can find a branching ratio $\mathcal{B}(\phi(3S)\to
K^{*}\bar{K}^{*})=7\times 10^{-3}$, which is obviously smaller than those of
other open-strange decay channels. The reason for this is that there exists a
node effect for the overlap integral among the wave functions of initial
$\phi(3S)$ state and final state $K^{*}\bar{K}^{*}$. From Fig. 2(h), it can be
seen that these resonance contributions in process $e^{+}e^{-}\to
K^{*+}K^{*-}$ are of the order of several pb, which is far lower than a direct
continuum. Therefore, this is a strong evidence to support our explanation to
the vector structure around 2.2 GeV observed in open-strange processes. In
conclusion, our theoretical analysis can well clarify the present puzzling
situation in understanding the cross section of open-strange processes from
$e^{+}e^{-}$ annihilation without adding any unknown resonance contributions,
which means that this work provides a new opinion to construct the light
vector meson family and understand the $Y(2175)$.

Finally, we would like to suggest our experimental colleagues to search for a
predicted new resonance structure $\phi(4S)$ around 2.5 GeV, whose signal can
be discovered by several typical open-strange processes $e^{+}e^{-}\to
K\bar{K}$, $e^{+}e^{-}\to K\bar{K}^{*}+c.c.$, and $e^{+}e^{-}\to
K\bar{K}(1460)+c.c.$ as presented in Fig. 2, especially for $KK(1460)$
channel, where the experimental data on relevant energy region is still
lacking. This should be an interesting research topic on the future BESIII and
BelleII experiments.

Table 6:  The fitted parameters in the combined analysis to the experimental data of seven open-strange processes from $e^{+}e^{-}$ collisions. The listed phase angles $\theta_{\phi}$ are in units of radian.  Processes | $f_{\mathrm{Dir}}$ |  $a$ ($\mathrm{GeV}^{-2}$) | $\theta_{\phi(1S)}$ | $\theta_{\phi(2S)}$ | $\theta_{\phi(1D)}$ | $\theta_{\phi(3S)}$ | $\theta_{\phi(2D)}$ | $\theta_{\phi(4S)}$ |
---|---|---|---|---|---|---|---|---|---
$e^{+}e^{-}\to K^{+}{K}^{-}$ | $-0.29$ | $0.33$ | $6.09\pm 0.04$ | $0.52\pm 0.06$ | $\cdots$ | $2.54\pm 0.02$ | $5.77\pm 0.05$ | $2.80\pm 0.05$ |
$e^{+}e^{-}\to K\bar{K}^{*}+c.c.$ | $1.95\pm 0.07$ | $0.85\pm 0.01$ | $\cdots$ | $3.51\pm 0.10$ | $\cdots$ | $2.74\pm 0.17$ | $1.32\pm 0.41$ | $4.84\pm 0.38$ |
$e^{+}e^{-}\to K^{*+}{K}^{*-}$ | $-2.68\pm 0.05$ | $0.89\pm 0.01$ | $\cdots$ | $\cdots$ | $\cdots$ | $2.14\pm 0.47$ | $2.57\pm 0.36$ | $3.05\pm 0.49$ |
$e^{+}e^{-}\to K_{1}(1270)^{+}{K}^{-}$ | $0.78\pm 0.07$ | $0.22\pm 0.01$ | $\cdots$ | $\cdots$ | $2.47\pm 0.52$ | $5.02\pm 0.63$ | $4.87\pm 1.10$ | $1.28\pm 0.95$ |
$e^{+}e^{-}\to K_{1}(1400)^{+}{K}^{-}$ | $0.87\pm 0.16$ | $0.28\pm 0.04$ | $\cdots$ | $\cdots$ | $\cdots$ | $4.62\pm 0.21$ | $5.79\pm 0.65$ | $6.28\pm 0.33$ |
$e^{+}e^{-}\to K_{2}^{*}(1430)\bar{K}+c.c.$ | $1.09\pm 0.11$ | $0.80\pm 0.02$ | $\cdots$ | $\cdots$ | $\cdots$ | $3.54\pm 0.10$ | $1.11\pm 0.51$ | $4.67\pm 0.24$ |
$e^{+}e^{-}\to K(1460)^{+}{K}^{-}$ | $-0.12\pm 0.22$ | $0.46\pm 0.35$ | $\cdots$ | $\cdots$ | $\cdots$ | $6.22\pm 0.18$ | $5.55\pm 0.64$ | $6.26\pm 1.30$ |

##  V Summary

Recently, the BESIII Collaboration performed the precise measurements of cross
sections for open-strange processes $e^{+}e^{-}\to K^{+}K^{-}$ Ablikim:2018iyx
and $e^{+}e^{-}\to K^{+}K^{-}\pi^{0}\pi^{0}$ Ablikim:2020pgw , where a clear
structure around 2.2 GeV was observed. This is the first certain evidence for
the existence of the vector structure around 2.2 GeV in open-strange reaction
processes. The experimental resonance masses and widths of this newly observed
vector structure are found to be consistent with those of the $Y(2175)$
firstly reported in hidden-strange reaction $e^{+}e^{-}\to\phi
f_{0}(980)\to\phi\pi^{+}\pi^{-}$ by the BaBar Collaboration in 2006
Aubert:2006bu . Thus, this simple comparison seems to imply that this
structure around 2.2 GeV may be the same state as $Y(2175)$. In the past
years, the popular theoretical explanations of $Y(2175)$ mainly include hybrid
$s\bar{s}g$ Ding:2006ya , vector strangeonium state $\phi(3S)$ Barnes:2002mu ;
Pang:2019ttv , and $\phi(2D)$ Wang:2012wa ; Ding:2007pc ; Pang:2019ttv , which
are just favored by the open-strange channels. However, the theoretical
studies on their decay behaviors Ding:2006ya ; Barnes:2002mu ; Pang:2019ttv ;
Wang:2012wa ; Ding:2007pc ; Close:1994hc ; Close:2003mb ; Zhu:1998sv ;
Zhu:1999wg  definitely indicate that it is difficult to understand the
experimental data of these open-strange reactions Ablikim:2018iyx ;
Ablikim:2020pgw  either under the hybrid or strangeonium assignment to the
$Y(2175)$. This means that there are no appropriate theoretical pictures to
convincingly explain this newly observed vector structure around 2.2 GeV.

In order to clarify this puzzling situation, we have pointed out that it is
not an easy task to analyze the cross sections of the open-strange processes
from $e^{+}e^{-}$ annihilations, where both highly excited and light vector
mesons $\rho$, $\omega$, and $\phi$ states may have significant contributions.
Thus, a simple Breit-Wigner fit to the signal around 2.2 GeV observed in the
cross sections of open-strange processes must be a very rough treatment. Based
on this motivation, in this work, we have performed a combined analysis to the
cross sections of seven reported open-strange reactions of $e^{+}e^{-}\to
K^{+}{K}^{-}$ Ablikim:2018iyx ; BaBar:2013jqz , $e^{+}e^{-}\to
K\bar{K}^{*}+c.c.$ Aubert:2007ym , $e^{+}e^{-}\to K^{*+}K^{*-}$
Ablikim:2020pgw , $e^{+}e^{-}\to K_{1}(1270)^{+}K^{-}$ Ablikim:2020pgw ,
$e^{+}e^{-}\to K_{1}(1400)^{+}K^{-}$ Ablikim:2020pgw , $e^{+}e^{-}\to
K_{2}^{*}(1430)\bar{K}+c.c.$ Aubert:2007ym  and $e^{+}e^{-}\to
K(1460)^{+}K^{-}$ Ablikim:2020pgw  by introducing the light vector meson
contributions, which is supported by the study of hadron spectroscopy. Here,
we have employed an unquenched relativized potential model Pang:2017dlw ;
Song:2015nia ; Song:2015fha ; Wang:2016krl ; Wang:2019mhs ; Wang:2020prx ;
Wang:2018rjg  to study the mass spectra and wave functions of light vector
meson states. By taking the exact numerical wave functions from the potential
model as input, the di-lepton widths and the branching ratios of open-strange
decay channels of light vector mesons can be estimated without any parameter
dependence. Based on these available theoretical results, the cross sections
of the seven open-strange reactions are found to be well described by
considering the direct production and the light vector meson contributions.
Furthermore, we have demonstrated that the observed vector structure around
2.2 GeV in these open-strange channels does not correspond to a single
resonance and it can be identified as an interference signal from highly
excited strange quarkonium states $\phi(3S)$ and $\phi(2D)$, by which the
present puzzling situations can be naturally clarified. From this practical
example in this work, we have actually provided a new perspective to construct
the light vector meson family and understand the $Y(2175)$, which can continue
to be tested in hidden-strange channels and other related processes. These
should be helpful to solve the messy research situation of light vector mesons
around 2.2 GeV thoroughly in the future.

In the full hadron spectrum, the light meson spectroscopy has a particular
position because there exist abundant experimental data in the energy region
of light mesons. In addition to the intriguing vector structure around 2.2
GeV, we have also predicted a new strange quarkonium structure $\phi(4S)$ near
$\sqrt{s}=2.5$ GeV, whose mass and total width are predicted to be
$m_{\phi(4S)}=2485\pm 5~{}\mathrm{MeV}$ and $\Gamma_{\phi(4S)}=165\pm
3~{}\mathrm{MeV}$, respectively. The establishment of this particle in
experiments is very important for further constructing the light vector meson
family. This should be a challenging task to the experimentalist community and
is worth working in the future.

##  Appendix A A concise review of the unquenched relativized potential model

The Hamiltonian depicting the interaction between quark and antiquark in the
unquenched relativized potential model is described as Pang:2017dlw ;
Song:2015nia ; Song:2015fha ; Wang:2016krl ; Wang:2019mhs ; Wang:2020prx ;
Wang:2018rjg ; Godfrey:1985xj

$\displaystyle\widetilde{H}=\widetilde{H}_{0}+\widetilde{V}_{\mathrm{eff}}(\mathbf{p},\mathbf{r})$
(11)

with

$\displaystyle\widetilde{H}_{0}=\sqrt{m_{q}^{2}+\mathrm{p^{2}}}+\sqrt{m_{\bar{q}}^{2}+\mathrm{p^{2}}},$
(12)

where $m_{q}$ and $m_{\bar{q}}$ denote the constituent masses of light quark
and antiquark, respectively. In our calculation, the quark masses
$m_{u}=m_{d}=0.22$ GeV and $m_{s}=0.424$ GeV are taken. The effective
potential of $q\bar{q}$ interaction contains three parts, i.e.,

$\displaystyle\widetilde{V}_{\mathrm{eff}}(\mathbf{p},\mathbf{r})=\widetilde{H}^{\mathrm{conf}}+\widetilde{H}^{\mathrm{so}}+\widetilde{H}^{\mathrm{hyp}},$
(13)

where the confinement interaction $\widetilde{H}^{\mathrm{conf}}$ includes a
short $\gamma^{\mu}\otimes\gamma_{\mu}$ one-gluon-exchange interaction and an
unquenched confining interaction. In the nonrelativistic limit, the effective
potential can be written as the standard nonrelativistic expression, i.e.,

$\displaystyle
V_{\mathrm{eff}}(\mathbf{p},\mathbf{r})=H^{\mathrm{conf}}+H^{\mathrm{so}}+H^{\mathrm{hyp}}$
(14)

with

$\displaystyle\begin{split}H^{\mathrm{conf}}&=\left[-\frac{3}{4}\left(c+\frac{b(1-e^{-\mu
r})}{\mu}\right)+\frac{\alpha(r)}{r}\right]\
(\bm{F_{q}}\cdot\bm{F_{\bar{q}}})\\\ &=S(r)+G(r),\end{split}$ (15)

where $H^{\mathrm{conf}}$ contains the spin-independent color screened
confinement $S(r)=c+b(1-e^{-\mu r})/\mu$ and Coulomb-type interaction
$G(r)=\alpha(r)/r$, and $\bm{F}$ is related to the Gell-Mann matrices in color
space, and $\langle\bm{F_{q}}\cdot\bm{F_{\bar{q}}}\rangle=-4/3$ is taken in
meson system. The third term $H^{\mathrm{hyp}}$ is the color-hyperfine
interaction and reads

$\displaystyle\begin{split}H^{\mathrm{hyp}}=&-\frac{\alpha_{s}(r)}{m_{q}m_{\bar{q}}}\left[\frac{8\pi}{3}\mathbf{S_{q}}\cdot\mathbf{S_{\bar{q}}}\delta^{3}(\mathbf{r})\right.\\\
&\left.+\frac{1}{r^{3}}\left(\frac{3\mathbf{S_{q}}\cdot\mathbf{r}\mathbf{S_{\bar{q}}}\cdot\mathbf{r}}{r^{2}}-\mathbf{S_{q}}\cdot\mathbf{S_{\bar{q}}}\right)\right]\
\left(\bm{F_{q}}\cdot\bm{F_{\bar{q}}}\right).\end{split}$ (16)

The second term $H^{\mathrm{so}}=H^{\mathrm{so(cm)}}+H^{\mathrm{so(tp)}}$ in
Eq. (14) is the spin-orbit interaction with the color magnetic term resulting
from one-gluon-exchange

$\displaystyle
H^{\mathrm{so(cm)}}=-\frac{\alpha_{s}(r)}{r^{3}}\left(\frac{1}{m_{q}}+\frac{1}{m_{\bar{q}}}\right)\
\left(\frac{\mathbf{S_{q}}}{m_{q}}+\frac{\mathbf{S_{\bar{q}}}}{m_{\bar{q}}}\right)\cdot\bm{L}\left(\bm{F_{q}}\cdot\bm{F_{\bar{q}}}\right)$
(17)

and Thomas precession term

$\displaystyle H^{\mathrm{so(tp)}}=-\frac{1}{2r}\frac{\partial
H^{\mathrm{conf}}}{\partial r}\
\left(\frac{\mathbf{S_{q}}\cdot\bm{L}}{m_{q}^{2}}+\frac{\mathbf{S_{\bar{q}}}\cdot\bm{L}}{m_{\bar{q}}^{2}}\right).$
(18)

In the above expressions, $\mathbf{S_{q}}$ and $\mathbf{S_{\bar{q}}}$ denote
the spin of light flavor quark and antiquark, respectively, while $\bm{L}$ is
the orbital momentum between two quarks inside the meson.

The relativistic effect is introduced by two ways (see Ref. Godfrey:1985xj
for more details). First, considering the effects of internal motion inside a
light hadron and the non-locality interactions between light quark and
antiquark, a smearing function could be introduced, i.e.,

$\displaystyle\rho\left(\bm{r}-\bm{r^{\prime}}\right)=\frac{\sigma_{12}^{3}}{\pi^{3/2}}e^{-\sigma_{12}^{2}(\bm{r}-\bm{r^{\prime}})^{2}}$
(19)

with

$\displaystyle\begin{split}\sigma_{12}^{2}=&\sigma_{0}^{2}\left[\frac{1}{2}+\frac{1}{2}\left(\frac{4m_{q}m_{\bar{q}}}{(m_{q}+m_{\bar{q}})^{2}}\right)^{4}\right]\\\
&+s^{2}\left(\frac{2m_{q}m_{\bar{q}}}{m_{q}+m_{\bar{q}}}\right)^{2},\end{split}$
(20)

where $\sigma_{0}=1.8$ GeV and $s=3.88$ GeV are constant. Hence, the
unquenched confinement potential $S(r)$ and one-gluon exchange interaction
$G(r)$ become a smeared potential $\widetilde{S}(r)$ and $\widetilde{G}(r)$ by
the general transformation

$\displaystyle\widetilde{f}(r)=\int
d^{3}r^{\prime}\rho(\bm{r}-\bm{r^{\prime}})f(r^{\prime}).$ (21)

In addition, a relativized potential should depend on quark momentum, which
can be taken into account by introducing momentum-dependent factor. So the
Coulomb term $\widetilde{G}(r)$ and the contact, tensor, vector spin-orbital
and scalar spin-orbital potential $\widetilde{V}_{i}(r)$ could be modified as

$\displaystyle\widetilde{G}(r)\rightarrow\left(1+\frac{p^{2}}{E_{q}E_{\bar{q}}}\right)^{1/2}\widetilde{G}(r)\left(1+\frac{p^{2}}{E_{q}E_{\bar{q}}}\right)^{1/2},$
$\displaystyle\frac{\widetilde{V}_{i}(r)}{m_{q}m_{\bar{q}}}\rightarrow\left(\frac{m_{q}m_{\bar{q}}}{E_{q}E_{\bar{q}}}\right)^{1/2+\epsilon_{i}}\frac{\widetilde{V}_{i}(r)}{m_{q}m_{\bar{q}}}\left(\frac{m_{q}m_{\bar{q}}}{E_{q}E_{\bar{q}}}\right)^{1/2+\epsilon_{i}},$
(22)

where $E_{q}$ ($E_{\bar{q}}$) is the energy of quark (antiquark), and
$\epsilon_{i}$ denotes a momentum correction parameter with a different type
of interactions in Eqs. (16)-(18). One may see that the momentum-dependent
factor returns to unity in the nonrelativistic limit.

By diagonalizing the Hamiltonian in Eq. (11) with a series of the SHO basis,
the obtained eigenvalues and eigenvectors can correspond to the meson mass and
wave function, respectively. In the momentum space, the general form of an SHO
wave function is

$\displaystyle\begin{split}\Psi_{nLM_{L}}(\mathbf{p})=&R_{nL}(p,\beta)Y_{LM_{L}}(\mathbf{\Omega}_{p})\end{split}$
(23)

with

$\displaystyle\begin{split}R_{nL}(p,\beta)=&\frac{(-1)^{n}(-i)^{L}}{\beta^{3/2}}\;e^{-\frac{p^{2}}{2\beta^{2}}}\sqrt{\frac{2n!}{\Gamma(n+L+3/2)}}\left(\frac{p}{\beta}\right)^{L}\\\
&\times L_{n}^{L+1/2}\left(\frac{p^{2}}{\beta^{2}}\right),\end{split}$ (24)

where $R_{nL}(p,\beta)$ is a radial wave function of a harmonic oscillator,
and $Y_{LM_{L}}(\mathbf{\Omega}_{p})$ is a spherical harmonic function and
$L_{n}^{L+1/2}(x)$ is the associated Laguerre polynomial.

##  Appendix B  Quark pair creation model

The quark pair creation model Micu:1968mk ; LeYaouanc:1977gm  is usually
applied to study the OZI-allowed two-body strong decays of a hadronic state,
which are absolutely dominant decay modes for the meson or baryon state above
the decay threshold. In the following, we will give a brief introduction of
the QPC model. When a meson decays, a quark-antiquark pair created from the
vacuum with the quantum number $J^{PC}=0^{++}$ has a connnection with
antiquark and quark inside the initial meson to produce two final mesons. The
decay matrix element of this process $A\to BC$ can be written as $\left\langle
BC|\mathcal{T}|A\right\rangle=\delta^{3}(\mathbf{P}_{B}+\mathbf{P}_{C})\times\mathcal{M}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}(\mathbf{P})$,
where the transition operator $\mathcal{T}$ represents a quark-antiquark pair
creation from the vacuum. Taking a particle $A$ as an example, the wave
function of the mock state $(A,B,C)$ is defined as Micu:1968mk ;
LeYaouanc:1977gm

$\displaystyle|A(n_{A}^{2S_{A}+1}L_{AJ_{A}M_{J_{A}}})(\bm{P}_{A})\rangle\equiv\sum\limits_{M_{L_{A}},M_{S_{A}}}\langle
L_{A}M_{L_{A}}S_{A}M_{S_{A}}|J_{A}M_{J_{A}}\rangle$
$\displaystyle\times\sqrt{2E_{A}}\int
d^{3}\bm{p}_{A}\psi_{n_{A}L_{A}M_{L_{A}}}(\bm{p}_{A})\chi_{S_{A}M_{S_{A}}}^{12}\phi_{A}^{12}\omega_{A}^{12}$
$\displaystyle\times\left|q_{1}\left(\frac{m_{1}}{m_{1}+m_{2}}\bm{P}_{A}+\bm{p}_{A}\right)\bar{q}_{2}\left(\frac{m_{2}}{m_{1}+m_{2}}\bm{P}_{A}+\bm{p}_{A}\right)\right\rangle,$
(25)

where $m_{1}$ and $m_{2}$ are masses of quark $q_{1}$ and antiquark
$\bar{q}_{2}$, respectively, and $n_{A}$ is the radial quantum number of a
meson $A$, and $\bm{S}_{A}$ and $\bm{L}_{A}$ are spin
$\bm{S}_{q}+\bm{S}_{\bar{q}}$ and relative orbital angular momentum between
$q_{1}$ and $\bar{q_{2}}$, respectively. $\bm{J}_{A}=\bm{S}_{A}+\bm{L}_{A}$ is
the total spin, while $\bm{P}_{A}=\bm{p}_{1}+\bm{p}_{2}$ and $E_{A}$ are
center-of-mass (CM) momentum and energy, respectively.
$\bm{p}_{A}=\frac{m_{1}\bm{p_{1}}-m_{2}\bm{p_{2}}}{m_{1}+m_{2}}$ denotes the
relative momentum between $q_{1}$ and $\bar{q}_{2}$.
$\chi_{S_{A}M_{S_{A}}}^{12}$, $\phi_{A}^{12}$, $\omega_{A}^{12}$, and
$\psi_{n_{A}L_{A}M_{L_{A}}}(\bm{p}_{A})$ are the spin, flavor, color, and
spatial wave function of a meson $A$, respectively. The total decay width of
$A\to BC$ in the center-of-mass (CM) frame is given by

$\displaystyle\Gamma_{A\to
BC}=\frac{\pi}{4}\frac{|\mathbf{P}|}{m_{A}^{2}}\sum\limits_{JL}\left|\mathcal{M}^{JL}(\mathbf{P})\right|^{2}.$
(26)

Here, $\mathbf{P}=\mathbf{P}_{B}=-\mathbf{P}_{C}$. $\bm{L}$ and $\bm{J}$
denote the relative orbital angular and total spin momentum between final
states $B$ and $C$, respectively. $\mathcal{M}^{JL}(\mathbf{P})$ is the
partial wave amplitude, which can be directly related to the helicity
amplitude $\mathcal{M}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}(\mathbf{P})$ according to
the Jacob-Wick formula Jacob:1959 . In the CM frame the specific form of
$\mathcal{M}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}(\mathbf{P})$ can be written as

$\displaystyle\begin{split}\mathcal{M}^{M_{J_{A}}M_{J_{B}}M_{J_{C}}}(\mathbf{P})=&\gamma\sqrt{8E_{A}E_{B}E_{C}}\sum\limits_{M_{L_{A}},M_{S_{A}},M_{L_{B}},M_{S_{B}},M_{L_{C}},M_{S_{C}}}\langle
L_{A}M_{L_{A}}S_{A}M_{S_{A}}|J_{A}M_{J_{A}}\rangle\;\langle
L_{B}M_{L_{B}}S_{B}M_{S_{B}}|J_{B}M_{J_{B}}\rangle\\\ &\times\langle
L_{C}M_{L_{C}}S_{C}M_{S_{C}}|J_{C}M_{J_{C}}\rangle\langle
1m1-m|00\rangle\;\langle\chi_{S_{B}M_{S_{B}}}^{14}\chi_{S_{C}M_{S_{C}}}^{32}|\chi_{S_{A}M_{S_{A}}}^{12}\chi_{1-m}^{34}\rangle\\\
&\times\left[\langle\phi_{B}^{14}\phi_{C}^{32}|\phi_{A}^{12}\phi_{0}^{34}\rangle\mathcal{I}(\mathbf{P},m_{2},m_{1},m_{3})+(-1)^{1+S_{A}+S_{B}+S_{C}}\langle\phi_{B}^{32}\phi_{C}^{14}|\phi_{A}^{12}\phi_{0}^{34}\rangle\mathcal{I}(\mathbf{-P},m_{2},m_{1},m_{3})\right],\end{split}$
(27)

where $\gamma$ reflects the creation possibility of a quark pair
$q_{3}\bar{q_{4}}$ from the vacuum, which is generally considered to be a
universal constant for the decay of a specific meson system and can be
determined by the relevant experimental data. It needs to be mentioned that
the creation strength for the $s\bar{s}$ pair creation is different from that
of the $u\bar{u}+d\bar{d}$ pair, where there exists the relation
$\gamma_{s}=\gamma_{u}/\sqrt{3}$ LeYaouanc:1977gm . Additionally, $m_{3}$ is
the mass of constituent quark $q_{3}$ and $\phi_{0}$ is the flavor wave
function of $q_{3}\bar{q_{4}}$ pair. The expression of the momentum space
integral $\mathcal{I}(\mathbf{P},m_{2},m_{1},m_{3})$ reads as

$\displaystyle\begin{split}\mathcal{I}(\mathbf{P},m_{2},m_{1},m_{3})=&\int
d^{3}\bm{p}\psi_{n_{B}L_{B}M_{L_{B}}}^{*}(\frac{m_{3}}{m_{1}+m_{3}}\mathbf{P}+\bm{p})\\\
&\times\psi_{n_{C}L_{C}M_{L_{C}}}^{*}(\frac{m_{3}}{m_{2}+m_{3}}\mathbf{P}+\bm{p})\\\
&\times\psi_{n_{A}L_{A}M_{L_{A}}}(\mathbf{P}+\bm{p})\mathcal{Y}_{1}^{m}(\bm{p}),\\\
&\end{split}$ (28)

where $\mathcal{Y}_{1}^{m}(\bm{p})$ denotes the solid harmonic polynomial.

## ACKNOWLEDGEMENTS

The authors would like to thank Dr. Wen-Biao Yan and Ya-Teng Zhang for useful
discussions. 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, and the National Natural Science Foundation of
China under Grant No. 12047501.

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